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Metabolic adaptation to vitamin auxotrophy by leaf-associated bacteria
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Birgitta Ryback1,
Miriam Bortfeld-Miller1 &
Julia A. Vorholt ORCID: orcid.org/0000-0002-6011-49101
The ISME Journal volume 16, pages 2712–2724 (2022)Cite this article
Auxotrophs are unable to synthesize all the metabolites essential for their metabolism and rely on others to provide them. They have been intensively studied in laboratory-generated and -evolved mutants, but emergent adaptation mechanisms to auxotrophy have not been systematically addressed. Here, we investigated auxotrophies in bacteria isolated from Arabidopsis thaliana leaves and found that up to half of the strains have auxotrophic requirements for biotin, niacin, pantothenate and/or thiamine. We then explored the genetic basis of auxotrophy as well as traits that co-occurred with vitamin auxotrophy. We found that auxotrophic strains generally stored coenzymes with the capacity to grow exponentially for 1–3 doublings without vitamin supplementation; however, the highest observed storage was for biotin, which allowed for 9 doublings in one strain. In co-culture experiments, we demonstrated vitamin supply to auxotrophs, and found that auxotrophic strains maintained higher species richness than prototrophs upon external supplementation with vitamins. Extension of a consumer-resource model predicted that auxotrophs can utilize carbon compounds provided by other organisms, suggesting that auxotrophic strains benefit from metabolic by-products beyond vitamins.
Coenzymes are essential for cellular metabolism. They form the core of many enzyme-catalyzed reactions (e.g., redox reactions, transaminations, and carbon–carbon bond formation), and act as carriers for one-carbon units and organic acids [1]. Organisms with defects in their biosynthesis, auxotrophs, must obtain coenzymes or coenzyme precursor molecules, i.e., vitamins by external supply. Because they rely on the supply of vitamins in their environment to enable growth, coenzyme instability would exacerbate the need for external supply. It has recently been shown that the carbon backbone of coenzymes is stable in vivo [2]. In fact, coenzyme longevity, i.e., the absence of intracellular degradation and rebuilding, is an inherent property that might have been selected for evolutionarily [2, 3] and can be considered a prerequisite for the proliferation of auxotrophic organisms. Auxotrophic requirements can evolve rapidly in laboratory evolution experiments in nutrient-rich media as shown in Escherichia coli populations [4]. Auxotrophic strains have also been isolated from nature and are often the result of gene loss [5,6,7]. However, it is currently unclear how common auxotrophies are in bacteria in most environments. A number of recent studies have reported predictions on nutritional requirements directly from genomic data using either genome-scale metabolic models or identifying gaps in genomes without experimental validation [8,9,10,11,12,13] or focusing the experimental validation to few strains [14,15,16]. These studies have been used to predict interaction networks of microbial communities, which in turn have served to infer metabolic cross-feeding in bacterial communities [17]. The frequency of auxotrophies for coenzymes, nucleobases, and amino acids in sequenced bacteria has been estimated in computational analyses and could be as high as 75% [13], although these analyses likely exhibit a high false positive rate when compared to experimentally obtained data [18].
Auxotrophs require an external supply of vitamins, which raises the question of physiological or other adaptations of auxotrophs, including vitamin storage. It has long been known that auxotrophic lactic acid bacteria maintain growth after the removal of vitamins [19]. Although intracellular storage of elemental compounds such as carbon, phosphorus, and nitrogen has been shown [20,21,22], the storage of small molecules—including vitamins and coenzymes—is currently not well established. It is also not clear how widespread potential storage is in environmental bacteria. There is a renewed interest to assess storage of diverse compounds due to the ecological implications of microbes' ability to maintain growth without external access to resources [23]. Prototrophic E. coli strains that were mutated in genes involved in biosynthesis of coenzymes, i.e., "artificial" auxotrophic mutants, have a storage capacity for only about one doubling after removal of the essential vitamin [2]. This two-fold excess pool of coenzymes is probably the minimum requirement to keep metabolism robust to cell expansion during cell division. These results led us to hypothesize that for auxotrophs found in nature where coenzyme supply may be erratic, different strategies to cope with nutrient dependence and coenzyme storage may have evolved. Loss of biosynthesis of one compound could also affect a cell's function in a more systemic way. Direct physiological consequences resulting from loss of biosynthesis have been studied in model organisms such as E coli, Bacillus subtilis, and Acinetobacter baylyi and include fitness benefit [4, 13, 24] and cross-feeding potential [25, 26]. These studies show that, as long as the required nutrient is available, auxotrophs outcompete the wild-type. It can be anticipated that given sufficient evolutionary time, other traits might be selected for to compensate for the loss of a biosynthetic ability. This loss of an essential function, e.g., a particular biosynthetic pathway, may then extend to other compounds that are available in the now obligatory niche, as well as to features that help deal with nutrient shortage. The latter could involve, for example, optimizing the use of auxotrophic coenzymes in metabolic pathways, increased vitamin storage, investing more energy into repair systems, or optimizing growth and nutrient uptake rates [9, 27,28,29].
Here, we first investigate the occurrence of auxotrophies in the At-LSPHERE collection, which consists of 224 strains isolated from leaves of Arabidopsis thaliana plants growing in nature [30]. The strains had been isolated on vitamin and amino acid replete medium, which allows for systematic nutrient drop-out experiments to examine auxotrophies within the strain collection. We then address whether auxotrophic strains maintain a higher coenzyme storage than prototrophic strains and explore genomic traits that co-occur (and potentially co-evolved) with auxotrophy. Finally, we examine vitamin acquisition as well as auxotrophic fitness in co-culture experiments.
Vitamin storage in auxotrophic bacteria
To identify auxotrophic strains for this study, we screened a representative 224-member strain collection (At-LSPHERE) [30]. Briefly, triplicates of each strain were cultivated in 96-well plates filled with liquid media supplemented with vitamins or amino acids, with both types of compounds, or without supplements. High-throughput optical density (OD600) measurements were used as a readout for growth. We found that out of the 156 strains that grew in the medium, 50% (78) required supplements for growth and were therefore likely auxotrophic (Supplementary Fig. 1). Auxotrophy was especially common in Actinobacteria and Alphaproteobacteria, of which more than half of the tested strains were auxotrophic. From these 78 strains, we selected 50 putative auxotrophs for further characterization in individual drop-out media and found that most vitamin auxotrophies were for biotin, thiamine, niacin, and pantothenate for which we found 10 or more auxotrophs each (Fig. 1a; Supplementary Fig. 1).
Fig. 1: Vitamin storage in auxotrophic strains.
a Co-occurring vitamin auxotrophies as inferred from lack of growth without a given vitamin in a screen on drop-out media for 50 strains. Niacin, pantothenate, biotin, and thiamine were the most commonly required vitamins. Edge thickness correspond to the number of strains in which the two putative auxotrophies co-occur. PABA = para-aminobenzoic acid. (Source data in Source Data 1.) b Representative dynamic vitamin depletion experiments. Growth is presented as doublings over time. At t = 0, a switch from preculture supplemented with all four vitamins to a vitamin-deplete medium (coloured lines) or, as a control, supplemented medium (black line) was performed. If a strain was auxotrophic for a given vitamin, it depleted off that vitamin after some time and stopped growing; for example, Aureiomonas Leaf324 was found auxotrophic for all four vitamins. The time point at which vitamin-depleted and supplemented cultures deviated are visualized over the lineplot for all three replicates separately (see Materials and Methods for details). (Source data in Source Data 2.) c Auxotrophy heatmap. Light color refers to compounds without which a strain's growth deviates from the supplemented control and thus the strain is auxotrophic for. For example, for Aureiomonas Leaf324, all 4 depletion columns are light grey as its growth ceased without each of the compounds individually. (Source data in Source Data 3.) d Coenzyme storage in auxotrophic strains (interquartile range and median). The number of doublings in the absence of the indicated vitamin corresponds to the storage capacity. (Source data in Source Data 3).
Source Data 1Source Data 2Source Data 3Source Data 3.
Although it is useful to characterize nutrient requirements of many strains in parallel, the application of a screen to elucidate the nutrient requirements of individual strain is limited due to linear range of plate readers, which could lead to false predictions. To overcome this limitation and for further analysis, we selected 22 strains from various phyla that did not form aggregates upon growth in liquid media, which we ensured visually and by confirming that their colony-forming unit counts were within the expected range (order of 1e9/ml) (Supplementary Table 1; see selected strains in Supplementary Table 2). We cultivated each strain in semi-continuous batch cultures and monitored OD600 at 10 minute intervals, diluting each well with fresh medium after every two doublings to ensure continued exponential growth. We used these data to examine growth of each of the strains on glucose minimal media supplemented with biotin, niacin, thiamine, and pantothenate, and observed that all strains grew in the presence of these four vitamins (black trajectories in Fig. 1b and Supplementary Fig. 2). Subsequently, we performed growth experiments by switching cells to vitamin-free media (see Methods), thereby exposing the cells to vitamin depletion. We classified a given strain as an auxotroph if its growth in a vitamin drop-out medium deviated from the supplemented control cultures (scatterplot overlay in Fig. 1b and Supplementary Fig. 2). All 22 strains were thereby confirmed to be auxotrophs in this experiment (Fig. 1c). Specifically, each strain was auxotrophic for two vitamins on average, and 80% of the strains were auxotrophic for multiple vitamins.
Inside cells, vitamins are processed into coenzymes. Coenzymes are catalytically highly reactive, yet stable [2]. Because they are not consumed in metabolic reactions and are only synthesized to compensate dilution by growth, coenzymes or their precursors are potentially storable in cells. To assess the potential for intracellular vitamin storage in the selected auxotrophic strains, we used the semi-continuous growth data to calculate the total number of doublings each strain achieved after vitamin deprivation before deviating from exponential growth as determined by comparison to supplemented control cultures (Fig. 1b; Supplementary Fig. 2). These values can thus serve as indicators of vitamin storage. We found that most strains had a vitamin storage allowing for 1–2 doublings, which was similar to that of E. coli mutants [2]. However, we also observed that some auxotrophic strains maintained exponential growth for a prolonged period of time after the removal of biotin, translating to 4–5 doublings using the biotin reserves (16–32 fold excess pool). One strain, Chryseobacterium Leaf201, had a reservoir that allowed for 9 doublings (512 fold excess) after vitamin withdrawal (Fig. 1d). We also exemplarily confirmed for Rhizobium Leaf68 that the growth arrest coincides with intracellular depletion of the coenzyme (Supplementary Fig. 3).
Gene absence underlies auxotrophy
Knowing vitamin requirements for 22 validated strains, we next set out to identify the potential genetic basis for the observed auxotrophies by analysing the vitamin biosynthetic pathways using Clusters of Orthologous Groups of Proteins (COG) annotation. As prototrophic controls, we studied additional 13 strains that we confirmed to grow on minimal medium without supplements (Supplementary Fig. 4). First, we asked if we could have predicted the observed auxotrophy from genomic data alone. Following previous reports in which the percentage of biosynthetic genes absent from a pathway has been used as a marker of auxotrophy [12,13,14,15, 31], we studied the fraction of biosynthetic COG terms present on all four coenzyme synthesis pathways. All strains, including prototrophs, lacked many biosynthetic steps in the pathways for all four studied coenzymes (biotin, thiamine, coenzyme A, NAD), and we did not observe systematically fewer pathway steps in auxotrophs (Fig. 2). We then asked whether inferring auxotrophies from the presence/absence of individual genes would result in improved prediction of experimentally validated auxotrophies. To this end, we compared the frequency at which each COG term of a vitamin biosynthesis pathway was present for the auxotrophic versus prototrophic group using a Chi-squared test. In this way, we identified 1–3 absent genes sufficient to explain each auxotrophy (Table 1). For more detail on the nature of the specific reactions, see Supplementary Note 1.
Fig. 2: Identification of mutations underlying auxotrophy. Source data in Source Data 4.
a Fraction of COG terms of each coenzyme biosynthesis pathway (rows) present in each strain (columns); strains are color-coded by phylum and clustered by percentage of pathway present. Black rectangles indicate that the corresponding strain was found to be auxotrophic for the coenzyme in this study (Fig. 1).
Table 1 Genes mapping to COG terms that are significantly less prevalent in auxotrophs vs. prototrophs.
After identifying 1–3 genes likely responsible for each observed auxotrophy, we investigated whether the findings also applied to the remaining auxotrophic strains in the At-LSPHERE strain collection (Supplementary Fig. 1c and Fig. 1a). To achieve this, we wanted to create robust classifiers for each of the four observed auxotrophic requirements. We first generated a sample of 63 auxotrophic strains, for 35 of which were validated above and additional 28 strains for which predictions from Supplementary Fig. 1c were used as the target variable. Secondly, we trained a decision tree classifier based on either the 1–3 features selected in the chi-squared analysis (Table 1) or equally many randomly selected features. These models reached performance metrics of 80–100% (Supplementary Fig. 5). We used the models to predict auxotrophy for 139 strains from the At-LSPHERE collection for which growth was observed in the supplemented medium (and therefore are comparable in physiology), and for which COG annotations were available (Supplementary Table 3). We also used these features to reinvestigate auxotrophy at the taxonomic level. The COG terms we identified to be absent in the analysis reproduced the estimated phylogenetic enrichment for auxotrophy in Actinobacteria and Alphaproteobacteria closely related to Rhizobium spp. (Supplementary Fig. 6; compare to Supplementary Fig. 1).
Genome reduction in auxotrophic strains is mainly non-specific
Genome reduction has previously been described in environmental auxotrophic bacteria [32]. We thus addressed genome reduction and potential concomitant gene loss events for the 139 strains for which auxotrophy predictions were possible based on functional genomic annotation and observed growth in a defined medium (Supplementary Fig. 6). Indeed, we found significantly smaller genomes in auxotrophs when compared to prototrophs (Fig. 3a). This effect was nonetheless class/phylum dependent: Genome reduction was significant in Actinobacteria (19% smaller genomes), whereby a high fraction (>50%) of strains are auxotrophs (Supplementary Fig. 1b). As there are only six species of Firmicutes within the analysed strains, drawing general conclusions is challenging. However, the four auxotrophs exhibit a genome that was only half the size of the two prototrophic strains and also feature fewer open reading frames (Supplementary Fig. 7). The auxotrophic strains identified here required on average two amino acids and two vitamins (Fig. 1, Supplementary Fig. 1). Therefore, the observed loss of biosynthetic genes would account for 0.25% smaller genomes in auxotrophs (for a bacterium that encodes on average 5,000 genes; estimated based on Fig. 3a). Thus, our observation that auxotrophs have smaller genomes provokes the question if there are other auxotrophy-associated biological processes or pathways that explain the genome reduction.
Fig. 3: Genome-wide comparative genome analysis between prototrophs and auxotrophs.
Genomes obtained from RefSeq with the accession numbers in Source Data 5. In all panels, 139 strains that grew in liquid cultures (Supplementary Fig.1) and for which COG annotations were available were included. Auxotrophy status was predicted using the models in Fig. 2 and Table 1. a Genome sizes between auxotrophs and prototrophs. b Number of reactions that require each vitamin-derived coenzyme (x-axis) in strains that are auxotrophic vs. prototrophic for each coenzyme (in separate plots). For example, the first plot at the top left shows a comparison of coenzyme usage of 6 coenzymes between biotin auxotrophs and prototrophs. Metabolic models were obtained by executing CarveMe [33] on the RefSeq accession numbers. c Pathways on which >20% of genes were differentially present between auxotrophs and prototrophs for each vitamin. Here, "underrepresented" refers to less abundant in auxotrophs.
Source Data 5.
A conceivable genomic adaptation that would link auxotrophy to genome reduction is that auxotrophic strains have evolved to be less dependent on coenzymes that they cannot synthesize. However, the opposite scenario is also plausible: if the vitamin is freely available in the environment, there could be a positive selection pressure to prefer coenzymes from costless vitamins. To investigate whether there is a correlation (positive or negative) between auxotrophy for a given coenzyme and use of the same in terms of number of enzymes, we created genome-scale metabolic models for all individual strains (GEM) [33]. Overall, the number of reactions in metabolic models were similar in both groups (Supplementary Fig. 7), indicating that auxotrophs do not have streamlined metabolic networks. We set out to investigate whether auxotrophic strains preferentially use one coenzyme over the other in their enzymatic reactions; in particular, niacin auxotrophs may potentially use FAD or FMN dependent enzymes instead of NAD(P) dependent ones. Using GEMs, we computed the number of enzymes that require each coenzyme for both auxotrophs and prototrophs to measure the degree of dependency that a strain has on the corresponding vitamin. Although GEMs do not include information about expression levels of the individual enzymes in question (and therefore potential differential requirement of the enzyme), their stoichiometric constraints and network structure provide greater confidence for gene expression than genomic analysis alone. We found that for most coenzymes, auxotrophs and prototrophs had the same number of reactions (Fig. 3b). Niacin auxotrophs had 10% more enzymes using NAD(H) and 20% less enzymes using FAD(H) (p-value <0.05, χ2 test), indicating a preference to use the coenzyme for which the biosynthesis is lost. For biotin, auxotrophic strains had 15% more biotin-dependent enzymes (p-value <0.05, χ2 test). Overall, this analysis shows that the genome reduction observed in auxotrophic strains cannot systematically be explained by fewer enzymes that require the respective coenzyme. As such, metabolic capacity may not be reduced in auxotrophs, and auxotrophs may have allocated their metabolic reactions to using the essential vitamins.
Apart from direct impact on the use of coenzymes, we reasoned that auxotrophic strains might have evolved other traits indirectly related to coenzyme usage, either prior to or after becoming auxotrophs. Such traits could entail loss of function events in other biosynthesis pathways, as well as loss of genes that are not essential in the confined niche, such as utilization of a carbon source. We therefore, sought to address metabolic reallocation more generally. To find genes whose presence correlate with auxotrophy, we performed a genome-wide functional genomics analysis. We first evaluated whether each COG term was differentially abundant between auxotrophs and prototrophs, then mapped the significantly different COG terms to pathways. Significance was determined by χ2 test and multiple testing correction was performed with FDR. In this way, we identified a set of conserved differences between auxotrophs and prototrophs (Fig. 3c, Supplementary Table 4). Overall, the pattern was dominated by loss of function in auxotrophs. Setting the cut-off of differentially abundant COG terms to 20% of all pathway genes, we recovered the biosynthetic pathway for each known auxotrophy, which confirmed that this approach can capture biologically meaningful differences between the groups. Other gene absences entailed loss of enzymatic function in energy, especially glucose metabolism as well as other biosynthetic pathways such as amino acids and other coenzymes beyond the ones tested here, whereas salvage systems were more prevalent in the auxotrophic group.
Auxotrophs are complemented by vitamins from co-cultivated species
Co-culture experiments can be used to provide insights into bacterial interactions in microbial communities [34,35,36,37]. Here, we asked whether auxotrophs access vitamins from co-cultivated prototrophs. To this end, we selected phylogenetically representative vitamin auxotrophic (n = 10) and prototrophic (n = 10) strains for a co-culture experiment (Fig. 4a). We conducted a serial dilution experiment of the strain mix (3 biological replicates each in 3 technical replicates) in minimal medium supplemented with 20 mM glucose and all 20 proteinogenic amino acids (100 μM each). The media differed only in whether they were supplemented with vitamins. (Fig. 4b, c). The cultures were analysed at regular intervals by 16S rRNA gene sequencing to determine the relative composition of the bacterial species. Growth in both conditions was comparable (Fig. 4c). We observed that the relative abundance profile between media was similar in both groups, i.e., auxotrophs and prototrophs, and independent of whether vitamins were supplemented or not (Fig. 4d, gray vs orange columns for each strain; Supplementary Fig. 8). The only auxotrophic species that benefited from supplemented vitamins was Pedobacter Leaf132 (p-value 0.003, rm-ANOVA). Two auxotrophic strains, Rhizobium Leaf202 and Arthrobacter Leaf137, had higher relative abundances in cultures where vitamins were not supplemented (p values <0.01 rm-ANOVA). This analysis thus shows that lack of external vitamin supplementation does not prevent the growth of auxotrophs when cultivated in a community with prototrophic strains and suggests vitamin cross-feeding.
Fig. 4: Co-culture experiment to determine whether auxotrophic strains are able to access vitamins from prototrophs. Source data in Source Data 6, 7.
a Selection of strains for co-cultures. A total of 20 strains were chosen to represent the phylogenetic diversity in the A. thaliana leaf microbiota. Clustering is based on full-length 16S rRNA gene sequences. b Strains shown in panel A were mixed together in equal numbers (see Methods for details) in three biological inocula. Each inoculum was used to inoculate three technical replicates under two growth regimes: minimal medium with and without vitamins, resulting in 18 cultures. All 20 proteinogenic amino acids were added to all cultures (100 µM). c Growth measurements of the cultures from b Cultures were grown for a total of 120 h, and diluted at 24, 48, and 72 h. At 9, 24, 48, 72, 96, and 120 h (indicated as dots), samples for 16S rRNA gene sequencing (cells) and LC/MS (supernatants) were taken. d Relative abundance of all 20 strains (columns) in the co-culture experiment. For each strain, the average of the three technical replicates for each biological replicate is shown as the change of relative abundance compared to the previous time point, first in minimal medium with vitamins (columns labelled with orange boxes), then in minimal medium without vitamins (columns labelled with grey). For each strain, the three replicates in the same condition are separated by thin white lines and the two conditions are separated by thick gray lines. The 20 strains are separated by thick black lines. Strains are clustered by phylogeny as computed from 16S rRNA gene alignment.
To assess the presence of cross-feeding, we examined the spent media from the co-cultures mentioned above (Fig. 5). We measured the media without bacteria (0 h time point) as well as supernatants after 9 h and 24 h of co-culture. We first compared the vitamin pools in the two conditions (± vitamins). When vitamins were supplemented, niacin, pantothenate, and thiamine concentration decreased between 0 h and 9 h, and further between 9 h and 24 h, suggesting net uptake by the bacteria grown in the medium (orange series in Fig. 5a). Decrease in the vitamin pools of these four vitamins was expected considering that half of the strains inoculated were auxotrophic for one or more of them. Biotin concentration, however, did not decrease during the experiment (orange series in the first panel in Fig. 5a) despite almost all of the strains being auxotrophic for it, indicating either small biotin uptake rates by auxotrophs, or that prototrophic strains secreted biotin at rates comparable to the uptake of biotin by auxotrophic strains. We also observed uptake of pyridoxal, for which no strain was auxotrophic. Riboflavin concentration was below the detection limit in the bacteria-free media, but it consistently accumulated in the medium during bacterial growth, suggesting that at least one of the strains in the co-culture secreted more riboflavin than others took up. In the vitamin-deplete medium, we found an accumulation of niacin, pantothenate, and pyridoxal at 9 h, followed by a decrease in these vitamins at 24 h, implying that the bacteria first secreted these vitamins into the medium and subsequently took them up (Fig. 5a, grey series). Thiamine and biotin pools remained below the detection limit. Amino acids that were supplemented in both media (alongside with glucose as the single main carbon source), decreased in all co-cultures when compared to the pure media, indicating net uptake from the medium (Fig. 5b). The majority of the provided glutamate (an attractive carbon and nitrogen source [38]) was still available after 9 h of co-culture and mainly consumed during the stationary phase.
Fig. 5: Exometabolomics for co-cultures. Source data in Source Data 8.
a Time series of vitamin consumption in co-cultures as measured from co-culture supernatants. For each vitamin, the log10 of total ion current normalized peak area is first shown in vitamin-supplemented cultures, and then for vitamin-deplete cultures. b Amino acids measured from the co-culture supernatants. Data are log10 transformed peak areas normalized to the signal from medium which was supplemented with amino acids in both media (+ and –vitamins). Each cell in the heatmap represents the average of 9 replicates.
Auxotrophs benefit from co-cultures beyond vitamin supply
Auxotrophic mutants often have a fitness advantage over prototrophic ancestral strains in pairwise competition experiments [13, 24, 39]. In the following analysis, we assessed whether this also holds true in a community context of natural auxotrophs and prototrophs. Specifically, we predicted species richness from the growth rate and yield of each strain in individual cultures and compared these predictions to experimentally obtained data (Fig. 4). We conducted the analysis on relative abundance data in minimal medium supplemented with vitamins (orange columns in Fig. 4d) by comparing the species richness (number of detected species divided by the number of species originally introduced to the co-culture) over time in the two groups: auxotrophs and prototrophs. We found that two-thirds of the strains were undetected and hence interpreted as lost after the first two dilution cycles (after 24 h and 48 h) before eventually stabilizing at 72 h (Fig. 6a). More strains were lost in the prototrophic group: three-quarters of prototrophs were undetected after 48 h, significantly less than auxotrophs whereby only half were undetected (p ≪ 0.001, rm-ANOVA, Fig. 6a). To investigate whether the statistically different outcome between auxotrophs and prototrophs was to be expected, we attempted to predict species richness based on growth in monocultures. To this end, growth parameters were experimentally determined in the same vitamin-supplemented medium used for the co-culture experiments (Fig. 6b). Using these data, we parametrized a consumer-resource model and simulated the expected abundance of each strain in the vitamin-supplemented co-cultures. Briefly, the glucose concentration was iteratively updated to simulate eventual carbon limitation in a batch culture, growth rates were updated according to Monod kinetics, and no interaction between species was included (see Methods for modelling details). From the simulated abundances, we calculated the theoretical species richness. We found that in the prototrophic group, the simulation qualitatively captured the experimentally observed species richness (light blue line compared to the dark blue line in Fig. 6c), and correctly predicted the final species richness. All auxotrophic species were lost in the simulation (light red line compared to the dark red line in Fig. 6c), as expected due to their lower growth rates and yields compared to the prototrophic strains (Fig. 6b).
Fig. 6: Auxotrophs benefit from co-cultures beyond vitamin supply.
a Species richness in co-cultures cultivated in vitamin-supplemented medium. Source data in Source Data 6. b Experimentally determined growth rates and yields (=carrying capacity) for each strain growing in the vitamin-supplemented medium. Source data for growth rates in Source Data 9 and for yields in Source Data 10. c–f Species richness as modelled by consumer-resource models. Error bars for the models were obtained via sensitivity analyses as follows. In c and d, the threshold for a bacterium to count as "present" was varied from 0.1% to 1% of total abundance. In e, the threshold in growth rate for bacteria that were set to secrete the second carbon source was varied between 0.3 h−1 (n = 15) to 0.6 h−1 (n = 3). In f, the efficacy of auxotrophs to preferentially use the second carbon source was varied from 2 to 5 fold that of prototrophs'. Source data in Source Data 6.
Source Data 6Source Data 9Source Data 10.
The apparent contradiction between the consumer-resource model and the experimental data prompted us to explore whether an extension of the models could capture the observed species richness. We began by implementing a vitamin cross-feeding term, which was not included in the initial model. From the exometabolomics measurements (Fig. 5), we learned that prototrophs secrete vitamins into the medium, thereby at least transiently increasing its vitamin concentration. Moreover, 16S rRNA gene sequencing of samples derived from cultures to which no vitamins were added indicated that the observed increase in vitamin concentration during co-cultivation with prototrophs was sufficient to support growth of the auxotrophs (Fig. 4). If the vitamin concentration in the growth medium became low enough to limit growth prior to carbon limitation, these extra vitamins could have resulted in an increase in yield. To this end, we acquired growth data in a range of physiologically relevant vitamin concentrations (Supplementary Fig. 9). The observed differences in both growth rate and yield were generally small. Nonetheless, we modified the growth of each strain in the consumer resource model from each strains' growth parameters at 1 µM vitamins to those when grown in 10 µM vitamins, thereby reducing potential early growth arrest. We observed that this simulated increase in vitamin concentration did not result in increase of predicted auxotrophic species richness (Fig. 6d).
Consumer resource models with vitamin cross-feeding did not capture the experimentally observed difference in species richness (Fig. 6d). An alternative scenario for the high richness in the auxotrophic group of the co-culture is that they may utilize metabolic by-products as carbon sources. It is known that even under aerobic conditions, many fast-growing bacteria ferment their carbon source, resulting in secretion of by-products such a pyruvate, acetate, or succinate [40, 41]. Thus, changes in community composition during stationary phase (between 9 h and 24 h as well as 96 h and 120 h) is likely due to (co-)consumption of metabolic by-products after the main carbon source glucose is consumed. To address this scenario, we introduced a second carbon source, effectively mimicking secretion of a compound such as acetate by fast-growing prototrophic strains (Fig. 6b) into the model. We then allowed all slow-growing strains to grow on this newly available resource at rates proportional to their growth rate on glucose. As auxotrophs were enriched in the slow-growing group, they were able to preferentially use the by-product, while giving the prototrophic group an equal chance at growing on the secreted substrate given that the boundary condition of slow growth was met (see Supplementary Fig. 10 for details about concentration and effect to total cell number). Adding the carbon cross-feeding term indeed improved the predictions in comparison to the original model; both groups lose in species richness but neither group is entirely lost, and the species richness stabilises eventually (Fig. 6e). This model, however, still predicts the prototrophic group to have a higher final species richness. We, therefore, gave auxotrophs preferential access to the newly available carbon source by increasing their substrate affinity. In this scenario, the models captured the experimentally determined qualitative dynamics (Fig. 6f). Taken together, this analysis suggests that increased ability for carbon source cross-feeding determines the success of auxotrophic strains in co-cultures.
Coenzymes are not only ubiquitous but also generally conserved across all domains of life. They expand the catalytic toolbox of proteins and act as carrier molecules for organic acids and one-carbon units [1]. Here, we characterized the auxotrophic requirements of environmental bacteria isolated from the phyllosphere of A. thaliana and explored genomic and physiological traits that co-occur with the inability to synthesize vitamins. We found that vitamin auxotrophy was common for biotin, niacin, pantothenate, and thiamine and that auxotrophy was systematically associated with the absence of specific genes on the respective biosynthetic pathways. Although other studies have reported the absence of some of the genes identified in this study in other auxotrophs [7, 8], further studies will be required to assess whether auxotrophy is generally caused by the loss of these few specific genes. Furthermore, our experimental and bioinformatics analyses defined a set of auxotroph-specific functions as well as cross-feeding and storage of coenzymes in diverse naturally auxotrophic bacterial isolates.
After a strain loses a biosynthetic pathway, it becomes dependent on an external source of that nutrient. Therefore, its niche will be limited to those with at least occasional access to this resource. Hence, populations of strains may lose the ability to synthesize all compounds that the enforced niche can supply. Indeed, we observe multiple auxotrophies in four out of five confirmed auxotrophs (Fig. 1). Cross-feeding of nutrients was previously shown to promote coexistence [42]. In a two-strain system, a "beneficiary" strain (loss-of-synthesis mutant) did not outcompete the "helper" strain that supplied the nutrient. If a third strain that supplies the same nutrient was involved, the beneficiary outcompeted the helper strain; thus, one exchanged metabolite may only stabilize a community consisting of two strains. Communities found in nature may have hundreds or thousands of members. Therefore, multiple auxotrophies are required by each auxotrophic strain for cross-feeding interactions to stabilize large communities. As we observe multiple auxotrophies in 80% of auxotrophs, they may contribute to the observed stability of the community [30, 43] by cross-feeding. Therefore, we suggest that the identity of the required compound is dictated by the niche, but the number of auxotrophies per strain is rather an inherent property of the community. Although largely oligotrophic, there are detectable amounts of carbon sources (glucose, methanol, sucrose, fructose) [44], amino acids, and even vitamins on leaf surfaces, made available by either the plant host itself of the bacteria inhabiting the niche [6, 45, 46]. Auxotrophies for metabolically costly vitamin B12 (which is not used in plant metabolism) were not observed (Supplementary Fig. 11).
The genomes of bacterial populations change in response to selection pressure set by their environment and as random effects, e.g., genetic drift. For example, biosynthesis of diverse compounds requires allocation of cellular resources in terms of energy (mainly ATP and redox coenzymes) and protein synthesis, i.e., a cost. Provided that the product of such biosynthetic process is available in the environment, those organisms that lose the biosynthetic ability for that compound avoid the cost. In a number of studies, it was argued that by avoiding the cost of biosynthesis, cells gain a fitness advantage, which allows for selective advantage of loss of biosynthesis—this theory is called the Black Queen Hypothesis [4, 13, 39]. However, the loss of biosynthesis may also be neutral and caused by genetic drift. Smaller genomes are often observed in bacteria isolated from rich growth environments, and is strongly linked to endosymbiotic and parasitic lifestyles [32, 47,48,49]. It is however not possible to infer the mechanism between selection and genetic drift based on observed genome reduction.
Genome reduction can occur in two ways: 1. general genome reduction and 2. loss of specific genes [42]. In this study, we observed both specific and general absence of genes. However, most of the genome reduction fell under the first category as we observed up to 19% smaller genomes in auxotrophs, less than 1% of which can be explained by the lack of the genes directly related to auxotrophy (Figs. 2 and 3). We also observed genomic and metabolic reallocation, especially in niacin auxotrophs (Fig. 3b, c). Our analyses also indicate that auxotrophic bacteria have more enzymes that require the vitamin they are auxotrophic for (especially biotin and NAD) than prototrophs (Fig. 3b). This observation does not imply larger turnover or need for the coenzyme but rather how many reactions could be disturbed by limitation of the vitamin and therefore the impact that vitamin limitation would have on the metabolic network. Whether the requirement itself remains constant or even decreased depends on the extent to which the genes are translated. We hypothesize that the degree of dependency (here defined by the number of enzymes using the coenzyme) might not impact auxotrophs as much as it affects prototrophs because prototrophs might minimize their dependency on the coenzyme due to cost of vitamin biosynthesis.
The proportion of biosynthetic genes present in a pathway is often used to classify bacteria to auxotrophs or prototrophs [10, 13, 16]. Our results are in line with previously published work [18], suggesting that predicting auxotrophy based on the frequency of genes results in inaccurate predictions. We also observed that, although random gaps in genomes in general and coenzyme biosynthetic pathways in particular are frequently observed, lack of 1-3 specific coenzyme biosynthesis genes per pathway is strongly associated with observed auxotrophy (Fig. 2; Table 1). Loss of some of the genes we identified here have already been associated with auxotrophy before [7, 8]. It remains an open question whether auxotrophy is universally caused by the loss of a few specific genes. Prediction accuracy of auxotrophies based on genomic data might thus be improved by, for example, giving a higher weight to genes that are known to be commonly lost in auxotrophs. In this study, we could predict auxotrophy with 80–100% accuracy, recall, and precision by training models with the selected features (Table 1).
We observed a higher robustness to biotin limitation in some auxotrophic strains (Fig. 1), which raises the question about the storage mechanism. In mammals, biotin storage has been linked to the mitochondrial acetyl-CoA carboxylase [50]. In biotin auxotrophic yeasts, histone biotinylation has been suggested to serve as potential biotin storage in auxotrophic Candida albicans, which is in contrast to biotin prototrophic yeast Saccharomyces cerevisiae that does not biotinylate its histones [51]. At its simplest, the storage could be unspecific and distributed among proteins that covalently bind biotin as the prosthetic group. This hypothesis is supported by the observation that biotin auxotrophs have more biotin-requiring enzymes in their metabolic networks (Fig. 3b). Whether elevated number of biotin-dependent enzymes is a storage strategy in bacteria remains to be tested, but is supported by a recent study that showed biotin-binding rhizavidin to serve in biotin storage for Rhizobium spp [52].
Auxotrophy arises frequently in evolution, and in this study, we identified auxotrophs in all major phyla within the A. thaliana leaf microbiota. Further, in this study (Figs. 4–6, Supplementary Fig. 8), we observe, in agreement with other studies, that auxotrophs persist even in vitamin-free co-cultures with prototrophs [12, 14, 16]. These observations invoke the question, whether auxotrophs have a designated role in microbial community assembly. Metabolic dissimilarity, at least theoretically and in the case of some amino acids, may drive cross-feeding [26]. Our results support this idea as we find auxotrophs and prototrophs to be metabolically and phylogenetically different from each other. In this study, we confirmed that auxotrophs maintain their population well in the absence of vitamins given that they are cultivated together with prototrophic strains that provide them vitamins. Therefore, we hypothesise that auxotrophs remain in growth arrest until enough vitamins are secreted to resume growth, and that this lifestyle shifts the selection pressure to efficient consumption of metabolic by-products released by other strains.
We found that auxotrophic strains were more successful in vitamin-supplemented co-cultures than expected from their growth in individual cultures. A consumer-resource model parametrized by experimentally obtained growth rate and yield data with a carbon cross-feeding term sufficiently captured this observation. As growth of auxotrophic strains cannot be resumed until prototrophs have secreted enough vitamins, preferentially using a carbon source secreted alongside with the vitamin could give auxotrophs a competitive edge. Our findings and inference are congruent with previous results where stable coexistence was explained by implementing a non-specific carbon cross-feeding term in a consumer-resource model framework, and that growth on such metabolic by-products is often comparable to growth on primary carbon source such as glucose [35]. Our framework suggests that not only does such metabolic facilitation promote coexistence, but preferential consumption of carbon by-products may also represent a viable strategy to avoid competitive exclusion. Taken together, our results suggest that auxotrophy is a part of a lifestyle that specializes in consumption of metabolic products of other bacteria and can therefore be beneficial for free-living bacteria that are a part of a microbial community.
Strains and growth assays
All At-LSPHERE strains used in this study were previously published [30]. All growth assays were performed at 28 °C for At-LSPHERE strains and 37 °C for E. coli [53] strains. The turbidity of shake flask cultures was determined by measuring the optical density at 595 nm (="OD600") in semi-micro cuvettes (Bio-Greiner) using a Biophotometer Plus (Eppendorf). Samples were diluted as appropriate to keep the OD readings in the linear range. For batch cultures with continuous OD monitoring for 96 well plates (TPP flat bottom), Tecan Infinite M200 Pro was used at 1 mm amplitude orbital shaking.
The medium base was a phosphate buffer (2.4 g/l K2HPO4, 2.08 g/l NaH2PO4·2H2O) with mineral salts (1.62 g/l NH4Cl, 0.2 g/l MgSO4·7H20). Glucose was added at 20 mM and pyruvate at 40 mM. For all media components, 10x stocks were prepared, dissolved in ddH2O and filter sterilized. Vitamins were supplemented when indicated in the following concentrations: D-pantothenic acid hemi calcium salt 1.05 µM, biotin 0.41 µM, riboflavin 1.06 µM, thiamine · HCl 1.19 µM, pyridoxal · HCl 0.98 µM, p-amino benzoic acid 1.09 µM, cobalamine 0.14 µM, lipoic acid 0.24 µM, niacin 1.22 µM, folic acid 0.23 µM. All 20 proteinogenic L-amino acids were supplemented at 0.1 mM each, except L-tryptophan and L-tyrosine were supplemented at 0.05 mM. 1000x stocks were prepared for all other vitamins and amino acids except for riboflavin and tryptophan for which 100x stocks were prepared.
Growth analysis
Linear regression on ln transformed data was carried out using linear_regression function from sklearn. Datapoints that lied outside of log-linear scale were omitted. Growth rates were only calculated for cultures that performed at least 2 doublings in exponential phase.
In depletion experiments (see Fig. 1 and Supplementary Fig. 2), growth of cells cultivated on vitamin-free media was defined to be deviating from the supplemented control if the following criteria were fulfilled. There had to be at least a 0.25 doubling difference in the total number of doublings in the vitamin-free and vitamin-supplemented cultures. Additionally, in at least two out of the three replicates the growth rate after the 0.25 doubling separation had to be reduced by 25%; in the remaining replicate, a 10% reduction was considered acceptable. The storage was defined as the number of doublings a strain performed without the vitamin at this time point. Solid R2A medium was obtained from Sigma.
Auxotrophy screens
In order to screen all 224 bacteria in the At-LSPHERE collection [30] for growth in 4 media (+vitamins and amino acids, −supplements, +amino acids and +vitamins), each strain was inoculated in liquid R2A from a solid R2A plate and let grow to stationary phase (24 h). Then, 10 µl of stationary-phase pre-culture was inoculated in 190 µl of each of the 4 media for an overnight pre-culture at 28 °C on a 220 rpm shaker in three biological replicates. Main cultures were inoculated in the same fashion, and OD measurements were taken at two time points: t = 0 and t = 24 h. The OD was normalized to the OD in control (cell-free, medium-filled) wells. For the drop-out screen with 50 strains, same protocol was repeated with a few changes. Here, 5 µl of stationary-phase pre-culture on R2A was inoculated in 45 µl of each of the 4 media (+vitamin and amino acids, −supplements, +vitamins, +amino acids) and each of the 30 drop-out media (each individual compound in Source Data 1) for the overnight pre-culture and again for the main culture. Every 24 h, a 5 µl of stationary-phase culture was transferred to solid R2A plates and colony color and morphology were compared to known characteristics of each strain to exclude contaminations.
High-throughput depletion experiments
Each strain was inoculated from solid R2A plates on a 96 well plate in biological triplicates and five technical replicates in a medium supplemented with 20 amino acids and 4 vitamins: thiamine, niacin, biotin, and pantothenate at 200 µl culture volume. Main cultures were prepared next day by diluting 10 µl of each cell culture into fresh media. After an overnight culture, late exponential stage was reached. The five technical replicates were pooled together into a sterile 2 ml Eppendorf tube and washed twice by centrifuging the cell suspension tube (10,000 rpm, 2 min), removing supernatant, and dissolving the cell pellet in 1.5 ml of sterile MgCl2. After the second wash, cells were collected via a centrifugation step as before, and the resulting pellets were dissolved in 50 µl of MgCl2. 5 µl of the resulting cell suspension was inoculated in 195 µl of five media that contained all 20 amino acids and differed in vitamin supplements. One of the media was a control medium, and it included all four vitamins as the preculture medium. The other four media each lacked one of the following vitamins: thiamine, biotin, niacin, or pantothenate. Growth was then monitored on a Tecan plate reader (for details see "Growth assays"). After every 2 doublings, cultures were diluted as appropriate to maintain exponential growth in the plate reader's linear range.
Sample preparation for LC/MS analysis of depletion experiments
Rhizobium Leaf68 was cultivated in 20 ml culture in a medium containing 20 amino acids and biotin, niacin, and pantothenate. In late-exponential phase (OD~1), cells were collected via centrifugation (10,000 rpm, 2 minutes), followed by two rounds of washing and pelleting with 20 ml of sterile MgCl2 and 10,000 rpm for 2 min. Afterwards, cells were collected via centrifugation and dissolved in 1 ml of MgCl2. Cultures were inoculated in four shake flask cultures with 100 µl of this inoculum: one of the cultures was supplemented with all 3 vitamins, and a drop-out medium for each vitamin, respectively. OD was monitored once per doubling, and the intracellular metabolome was sampled as follows. Cultures were kept at 28 °C in a shaking water bath, and sample volume was determined based on the OD (1/OD ml). The determined volume of cell suspension was pipetted onto a filter standing atop a suction flask to remove the medium. Cells standing on filter were washed with 10 ml of dH2O that was kept at 28 °C in water bath, and the washed filter was subsequently transferred to a Schott flask containing 8 ml of cold acidified (−20 °C) quenching solution (3 parts MeCN:1 part MeOH:1 part 0.05 M formic acid). After 10 minutes, the quenching solution was transferred to a 50 ml Falcon tube and lyophilized at −50 °C overnight. The resulting powder was dissolved in 250 µl of pre-cooled dH2O, and kept in −80 °C until analysis.
LC/MS analysis
LC separation was achieved with a Thermo Ultimate 3000 UHPLC system (Thermo Scientific) at a flow rate of 500 µl min–1. Two different separation methods were applied. First separation was achieved by hydrophilic interaction (HILIC; Aquity UHPLC BEH Amide column [100 × 2.1 mm, 1.7 µm particle sizes; Waters]) as described in [54]. For HILIC analysis, 50 µl of the aqueous sample was dried (SpeedVac) and dissolved in MeCN. The C18 reversed phase (C18RP) separation was achieved using a Kinetex XB-C18 column (particle size 1.7 µm, pore size 100 Å; dimensions 50 × 2.1 mm2, Phenomenex) as described elsewhere [55]. For mass analysis, LC instrument was coupled to a Thermo QExactive plus instrument (Thermo Fisher Scientific), and the mass spectrometer was operated both positive and negative FTMS mode at mass resolution of 30,000 (m/z = 400). Heated electro spray ionization (ESI) probe was used applying the following source parameters: vaporizer 350 °C; aux gas 5; ion spray voltage +3.5 kV, sheath gas, 50; sweep gas, 0; radio frequency level, 50.0; capillary temperature, 275 °C. To analyze the data, targeted extraction of ion chromatograms was conducted using emzed2 [56]. Retention time windows were determined based on chemical standards, and selected windows were normalized to background signal.
Co-culture experiments
As a pre-experiment, CFU counts per unit of OD were determined using a dilution plating method. Colonies were counted from a dilution that allowed for determination from 5 µl dots (Supplementary Table 5). All 20 strains were then inoculated from solid R2A plates into liquid medium containing 20 amino acids and 10 vitamins and grown overnight in biological triplicates in 10 ml culture volume. After the pre-culture, all strains were in stationary phase. The OD of each culture was measured and adjusted to 5e8 cells per ml. These adjusted cultures were then combined into an inoculum where each strain had an abundance of approximately 1/20. The inoculum was washed as described in section "Sample preparation for LC/MS analysis of depletion experiments". Five sequencing samples were taken from each inoculum, and three technical replicates in both vitamin-supplemented and vitamin-free media were inoculated from each inoculum. Culture volumes were 20 ml, and cultures were allowed to grow at 28 °C with 200 rpm orbital shaking. At 9, 24, 48, 72, 96, and 120 h, samples for 16S rRNA gene sequencing and exometabolomics were taken as follows. For 16S rRNA gene sequencing, a 1 ml sample was transferred to a DNA extraction tube from FastDNA SPIN Kit for Soil. The tubes were centrifugated (10,000 rpm, 5 minutes, 4 °C), supernatants were removed, and the tubes with cell pellets were frozen and kept at -80 °C until extraction. For exometabolomics, 1 ml of culture was transferred to a 2 ml Eppendorf tube and centrifugated (10,000 rpm, 5 min, 4 °C). 200 µl of supernatants were stored in two technical replicates on a 96 well plate and stored at −20 °C until analysis (see "LC/MS analysis" for details).
16S rRNA gene amplicon library preparation and sequencing
DNA was extracted using the FastDNA SPIN Kit for Soil (MP Biomedicals) following the manufacturer's instructions. The samples were transferred to DNA low-binding 96-well plates (Frame Star 96, semiskirted), the DNA concentration was quantified using double-stranded DNA QuantiFluor (Promega) and normalized to 1 ng µl−1. The 16S rRNA gene amplicon library was generated as follows. PCR amplification, clean-up, and barcoding PCR were performed as in [57]. DNA concentration was determined as above, and each well was normalized to 1 ng µl−1. Equal volume from each well was then combined into a pooled 16S rRNA gene amplicon library, and the library was cleaned twice with AMPure magnetic beads using a bead-to-DNA ratio of 0.9 to remove small DNA fragments. The amplicon length distribution of the library was assessed on a 2200 TapeStation using HS D1000 (Agilent), resulting in an effective library size of 554–643 bp. Sequencing was performed for 12 pM samples on a MiSeq desktop sequencer (Illumina) at the Genetic Diversity Centre Zurich using the MiSeq reagent kit v.3 (paired end, 2 × 300 bp, 600 cyc PE). Denaturation, dilution, and addition of 15% PhiX to the DNA library were performed according to the manufacturer's instructions. Custom sequencing primers were used as described previously [30].
Comparative genomics analyses
The genome of each strain was obtained querying RefSeq with the accession numbers in Source Data 5. Genes were mapped to COG terms using eggnog mapper and all strains were then labelled as either auxotroph or prototroph iteratively for each of the following compounds: biotin, thiamine, pantothenate, niacin. The analysis was restricted to COG terms for which a pathway mapping is provided (https://www.ncbi.nlm.nih.gov/research/cog/pathways/). A contingency table was then built based on the presence of each COG term in auxotrophs and prototrophs respectively. A χ2 test was performed on the contingency table, and p-values as well as information about which group had the higher presence of each COG term was stored. Resulting p values were subsequently corrected using the Benjamini-Hochberg method. For significantly different (q-value < 0.05) COG terms, a functional annotation was retrieved from COG database API (https://www.ncbi.nlm.nih.gov/research/cog/api/cog/) and mapped to biological process via KEGG (available from Biopython). The effectiveness of multiple testing correction was confirmed by generating random models by shuffling the auxotrophy/prototrophy labels. No COG term was significantly differently abundant between randomly assigned auxotrophs and prototrophs.
To create models for predicting auxotrophy from COG annotations, we trained decision tree classifiers based on the feature selection process presented above, or randomly chosen COG terms from each pathway (500 randomizations). First, we appended the dataset of 35 validated strains with additional 28 strains from the drop-out screen presented in Supplementary Fig. 1 to decrease sampling bias. This total set of 63 was divided to balanced training and testing datasets using 40% of the data for testing using "train_test_split" and decision tree classifiers were generated with "DecisionTreeClassifier" from sklearn library with a maximum depth of three nodes.
Consumer-resource models
Consumer-resource models were applied to analyze the success of auxotrophs in co-culture experiments. Consumer-resource models are representations of an organism's abundance as a function of its ability to consume a given resource. Here, we applied a consumer-resource model to determine the abundance of each bacterial strain using its experimentally observed CFU count (Supplementary Table 5), growth rate (μ), and yield. The carrying capacity C for each strain was determined by multiplying its experimentally-determined maximum yield by the CFU count per OD (CFU/OD column in Supplementary Table 5). All parameters are clarified in Supplementary Table 6.
Default model
At t = 0, each strain's abundance (CFU/ml) was set to its experimentally-observed abundance in the beginning of the experiment (the CFU/ml column in Supplementary Table 7) divided by 4 (as the strains were first mixed together, diluting each strain's abundance by a factor of 20, the inoculum was inoculated in 1/200 into fresh medium, and finally multiplied by 1000 to estimate the CFU/ml).
The concentration of each strain was then updated for each time point. Here, the growth rate of the strain was first updated according to Monod kinetics
$$\mu _{Strain} = \mu _{Strain,\,\max } \ast \frac{{\left[ S \right]}}{{\left[ S \right] + K_S}}$$
Where substrate affinity Ks was assumed to be proportional to the strain's yield (in terms of OD) and calculated by dividing the maximum yield of all strains by each strain's yield. For each strain, an ordinary differential equation was formulated to describe change in that strain's concentration:
$$\frac{{d\left[ {Strain} \right]}}{{dt}} = \mu _{Strain} \ast \left[ {Strain} \right] \ast \frac{{1 - \left[ {Strain} \right]}}{{C_{Strain}}}$$
Where Cstrain is the carrying capacity or yield of that strain. These strain-specific equations were coupled to an ODE describing change in glucose concentration where [Glucose] was initially set to 20 mM.
$$\frac{{d\left[ {Glucose} \right]}}{{dt}} = sum\left( {\frac{{10 \ast \overline {\mu _{Strain}} \ast \overline {[Strain]} }}{{\overline {C_{Strain}} }}} \right) \ast \left[ {Glucose} \right]$$
Where the first part under the sum estimates each's strains glucose consumption rate from its growth parameters (growth rate multiplied by 10). The total glucose consumption is then estimated from the consumption rates and number of cells scaled to carrying capacity. The unit under the sum is h−1. The model was then solved using odeint solver from scipy.
At t∈{24,48,72,96} a dilution step (1/200) was simulated by the following:
$$\overline {\left[ {Strain} \right]} = \overline {[Strain]} /200$$
$$\left[ S \right] = 20$$
$$\mu _{strain} = \mu _{strain,\,\max }$$
Extension 1: Vitamin cross-feeding
The carrying capacity C was determined for each strain separately as described above. Based on data presented in Fig. 5, we estimated that vitamin concentration increased maximally 10-fold in the presence of prototrophic strains. Since 1 µM vitamins were supplemented to cultures, we repeated the simulation described above with the carrying capacity CStrain for each strain based on their experimentally determined yield with 10 µM vitamins (Supplementary Fig. 9). For strains with lacking data, the average increase in carrying capacity was used instead.
Extension 2: Carbon cross-feeding
The second carbon source was simulated by setting a secretion flux only for strains whose growth rate was greater than a given threshold (varied from 0.3 −h to 0.6 h−1). The secretion flux was scaled down by 50% from the glucose uptake rate. At t = 0, S2 = 0.
The secretion of S2 into the medium by fast-growing strains was therefore controlled by
$$r_{S_2} = \left\{ {\begin{array}{*{20}{c}} {r_{glc,\,strain} \ast 0.5,}&\mu _{strain} \ge threshold \\ \hfill0,&{\mu _{strain} \, < \, threshold} \end{array}} \right.$$
The change in concentration of [S2] was then
$$\frac{{d\left[ {S_2} \right]}}{{dt}} = sum\left( {\frac{{10 \ast \overline {r_{s2,\,Strain}} \ast \overline {\left[ {Strain} \right]} }}{{\overline {C_{Strain}} }}} \right) \ast \left[ {S_2} \right]$$
The formula for concentration and production rate of S2 led to biologically realistic concentration range (~5 mM; Supplementary Fig. 10). On the receiving end, the growth rates for S2 were estimated as follows
$$\mu _{2,\,Strain} = \mu _{Strain,\,\max } \ast Scaling\_factor \ast \frac{{\left[ {S_2} \right]}}{{\left[ {S_2} \right] + K_{S_2}}}$$
For simulations in which the auxotrophs and prototrophs were equally efficient, the Scaling_factor parameter was set to 1/3. In order to make auxotrophs more efficient, the scaling factor was set to 1 for auxotrophs and kept at 1/3 for prototrophs. The sensitivity for this parameter was tested by setting it in range from 1/2 to 2. The concentration of each strain was simulated as follows:
$$\frac{{d\left[ {Strain} \right]}}{{dt}} = \mu _{Strain,\,glu\cos e} \ast \left[ {Strain} \right] \ast \frac{{1 - \left[ {Strain} \right]}}{{C_{Strain}}} + \mu _{2,\,Strain,} \ast \left[ {Strain} \right] \ast \frac{{1 - \left[ {Strain} \right]}}{{C_{S2,\,Strain}}}$$
Where \(C_{S_{2,\,Strain}}\)was set to 70. This parameter was set based on the finding that growth on metabolic by-products can be comparable to growth on glucose [49].
The simulations were repeated for time t∈{1,2,3,…,120}. At t∈{24,48,72,96}, dilution was simulated as described above.
Analysis software and statistical analysis
Unless otherwise stated, all analyses were performed on a Windows machine running Python 3.8 via Anaconda3 using custom scripts. Data were handled in pandas dataframes (V 1.1.3), for numerical computing numpy library (V 1.20.1) was used, and linear regression and multiple testing correction were performed via the sklearn and statsmodels (V 0.23.0 and V 0.12.0, respectively) implementations. For statistical testing, scipy (V 1.5.2) implementations were used. API's were queried via requests (V 2.22.0) and KEGG via Biopython (V 1.76). Metabolic models were generated using CarveMe [33] (V 1.2.2) following the published tutorial (https://carveme.readthedocs.io/en/latest/usage.html). For continuous data, t-test or Welch's t test was performed depending on variance within each group. For categorical data, χ2 tests were conducted. The number of replicates (n) and the type of test conducted can be found in respective Figure caption.
Relevant code used to generate consumer resource models are available as Supplementary material. Raw LC/MS metabolomics data for supernatants and depletion experiments are available in MetaboLights https://www.ebi.ac.uk/metabolights/index (Accession ID MTBLS5309). Raw sequencing data are available in European Nucleotide Archive https://www.ebi.ac.uk/ena/browser/home?show=reads (Accession ID PRJEB55397).
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We thank M. Schäfer for assistance in the auxotrophy screen as well as critical reading of the manuscript, C. Vogel for providing the COG annotations and evaluation of the phylogenetic analyses, S. Pfeilmeier for help in experimental design of the co-culture experiments, D. Demaj and P. Kirner for assistance in 16S rRNA gene library preparation, T. Stewart and J. Kurmann for experimental assistance with the vitamin titration experiments. We thank L. Hemmerle, P. Kiefer, and P. Keller for helpful discussions, and A. Pacheco for critical reading of the manuscript. We thank the Genetic Diversity Centre Zurich and especially S. Kobel for technical assistance and expertise with regards to sequence data, and C. Field for demultiplexing sequence data. This work was supported by the Swiss National Science Foundation (grant no. 310030B_201265).
Open access funding provided by Swiss Federal Institute of Technology Zurich.
Institute of Microbiology, ETH Zurich, 8093, Zurich, Switzerland
Birgitta Ryback, Miriam Bortfeld-Miller & Julia A. Vorholt
Birgitta Ryback
Miriam Bortfeld-Miller
Julia A. Vorholt
BR and JAV designed the study. BR conducted the experiments, analyses, and modelling. BR and MB-M performed co-culture experiments, sampling, and 16S rRNA gene amplicon library preparation and sequencing. BR and JAV wrote the manuscript.
Correspondence to Julia A. Vorholt.
Supplementary Tables
Python Consumer resource models
Ryback, B., Bortfeld-Miller, M. & Vorholt, J.A. Metabolic adaptation to vitamin auxotrophy by leaf-associated bacteria. ISME J 16, 2712–2724 (2022). https://doi.org/10.1038/s41396-022-01303-x
Revised: 13 July 2022
Issue Date: December 2022
Reply to: Erroneous predictions of auxotrophies by CarveMe
Daniel Machado
Kiran R. Patil
Nature Ecology & Evolution (2022) | CommonCrawl |
For how many integers $n$ where $2 \le n \le 100$ is $\binom{n}{2}$ odd?
$\binom{n}{2} = \frac{n(n-1)}{2}$. In order for this fraction to be odd, neither $n$ nor $n-1$ can be divisible by $4$, because only one of $n$ and $n-1$ can be even. There are $25$ integers where $n$ is divisible by $4$, namely the multiples of $4$ from $4$ to $100$. There are $24$ integers where $n-1$ is divisible by $4$. We can obtain these integers by incrementing all the multiples of $4$ by $1$, but we must not include $100$ since $100+1 = 101 > 100$. Therefore, there are $49$ invalid integers, so there are $99 - 49 = \boxed{50}$ valid integers. | Math Dataset |
scAI: an unsupervised approach for the integrative analysis of parallel single-cell transcriptomic and epigenomic profiles
Suoqin Jin1 na1,
Lihua Zhang1,2 na1 &
Qing Nie ORCID: orcid.org/0000-0002-8804-33681,2,3
Genome Biology volume 21, Article number: 25 (2020) Cite this article
Simultaneous measurements of transcriptomic and epigenomic profiles in the same individual cells provide an unprecedented opportunity to understand cell fates. However, effective approaches for the integrative analysis of such data are lacking. Here, we present a single-cell aggregation and integration (scAI) method to deconvolute cellular heterogeneity from parallel transcriptomic and epigenomic profiles. Through iterative learning, scAI aggregates sparse epigenomic signals in similar cells learned in an unsupervised manner, allowing coherent fusion with transcriptomic measurements. Simulation studies and applications to three real datasets demonstrate its capability of dissecting cellular heterogeneity within both transcriptomic and epigenomic layers and understanding transcriptional regulatory mechanisms.
The rapid development of single-cell technologies allows for dissecting cellular heterogeneity more comprehensively at an unprecedented resolution. Many protocols have been developed to quantify transcriptome [1], such as CEL-seq2, Smart-seq2, Drop-seq, and 10X Chromium, and techniques that measure single-cell chromatin accessibility (scATAC-seq) and DNA methylation have also become available [2]. More recently, several single-cell multiomics technologies have emerged for measuring multiple types of molecules in the same individual cell, such as scM&T-seq [3], scNMT-seq [4], scTrio-seq [5], sci-CAR-seq [6], and scCAT-seq [7]. The resulting single-cell multiomics data has potential of providing new insights regarding the multiple regulatory layers that control cellular heterogeneity [8, 9].
Gene expression is often regulated by transcription factors (TFs) via interaction with cis-regulatory genomic DNA sequences located in or around target genes [10, 11]. Epigenetic modifications, including changes in chromatin accessibility and DNA methylation, play crucial roles in the regulation of gene expression [12, 13]. Many tools have been developed for the integrative analysis of transcriptomic and epigenomic profiles in bulk samples [14,15,16]. For example, Zhang et al. integrated the analysis of bulk gene expression, DNA methylation, and microRNA expression using joint nonnegative matrix factorization [16]. Argelaguet et al. [17] presented MOFA, a generalization of principal component analysis (PCA) which is applicable to both bulk and single-cell datasets [18, 19].
Single-cell multiomics data are inherently heterogenous and highly sparse [9]. Although many integration methods initially developed for bulk data might be applicable to such data, it has become increasingly clear that new and different computational strategies are required due to unique characteristics of single-cell data [9]. In particular, scATAC-seq data are extremely sparse (e.g., over 99% zeros in sci-CAR-seq) and nearly binary [20], thus making it difficult to reliably identify accessible (or methylated) regions in a cell.
A growing number of methods have been developed for scRNA-seq data integration [21,22,23]. However, only few methods have been proposed for integrating multiomics profiles, and these methods were designed for data measured in different cells (i.e., not the same single cells) but sampled from the same cell population [22,23,24,25]. MATCHER used a Gaussian process latent variable model to compute the "pseudotime" for every cell in each omics layer and to predict the correlations between transcriptomic and epigenomic measurements from different cells of the same type [24]. A coupled nonnegative matrix factorization method performed clustering of single cells sequenced by scRNA-seq and scATAC-seq through constructing a "coupling matrix" for regulatory elements and gene associations [25]. Recently, Seurat (version 3) [22] and LIGER [23] were developed for integrating scRNA-seq and single-cell epigenomic data. Both of these methods first transform the epigenomic data into a synthetic scRNA-seq data through estimating a "gene activity matrix," and then identify "anchors" between this synthetic data and scRNA-seq data through aligning them in a low-dimensional space. The gene activity matrix is created by simply summing all counts within the gene body +2 kb upstream. Such strategy may introduce improper synthetic data due to complex transcriptional regulatory mechanisms between gene expression and chromatin accessibility. The improper synthetic data may further lead to imperfect alignment when they are applied to parallel transcriptomic and epigenomic profiles, and likely affect downstream analysis. Moreover, the inference of interactions between transcriptomics and epigenetics often requires both measurements from the same single cell [8].
Here, we present a single-cell aggregation and integration (scAI) approach to integrate transcriptomic and epigenomic profiles (i.e., chromatin accessibility or DNA methylation) that are derived from the same cells. Unlike existing integration methods [16, 17, 22, 24,25,26], scAI takes into consideration the extremely sparse and near-binary nature of single-cell epigenomic data. Through iterative learning in an unsupervised manner, scAI aggregates epigenomic data in subgroups of cells that exhibit similar gene expression and epigenomic profiles. Those similar cells are computed through learning a cell-cell similarity matrix simultaneously from both transcriptomic and aggregated epigenomic data using a unified matrix factorization model. As such, scAI represents the transcriptomic and epigenomic profiles with biologically meaningful low-rank matrices, allowing identification of cell subpopulations; simultaneous visualization of cells, genes, and loci in a shared two-dimensional space; and inference of the transcriptional regulatory relationships. Through applications to eight simulated datasets and three published datasets, and comparisons with recent multiomics data integration methods, scAI is found to be an efficient approach to reveal cellular heterogeneity by dissecting multiple regulatory layers of single-cell data.
Overview of scAI
To deconvolute heterogeneous single cells from both transcriptomic and epigenomic profiles, we aggregate the sparse/binary epigenomic profile in an unsupervised manner to allow coherent fusion with transcriptomic profile while projecting cells into the same representation space using both the transcriptomic and epigenomic data. Using the normalized scRNA-seq data matrix X1 (p genes in n cells) and the single-cell chromatin accessibility or DNA methylation data matrix X2 (q loci in n cells) as an example, we infer the low-dimensional representations via the following matrix factorization model:
$$ {\min}_{W_1,{W}_2,H,Z\ge 0}\alpha {\left\Vert {X}_1-{W}_1H\right\Vert}_F^2+{\left\Vert {X}_2\left(Z\circ R\right)-{W}_2H\right\Vert}_F^2+\lambda {\left\Vert Z-{H}^TH\right\Vert}_F^2+\gamma \sum \limits_j{\left\Vert {H}_{.j}\right\Vert}_1^2, $$
where W1 and W2 are the gene loading and locus loading matrices with sizes p × K and q × K (K is the rank), respectively. Each of the K columns is considered as a factor, which often corresponds to a known biological process/signal relating to a particular cell type. \( {W}_1^{ik} \) and \( {W}_2^{ik} \) are the loading values of gene i and locus i in factor k, and the loading values represent the contributions of gene i and locus i in factor k. H is the cell loading matrix with size K × n (H.j is the jth column of H), and the entry Hkj is the loading value of cell j when mapped onto factor k. Z is the cell-cell similarity matrix. R is a binary matrix generated by a binomial distribution with a probability s. α, λ, γ are regularization parameters, and the symbol ∘ represents dot multiplication. The model aims to address two major challenges simultaneously: (i) the extremely sparse and near-binary nature of single-cell epigenomic data and (ii) the integration of this binary epigenomic data with the scRNA-seq data, which are often continuous after being normalized.
Aggregation of epigenomic profiles through iterative refinement in an unsupervised manner
To address the extremely sparse and binary nature of the epigenomic data, we aggregate epigenomic data of similar cells based on the cell-cell similarity matrix Z, which is simultaneously learned from both transcriptomic and epigenomic data iteratively. Epigenomic data can be simply aggregated by X2Z. However, this strategy may lead to over-aggregation, for example, in one subpopulation, similar cells exhibit almost the same aggregated epigenomic signals, which improperly reduces the cellular heterogeneity. To reduce such over-aggregation, a binary matrix R, generated from a binomial distribution with probability s, is utilized for randomly sampling of similar cells. After normalizing H with the sum of each row equaling 1 in each iteration step and Z°R with the sum of each column equaling 1, then the aggregated epigenomic profiles are represented by X2(Z ∘ R). The ith column of X2(Z ∘ R) represents the weighted combination of epigenomic signals from some cells similar to the ith cell. These strategies not only enhance epigenomic signals, but also maintain cellular heterogeneity within and between different subpopulations.
Integration of binary and count-valued data via projection onto the same low-dimensional space
Through aggregation, the extremely sparse and near-binary data matrix X2 is transformed into the signal-enhanced continuous matrix X2(Z ∘ R), allowing coherent fusion with transcriptomic measurements (Fig. 1a). These two matrices are projected onto a common coordinate system represented by the first two terms in the optimization model (Eq. (1)). In this way, cells are mapped onto a K-dimensional space with the cell loading matrix H, and the cell-cell similarity matrix Z is approximated by H′H, as represented by the third term in Eq. (1). The sparseness constraint on each column of H is added by the last term of Eq. (1).
Overview of scAI. a scAI learns aggregated epigenomic profiles and low-dimensional representations from both transcriptomic and epigenomic data in an iterative manner. scAI uses parallel scRNA-seq and scATAC-seq/single cell DNA methylation data as inputs. Each row represents one gene or one locus, and each column represents one cell. In the first step, the epigenomic profile is aggregated based on a cell-cell similarity matrix that is randomly initiated. In the second step, transcriptomic and aggregated epigenomic data are simultaneously decomposed into a set of low-rank matrices. Entries in each factor (column) of the gene loading matrix (gene space), locus loading matrix (epigenomic space), and cell loading matrix (cell space) represent the contributions of genes, loci, and cells for the factor, respectively. In the third step, a cell-cell similarity matrix is computed based on the cell loading matrix. These three steps are repeated iteratively until the stop criterion is satisfied. b scAI ranks genes and loci in each factor based on their loadings. For example, four genes and loci are labeled with the highest loadings in factor 3. c Simultaneous visualization of cells, marker genes, marker loci, and factors in a 2D space by an integrative visualization method VscAI, which is constructed based on the four low-rank matrices learned by scAI. Small filled dots represent the individual cells, colored by true labels. Large red circles, black filled dots, and diamonds represent projected factors, marker genes, and marker loci, respectively. d The regulatory relationships are inferred via correlation analysis and nonnegative least square regression modeling of the identified marker genes and loci. An arch represents a regulatory link between one locus and the transcription start site (TSS) of each marker gene. The arch colors indicate the Pearson correlation coefficients for gene expression and loci accessibility. The red stem represents the TSS region of the gene, and the black stem represents each locus
Downstream analysis using the inferred low-dimensional representations
scAI simultaneously decomposes transcriptomic and epigenomic data into multiple biologically relevant factors, which are useful for a variety of downstream analyses (Fig. 1b–d). (1) The cell subpopulations can be identified from the cell loading matrix H using a Leiden community detection method (see the "Methods" section). (2) The genes and loci in the ith factor are ranked based on the loading values in the ith columns of W1 and W2 (see Fig. 1b and the "Methods" section). (3) To simultaneously analyze both gene and loci information associated with cell states, we introduce an integrative visualization method, VscAI. By combining these learned low-rank matrices (W1, W2, H, and Z) with the Sammon mapping [27] (see the "Methods" section), VscAI simultaneously projects genes and loci that separate the cell states into a two-dimensional space alongside the cells (Fig. 1c). (4) Finally, the regulatory relationships between the marker genes and the chromosome regions in each factor or cell subpopulation are inferred by combining the correlation analysis and the nonnegative least square regression modeling of gene expression and chromatin accessibility (see Fig. 1d and the "Methods" section). Overall, these functionalities allow the deconvolution of cellular heterogeneity and reveal regulatory links from transcriptomic and epigenomic layers.
Model validation and comparison using simulated data
To evaluate scAI, we simulated eight single-cell datasets with the sparse count data matrix X1 and the sparse binary data matrix X2 (i.e., paired scRNA-seq and scATAC-seq). To recapitulate the properties of the single-cell multiomics data (e.g., a high abundance of zeros and binary epigenetic data), we generated bulk RNA-seq and DNase-seq profiles from the same sample with MOSim [28]. Then, we added the effects of dropout and binarized the data. A detailed description of the simulation approach and the simulated data are shown in Additional file 1: Supplementary methods (Simulation datasets) and Additional file 2: Table S1. These datasets encompass eight scenarios with different transcriptomic/epigenomic properties: different sparsity levels (dataset 1), different noise levels (dataset 2), missing clusters in the epigenomic profiles (i.e., clusters defined from gene expression do not reflect epigenetic distinctions) (dataset 3), missing clusters in the transcriptomic profiles (i.e., clusters defined from epigenetic profile do not reflect gene expression distinctions) (dataset 4), discrete cell states (dataset 5), a continuous biological process (dataset 6), imbalanced cluster sizes with the same number of clusters defined from both transcriptomic and epigenomic profiles (dataset 7), and imbalanced cluster sizes with missing clusters in the epigenomic profiles (dataset 8).
First, we compared the visualization of cells using the scRNA-seq data, scATAC-seq data, and aggregated scATAC-seq data, respectively (Fig. 2a). Due to the inherent sparsity and noise in the data, the cells were not well separated in the scRNA-seq data and the scATAC-seq data using Uniform Manifold Approximation and Projection [29] (UMAP) (Fig. 2a) and t-SNE (Additional file 2: Figure S1), in particular for datasets 5 and 6. However, the cell subpopulations were clearly distinguishable in the low-dimensional space when using the aggregated scATAC-seq data generated by scAI for all eight different scenarios (Fig. 2a). In addition, the cell subpopulations were well separated when visualized by VscAI, which embedded cells in two dimensions by leveraging the information from both scRNA-seq and scATAC-seq data (Fig. 2b). For dataset 3 and dataset 4, in which one cluster was missing in either the transcriptomic or the epigenomic data alone, scAI was able to reveal all the anticipated clusters. For example, in dataset 4, only four clusters were revealed in the scRNA-seq data, but five clusters were embedded in the scATAC-seq data (the fourth row of Fig. 2a). Without the addition of the scATAC-seq information, four clusters were detected (Additional file 2: Figure S2), whereas the integration of both the scRNA-seq and the scATAC-seq data revealed five clusters. In the first five datasets, the cell states are discrete whereas dataset 6 depicts a continuous transition process at five different time points. The continuous transitions in these five cell states were well characterized by scAI with the aggregated scATAC-seq data but could not be captured by using only the sparse scATAC-seq data with UMAP (the sixth row of Fig. 2a) and t-SNE (Additional file 2: Figure S1). For the datasets 7 and 8 with imbalanced cluster sizes, scAI accurately revealed all the expected clusters. In particular, three cell clusters were observed in the low-dimensional space of both scATAC-seq and aggregated scATAC-seq data in the dataset 8 (the eighth row of Fig. 2a). However, five cell clusters were well distinguished after integrating with scRNA-seq data, as shown in the VscAI space (the eighth row of Fig. 2b).
Performance of scAI and its comparison with MOFA using eight simulated datasets. a 2D visualization of cells by applying UMAP to scRNA-seq, scATAC-seq, and aggregated scATAC-seq data obtained from scAI. Each row shows one example of each scenario from the simulated datasets. Cells are colored based on their true labels. b Cells are visualized by VscAI. c Accuracy of scAI (evaluated by AUC) in reconstructing cell loading (blue color), gene loading (orange color), and locus loading (yellow color) matrices, respectively. For each scenario, we generated a set of simulated data using five different parameters, which are indicated on the x-labels. The numbers outside and inside the brackets represent the parameters in the simulated scRNA-seq and scATAC-seq data, respectively. We applied scAI to each dataset 10 times with different seeds and then calculated the average AUCs with respect to the ground truth of the loading matrices. Datasets 5 and 6 were generated based on real datasets, which do not have ground truth of the gene/locus loading matrices. d Variance explained by each latent factor (LF) using MOFA. e Comparison of the accuracy (evaluated by normalized mutual information, NMI) of scAI and MOFA in identifying cell clusters
Next, we used the area under receiver operating characteristic curve (AUC) to quantitively evaluate the accuracy of scAI in reconstructing cell loading matrix H, gene loading matrix W1, and locus loading matrix W2, which were used for identifying cell clusters, factor-specific genes, and loci in the downstream analyses, respectively. scAI was found to perform robustly and accurately with different sparsity levels and noise levels (Fig. 2c). For example, even with the sparsity levels of X1 and X2 at 98% and 99.6% in dataset 1, and 79.4% and 97.5% in dataset 5, scAI was able to reconstruct these loading matrices with high accuracy (Fig. 2c).
Moreover, to study whether stronger noise or the initial data with less discriminative patterns have effects on the performance of scAI, we added stronger noise and sparsity levels, and also made the initial data less discriminative among clusters by increasing the parameter value coph, on the simulation dataset 8. We found that the noise levels and parameter coph values have little effects on the reconstructed loading matrices. The sparsity level affects the performance if it is larger than some threshold (e.g., the sparsity of scRNA-seq and scATAC-seq data is larger than 98.9% and 99.5%, respectively), as shown in Additional file 2: Figure S3.
Finally, we applied MOFA [17], a multiomics data integration model designed for bulk data and single-cell data, to the eight datasets (Fig. 2d, e). MOFA decomposes multiomics data matrices into several weight matrices and one factor matrix using a statistically generalized principal component analysis method. For all the datasets except for dataset 7, the factors learned by MOFA only accounted for the variability of the scRNA-seq data, and could not capture the variance in the scATAC-seq data (Fig. 2d). We compared scAI with MOFA on cell clustering (Fig. 2e), finding MOFA does not perform as well as scAI for these simulation datasets (Fig. 2e).
The analysis on simulation data indicates scAI's potential in aggregating scATAC-seq data, identifying important genes and loci, and uncovering discrete and continuous cell states in single-cell transcriptomic and epigenomic data with inherently high sparsity and noise levels.
Identifying subpopulations with subtle transcriptomic differences but strong chromatin accessibility differences
To evaluate scAI in capturing cell subpopulations in complex tissues, we analyzed 8837 cells from mammalian kidney using the paired chromatin accessibility and transcriptome data [6]. In a previous study, a semi-supervised clustering method was applied to the scRNA-seq data, and then, aggregated epigenomic profiles were generated based on the identified cell clusters [6]. As such, the cellular heterogeneity induced by epigenetics was unable to be captured in this method.
scAI identified 17 subpopulations with either distinct gene expression or chromatin accessibility profiles with the default resolution parameter equaling 1 (see the "Methods" section; Fig. 3a, b, d; Additional file 1). Compared to the original findings [6], our integrative analysis of transcriptomic and chromatin accessibility profiles indicated that the known cell types such as Collecting Duct Principal Cells (CDPC) were much more heterogeneous. We identified two subpopulations of CDPC (C9 and C12, Additional file 2: Figure S4a) that were captured by factor 2 and factor 8, respectively (Fig. 3c). Gene loading analysis of these two factors revealed that Fxyd4 and Frmpd4 are the specific markers of C9, while Egfem1 and Calb1 are the specific makers of C12 (Fig. 3c, and Additional file 2: Figure S4b and c). Importantly, while some identified subpopulations showed only subtle differences in their transcriptomic profiles, they exhibited distinct patterns in their epigenomic profiles (Fig. 3b, d). For example, C2 and C7 (subpopulations of proximal tubule S3 cells (type 1)), and C8 and C10 (subpopulations of proximal tubule S1/S2 cells) have similar gene expression profiles (Fig. 3b), but, exhibit strong differential accessibility patterns (Fig. 3e). The average signals of each locus across cells in each subpopulation are significantly different (Fig. 3e).
Identifying new epigenomics-induced subpopulations by simultaneously analyzing transcriptomic and epigenomic profiles in mouse kidney. a UMAP visualization of cells, which are colored by the inferred subpopulations. b Heatmap of differentially expressed genes. For each cluster, the top 10 marker genes and their relative expression levels are shown. Selected genes for each cluster are color-coded and shown on the right. c UMAP plots show the cell cluster-specific patterns of the identified factors (left), and the ranking plots show the top marker genes in the corresponding factors (right). In the projected factor pattern plots, cells are colored based on the loading values in the factor from the inferred cell loading matrix. In gene ranking plots, genes are ranked based on the gene scores in the factor from the gene loading matrix. Labeled genes are representative markers. d Heatmap showing the relative chromatin accessibility of cluster-specific loci. e Heatmap of the raw chromatin accessibility of individual cells (left) and the average chromatin accessibility of cell clusters (including C2, C7, C8, and C10) (right) using differential accessible loci among the cell clusters. f Regulatory information of eight identified cell clusters. The identities of these subpopulations were shown on the most left
To further characterize these differential accessible loci and identify the specific transcriptional regulatory mechanisms of these epigenetics-induced subpopulations, we performed gene ontology enrichment and motif discovery analysis using GREAT and HOMER, respectively (Fig. 3f). Notably, for the two subpopulations C8 and C10 of proximal tubule S1/S2 cells, the C8-specific accessible loci were related to the chromatin binding and histone deacetylase complex, and were further enriched for binding motifs of MAFB and JUNB, both of which are known regulators of proximal tubule development [30]. Differential accessible loci of C10 were enriched in VEGFR signaling pathway, consistent with the role in the maintenance of tubulointerstitial integrity and the stimulation of proximal tubule cell proliferation [31].
Moreover, we applied chromVAR [32] to analyzing the differential accessible loci between C2 and C7, and C8 and C10, respectively. chromVAR calculates the bias corrected deviations in accessibility. For each motif, there is a value for each cell, which measures how different the accessibility for loci with that motif is from the expected accessibility based on the average of all the cells. By performing hierarchical clustering of the calculated deviations of top 30 most variable TFs, we found that these TFs were divided into 2 clusters, and each TF cluster was specific to 1 particular cell subpopulation, which was found to be consistent with the clustering by scAI (Additional file 2: Figure S5).
Revealing underlying transition dynamics by analyzing transcription and chromatin accessibility simultaneously
Next, we applied scAI to data from lung adenocarcinoma-derived A549 cells after 0, 1, and 3 h of 100 nM dexamethasone (DEX) treatment, including scRNA-seq and scATAC-seq data from 2641 co-assayed cells [6]. scAI revealed two factors, where factor 1 was enriched with cells from 0 h and factor 2 was enriched with cells from 3 h (Fig. 4a). Factor-specific genes and loci were identified by analyzing the gene and locus loading matrices (Fig. 4b). Among them, known markers of glucocorticoid receptor (GR) activation [33,34,35] (e.g., CKB and NKFBIA) were enriched in factor 2, and markers of early events after treatment [36] (e.g., ZSWIM6 and NR3C1) were enriched in factor 1. We collected TFs of these known markers from hTFtarget database (http://bioinfo.life.hust.edu.cn/hTFtarget/). Interestingly, the TF motifs, such as FOXA1 [37], CEBPB [38], CREB1, NR3C1, SP1, and GATA3 [39], also had high enrichment scores in the inferred factors (Fig. 4c), in agreement with that these motifs are key transcriptional factors of GR activation markers [40]. Particularly, CEBPB binding was shown positively associated with early GR binding [41], and GR binds near CREB1 binding sites that makes enhancer chromatin structure more accessible [42]. In the low-dimensional space visualized by VscAI, markers of early events, such as ZSWIM6 and NR3C1, were located near cells from 0 h, while markers of GR activation, such as CKB, NKFBIA, and ABHD12, were located near cells from 3 h (Fig. 4d), providing a direct and intuitive way to interpret the data.
Revealing cellular heterogeneity and regulatory links of dexamethasone-treated A549 cells. a Heatmap of the cell loading matrix H obtained by scAI. Cells are ordered and divided into early, transition, and late stages based on hierarchical clustering. The bar at the bottom indicates the collection time of each cell. b Genes are ranked in each factor based on gene scores calculated from gene loading matrix, in which the known markers are indicated. c Loci are also ranked based on locus scores from locus loading matrix, in which the motifs and the corresponding logo of some TFs of the known marker genes are indicated. The binding TFs of the known marker genes and the chromosome loci of these motifs were found from hTFtarget database. d Visualization of cells by VscAI. Known marker genes (left panel) and motifs related with these marker genes (right panel) were projected onto the same low-dimensional space. The same motifs such as SMAD3 and NR3C1 are shown in two opposite positions, as they are enriched in different loci. These loci were located within 10 kb of marker genes' regulatory regions, which were extracted from the database (http://bioinfo.life.hust.edu.cn/hTFtarget/) in lung tissue. Here, we visualized the motifs instead of individual loci for easier understanding. e The fold enrichment (FE) values of inferred regulatory links of the known markers, which were validated by the hTFtarget database. f Inferred regulatory links of gene ABHD12 for each factor and the epigenome browser visualization of DNase-seq data and NR3C1 ChIP-seq data derived from chromatin regions near TSS of ABHD12. The red stem represents the TSS region of the gene, and the black stem represents each locus. Five regulators which correspond to the inferred regulatory links were indicated. The gray region shows the distinct regulatory links (regulated by NR3C1) between 0 h and 1 and 3 h. g The pseudotemporal chromatin accessibility trajectory was inferred with the aggregated scATAC-seq data. Cells were visualized in the first two diffusion components (DCs). The gray line is the fitted principal curve. Bottom: the percentages of cells at the three time points during the inferred pseudotime, which was divided into 10 bins. h Inferred pseudotime of three key genes. The black line indicates the fitted expression levels using cubic splines. i Left: "Rolling wave" plot shows the normalized smoothed accessibility data for the pseudotime-dependent accessible loci clustered into two groups. Middle: the normalized smoothed gene expression data for the pseudotime-dependent genes along the inferred accessibility trajectory using the aggregated scATAC-seq data. Loci and genes are ordered based on the onset of activation. Right: the corresponding gene dynamics along the cellular trajectory inferred only using scRNA-seq data
To systematically assess the top ranked genes and loci in the identified factors, we performed pathway enrichment analysis of genes with MSigDB [43] and loci with GREAT [44]. As expected, several processes relevant to GR activation were uncovered, such as the "neurotrophin signaling pathway," a pathway previously reported to have a direct effect on GR function [45]. The "Fc epsilon RI signaling pathway" was enriched in factor 2 (Additional file 2: Figure S6a), which is in good agreement with that the reduction of Fc epsilon RI levels might be one of the favorable anti-allergic functions of glucocorticoids in mice [46]. Furthermore, processes such as "genes involved in glycogen breakdown (glycogenolysis)," "genes involved in glycerophospholipid biosynthesis," and "pentose and glucuronate interconversions" were enriched in the nearby genes of the factor-specific loci (Additional file 2: Figure S6b).
While the DEX treatment of A549 cells is known to increase both transcription and promoter accessibility for markers of GR activation [6], little is known on the regulatory relationships. We inferred regulatory links between cis-regulatory elements and target marker genes using perturbation-based correlation analysis and further identified bounded TFs that regulate target marker genes using nonnegative least square regression (see the "Methods" section). To assess the accuracy of the inference, we evaluated whether these regulatory relationships were enriched in an independent database of TF-target relationships for human (hTFtarget, http://bioinfo.life.hust.edu.cn/hTFtarget/) (see the "Methods" section). Encouragingly, high enrichment of the inferred regulatory relationships for the key markers of GR activation was observed (Fig. 4e), and the inferred regulatory relationships were able to be validated using ChIP-seq and DNase-seq data from ENCODE (https://www.encodeproject.org/). For the GR activation marker ABHD12 that was highly enriched in factor 2, we identified distinct regulatory links between factor 1 (enriched with cells from 0 h) and factor 2 (enriched with cells from 3 h). Among its regulators, the glucocorticoid receptor NR3C1 was revealed in factor 2 (Fig. 4f). Visualizing the chromatin signals of ChIP-seq data of NR3C1 and DNase-seq data using WashU Epigenome Browser (https://epigenomegateway.wustl.edu/browser), we found that most cis-regulatory elements are located in the open regions of the DNase-seq data, and that NR3C1 exhibits signals within 50 kb of the transcription start site (TSS) of ABHD12 at 1 and 3 h but no signals at 0 h in the ChIP-seq data. This is consistent with our prediction on the regulation between NR3C1 and ABHD12 existing in factor 2, but not in factor 1.
scAI provides an unsupervised way to aggregate sparse scATAC-seq data from similar cells through iterative refinement, which facilitates and enhances the direct analysis of scATAC-seq data. We next assess the performance of the aggregated scATAC-seq data in comparison with the raw scATAC-seq or scRNA data, in terms of the identification of cell states, the low-dimensional visualization of cells, and the reconstruction of the pseudotemporal dynamics. The previous study [6] identified two clusters that comprised a group of untreated cells and a group of DEX-treated cells, in which treated cells collected from 1 and 3 h form one cluster. Our analysis recovered three cell states, including an early state enriched by cells from 0 h, a transition state enriched by cells from 1 h, and a late state enriched by cells from 3 h (Fig. 4a). Due to the high sparsity (96.8% for scRNA-seq and 99.2% for scATAC-seq) and the near-binary nature of the scATAC-seq data, dimension reduction methods, such as t-SNE, were found to fail to distinguish the different cell states (Additional file 2: Figure S6c). However, scAI uncovered distinct cell subpopulations, as seen in the low-dimensional space, based on the aggregated data (Additional file 2: Figure S6c).
We next study the pseudotemporal dynamics of A549 cells using our previously developed method scEpath [47]. Compared to the trajectory inferred using only the scRNA-seq data, which lacks well-characterized GR activation trends for cells measured at three different time points (Additional file 2: Figure S6d), a clear and consistent trajectory was inferred when using the aggregated scATAC-seq data (Fig. 4g, h). We identified pseudotime-dependent genes and loci that were significantly changed along the inferred trajectories. The pseudotemporal dynamics of these genes along the trajectory inferred using only the scRNA-seq data were found to be discontinuous, in contrast to the aggregated scATAC-seq data obtained from scAI led to continuous trajectory (Fig. 4i). Previously, we used the measure scEnergy to quantify the developmental process [47]. Here, we found no significant differences in the single-cell energies between different time points when only using the scRNA-seq data. However, significantly decreased scEnergy values were seen during treatment according to the aggregated scATAC-seq data (Additional file 2: Figure S6e).
Overall, the aggregated scATAC-seq data by scAI can better characterize the dynamics of DEX treatment, and scAI suggests new mechanisms regarding the GR activation process in DEX-treated A549 cells, including a transition state and differential cis-regulatory relationships.
Uncovering coordinated changes in the transcriptome and DNA methylation along a differentiation trajectory
To study data with simultaneous single-cell methylome and transcriptome sequencing [3, 8, 48], we applied scAI to a dataset obtained from 77 mouse embryonic stem cells (mESCs), including 13 cells cultured in "2i" media and 64 serum-grown cells, which were profiled by parallel single-cell methylation and transcriptome sequencing technique scM&T-seq [3]. The DNA methylation levels were characterized in three different genomic contexts, including CpG islands, promoters, and enhancers, which are usually linked to transcriptional repression [49, 50].
Because DNA methylation data are sparse and binary, direct dimensional reduction may fail to capture cell subpopulations (Fig. 5a). scAI was able to distinguish cell subpopulations after aggregation (Fig. 5a), showing three subpopulations, C1, C2, and C3. Among them, C3 was captured by factor 1 with cells cultured in "2i" media and a few serum-grown cells, while C1 and C2 were captured by factors 2 and 3, respectively, with other serum-grown cells (Additional file 2: Figure S7).
Uncovering coordinated changes between the transcriptome and DNA methylation within an embryonic stem cell differentiation trajectory. a Comparisons of principal component analyses (PCA) of scRNA-seq data, single-cell DNA methylation data, and aggregated single-cell DNA methylation data learned by scAI. Cells are colored based on the cell subpopulations identified using scAI. Marker shapes denote the culture conditions. b Heatmap of the expression level and methylation level of cluster-specific marker genes (left), loci (middle), and loci within 500 kb of the TSS of marker genes (right) in the three cell clusters. c Genes and loci are ranked based on their enrichment scores in each factor. Labeled genes (top) are known pluripotency markers or differentiation markers. Labeled loci (bottom), including CpG sites, enhancers, and promoters, are located within 500 kb of the TSS of marker genes in each factor. d VscAI visualization of cells and known pluripotency markers and differentiation markers derived from transcriptome (top) and DNA methylation (bottom) data
Based on the top gene and locus loadings in each factor, we identified 688, 877 and 422 marker genes and 2164, 953 and 4461 differential methylated loci in C1, C2, and C3, respectively, with distinct gene expression and methylation patterns among these three groups (Fig. 5b). Moreover, methylation levels of loci near marker genes also showed group-specific patterns (Fig. 5b). Several known pluripotency markers (e.g., Essrb, Tcl1, Tbx3, Fbxo15, and Zpf42) exhibited the highest gene enrichment scores in factor 1 but the lowest gene enrichment scores in factors 2 and 3. In contrast, differentiation markers, such as Krt8, Tagln, and Krt19, exhibited higher gene enrichment scores in factor 3 but lower enrichment scores in factors 1 and 2 (Fig. 5c). Factor 2 exhibited an intermediate state with a relatively low expression level of both pluripotency and differentiation markers. Interestingly, several new marker genes of this intermediate state were observed, such as Fgf5, an early differentiation marker involved in neural differentiation in human embryonic stem cells [51]. Factor-specific loci located in the CpG, promoter, and enhancer regions of marker genes are also shown in Fig. 5c. The pluripotency markers Essrb and Tcl1 had higher gene enrichment scores, and their corresponding CpG, promoter, and enhancer regions had higher locus enrichment scores in factor 1. This relationship is consistent with the fact that some DNA methylation located in the CpG, promoter, and enhancer regions exhibit a negative relationship with the expression level of target genes.
A continuous differentiation trajectory, which was characterized by the differentiation of naïve pluripotent cells (NPCs) into primed pluripotent cells and ultimately into differentiated cells (DCs), was observed using VscAI (Fig. 5d). Additionally, the embedded genes and factors showed how specific genes and factors contribute to the differentiation trajectory. For example, pluripotency markers, such as Zpf42, Tex19.1, Fbxo15 Morc1, Jam2, and Esrrb [52, 53], were visually close to factor 1, while differentiation markers, such as Krt19 and Krt8 [54], were close to factor 3 (Fig. 5d). Interestingly, although both pluripotency and differentiation markers were not highly expressed in the early differentiated state in factor 2, some methylated loci of these markers (e.g., CpG regions of Zfp42 and Tex19.1, enhancer region of Jam2 and Tcl1, and promoter region of Anxa3) were enriched in factor 2 (Fig. 5d). These observations might be because their other regions (CpG, enhancer, or promoter) are methylated or DNA methylation is not the main driven force for transcriptional silencing. Overall, scAI shows coordinated changes between transcriptome and DNA methylation along the differentiation process.
Comparison with three multiomics data integration methods
We next compared scAI with three recent single-cell integration methods, MOFA [17], Seurat (version 3) [22], and LIGER [23], on A549 and kidney datasets. Similar to the observations on the simulation datasets (Fig. 2d), MOFA cannot capture the variations in the scATAC-seq data as the variances explained by the learned factors in the scATAC-seq data were nearly zero (Additional file 1: Supplementary methods (Details of data analysis by MOFA) and Additional file 2: Figure S8a-e). While Seurat and LIGER were designed for connecting cells measured in different experiments, we applied them to the two co-assayed single-cell multiomics data to test whether they are able to make links between co-assayed cells. We assessed the comparison using two metrics: (a) entropy of batch mixing and (b) silhouette coefficient. The entropy of batch mixing measures the uniformity of mixing for two samples in the aligned space [55], for which scRNA-seq and scATAC-seq profiles were treated as two batches, and a higher entropy value means better alignment. The silhouette coefficient quantifies the separation between cell groups using distance matrices calculated from a low-dimensional space [55], for which cell group labels were taken from the original study [6] and a higher silhouette coefficient indicates better preservation of the differences and structures between different cell groups.
The t-SNE analysis shows the co-assayed cells were aligned better by LIGER than Seurat when the two methods were applied to A549 dataset (Fig. 6a). This observation is further confirmed by computing the entropy of the batch mixing based on the aligned t-SNE space. We also computed the entropy of perfect alignment (i.e., the t-SNE coordinates of each pair of co-assayed cells are the same), and found that LIGER showed higher entropy value than Seurat, but lower entropy than the perfect alignment (Fig. 6a). In addition, we explored the quality of time point-based grouping of cells on the t-SNE space. Cells from 1 and 3 h were mixed together on the t-SNE space generated by Seurat, while there was a gradual change of cells from 0 to 3 h on the t-SNE space generated by LIGER (Fig. 6b). We also performed t-SNE on the cell loading matrix inferred by scAI (Additional file 2: Figure S8f), and found that scAI was able to capture the gradual change of cells transitioning from 0 to 3 h. Quantitatively, scAI produced significantly higher silhouette coefficients than those from both Seurat and LIGER (Fig. 6b).
Comparisons with multiomics data integration methods. a t-SNE visualizations of scRNA and scATAC-seq data from co-assayed A549 cells, colored by measurements (RNA vs. ATAC) after integration with Seurat (left) and LIGER (middle). Right panel: comparisons of alignment score (quantified by the entropy of batch mixing) from perfect alignment (termed as gold-standard) with that computed from the aligned t-SNE space using Seurat and LIGER. p values are from the Wilcoxon rank-sum test. b Cells are colored by the data collection times. Right panel: comparisons of silhouette coefficient computed from the t-SNE coordinates of each cell generated by scAI with that computed from the aligned t-SNE space using Seurat and LIGER. c, d UMAP visualizations of scRNA and scATAC-seq data from co-assayed mouse kidney cells colored by measurements (RNA vs. ATAC) (c) and published cell labels (d) after integration with Seurat and LIGER. The alignment score and silhouette coefficient were also shown
In the kidney dataset, by computing the entropy of the batch mixing based on the aligned UMAP space, we observed significantly lower entropy of Seurat and LIGER than that of the perfect alignment (Fig. 6c). We then also calculated the silhouette coefficient using the UMAP space for all three methods (Fig. 6d and Additional file 2: Figure S8f). Again, significantly higher silhouette coefficients were observed in scAI, in comparison with those in Seurat and LIGER (Fig. 6d). Together, these results suggest that integration methods designed for measurements in different cells (e.g., Seurat and LIGER) may not accurately identify correspondences between the co-assayed cells, leading to errors in downstream analysis, and the integration of parallel single-cell omics data needs specialized methods, such as scAI, to deal with the epigenomic data with inherently high sparsity and to better preserve intrinsic differences between cell subpopulations.
Comparison with methods using single omics data
To evaluate the significance of the parallel profiling of multiomics over single omics data, we compared scAI with methods that use only transcriptomic data or only epigenomic data on both simulation and real datasets. Specifically, we compared scAI with two methods designed for only scRNA-seq data, including Seurat and SC3 [56], and two methods designed for only scATAC-seq data, including Signac (https://satijalab.org/signac/) and scABC [57]. On simulation datasets, we evaluated the performance of cell clustering using normalized mutual information (NMI). On real datasets, we compared the clustering based on those four methods with prior labels using UMAP.
On simulation datasets, we observed comparable NMI values between scAI and SC3, but slightly lower values of Seurat (Additional file 2: Figure S9). For the clustering of scATAC-seq data, both Signac and scABC showed significantly lower NMI values compared to those by scAI using both scRNA-seq and scATAC-seq data. On A549 real datasets, by visualizing cells in UMAP, we found that both Seurat and SC3 were unable to detect the transition stage and distinguish cells from 1 and 3 h. Cell clusters identified by Signac and scABC using scATAC-seq data alone were found to be inconsistent with the prior labels (Additional file 2: Figure S10a). On kidney dataset, Seurat was unable to distinguish the DCTC cells and CDPC cells, and Signac and scABC were also producing clusters inconsistent with prior labels (Additional file 2: Figure S10b). On mESC dataset, while both Seurat and SC3 correctly identified the cell subpopulations, clusters identified by Signac and scABC also mixed together in UMAP (Additional file 2: Figure S10c). Overall, scAI is able to consistently identify the expected clusters and also the clusters with subtle transcriptomic differences but strong chromatin accessibility differences (as shown in kidney dataset), showing the importance of integrating parallel single-cell multiomics data.
A key challenge in analyzing single-cell multiomics data is to integrate and characterize multiple types of measurements coherently in a biologically meaningful manner. Often, different components in such multiomics measurements exhibit fundamentally different features, for example, some data are binary and inherently sparse whereas the other are more akin to a continuous distribution after normalization [9]. We presented an unsupervised method, scAI, for integrating scRNA-seq data and single-cell chromatin accessibility or DNA methylation data obtained from the same single cells. scAI learned three sets of low-dimensional representations of high-dimensional data: the gene, locus, and cell loading matrices describing the relative contributions of genes, loci, and cells in the inferred factors, and the cell-cell similarity matrix used for aggregating sparse epigenomic data. These learned low-rank matrices allow direct identification of cell subpopulations/states and the associated marker genes or loci that characterize each subpopulation, and provide a convenient visualization of cells, genes, and loci in the same low-dimensional space. Simultaneous analyses of the gene and locus loading matrices enable inference of the regulatory relationships between the transcriptome and the epigenome. Together, scAI provides an effective and biologically meaningful way to dissect heterogeneous single cells from both transcriptomic and epigenomic layers.
The sparse and binary nature of single-cell ATAC-seq or DNA methylation data poses a computational challenge in analysis. Aggregation has been a primary method for analyzing such data [20]. For example, Cicero, an algorithm used for predicting cis-regulatory DNA interactions from scATAC-seq data, aggregates similar cells using a k-nearest neighbors approach based on a reduced dimensional space (e.g., t-SNE and DDRTree) [58]. However, as shown in our simulated data and real co-assayed data, dimensional reduction techniques often fail to capture cell similarity from the chromatin accessibility or DNA methylation profiles. To deal with this difficulty, scAI first combines sparse epigenomic profiles from subgroups of cells that exhibit similar gene expression and epigenomic profiles. These similar cells are analyzed by learning a cell-cell similarity matrix based on a matrix factorization model. The differences between such learned similarity matrix and the similarity matrix computed using only scRNA-seq or only aggregated scATAC-seq data were also investigated (Additional file 1 (Comparison of cell-cell similarity matrix) and Additional file 2: Figure S11 and Figure S12). Our iterative and unsupervised approach combines information from multiple-omics layers by taking advantages of the strengths in optimization models.
To investigate whether scAI might make epigenomic data seemingly more distinct than they actually are, we employed the following two strategies on simulation datasets. Firstly, we compared the aggregated scATAC-seq data obtained from scAI with the raw ATAC-seq data prior to making them sparse and binarization (termed as bulk ATAC-seq data hereafter) in two ways: the direct visualization of loci patterns using heatmap and the low-dimensional visualization of cells using UMAP. The bulk ATAC-seq data and the aggregated scATAC-seq data were found to exhibit the same loci patterns (Additional file 2: Figure S13a). Both bulk ATAC-seq and aggregated scATAC-seq data were found to be distinct across clusters (Additional file 2: Figure S13b). These observations were consistent across all the eight simulation datasets. Secondly, we randomly permuted scATAC-seq data across all cells before applying scAI to the scRNA-seq data and the permuted scATAC-seq data. We found that the aggregated permuted scATAC-seq data were still distinct across clusters in some cases in UMAP (Additional file 2: Figure S13c), partly because there were still differential accessibility patterns across these clusters after permutation (Additional file 2: Figure S13d). Next, we considered an extreme case where all the values of scATAC-seq data are equal and found aggregated scATAC-seq data did not produce any artificial clusters, partly due to our normalization strategy in which scAI aggregates scATAC-seq profile after normalizing Z°R with the sum of each column equaling 1. On the other hand, scAI is able to identify cell clusters with high accuracy on all simulation datasets (Fig. 2e). Our analysis suggests scAI robustly maintains cellular heterogeneity within and between different subpopulations when it enhances epigenomic signals.
To investigate whether scAI introduces high portion of false positives during differential accessibility analysis using aggregated scATAC-seq data, we calculated the percentage of false positive differential accessible loci based on the aggregated scATAC-seq data by comparing them to the differential accessible loci identified using the bulk ATAC-seq data. Specifically, the percentage of false positives was defined as the percentage of differential accessible loci that were not in the set of differential accessible loci identified using the bulk ATAC-seq data. We adopt the Wilcoxon rank sum test for accessibility of cells in each subpopulation and the remaining cells. We found that the percentages of false positive differential accessible loci were less than 7% on simulation datasets (Additional file 2: Figure S14). A direct visualization for the datasets 7 and 8 with imbalanced cluster size shows consistent loci patterns and highly overlapped differential accessible loci between the aggregated scATAC-seq data and bulk ATAC-seq data (Additional file 2: Figure S15). These results suggest that the aggregation strategy has a good control of false positives for differential accessibility analysis.
The single-cell multiomics data are sparse and have large amounts of missing values. The scRNA-seq data have two states: non-zero and zero values. The zero values might be either non-expressed values or due to dropout events [59]. The single-cell methylation data have three states: methylated, unmethylated, and missing values. While replacing missing values by zeros and adopting a model that can potentially impute the missing values, a strategy used in scAI, might improve downstream analysis due to the fact that the large portions of missing values contain true zero values, such approach likely has several limitations. First, it might introduce false signals when the missing values might actually correspond to non-zero signals. Second, such approach cannot distinguish methylated and missing states for the DNA methylation data. One way to address such difficulty is to throw away the missing values, which is particularly useful for the methylation data (e.g., scM&T-seq [3]) because it allows to distinguish methylated and missing values. One powerful approach is to use probabilistic models, such as MOFA [17] and its successor MOFA+ [19], which do not include those missing value regions when computing the likelihood. In principle, we can throw away the missing values in scAI by incorporating a binary matrix into the second term of our model (Eq. (1)), an approach similar to incomplete nonnegative matrix factorization model [60].
Comparing with recent methods, such as MOFA [17], Seurat [22], and LIGER [23], scAI is able to capture cell states with higher accuracy for the multiomics data in which only gene expression or chromatin accessibility may be discriminated between cell states, for example, to uncover novel cell subpopulations with distinct epigenomic profiles but similar transcriptomic profiles, as seen in the kidney dataset. Such capability of identifying cell subpopulation exhibiting only distinct epigenetic profiles will facilitate further analysis of epigenetics in controlling cell fate decision and may help to reveal important transcriptional regulatory mechanism [61]. Similar to uncovering new cell subpopulation, scAI can uncover new cell transition states induced by epigenetics as seen in the analysis of the dexamethasone-treated A549 cell dataset [6], and identify co-regulations coordinated between transcriptome and DNA methylation, as seen in the mESC dataset.
For the methods (e.g., Seurat and LIGER) that are designed for integrating single-cell data measured in different cells, in principal, they can be applied to the parallel single-cell multiomics data. However, we found that these two methods yield deficient alignment between co-assayed cells, as seen in the A549 and kidney datasets. Such alignment errors might affect downstream analysis such as inferring regulatory links. Moreover, these two methods, unlike scAI, need to transform other types of features such as chromatin accessibility or DNA methylation into gene level, which leads to limited resolution and cannot make full use of epigenomic information. As parallel single-cell multiomics data becomes more widely available, methods like scAI will be essential to make sense of this new type of data.
Parallel single-cell sequencing provides a great opportunity to infer the regulatory links between transcriptome and epigenome [9]. In this study, the regulatory links between chromatin regions and marker genes were inferred by combining the correlation analysis and the nonnegative least square regression, as seen in the A549 dataset. Because many factors such as chromatin regulators, histone modification, and the microenvironment can affect the transcriptional regulation [62], more complex and accurate models are needed to improve the accuracy of regulatory relationship inference. While it remains to be done, scAI provides a computational tool for integrating parallel single-cell omics data, including visualization, clustering, differential expression/chromatin accessibility analysis, and regulatory relationship inference.
Here, we present scAI, which is one of the first computational methods for the integrative analysis of single-cell transcriptomic and epigenomic profiles that are measured in the same cell. scAI was shown to be an effective tool to characterize multiple types of measurements in a biologically meaningful manner, dissect cellular heterogeneity within both transcriptomic and epigenomic layers, and understand transcriptional regulatory mechanisms. Due to rapid development of single-cell multiomics technologies, scAI will facilitate the integrative analysis of the current and upcoming multiomics data profiled in the Human Cell Atlas as well as the Pediatric Cell Atlas [63].
Optimization algorithm for scAI
The optimization problem (Eq. (1)) is solved by a multiplicative update algorithm, which updates variables W1, W2, H, and Z iteratively according to the following equations (Additional file 1: Supplementary methods (Details of scAI) and Additional file 2: Figure S16):
$$ {W}_1^{ij}\leftarrow {W}_1^{ij}\frac{{\left({X}_1{H}^T\right)}^{ij}}{{\left({W}_1H{H}^T\right)}^{ij}} $$
$$ {W}_2^{ij}\leftarrow {W}_2^{ij}\frac{{\left({X}_2\left(Z\circ R\right){H}^T\right)}^{ij}}{{\left({W}_2H{H}^T\right)}^{ij}} $$
$$ {H}^{ij}\leftarrow {H}^{ij}\frac{{\left(\alpha {W}_1^T{X}_1+{W}_2^T{X}_2\left(Z\circ R\right)+\lambda H\left(Z+{Z}^T\right)\right)}^{ij}}{{\left(\left(\alpha {W}_1^T{W}_1+{W}_2^T{W}_2+2\lambda H{H}^T+\gamma e{e}^T\right)H\right)}^{ij}} $$
$$ {Z}^{ij}\leftarrow {Z}^{ij}\frac{{\left(\left({X}_2^T{W}_2H\right)\circ R+\lambda {H}^TH\right)}^{ij}}{{\left(\left({X}_2^T{X}_2\left(Z\circ R\right)\right)\circ R+\lambda Z\right)}^{ij}}, $$
where \( {W}_I^{ij},I=1,2 \) represent the entry in the ith row and jth column of W1 (p × K) and W2 (q × K). Hij and Zij represent the ith row and the jth column of H (K × n) and Z (n × n). e (K × 1) represents a vector with all elements being 1. In each iteration step, H is scaled with the sum of each row equaling 1.
In this algorithm, we initialize W1, W2, H, and Z using a 0–1 uniform distribution and generate a binary matrix R using a Bernoulli distribution with a probability s. α and λ are parameters to balance each term, and γ is a parameter to control sparsity of each row of H. The default values for those parameters are as follows: s = 0.25, α = 1, λ = 10,000, and γ = 1. The rank K is determined by a stability-based method [28] (Additional file 1: Supplementary methods (Rank selection) and Additional file 2: Figure S17 and Figure S18). Since H is scaled by row, the entry of matrix H is less than 1. Thus, the magnitude of the third term is small and λ usually is large to ensure the importance of this term. The parameter α is set to be small because the magnitude of this term is usually relatively large, which does not mean that W1 and W2 are not important in the model. The parameters used in all the datasets are summarized in Additional file 2: Table S2. Robustness analysis on the parameter indicates that the overall performance of scAI is relatively robust to choices of parameter values within certain ranges (Additional file 1: Supplementary methods (Robustness analysis) and Additional file 2: Figure S19).
Identification of cell subpopulations
From transcriptomic and epigenomic profiles, scAI projects cells into a cell loading matrix H, which is a low-dimensional representation of both profiles. The subpopulations are then identified by clustering through H using the Leiden community detection method [64]. Specifically, a shared nearest neighbor (SNN) graph is first constructed by calculating the k-nearest neighbors (20 by default) for each cell based on the matrix H. Then, the fraction of shared nearest neighbors between the cell and its neighbors is used as weights of the SNN graph. Next, we identify cell subpopulations by applying the Leiden algorithm [64] to the constructed SNN graph with a default resolution parameter setting of 1.
Identification of cell subpopulation-specific marker genes and epigenomic features
After determining the cell subpopulations, we adopt a likelihood-ratio test for gene expression of cells in the kth cell subpopulation and cells not in the kth cell subpopulation. Genes are considered as the kth cell subpopulation-specific marker genes if (i) the p values are less than 0.05, (ii) the log fold-changes are higher than 0.25, and (iii) the percentage of cells with expression in the kth cell subpopulation is higher than 25%. Cell subpopulation specific-epigenomic features are identified using a similar approach.
Visualization of cells, genes, and loci in a 2D space
scAI simultaneously decomposes gene expression matrix and accessibility or methylation matrix into a set of low-rank matrices, including the gene loading matrix W1, locus loading matrix W2, cell loading matrix H, and cell-cell similarity matrix Z. Based on these inferred low-dimensional representations, we simultaneously visualize cells, genes, and loci in a single two-dimensional space using similarity weighted nonnegative embedding [65]. Specifically, we first compute the coordinates of the inferred factors. H is smoothed by the similarity matrix Z using Hs = H × Z. Then, we compute pairwise similarity matrix S between factors (rows of Hs) by cosine distance. The similarity matrix S is converted into a distance matrix D according to \( D=\sqrt{2\left(1-S\right)}. \) The Sammon mapping method [27] is then used to project the distance matrix D onto a two-dimensional space (a matrix with K rows (K is the number of factors) and 2 columns). The values in this two-dimensional matrix are scaled (ranging from zero to one) to obtain the coordinates of factor C according to C = (Ckx, Cky), where Ckx and Cky represent the x and y coordinates of the kth factor.
Next, we compute the coordinates of cell j (E = (Ejx, Ejy)) in the two-dimensional space according to:
\( {E}_{jx}=\frac{\sum_k{\left({H}^{kj}{C}_{kx}\right)}^{\alpha }}{\sum_k{\left({H}^{kj}\right)}^{\alpha }},{E}_{jy}=\frac{\sum_k{\left({H}^{kj}{C}_{ky}\right)}^{\alpha }}{\sum_k{\left({H}^{kj}\right)}^{\alpha }}, \)
where the parameter α controls how tight the allowed embedding is between the cells and the factors. The reasonable value range is from 1 to 2. Large values move the cells closer to the factors, while it may distort the data when α is higher than 2. α = 1.9 is used as default. The coordinates of cells E are further smoothed by the similarity matrix Z using Es = E × Z and then are used for visualization.
Finally, we embed the marker genes and loci into the same two-dimensional space according to W1 and W2 as follows:
\( {F}_{jx}^I=\frac{\sum_k{\left({W}_I^{jk}{C}_{kx}\right)}^{\alpha }}{\sum_k{\left({W}_I^{jk}\right)}^{\alpha }},{F}_{jy}^I=\frac{\sum_k{\left({W}_I^{jk}{C}_{ky}\right)}^{\alpha }}{\sum_k{\left({W}_I^{jk}\right)}^{\alpha }}, \)
where I = 1,2 represents the embedding of genes and loci, respectively. Accordingly, using this integrative dimension-reduction approach, the marker genes and loci that separate cell states alongside the cells can be visualized together to help interpretation of multiomics data in an intuitive way.
Identification of factor-specific marker genes and epigenomic features
Using scAI, we obtain gene loading and locus loading matrices, W1 and W2, and the values in each column of W1 and W2 are respectively used to identify the genes and epigenomic features associated with each factor. To rank the gene i in factor k, we define a gene score:\( {S}_1^{ik}={W}_1^{ik}/\sum \limits_j{W}_1^{jk} \). Similarly, we rank the loci in each factor by defining a locus score based on W2.
To identify factor-specific marker genes and epigenomic features, we divide the genes and loci into two groups for each factor. The z-score is computed for each entry in each column of W1 and W2: \( {z}_1^{ik}=\left({W}_1^{ik}-{\mu}_1^j\right)/{\sigma}_1^k \) and \( {z}_2^{ik}=\left({W}_2^{ik}-{\mu}_2^j\right)/{\sigma}_2^k \), where \( {\mu}_1^k,{\mu}_2^k \) are the average values of the kth column in W1 and W2, respectively, and \( {\sigma}_1^k,{\sigma}_2^k \) are the corresponding standard deviations. Let AGk and ALk represent the sets of candidate genes and loci, respectively, associated with the kth factor if \( {z}_1^{ik},{z}_2^{ik} \) are greater than T (0.5 by default). Smaller T value gives more features that might contain redundant information, whereas larger T value might leave key features out. We also divide the cells into two groups for each factor using the similar method. In more detail, we compute the z-score for each entry in each row of the cell loading matrix H by zkj = (Hkj − μk)/σk. If zkj is greater than T, cell j is assigned to \( {C}_1^k \); otherwise, it is assigned to \( {C}_2^k \). Next, using a Wilcoxon rank-sum test for the candidate genes in AGk in cells in \( {C}_1^k \) and \( {C}_2^k \), we statistically test the differences of the candidate genes in the different cell groups. Candidate genes are considered as factor-specific marker genes if (i) the p values are less than 0.05, (ii) the log fold-changes are higher than 0.25, and (iii) the percentage of cells with expression in \( {C}_1^k \) is greater than 25%. Factor-specific epigenomic features are identified using the similar approach.
Inference of factor-specific transcriptional regulatory relationships
Once the factor-specific marker genes and loci are determined, we next infer the regulatory links between them. For factor k, the two sets AGk and ALk consist of the identified factor-specific marker genes and loci, respectively. For a gene gi in AGk, we select a locus set \( {L}_k^i\left(\subseteq A{L}_k\right) \), which includes loci within 500 kb of the transcription start site (TSS) of gi, as candidate regulatory regions for a gene gi. To determine whether the expression level of gi is influenced by the accessible status of the candidate regions in \( {L}_k^i \), we use a perturbation approach based on the correlations between the expression level and accessibility. In this approach, first, we compute the Pearson correlation P1 between the gi expression level and the accessibility of each locus in \( {L}_k^i \) in all cells. Second, we perturb the gi expression levels by setting its expression in cells in cell group \( {C}_1^k \) to 0 and then compute the weighted correlation P2 between the perturbed gi expression level and the accessibility of \( {L}_k^i \) in all cells with Hk. as its weight, where Hk. represents the kth row of H. Third, we set the accessibility of \( {L}_k^i \) in cells in cell group \( {C}_1^k \)to 0 and then compute the weighted correlation P3 between the original gi expression level and the perturbed accessibility of \( {L}_k^i \) in all cells with Hk.. Finally, we compute the differential correlation according to dP1 = ∣ P1 − P2 ∣ , dP2 = ∣ P1 − P3∣. The regulatory links between gene gi and loci \( {l}_k^{s_i}\subseteq {L}_k^i \) are indicated if the differential correlation of dP1 or dP2 is greater than the average value of P1 and the original correlation P1 is greater than the average value of P1.
For the identified regulatory links between genes and loci, to determine which transcription factors (TFs) regulate each gene gi, we first identified TF motifs enriched in the loci set \( {l}_k^{s_i} \) using chromVAR [32]. When running chromVAR using default parameters, the raw scATAC-seq data matrix of all loci was used as an input. Then, we regressed the gene expression level \( {E}_{C_1^k}^i \) of each gene across cells in \( {C}_1^k \) with that of the identified TFs \( {E}_{C_1^k}^{i_{TF}} \) using nonnegative least squares regression, i.e., \( {\hat{\beta}}^i=\arg {\min}_{\beta^i}{\left\Vert {E}_{C_1^k}^{i_{TF}}-{E}_{C_1^k}^i{\beta}^i\right\Vert}_2^2,s.t.{\beta}^i\ge 0 \). Regulatory relationships were inferred if the regression coefficients \( {\hat{\beta}}^i \) of the TFs were greater than zero.
Validation of the inferred regulatory relationships
To validate the inferred regulatory relationships in A549 dataset, we collected all TFs that regulate the marker genes (ABHD12, BASP1, CDH16, CKB, NFKBIA, NR3C1, PER1, SCNN1A, and TXNRD1) from the hTFtarget database (http://bioinfo.life.hust.edu.cn/hTFtarget/), which curated a comprehensive TF-target regulation from various ChIP-seq datasets of human TFs from NCBI Sequence Read Archive (SRA) and ENCODE databases. We take ABHD12 as an example to compute the fold enrichment of the inferred regulatory relationships in this database. Among the total 374 collected TFs in chromVAR, 92 TFs are found to regulate ABHD12 in hTFtraget. Among our identified 12 TFs of ABHD12 using chromVAR, 7 TFs are found to regulate ABHD12 in hTFtraget. Thus, the fold enrichment of our predicted regulations of ABHD12 is calculated by (7/12)/(92/374) = 2.37. A fold enrichment value greater than 1 indicates an over-representation of the inferred regulations in the database.
Datasets and preprocessing
The kidney and A549 datasets were downloaded from GSM3271044 and GSM3271045, and GSM3271040 and GSM3271041, respectively. The preprocessed mESC dataset was obtained from a previous study [17]. The detailed description of these datasets and their preprocessing were shown in Additional file 1: Supplementary methods (Details of datasets and preprocessing).
Feature selection
Two feature selection methods were used in this study. If the cell groups were known (e.g., at the time of data collection), the most informative genes were selected using a Wilcoxon rank-sum test with the same parameters as in the identification of factor-specific features. For example, for the scRNA-seq data in the A549 dataset, we identified the differentially expressed genes at different time points and used these genes as informative genes for the downstream analyses. For other datasets, the average expression of each gene and the Fano factor were first calculated. The Fano factor, defined as the variance divided by the mean, is a measure of dispersion. Next, the average expression of all genes was binned into 20 evenly sized groups, and the Fano factor within each bin was normalized using z-score. Then, genes with normalized Fano factors larger than 0.5 and average expressions larger than 0.01 were selected. Moreover, we also selected genes with larger Gini index values [66]. GiniClust R package was run with default parameters. Briefly, genes whose normalized Gini index is significantly above zero (p value < 0.0001) are labeled high Gini genes and selected for further analysis. For the kidney dataset, we selected the informative genes using the second method and loci that were within 50 kb of the TSS of these informative genes.
Method comparisons on three datasets
We compare the performance of scAI with three other methods, including MOFA [17], Seurat (version 3) [22], and LIGER [23]. MOFA takes normalized scRNA and scATAC-seq data as inputs, then infers latent factors using a generalized PCA and assesses the proportion of variance explained by each factor in each type of data. Seurat derives a "gene activity matrix" from the peak matrix of the scATAC-seq data by simply summing all counts within the gene body + 2 kb upstream, representing a synthetic scRNA-seq dataset to leverage for integration. Seurat then co-embeds the scRNA-seq and scATAC-seq cells in the same low-dimensional space by identifying "anchors" between the ATAC-seq and RNA-seq datasets. Since LIGER does not provide specific functions for integrating scRNA-seq and scATAC-seq or DNA methylation data, we used scRNA-seq data and the inferred "gene activity matrix" from Seurat as inputs for integrative analysis. The detailed description of how these comparisons were performed is available in Additional file 1: Supplementary methods (Details of method comparisons on three datasets).
Based on the first two dimensions of t-SNE or UMAP, we quantify the alignment score of the scRNA-seq and scATAC-seq cells using entropy of batch mixing, and assess the separation of the cell groups using silhouette coefficient. These two evaluation metrics were defined in [55]. The detailed description is available in Additional file 1: Supplementary methods (Details of method comparisons on three datasets).
scAI is implemented as both MATLAB and R packages, which are freely available under the GPL-3 license. Source codes as well as the workflows of simulation and real datasets have been deposited at the GitHub repository (MATLAB package: https://github.com/amsszlh/scAI [67] and R package: https://github.com/sqjin/scAI) [68].
The datasets analyzed in this study are available from the Gene Expression Omnibus (GEO) repository under the following accession numbers: GSM3271044 and GSM3271045 [6], GSM3271040 and GSM3271041 [6], and GSE74535 [3].
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Review history
The review history is available as Additional file 3.
Barbara Cheifet was the primary editor on this article and handled its editorial process and peer review in collaboration with the rest of the editorial team.
This work was supported by a NSF grant DMS1763272, a grant from the Simons Foundation (594598, QN), and NIH grants U01AR073159, R01GM123731, and P30AR07504.
Suoqin Jin and Lihua Zhang contributed equally to this work.
Department of Mathematics, University of California, Irvine, CA, 92697, USA
Suoqin Jin, Lihua Zhang & Qing Nie
The NSF-Simons Center for Multiscale Cell Fate Research, University of California, Irvine, CA, 92697, USA
Lihua Zhang & Qing Nie
Department of Developmental and Cell Biology, University of California, Irvine, CA, 92697, USA
Qing Nie
Suoqin Jin
Lihua Zhang
LZ, SJ, and QN conceived the project. LZ and SJ contributed equally to this work. LZ and SJ conducted the research. QN supervised the research. LZ, SJ, and QN contributed to the writing of the manuscript. All authors read and approved the final manuscript.
Correspondence to Qing Nie.
Supplementary Methods.
Supplementary Figures and Tables.
Review history.
Jin, S., Zhang, L. & Nie, Q. scAI: an unsupervised approach for the integrative analysis of parallel single-cell transcriptomic and epigenomic profiles. Genome Biol 21, 25 (2020). https://doi.org/10.1186/s13059-020-1932-8
Integrative analysis
Single-cell multiomics
Simultaneous measurements
Sparse epigenomic profile | CommonCrawl |
\begin{definition}[Definition:Well-Defined/Operation]
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\RR$ be a congruence for $\circ$.
Let $\circ_\RR$ be the operation induced on $S / \RR$ by $\circ$.
Let $\struct {S / \RR, \circ_\RR}$ be the quotient structure defined by $\RR$, where $\circ_\RR$ is defined as:
:$\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$
Then $\circ_\RR$ is '''well-defined (on $S / \RR$)''' {{iff}}:
:$x, x' \in \eqclass x \RR, y, y' \in \eqclass y \RR \implies x \circ y = x' \circ y'$
\end{definition} | ProofWiki |
Low-cost production and application of lipopeptide for bioremediation and plant growth by Bacillus subtilis SNW3
Aiman Umar1,
Aneeqa Zafar1,
Hasina Wali1,2,
Meh Para Siddique4,
Muneer Ahmed Qazi1,3,
Afshan Hina Naeem1,
Zulfiqar Ali Malik3 &
Safia Ahmed ORCID: orcid.org/0000-0001-9773-64331
At present time, every nation is absolutely concern about increasing agricultural production and bioremediation of petroleum-contaminated soil. Hence, with this intention in the current study potent natural surfactants characterized as lipopeptides were evaluated for low-cost production by Bacillus subtilis SNW3, previously isolated from the Fimkessar oil field, Chakwal Pakistan. The significant results were obtained by using substrates in combination (white beans powder (6% w/v) + waste frying oil (1.5% w/v) and (0.1% w/v) urea) with lipopeptides yield of about 1.17 g/L contributing 99% reduction in cost required for medium preparation. To the best of our knowledge, no single report is presently describing lipopeptide production by Bacillus subtilis using white beans powder as a culture medium. Additionally, produced lipopeptides display great physicochemical properties of surface tension reduction value (SFT = 28.8 mN/m), significant oil displacement activity (ODA = 4.9 cm), excessive emulsification ability (E24 = 69.8%), and attains critical micelle concentration (CMC) value at 0.58 mg/mL. Furthermore, biosurfactants produced exhibit excellent stability over an extensive range of pH (1–11), salinity (1–8%), temperature (20–121°C), and even after autoclaving. Subsequently, produced lipopeptides are proved suitable for bioremediation of crude oil (86%) and as potent plant growth-promoting agent that significantly (P < 0.05) increase seed germination and plant growth promotion of chili pepper, lettuce, tomato, and pea maximum at a concentration of (0.7 g/100 mL), showed as a potential agent for agriculture and bioremediation processes by lowering economic and environmental stress.
Environmental pollution due to petroleum products such as crude oil, diesel, and gasoline is of major ecological concern nowadays (Jimoh and Lin 2019). Major health problems in humans and animals are occurred due to the release of petroleum and its by-products in a terrestrial and aquatic ecosystem because of having mutagenic, carcinogenic, and teratogenic effects (Yadav et al. 2016). Petroleum-derived pollutants result in the limitation of phosphorus, iron, and nitrogen availability in agricultural soil (Nogueira et al. 2011). In today's challenging world enhanced agricultural productivity is the need of the hour to encounter human food demands. However, equally alarming is the damage of agricultural land by pollutants that needs bioremediation strategies. Hence, researchers must focus on remediation of all these issues. Biosurfactants are amphiphilic secondary metabolites that exhibit surface-active properties produced by bacteria, fungi, and yeast (Santos et al. 2016). Biosurfactant-producing microorganisms enhance plant growth through improvement in plant immunity against organic contaminants in the environment. Furthermore, they are also efficient in alleviating stress responses in plants along with strengthening plant growth and development (Almansoory et al. 2019). The surfactin preferably and to the lower extent fengycin, lipopeptides are capable to provoke defense responses that generate signaling molecules for activation of induced systemic resistance (ISR) in plants (Ongena et al. 2007). Other lipopeptides that are reported for induction of plant defense response includes iturin (Yamamoto et al. 2015), mycosubtilin (Farace et al. 2015), bacillomycin D (Wu et al. 2018), and sessilin and orfamide (D'aes et al. 2014). One of the positive influences of the use of lipopeptides in agriculture is its biocompatibility with living organisms (Ławniczak et al. 2013). Hence, to minimize the initial dose of fertilizers by seed stimulation strategies and its equal distribution in the soil is made possible through biosurfactants (Krawczyńska et al. 2012). Many researchers verified that plant growth-promoting rhizobacteria (PGPR) positively enhance plant development after association with the hydrocarbon-degrading bacteria in the contaminated soil (Pawlik et al. 2017). Different plant growth-promoting traits include phosphate solubilization, siderophore production, hydrogen cyanide (HCN) production, indole acetic acid (IAA) production, and systemic resistance induction (Benaissa 2019). Hence, for employing biosurfactants in agriculture, bioremediation and its application in other fields the reduction in cost needed for production are of absolute concern (Jimoh and Lin 2019). Increase in awareness among public about the use of environment-friendly and sustainable green products demand new strategies development to cut down the production cost for replacement of toxic synthetic surfactants with biosurfactants (Shaban and Abd-Elaal 2017). Biosurfactants with numerous useful applications provide growing interest in diverse industrial sectors including food, medicine, cosmetics, and agriculture (Patil et al. 2014). However, the production cost is still high that depends on the availability of raw materials and downstream processing for scaleup at the industrial level (Akbari et al. 2018). Raw materials used for biosurfactant production accounts for about 50% of the final production cost (Rufino et al. 2007). Better choice of raw material is a way to cut down the budget and make the process economically feasible (Jimoh and Lin 2019; Mukherjee et al. 2006). Unlike synthetic surfactants that are produced from petroleum feed stock, biosurfactants could be produced using waste materials like agriculture waste (wheat bran), brewery waste, and food waste by-products (potato peels and waste frying oil) that not only reduce cost but also helps in waste disposal in environment-friendly manner (Moshtagh et al. 2018; Vea et al. 2018). In the present study, we used potato peels powder, waste frying oil, molasses, and white beans powder as a low-cost substrate for biosurfactant production. Hence, with all the above intentions the current study was conducted to produce stable potent biosurfactants employing various cost-effective renewable resources and to evaluate the potential of produced lipopeptides for detoxification and management of crude oil contaminated soil and to promote plant growth and development.
All chemicals used in the study werepurchased from Sigma-Aldrich (Merck KGaA, Darmstadt, Germany), and are of analytical grade. The standard surfactin (≥ 98% purity) used as a reference for lipopeptides characterization in this study was obtained from Sigma-Aldrich. Fertilizers (NPK; 20-10-10) used in the study were bought from Agro-chemicals, Fertilizers (TAK Agro Brand). Antibiotics used in the current study were purchased from Werrick Pharmaceuticals Pakistan. Crude oil used was collected from Pakistan petroleum limited.
Microorganism and culture conditions
In the current study, Bacillus subtilis SNW3 (Genbank Acc. No. JX534509.1), obtained from Microbiology Research Lab, Quaid-i-Azam University, Islamabad, was previously identified and isolated from contaminated soil of Fimkessar oil field, Chakwal, Pakistan (Malik and Ahmed 2012). This strain SNW3 also referred to as QVS1 is deposited with the Belgian Coordinated Collections of Microorganisms BCCM/LMG, Ghent, Belgium, under Accession Number "LMG P-30406". The bacterial sample was cultured on nutrient agar plates (Yeast extract 2.0; Beef extract 1.0; Peptone 5.0; Sodium chloride 5.0; Agar 15 g/L) incubated for 24 h at 30°C to obtain separate pure colonies, stored for regular use at 4°C and sub-cultured before use. The strain was preserved at − 80°C in nutrient broth (Peptone, 5; Meat extract, 1; Yeast extract, 2.0 and sodium chloride g/L) supplemented with 30% glycerol.
Cost-effective substrates for biosurfactant production
For low cost biosurfactant production various cost-effective substrates were evaluated that includes: potato peels powder (total carbohydrate 68.7%; starch 25%; protein 18%; non-starch polysaccharide 30%; acid-soluble and acid-insoluble lignin 20% and nitrogen 1.3%) (Liang et al. 2014), molasses (total sugars 62.3%, sucrose 48.8%, starch 0.33% and ash 13.1%) (Palmonari et al. 2020), white beans powder (protein 15.62%; carbohydrates 60.47%; lipids 2.13%; crude fibre 14.15%) (Alayande et al. 2012), waste frying oil (palmitic acid 15.86%; oleic acid 29.83%; stearic acid 4.87% and linoleic acid 28.85%) (Banani et al. 2015) and nitrogen sources: sodium nitrite, urea and ammonium nitrate while, conventional media yeast extract (protein 62.5%; sugar 2.90 %; fat 0.10 %; ash 9.50 %) was used as control. Each carbon source listed above was designed to use individually, then selected substrates were used in different combinations to achieve an optimized medium composition. Molasses used in current study was obtained from Chashma Sugar Mills Limited in Dera Ismail Khan (Pakistan). Potato peels and waste frying oil were obtained from café located at Quaid-i-Azam University Islamabad (Pakistan). Whereas white beans were obtained from National Agricultural Research Council (NARC) Islamabad Pakistan.
Cost-effective substrates were categorized through various methods: soluble total organic nitrogen analysis through the Kjeldahl method (Toledo et al. 2018), soluble total organic carbon with a TOC analyzer (Multi N/C 3100, Analytic Jena), and Dumas method for total organic carbon (TOC) and total organic nitrogen in the solid fraction by applying (LECO, TruSpec CHN) tool (Munera-Echeverri et al. 2020).
Inoculum
Bacillus subtilis SNW3, streaked and stored on nutrient agar plates at 4°C was used for inoculum preparation. A loop full of culture from a single isolated colony on plate added in 100 mL nutrient broth (Peptone, 5; Meat extract, 1; Yeast extract, 2.0 and sodium chloride, 5 g/L) incubated at 30°C for 48 h then seed culture from the nutrient broth was used as inoculum for all experiments.
Production optimization, extraction, and partial purification of biosurfactant
The strain Bacillus subtilis SNW3 was grown on conventional yeast extract media (2% w/v) and mineral salt medium (MSM) as described by (Abouseoud et al. 2008; Rastogi et al. 2021) of given composition (g/L: KH2PO4, 2.0; K2HPO4, 4.0; FeSO4·7H2O, 0.025; MgSO4·7H2O, 1.0; KCl, 0.2; NaCl, 5.0; CaCl2·2H2O, 0.02; and trace elements solution with composition of MnSO4·4H2O, 1.78; ZnSO4·7H2O, 2.32; CuSO4·5H2O, 1.0; H3BO3, 0.56; KI, 0.66 and NH4MoO4·2H2O, 0.39). Different environmental process parameters significant for biosurfactant production were evaluated using above mentioned media at various range of temperature (15, 30, 37 and 50°C), pH (2, 4, 6, 8, 10, 12), agitation speed (0, 150 and 250 rpm) and inoculum size (0.5, 1, 1.5, 2 and 2.5). Initially, three nitrogen sources (urea, sodium nitrate, and ammonium nitrate) and four cost-effective substrates (white beans powder, potato peels powder, waste frying oil, and molasses) were tested separately. After that for different combinations, the selected nitrogen source and cost-effective substrates added with MSM were used in various combined media compositions. Optimization of substrate and culture conditions are given in (Table 1). Yeast extract as the most preferable substrate for biosurfactant production was used as control media (Qazi et al. 2013). The designed experiments for substrate evaluation were run with 100 mL media in 250 mL Erlenmeyer flask with pH adjusted to 7.0 ± 0.2 and kept in a shaker for 96 h of incubation at 30°C and 150 rpm. The cell-free supernatant obtained after centrifugation at 12,000 rpm was acidified up to pH 2.0 with 1 M hydrochloric acid (HCL) and kept overnight at 4 °C. For bacterial biomass production, the collected pallet was washed with saline solution (0.9% w/v NaCl), oven-dried at 100°C, and weighted (Guerfali et al. 2020).
Table 1 Optimization of low-cost substrate and culture conditions for lipopeptide production by Bacillus subtilis SNW3
Dry weight of lipopeptides
For crude biosurfactants, pelleted precipitates were extracted with chloroform/methanol (2:1) and concentrated by rotary evaporation (Marchut-Mikolajczyk et al. 2018). After that concentrate was poured into a pre-weighted sterile beaker. The crude lipopeptides were oven-dried at 60 °C for 24 h. Plates were weighted after drying (Anandaraj and Thivakaran 2010). The following formula was used to calculate the dry weight of crude lipopeptide extract produced by Bacillus subtilis SNW3 under optimized substrate and culture conditions. Dry weight of lipopeptide produced = (Weight of plate containing dried lipopeptide − Empty plate weight).
Assessment of biosurfactant production
For quantitative analysis of biosurfactant production, various assays were used that include surface tension measurement (SFT), oil displacement assay (ODA), and emulsification index (E24). The sample used for analysis was in the form of cell-free supernatant (CFS).
Oil displacement activity (ODA)
For estimation of biosurfactant production oil displacement activity (ODA) was performed according to the method of Yalçın et al. (2018). Briefly, 20 µL of crude oil was add on the surface of 40 mL distilled water in the petri dish. The cell-free supernatant (CFS) of 10 µL was placed gently on a uniform crude oil layer formed on distilled water. Oil layer was displaced, and clear zone diameter was measured in centimetre (cm). Production medium without inoculum was used as negative control. Clear zone formation indicates biosurfactant presence in CFS.
Emulsification index (E24)
Emulsification index (E24%) was used to estimate the emulsifying capacity of the biosurfactant, performed through protocol of Ferhat et al. (2011) with minor modifications. In short, kerosene oil (2 mL) and an equal volume of cell-free supernatants were added in the test tube and mixed for 2 min on the vertex mixer. After that, these test tubes were kept undisturbed for 24 h at room temperature. The percent emulsification was measured using given formula where heights were calculated in centimetres (cm).
$${\text{E}}_{24} \left( \% \right) = \frac{{{\text{Height}}\;{\text{of}}\;{\text{the}}\;{\text{emulsion}} \left( {{\text{cm}}} \right)}}{{{\text{Total}}\;{\text{height}}\;{\text{of}}\;{\text{the}}\;{\text{solution}} \left( {{\text{cm}}} \right)}} \times 100.$$
Surface tension (SFT) measurement
For quantitative analysis of biosurfactant produced by Bacillus subtilis SNW3 surface tension (SFT) of the cell-free supernatant was measured in mN/m by using KRUSS K20 digital Tensiometer (Kruss GmbH, Hamburg, Germany). SFT measurement was performed at room temperature by Wilhelmy plate through according to the protocol given by the manufacturer. Cell-free supernatant of 25 to 30 mL was put on the tensiometer platform in a glass cup. Wilhelmy plate was sterilized before use, adjusted on the tensiometer, submerged in the broth followed by surface tension measurement (Novikov et al. 2017).
Structural characterization of lipopeptide by thin-layer chromatography (TLC) and Fourier transform infrared spectroscopy (FTIR)
For thin-layer chromatography (TLC) and Fourier transform infrared spectroscopy (FTIR) analysis, extracted form of crude biosurfactant was used while surfactin a class of lipopeptide (from sigma) was taken as standard for initial characterization. Crude biosurfactant components were separated on Silica coated aluminum plates, silica gel 60 F254, MERCK Germany using chloroform: methanol: acetic acid (85:10:5, v/v) visualized under the wavelength of 254 and 365 nm to find retention factor (Rf) (Joy et al. 2017). For determination of chemical nature of bonds and, functional groups present in the crude form of biosurfactant produced FTIR analysis was performed. 10 mg of crude biosurfactant was loaded and the spectrum was observed at the range of 4500–450 cm−1 using Tensor 27 (Bruker) FTIR spectrophotometer, equipped with ZnSe ATR (Marchut-Mikołajczyk et al. 2019).
Determination of critical micelles concentration (CMC) and critical micelle dilution (CMD) of lipopeptides
The CMC of the produced biosurfactant was analyzed through change in surface tension reduction values with varying concentrations of 0.06 to 1.24 mg/mL prepared in demineralized water (Datta et al. 2018). For critical micelle dilution cell, free supernatant was diluted10-folds up to three levels (i.e. 10×, 100×, and 1000×) named as CMD−1, CMD−2, and CMD−3, respectively. Surface tension reduction value was analyzed by Wilhelmy plate method using KRUSS K20 digital Tensiometer (Kruss GmbH, Hamburg, Germany), performed at room temperature (Campos et al. 2019).
Lipopeptides stability studies
To elucidate the thermal stability of lipopeptide, thestandard solutions of crude biosurfactant were prepared at a concentration of 600 mg/L and incubated at different temperatures (20–121 °C) for 1 h. Furthermore, a stability test of the produced lipopeptide at saline conditions was performed. Different concentrations of sodium chloride NaCl (1–10%) were added to the biosurfactant solutions and incubated at 30 °C for 1 h. To determine pH effect on lipopeptide activity different buffer solutions were prepared and added to the biosurfactant standard solution, adjusted to pH 1–5 using citrate–phosphate buffer, pH 7 using phosphate buffer, and pH 9–11 using carbonate-bicarbonate buffer solutions, incubated for 30 min. The stability of the produced lipopeptide was checked through surface tension reduction value of each treated sample (Goswami and Deka 2019).
Functional characterization of the lipopeptides
Lipopeptide screening for antimicrobial activity
Lipopeptide for its antimicrobial potential was assessed through well diffusion assay as mentioned before Singh et al. (2014). Mueller–Hinton agar plates were prepared to contain Escherichia coli ATCC 25922 and Salmonella typhi ATCC 14028. The crude lipopeptide (10 mg/mL), ciprofloxacin and clarithromycin (1 mg/mL), and biosurfactant in addition with antibiotics (5:0.5 mg/mL) were poured at a concentration of 100 µL and kept at 37°C for 24 h of incubation. To determine the bacterial sensitivity to lipopeptide, the diameter of inhibition zone (mm) was measured according to Clinical Laboratory Standards Institute (Wayne 2002). To investigate the additive effect of lipopeptide with antibiotics any increase in the diameter of the inhibition zone was measured as compared to antibiotics. Antibiotics without biosurfactants were used as a positive control (Ekprasert et al. 2020).
Exploration of lipopeptides for seeds germination and plant growth
The seeds of tomato (Solanum Lycopersicum), pea (Pisum sativum), chili pepper (Capsicum annuum), and lettuce (Lactuca sativa) were collected from National Agricultural Research Centre (NARC) Islamabad, Pakistan. Obtained seeds were surface sterilized with 10% Na–hypochlorite for 20 min and then washed with sterile distilled water before use. The crude lipopeptide extract produced with cost-effective optimized media was used in this assay. The seed germination experiment was conducted in a petri plate containing 40 seeds positioned in filter paper and cotton soaked with four different concentrations (0.1, 0.3, 0.5, and 0.7 g/100 mL) of crude lipopeptide solution. Distilled water100% v/v was used as a control. These plates were kept in yellow light at 25 °C for 7 days after that relative seed germination (G, %): [No. of seeds germinated (treatment)/No. of seeds germinated (control) × 100] was calculated. Following the germination test seeds treated with lipopeptide solution were transferred in pots (seeds without pre-treatment with biosurfactant were used as control) and kept in a greenhouse with temperature maintained between 20 and 22 °C. Furthermore, for plant growth stimulation crude lipopeptide solution was added in pots at a concentration (0.1, 0.3, 0.5, and 0.7 g/100 mL) dissolved in distilled water thrice with 10 days interval while in control pots pure water was added. The emergence of plant seedlings was tested and checked for the morphological characteristic of plants like shoot length (mm), root length (mm), and dry weight (g) of plants after 40 days (Huang et al. 2017).
Bioremediation assay in shake flask fermentation
The biodegradation efficiency of crude oil by Bacillus subtilis SNW3 was analyzed as illustrated by Varjani and Upasani (2016) with minor changes. An aliquot of 2 mL pre cultured Bacillus subtilis SNW3 was transferred into 250 mL of Erlenmeyer flask containing 100 mL mineral salt media and different concentrations 0.5, 1, 1.5, and 2% (v/v) of crude oil. For monitoring of abiotic loss of the crude oil, an uninoculated media was used as control. All these flasks were incubated for 21 days at 200 rpm and 35C. The bacterial growth in crude oil was detected through measurement of the absorbance at (OD600 nm) through spectrophotometer while SFT was measured by tensiometer. To estimate the residual crude oil in media, crude oil was extracted with hexane, left for evaporation in a pre-weight clean beaker. For quantification of remaining crude oil after degradation gravimetric analysis was performed at different time intervals by following the formula proposed by Patowary et al. (2017).
$${\text{Hydrocarbon}}\;{\text{degradation }}\% = {\text{Amount}}\;{\text{of}}\;{\text{crude}}\;{\text{oil}}\;{\text{degraded}}/{\text{Amount}}\;{\text{of}}\;{\text{crude}}\;{\text{oil}}\;{\text{added}}\;{\text{in}}\;{\text{the}}\;{\text{media}} \times 100.$$
Bioremediation of crude oil in the soil through various design treatments
In this assay bioremediation of crude oil contaminated soil was monitored by collecting garden soil from Quaid-i-Azam University Islamabad. Biosurfactant suitability for removing hydrophobic pollutants from soil was analyzed by collecting 5–10 cm deep topsoil while following the protocol of Okop et al. (2012) and transported in a clean container to the Microbiology laboratory of Quaid-i-Azam University Islamabad Pakistan. The soil collected was air dried and sieved with a 2 mm sieve after that 5% of crude oil was sprayed on the soil to pollute soil homogenically. The polluted soil was left undisturbed for 5 days and then divided into 200 g of equal parts and dispensed in pots. These pots were left undisturbed in the open air for a week. Then for conducting bioremediation experiments various designed treatments were established added twice throughout the remediation period: (T0) addition of distilled water as control, (T1) cell-free supernatant obtained after 96 h of incubation with maximum biosurfactant produced by using optimized cost-effective substrate media, (T2) addition of active culture of Bacillus subtilis SNW3 (2% inoculum size) cultured in nutrient broth kept in shaker incubator at 30 °C, 150 rpm for 24 h of incubation, (T3) effect of two additives was assessed for bioremediation i.e. T1 + T2, (T4) addition of tween 80 (T5) addition of fertilizer (NPK; 20-10-10) to analyse the effect of fertilizer on bioremediation in comparison to the produced lipopeptide (Pelletier et al. 2004) and (T6) an additional control containing autoclaved soil and 5% crude oil (w/v) was used to examine the crude oil degradation in the soil. Detailed information about all the above treatments is shown in (Table 2). The soil content of each pot was tilled twice a week for aeration with moisture maintenance at 60 % and temperature of 28–30°C, providing all those conditions that are appropriate for crude oil-degrading microbes present in the soil (Agamuthu et al. 2013). After that soil samples of 10 g were collected from different areas of the plastic pots at the 30, 60, 90th day and were gravimetrically determined using the formula given by Ganesh and Lin (2009).
Table 2 Different design treatments for removal of crude oil from contaminated soil
The obtained results were analyzed statistically with the use of STATISTICA software, one-way ANOVA (version 8.1). The difference between obtained results was analyzed by using Tukey's test to find individual and control mean ± standard deviation. Significance value was set at p = 0.05 and p-values ≤ 0.05 were considered significant.
Substrate screening and optimization studies for lipopeptide production
Potato peels powder, molasses, white beans powder, and waste frying oil were evaluated as cheap media for lipopeptide production by Bacillus subtilis SNW3. Optimized results for culture conditions with 2% yeast extract media showed 30 °C as optimum for maximum lipopeptide production with an ODA value of 1.26 cm While other optimized cultural conditions were with 1% inoculum size, 150 rpm, and pH of 6 (Additional file 1: Fig. S1). At the end of the fermentation process, the obtained ODA values were 1.3, 2.4, 0.9, and 1.8 cm for potato peels powder, white beans powder, sugar cane molasses, and waste frying oil media respectively with 2% w/v concentration. Additionally, the surface tension reduction values observed for all four biosurfactant solutions were reduced from 72 mN/m to 41.3 (potato peels powder), 33.6 (sugar cane molasses), 41 (white beans powder) and 38.2 mN/m (waste frying oil). Though good emulsification values of all these biosurfactant solutions were obtained to about 55 to 57% (Fig. 1a). In the current study among nitrogen sources tested preferably urea act as a good nitrogen source that showed surface tension reduction of 31.4 mN/m and ODA value of 2 cm (Fig. 1b). It was observed that white beans powder and waste frying oil gave significant oil displacement value. the The characterization of substrate samples in terms of total organic carbon and nitrogen content of the yeast extract, white beans powder and potato peels powder used in this study is presented in Additional file 1: Table S1. It has been also found that combination of carbon sources enhances biosurfactant synthesis. The final optimized cost effective media was [white beans powder (6% w/v) + waste frying oil (1.5% w/v) + urea (0.1% w/v)] with significant lipopeptide yield indicated ODA of 4.9 cm, emulsification index of 69.8% and surface tension reduction value up to 28.8 mN/m (Fig. 1c; Additional file 1: Fig. S5a, b).
SFT, E24 and ODA values for lipopeptide production by Bacillus subtilis SNW3 a with alternative carbon sources used individually, b different nitrogen sources, c with a combination of carbon and nitrogen energy sources and d production analysis of surfactin under optimized conditions with yeast extract as a reference, in shake flask fermentation at 30 °C. P.P.P: potato peels powder; W.B.P: white beans powder; Mol.: molasses; W.F.O: waste frying oil; Y.E: yeast extract
We aimed the use of alternative non-conventional media for lipopeptide production with optimized media gave 1.17 g/L of crude lipopeptide that was almost double from 0.56 g/L with yeast extract control media, with biomass yield of 4.6 and 3.2 g/L respectively (Fig. 1d). The cost effective usage of media component showed that on average 1 kg of white beans powder with 240 mL of waste frying oil and 640 g of yeast extract media would be enough for preparing 16 L of fermentation media that gave 1.17 g/L of surfactin production.
Thin layer chromatography analysis
Characterization of crude biosurfactant produced with final optimized media by Bacillus subtilis SNW3 was carried out by thin-layer chromatography (TLC) and Fourier transform infrared spectroscopy (FTIR). Results obtained by TLC indicate the lipopeptide nature of the biosurfactant product with most prominentband observed against standard surfactin having retention factor (Rf) value of 0.68 as illustrated in (Fig. 2c, d).
FTIR spectrum and TLC profile of crude lipopeptide produced by Bacillus subtilis SNW3 in comparison to standard surfactin show as a FTIR of standard surfactin, b FTIR of crude lipopeptide extract, c TLC profile of crude lipopeptide extract, d TLC profile of crude lipopeptide in comparison to standard surfactin
Structural characterization of biosurfactant produced by FTIR
The FTIR spectra represent the presence of carboxylic functional groups and aliphatic amines the characteristic of the lipopeptide nature of biosurfactant produced. The FTIR spectra show a sharp peak at 1023 cm−1 and 972 cm−1 that corresponds to the presence of C–N aliphatic amines in standard and crude biosurfactant (Fig. 2a, b). The peaks in FTIR spectra at 1450 and 1130 suggest the presence of stretching bands between carbon atoms and hydroxyl groups. The absorbance appears at 1762 cm−1 and 1757 cm−1 attributed to the vibrations due to the ester carbonyl group of peptide components. The peaks observed in FTIR spectra at 2942 and 2926 corresponds to the presence of C–H bands (CH2–CH3 stretching). Another peak ranging from 3500 to 3200 cm−1 indicated the presence of alcohols and phenols (O–H stretch, H-bond). The spectra presented in the current study in comparison to standard surfactin from sigma suggested the presence of peptide moiety and aliphatic groups, a distinctive feature of lipopeptides nature of biosurfactant produced by Bacillus subtilis SNW3.
In this study, we observed that lipopeptides produced by Bacillus subtilis SNW3 showed antimicrobial and synergistic effects with antibiotics against Escherichia coli and Salmonella typhi. In the case of lipopeptide alone, the growth of E. coli was affected more as compared to S. typhi. However, in combination with antibiotics, the results obtained showed an increase in zone of inhibition from 18 to 30 mm for E. coli and 42 to 45 mm for S. typhi containing lipopeptide plus ciprofloxacin (Fig. 3a). Thus, the addition of lipopeptide renders bacteria more sensitive to ciprofloxacin used in the case of E. coli. Similarly, the inhibition zone around lipopeptide plus clarithromycin increased from 20 to 30 mm and 19 to 25 mm for E. coli and S. typhi (Fig. 3b). In general, lipopeptide showed antibacterial and an additive effect while used in combination with antibiotics, shown in Additional file 1: Fig. S2.
Antibiogram of crude lipopeptide extract tested with antibiotics a ciprofloxacin and b clarithromycin, against Escherichia coli and Salmonella typhi
Critical micelle concentration (CMC) and critical micelle dilution (CMD) determination
The crude biosurfactant obtained from Bacillus subtilis SNW3 which was dissolved in distilled water at different concentrations showed a reduction in surface tension of water from 72 to 36 mN/m with an increase in lipopeptide concentration. Lipopeptide produced seems to be more competent exhibited a CMC at 0.5 mg/mL, with surface tension reduction of 36 mN/m (Additional file 1: Fig. S3a). The surface tension values remain stable with an SFT value of 29 mN/m to 32 mN/m after making threefold dilutions showing effective lipopeptide concentration in the medium (Additional file 1: Fig. S3b).
Stability studies
The applicability of biosurfactant produced depends on its behavior shows at different conditions of temperature, pH, and salinity (Gudina et al. 2010). The lipopeptide produced during the current study was found to be more stable after exposure to various temperatures ranges since no significant difference was detected for surface tension reduction values from 20 to 121 °C. The favorable surface tension reduction values were observed over a pH range of 1 to 11, although in between pH 5 to 7 lipopeptides produced was found to be more stable (Fig. 4). At pH 1, surface tension value raised slightly up to 35 mN/m that means that produced lipopeptides possess stability at acidic conditions but more effectively stable at alkaline conditions. Besides this, lipopeptide exhibit stablity over a wide range of salt concentrations 1 to 8%, an increase in SFT value at 10% NaCl concentration.
Stability of crude lipopeptide extract against various environmental factors like temperature ranges 20–121 °C, NaCl conc. 1–10% (w/v) and pH ranges 1–11. DW distilled water, Temp temperature, NaCl sodium chloride
Effect of lipopeptide on plant growth promotion
In this study, Solanum lycopersicum (tomato), Pisum sativum (pea), Capsicum annuum (chili pepper), and Lactuca sativa (lettuce) were examined to demonstrate the effects of biosurfactants, that showed significantly (P < 0.05) better effects on seed germination and plant growth promotion. Statistical data about seeds germination and plants growth obtained is shown in Additional file 1: Table S2. The best results for germination were obtained at higher concentrations (0.7 mg/mL) of lipopeptide tested. Among all seeds tested significant (P < 0.05) stimulation was observed for chili pepper seeds, showed almost double 51.7% germination at 0.5 g/100 mL in comparison to control 21.6% with MilliQ water. Similarly, tomato seeds showed 68.75% germination at 0.7 g/100 mL in comparison to control water (56.25%), while pea and lettuce seeds were affected to some extent as shown in (Additional file 1: Fig. S4a).
The applied biosurfactant treatments also augmented the dry biomass of plants. The plants that arose after treating with different concentrations of lipopeptide displayed higher biomass in comparison to control. Our current findings showed that a significant (P < 0.05) increase in weight was observed for chili pepper and lettuce at 0.7 g/100 mL of lipopeptide. Biomass exhibited by chilli and lettuce was 0.21 g and 0.25 g respectively, that is four times increase in relative to control 0.06 g of the seedling. Although for pea and tomato similarly a positive effect was noted with the addition of 0.7 g/100 mL of lipopeptide that significantly increase (P < 0.05) dry biomass almost double in relative to control (Additional file 1: Fig. S4b). Interestingly, in the present study positive effect of lipopeptide on dry biomass was observed for all seeds tested but maximum for chili pepper and lettuce.
Almost all surfactin concentrations tested showed an immense effect on root elongation. The plants treated with a higher concentration of lipopeptide 0.7 g/100 mL enhanced the root growth at maximum. Biosurfactant treatment signifies (P < 0.05) better elongation of seedling roots in lettuce, pea, and chili pepper almost two times greater than control. The tomato seedlings treated with lipopeptide also showed an increase in root development (Additional file 1: Fig. S4c).
All lipopeptide concentrations tested showed significant effect on plant growth parameters. The chili pepper plants showed a significant (P < 0.05) difference in height 8.06 mm after treatment with 0.7 g/100 mL of lipopeptides almost double as compared to control (Additional file 1: Fig. S4d). Whereas lettuce plants showed a gradual increase in height with an increase in lipopeptide concentration. The effects of lipopeptide on seed germination and plant growth promotion are shown in Fig. 5.
Effect of crude lipopeptide extract on seed germination of A lettuce, B tomato, C beans, and D chili The portion (A–D1) show untreated control plants and (A–D2) for plants treated with lipopeptide extract at 0.7 g/mL for 40 days
Bioremediation assay using shake flask fermentation
Biosurfactants are used to emulsify hydrocarbons with the reduction in surface tension, enhancement of water solubility, and increasing oil displacement from soil particles (Andrade Silva et al. 2014; Geetha et al. 2018). In the present study, the pattern for Bacillus subtilis SNW3 growth on crude oil and MSM revealed that there was an increase in microbial growth up to 13 days of incubation, whereas after that decline in growth was recorded. Observable effects on growth were observed with 1 and 1.5% of crude oil used. With increase in microbial growth the SFT value of the culture medium reduced from 72 to 29 mN/m, indicates the lipopeptide production by Bacillus subtilis SNW3 (Fig. 6a). The simultaneous microbial growth and crude oil biodegradation in culture broth media demonstrate utilization of various components of crude oil by Bacillus subtilis SNW3 (Patowary et al. 2017). However, we observed that maximum degradation 86% was achieved with 1.5% crude oil as compared to control (Fig. 6b; Additional file 1: Fig. S5c, d). Schematic diagram showing bacterial strain activity in degradation of crude oil recalcitrant hydrocarbons with simultaneously lipopeptide production (Additional file 1: Fig. S6).
The growth of a Bacillus subtilis SNW3 on crude oil and MSM with surface tension reduction values for 21 days, plane lines for OD600 and dotted lines for SFT, and b quantity of crude oil degraded (%) by Bacillus subtilis SNW3 while growing on crude oil and MSM for 21 days
The current study revealed that lipopeptide produced by Bacillus subtilis SNW3 effectively removes crude oil from the soil. After applying different strategies, the residual crude oil content of each treatment showed different extents of biodegradation. It was revealed that the combined strategy used in T3 gave measurable remediation of crude oil as compared to other treatments tested. There was a gradual increase in bioremediation capacity with time, maximum after 90 days. T3 treated with Bacillus subtilis SNW3 cultured microorganisms and CFS containing lipopeptides (80.2%), show a significant difference from T0 control (11.6%) with distilled water. The better bioremediation results (73.2%) were obtained in T1 with the addition of lipopeptide than those obtained by T2 (63.8%) with the addition of biosurfactant producing Bacillus subtilis SNW3.
In the current study while making a comparison for bioremediation with chemical compounds it was observed that in T4, the addition of Tween 80 showed 65.4% degradation lower than those treated with biosurfactants. The oil reduction results obtained for T5 with fertilizer showed (32.6%) while the lowest degradation (5.4%) was observed for T6 using autoclaved soil (Fig. 7).
The percent degradation of crude oil contaminated soil through various design treatments from T1–T6 for 90 days
Biosurfactant production by using cost-effective substrates produced by Bacillus subtilis was previously studied by many researchers (Secato et al. 2016). An easy way to achieve cost-effective bioprocesses for biosurfactant production is by using a low-cost substrate. The use of waste frying oil as a sole source of carbon and energy for lipopeptide production by two Bacillus strains were previously reported by Md Badrul Hisham et al. (2019) that gave surface tension reduction values up to 36 mN/m. Our results showed that Bacillus subtilis SNW3 growing on 2% waste frying oil gave 38 mN/m reduction value are in line with previous findings. De Lima et al. (2009) reported rhamnose production by Pseudomonas aeruginosa PACL strain cultivating on waste frying soybean oil with 100% emulsification index, surface tension reduction up to 26.0 mN/m, and concentration of 3.3 g/L while in the current study 56.3% emulsification was observed with 2% waste frying oil. Research conducted by Abdel-Mawgoud et al. (2008) investigated surfactin production in a cost-effective manner with the use of 16% molasses and other trace elements that produce a surfactin yield of 1.12 g/L. However, it is also stated in many studies that the presence of hydrophobic substrate is essential for the production of biosurfactants (Santos et al. 2016). According to literature different types of oils e.g., vegetable oils, waste cooking oil, glycerol, glucose, and diesel were screened out for biosurfactant production by fungal species M. circinelloides that showed 11.7 cm ODA with the use of waste cooking oil as carbon source. In another study conducted by Hasanizadeh et al. (2017) for biosurfactant production showed maximum biosurfactant production with the use of 8% (v/v) waste cooking oil as a carbon source. Nitrogen is considered an important component of the medium used for biosurfactant production (Wu et al. 2008). In the literature, for higher biosurfactant yields number of nitrogenous compounds are listed that include yeast extract (Rodrigues et al. 2006a), beef extract, urea (Elazzazy et al. 2015), peptone, and meat extract (Gudiña et al. 2011). Urea was considered as a cheaper nitrogen source for significant lipopeptide production in comparison to sodium nitrate (Elazzazy et al. 2015; Farace et al. 2015; Ghribi and Ellouze-Chaabouni 2011). Yeast extract has been extensively selected in several studies (Marcelino et al. 2019). The yeast extract was used as a control medium as previously yeast extract was found as the most preferable substrate for significant biosurfactant production (Qazi et al. 2013). For instance, L. paracasei A20 preferred yeast extract as the significant medium for biosurfactant production followed by meat extract, while peptone was chartagorised as least important component of the medium (Gudiña et al. 2011).
It was reported previously that limitation in nitrogen concentration results in higher biosurfactant yield (Wu et al. 2019). The high carbon to nitrogen ratio (C/N) of the production medium (i.e., low level of nitrogen) limits bacterial growth, promoting cell metabolism for metabolites production (Nurfarahin et al. 2018). P. aeruginosa LBM10 in a culture medium containing soy bean oil as carbon source and NaNO3 as the source of nitrogen produce significant biosurfactant yield. Media composition with low nitrogen level produce (1.42 g/L) as compared to higher nitrogen concentrations with biosurfactant yield of 0.94 g/L (Prieto et al. 2008). The Kjeldahl or Dumas methods used for the evaluation of the crude protein in foods determine the total organic nitrogen of foods (Chandra-Hioe et al. 2018). The nitrogen content determination is crucial for the analysis of crude protein content. However, these provide rough assumptions as to the relative nitrogen and amino acid content differ between food proteins (Mæhre et al. 2018).
In the current study substrates were used in combination to increase biosurfactant production by reduce the price of culture media (Rufino et al. 2008). Slivinski et al. (2012) reported the use of okara obtained after processing ground soybeans as a substrate for surfactin by Bacillus pumilus UFPEDA 448. Zhu et al. (2013) also investigated the use of soybean flour as a substrate for surfactin production by Bacillus amyloliquefaciens XZ-173. To the best of our knowledge, for economical biosurfactant production, only a few studies are conducted by using soybean, but no single study is present that investigated the use of white beans powder as a substrate for low-cost production. The concentration of crude lipopeptide produced was (about 1.17 g/L), closed to other reported values for biosurfactant production using cost-effective substrates. The cost required for the preparation of one liter of optimized low-cost media in the current study is 0.078 EUR, which is just 0.8% of one-liter synthetic yeast extract media cost 10.5 EUR. Hence utilizing these cost-effective nonconventional media instead of synthetic yeast extract contribute to a 99% reduction in cost required for medium preparation.
It was reported by Samak et al. (2020) that 30°C is the optimum temperature for biosurfactant production that is in correspondence with results obtained in the current study. Our findings showed that no significant lipopeptide was produced at a static condition that might be due to lack of oxygenation (Santos et al. 2014). In a previous study conducted by Hemlata et al. (2015) for biosurfactant production by Stenotrophomonas maltophilia NBS-11 showed maximum production at pH 7. Urea and ammonium nitrate have been already used and reported in the literature as a very cost-effective nitrogen source for biosurfactant production by Candida spp. (Alwaely et al. 2019).The study conducted by Medeot et al. (2017) showed a high yield of biosurfactants (1.7 mg/mL) while using NH4NO3 and glucose as substrate for production by Bacillus amyloliquefaciens MEP218. In the same way, the combination of sucrose and NH4NO3 was reported by Fernandes et al. (2016), gave high yield of biosurfactant (0. 2 g/L) by Bacillus subtilis RI4914. Moreover, a study conducted for biosurfactant production showed optimum yield with 0.3% sodium nitrate by Pseudoxanthomonas sp. G3 (Purwasena et al. 2020). Current findings also showed maximum lipopeptide yield while using carbon nitrogen substrates in combination.
Surfactin acts as quorum-sensing molecule that provides a potential tool for the regulation of fermentation (Gupta et al. 2017), while carbon metabolism regulates the balance between the products and biomass. Biosurfactant production occurs frequently during a stationary stage of the cell growth after depletion of the nitrogen source (Onwosi and Odibo 2012). As a result of the current study, the final optimized carbon–nitrogen combination of the media significantly produced maximum biosurfactant with biomass and crude lipopeptide yield of 4.6 and 1.17 g/L respectively. Recently, Phulpoto et al. (2020) reported glycerol and NH4NO3 as combined C/N media for surfactin production that significantly produce biomass and crude biosurfactant yield of 4140 and 1255 mg/L, respectively. Similarly, it was reported by Lu et al. (2016), that for fengycin production through Bacillus amyloliquefaciens fmb-60 biomass yield could be a significant factor. In Bacillus amyloliquefaciens BZ-6 (Liu et al. 2012), Bacillus amyloliquefaciens MEP218 (Medeot et al. 2017), Bacillus amyloliquefaciens fmb-60 (Lu et al. 2016) and Bacillus subtilis strains (Makkar et al. 2011), a direct correlation was reported between lipopeptide production and biomass yield. Primary characterization for biosurfactant produced by Bacillus subtilis SNW3 was carried out by using TLC using surfactin from sigma as standard. Here, our results for TLC of crude biosurfactant sample indicated lipopeptide nature of biosurfactant product through the presence of surfactin with an Rf value of 0.68. These findings are consistent with other reported studies, where Rf value of 0.76 was observed by Ramyabharathi et al. (2018) for surfactin produced by Bacillus subtilis. Results obtained from previous findings also showed Rf values of 0.09, 0.3, and 0.75 for fengycin, iturin, and surfactin respectively using Bacillus subtilis UMAF6619, UMAF6614, UMAF8561, UMAF6639, and Bacillus amyloliquefaciens PPCB004 (Arrebola et al. 2010). FTIR results obtained were following TLC. The chemical structure of biosurfactant produced by Bacillus subtilis SNW3 was revealed by analyzing the crude extract using fourier transform infrared spectroscopy. FTIR analysis of crude biosurfactant produced by Bacillus subtilis SNW3 showed that it contains alcohols and carboxylic acids (lipids) and peptide moieties (proteins) that indicate lipopeptide nature of biosurfactant. A similar pattern of FTIR aliphatic and peptide moieties was reported for the presence of lipopeptides by Kiran et al. (2017). The observed pattern of IR spectrum was very similar to the spectrum obtained by de Faria et al. (2011) who reported the appearance of the stretch at 1721 cm−1 that indicates the presence of lactone carbonyl group. Similar FTIR absorption spectra were reported in the literature for lipopeptide (Pereira et al. 2013).
Lipopeptide produced during the current study not only provides potential antibacterial activity but also renders bacteria more susceptible to the available antibiotics. Biosurfactants could be a suitable substitute for antimicrobial compounds and synthetic medicines and might be used as efficient therapeutic agents (Gudiña et al. 2013). The antimicrobial effect of biosurfactants is due to their potential to form pores inside cell membranes (Gudiña et al. 2010) this characteristic might increase the effectiveness of antibiotics. Sambanthamoorthy et al. (2014) revealed antimicrobial activities against A. baumannii, E. coli, and S. aureus at a concentration of 25–50 mg/mL. In our findings lipopeptide showed antimicrobial effect at a lower concentration of 10 mg/mL that showed more effectiveness of the lipopeptide product. These results are consistent with previous studies which suggest a synergistic effect of biosurfactant with antibiotics (Joshi-Navare and Prabhune 2013; Rivardo et al. 2011). The promising feature of natural antimicrobial peptides is their low toxicity and slow microbial resistance emergence rate as compared to the current antibiotics (Wang et al. 2019). Our findings suggested that lipopeptides could extend the clinical use of current antibiotics. Domhan et al. (2018) reported lipopeptides as novel antimicrobial agents against resistant microbial pathogens with favorable pharmacokinetics and enhanced antibacterial activity.
The critical micelle concentration (CMC) is the minimum biosurfactant concentration needed to achieve the lowest surface tension value at which micellar aggregates formation starts (Ma et al. 2016). CMC is an important characteristic of surface-active agents for evaluation of their interfacial activity (Zhou et al. 2019b). The CMC of crude lipopeptide produced by Bacillus subtilis SNW3 was found to be ≤ 0.58 mg/mL, significant as compared to 2.7 mg/mL (Ghasemi et al. 2019). These results were also efficient as compared to commonly used synthetic surfactants sodium dodecyl sulfate (SDS) that attains CMC value at 2100 mg/L (Chen et al. 2006).
After biosurfactant production purification strategies account for near 60% of the total cost (Banat et al. 2010). Taking into consideration the industrial economic value, most the biosurfactants are required either in crude form or in form of broth preparations (Banat et al. 2010). Lipopeptide produced by Bacillus subtilis SNW3 exhibits excellent stability over an extensive range of pH (1–11), salinity (1–8%), temperature (20–121°C), and even after autoclaving. The decrease in stability of biosurfactants at acidic conditions might be due to the protonation of negative polar ends of surfactin molecules (Gogoi et al. 2016). A previous study conducted by Purwasena et al. (2019) showed biosurfactant stability with emulsification at a high temperature of 120 °C, pH of 4–10, and NaCl concentration of 10% (w/v) that are in accordance with the current study. Moussa et al. (2013) showed biosurfactant stability at 20–120°C produced by Bacillus methylothrophicus and Rhodococcus equi strains. Different studies showed reduced biosurfactant stability under alkaline conditions. Several other studies have been reported about the stability of the biosurfactants at high salinity and temperature (Rodrigues et al. 2006b; Gudina et al. 2010). Significant stability of lipopeptide was found at high salinity and temperature and our findings were inconsistent with previous studies (Das and Kumar 2018; Hentati et al. 2019; Purwasena et al. 2019). Surfactin with excellent stability at wide range of temperature, pH and salinity widens its applicability in several industrial sectors e.g. food, pharmaceuticals, detergents, agricultural and bioremediation (Fenibo et al. 2019).
In the modern agricultural field use of bacterial biosurfactants plays an important role as they are eco-friendly and affordable (Hafeez et al. 2019; Muthusamy et al. 2008). Lipopeptides derived from bacterial strains are eco-friendly, less toxic, with more stability in harsh environments and highly biodegradable as compared to their synthetic counterparts (Lima et al. 2011). The genera Bacillus and Pseudomonas proved as e major producers of biosurfactant molecules (Hussain and Khan 2020; Zhou et al. 2019a). Lipopeptide produced by Bacillus subtilis SNW3 had a noticeable effect on seed germination and plant growth promotion that becomes more prominent with the increase in concentration. Our findings showed significant increase in seed germination of all four species tested most prominent for chilli and tomato. This increase in germination might be due to the reason that biosurfactant increases the permeability of seed coat to water that indirectly makes quicker the metabolic processes inside seeds (Kaur et al. 2017). The applied lipopeptide treatments augmented the dry biomass of all seeds tested, maximum for chilli pepper and lettuce. Similar results with an increase in plant biomass were observed by Liu et al. (2014). This increase in plant biomass might be due to the enhanced production of phytohormones and improved mineral solubilization in soil (Das and Kumar 2016). Almost all biosurfactant concentrations tested showed an immense effect on root elongation with better elongation in lettuce, pea and chilli. Enhanced plants root elongation could be due to decrease strength of wrapping tissues and seed coating that favors root development (da Silva et al. 2015). Another reason for the increase in root development by applying biosurfactants could be due to minimizing anaerobiosis conditions in the soil (Shukry et al. 2013). In current study significant effect was observed on plant growth promotion more prominent for chilli plants as compared to control. It was demonstrated by Cawoy et al. (2014) that surfactin by Bacillus isolates provokes concentration dependent induce systemic resistance (ISR). Surfactin act as a signaling molecule that provokes cannibalism and the formation of a matrix (López et al. 2009b). The improved plant development with biosurfactants ciould be incresae nutrients bioavailability and emulsification of hydrophobic compounds (Marchut-Mikolajczyk et al. 2018). Several researchers have reported the biosurfactant effect on seed germination, but to our knowledge, no previous study is available about current vegetable plants. Biopreparations are widely used nowadays for the enhanced seed quality and improved plant germination in contaminated soil (Mukherjee et al. 2006). However, some research gaps are still required to be filled about mechanisms followed by biosurfactants concerning enhanced growth and development of plants.
Lipopeptides are mostly applied in the biomedical field and only a few reports are present that showed the success story of lipopeptides in bioremediation of oil-polluted environments. In recent years, the use of biosurfactants for the treatment of oil-contaminated soil is increased (Karlapudi et al. 2018). Indigenous microbes that are normally present in oil-contaminated soil are mainly involved in the biodegradation of oil pollutants (Iwai et al. 2011; Lee et al. 2018). Through introducing biosurfactant-producing bacteria in the contaminated environment results in enhanced bioremediation utilizing solubilization, mobilization, and emulsification of hydrocarbons (Nievas et al. 2008). Crude oil is a complex mixture of aliphatic and aromatic hydrocarbons that inhibits the uptake of carbon sources required for metabolism and growth (Das and Chandran 2011). Many reports are present about the efficacy of biosurfactants produced by Bacillus species in oil recovery methods, bioremediation processes and industrial sectors (Greenwell et al. 2016; Ismail et al. 2013; Pereira et al. 2013).
Data obtained from our findings suggested Bacillus subtilis SNW3 as a potential bioremediation agent as compared to previously reported biodegradation studies (Sathishkumar et al. 2008). Bioremediation experiments conducted by Kumari et al. (2012) showed degradation percent of 49.5 and 60.6% for total petroleum hydrocarbons (TPH) by Rhodococcus sp. NJ2 and Pseudomonas sp. BP10 respectively. In our study more significant biodegradation of 86% was observed after 21 days. Al-Wasify and Hamed (2014) explained that P. aeruginosa reveal about 77.8% of maximum degradation after an incubation period of 28 days. Studies reported bioremediation of 49–54% for crude oil-polluted environments (Bordoloi and Konwar 2008) and more than 85% for diesel oil-contaminated sand (Silva et al. 2010).
Biosurfactant produced by S. marcescens UEO15 confirmed 59% and 78% degradation of kerosene and crude oil, in comparison of 25% and 10% with distilled water used as control (Elemba et al. 2010). Our study showed 86% efficacy in comparison to previous findings suggesting it as a more suitable bioremediation component in environmental sectors.
Tween 80 is suitable for remediation of contaminated soil because of its low cost as compared to other non-ionic surfactants (Bautista et al. 2009), most successfully reported for polycyclic aromatic hydrocarbons PAHs (Gong et al. 2015). In the current study, T4 showed a 65.4% degradation rate that might be due to acidic conditions of soil that is unsuitable for microbial growth (Liu et al. 2010). Our findings suggested that treatments with biosurfactants enhanced the degradation rate employed it as a better bioremediation agent. Mishra and Singh (2012) reported that among degradative enzymes alkane hydroxylase produced by Rhodococcus sp. NJ2 and P. aeruginosa PSA5 result in degradation of n-hexadecane. Genes involved in the production of these degradative enzymes are reported in previous studies (de Gonzalo et al. 2016). Previous studies have reported the fertilizer as treatment to check the effects of NPK on the biodegradation of hydrocarbons (Pelletier et al. 2004). Nutrients especially phosphorus, nitrogen, and in some cases, iron are very essential ingredients for successful biodegradation of hydrocarbon pollutants (Adams et al. 2015).
Moreover, bioremediation technology is considered to be non-invasive and quite cost-effective (Azubuike et al. 2016; Kumar and Yadav 2018). Biodegradation through microorganisms signifies one of the principal mechanisms by which petroleum and other hydrocarbon pollutants can be removed from the environment (Al-Hawash et al. 2018; Das and Chandran 2011) and is cheaper than other remediation technologies (da Rocha Junior et al. 2019).
The current study demonstrated the use of cost-effective media for lipopeptide production by Bacillus subtilis SNW3. The possibility of utilizing waste frying oil in combination with white beans might be proved to be efficient to substitute yeast extract media and worthwhile for its industrial-scale production. The lipopeptides obtained exhibited potential emulsifying and surface tension reducing capabilities with strong stability at a wide range of pH, temperature, and salinity. In addition, lipopeptide produced showed higher potential for seed germination and plant growth promotion of Capsicum annuum, Lactuca sativa, Solanum Lycopersicum, Pisum sativum, and removal of crude oil from contaminated soil, suggesting its potential applications in environmental and agriculture sectors.
The data used to support the findings of this study are available from the corresponding author upon request.
ATCC:
American type culture collection
BCCM:
Belgian coordinated collections of microorganisms
CMC:
Critical micelle concentration
CMD:
Critical micelle dilution
CFS:
Cell-free supernatant
E24:
Emulsification ability
FTIR:
Fourier-transform infrared spectroscopy
HCN:
Hydrogen cyanide
IAA:
Indole acetic acid
ISR:
Induced systemic resistance
Mol.:
MSM:
Mineral salt medium
NaCl:
NARC:
National Agricultural Research Council
Optical density
ODA:
Oil displacement activity
P.P.P:
Potato peels powder
PAHs:
Polycyclic aromatic hydrocarbons
PGPR:
Plant growth-promoting rhizobacteria
Rf:
Retention factor
SDS:
Sodium dodecyl sulfate
SFT:
TLC:
TOC:
Total organic carbon
Total organic nitrogen
TPH:
Total petroleum hydrocarbons
W.B.P:
White beans powder
W.F.O:
Waste frying oil
Y.E:
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We acknowledge NARC (National Agriculture Research Council) Pakistan for providing seeds for the experiment.
No funding from externals is received for this research.
Department of Microbiology, Faculty of Biological Sciences, Quaid-i-Azam University, Islamabad, 45320, Pakistan
Aiman Umar, Aneeqa Zafar, Hasina Wali, Muneer Ahmed Qazi, Afshan Hina Naeem & Safia Ahmed
Department of Microbiology, University of Balochistan, Quetta, 87300, Pakistan
Hasina Wali
Department of Microbiology, Shah Abdul Latif University, Khairpur, Sindh, 66111, Pakistan
Muneer Ahmed Qazi & Zulfiqar Ali Malik
Department of Psychology, Quaid-i-Azam University, Islamabad, 45320, Pakistan
Meh Para Siddique
Aiman Umar
Aneeqa Zafar
Muneer Ahmed Qazi
Afshan Hina Naeem
Zulfiqar Ali Malik
Safia Ahmed
AU and SA: conception and study design; AU and AZ: carry out experimental work of study; AU, MPS and HW, AHN: testing and data analysis of the study; SA: supervision; AU: draft the manuscript; AU, MAQ, ZAM, and SA: revised the manuscript. All authors read and approved the final manuscript.
Correspondence to Safia Ahmed.
This article does not contain any human and animal studies performed by any of the authors.
Effect of cultural conditions on lipopeptide production by Bacillus subtilis SNW3. (a) Temperature (b) inoculum (c) agitation and (d) pH, error bars represent ± standard deviation of triplicate values. Figure S2. Antibacterial activity of lipopeptide produced by Bacillus subtilis SNW3 against Escherichia coli and Salmonella typhi by using agar well diffusion assay. The error bars represent ± standard deviation of triplicate values. Figure S3. Lipopeptide characterization by (a) critical micelles concentration (CMC) and (b) critical micelles dilution (CMD); produced by Bacillus subtilis SNW3 in relation to SFT measurement under optimized conditions. Figure S4. Effect of crude lipopeptide extract obtained from Bacillus subtilis SNW3 on (a) germination of seeds (b) dry biomass of plant (c) root length and (d) height of plants. The error bars represent ± standard error of triplicate values. Figure S5. Results obtained for lipopeptide produced on optimized media (a) oil displacement activity in crude oil, (b) Emulsification activity (E24) up to 70%) (c) screening of Bacillus subtilis SNW3 for growth and bioremediation with 1.5% crude oil in uninoculated control and sample after 21 days (d) extraction of crude oil media with hexane after 21 days from uninoculated control and sample through gravimetric analysis. Figure S6. Schematic diagram showing bacterial strain activity in degradation of crude oil recalcitrant hydrocarbons with simultaneously lipopeptide production. Table S1. Analysis of total organic carbon (TOC) and total organic nitrogen (TON) content of the substrate tested. Table S2. Statistical Mean (M), Std. Deviation (SD), Std. Error (SE) and P value for relative seed germination, dry biomass, root length and plant height at four different concentrations of lipopeptide produced by Bacillus subtilis SNW3 used for four different plant species.
Umar, A., Zafar, A., Wali, H. et al. Low-cost production and application of lipopeptide for bioremediation and plant growth by Bacillus subtilis SNW3. AMB Expr 11, 165 (2021). https://doi.org/10.1186/s13568-021-01327-0
Ecotoxicity
Bacillus subtilis SNW3
Lipopeptide
Bioremediation
Plant growth promotion | CommonCrawl |
\begin{document}
\title{Rainbow Hamiltonicity in uniformly coloured perturbed graphs}
\begin{abstract} \setlength{\parskip}{
amount} \setlength{\parindent}{0pt} \noindent
We investigate the existence
of a rainbow Hamilton cycle
in a uniformly edge-coloured randomly perturbed graph.
We show that for every \(\delta \in (0,1)\) there exists $C = C(\delta) > 0$ such that the following holds.
Let $G_0$ be an $n$-vertex graph with minimum degree at least $\delta n$
and suppose that each edge of the union
of $G_0$, with the random graph $\gnp{n}{C/n}$ on the same vertex set, gets a colour in \([n]\) independently and uniformly at random.
Then, with high probability, $G_0 \cup \gnp{n}{C/n}$ has a rainbow Hamilton cycle.
This improves a result of Aigner-Horev and Hefetz, who proved the same when the edges are coloured uniformly in a set of $(1 + \varepsilon)n$ colours. \end{abstract}
\section{Introduction}
Given \(\delta \in (0,1)\), let \(\mathcal{G}_{\delta,n}\) be the collection of graphs on vertex set $[n]$ with minimum degree at least $\delta n$. Determining the minimum \(\delta\) that guarantees that every member of \(\mathcal{G}_{\delta,n}\) contains a given spanning subgraph is a central theme in extremal combinatorics. The prototypical example is Dirac's theorem~\cite{dirac}, which says that the minimum \(\delta\) such that every member of \(\mathcal{G}_{\delta,n}\) is Hamiltonian is \(1/2\). On the other hand, one of the main pursuits of probabilistic combinatorics is understanding the minimum \(p\) such that \(\gnp{n}{p}\), the binomial random graph on $[n]$ with edge probability \(p\), contains a given subgraph \emph{with high probability}\footnote{ We say that a sequence of events \((A_n)_{n \in \mathbb{N}}\) holds \emph{with high probability} if \(\prob{A_n} \rightarrow 1\) as \(n\rightarrow \infty\). }. Following the breakthrough of P\'osa~\cite{posa}, it was proven in~\cite{komlos83,korshunov1977}
that $\gnp{n}{p}$ is Hamiltonian with high probability, if it has minimum degree at least $2$ with high probability, showing that the threshold $p$ for Hamiltonicity is $(1 + o(1))\log n / n$.
As an interpolation between the two models, Bohman, Frieze and Martin~\cite{bohman} introduced the \emph{perturbed graph} model.
Given a fixed $\delta > 0$, this is defined as \(G_0 \cup \gnp{n}{p}\), where \(G_0 \in \cG_{\delta,n}\), i.e.\ this is the union of some graph on vertex set $[n]$ with minimum degree at least $\delta n$, and the random graph $\gnp{n}{p}$ on the same vertex set. In~\cite{bohman} the authors showed that there exists \(C\), depending only on \(\delta\), such that for all \(G_0 \in \mathcal{G}_{\delta,n}\), the perturbed graph \( G_0 \cup \gnp{n}{C/n}\) is with high probability Hamiltonian. That is, for every graph with linear minimum degree, adding linearly many random edges results in a graph that is with high probability Hamiltonian. This is best possible for all $\delta \in (0, 1/2)$ up to the value of \(C\), since the complete bipartite graph with parts of size \(\delta n\) and \((1 - \delta)n\) requires \(\Omega(n)\) edges to be Hamiltonian. (When $\delta \ge 1/2$ no random edges are needed, due to Dirac's theorem.) By now there is a sizeable literature on the perturbed model; see e.g.\ \cite{rainbow-perturbed-trees,anastos-first,kriv-kwan-sudakov,shoham-large-cliques,shoham-small-cliques,bottcher2019universality,bottcher2023triangles,krivelevich2006smoothed}.
In this paper we consider a rainbow variant of the above result. A subgraph \(H\) of an edge coloured graph \(G\) is called \emph{rainbow} if no two edges of \(H\) share a colour. For a finite set of colours \(\mathcal{C}\), a graph \(G\) is \emph{uniformly coloured} in \(\mathcal{C}\) if each edge of \(G\) gets a colour in \(\mathcal{C}\) independently and uniformly at random.
The problem of finding rainbow subgraphs of uniformly coloured graphs is well studied, in particular for \(\gnp{n}{p}\) \cite{cooper-frieze,ferber-kriv,frieze-loh,ferber-optimal-colours,frieze-bal-optimal-colours}.
The problem of finding rainbow subgraphs in the uniformly coloured perturbed graph \(G \sim G_0 \cup \gnp{n}{p}\), where \(G_0 \in \mathcal{G}_{\delta,n}\), was first considered more recently~\cite{shoham-small-cliques,shoham-large-cliques,rainbow-perturbed-trees, elad-hefetz,anastos-first}. In particular, the problem of containing a rainbow Hamilton cycle was first addressed by Anastos and Frieze~\cite{anastos-first}, who showed that if the number of colours is at least about \(120 n\), then \(G\sim G_0 \cup \gnp{n}{C/n}\) has with high probability a rainbow Hamilton cycle, for \(C\) depending only on \(\delta\) and all $G_0 \in \mathcal{G}_{\delta, n}$. Aigner-Horev and Hefetz~\cite{elad-hefetz} improved this result by showing that, at the same edge probability in the random graph,
\(n + o(n)\) colours suffice. We prove that the optimal number of colours suffices.
\begin{theorem}\label{main-thm} For any \(\delta \in (0,1)\) there exists \(C>0\) such that the following holds. For \(G_0 \in \cG_{\delta,n}\), let
\(G \sim G_0\cup \gnp{n}{C/n}\) be uniformly coloured in \([n]\).
Then with high probability \(G\) contains a rainbow Hamilton cycle. \end{theorem}
As explained above, our result has the optimal edge probability, up to the dependence of $C$ on $\delta$, for $\delta \in (0,1/2)$.
The paper is structured as follows. In \Cref{section:proof-sketch} we sketch the proof of \Cref{main-thm}. In \Cref{section:proof-main-result} we prove \Cref{main-thm} assuming \Cref{lemma:main}, the key lemma of the paper. Next in \Cref{section:prelims} we state and prove some preliminary results that we need. In \Cref{section:absorber} we prove the existence of `gadgets' which underpin \Cref{lemma:main}. In \Cref{section:proof-main-lemma} we prove \Cref{lemma:main}.
Throughout the paper, we will assume that \(n\) is sufficiently large. Asymptotic notation hides absolute constants: if for some \(x, \varepsilon, n>0\) we write \(x=O(\varepsilon n)\), then there is an absolute constant \(C>0\), which does not depend on \( x, \varepsilon, n\) or any other parameters, such that \(x\le C \varepsilon n\). We write \(x \ll y\) if \(x<f(y)\) for an implicit positive increasing function $f$. We denote the set of colours on the edges of a graph \(H\) by \(\mathcal{C}(H)\), and say that a graph \(H\) is \emph{spanning} in a colour set \(\mathcal{C}'\) if \(\mathcal{C}(H) = \mathcal{C}'\). Finally, we let \(\mindegG\) be an arbitrary member of \(\mathcal{G}_{\delta,n}\), which is the family of $n$-vertex graphs with minimum degree at least $\delta n$.
\newcommand{P_{\text{abs}}}{P_{\text{abs}}}
\section{Proof sketch} \label{section:proof-sketch}
Our proof uses the \emph{absorption method}. This method is typically applicable when one searches for a spanning subgraph, and involves two stages: finding an almost spanning subgraph; and dealing with the remainder, by having a `special' set of vertices, put aside at the beginning, that can cover any sufficiently small set of vertices.
This is done in \Cref{lemma:main}, which says the following: with high probability there exists a rainbow path \(P_{\text{abs}}\) such that, for any sets of vertices \(V'\) and colours \(\mathcal{C}'\) with \(\abs{V'} = \abs{\mathcal{C}'}\), disjoint from the vertices and colors of \(P_{\text{abs}}\), there exists another rainbow path \(Q\) with vertex set \(V' \cup V(P_{\text{abs}})\) and colours \(\mathcal{C}' \cup \mathcal{C}(P_{\text{abs}})\), whose ends can be any vertices in \(V'\).
We now sketch the proof of \Cref{lemma:main}. We first put aside a subset of the vertices and a subset of the colours, which are typically called the `reservoir' (also called `flexible set'), that have the following property: for any sets of vertices and colours \(V', \mathcal{C}'\) of the same small size (much smaller than the reservoir) which are disjoint from the reservoir, we can find a rainbow path \(P_0\) that uses \(V'\), \(\mathcal{C}'\) and a \(\Theta(\abs{V'})\) subset of the vertices and colours in the reservoir.
Then the question is how to cover the rest of the reservoir; to this end, we build an `absorbing structure' (\(P_{\text{abs}}\) above) which has the following property: it can `absorb' any subset of vertices and colours of the same size of the reservoir in a rainbow path \(P_{\text{abs}}'\). Then combining \(P_{\text{abs}}'\) and \(P_0\) gives \(Q\).
The path \(P_{\text{abs}}\) and the `absorbing structure' in which it resides are built by putting together several `absorbing gadgets', graphs on \(\Theta(1)\) vertices with the following property:
each gadget has two paths with the same endpoints such that one avoids a designated pair of a vertex and a colour in the reservoir, and the other one `absorbs' the same pair; see \Cref{fig:absorber}. The construction of the gadgets is done in \Cref{section:absorber}. This absorbing structure was introduced by Gould, Kelly, K\"uhn and Osthus~\cite{kuhn-osthus} for constructing rainbow Hamilton paths in random optimal colourings of the complete graph, and is based on ideas of Montgomery \cite{montgomery}.
For finding an almost spanning rainbow path we use a rainbow version of depth first search~\cite{ferber-kriv, elad-hefetz}, which was used for the same problem in~\cite{elad-hefetz}.
\section{Proof of \Cref{main-thm}} \label{section:proof-main-result}
In this section we prove the main theorem, \Cref{main-thm}. We will use \Cref{lemma:main} below, which we prove in \Cref{section:proof-main-lemma}.
\begin{lemma} \label{lemma:main}
Let \(\delta, \gamma, \eta \in (0,1)\) and \(C>0\) be constants such that
\(
C^{-1} \ll \eta \ll \gamma \ll \delta.
\)
Let \(G \sim \gdn{\delta}{n}\cup \gnp{n}{C/n}\) be uniformly coloured in \(\mathcal{C} = [n]\).
Then, with high probability,
\(G\) has a rainbow path \(P_{\text{abs}} \) of length at most
\(\gamma n\) with the following property.
For any
\(V' \subseteq V\setminus V(P_{\text{abs}})\),
\(\mathcal{C} ' \subseteq \mathcal{C} \setminus \mathcal{C}(P_{\text{abs}}) \) with
\(2 \le \abs{V'} = \abs{\mathcal{C}'}\le \eta n\)
and distinct \(x,y\in V'\),
there exists a path \(Q\) such that
\begin{itemize}
\item \(Q\) has ends \(x,y\),
\item \(V(Q) = V(P_{\text{abs}}) \cup V'\),
\item \(\mathcal{C}(Q) = \mathcal{C}(P_{\text{abs}}) \cup \mathcal{C}'\).
\end{itemize}
\end{lemma}
The next lemma is a rainbow version of a commonly used consequence of the depth first search algorithm~\cite{dfs-pseudorandom}, which we will use to find an almost spanning rainbow path. This lemma was used in~\cite{elad-hefetz} for the same problem.
\begin{lemma}[Prop.\ 2.1~\cite{elad-hefetz}; Lem.\ 2.17~\cite{ferber-kriv}] \label{lemma:rdfs-lemma}
Let \(G\) be a graph with its edges coloured in a set \(\mathcal{C}\).
If for any two disjoint sets of vertices \(X,\, Y\) of size \(k\) we have
\(
\abs{\mathcal{C}{(E(X,Y))}} \ge \abs{V(G)},
\)
then \(G\) has a rainbow path of length at least \(\abs{V(G)} -2k + 1\). \end{lemma}
The next lemma can easily be proved using Chernoff's bound (cf.~\Cref{thm:Chernoff}). \begin{lemma} \label{lemma:rdfs-cond-holds}
Let \(\alpha \in (0,1)\) and \(C>0\) be constants with \(C^{-1} \ll \alpha\).
Let \(G \sim \gnp{n}{C/n}\) be uniformly coloured in \(\mathcal{C} = [n]\).
Then, with high probability, for any two disjoint sets of vertices \(X,\, Y\) of size \(\alpha n\) we have
\(
\abs{\mathcal{C}{(E(X,Y))}} \ge (1-\alpha)n.
\) \end{lemma}
Our main theorem now follows easilly.
\begin{proof}[Proof of Theorem~\ref{main-thm}] Let \(\eta, \gamma\) be constants such that \(
C^{-1} \ll \eta \ll \gamma \ll \delta. \) By \Cref{lemma:main,lemma:rdfs-cond-holds} we may assume that there exists a path \(P_{\text{abs}}\) with the properties in \Cref{lemma:main}; and that for any disjoint \(X,Y\subseteq V\) of size \(k=\eta n/4\) we have \(\abs{\mathcal{C}(E(X,Y))} \ge n - k\). Let $\mathcal{C}_2 = \mathcal{C} \setminus \mathcal{C}(P_{\text{abs}})$, let \(V' \subseteq V \setminus V(P_{\text{abs}})\) be an arbitrary set of size \(k\), and let \(V_2 = V\setminus \left( V' \cup V(P_{\text{abs}})\right)\). So $\abs{\mathcal{C}_2} \ge n - \gamma n$ and $\abs{V_2} = \abs{\mathcal{C}_2} - k - 1$. Then, for every disjoint $X, Y \subseteq V_2$ of size $k$, \[
\abs{\mathcal{C}(E(X,Y)) \cap \mathcal{C}_2} \ge |\mathcal{C}_2| - k \ge \abs{V_2}. \]
Therefore, in the spanning subgraph of \(G[V_2]\) whose edges are edges in \(G\) coloured in \(\mathcal{C}_2\), by \Cref{lemma:rdfs-lemma} there exists a rainbow path \(P_2\) of length at least \(\abs{V_2} - 2k + 1\). Hence \(V_2' = V(G) \setminus \left( V(P_{\text{abs}}) \cup V(P_2) \right)\) has size between $k$ and \(3k \le \eta n - 2\) and \(\mathcal{C}_2 ' = \mathcal{C} \setminus \left( \mathcal{C}(P_{\text{abs}}) \cup \mathcal{C}(P_2) \right)\) has size \(\abs{V_2'}+2 \). Let \(x,y\) be the endpoints of \(P_2\). Then by the property of \(P_{\text{abs}}\) there exists a rainbow path \(Q\) spanning in \(V(P_{\text{abs}}) \cup V_2' \cup \{x,y\}\) and \(\mathcal{C}(P_{\text{abs}})\cup \mathcal{C}_2'\) with endpoints \(x,y\). Then \(P_2 \cup Q\) is a rainbow Hamilton cycle. \end{proof}
\section{Preliminaries} \label{section:prelims} In this section we collect three preliminary results that we need: the Chernoff bound, cf. \Cref{thm:Chernoff}; that random sparse subgraph of dense hypergraphs have large matchings, cf. \Cref{lemma:linear-matching}; and that in the perturbed graph, between any two vertices, there is a large rainbow collection of paths of length three, cf. \Cref{lemma:con}.
\begin{theorem}[{Chernoff Bound, \cite[eq.\ (2.8) and Theorem 2.8]{jlr}}] \label{thm:Chernoff}
For every \(\varepsilon>0\) there exists \(c_\varepsilon>0\) such that the following holds.
Let \(X\) be the sum of mutually independent indicator random variables and write \(\mu = \expect{X}\).
Then
\[
\prob{\abs{X- \mu} \ge \varepsilon \mu} \le 2{\rm e}^{-c_\varepsilon\, \mu}.
\]
\end{theorem}
The next lemma, despite its technical appearance, proves the following straightforward statement: quite \emph{sparse} random subgraphs of dense hypergraphs contain, with high probability, a matching of linear size. \begin{lemma} \label{lemma:linear-matching}
Let \(\alpha, c, c'>0\) and \(r\ge 2\) be an integer such that \(c' \ll \alpha, c, r\).
Let \(\mathcal{H}\) be an \(r\)-uniform hypergraph on \(n\) vertices with at least \(\alpha n^r\) edges.
Let \(\mathcal{H}_m\) be the random subgraph of \(\mathcal{H}\) that consists of \(m=cn\) edges of \(\mathcal{H}\), chosen with replacement and uniformly at random.
Then, with probability at least \(1-{\rm e}^{-\frac{c \alpha n}{4}}\), the hypergraph \(\mathcal{H}_m\) has a matching of size at least \(c' n\).
Let \(\mathcal{H}_p\) be the random subgraph of \(\mathcal{H}\), where we keep each edge independently with probability \(p = c n^{-r+1}\). Then with probability at least
\(1-{\rm e}^{-\frac{c \alpha n}{2r}}\), the hypergraph \(\mathcal{H}_p\) has a matching of size at least \(c' n\). \end{lemma} \begin{proof} Write \(\beta(\mathcal{G})\) for the size of the largest matching of a hypergraph \(\mathcal{G}\).
It is not hard to see that \(\mathcal{H}\) contains an induced subgraph of minimum degree at least \(\alpha n^{r-1}\). Hence, without loss of generality, we may assume that \(\mathcal{H}\) has minimum degree at least \(\alpha n^{r-1}\).
We first prove the result for \(\mathcal{H}_m\). Suppose \(\beta(\mathcal{H}_m) < c' n\), and let \(M\) be a maximal matching. Then \(S = V(\mathcal{H})\setminus V(M)\) is an independent set in \(\mathcal{H}_m\) and \(\abs{S} \ge (1-rc')n\). By the minimum degree condition of \(\mathcal{H}\), the number of edges with all vertices in \(S\) is at least \(
\frac{1}{r}|S| (\alpha n^{r-1} - rc'n^{r-1}) \ge \frac{1}{r} (1-rc')\, (\alpha - c') n^r. \)
This gives \begin{align*} \prob{S \text{ is independent}} &= \left(1- \frac{e(\mathcal{H}[S])}{e(\mathcal{H})}\right)^m \\ &\le \exp \left( - cn \cdot \frac{\frac{1}{r} (1-rc')\, (\alpha - rc') n^r}{\binom{n}{r}} \right) \\ &\le \, \exp \left( - \frac{1}{2}(r-1)!\, c\, (1-rc')\, (\alpha - rc') n \right) \end{align*} Then, since \(rc'<1/2\), the number of \(S\subseteq V\) with \( \abs{S} \ge (1-rc')n\) is at most \[ n\binom{n}{(1-rc')n} = n\binom{n}{rc' n} \le {\rm e}^{2 rc' n} (rc')^{-rc' n}. \] Thus by the union bound \( \prob{\beta(\mathcal{H}_m) < c' n} \le \exp \left( f_{c,r,\alpha} (c') n \right), \) where \[ f_{c,r,\alpha}(c') = 2rc' - rc' \ln (rc') - \frac{(r-1)!}{2} c (1-rc') (\alpha - rc') \] Since \(f_{c,r,\alpha} (c')\) is continuous near \(0\) and \( f_{c,r,\alpha} (c') \rightarrow 0 -0 - \frac{(r-1)!}{2}\, c\, \alpha <0 \) as \(c' \rightarrow 0\), for \(c' = c'(c,r,\alpha)\) sufficiently small \(f_{c,r,\alpha} (c') \le - \frac{(r-1)!}{4}\, c\, \alpha \le - \frac{1}{4} c \alpha \), which gives the first part of the lemma.
For the second part of the lemma observe that the same argument works: with \(S\) as above, in \(\mathcal{H}_p\) we have \[ \prob{S \text{ is independent}} = (1-p)^{e(\mathcal{H}[S])} \le \exp \left( - c n^{-r+1} \cdot \frac{1}{r} (1-rc')\, (\alpha - rc') n^r \right) \] and a similar calculation as above shows that the probability there is such an \(S\) is at most \({\rm e}^{-\frac{c\alpha n}{2r}}\).
\end{proof}
\newcommand{\lambda_{\text{con}}}{\lambda_{\text{con}}} \newcommand{\alpha_{\text{con}}}{\alpha_{\text{con}}}
\begin{lemma}[Triangles and Short Paths] \label{lemma:con}
Let \(0<\delta<1\), \(q, C>0\) and \(\rho, \lambda_{\text{con}} \ll \delta, q, C\).
Let \(\mathcal{C}\) be a set of colours of size \(qn\).
Let \(G\sim \gdn{\delta}{n} \cup \gnp{n}{C/n}\) be uniformly coloured in \(\mathcal{C}\).
Then, with probability at least \(1-{\rm e}^{-\lambda_{\text{con}} n }\), the following holds.
For any \(u,v \in V(G)\)
there is a matching \(M\) of size at least
\(\rho n\) such that the colours of the edges \(ux,xy,vy\), for \(xy \in M\), are all distinct.
\end{lemma} \begin{proof} \renewcommand{\mindegG}{\gdn{\delta}{n}} Fix \(u,v \in V\). Let $\rho_1$ be a constant such that \( \rho, \lambda_{\text{con}} \ll \rho_1 \ll C, \delta,q \).
By the minimum degree assumption, there exist disjoint subsets \(N_u \subseteq N_{\mindegG}(u)\), \(N_v \subseteq N_{\mindegG}(v)\), of size \(\delta n/2\). Consider the bipartite graph with bipartition \((N_u, N_v)\) and edges \[ \left\{\, zw \in E(\gnp{n}{C/n}):\: z\in N_u,\ w \in N_v \right\}. \] This is a random subgraph of the complete bipartite graph, with each part having order \(\delta n/2\), and edge probability \( C/n. \) Hence, by \Cref{lemma:linear-matching}, with probability \(1 - {\rm e}^{-\Omega( C\delta n)}\), there is matching \(M\) of size \(\rho_1 n\).
For each \(zw\in M\), reveal whether the path \(uzwv\) is rainbow, without exposing the colours. Then each \(uzwv\) is rainbow independently with probability \(1-o(1)\). Hence by Chernoff's bound (\Cref{thm:Chernoff}), with probability \(1-{\rm e}^{-\Omega(\rho_1 n)}\), there is \(M'\subseteq M\) with \(\abs{M'} \ge \rho_1 n /2\) such that each \(uzwv\) is rainbow, for all \(zw \in M'\).
Let \( \mathcal{P} = \{uzwv: zw \in M'\}\). It remains to show we can find a large \(M^{\prime \prime} \subseteq M'\) such that the collection \( \mathcal{P}' = \{uzwv: zw \in M^{\prime \prime}\}\) is rainbow.
Now reveal the colours on the edges in \(\mathcal{P}\). By symmetry, each triple of distinct colours in \(\mathcal{C}\) is equally likely to appear in \(\mathcal{P}\). Hence \(\mathcal{P}\) corresponds to selecting uniformly at random with replacement \(\abs{\mathcal{P}} \ge \rho_1 n/2\) edges from the complete 3-graph with vertex set \(\mathcal{C}\). Thus, by \Cref{lemma:linear-matching},
with probability \( 1- {\rm e}^{-\Omega(\rho_1 q n)}\), there exists \(M^{\prime \prime}\subseteq M'\) of size \(\rho n\) so that the colours of \( \mathcal{P}' = \{ uxyv : xy \in M^{\prime \prime}\}\) form a matching in the complete 3-graph on \(\mathcal{C}\) i.e.\ \(\mathcal{P}'\) is rainbow.
The probability this fails for some pair \(u,v\) is, by the union bound, at most \[ n^2 \cdot \left( {\rm e}^{-\Omega( C\delta n)} + {\rm e}^{-\Omega(\rho_1 n)} + e^{-\Omega(\rho_1 q n)}\right) \le {\rm e}^{-\lambda_{\text{con}} n}, \] proving the lemma. \end{proof}
\section{Finding absorbers} \label{section:absorber} \newcommand{\cols_{\text{gadget}}}{\mathcal{C}_{\text{gadget}}} \newcommand{q_{\text{gadget}}}{q_{\text{gadget}}}
In this section we prove \Cref{lemma:absorbers-exist}, which asserts that for any vertex \(v\), colour \(c\) and any small (but linear in size) set of forbidden vertices and colours, we can find an `absorber' (cf. \Cref{def:absorber}) for \(v,c\). These absorbers are the building blocks for \(P_{\text{abs}}\) in \Cref{lemma:main}. To construct these absorbers we will need to find a rainbow 4-cycle containing a given colour \(c\), and none of the forbidden vertices and colours. This is the most technical part of our proof, and is done in \Cref{lemma:squares}.
\begin{definition}[Absorber] \label{def:absorber}
Let \(v\) be a vertex and \(c\) a colour.
A \emph{\((v,c)\)-absorber}
is a graph $A_{v,c}$ with \(v\in V(A_{v,c})\) and \(c\in \mathcal{C}(A_{v,c})\) that has two paths \(P, P'\) with the following properties.
\begin{itemize}
\item They are rainbow.
\item They have the same endpoints.
\item
\(P\) is spanning in \(V(A_{v,c})\) and
\( V(P') = V(P)\setminus \{v\} = V(A_{v,c})\setminus \{v\} \).
\item
\(P\) is spanning in \(\mathcal{C}(A_{v,c})\) and
\(
\mathcal{C}(P')
= \mathcal{C}(P)\setminus\{c\}
= \mathcal{C}(A_{v,c})\setminus \{c\}
\).
\end{itemize}
We call \(P\) the \emph{\((v,c)\)-absorbing path}
and \(P'\) the \emph{\((v,c)\)-avoiding path}.
The \emph{internal vertices} of \(A_{v,c}\) are \(V(A_{v,c})\setminus \{v\}\) and the \emph{internal colours} are \(\mathcal{C}(A_{v,c}) \setminus \{c\}\).
For the sake of concreteness, we will refer to one of the endpoints of the paths as the \emph{first vertex} of the absorber and the other one as the \emph{last vertex}. \end{definition}
\begin{lemma} \label{lemma:absorbers-exist}
Let \(0<\delta<1\), \(C>0\) and
\( C^{-1} \ll \nu \ll \delta\).
Let \(G\sim \gdn{\delta}{n} \cup \gnp{n}{C/n}\) be uniformly coloured in \(\mathcal{C}=[n]\).
Then with high probability
the following holds.
For any \(v\in V(G)\) and \(c \in \mathcal{C}\)
and for all \(V' \subseteq V(G)\) and \(\mathcal{C}' \subseteq \mathcal{C}\)
that have size at least \((1-\nu )n\),
there exists a \((v,c)\)-absorber on 11 vertices with internal vertices in \(V'\) and internal colours in \(\mathcal{C}'\). \end{lemma}
Our absorbers will consist of the union of a triangle, a 4-cycle and two paths of length three between opposite vertices of the cycle and between a vertex in the triangle and a vertex in the 4-cycle. We require the colours of the triangle to match the internal colours of the square. See \Cref{fig:absorber}.
\begin{figure}
\caption{At the top is a \((v,c)\)-absorber. At the bottom the first figure shows the \((v,c)\)-absorbing path and the second figure the \((v,c)\)-avoiding path.}
\label{fig:absorber}
\end{figure}
\subsection{Finding squares} \label{subsect:squares} \newcommand{\alpha'}{\alpha'} \newcommand{\beta}{\beta}
We will use the following theorem, due to Fox and Sudakov~\cite{fox-sudakov}, that is based on the dependent random choice method.
\begin{theorem}[Theorem 3.1~\cite{fox-sudakov}; see also Prop.\ 5.3~\cite{fox-sudakov-DRC-survey}] \label{thm:DRC-strong} Let \(\alpha, \alpha'>0\) be constants such that \(\alpha' \ll \alpha\). Let $G$ be a bipartite graph of order \(n\) with bipartition \((A, B)\) and $e(A,B) \geq \alpha n^2$. Then there are $A' \subseteq A$, $B' \subseteq B$ such that for all \(a \in A', b \in B'\), the number of paths of length three between \(a\) and \(b\) in \(G[A',B']\) is at least \(\alpha' n^2\). \end{theorem}
\newcommand{\beta_{\text{DRC\:}}}{\beta_{\text{DRC\:}}} \newcommand{G_{\text{DRC\:}}}{G_{\text{DRC\:}}} \newcommand{E_{\text{DRC\:}}}{E_{\text{DRC\:}}} \newcommand{\text{DRC\:}}{\text{DRC\:}}
\begin{lemma} \label{lemma:DRC-bipartite-graph} Let \(\alpha, \beta>0\) be constants such that \(\beta \ll \alpha\). Let $G$ be a bipartite graph on $n$ vertices with bipartition \((A, B)\) and $e(A,B) \geq \alpha n^2$. Then there exist disjoint sets \(A_1, A_2\subseteq A\), \(B_1, B_2 \subseteq B\), such that for any \(a\in A_1, b\in B_1\), the number of paths of length three between \(a,b\) with internal vertices in \(A_2, B_2\) is at least \(\beta n^2\). Moreover, the minimum degree of \(G[A_1, B_1]\) is at least \(\beta n\). \end{lemma} \begin{proof} Let \(\alpha'\) satisfy \(\beta \ll \alpha' \ll \alpha \) and let \(A', B'\) be given by Theorem~\ref{thm:DRC-strong}. Let \((A_1, A_2)\) be a random partition of \(A'\), and \((B_1,B_2)\) be a random partition of \(B'\), i.e.\ each \(a\in A'\) lies in \(A_1\) independently with probability \(1/2\), and similarly for \(B_1\).
Then, since \(G[A',B']\) has minimum degree at least \(\alpha' n\), for each \(a\in A'\) the expected number of neighbours of \(a\) in \(B_1\) is at least \(\alpha' n/2\); the same is true for the number of neighbours of \(b\in B'\) in \(A_1\). Hence from Chernoff's bound, for any \(a\in A', b\in B'\), \[ \prob{ \abs{N(a) \cap B_1} \ge \alpha' n /3}, \, \prob{ \abs{N(b) \cap A_1} \ge \alpha' n /3} \ge 1- {\rm e}^{-\Omega(\alpha' n)}. \] Consider a pair \(a\in A', b\in B'\). Notice that the number of paths of length three between \(a,b\) in \(G[A',B']\) is equal to the number of edges between \(G[N(a), N(b)]\). From \Cref{thm:DRC-strong}, the number of edges of \(G[N(a)\cap B', N(b) \cap A']\) is at least \(\alpha' n^2\), hence there are \(B_{a,b} \subseteq N(a)\cap B'\), \(A_{a,b} \subseteq N(b)\cap A'\) such that \(G[A_{a,b}, B_{a,b}]\) has minimum degree at least \(\alpha' n\). Then the expected number of neighbours of each \(a'\in A_{a,b}\) in \(B_{a,b} \cap B_2\) is at least \(\alpha' n /2\), so by Chernoff's bound, for \(a'\in A_{a,b}\), \[ \prob{ \abs{N(a') \cap B_{a,b} \cap B_2} \ge \alpha' n/3 } \ge 1- {\rm e}^{-\Omega(\alpha' n)}. \] Similarly, for \(b'\in B_{a,b}\), \[ \prob{ \abs{N(b')\cap A_{a,b} \cap A_2} \ge \alpha' n/3 } \ge 1- {\rm e}^{-\Omega(\alpha' n)}. \] Hence, for each \(a\in A', b\in B'\), the probability that the minimum degree of \(G[A_{a,b}\cap A_2, B_{a,b} \cap B_2]\) is less than \(\alpha' n/3\) is, by the union bound, at most \[ \abs{A_{a,b}} {\rm e}^{-\Omega(\alpha' n)} + \abs{B_{a,b}} {\rm e}^{-\Omega(\alpha' n)} \le {\rm e}^{-\Omega(\alpha' n)}. \]
Moreover, the number of paths of length three between \(a,b\) with internal vertices in \(A_2, B_2\) is \[ e(N(a) \cap B_2, N(b) \cap A_2) \ge e(N(a) \cap B_2 \cap B_{a,b}, N(b) \cap A_2 \cap A_{a,b}), \] which is at least the square of the minimum degree of \(G[N(a) \cap B_2 \cap B_{a,b}, N(b) \cap A_2 \cap A_{a,b}]\). Hence, the probability that the number of paths of length three between \(a\in A', b\in B'\) with internal vertices in \(A_2, B_2\) is less than \(\alpha'^2 n^2 /9\) is at most \({\rm e}^{-\Omega(\alpha' n)}\).
By the union bound over \(a\in A', b\in B'\) and pairs \((a,b) \in A' \times B'\) the probability that a random partition fails to satisfy the lemma, with \(\beta = \alpha'^2/9\), is at most \( n {\rm e}^{-\Omega(\alpha' n)} + n^2 {\rm e}^{-\Omega(\alpha' n)}<1\). Thus there exists a partition as desired. \end{proof}
\newcommand{\lambda_{\text{sq}}}{\lambda_{\text{sq}}} \begin{lemma} \label{lemma:squares} Let \(\delta, q_1, q_2, \lambda_{\text{sq}}\) be constants such that \(0<\delta<1\), \(0<q_2<q_1\) and \(0<\lambda_{\text{sq}} \ll \delta, q_2\).
Let \(\mathcal{C}\) be a set of colors with \( \abs{\mathcal{C}}= q_1 n\) and \(\mathcal{C}_0 \subseteq \mathcal{C}^3\) be a collection of colour triples that are pairwise disjoint, with \(\abs{\mathcal{C}_0} = q_2 n\). Let \(G\) be a graph of order \(n\) and minimum degree at least \(\delta n\) which is uniformly coloured in \(\mathcal{C}\). Then, with probability at least \(1-{\rm e}^{-\lambda_{\text{sq}} n}\), the following holds. For any \(c \in \mathcal{C}\) there exists a 4-cycle in \(G\) coloured \((c_1, c_2, c_3,c)\), for some \((c_1,c_2,c_3) \in \mathcal{C}_0\). \end{lemma}
\begin{proof}
\newcommand{\gamma}{\gamma}
\newcommand{\mclb_3}{\gamma_3}
\newcommand{G_{\text{DRC}, c\,}}{G_{\text{DRC}, c\,}}
Let $\beta, \gamma_1, \mclb_3$ be constants such that
\(\beta \ll \delta\) and
\(\lambda_{\text{sq}} \ll \mclb_3 \ll \gamma_1 \ll \gamma \ll \beta, q_1^{-1} \).
Fix \(c\in \mathcal{C}\).
By passing to a bipartite subgraph of \(G\) with at least \(e(G)/2\) edges,
from \Cref{lemma:DRC-bipartite-graph} there exist disjoint
\(A_1,B_1,A_2,B_2 \subseteq V(G)\) such that the bipartite graph
\(G[A_1, B_1]\) has minimum degree at least \(\beta n\), and for all \(a\in A_1, b\in B_1\), the number of edges in \(G[N(a) \cap B_2, N(b) \cap A_2]\) is at least \(\beta n^2\).
We will reveal the colours of the edges in $G[A_1 \cup A_2, B_1 \cup B_2]$ in the order
\(E(A_1,B_1), E(A_1,B_2), E(A_2,B_1), E(A_2,B_2)\).
Since each edge of \(G[A_1, B_1]\) is coloured \(c\) independently with probability \((q_1 n)^{-1}\),
by \Cref{lemma:linear-matching}, with probability at least
\(1-{\rm e}^{- \Omega(\lambda_{\text{sq}} n)}\),
there exists a matching
\(M \subseteq G[A_1, B_1]\) of size at least \(\gamma n\) with all edges coloured \(c\).
For \((c_1, c_2, c_3) \in \mathcal{C}_0\) say an edge \(e\in E(A_2,B_2)\) is \emph{good} for \((c_1, c_2, c_3)\), if, when \(\mathcal{C}(e) = c_2\), it completes a 4-cycle coloured
\((c_1,c_2,c_3,c)\) with vertices in $A_1, B_1, A_2, B_2$ (in this order). Notice that, this definition does not depend on the colours of the edges in $G[A_2, B_2]$.
Let
\[
F(c_1,c_2,c_3) := \{ e\in E(A_2,B_2): e \text{ is good for } (c_1,c_2,c_3) \}.
\]
\begin{claim} \label{claim:good-edges}
Fix \((c_1,c_2,c_3) \in \mathcal{C}_0\).
With probability at least
\(1-{\rm e}^{- \Omega(\lambda_{\text{sq}} n)}\),
\(
\abs{F(c_1,c_2,c_3)} \ge \mclb_3 n.
\)
\end{claim}
\begin{proof}
Let \(ab \in M\).
Since \(e(N(a) \cap B_2, N(b) \cap A_2) \ge \beta n^2\), there exist \(A_{ab} \subseteq N(b) \cap A_2\),
\(B_{ab} \subseteq N(a) \cap B_2\)
such that \(G[A_{ab}, B_{ab}]\) has minimum degree at least \(\beta n\).
Let \(G'\) be the spanning subgraph of \(G\) such that
\(xy \in E(G')\) if and only if the following holds:
\begin{itemize}
\item If \(xy \in E_{G}(A_1, B_1)\) then \(xy \in M\).
\item If \(xy \in E_{G}(A_1,B_2)\) then \(x \in V(M) \cap A_1\) and \(y \in B_{x M(x)}\), where \(M(x)\) is the neighbour of \(x\) in \(M\).
\item If \(xy \in E_{G}(A_2, B_1)\) then \( y\in V(M) \cap B_1\) and \(x \in A_{M(y)y}\).
\item If \(xy \in E_{G}(A_2,B_2)\) then \(xy \in E_G(A_{e}, B_{e})\) for some \(e \in M\).
\end{itemize}
Since \(G\) is 4-partite with parts \(A_1,B_1,A_2,B_2\), this exhausts all possible edges of \(G'\).
Then the number of edges of \(G'[A_1, B_2]\) is at least
\(\sum_{a \in A_1 \cap V(M)} \abs{B_{ab}} \ge \gamma \beta n^2\).
Moreover, each edge is coloured \(c_1\) independently with probability \((q_1 n)^{-1}\).
Therefore, by \Cref{lemma:linear-matching},
with probability at least
\(1-{\rm e}^{-\Omega(\lambda_{\text{sq}} n)}\),
there is a matching \(M_1\) in \(G'[A_1, B_2]\) coloured \(c_1\) that has size at least \(\gamma_1 n\).
Finally, we will find a large matching \(M_3\) coloured \(c_3\) which, along with \(M_1\) and \(M\) will give us a large number of good edges for \((c_1,c_2,c_3)\).
To this end, let \(G^{\prime \prime}\) be the spanning subgraph of \(G'\) such that \(xy \in E(G^{\prime \prime})\) if and only if the following holds:
\begin{itemize}
\item if \(xy \in E_{G'}(A_1, B_1)\) then \(xy \in M\) and \(x\in V(M) \cap V(M_1)\).
\item If \(xy \in E_{G'}(A_1,B_2)\) then \(xy \in M_1\).
\item If \(xy \in E_{G'}(A_2, B_1)\) then
\(y\in V(M) \cap B_1\), \(M(y) \in V(M_1)\), and
\(x \in A_{M(y)y} \cap N_{G'}\left(M_1(M(y))\right) \).
\item If \(xy \in E_{G'}(A_2,B_2)\) then
there exists \(ab \in M\) such that \(y=M_1(a)\) and \(x\in A_{ab}\).
\end{itemize}
Again, this exhausts all possibilities for the edges of \(G^{\prime \prime}\).
Then the number of edges of \(G^{\prime \prime}[A_2,B_1]\) is at least
\[
\sum_{ab \in M:\, a\in V(M_1) \cap A_1} \abs{N_{G'}(M_1(a))\cap A_{ab}}
\ge \gamma_1 \beta n^2,
\]
where we use that for all \(ab\in M\) the minimum degree of
\(G^{\prime \prime}[A_{ab}, B_{ab}]\) is at least \(\beta n\).
Moreover, each edge of \(G^{\prime \prime}[A_2,B_1]\) is coloured \(c_3\) independently with probability \((q_1n)^{-1}\).
Hence, by \Cref{lemma:linear-matching}, with probability
at least
\(1-{\rm e}^{-\Omega(\lambda_{\text{sq}} n)}\),
there exists a matching \(M_3\) in \(G^{\prime \prime}[A_2,B_1]\) coloured \(c_3\) that has size
at least \(\gamma_3 n\).
Let
\[
F_0(c_1,c_2,c_3):= \{ xy \in E_{G^{\prime \prime}}(A_2,B_2): x \in V(M_3) \cap A_2 \ \}.
\]
Notice from the definition of \(G^{\prime \prime}\) that every \(x\in A_2\) has a neighbour in \(B_2\), hence
\(\abs{F_0(c_1,c_2,c_3)} \ge \abs{M_3}\).
Moreover, if \(xy \in F_0(c_1,c_2,c_3)\), then, by the definition of \(G^{\prime \prime}\),
there are \(a\in A_1, b \in B_1\) such that
\(a b \in M\), \(ay \in M_1\), \(xb \in M_3\); i.e.\
\(\mathcal{C}(ab) = c\),
\(\mathcal{C}(ay) = c_1\),
\(\mathcal{C}(xb) = c_3\).
Therefore, \(xy\) is a good edge for \((c_1,c_2,c_3)\).
Thus \(F_0(c_1,c_2,c_3) \subseteq F(c_1,c_2,c_3)\),
so
\(\abs{F(c_1,c_2,c_3)} \ge \abs{F_0(c_1,c_2,c_3)} \ge \abs{M_3} \ge \gamma_3 n\) and the claim
follows.
\end{proof}
By the union bound over \((c_1, c_2, c_3) \in \mathcal{C}_0\), for which there are \(q_2 n\) choices,
\Cref{claim:good-edges} implies that
with probability
at least
\(1-{\rm e}^{-\Omega(\lambda_{\text{sq}} n)}\),
for each \((c_1, c_2, c_3) \in \mathcal{C}_0\),
\(\abs{F(c_1,c_2,c_3)} \ge \mclb_3 n\).
Let
\[
F'(e) := \{c_2 \in \mathcal{C}:
\text{there exist \(c_1, c_3\) such that }
(c_1,c_2,c_3) \in \mathcal{C}_0 \text{ and } e \in F(c_1, c_2, c_3)\}.
\]
Then, using that no two triples in \(\mathcal{C}_0\) share a colour we have
\[
\sum_{e \in E(A_2,B_2)} \abs{F'(e)} = \sum_{(c_1,c_2,c_3) \in \mathcal{C}_0} \abs{F(c_1,c_2,c_3)}
\ge
\abs{\mathcal{C}_0} \mclb_3 n
= q_2 \mclb_3 n^2.
\]
Now we reveal the colours of \(E(A_2,B_2)\).
For \(e\in E(A_2,B_2)\) let \(A_e\) be the event that \(e\) gets a good colour i.e.\
\(\mathcal{C}(e) \in F'(e)\).
Then \(\prob{A_e} = \abs{ F'(e)} /q_1 n \).
Each edge is coloured independently,
so the events \(A_e\) are mutually independent.
Hence, the probability that no
\(e\in E(A_2,B_2)\) gets a good colour is
\begin{align*}
\prod_{e \in E(A_2,B_2)} \left(1- \prob{A_e}\right)
& \le \exp \left(
-\sum_{e \in E(A_2,B_2)} \prob{A_e}
\right) \\
& =
\exp \left(-\sum_{e \in E(A_2, B_2)} \frac{|F'(e)|}{q_1n}\right)
\le
\exp \left( -\frac{q_2 \mclb_3 n}{q_1} \right).
\end{align*}
Hence, with probability at least
\(1-{\rm e}^{ -\frac{q_2 \mclb_3 n}{q_1}}\),
at least one edge gets a good colour, i.e.\ there exists $e \in E(A_2, B_2)$ such that $\mathcal{C}(e) \in F'(e)$, as required for the lemma.
The above fails for some colour \(c\) with probability at most
\[
q_1 n {\rm e}^{-\Omega(\lambda_{\text{sq}} n)} \le {\rm e}^{\lambda_{\text{sq}} n},
\]
proving the lemma. \end{proof}
\subsection{Proof of Lemma~\ref{lemma:absorbers-exist}}
\begin{proof}[Proof of \Cref{lemma:absorbers-exist}] \renewcommand{\mindegG}{\gdn{\delta}{n}}
\newcommand{\lambda}{\lambda}
\newcommand{\prblem}{\lambda}
\newcommand{\prblem}{\lambda}
Let $\rho, \rho'$ be constants satisfying
\(C^{-1} \ll \nu \ll \lambda, \rho, \rho' \ll \delta\).
Fix \(v\in V(G),\, c\in \mathcal{C}\) and
\(V' \subseteq V(G),\, \mathcal{C}' \subseteq \mathcal{C}\) of size at least $(1 - \nu)n$.
\newcommand{\mindegGprm}{\gdn{\delta/2}{n'}}
\newcommand{\randomGprm}{\gnp{n'}{C/2n'}}
For the next claim, it is useful to refer to \Cref{fig:absorber}.
\begin{claim} \label{claim:sq-tr}
With probability \(1-{\rm e}^{-\Omega(\prblem n)}\),
there exist a 4-cycle \(K=xyzw\) and a triangle \(T=vuu'\) in \(G[V']\) such that
\(\mathcal{C}(yz) = c\),
\(\mathcal{C}(xw) = \mathcal{C}(u u')\),
\(\mathcal{C}(xy) = \mathcal{C}(vu')\),
\(\mathcal{C}(zw) = \mathcal{C}(vu)\).
\end{claim}
\begin{proof}
Let \((V_{\triangle}, V_{\square})\) be a random partition of \(V'\).
Then from Chernoff's bound, a union bound over \(v\in V'\), and $\nu \ll 1$, with probability \(1-{\rm e}^{-\Omega(\delta n)}\), the graphs \(\gdn{\delta}{n}[V_{\triangle}]\), \(\gdn{\delta}{n}[V_{\square}]\) have minimum degree at least \(\delta n /3\) and \(\abs{V_{\triangle}}, \abs{V_{\square}} \ge n/3\).
First reveal the random edges and colours of $G[V_{\triangle}]$.
Then, by \Cref{lemma:con},
with probability \(1-{\rm e}^{-\Omega(\prblem n)}\),
there is a collection
\(\Delta_v\) of \(\rho n\) rainbow triangles that pairwise intersect only on \(v\); are pairwise colour-disjoint; and \(V(\Delta_v) \subseteq V_{\triangle} \cup \{v \} \).
Let
$\mathcal{C}_v$ be the collection of colour triples $(\mathcal{C}(vu), \mathcal{C}(uu'), \mathcal{C}(vu'))$ with $vuu' \in \Delta_v$, whose three colours are in $\mathcal{C}'$. Then $\abs{\mathcal{C}_v} \ge \rho n - 3\nu n \ge (\rho/2)n$.
Next reveal the colours of edges in $G[V_{\square}]$.
By setting \(\mathcal{C}_0 = \mathcal{C}_v\) in \Cref{lemma:squares}, it follows that with probability \(1-{\rm e}^{-\Omega(\prblem n)}\)
there exists a 4-cycle \(xyzw\) and a triangle \(vuu' \in \Delta_v\) with colours in $\mathcal{C}_v$, such that
\(\mathcal{C}(yz) = c\),
\(\mathcal{C}(xw) = \mathcal{C}(u u')\),
\(\mathcal{C}(xy) = \mathcal{C}(vu')\),
\(\mathcal{C}(wz) = \mathcal{C}(vu)\).
We fail to find a triangle or square as required with probability at most \({\rm e}^{-\Omega(\prblem n)}\).
\end{proof}
By \Cref{lemma:con}, with probability \(1-{\rm e}^{-\Omega(\prblem n)}\), for every \(u,v \in V'\) there are \(\rho' n\) rainbow paths of length three between \(u,v\) which are pairwise colour disjoint and internally vertex disjoint.
Hence, with probability
\(1-{\rm e}^{-\Omega(\prblem n)}\), this and the conclusion of \Cref{claim:sq-tr} hold simultaneously.
Then, using $\nu \ll \rho'$, there exists two colour- and vertex-disjoint rainbow paths $P_1, P_3$ of length $3$ such that: $P_1$ has endpoints $u_2, w$; $P_3$ has endpoints $x,z$; the interiors of $P_1, P_2$ are in $V' \setminus (V(K) \cup V(C))$; and the colours of $P_1, P_2$ are in $\mathcal{C}' \setminus (\mathcal{C}(K) \cup \mathcal{C}(T))$.
Then the graph $A_{v,c}$, defined as
\[
A_{v,c} = K\, \cup\, T \,\cup\, P_1\, \cup\, P_2,
\]
is a \((v,c)\)-absorber: the
\((v,c)\)-absorbing path is
\(u v u' P_1 w x P_2 z y\)
and the \((v,c)\)-avoiding path is
\(u u' P_1 w z P_2 x y\), and it is straightforward to check they satisfy~\Cref{def:absorber}.
Clearly \(A_{v,c}\) has 11 vertices.
The number of \(V' \subseteq V\) of size at least \((1-\nu) n\) is at most
\(n \binom{n}{\nu n} = {\rm e}^{O\left(\nu \log \nu\right) n}\),
and the same bound holds for the number of
\(\mathcal{C}'\subseteq \mathcal{C}\) of the same size.
Using
\(\nu \ll \prblem\),
the probability we fail to find an absorber for some \(v,\, c,\, V',\, \mathcal{C}'\) is by the union bound at most
\[
n^2{\rm e}^{O\left(\nu\log\nu \right)n} \cdot {\rm e}^{-\Omega(\prblem n)}
\le n^{-2}.\qedhere
\]
\end{proof}
\section{Proof of Lemma~\ref{lemma:main}} \label{section:proof-main-lemma}
To cover an arbitrary small subset of the vertices using \(P_{\text{abs}}\) into a rainbow path \(Q\) we need the following lemma, which asserts that for any two vertices and a color we can connect them with a short rainbow path through a random subset of the vertices. \begin{lemma}[Flexible sets]\label{lemma:cover} Let \(\zeta, \mu, \delta \in (0,1)\) and \(C>0\) be constants such that \(C^{-1} \ll \zeta\ll \mu \ll \delta\). Let \(G\sim \gdn{\delta}{n} \cup \gnp{n}{C/n}\). Then there exist \(V_{\text{flex}\,}\subseteq V\), \(\cols_{\text{flex}} \subseteq \mathcal{C}\) of size \(2 \mu n\) such that with high probability the following holds. For all \(u,v \in V\), \(c\in \mathcal{C}\), and \(V_{\text{flex}\,}' \subseteq V_{\text{flex}\,}\), \(\cols_{\text{flex}}' \subseteq \cols_{\text{flex}}\) of size at least \((2\mu - \zeta)n\), there exists a rainbow path of length seven with endpoints \(u,v\), internal vertices in \(V_{\text{flex}\,}'\) and colours in \(\cols_{\text{flex}}' \cup \{c\}\), that contains the colour \(c\). \end{lemma}
\begin{proof}
Let $\gamma$ be a constant satisfying $C^{-1} \ll \zeta \ll \gamma \ll \mu \ll \nu \ll \delta$.
For a colour $c$, let $M_c$ be a largest matching of colour $c$ in $G$, and for distinct vertices $u,v$, let $\mathcal{P}_{u,v}$ be a largest collection of pairwise vertex- and colour-disjoint rainbow paths of length three with endpoints $u,v$.
By \Cref{lemma:linear-matching,lemma:con}, with probability $1 - e^{-\gamma n}$, we have $|M_c| \ge \gamma n$ and $|\mathcal{P}_{u,v}| \ge \gamma n$ for every colour $c$ and distinct vertices $u,v$.
\def \Vrand {V'}
\def \Crand {\mathcal{C}'}
Let $\Vrand$ be a random subset of $V$, obtained by including each vertex independently with probability $\mu$, and let $\Crand$ be a random subset of $\mathcal{C}$, obtained by including each colour independently with probability $\mu$.
Then, by Chernoff and union bounds, with high probability,
the following properties hold.
\begin{itemize}
\item
$|\Vrand|, |\Crand| \le 2\mu n$,
\item
at least $\frac{1}{2} \mu^2 \gamma n$ edges in $M_c$ have both endpoints in $\Vrand$, for every $c \in \mathcal{C}$,
\item
at least $\frac{1}{2} \mu^5 \gamma n$ paths in $\mathcal{P}_{u,v}$ have their interior vertices in $\Vrand$ and all colours in $\Crand$, for all distinct $u,v \in V$.
\end{itemize}
Suppose that all three properties hold, and let $V_{\text{flex}}$ be a subset of $V$ that contains $\Vrand$ and has size $2\mu n$ and let $\cols_{\text{flex}}$ be a subset of $\mathcal{C}$ that contains $\Crand$ and has size $2\mu n$.
We show that these sets satisfy the requirements of the lemma. Indeed, fix $u,v, c$ and $V_{\text{flex}}',\cols_{\text{flex}}'$ as in the lemma.
Then, as $\zeta \ll \mu, \gamma$, there is an edge $e = xy \in M_c$ with both ends in $V_{\text{flex}}'$.
Similarly, there are paths $P_1 \in \mathcal{P}_{u,x}, P_2 \in \mathcal{P}_{y,v}$ that are vertex- and colour-disjoint, their interiors are in $V_{\text{flex}}'$, and their colours are in $\cols_{\text{flex}}' \setminus \{c\}$. Then $P_1 \cup e \cup P_2$ is a path that satisfies the requirements of the lemma. \end{proof}
\newcommand{\remove}[1]{} \remove{ \begin{proof} \newcommand{\flxcnst'_1}{\mu'_1} \newcommand{\flxcnst'_2}{\mu'_2} \newcommand{\flxcnst'_3}{\mu'_3}
Let \[ C^{-1} \ll \zeta \ll \mu'\ll \flxcnst'_3 \ll \flxcnst'_2 \ll \flxcnst'_1 \ll \mu \ll \nu \ll \delta \] be constants. We will show that there exist \(V_{\text{flex}\,}, \cols_{\text{flex}}\) and some constant \(\mu'\) with the following property: with probability at least \(1-n^{-2}\), for all \(u,v,c\), there exist \(\mu' n\) paths of length five between \(u,v\) which are rainbow, with colours in \(\cols_{\text{flex}}\), the third edge coloured \(c\) and internal vertices in \(V_{\text{flex}\,}\). Moreover, these paths are internally vertex disjoint, and the only colour they share is \(c\). Then since \(\zeta \ll \mu '\), one of these paths will have all its colours in \(\cols_{\text{flex}}' \cup \{c\}\) and internal vertices in \(V_{\text{flex}\,}'\).
Let \(V_{\text{flex}\,}^1, V_{\text{flex}\,}^2\) be two disjoint random subsets of \(V\) of size \(\mu n\). Let \(\cols_{\text{flex}}\) an arbitrary subset of \(\mathcal{C}\) of size \(2 \mu n\). Then for each \(x\in V\), \( \expect{\abs{ N_{\mindegG}(x) \cap V_{\text{flex}\,}^i}} = \mu \abs{ N_{\mindegG}(x)} \ge \mu \delta n, \) for \(i=1,2\). Hence the hypergeometric version of Chernoff's bound (\Cref{thm:Chernoff}) and the union bound over \(x\in V\) and \(i=1,2\) yield that with probability \(1-{\rm e}^{-\Omega(\delta \mu n)}\), for all \(x\in V\), \(\abs{N_{\mindegG}(x) \cap V_{\text{flex}\,}^{i}} \ge \delta \mu n/2\), where \(i=1,2\). Hence there exist disjoint sets \(V_{\text{flex}\,}^1, V_{\text{flex}\,}^2\) of size \(\mu n\) such that for all \(x\in V\), \(\abs{N_{\mindegG}(x) \cap V_{\text{flex}\,}^{i}} \ge\delta \mu n/2\), i.e.\ \(\mindegG[V_{\text{flex}\,}^1], \mindegG[V_{\text{flex}\,}^1]\) have minimum degree \(\delta \mu n/2\).
Fix \(u,v,c\). Reveal the colouring of \(\mindegG[V_{\text{flex}\,}^1]\). Since each edge of \(\mindegG[V_{\text{flex}\,}^1]\) is coloured \(c\) independently with probability \(n^{-1}\), by \Cref{lemma:linear-matching} with probability \(1-{\rm e}^{-\Omega(\delta \mu n)}\) there exists a matching \(M_c\) of size \(\flxcnst'_1 n\) coloured \(c\).
Let \( N_u \subseteq N_{\mindegG[V_{\text{flex}\,}^2]}(u) \), \(N_v \subseteq N_{\mindegG[V_{\text{flex}\,}^2]}(v) \) be disjoint and of size \( \delta \mu n/4\). Let \(V_{c,u}\) be a vertex cover of \(M_c\), and reveal the edges of \(\gnp{n}{C/n}\) with one endpoint in \(N_u\) and another in \(V_{c,u}\). By Lemma~\ref{lemma:linear-matching}, with probability \(1-{\rm e}^{-\Omega(C \flxcnst'_1 n)}\), there exists a matching in \(\gnp{n}{C/n}\) between \(N_u\) and \(V_{c,u}\) of size \(\flxcnst'_2 n\) (using that \(\flxcnst'_1 \ll \delta, \mu\)). Similarly, by another application of Lemma~\ref{lemma:linear-matching}, with probability \(1-{\rm e}^{-\Omega(C \flxcnst'_2 n)}\), there is matching between the \(M_c\)-neighbours of the subset of \(V_{c,u}\) in the matching and \(N_v\), which has size \(\flxcnst'_3 n\). Thus, with probability \(1-{\rm e}^{-\Omega(C \flxcnst'_2 n)}\), we have a collection \(\mathcal{P}(u,v,c)\) of paths of length 5 with \(\abs{\mathcal{P}(u,v,c)} \ge \flxcnst'_3 n\), which have endpoints \(u,v\), are internally vertex disjoint, their third edge is coloured \(c\), and all internal vertices are in \(V_{\text{flex}\,}\).
Now reveal whether each \(P \in \mathcal{P}(u,v,c)\) is rainbow, and the colours of all but the third edge lie in \(\mathcal{C}'\), without revealing the colours of the edges (other than the third edge). This is event holds independently for each path with probability at least \( (2\mu - \zeta)^4/2\), since \(\mathcal{C}' \ge (2\mu - \zeta)n\). Therefore from Chernoff's bound (\Cref{thm:Chernoff}), with probability \(1-{\rm e}^{-\Omega((2\mu - \zeta)^4 \flxcnst'_3 n)}\), there exists a \(\mathcal{P}'(u,v,c) \subseteq \mathcal{P}(u,v,c) \) with \(\abs{\mathcal{P}'(u,v,c) } \ge (\mu - \zeta)^4 \flxcnst'_3 n\), that is rainbow with colours in \(\cols_{\text{flex}}\).
Moreover, we claim that there is a subcollection of paths \(\mathcal{P}^{\prime \prime}(u,v,c) \subseteq \mathcal{P}^{\prime}(u,v,c)\), with \( \abs{\mathcal{P} ^{\prime \prime} (u,v,c)} \ge \mu' n\), such that no two paths in \(\mathcal{P}^{\prime \prime}(u,v,c)\) share a colour other than \(c\). This follows from Lemma~\ref{lemma:linear-matching}: consider the 4-uniform hypergraph with vertex set \(\mathcal{C}' \setminus \{ c\} \) where \(\{c_1,c_2,c_3,c_4 \}\) is an edge if and only if there is a path in \(\mathcal{P}^{\prime}(u,v,c)\) with these colours. By symmetry, every edge is equally likely, and there are \( \abs{\mathcal{P}^{\prime}(u,v,c)} \ge (\mu - \zeta)^4 \flxcnst'_3 n\) edges selected with repetition. Hence Lemma~\ref{lemma:linear-matching} implies that with probability \(1-{\rm e}^{-\Omega((\mu - \zeta)^4 \flxcnst'_3 n)}\) there is a matching of size \(\mu' n\), i.e.\ a collection \(\mathcal{P}^{\prime \prime}(u,v,c)\) of paths that share no colour other than \(c\).
This fails with probability at most \( n^{-5} \), so by the union bound over \(u,v,c\), we have the desired property with probability at least \(1-n^{-2}\). \end{proof} }
We will put together several \( (v,c) \)-absorbers to construct the paths in Lemma~\ref{lemma:main}, by having a \((v,c)\)-absorber for each edge of a bipartite graph which has the following property. This follows an idea introduced by Montgomery~\cite{montgomery}, which was adapted to the rainbow setting by Gould, Kelly, K\"uhn and Osthus~\cite{kuhn-osthus}.
\begin{definition}[Def.\ 3.3,~\cite{kuhn-osthus}] \label{def:robustly-matchable} Let $H$ be a balanced bipartite graph with bipartition $(A,B)$.
We say $H$ is \emph{robustly matchable} with respect to $A^{\prime},\, B^{\prime}$, for some \(A^{\prime} \subseteq A\) and \(B^{\prime} \subseteq B\) of equal size, if for every pair of sets $X\subseteq A^{\prime}, Y\subseteq B^{\prime}$ with $\abs{X} = \abs{Y} \leq \abs{A^{\prime}}/2,$ there is a perfect matching in $H[A\setminus X, B\setminus Y]$. We call \(A', B'\) the \emph{flexible sets} of \(H\). \end{definition}
\begin{proposition}[Lemma 4.5, \cite{kuhn-osthus}] \label{prop:rmbg} For every large enough $m\in \mathbb{N}$, there exists a $256$-regular bipartite graph with bipartition $(A,B)$ and $\abs{A} = \abs{B} = 7m$, which is robustly matchable with respect to some $A^{\prime} \subseteq A, B^{\prime} \subseteq B$ with $\abs{A^{\prime}} = \abs{B^{\prime}} = 2m$. \end{proposition} \begin{proof}[Proof of \Cref{lemma:main}]
Let \(\zeta, \mu, \nu \in (0,1)\) be constants such that
\[
C^{-1} \ll
\eta
\ll \zeta\ll \mu\ll \nu \ll \delta.
\]
Let \(V_{\text{flex}\,},\) \(\cols_{\text{flex}}\) be the sets given by \Cref{lemma:cover} that have size \(2\mu n\).
By the union bound, the conclusions of \Cref{lemma:con,lemma:cover,lemma:absorbers-exist} hold simultaneously with high probability.
Assume they all hold.
Let \(V_{\text{buf}\,},\, \cols_{\text{buf}\,}\) be arbitrary subsets of \(V\setminus V_{\text{flex}\,},\, \mathcal{C}\setminus \cols_{\text{flex}}\) of size \(5 \mu n\).
Let \(H\) be a bipartite graph on
\((V_{\text{flex}\,} \cup V_{\text{buf}\,},\, \cols_{\text{flex}} \cup \cols_{\text{buf}\,})\)
that is isomorphic to a graph as in \Cref{prop:rmbg} such that \(V_{\text{flex}\,},\, \cols_{\text{flex}}\) are the flexible sets.
\begin{claim} \label{claim:absorbing-template}
There is collection of absorbers $A_{v,c}$ on $11$ vertices and rainbow paths $P_{v,c}$ of length three, for each edge $vc$ in $H$, with the following properties:
the internal vertices of $A_{v,c}$ and of $P_{v,c}$ are pairwise disjoint and disjoint of $V_{\text{flex}\,} \cup V_{\text{buf}\,}$;
the internal colours of $A_{v,c}$ and the colours of $P_{v,c}$ are pairwise disjoint and disjoint of $\cols_{\text{flex}} \cup \cols_{\text{buf}\,}$; and for some ordering of the edges of $H$, the path $P_{v,c}$ starts with the last vertex of $A_{v',c'}$ and ends with the first vertex of $A_{v,c}$, where $v'c'$ is the predecessor of $vc$ in the ordering (so we can ignore $P_{vc}$ for the first edge $vc$).
\end{claim}
\begin{proof}
Let \(H_0\) be a maximal subgraph of
\(H\) with some ordering of its edges, for which we can find a collection of absorbers and paths as in the claim.
Suppose for contradiction \(H_0 \neq H\) and let
\(v_1 c_1 \in E(H\setminus H_0)\) and \(v_0 c_0\) be the last edge of \(H_0\) in the ordering, that has absorber \(A_{v_0, c_0}\).
Let \(V_0, \mathcal{C}_0\) be the union of the vertices and colours spanned by the absorbers for \(E(H_0)\), the paths connecting them, and
\(V_{\text{flex}\,} \cup V_{\text{buf}\,}\),
\(\cols_{\text{flex}} \cup \cols_{\text{buf}\,}\).
Then, since each absorber has $11$ vertices
and each path connecting consecutive absorbers has $4$ vertices, we have
\[
\abs{V_0}, \abs{\mathcal{C}_0} = O(e(H_0)) = O(\mu n) < \nu n/2,
\]
where
for the inequality we used that \(\mu \ll \nu\).
Hence by \Cref{lemma:absorbers-exist} there exists a \((v_1, c_1)\)-absorber \(A_{v_1,c_1}\) on 11 vertices
with internal vertices and internal colours disjoint from
\(V_0\) and \(\mathcal{C}_0\).
Moreover,
by \Cref{lemma:con} there exists a rainbow path $P_{v_1, c_1}$ of length three
between the last vertex of \(A_{v_0, c_0}\) and the first vertex of \(A_{v_1,c_1}\), with internal vertices disjoint from
\(V_0 \cup V(A_{v_1, c_1})\) and colours disjoint from \(\mathcal{C}_0 \cup \mathcal{C}(A_{c_1, c_1})\).
Then the subgraph of \(H\) with edges
\(E(H_0) \cup \{ v_1 c_1 \}\) satisfies the conditions of the claim and properly contains
\(H_0\), contradicting the maximality of \(H_0\).
\end{proof}
We can now define \(P_{\text{abs}}\).
Since \(H\) is regular bipartite, it has a perfect matching \(M\).
For \(vc \in E(H)\), let \(P_{M}(vc)\) be the
\((v,c)\)-absorbing path of \(A_{v,c}\), if \(vc\in E(M)\), and the avoiding path otherwise.
Let \(P_{vc} = \emptyset\) if $vc$ is the first edge, and otherwise let $P_{vc}$ be as in \Cref{claim:absorbing-template}.
Set
\[
P_{\text{abs}} = \bigcup_{vc \in E(H)} (P_{M}(vc) \cup P_{vc}).
\]
Then \(P_{\text{abs}}\) uses each \(v\in V_{\text{flex}\,} \cup V_{\text{buf}\,}\) and \(c \in \cols_{\text{flex}} \cup \cols_{\text{buf}\,}\) precisely once;
any other vertex and colour in \(P_{\text{abs}}\) is also used, by construction, precisely once.
Therefore
\(P_{\text{abs}}\)
is a rainbow path that is spanning in
\(\bigcup_{vc \in E(H)} (V(A_{vc}) \cup V(P_{vc}))\)
and \(\bigcup_{vc \in E(H)}(\mathcal{C}(A_{vc}) \cup \mathcal{C}(P_{vc}))\)
with endpoints the first vertex \(w\) of the first absorber and the last vertex \(w'\) of the last absorber.
We will now show how to construct \(Q\), given $V' \subseteq V \setminus V(P_{\text{abs}})$ and $\mathcal{C}' \subseteq\mathcal{C} \setminus \mathcal{C}(P_{\text{abs}})$ of size between $2$ and $\eta n$, with endpoints \(x,y \in V'\).
Let \(c_0 \in \mathcal{C}'\).
From \Cref{lemma:cover}, there exists a rainbow path \(Q_1\) with endpoints
\( w,x \), internal vertices in \(V_{\text{flex}\,}\) and colours in \(\cols_{\text{flex}} \cup \{c_0\}\), which includes the colour $c_0$ and has length 7.
\begin{claim}
There exists a rainbow path \(Q_2\) between \(w', y\), with internal vertices
\(V_{\text{flex}\,}^{\prime \prime} \cup (V'\setminus x)\),
and colours \(\cols_{\text{flex}}^{\prime \prime} \cup (\mathcal{C}'\setminus c_0)\),
for some \(V_{\text{flex}\,}^{\prime \prime} \subseteq V_{\text{flex}\,} \setminus V(Q_1)\),
\(\cols_{\text{flex}}^{\prime \prime} \subseteq \cols_{\text{flex}} \setminus \mathcal{C}(Q_1)\)
with
\(\abs{V_{\text{flex}\,}^{\prime \prime}} = \abs{\cols_{\text{flex}}^{\prime \prime}} \le \mu n - 7\).
\end{claim}
\begin{proof}
The Claim will follow by applying greedily \Cref{lemma:cover} to cover \(V'\setminus x\), \(\mathcal{C}'\setminus c_0\) using
\(V_{\text{flex}\,} \setminus V(Q_1)\), \(\cols_{\text{flex}} \setminus \mathcal{C}(Q_1)\)
in a rainbow path with endpoints \(w'\) and \(y\).
Fix a linear order of \(V'\setminus x\) with \(y\) the last vertex.
Let \(P_0\) be a longest path from \(w'\) to a vertex in $V' \setminus x$
among all rainbow paths
that start at \(w'\) and satisfy the following:
if \(u,v \in V(P_0) \cap (V'\setminus x)\) and \(u<v\), then \(u\) appears before \(v\) on \(P_0\);
every seventh vertex on \(P_0\) lies in \(V'\setminus x\), and all other vertices are in
\(\{w'\} \cup V_{\text{flex}\,}\setminus V(Q_1)\);
between consecutive vertices in \(V'\setminus x\), and between \(w'\) and the first vertex in \(V'\setminus x\), there is
exactly one edge with colour in $\mathcal{C}'\setminus c_0$, and all other edges have colours in \(\cols_{\text{flex}}\setminus \mathcal{C}(Q_1)\).
Let \(z\) be the last vertex of \(P_0\).
If \(z = y\) we are done so suppose otherwise, and let \(z'\in V'\setminus x\) be the vertex after \(z\) in the order.
Since, by construction,
\( \abs{V(P_0) \cap V'} = \abs{\mathcal{C}(P_0) \cap \mathcal{C}'} \),
there is also \(c_1 \in \mathcal{C}'\setminus \mathcal{C}(P_0)\).
Let \(V_{\text{flex}\,}' = V_{\text{flex}\,} \setminus (V(P_0) \cup V(Q_1))\),
\(\cols_{\text{flex}}' = \cols_{\text{flex}} \setminus (\mathcal{C}(P_0)\cup \mathcal{C}(Q_1))\).
Since
\(
\abs{P_0} \le 7\abs{V'} \le 7 \eta n,
\)
and \(Q_1\) has length 7,
using \(\eta \ll \zeta\),
it follows that
\(
\abs{V_{\text{flex}\,}'} = \abs{\cols_{\text{flex}}'} \ge (2\mu -\zeta)n.
\)
Hence by \Cref{lemma:cover} there is a rainbow path \(P_1\) between
\(z, z'\) of length 7, that contains an edge with colour \(c_1\),
and whose internal vertices and other colours are in \(V_{\text{flex}\,}', \cols_{\text{flex}}'\).
Then \(P_1 \cup P_0\) contradicts the maximality of \(P_0\).
Let \(V_{\text{flex}\,}^{\prime \prime} = V(P_0) \cap V_{\text{flex}\,}\),
\(\cols_{\text{flex}} ^{\prime \prime} = \mathcal{C}(P_0) \cap \cols_{\text{flex}}\).
Then
\(
\abs{V_{\text{flex}\,}^{\prime \prime}} = \abs{\cols_{\text{flex}} ^{\prime \prime}} < \abs{P_0} \le \zeta n < \mu n -7,
\)
so we can take $Q_2 = P_0$.
\end{proof}
Let $V_{\text{flex}\,}''' = (V(Q_1) \cup V(Q_2)) \cap V_{\text{flex}\,}$ and $\cols_{\text{flex}}''' = (\mathcal{C}(Q_1) \cup \mathcal{C}(Q_2)) \cap \cols_{\text{flex}}$.
Then
we have
$\abs{V_{\text{flex}\,}'''} = \abs{\cols_{\text{flex}}'''} \le \mu n$.
Hence by choice of $H$ there is a matching $M'$ between $V_{\text{flex}\,} \setminus V_{\text{flex}\,}'''$ and $\cols_{\text{flex}} \setminus \cols_{\text{flex}}'''$.
As before, for \(vc\in E(H)\)
let \(P_{M'}(vc)\) be the \((v,c)\)-absorbing path of \(A_{v,c}\) if \(vc\in E(M')\) and the avoiding path otherwise.
Let
\[
P_{\text{abs}}' = \bigcup_{vc\in E(H)} (P_{M'}(vc)\ \cup P_{vc}).
\]
Then \(P_{\text{abs}}'\) is a rainbow path that is spanning in
\(V(P_{\text{abs}})\setminus V_{\text{flex}\,}'''\)
and
\(\mathcal{C}(P_{\text{abs}}) \setminus \cols_{\text{flex}}'''\)
with endpoints \(w,w'\).
Therefore
\(Q = Q_1 \cup P_{\text{abs}}' \cup Q_2\) is a rainbow path, spanning in \(V(P_{\text{abs}}) \cup V'\) and \(\mathcal{C}(P_{\text{abs}}) \cup \mathcal{C}'\) and has endpoints \(x,y\). \end{proof}
\end{document} | arXiv |
Introduction to the A7000 Textbook
Chapter 1 Science and the Universe: A Brief Tour
1.1 The Nature of Astronomy
1.3 The Laws of Nature
1.4 Numbers in Astronomy
1.5 Consequences of Light Travel Time
1.6 A Tour of the Universe
1.7 The Universe on the Large Scale
1.8 The Universe of the Very Small
1.9 A Conclusion and a Beginning
8.0 Thinking Ahead
8.1 The Global Perspective
8.2 Earth's Crust
8.3 Earth's Atmosphere
8.4 Life, Chemical Evolution, and Climate Change
8.5 Cosmic Influences on the Evolution of Earth
Chapter 2 Observing the Sky: The Birth of Astronomy
2.1 The Sky Above
2.2 Ancient Astronomy Around the World
2.3 Astronomy of the First Nations of Canada
2.4 Ancient Babylonian, Greek and Roman Astronomy
2.5 Astrology and Astronomy
2.6 The Birth of Modern Astronomy – Copernicus and Galileo
2.7 For Further Exploration, Websites
2.8 Collaborative Group Activities
2.9 Questions and Exercises
Chapter 3 Orbits and Gravity
3.1 The Laws of Planetary Motion
3.2 Newton's Great Synthesis
3.4 Orbits in the Solar System
3.5 Motions of Satellites and Spacecraft
3.6 Gravity with More Than Two Bodies
3.7 For Further Exploration
Chapter 4 Earth, Moon, and Sky
4.1 Earth and Sky
4.2 The Seasons
4.3 Keeping Time
4.4 The Calendar
4.5 Phases and Motions of the Moon
4.6 Ocean Tides and the Moon
4.7 Eclipses of the Sun and Moon
Chapter 7 Other Worlds: An Introduction to the Solar System
7.1 Overview of Our Planetary System
7.2 Composition and Structure of Planets
7.3 Dating Planetary Surfaces
7.4 Origin of the Solar System
Chapter 9 Cratered Worlds
9.1 General Properties of the Moon
9.2 The Lunar Surface
9.3 Impact Craters
9.4 The Origin of the Moon
9.5 Mercury
9.6 Key Concepts and Summary, Further Explorations
9.7 Collaborative Group Activities and Exercises
Chapter 10 Earthlike Planets: Venus and Mars
10.0 Thinking Ahead
10.1 The Nearest Planets: An Overview
10.2 The Geology of Venus
10.3 The Massive Atmosphere of Venus
10.4 The Geology of Mars
10.5 Water and Life on Mars
10.6 Divergent Planetary Evolution
10.7 Collaborative Group Activities and Exercises
Chapter 11 The Giant Planets
11.1 Exploring the Outer Planets
11.2 The Giant Planets
11.3 Atmospheres of the Giant Planets
Chapter 12 Rings, Moons, and Pluto
12.1 Ring and Moon Systems Introduced
12.2 The Galilean Moons of Jupiter
12.3 Titan and Triton
12.4 Pluto and Charon
12.5 Planetary Rings
12.6 Summary, Further Exploration, Websites
Chapter 13 Comets and Asteroids: Debris of the Solar System
13.2 Asteroids and Planetary Defense
13.3 The "Long-Haired" Comets
13.4 The Origin and Fate of Comets and Related Objects
13.5 Key Concepts and Summary, Further Explorations, Websites
13.1 Asteroids
Chapter 14 Cosmic Samples and the Origin of the Solar System
14.2 Meteorites: Stones from Heaven
14.3 Formation of the Solar System
14.4 Comparison with Other Planetary Systems
14.5 Planetary Evolution
14.6 Collaborative Activities, Questions and Exercises
Chapter 15 The Sun: A Garden-Variety Star
15.1 The Structure and Composition of the Sun
15.2 The Solar Cycle
15.3 Solar Activity above the Photosphere
15.4 Space Weather
Chapter 16 The Sun: A Nuclear Powerhouse
16.1 Sources of Sunshine: Thermal and Gravitational Energy
16.2 Mass, Energy, and the Theory of Relativity
16.3 The Solar Interior: Theory
16.4 The Solar Interior: Observations
Chapter 17 Analyzing Starlight
17.1 The Brightness of Stars
17.2 Colors of Stars
17.3 The Spectra of Stars (and Brown Dwarfs)
17.4 Using Spectra to Measure Stellar Radius, Composition, and Motion
Chapter 18 The Stars: A Celestial Census
18.1 A Stellar Census
18.2 Measuring Stellar Masses
18.3 Diameters of Stars
18.4 The H–R Diagram
18.5 Collaborative Group Activities, Questions and Exercises
Chapter 19 Celestial Distances
19.2 Surveying the Stars
19.3 Variable Stars: One Key to Cosmic Distances
19.4 The H–R Diagram and Cosmic Distances
Chapter 20 Between the Stars: Gas and Dust in Space
20.1 The Interstellar Medium
20.2 Interstellar Gas
20.3 Cosmic Dust
20.4 Cosmic Rays
20.5 The Life Cycle of Cosmic Material
20.6 Interstellar Matter around the Sun
Chapter 21 The Birth of Stars and the Discovery of Planets outside the Solar System
21.1 Star Formation
21.2 The H–R Diagram and the Study of Stellar Evolution
21.3 Evidence That Planets Form around Other Stars
21.4 Planets beyond the Solar System: Search and Discovery
21.5 Exoplanets Everywhere: What We Are Learning
21.6 New Perspectives on Planet Formation
Chapter 22 Stars from Adolescence to Old Age
22.1 Evolution from the Main Sequence to Red Giants
22.2 Star Clusters
22.3 Checking Out the Theory
22.4 Further Evolution of Stars
22.5 The Evolution of More Massive Stars
Chapter 23 The Death of Stars
23.1 The Death of Low-Mass Stars
23.2 Evolution of Massive Stars: An Explosive Finish
23.3 Supernova Observations
23.4 Pulsars and the Discovery of Neutron Stars
23.5 The Evolution of Binary Star Systems
23.6 The Mystery of the Gamma-Ray Bursts
Chapter 24 Black Holes and Curved Spacetime
24.1 Introducing General Relativity
24.2 Spacetime and Gravity
24.3 Tests of General Relativity
24.4 Time in General Relativity
24.6 Evidence for Black Holes
24.7 Gravitational Wave Astronomy
Chapter 25 The Milky Way Galaxy
25.1 The Architecture of the Galaxy
25.2 Spiral Structure
25.3 The Mass of the Galaxy
25.4 The Center of the Galaxy
25.5 Stellar Populations in the Galaxy
25.6 The Formation of the Galaxy
Chapter 26 Galaxies
26.1 The Discovery of Galaxies
26.2 Types of Galaxies
26.3 Properties of Galaxies
26.4 The Extragalactic Distance Scale
26.5 The Expanding Universe
Chapter 27 Active Galaxies, Quasars, and Supermassive Black Holes
27.1 Quasars
27.2 Supermassive Black Holes: What Quasars Really Are
27.3 Quasars as Probes of Evolution in the Universe
BCIT Astronomy 7000: A Survey of Astronomy
By the end of this section, you will be able to:
Explain how and why massive stars evolve much more rapidly than lower-mass stars like our Sun
Discuss the origin of the elements heavier than carbon within stars
If what we have described so far were the whole story of the evolution of stars and elements, we would have a big problem on our hands. We will see in later chapters that in our best models of the first few minutes of the universe, everything starts with the two simplest elements—hydrogen and helium (plus a tiny bit of lithium). All the predictions of the models imply that no heavier elements were produced at the beginning of the universe. Yet when we look around us on Earth, we see lots of other elements besides hydrogen and helium. These elements must have been made (fused) somewhere in the universe, and the only place hot enough to make them is inside stars. One of the fundamental discoveries of twentieth-century astronomy is that the stars are the source of most of the chemical richness that characterizes our world and our lives.
We have already seen that carbon and some oxygen are manufactured inside the lower-mass stars that become red giants. But where do the heavier elements we know and love (such as the silicon and iron inside Earth, and the gold and silver in our jewelry) come from? The kinds of stars we have been discussing so far never get hot enough at their centers to make these elements. It turns out that such heavier elements can be formed only late in the lives of more massive stars.
Making New Elements in Massive Stars
Massive stars evolve in much the same way that the Sun does (but always more quickly)—up to the formation of a carbon-oxygen core. One difference is that for stars with more than about twice the mass of the Sun, helium begins fusion more gradually, rather than with a sudden flash. Also, when more massive stars become red giants, they become so bright and large that we call them supergiants. Such stars can expand until their outer regions become as large as the orbit of Jupiter, which is precisely what the Hubble Space Telescope has shown for the star Betelgeuse (see [link]). They also lose mass very effectively, producing dramatic winds and outbursts as they age. [link] shows a wonderful image of the very massive star Eta Carinae, with a great deal of ejected material clearly visible.
Eta Carinae.
Figure 1. With a mass at least 100 times that of the Sun, the hot supergiant Eta Carinae is one of the most massive stars known. This Hubble Space Telescope image records the two giant lobes and equatorial disk of material it has ejected in the course of its evolution. The pink outer region is material ejected in an outburst seen in 1843, the largest of such mass loss event that any star is known to have survived. Moving away from the star at a speed of about 1000 km/s, the material is rich in nitrogen and other elements formed in the interior of the star. The inner blue-white region is the material ejected at lower speeds and is thus still closer to the star. It appears blue-white because it contains dust and reflects the light of Eta Carinae, whose luminosity is 4 million times that of our Sun. (credit: modification of work by Jon Morse (University of Colorado) & NASA)
But the crucial way that massive stars diverge from the story we have outlined is that they can start additional kinds of fusion in their centers and in the shells surrounding their central regions. The outer layers of a star with a mass greater than about 8 solar masses have a weight that is enough to compress the carbon-oxygen core until it becomes hot enough to ignite fusion of carbon nuclei. Carbon can fuse into still more oxygen, and at still higher temperatures, oxygen and then neon, magnesium, and finally silicon can build even heavier elements. Iron is, however, the endpoint of this process. The fusion of iron atoms produces products that are more massive than the nuclei that are being fused and therefore the process requires energy, as opposed to releasing energy, which all fusion reactions up to this point have done. This required energy comes at the expense of the star itself, which is now on the brink of death ([link]). What happens next will be described in the chapter on The Death of Stars.
Interior Structure of a Massive Star Just before It Exhausts Its Nuclear Fuel.
Figure 2. High-mass stars can fuse elements heavier than carbon. As a massive star nears the end of its evolution, its interior resembles an onion. Hydrogen fusion is taking place in an outer shell, and progressively heavier elements are undergoing fusion in the higher-temperature layers closer to the center. All of these fusion reactions generate energy and enable the star to continue shining. Iron is different. The fusion of iron requires energy, and when iron is finally created in the core, the star has only minutes to live.
Physicists have now found nuclear pathways whereby virtually all chemical elements of atomic weights up to that of iron can be built up by this nucleosynthesis (the making of new atomic nuclei) in the centers of the more massive red giant stars. This still leaves the question of where elements heavier than iron come from. We will see in the next chapter that when massive stars finally exhaust their nuclear fuel, they most often die in a spectacular explosion—a supernova. Heavier elements can be synthesized in the stunning violence of such explosions.
Not only can we explain in this way where the elements that make up our world and others come from, but our theories of nucleosynthesis inside stars are even able to predict the relative abundances with which the elements occur in nature. The way stars build up elements during various nuclear reactions really can explain why some elements (oxygen, carbon, and iron) are common and others are quite rare (gold, silver, and uranium).
Elements in Globular Clusters and Open Clusters Are Not the Same
The fact that the elements are made in stars over time explains an important difference between globular and open clusters. Hydrogen and helium, which are the most abundant elements in stars in the solar neighborhood, are also the most abundant constituents of stars in both kinds of clusters. However, the abundances of the elements heavier than helium are very different.
In the Sun and most of its neighboring stars, the combined abundance (by mass) of the elements heavier than hydrogen and helium is 1–4% of the star's mass. Spectra show that most open-cluster stars also have 1–4% of their matter in the form of heavy elements. Globular clusters, however, are a different story. The heavy-element abundance of stars in typical globular clusters is found to be only 1/10 to 1/100 that of the Sun. A few very old stars not in clusters have been discovered with even lower abundances of heavy elements.
The differences in chemical composition are a direct consequence of the formation of a cluster of stars. The very first generation of stars initially contained only hydrogen and helium. We have seen that these stars, in order to generate energy, created heavier elements in their interiors. In the last stages of their lives, they ejected matter, now enriched in heavy elements, into the reservoirs of raw material between the stars. Such matter was then incorporated into a new generation of stars.
This means that the relative abundance of the heavy elements must be less and less as we look further into the past. We saw that the globular clusters are much older than the open clusters. Since globular-cluster stars formed much earlier (that is, they are an earlier generation of stars) than those in open clusters, they have only a relatively small abundance of elements heavier than hydrogen and helium.
As time passes, the proportion of heavier elements in the "raw material" that makes new stars and planets increases. This means that the first generation of stars that formed in our Galaxy would not have been accompanied by a planet like Earth, full of silicon, iron, and many other heavy elements. Earth (and the astronomy students who live on it) was possible only after generations of stars had a chance to make and recycle their heavier elements.
Now the search is on for true first-generation stars, made only of hydrogen and helium. Theories predict that such stars should be very massive, live fast, and die quickly. They should have lived and died long ago. The place to look for them is in very distant galaxies that formed when the universe was only a few hundred million years old, but whose light is only arriving at Earth now.
Approaching Death
Compared with the main-sequence lifetimes of stars, the events that characterize the last stages of stellar evolution pass very quickly (especially for massive stars). As the star's luminosity increases, its rate of nuclear fuel consumption goes up rapidly—just at that point in its life when its fuel supply is beginning to run down.
After the prime fuel—hydrogen—is exhausted in a star's core, we saw that other sources of nuclear energy are available to the star in the fusion of, first, helium, and then of other more complex elements. But the energy yield of these reactions is much less than that of the fusion of hydrogen to helium. And to trigger these reactions, the central temperature must be higher than that required for the fusion of hydrogen to helium, leading to even more rapid consumption of fuel. Clearly this is a losing game, and very quickly the star reaches its end. As it does so, however, some remarkable things can happen, as we will see in The Death of Stars.
In stars with masses higher than about 8 solar masses, nuclear reactions involving carbon, oxygen, and still heavier elements can build up nuclei as heavy as iron. The creation of new chemical elements is called nucleosynthesis. The late stages of evolution occur very quickly. Ultimately, all stars must use up all of their available energy supplies. In the process of dying, most stars eject some matter, enriched in heavy elements, into interstellar space where it can be used to form new stars. Each succeeding generation of stars therefore contains a larger proportion of elements heavier than hydrogen and helium. This progressive enrichment explains why the stars in open clusters (which formed more recently) contain more heavy elements than do those in ancient globular clusters, and it tells us where most of the atoms on Earth and in our bodies come from.
For Further Exploration
Balick, B. & Frank, A. "The Extraordinary Deaths of Ordinary Stars." Scientific American (July 2004): 50. About planetary nebulae, the last gasps of low-mass stars, and the future of our own Sun.
Djorgovsky, G. "The Dynamic Lives of Globular Clusters." Sky & Telescope (October 1998): 38. Cluster evolution and blue straggler stars.
Frank, A. "Angry Giants of the Universe." Astronomy (October 1997): 32. On luminous blue variables like Eta Carinae.
Garlick, M. "The Fate of the Earth." Sky & Telescope (October 2002): 30. What will happen when our Sun becomes a red giant.
Harris, W. & Webb, J. "Life Inside a Globular Cluster." Astronomy (July 2014): 18. What would night sky be like there?
Iben, I. & Tutokov, A. "The Lives of the Stars: From Birth to Death and Beyond." Sky & Telescope (December 1997): 36.
Kaler, J. "The Largest Stars in the Galaxy." Astronomy (October 1990): 30. On red supergiants.
Kalirai, J. "New Light on Our Sun's Fate." Astronomy (February 2014): 44. What will happen to stars like our Sun between the main sequence and the white dwarf stages.
Kwok, S. "What Is the Real Shape of the Ring Nebula?" Sky & Telescope (July 2000): 33. On seeing planetary nebulae from different angles.
Kwok, S. "Stellar Metamorphosis." Sky & Telescope (October 1998): 30. How planetary nebulae form.
Stahler, S. "The Inner Life of Star Clusters." Scientific American (March 2013): 44–49. How all stars are born in clusters, but different clusters evolve differently.
Subinsky, R. "All About 47 Tucanae." Astronomy (September 2014): 66. What we know about this globular cluster and how to see it.
BBC Page on Giant Stars: http://www.bbc.co.uk/science/space/universe/sights/giant_stars. Includes basic information and links to brief video excerpts.
Encylopedia Brittanica Article on Star Clusters: http://www.britannica.com/topic/star-cluster. Written by astronomer Helen Sawyer Hogg-Priestley.
Hubble Image Gallery: Planetary Nebulae: http://hubblesite.org/gallery/album/nebula/planetary/. Click on each image to go to a page with more information available. (See also a similar gallery at the National Optical Astronomy Observatories: https://www.noao.edu/image_gallery/planetary_nebulae.html).
Hubble Image Gallery: Star Clusters: http://hubblesite.org/gallery/album/star/star_cluster/. Each image comes with an explanatory caption when you click on it. (See also a similar European Southern Observatory Gallery at: https://www.eso.org/public/images/archive/category/starclusters/).
Measuring the Age of a Star Cluster: https://www.e-education.psu.edu/astro801/content/l7_p6.html. From Penn State.
Life Cycle of Stars: https://www.youtube.com/watch?v=PM9CQDlQI0A. Short summary of stellar evolution from the Institute of Physics in Great Britain, with astronomer Tim O'Brien (4:58).
Missions Take an Unparalleled Look into Superstar Eta Carinae: https://www.youtube.com/watch?v=0rJQi6oaZf0. NASA Goddard video about observations in 2014 and what we know about the pair of stars in this complicated system (4:00).
Star Clusters: Open and Globular Clusters: https://www.youtube.com/watch?v=rGPRLxrYbYA. Three Short Hubblecast Videos from 2007–2008 on discoveries involving star clusters (12:24).
Tour of Planetary Nebula NGC 5189: https://www.youtube.com/watch?v=1D2cwiZld0o. Brief Hubblecast episode with Joe Liske, explaining planetary nebulae in general and one example in particular (5:22).
Collaborative Group Activities
Have your group take a look at the list of the brightest stars in the sky in Appendix J. What fraction of them are past the main-sequence phase of evolution? The text says that stars spend 90% of their lifetimes in the main-sequence phase of evolution. This suggests that if we have a fair (or representative) sample of stars, 90% of them should be main-sequence stars. Your group should brainstorm why 90% of the brightest stars are not in the main-sequence phase of evolution.
Reading an H–R diagram can be tricky. Suppose your group is given the H–R diagram of a star cluster. Stars above and to the right of the main sequence could be either red giants that had evolved away from the main sequence or very young stars that are still evolving toward the main sequence. Discuss how you would decide which they are.
In the chapter on Life in the Universe, we discuss some of the efforts now underway to search for radio signals from possible intelligent civilizations around other stars. Our present resources for carrying out such searches are very limited and there are many stars in our Galaxy. Your group is a committee set up by the International Astronomical Union to come up with a list of the best possible stars with which such a search should begin. Make a list of criteria for choosing the stars on the list, and explain the reasons behind each entry (keeping in mind some of the ideas about the life story of stars and timescales that we discuss in the present chapter.)
Have your group make a list of the reasons why a star that formed at the very beginning of the universe (soon after the Big Bang) could not have a planet with astronomy students reading astronomy textbooks (even if the star has the same mass as that of our Sun).
Since we are pretty sure that when the Sun becomes a giant star, all life on Earth will be wiped out, does your group think that we should start making preparations of any kind? Let's suppose that a political leader who fell asleep during large parts of his astronomy class suddenly hears about this problem from a large donor and appoints your group as a task force to make suggestions on how to prepare for the end of Earth. Make a list of arguments for why such a task force is not really necessary.
Use star charts to identify at least one open cluster visible at this time of the year. (Such charts can be found in Sky & Telescope and Astronomy magazines each month and their websites; see Appendix B.) The Pleiades and Hyades are good autumn subjects, and Praesepe is good for springtime viewing. Go out and look at these clusters with binoculars and describe what you see.
Many astronomers think that planetary nebulae are among the most attractive and interesting objects we can see in the Galaxy. In this chapter, we could only show you a few examples of the pictures of these objects taken with the Hubble or large telescopes on the ground. Have members of your group search further for planetary nebula images online, and make a "top ten" list of your favorite ones (do not include more than three that were featured in this chapter.) Make a report (with images) for the whole class and explain why you found your top five especially interesting. (You may want to check [link] in the process.)
1: Compare the following stages in the lives of a human being and a star: prenatal, birth, adolescence/adulthood, middle age, old age, and death. What does a star with the mass of our Sun do in each of these stages?
2: What is the first event that happens to a star with roughly the mass of our Sun that exhausts the hydrogen in its core and stops the generation of energy by the nuclear fusion of hydrogen to helium? Describe the sequence of events that the star undergoes.
3: Astronomers find that 90% of the stars observed in the sky are on the main sequence of an H–R diagram; why does this make sense? Why are there far fewer stars in the giant and supergiant region?
4: Describe the evolution of a star with a mass similar to that of the Sun, from the protostar stage to the time it first becomes a red giant. Give the description in words and then sketch the evolution on an H–R diagram.
5: Describe the evolution of a star with a mass similar to that of the Sun, from just after it first becomes a red giant to the time it exhausts the last type of fuel its core is capable of fusing.
6: A star is often described as "moving" on an H–R diagram; why is this description used and what is actually happening with the star?
7: On which edge of the main sequence band on an H–R diagram would the zero-age main sequence be?
8: How do stars typically "move" through the main sequence band on an H–R diagram? Why?
9: Certain stars, like Betelgeuse, have a lower surface temperature than the Sun and yet are more luminous. How do these stars produce so much more energy than the Sun?
10: Gravity always tries to collapse the mass of a star toward its center. What mechanism can oppose this gravitational collapse for a star? During what stages of a star's life would there be a "balance" between them?
11: Why are star clusters so useful for astronomers who want to study the evolution of stars?
12: Would the Sun more likely have been a member of a globular cluster or open cluster in the past?
13: Suppose you were handed two H–R diagrams for two different clusters: diagram A has a majority of its stars plotted on the upper left part of the main sequence with the rest of the stars off the main sequence; and diagram B has a majority of its stars plotted on the lower right part of the main sequence with the rest of the stars off the main sequence. Which diagram would be for the older cluster? Why?
14: Referring to the H–R diagrams in [link], which diagram would more likely be the H–R diagram for an association?
15: The nuclear process for fusing helium into carbon is often called the "triple-alpha process." Why is it called as such, and why must it occur at a much higher temperature than the nuclear process for fusing hydrogen into helium?
16: Pictures of various planetary nebulae show a variety of shapes, but astronomers believe a majority of planetary nebulae have the same basic shape. How can this paradox be explained?
17: Describe the two "recycling" mechanisms that are associated with stars (one during each star's life and the other connecting generations of stars).
18: In which of these star groups would you mostly likely find the least heavy-element abundance for the stars within them: open clusters, globular clusters, or associations?
19: Explain how an H–R diagram of the stars in a cluster can be used to determine the age of the cluster.
20: Where did the carbon atoms in the trunk of a tree on your college campus come from originally? Where did the neon in the fabled "neon lights of Broadway" come from originally?
21: What is a planetary nebula? Will we have one around the Sun?
Thought Questions
22: Is the Sun on the zero-age main sequence? Explain your answer.
23: How are planetary nebulae comparable to a fluorescent light bulb in your classroom?
24: Which of the planets in our solar system have orbits that are smaller than the photospheric radius of Betelgeuse listed in in [link]?
25: Would you expect to find an earthlike planet (with a solid surface) around a very low-mass star that formed right at the beginning of a globular cluster's life? Explain.
26: In the H–R diagrams for some young clusters, stars of both very low and very high luminosity are off to the right of the main sequence, whereas those of intermediate luminosity are on the main sequence. Can you offer an explanation for that? Sketch an H–R diagram for such a cluster.
27: If the Sun were a member of the cluster NGC 2264, would it be on the main sequence yet? Why or why not?
28: If all the stars in a cluster have nearly the same age, why are clusters useful in studying evolutionary effects (different stages in the lives of stars)?
29: Suppose a star cluster were at such a large distance that it appeared as an unresolved spot of light through the telescope. What would you expect the overall color of the spot to be if it were the image of the cluster immediately after it was formed? How would the color differ after 1010 years? Why?
30: Suppose an astronomer known for joking around told you she had found a type-O main-sequence star in our Milky Way Galaxy that contained no elements heavier than helium. Would you believe her? Why?
31: Stars that have masses approximately 0.8 times the mass of the Sun take about 18 billion years to turn into red giants. How does this compare to the current age of the universe? Would you expect to find a globular cluster with a main-sequence turnoff for stars of 0.8 solar mass or less? Why or why not?
32: Automobiles are often used as an analogy to help people better understand how more massive stars have much shorter main-sequence lifetimes compared to less massive stars. Can you explain such an analogy using automobiles?
Figuring for Yourself
33: The text says a star does not change its mass very much during the course of its main-sequence lifetime. While it is on the main sequence, a star converts about 10% of the hydrogen initially present into helium (remember it's only the core of the star that is hot enough for fusion). Look in earlier chapters to find out what percentage of the hydrogen mass involved in fusion is lost because it is converted to energy. By how much does the mass of the whole star change as a result of fusion? Were we correct to say that the mass of a star does not change significantly while it is on the main sequence?
34: The text explains that massive stars have shorter lifetimes than low-mass stars. Even though massive stars have more fuel to burn, they use it up faster than low-mass stars. You can check and see whether this statement is true. The lifetime of a star is directly proportional to the amount of mass (fuel) it contains and inversely proportional to the rate at which it uses up that fuel (i.e., to its luminosity). Since the lifetime of the Sun is about 1010 y, we have the following relationship:
$$T={10}^{10}\frac{M}{L}\phantom{\rule{0.2em}{0ex}}\text{y}$$
where T is the lifetime of a main-sequence star, M is its mass measured in terms of the mass of the Sun, and L is its luminosity measured in terms of the Sun's luminosity.
Explain in words why this equation works.
Use the data in [link] to calculate the ages of the main-sequence stars listed.
Do low-mass stars have longer main-sequence lifetimes?
Do you get the same answers as those in [link]?
35: You can use the equation in [link] to estimate the approximate ages of the clusters in [link], [link], and [link]. Use the information in the figures to determine the luminosity of the most massive star still on the main sequence. Now use the data in [link] to estimate the mass of this star. Then calculate the age of the cluster. This method is similar to the procedure used by astronomers to obtain the ages of clusters, except that they use actual data and model calculations rather than simply making estimates from a drawing. How do your ages compare with the ages in the text?
36: You can estimate the age of the planetary nebula in image (c) in [link]. The diameter of the nebula is 600 times the diameter of our own solar system, or about 0.8 light-year. The gas is expanding away from the star at a rate of about 25 mi/s. Considering that distance = velocity $×$ time, calculate how long ago the gas left the star if its speed has been constant the whole time. Make sure you use consistent units for time, speed, and distance.
37: If star A has a core temperature T, and star B has a core temperature 3T, how does the rate of fusion of star A compare to the rate of fusion of star B?
nucleosynthesis
the building up of heavy elements from lighter ones by nuclear fusion
Previous: 22.4 Further Evolution of Stars
Next: 23.0 Thinking Ahead
BCIT Astronomy 7000: A Survey of Astronomy by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. | CommonCrawl |
\begin{document}
\begin{frontmatter}
\title{A New Approach to Regular \!+\!& Indeterminate Strings}
\author[ufu]{Felipe A. Louza} \ead{[email protected]} \address[ufu]{Faculty of Electrical Engineering, Federal University of Uberl\!+\!^andia, Brazil} \author[macmaster]{Neerja Mhaskar\corref{mycorrespondingauthors}} \cortext[mycorrespondingauthors]{Corresponding author} \ead{[email protected]} \address[macmaster]{Department of Computing and Software, McMaster University, Canada}
\author[macmaster,kcl,murdoch]{W.~F.~Smyth} \ead{[email protected]} \address[kcl]{Department of Informatics, King's College London, UK} \address[murdoch]{School of Engineering \!+\!& Information Technology, Murdoch University, Australia}
\begin{abstract} In this paper we propose a new, more appropriate definition of \itbf{regular} and \itbf{indeterminate} strings. A \itbf{regular} string is one that is ``isomorphic'' to a string whose entries all consist of a single letter, but which nevertheless may itself include entries containing multiple letters. A string that is not regular is said to be \itbf{indeterminate}. We begin by proposing a new model for the representation of strings, regular or indeterminate, then go on to describe a linear time algorithm to determine whether or not a string $\s{x} = \s{x}[1..n]$ is regular and, if so, to replace it by a lexicographically least (lex-least) string \s{y} whose entries are all single letters. Furthermore, we connect the regularity of a string to the transitive closure problem on a graph, which in our special case can be efficiently solved. We then introduce the idea of a feasible palindrome array $\mbox{MP}$ of a string, and prove that every feasible $\mbox{MP}$ corresponds to some (regular or indeterminate) string. We describe an algorithm that constructs a string \s{x} corresponding to given feasible $\mbox{MP}$, while ensuring that whenever possible \s{x} is regular and if so, then lex-least. A final section outlines new research directions suggested by this changed perspective on regular and indeterminate strings. \end{abstract}
\begin{keyword} indeterminate string \sep degenerate string \sep generalized string \sep transitive closure \sep palindrome \sep maximal palindrome array \sep Manacher's condition \sep reverse engineering \sep algorithm \sep stringology
\end{keyword}
\end{frontmatter}
\section{Introduction} \label{sect-intro} Under various names and guises, the idea of a string as something other than a sequence of single letters has been discussed for almost half a century. In 1974 Fischer \!+\!& Paterson \cite{FP74} studied pattern-matching on strings \s{x} whose entries could be \itbf{don't-care} letters; that is, letters matching any single letter in the alphabet $\Sigma$ on which the string is defined, hence matching every position in \s{x}. (Thus the don't care letter $*$ would match every letter in a given alphabet $\Sigma = \!+\!{a,b,c\!+\!}$.) In 1987 Abrahamson \cite{A87} extended this model by considering pattern-matching on \itbf{generalized} strings whose entries could be non-empty subsets of $\Sigma$. (Thus entries $\!+\!{b\!+\!}$, $\!+\!{a,b\!+\!}$, $\!+\!{b,c\!+\!}$ would match each other due to the shared $b$.)
Both of these models have been intensively studied in this century, with interest fueled by applications, especially to computational biology, where as a byproduct of the DNA sequencing process and in other contexts, indeterminate letters such as $\!+\!{a,t\!+\!}$, $\!+\!{c,g\!+\!}$ or $\!+\!{a,c,g,t\!+\!}$ may arise. The properties of strings with don't care letters, also called \itbf{holes}, have been intensively studied by Blanchet-Sadri, culminating in a 2008 book \cite{BS08} on \itbf{strings with holes}, also called \itbf{partial words}, that summarized the contributions of her many previous research articles. In 2009 a periodicity lemma for partial words was formulated in \cite{SW09a}.
Meanwhile Iliopoulos and various collaborators (see, for example, \cite{IMR08,IRVV10,IKP17}) studied pattern-matching and other problems, especially related to bioinformatics applications, on generalized strings (now renamed \itbf{degenerate strings}, or even elastic degenerate strings). Another sequence of papers \cite{HS03,HSW05,HSW06,HSW08,SW09} studied other pattern-matching algorithms on generalized strings (renamed \itbf{indeterminate strings}, with \itbf{indeterminate letters} such as $\!+\!{a,b\!+\!}$ or $\!+\!{b,c\!+\!}$ in some positions). In addition, other aspects of indeterminate strings have been studied, such as their border arrays \cite{NRR12}, cover arrays \cite{BRS09}, prefix arrays \cite{ARS15,CRSW15} and suffix arrays \cite{BIST03,NRR12}. The computation of an indeterminate string from its prefix graph was recently considered in \cite{HRSS18}.
The traditional approach to defining an indeterminate string only relies on the scope of its letters irrespective of the matching relationship between them. Therefore, a string $\s{x}=a\!+\!{a,c\!+\!} b \!+\!{a, d\!+\!} bb$ can be seen as an indeterminate string, when in essence it can be reduced to an ``isomorphic'' string $\s{x'}=aababb$ that is regular (see Section~\ref{sect-indet} for definitions). It is evident that algorithms designed for indeterminate strings are slower and at times require more space for processing. Therefore, it is important to identify strings that are ``truly'' indeterminate. In this paper we redefine an indeterminate string in a context that we believe captures the idea in a more appropriate way --- at once more general and more precise.
The outline of the paper is as follows: in Section~\ref{sect-indet} we present the necessary definitions, including in particular a model for the representation of strings that efficiently incorporates both regular and indeterminate. In Section \ref{sect-alg}, we apply this model to describe a linear time algorithm that determines whether or not a given string is regular and, if so, returns an isomorphic lexicographically least (lex-least) regular string, whose entries are all single letters. We go on to present an interesting connection between strings defined using our approach and the transitive closure of a relation, showing that for some special cases of a relation, its transitive closure can be computed in quadratic time. In Section~\ref{sect-maxpal} we introduce a ``Reverse Engineering'' problem \cite{FLRSSY99,FGLRSSY02,CCR09,DFHSS18} for both regular and indeterminate strings (in our new formulation). We first show that every feasible palindrome array MP does indeed correspond to some string, regular or indeterminate. Then, following \cite{IIBT10}, we describe an algorithm that, given a feasible palindrome array MP, computes a lex-least regular string corresponding to MP, whenever that is possible; and, if not, returns a corresponding indeterminate string. Section~\ref{sect-probs} discusses open problems and questions arising from this new approach.
\section{Regular \!+\!& Indeterminate Strings} \label{sect-indet} A \itbf{letter} $\ell$ is a finite list of $s$ distinct \itbf{characters} $c_1,c_2,\ldots,c_s$, each drawn from a set $\Sigma$
of size $\sigma = |\Sigma|$ called the \itbf{alphabet}. In the case that $\Sigma$ is ordered, $\ell$ is said to be in \itbf{normal form} if its characters occur in the ascending order determined by $\Sigma$. The integer $s = s(\ell)$ is called the \itbf{scope} of $\ell$. If $s = 1$, $\ell$ is said to be \itbf{regular}, otherwise \itbf{indeterminate}. Two letters $\ell_1,\ell_2$ are said to \itbf{match}, written $\ell_1 \approx \ell_2$, if and only if $\ell_1 \cap \ell_2 \ne \emptyset$. In the case that matching $\ell_1$ and $\ell_2$ are both regular, we may write $\ell_1 = \ell_2$.
For $n \ge 1$, a \itbf{string} $\s{x} = \s{x}[1..n]$ is a sequence
$\s{x}[1],\s{x}[2],\ldots,\s{x}[n]$ of letters, where $n = |\s{x}|$ is the \itbf{length} of \s{x}, and every $i \in \!+\!{1,\ldots,n\!+\!}$ is a \itbf{position} in \s{x}. If every letter in \s{x} is in normal form, then \s{x} itself is said to be in \itbf{normal form}. A tuple $T = (i,j_1,j_2)$ of distinct positions $i,j_1,j_2$ in \s{x} such that $$\s{x}[j_1] \approx \s{x}[i] \approx \s{x}[j_2]$$ is said to be a \itbf{triple}. A triple $T$ is \itbf{transitive} if $\s{x}[j_1] \approx \s{x}[j_2]$, otherwise \itbf{intransitive}. If every triple $T$ in \s{x} is transitive, then we say that \s{x} is \itbf{regular}; otherwise, \s{x} is \itbf{indeterminate}. The \itbf{scope} of \s{x} is given by $$S(\s{x}) = \max_{i\in \!+\!{1,\dots ,n\!+\!}} s(\s{x}[i]).$$
Two strings \s{x} and \s{y} of equal length $n$ are said to be \itbf{isomorphic} if and only if for every $i,j \in$ $\!+\!{1,\dots,n\!+\!}$, \begin{equation} \label{rule2} \s{x}[i] \approx \s{x}[j] \Longleftrightarrow \s{y}[i] \approx \s{y}[j]. \end{equation} Note that isomorphic strings may superficially have quite different representations; for example, $\s{y} = \!+\!{a,b\!+\!}\!+\!{b,c\!+\!}\!+\!{a,c\!+\!}$ is isomorphic to $\s{x} = aaa$. Thus we see that the definition given here of regularity of a string \s{x} is more general, including many more strings, than the usual definition (scope $= 1$) given in the literature. However: \begin{lemm} \label{lemmreg} Every regular string is isomorphic to a string of scope 1. \end{lemm} \begin{proof} In \cite{CRSW15,HRSS18} and elsewhere it is observed that a regular string $\s{x}[1..n]$ is equivalent to a graph $\mathcal{G}$ on vertices $\!+\!{1,\dots,n\!+\!}$ which is a collection of cliques with the property that vertices $i$ and $j \ne i$ are in the same clique if and only if $\s{x}[i] \approx \s{x}[j]$. Now if, for every clique $\mathcal{C}$ of $\mathcal{G}$, we choose a unique regular letter $\ell_{\mathcal{C}}$ and, corresponding to every position $i$ in $\mathcal{C}$, make the assignment $\s{x}[i] \leftarrow \ell_{\mathcal{C}}$, we thus construct a string on regular letters equivalent to $\mathcal{G}$; that is, a string of scope 1.
\quad\qedbox46\newline\smallbreak \end{proof} We remark that constructing the graph $\mathcal{G}$ described in Lemma~\ref{lemmreg}, then checking whether or not it is a collection of cliques, is a trivial and obvious approach to determining the regularity of \s{x}. In this paper we propose more efficient and interesting algorithms.
The following result is an easy corollary of Lemma~\ref{lemmreg}: \begin{corollary} \label{lemmmap} Given a regular string $\s{x}[1..n]$, then, corresponding to every triple $(i, j_1, j_2)$, we can assign a regular letter to $\s{y}[i], \s{y}[j_1], \s{y}[j_2]$ in such a way that the resulting string $\s{y}[1..n]$ is isomorphic to $\s{x}[1..n]$.
\quad\qedbox46\newline\smallbreak \end{corollary} For example, $\s{y} = ccdcdd$ is isomorphic to $\s{x} = a\!+\!{a,c\!+\!} b \!+\!{a, d\!+\!} bb$.
\subsection{A Model for the Representation of Indeterminate Strings} The preceding result suggests a need for an algorithm to determine whether or not an apparently indeterminate string is nevertheless actually regular, even though, like the string $\s{x} = a\!+\!{a,c\!+\!} b \!+\!{a, d\!+\!} bb$ mentioned in the Introduction, it contains indeterminate letters. We describe such an algorithm in the next section, but first we must deal with the following issue: in order to design algorithms on indeterminate strings \s{x}, we need to have some practical representation of such strings --- in particular, of indeterminate letters. This requires us to deal in some realistic way with the fact that, on an alphabet of size $\sigma$, there are $2^{\sigma}\!+\!- 1$ possible nonempty distinct letters. For $\sigma = 2$, we have only $a, b, \!+\!{a,b\!+\!}$, but for $\sigma = 26$, dealing efficiently with $2^{26} - 1$ distinct letter sets is beyond current processing capabilities. Fortunately, in practice, such cases do not arise: there may be ``many'' letter sets but, for large $\sigma$, not $2^\sigma\!+\!- 1$ of them.
Thus we need a model of computation that deals efficiently with indeterminate letters even when their number is large --- though not ``too large''. For given $\sigma$, we shall accordingly assume an integer alphabet with letters $\ell \in \!+\!{1,2,\ldots,\sigma\!+\!}$ and we shall interpret letter $\ell = 0$ as denoting a hole or don't-care. Otherwise, $\ell > \sigma$ represents an indeterminate letter, that also determines the position $\sigma' = \ell\!+\!- \sigma$ in an array $I[1..\sigma^*]$, where $\sigma^*$ is the maximum number of indeterminate letters (in addition to the don't care) allowed to occur in \s{x}. Each entry $I[\sigma']$ is a pair $(s(\ell),loc)$ that gives the scope $s(\ell)$ of the indeterminate letter $\ell$ and the starting location $loc$ in an array $L$, consisting of integers in the range $\!+\!{1\dots\sigma\!+\!}$, of the subarray specifying the $s(\ell)$ regular letters that constitute $\s{x}[i]$: $$(\ell_1, \ell_2, \ldots, \ell_{s(\ell)}).$$ Here the $\ell_j, 1 \le j \le s(\ell)$, are distinct elements of $\Sigma$ in normal form, thus satisfying $\ell_1 < \ell_2 < \cdots < \ell_{s(\ell)}$. For instance, consider $\s{x}=aac\!+\!{a,c\!+\!}gta\!+\!{g,t\!+\!}\!+\!{a,c\!+\!}\!+\!{g,t\!+\!}$ over $\Sigma=\!+\!{a,c,g,t\!+\!}$. To represent it using the above encoding, we map $a \rightarrow 1, c \rightarrow 2, g \rightarrow 3, t \rightarrow 4$. Since we have two indeterminate letters in $\s{x}$, $\sigma^*=2$, $0\leq \ell \leq 6$, and $\!+\!{a,c\!+\!} \rightarrow 5$ and $\!+\!{g, t\!+\!} \rightarrow 6$. Further, $I=[(2,1), (2,3)]$, $L=[1,2,3,4]$, and the encoding of $\s{x}$ is $1125341656$.
To fix the ideas, we suppose here that $\ell$, hence every entry in array $L$, is constrained to one byte (8 bits) of storage. Thus, for example, to handle English text, we should suppose that $\sigma = 77$ (26 lower case letters, 26 upper case, 10 numerical digits, 14 punctuation symbols, and space), leaving available values $\ell \in 78..255$: $\sigma^* = 178$ distinct indeterminate letters. (Doubling storage for $\ell$ to two bytes permits 65,458 indeterminate English letters to be defined.)
Another example: for DNA strings on $\Sigma = \!+\!{a,c,g,t\!+\!}$, $\sigma = 4$, we could define $a \rightarrow 1, c \rightarrow 2, g \rightarrow 3, t \rightarrow 4$, use zero for the don't care and 5--14 for the other 10 possible combinations of letters: $$ac,ag,at,cg,ct,gt,acg,act,agt,cgt.$$ Then for DNA a half byte (4 bits) for $\ell$ would be sufficient to identify all the indeterminate letters, and the associated array $L$ would require at most 24 half bytes to represent them.
Other models to represent indeterminate strings have been proposed. For example, in~\cite{prochazka19} all the non-empty letters (both regular and indeterminate) over the DNA alphabet $\Sigma_{DNA}=\!+\!{A,C,G,T\!+\!}$ are mapped to the IUPAC symbols $\Sigma_{IUPAC}=\!+\!{A,C,G,T,R,Y,S,W,K,M,B,D,H,V,N\!+\!}$, to construct an isomorphic regular string of scope $1$ over $\Sigma_{IUPAC}$. Further, in~\cite{btt2013} each symbol in the DNA alphabet is represented as a 4-bit integer power of $2$ ($2^i$, with $i \in \!+\!{0,1,2,3\!+\!}$). A non-empty indeterminate letter over $\Sigma_{DNA}$ is represented as $\Sigma_{\!+\!{s \in \mathcal{P}(\Sigma)\!+\!}}s$. Then instead of using the natural order on integers it uses a Gray code~\cite{gray53} (also known as the reflected binary code) to order indeterminate letters over $\Sigma_{DNA}$. Note that with the Gray code two successive values differ only by one bit, such as 1100 and 1101, which enables minimizing the number of separate intervals associated with each of the four symbols of $\Sigma_{DNA}$. The models described here are of interest primarily for representing indeterminate strings over the DNA alphabet. However, the same approach is applicable to any alphabet.
\section{Determining the regularity of \s{x}} \label{sect-alg} In this section we discuss a function REGULAR (Figure~\ref{fig-regular}) that determines whether or not a given string \s{x} is regular. REGULAR deals with strings \s{x} of scope $S(\s{x}) > 1$, since otherwise, for $S(\s{x}) = 1$, the regularity of \s{x} is immediate. As a byproduct, in the case that \s{x} is regular, REGULAR computes the lex-least string \s{y} of scope 1 on an integer alphabet that is isomorphic to \s{x}.
Before getting into the details of REGULAR, we need to consider the overall strategy. An issue that arises is this: in order to determine whether or not $\s{x}[1..n]$ is regular, it is necessary to look at all triples to determine whether or not one of them is intransitive, a process that it seems must require $n\choose 2$ letter comparisons, thus $\mathcal{O}(S(\s{x})n^2)$ time. However, it suffices to apply REGULAR to a \itbf{reduced string} $\s{x_R} = \s{x_R}[1..m]$ consisting of exactly one occurrence of each of the $m$ distinct letters (both regular and indeterminate) in \s{x}. Thus, in view of our constraint on $\ell$, $m \le 256$. Clearly \s{x_R} is regular if and only if \s{x} is regular. Consequently triple-testing time reduces to $\mathcal{O}(m^2)$, where of course $m << n$. (Indeed, even for the extreme two-byte case mentioned above, with $m = 65,536$, $m^2$ is only 4 billion or so, thus still $\mathcal{O}(n)$ in many cases.)
\begin{figure}
\caption{This Boolean function determines whether ($REGULAR = \tt{true}$) or not ($REGULAR = \tt{false}$) a string $\s{x}[1..n]$ of scope $S(\s{x}) > 1$ on alphabet $\Sigma$ is regular. In the former case, on exit the string \s{y} is the lex-least regular string of scope 1 on an integer alphabet $\Sigma' = \!+\!{1,2,\ldots,\sigma'\!+\!}$ that is isomorphic to \s{x}. }
\label{fig-regular}
\end{figure}
The first step of REGULAR therefore computes \s{x_R}, a process performed in $\Theta(n)$ time by a single scan of \s{x}, using a Boolean array $found[1..256]$ to indicate whether ($found[j] = \tt{true}$) or not ($found[j] = \tt{false}$) the letter $j = \s{x}[i]$ has occurred previously in $\s{x}[1..i-1]$. Note that in this process, the letters are placed in \s{x_R} in the order of their first occurrence in \s{x}. Thus: \begin{claim} \label{claim-basic} In cases of practical interest --- that is, when $n$ greatly exceeds $m$ --- the replacement of $\s{x} = \s{x}[1..n]$ by the reduced string $\s{x_R} = \s{x_R}[1..m]$, where $m$ is the number of distinct letters in \s{x}, permits the regularity of \s{x} to be determined in $\mathcal{O}(n)$ time.
\quad\qedbox46\newline\smallbreak \end{claim} Indeed, in many cases that arise in practice, the status of \s{x} follows from conditions that are more easily evaluated. For instance, \s{x} is indeterminate if \begin{itemize}
\item it contains all regular letters in $\Sigma$ and at least one indeterminate letter; or
\item it contains a don't-care and at least two regular letters; or
\item it contains an indeterminate letter $\ell$ as well as any two characters of $\ell$. \end{itemize}
Having constructed $\s{x_R}$, REGULAR executes the Boolean function $regular_{min}$ (Figure~\ref{fig-regularmin}), which returns $\tt{true}$ if and only if $\s{x_R}$ is regular. In that case, $regular_{min}$ also outputs a string $\s{y_R}$ of scope 1 on an integer alphabet $\Sigma'$ that is isomorphic to \s{x_R}. As explained below, it is then straightforward to compute a lex-least string \s{y} of scope 1 isomorphic to \s{x}.
\begin{figure}
\caption{Determine whether ($regular_{min} = \tt{true}$) or not ($regular_{min} = \tt{false}$) a reduced string $\s{x_R}[1..m]$ of scope $S(\s{x_R}) > 1$ on alphabet $\Sigma$ is regular. If $\tt{true}$, on exit the string $\s{y_R}$ is the lex-least regular string of scope 1 on the integer alphabet $\Sigma' = \!+\!{1,2,\ldots,\sigma'\!+\!}$ that is isomorphic to \s{x_R}.}
\label{fig-regularmin}
\end{figure}
In the function $regular_{min}$, we first initialize each letter in $\s{y_R}[1..m]$ to $0$. Then we scan $\s{x_R}$ from left to right, using $\s{y_R}$ to record previous matches. During this scan the following condition holds as long as \s{x_R} is regular: \begin{equation}\label{e:condition_c}
\s{x_R}[i] \approx \s{x_R}[j] \Leftrightarrow \s{y_R}[i] = \s{y_R}[j] \wedge \s{y_R}[i] \neq 0. \end{equation} If at position $i \in \!+\!{1,\dots,m\!+\!}$, we find $\s{y_R}[i]=0$ --- that is, it was not part of a previous match --- we form the new character $\sigma' \leftarrow \sigma' \!+\!+ 1$. We then scan right in the strings $\s{x_R}[i+1..m]$ and $\s{y_R}[i+1..m]$ to see if condition (\ref{e:condition_c}) continues to hold. If it does not, we mark $\s{x_R}$ as indeterminate and exit; otherwise, whenever $\s{x_R}[j] \approx \s{x_R}[i]$ and $\s{y_R}[j] = 0$, we assign $\s{y_R}[j] \leftarrow \sigma'$.
In $regular_{min}$ the inner loops execute $\binom{m}{2}$ times, and checking if two letters match takes $\mathcal{O}(\sigma)$ time for $\s{x}$ in normal form, $\mathcal{O}(\sigma\log\sigma)$ otherwise. Therefore:
\begin{lemm} \label{lem:regularminc}
Algorithm $regular_{min}$ runs in $\mathcal{O}(m^2\sigma)$ time, where $m = |\s{x_R}|$ and $1\leq m \leq 2^\sigma$, when \s{x_R} is in normal form; otherwise, $\mathcal{O}(m^2 \sigma\log\sigma)$ time, where $\mathcal{O}(\sigma\log\sigma)$ is the maximum time needed to sort the characters in two letters.
\quad\qedbox46\newline\smallbreak \end{lemm} \begin{lemm} \label{lemm:yRy} When $\s{x}[1..n]$ is regular, the (lex-least) corresponding \s{y} can be computed from \s{y_R} in $\Theta(n)$ time. \end{lemm} \begin{proof} As noted above, the letters in $\s{x_R}[1..m]$ occur in order of their first appearance in \s{x}. Moreover, the letters in \s{y_R} are assigned in increasing order $1,2,\ldots,\sigma'$ of occurrence of the positions of new letters in $\s{x_R}$. Thus, to compute \s{y}, we first form $P_{\s{x_R}}[1..n]$, in which $P_{\s{x_R}}[j]$ is the position at which the letter $\s{x}[j]$ occurs in \s{x_R}. Therefore, the elements of $P_{\s{x_R}}$ are integers in the range $\!+\!{1 \dots m\!+\!}$. Clearly, $P_{\s{x_R}}[1..n]$ can be computed in $\Theta(n)$ time. Then for every $i \in \!+\!{1 \dots n\!+\!}$, we compute $$\s{y}[i] \leftarrow \s{y_R}\big[P_{\s{x_R}}[i]\big],$$ yielding a lex-least string of scope 1 isomorphic to \s{x}. \end{proof}
\begin{lemm} \label{lem:regularc} Algorithm REGULAR runs in $\Theta(n)$ time for a string $\s{x}[1..n]$ from a constant alphabet. \end{lemm} \begin{proof} The running time of REGULAR is equal to the total time required for (1) construction of \s{x_R} from \s{x}; (2) execution of $regular_{min}$; (3) construction of \s{y} from \s{y_R}.
As we have seen (Claim 1), (1) requires $\Theta(n)$ time, while Lemma~\ref{lem:regularminc} tells us that (2) requires $\mathcal{O}(m^2 \sigma\log\sigma)$ time. However, since $m$ and $\sigma$ are constants, (2) requires constant time. Further, as explained in Lemma~\ref{lemm:yRy}, \s{y} can be computed from \s{y_R} in $\Theta(n)$ time. Therefore, we conclude that the running time of REGULAR is $\Theta(n)$.
\quad\qedbox46\newline\smallbreak \end{proof}
At a cost of adding $\Theta(m^2)$ bits of additional storage, together with $\mathcal{O}(m^2\sigma)$ preprocessing time, the regularity of the reduced string \s{x_R} can actually be determined in $\mathcal{O}(m^2)$ time in all cases, as we now explain. Essentially, we replace \s{x_R} by a symmetric adjacency matrix $M^{\approx}_{\s{x_R}} = M^{\approx}_{\s{x_R}}[1..m,1..m]$ based on the matches in \s{x_R}: for $1 \le i,j \le m$, \begin{eqnarray} M^{\approx}_{\s{x_R}}[i,j] = \begin{cases} 1 & \mbox{ if } \s{x_R}[i] \approx \s{x_R}[j]; \!+\! 0 & \mbox{ otherwise}. \end{cases} \end{eqnarray} Since each match may require $\mathcal{O}(\sigma)$ time, this construction can be performed in $\mathcal{O}(m^2\sigma)$ time in the worst case.
Then, simply replacing references to \s{x_R} in function {\it regular\!+\!_{min}} by references to $M^{\approx}_{\s{x_R}}$, we obtain the $\mathcal{O}(m^2)$ function {\it regular\!+\!_{min}\!+\!_matrix} (Figure~\ref{fig-regularminM}).
\begin{figure}\label{fig-regularminM}
\end{figure}
Effectively, $M^{\approx}_{\s{x_R}}$ defines an undirected graph $G^{\approx}_{\s{x_R}} = (V,E)$ on $m$ vertices, where there exists an edge $(i,j) \in E$ if and only if $M^{\approx}_{\s{x_R}}[i,j] = 1$. The match relation $\approx$ is necessarily both \itbf{reflexive} ($M^{\approx}_{\s{w}}[i,i] = 1$) and \itbf{symmetric} ($M^{\approx}_{\s{w}}[i,j] = M^{\approx}_{\s{w}}[j,i])$ for any string \s{w}. If in addition our string $\s{w} = \s{x_R}$ is regular, it follows immediately from our definition that the match relation is transitive: $$M^{\approx}_{\s{x_R}}[i,j_1] = M^{\approx}_{\s{x_R}}[i,j_2] \Longleftrightarrow M^{\approx}_{\s{x_R}}[j_1,j_2] = M^{\approx}_{\s{x_R}}[i,j_1].$$ Thus for a regular string \s{x_R}, algorithms $regular_{min}$ and $regular_{min\!+\!_matrix}$ effectively compute the \itbf{transitive closure} of the match relation $\approx$ on \s{x_R} (equivalently, of the adjacent vertices in $G^{\approx}_{\s{x_R}}$); that is, they identify the maximum sets of positions that match in \s{x_R} (the cliques in $G^{\approx}_{\s{x_R}}$).
In the literature the Floyd-Warshall algorithms \cite{Cormen01, Floyd62} are the standard for computing the transitive closure on a graph $G = (V,E)$, with $|V| = m$. These algorithms execute in time $\mathcal{O}(m^3)$, and it is a long-standing open problem whether a faster algorithm exists. They are very general, not assuming the reflexive and symmetric conditions that hold in our case. Nevertheless, we have: \begin{lemm}\label{lem:tc} For a relation on $m$ objects that is reflexive and symmetric, the transitive closure can be computed in $\mathcal{O}(m^2)$ time.
\quad\qedbox46\newline\smallbreak \end{lemm}
\section{Strings from the Maximal Palindrome Array} \label{sect-maxpal} Given $\s{x} = \s{x}[1..n]$, a \itbf{substring} $\s{u} = \s{x}[i..j]$, $1 \le i \le j \le n$, of length $\ell = j\!+\!- i\!+\!+ 1$ is said to be a \itbf{palindrome} if $\s{x}[i\!+\!+ h] \approx \s{x}[j\!+\!- h]$ for every $h \in 0..\floor{\ell/2}$; a \itbf{maximal palindrome} if one of the following holds: $i = 1$, $j = n$, or $\s{x}[i\!+\!- 1] \not\approx \s{x}[j\!+\!+ 1]$. The \itbf{centre} of a palindrome \s{u} is at position $i \!+\!+ \frac{\ell\!+\!- 1}{2}$, which is not an integer for odd $\ell$.
Thus to facilitate discussion of the palindromic structure of \s{x}, we form the string $$\s{x^*}[1..m] = \!+\!# x_1\!+\!# x_2 \!+\!# \cdots \!+\!# x_n\!+\!#,$$ where $\!+\!# \not\in \Sigma$ and $m = 2n\!+\!+ 1$. Now every palindromic substring of \s{x} of length $\ell \ge 1$ maps into a palindromic substring of \s{x^*} of odd length $d = 2\ell\!+\!+ 1$, and every palindrome in \s{x^*} has an integral centre position. We call $d$ the \itbf{diameter} and $r = \floor{d/2}$ the \itbf{radius} of the palindrome in \s{x^*}.
We can now define the \itbf{maximal palindrome array} $\mbox{MP} = \mbox{MP}_{\s{x^*}}$ of \s{x^*}: for every $i \in \!+\!{1, \dots, m\!+\!}$, if $\s{x^*}[i] = \!+\!#$ and $\s{x^*}[i\!+\!- 1] \not\approx \s{x^*}[i\!+\!+ 1]$, then $\mbox{MP}[i] = 0$ (radius zero); otherwise, $\mbox{MP}[i]\geq 1$ is the radius of the maximal palindrome centred at position $i$. We assume that $\s{x^*}[0]=\s{x^*}[m+1]=\emptyset$. For example, $\mbox{MP}_{\s{x^*}}$ derived from $\s{x} = aabac$ is as follows: \begin{equation} \label{ex1} \begin{array}{rcccccccccc} \scriptstyle 1 & \scriptstyle 2 & \scriptstyle 3 & \scriptstyle 4 & \scriptstyle 5 & \scriptstyle 6 & \scriptstyle 7 & \scriptstyle 8 & \scriptstyle 9 & \scriptstyle 10 & \scriptstyle 11 \!+\! \s{x^*} = \!+\!# & a & \!+\!# & a & \!+\!# & b & \!+\!# & a & \!+\!# & c & \!+\!# \!+\! \mbox{MP}_{\s{x^*}} = 0 & 1 & 2 & 1 & 0 & 3 & 0 & 1 & 0 & 1 & 0 \end{array} \end{equation} The most general form of the palindrome array is given by \begin{equation} \label{generalform} \mbox{MP} = 0 i_2 i_3 \cdots i_{m-1} 0, \end{equation} where for every $j \in \!+\!{2,\dots,m-1\!+\!}$: \begin{itemize} \item[(a)] $i_j \in (1\!+\!- j\bmod 2)..\min(j\!+\!- 1,m\!+\!- j)$; \item[(b)] $i_j$ is odd if and only if $j$ is even. \end{itemize}
Any array satisfying (\ref{generalform}) is said to be \itbf{feasible}.
\begin{lemm} \label{lemm-exist} There exists a string corresponding to every feasible palindrome array. \end{lemm} \begin{proof} Suppose that a feasible palindrome array $\mbox{MP} = \mbox{MP}_{\s{x^*}}[1..m]$ is given for some odd positive integer $m$. We show how to construct a corresponding \s{x^*}. First, for every odd $c \in \!+\!{1,\dots,m\!+\!}$, we may of course assign $\s{x^*}[c] \leftarrow \!+\!#$. Suppose that the even positions $c$ in \s{x^*} are initially empty. For every $c \in \!+\!{3, \dots, m\!+\!- 2\!+\!}$ such that $\mbox{MP}[i] = r \ge 2$, add a unique character to each pair of positions $c\!+\!- k,c\!+\!+ k$ in \s{x^*}, where \begin{itemize}
\item[$\bullet$] ($c$ even, $r$ odd)
$k = 2,4,\ldots,r\!+\!- 1$;
\item[$\bullet$] ($c$ odd, $r$ even)
$k =1,3,\ldots,r\!+\!- 1$. \end{itemize} Finally, assign a unique character to each position that remains empty. The resulting (perhaps indeterminate) string will have MP as its palindrome array.
\quad\qedbox46\newline\smallbreak \end{proof} Applying the construction of Lemma~\ref{lemm-exist} to $\mbox{MP}_{\s{x^*}} = 0 1 2 1 0 3 0 1 0 1 0$ given in (\ref{ex1}) yields a string such as: \begin{equation} \label{ex2} \begin{array}{rcccccccccc} \scriptstyle 1 & \scriptstyle 2 & \scriptstyle 3 & \scriptstyle 4 & \scriptstyle 5 & \scriptstyle 6 & \scriptstyle 7 & \scriptstyle 8 & \scriptstyle 9 & \scriptstyle 10 & \scriptstyle 11 \!+\! \s{x^*} = \!+\!# & a & \!+\!# & \!+\!{a,b\!+\!} & \!+\!# & c & \!+\!# & b & \!+\!# & d & \!+\!# \!+\! \mbox{MP}_{\s{x^*}} = 0 & 1 & 2 & 1 & 0 & 3 & 0 & 1 & 0 & 1 & 0 \end{array} \end{equation} Note that, since the triple $(4,2,8)$ is intransitive ($\s{x}[2] \not\approx \s{x}[8]$), \s{x^*} is indeterminate. However, as we have seen in (\ref{ex1}), the regular string $\!+\!# a \!+\!# a \!+\!# b \!+\!# a \!+\!# c \!+\!#$
has the same palindrome array: a palindrome array can correspond to both a regular and an indeterminate string!
To each position $c \in \!+\!{1, \dots, m\!+\!}$ of a feasible $\mbox{MP}$ array we associate a pair of integers $(i,j)$ such that $i=c-\mbox{MP}[c] - 1$ and $j= c+ \mbox{MP}[c] + 1$; then, provided $0<i,j< m+1$, we must have $\s{x^*}[i] \not \approx \s{x^*}[j]$. We call this pair $(i,j)$ the \itbf{forbidden pair} with respect to $c$, and the characters at $\s{x^*}[i], \s{x^*}[j]$ \itbf{forbidden characters} with respect to $c$. For processing purposes, it will be convenient to assume that $\s{x^*}[0]=\s{x^*}[m+1]=\emptyset$. We denote by $\mbox{FP}$ the set of all forbidden pairs with respect to each centre $c \in \!+\!{1, \dots, m\!+\!}$. See (\ref{ex:nonrmp})
for an example.
Consider the feasible $\mbox{MP}$ array given in~(\ref{ex:nonrmp}). This array does not correspond to any regular string because the triple $(6,2,4)$ is intransitive ($\s{x^*}[2] \not \approx \s{x^*}[4]$), since the pair $(2,4)$ is a forbidden pair with respect to the centre $3$. Indeed, any triple of $\s{x^*}$ which includes a forbidden pair is intransitive, resulting in an indeterminate string. However, if a given $\mbox{MP}$ array does correspond to a regular string, we will say that it is \itbf{regular}.
\begin{equation} \label{ex:nonrmp} \begin{array}{rcccccc} c \qquad 1 & 2 & 3 & 4 & 5 & 6 & 7 \!+\! MP \qquad 0 & 1 & 0 & 3 & 2 & 1 & 0 \!+\! \s{x^*} \qquad \!+\!# & 1 & \!+\!# & \!+\!{2,3\!+\!} & \!+\!# & \!+\!{1,3\!+\!} & \!+\!# \!+\!
\mbox{FP} \qquad (0,2) & (0,4) & (2,4) & (0,8) & (2,8) & (4,8) & (6,8) \end{array} \end{equation}
In order to characterize MP arrays, it is useful to introduce Manacher's condition \cite{manacher:75}, restated in \cite{IIBT10}, and further restated here.
In $\mbox{MP} = \mbox{MP}_{\s{x^*}}$, we consider each centre $c$ of a palindrome of radius $r = \mbox{MP}[c]$, where for $c$ even, $r \ge 3$, and for $c$ odd, $r \ge 2$. Then we must have $\s{x^*}[c\!+\!- k] \approx \s{x^*}[c\!+\!+ k]$, where $1 \le k \le r$. We then have \begin{defn}\label{defM} [Manacher's condition] Let $r_{\ell} = \mbox{MP}[c\!+\!- k]$ and $r_r = \mbox{MP}[c\!+\!+ k]$, for each $1 \le k \le r$, where $r = \mbox{MP}[c]$. Every position $c$ in a regular palindrome array $\mbox{MP} = \mbox{MP}_{\s{x^*}}$ satisfies the following: \begin{itemize}
\item[(a)]
{\bf if\ } $r_{\ell} \ne r\!+\!- k$ {\bf then\ } $r_r = \min(r_{\ell},r\!+\!- k)$
{\bf else\ } $r_r \ge r_{\ell}$;
\item[(b)]
{\bf if\ } $r_r \ne r\!+\!- k$ {\bf then\ } $r_{\ell} = \min(r_r,r\!+\!- k)$
{\bf else\ } $r_{\ell} \ge r_r$. \end{itemize} \end{defn}
For completeness, we outline here Manacher's algorithm to construct an $\mbox{MP}$ array for a given string $\s{x^*}$ of scope $1$. The algorithm scans $\s{x^*}$ from left to right and evaluates centres to compute $\mbox{MP}[c]$. While evaluating a centre $c$ and each value of the range $k$, it checks whether $r_{\ell}=r-k$; if so, it stops evaluating centre $c$ and moves to evaluating centre $c\!+\!+ k$; otherwise it assigns $r_r \leftarrow \min(r_{\ell},r\!+\!- k)$ and continues evaluating the same centre $c$ and the next range value $k+1$.
In \cite{manacher:75} it is shown that, for every palindrome in string \s{x^*} such that $S(\s{x^*}) = 1$, Manacher's condition must hold. Thus, by Lemma~\ref{lemmreg}, Manacher's condition holds for every regular string; that is, for every string whose triples are all transitive.
On the other hand, note that in example (\ref{ex:nonrmp}), for $c = 4$ we have $$r = \mbox{MP}[4] = 3,\!+\! k = 1,\!+\! r_{\ell} = \mbox{MP}[3] = 0,\!+\! r_r = \mbox{MP}[5] = 2,$$ so that, by Definition~\ref{defM}(a), since $r_{\ell} \ne r\!+\!- k$, therefore $r_r = \min(0,2) = 0$ should hold, which is false. Thus for this indeterminate string, Manacher's condition does {\it not} hold.
The above suggests extending the construction of Lemma~\ref{lemm-exist} so as to yield, whenever possible, a regular string corresponding to a given palindrome array. See Figure~\ref{fig-construct}.
The procedure $construct$ takes a feasible $\mbox{MP}$ array as input and returns a lex-least regular string $\s{x^*}$ if $\mbox{MP}$ is regular; otherwise it returns an indeterminate string. This procedure is based on the Manacher's algorithm presented in~\cite{manacher:75}. To ensure that the resulting string, if regular, is a lex-least string, it maintains for each centre $c$ a set $\mbox{FS}[c]$ containing the indices of the forbidden characters.
\begin{figure}
\caption{Given a feasible palindrome array MP, $construct$ if possible produces a corresponding lex-least regular string \s{x^*} of scope 1 on an integer alphabet $\Sigma = \!+\!{1,2,.\ldots\!+\!}$. If such a string is not possible, then on exit $regular = \tt{false}$ and \s{x^*} is an indeterminate string on $\Sigma$ corresponding to MP. }
\label{fig-construct}
\end{figure}
\begin{figure}
\caption{Given centre $c$ and its radius $r$, $update\mbox{FS}$ updates the $\mbox{FS}$ array at the left position in the forbidden pair with respect to $c$.}
\label{fig-updateFS}
\end{figure}
In procedure $construct$ the output string is first initialized with $0$ at even positions and $\!+\!#$ at odd positions. Then $\s{x^*}[2]$ is set to $1$, as we are interested in the lex-least regular $\s{x^*}$, if it exists. Next we examine a centre $3 \leq c \leq m \!-\! 1$ and update the $\mbox{FS}[c]$ set based on its forbidden pair. If the string $\s{x^*}[1..c\!-\! 1]$ constructed so far is regular and the centre $c$ and range $k$ satisfy Manacher's condition, we continue to construct the regular string by copying the previously filled letter $\s{x^*}[c\!-\! k]$ to its corresponding matching position $c\!+\!+ k$. Whenever there is a choice of filling an empty position --- that is, when $\s{x^*}[c]=0$ --- the lex-least character which is not in the forbidden set of characters $\mbox{FS}[c]$ is chosen. Note that we use the same strategy as presented in Manacher's algorithm~\cite{manacher:75} to avoid evaluating every centre if $\mbox{MP}$ is regular. However, unlike Manacher's algorithm, instead of evaluating a new centre when $r_{\ell}=r-k$ and $r_r > r_{\ell}$, we record the next centre to be evaluated, and finish evaluating the current centre. After which we move to the recorded centre and check if Manacher's condition is satisfied.
Furthermore, to avoid invoking the $update\mbox{FS}$ procedure repeatedly on a centre $c$, we use a Boolean array $empty$ of length $m$, such that $empty[c] = true$ if and only if the centre $c$ has not been used to update the $\mbox{FS}$ array; otherwise, we set $empty[c] \leftarrow false$. If a given centre $c$ and range $k$ do not satisfy Manacher's condition, $regular$ is set to $false$ and every subsequent letter match including the current one is achieved by adding a new character to the letters at positions $c\!-\! k$ and $c\!+\!+ k$, if they are even. Furthermore, for the remainder of the $\mbox{MP}$ array all centres are examined.
\begin{lemm}\label{lemm-regconstx} Let $\s{x^*}$ be the string produced by procedure $construct$. Then $S(\s{x^*})=1 \Leftrightarrow \s{x^*}$ is regular. \end{lemm} \begin{proof} ($\Rightarrow$) Since $S(\s{x^*})=1$, the regularity of $\s{x^*}$ is immediate, as for all positions $1 \leq i,j \leq m$, $\s{x^*}[i] \approx \s{x^*}[j] \Leftrightarrow \s{x^*}[i] = \s{x^*}[j]$. Then any triple in $\s{x^*}$ is transitive by the associative property of equality. Therefore $\s{x^*}$ is regular.
($\Leftarrow$) $\s{x^*}$ is regular only if Manacher's condition holds for every centre $c$, radius $r$ and range $k$. In such a case $construct$ either copies existing letters of scope one or fills an empty position with a new character to produce $\s{x^*}$. Therefore $S(\s{x^*})=1$.
\quad\qedbox46\newline\smallbreak \end{proof}
\begin{thrm}\label{thrm-const} Let $\s{x^*}$ be the string produced by the procedure $construct$ on an input $\mbox{MP}$. Then $\s{x^*}$ is regular $\Leftrightarrow$ $\mbox{MP}$ is regular. \end{thrm} \begin{proof} ($\Rightarrow$) $\s{x^*}$ is regular only if Manacher's condition holds for every centre $c \in \!+\!{1, \ldots, m\!+\!}$, and for every corresponding radius $r$ and range $k$. Since Manacher's condition holds, it follows from the result in~\cite{manacher:75} that $\mbox{MP}$ is regular.
($\Leftarrow$) From~\cite{manacher:75}, MP is regular only if Manacher's condition holds. Since the condition holds, during the execution of $construct$ an empty position is filled either by copying previously occurring letters of scope one or by a new character. Thus the resulting string is of scope $1$, so that from Lemma~\ref{lemm-regconstx}, $\s{x^*}$ is regular.
\quad\qedbox46\newline\smallbreak \end{proof}
Using the two examples given below in (\ref{ex3}) and (\ref{ex4}), we illustrate the execution of procedure $construct$.
In (\ref{ex3}) the positions corresponding to all odd centres are filled with $\!+\!#$ and positions corresponding to all even centres are filled with $0$. Furthermore, $\mbox{FS}[1..m] \leftarrow \emptyset$, $empty[1..2] \leftarrow false$ and $empty[3..m] \leftarrow true$. Then for centre $c = 2$, $\s{x^*}[2] \leftarrow 1$, and for centre $c = 3$, $FS[4] \leftarrow \!+\!{2\!+\!}$ and $empty[3] \leftarrow false$. At centre $c=4$, $empty[4] \leftarrow false$, and since $r=3$ we check if Manacher's condition holds for $k=3,2,1$. As we do this, we also check whether $empty[c+k]$ is $true$; if so, we set it to $false$ and compute the $\mbox{FS}$ set w.r.t to these centres, thereby setting $\mbox{FS}[6]$ and $\mbox{FS}[8]$ to $\!+\!{4\!+\!}, \!+\!{4,6\!+\!}$ respectively. Since Mancher's condition holds, the setting at position $\s{x^*}[2]$ is copied into $\s{x^*}[6]$; next, since position $\s{x^*}[4]=0$, it is filled with the next lex-least character which is not equal to a character at any index position in the forbidden set $\mbox{FS}[4]=\!+\!{2\!+\!}$ --- that is, $2$.
Similarly for centre $c = 8$, $empty[8..15] \leftarrow false$, $\mbox{FS}[10] \leftarrow \!+\!{8\!+\!}, \mbox{FS}[12] \leftarrow \!+\!{8,10\!+\!}$. Since Manacher's condition holds for all $k=7, 6, \ldots, 1$, $\s{x^*}[2]$, $\s{x^*}[4]$ and $\s{x^*}[6]$ are copied into $\s{x^*}[14],\s{x^*}[12]$ and $\s{x^*}[10]$, respectively. Finally, since $\s{x^*}[8]=0$, the position is filled with the next lex-least character $3$ which is not equal to a character at any index position in the forbidden set $\mbox{FS}[8]=\!+\!{4,6\!+\!}$. The resulting string $\s{x^*}$ is regular and lex-least. \begin{small} \begin{equation} \label{ex3} \begin{array}{rcccccccccccccccc} \scriptstyle 1 & \scriptstyle 2 & \scriptstyle 3 & \scriptstyle 4 & \scriptstyle 5 & \scriptstyle 6 & \scriptstyle 7 & \scriptstyle 8 & \scriptstyle 9 & \scriptstyle 10 & \scriptstyle 11 & \scriptstyle 12 & \scriptstyle 13 & \scriptstyle 14 & \scriptstyle 15 \!+\! MP = 0 & 1 & 0 & 3 & 0 & 1 & 0 & 7 & 0 & 1 & 0 & 3 & 0 & 1 & 0 \!+\! \s{x^*} = \!+\!# & 1 & \!+\!# & 2 & \!+\!# & 1 & \!+\!# & 3 & \!+\!# & 1 & \!+\!# & 2 & \!+\!# & 1 & \!+\!# \!+\! FS = \emptyset & \emptyset & \emptyset & \!+\!{2\!+\!} & \emptyset & \!+\!{4\!+\!} & \emptyset & \!+\!{4, 6\!+\!} & \emptyset & \!+\!{8\!+\!} & \emptyset & \!+\!{8,10\!+\!} & \emptyset & \!+\!{12\!+\!} & \emptyset \!+\! \end{array} \end{equation} \end{small}
Observe that the $\mbox{MP}$ array in (\ref{ex4}) only differs from the $\mbox{MP}$ array in (\ref{ex3}) at position $12$. However, the $\mbox{MP}$ array in (\ref{ex4}) does not correspond to any regular string, since, as we shall see, it fails Manacher's condition for $c=8,\!+\! r=7$, and $k=4$. In (\ref{ex4}) $construct$ produces the same string as in (\ref{ex3}) up until $c=7$. Then at centre $c = 8$, $\s{x^*}[2]$ is first copied into $\s{x^*}[14]$. However, since for $k=4$ the $\mbox{MP}$ array fails Manacher's condition, a new lex-least character $3$ is added to the letters at positions $4$ and $12$. Then at positions $6$ and $10$, a new character $4$ is added to both $\s{x^*}[6]$ and $\s{x^*}[10]$. Finally, since $\s{x^*}[8]=0$, it is filled with another new character $5$. {\small \begin{equation} \label{ex4} \begin{array}{rcccccccccccccccc} \scriptstyle 1 & \scriptstyle 2 & \scriptstyle 3 & \scriptstyle 4 & \scriptstyle 5 & \scriptstyle 6 & \scriptstyle 7 & \scriptstyle 8 & \scriptstyle 9 & \scriptstyle 10 & \scriptstyle 11 & \scriptstyle 12 & \scriptstyle 13 & \scriptstyle 14 & \scriptstyle 15 \!+\! MP = 0 & 1 & 0 & 3 & 0 & 1 & 0 & 7 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \!+\! \s{x^*} = \!+\!# & 1 & \!+\!# & \!+\!{2,3\!+\!} & \!+\!# & \!+\!{1,4\!+\!} & \!+\!# & 5 & \!+\!# & 4 & \!+\!# & 3 & \!+\!# & 1 & \!+\!#\!+\! \end{array} \end{equation} }
We now discuss the running time complexity of $construct$. This is dependent on the input $\mbox{MP}$ array. If $\mbox{MP}$ is regular, then $construct$ yields a lex-least regular string in $\mathcal{O}(n)$ time; otherwise it yields an indeterminate string in $\mathcal{O}(n^2)$ time. We remark that to avoid having $\sigma$ in the running time, we use the strategy discussed in~\cite{IIBT10} to fill the lex-least character not in the list of forbidden characters at centre $c$. Here we briefly outline the strategy: The paper \cite{IIBT10} uses a bit vector $F$ of length $n$ where the element $F[h]$ corresponds to the $h$-th lex-least least character. Initially all elements of $F$ are set to $0$. If a character $\s{x}[j]$ is forbidden w.r.t the character $\s{x}[i]$, and if $\s{x}[j]$ is the $h$-th lex-least character, then $F[h]$ is set to $1$. After finding all forbidden characters for $\s{x}[i]$ and setting the $F$ vector appropriately, the lex-least character for $\s{x}[i]$ is set equal to the $k$-th lex-least character, where $k$ is the first (or smallest) index such that $F[k]=0$.
If $\mbox{MP}$ is regular, then similar to Manacher's algorithm, not every centre is evaluated to yield a regular string. Suppose $c$, $1 \leq c \leq m$, is the least centre for which the Manacher's test is performed and is satisfied. Then, two cases arise. First, there exists a $k$ (in $construct$ we consider the minimum $k$), $1\leq k \leq r$, where the following subcondition of Manacher's condition (Definition~\ref{defM}) is satisfied: $r_{\ell} = r\!-\! k$ and $r_r > r_{\ell}$. Second, when the subcondition is not satisfied, all the palindromes that arise at centres $c'\leq c\!+\!+ r$ are contained in the palindrome at $c$ and also satisfy Manacher's condition. Therefore, to avoid duplication the next centre to be evaluated is $c\!+\!+r\!+\!+1$.
In the former case, however, the palindrome at centre $c+k$ extends beyond the right edge of the palindrome centred at $c$. Therefore, $c\!+\!+ k$ is the next centre to be evaluated. However, since $c$ satisfies Manacher's, a shorter palindrome of length $r\!-\! k$ at centre $c\!+\!+k$ also occurs at centre $c\!-\! k$. Therefore, we only need to check the positions occurring after the first $r\!-\! k$ positions on both sides of $c\!+\!+k$.
In both these cases a position in the $\mbox{MP}$ array is visited at most twice. Therefore, the running time of $construct$ to yield a regular string is $\mathcal{O}(n)$.
If $\mbox{MP}$ is not regular, after the first centre that fails Manacher's test, every centre is evaluated, and as a result $construct$ yields an indeterminate string in $\mathcal{O}(n^2)$ time. However, note that the running time for an $\mbox{MP}$ array that is not regular largely depends on the centre that fails the Manacher's test. If the centre lies close to the end of the array, then the indeterminate string is constructed in $\mathcal{O}(n)$ time; otherwise it takes $\mathcal{O}(n^2)$ time. Thus we have:
\begin{thrm}\label{thrm-rtconstruct} Given a regular MP array of length $m = 2n\!+\!+ 1$, procedure $construct$ yields a lex-least regular string in $\mathcal{O}(n)$ time. \end{thrm}
\begin{thrm}\label{thrm-rtconstruct} Given a MP array of length $m = 2n\!+\!+ 1$ that is not regular, procedure $construct$ yields an indeterminate string in $\mathcal{O}(n^2)$ time. \end{thrm}
We remark that our algorithm $construct$ is similar to the approach presented in~\cite[Algorithm~2]{IIBT10} to construct a regular string if $\mbox{MP}$ is regular. The key differences between $construct$ and their approach is that their algorithm sometimes verifies an $\mbox{MP}$ array which is not regular to be regular, and therefore outputs an incorrect lex-least regular string $\s{w}$. To correct this, they execute Manacher's algorithm~\cite[Algorithm~1]{IIBT10} on $\s{w}$ to compute its palindrome array $\mbox{MP}_{\s{x}}$. Then they output $\s{w}$ if and only if $\mbox{MP}_{\s{x}} = \mbox{MP}$. We on the other hand use a direct approach --- our algorithm outputs a regular string if and only if the input $\mbox{MP}$ array is regular. Furthermore, if $\mbox{MP}$ is not regular, their algorithm simply exits, while $construct$ on the other hand yields an indeterminate string.
Consider the example shown in Figure~\ref{ex5}. For the given $\mbox{MP}$ array, $construct$ produces the string $\s{x^*}$. Observe that not all the centres $4-11$ satisfy Manacher's condition. Furthermore, if a centre $c$ and a range $k$ fail Manacher's condition, it is not necessarily the case that the centre $c+k$ also fails it. For example, for centre $c=8$, and range $k=3$, Manacher's condition fails; however, the centre $c+k=11$ satisfies it.
\begin{figure}
\caption{Given an $\mbox{MP}$ array that is not regular, procedure $construct$ produces the indeterminate string $\s{x^*}$.}
\label{ex5}
\end{figure}
As can be observed in the example shown in Figure~\ref{ex5}, the number of centres that fail Manacher's condition is of order $n$, where $n$ is the length of the string. Therefore, we are forced to evaluate $\mathcal{O}(n)$ centres in this example. Hence we state the following:
\begin{conj}\label{lem:con} To construct an indeterminate string over an MP array of length $m = 2n\!+\!+ 1$ that is not regular requires at least $\Omega(n^2)$ time.
\end{conj}
\section{Conclusion and Open Problems} \label{sect-probs}
In this paper we have characterized regular strings as those whose triples are all transitive; the strings for which this is not true are called indeterminate. A new approach based on the transitivity of matching among the regular letters of a string enables us to precisely define strings and eliminate false positive cases seen in the traditional definition of indeterminate strings. Based on this new definition, we present a linear time algorithm REGULAR that checks the regularity of the given string $\s{x}$ by checking the regularity of its corresponding reduced string $\s{x_R}$, and as a byproduct yields a lex-least regular string that is isomorphic to $\s{x}$ whenever $\s{x}$ is regular.
To check the regularity of $\s{x_R}$, we present two algorithms $regular_{min}$ and $regular_{min\!+\!_matrix}$, which illustrate an interesting connection between the matching relation defined on the letters of $\s{x}$ and the computation of the relation's transitive closure. Many algorithms exist to compute the transitive closure of a relation in $\mathcal{O}(n^3)$ time. However, it is a long standing open problem whether this complexity can be reduced. We show that for important special cases, the transitive closure can be computed in $\mathcal{O}(n^2)$ time using the $regular_{min\!+\!_matrix}$ function.
Furthermore, we consider the reverse engineering problem of constructing a string (regular or indeterminate) from the given palindrome array. We present the algorithm $construct$ to output a lex-least regular string, if it exists, in $\mathcal{O}(n)$ time; otherwise the algorithm returns an indeterminate string in $\mathcal{O}(n^2)$ time. An immediate question arises whether an algorithm faster than $construct$ exists to output an indeterminate string. It would also be interesting to develop an algorithm that yields an indeterminate string over a minimum alphabet, corresponding to an $\mbox{MP}$ array that is not regular. Furthermore, although the running time complexity of REGULAR is $\mathcal{O}(n)$, the constant hidden in it could in extreme cases be rather large -- is it possible to reduce the size of this constant?
More generally, a new definition of regularity in strings may affect the relationship between strings and the data structures mentioned in the Introduction that facilitate their use: border array, cover array, prefix array, and suffix array. A string containing indeterminate letters may now actually be regular, provided all its triples are transitive: can these important arrays be computed accordingly in linear time?
Conversely, what effect does this new definition have on the ``reverse engineering'' problem related to these arrays? We have presented in Section~\ref{sect-maxpal} an algorithm that reverse engineers the palindrome array to a regular string, whenever possible. There exist algorithms to reverse engineer the other array noted above --- can they be modified/extended to yield equivalent results? Can the time requirements of these algorithms be reduced to near-linear, except in pathological cases?
\section*{Acknowledgments} We thank the reviewers for carefully reading the manuscript and providing insightful comments which led to substantial improvements in the manuscript.
\paragraph{Funding:} Louza was supported by Grant $\!+\!#$2017/09105-0 from the S\!+\!~ao Paulo Research Foundation (FAPESP). Mhaskar and Smyth were supported by Grant 36797 from the Natural Sciences \!+\!& Engineering Research Council of Canada (NSERC).
\end{document} | arXiv |
George Pirie (mathematician)
George Pirie (19 July 1843 – 21 August 1904) was a Scottish mathematician, mathematical scientist, and Reverend in the Church of Scotland. He was an expert in the field of dynamics and the approximation of π.
Early life and education
Pirie was the son of Very Rev William Robinson Pirie and his wife, Margaret Chalmers Forbes, daughter of Very Rev Lewis William Forbes. He was born in the manse at Dyce near Aberdeen on 19 July 1843.[1]
He was educated at Aberdeen Grammar School. He then studied Mathematics and Physics (then called Natural Philosophy) at The University of Aberdeen under Prof David Thomson. He graduated MA in 1862. He then won a place at Queens' College, Cambridge. He was Fifth Wrangler in the Mathematical Tripos of 1866.[2]
Career
Pirie was a Fellow of Queens' College from 1866 to 1888.[2] He lectured at Cambridge from 1866 and in 1878 was offered the chair as Professor of Mathematics at Aberdeen University in replacement of Prof Frederick Fuller.[3]
Pirie was elected a member of the London Mathematical Society in November 1881.[4] He also took part in the activities of the Aberdeen Branch of the Royal Scottish Geographical Society.[5][6] Since 1881 he was also a member of the Aberdeen Philosophical Society, and served as a Vice-President of the society in 1900.[7]
Personal life
He lived at 33 College Bounds close to King's College in Old Aberdeen.[8]
He was married to Anne Elizabeth Reid daughter of Rev Reid of Auchindoir.
Their children included Margaret Forbes Pirie (1884–1963) and Elizabeth Mary Pirie (1888–1971).[9]
He died unexpectedly at Braemar on 21 August 1904, from heart failure.[10] He is buried next to his parents on the north side of St Machar's Cathedral in Old Aberdeen.
Recognition
He was awarded an honorary doctorate (LL.D) by St. Andrews University in 1896.[2][11]
Publications
• George Pirie (1875). Lessons on Rigid Dynamics. London: MacMillan and Co. p. 20. Lessons on Rigid Dynamics.[12]
• A Short Account of the Principal Geometrical Methods of Approximating to the Value of π[13]
References
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| Wikipedia |
\begin{definition}[Definition:Module Structure of Polynomial Ring]
Let $R$ be a commutative ring with unity.
Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$.
The '''$R$-module structure''' on $R \sqbrk X$ is the module structure as an algebra over $R$.
\end{definition} | ProofWiki |
\begin{document}
\begin{abstract} We have solved a number of holonomic PDEs derived from the Bessel modules which are related to the generating functions of classical Bessel functions and the difference Bessel functions recently discovered by Bohner and Cuchta. This $D$-module approach both unifies and extends generating functions of the classical and the difference Bessel functions. It shows that the algebraic structures of the Bessel modules and related modules determine the possible formats of Bessel's generating functions studied in this article. As a consequence of these $D$-modules structures, a
number of new recursion formulae, integral representations and new difference Bessel polynomials have been discovered. The key ingredients of our argument involve new transmutation formulae related to the Bessel modules and the construction of $D$-linear maps between different appropriately constructed submodules. This work can be viewed as $D$-module approach to Truesdell\rq{}s $F$-equation theory specialised to Bessel functions. The framework presented in this article can be applied to other special functions. \end{abstract}
\maketitle \setcounter{tocdepth}{2} \tableofcontents
\section{Introduction} Generating functions play important roles in both pure and applied mathematics \cite{FS_2009, Wilf_2006}. In particular, generating functions of Bessel functions play pivotal roles in the mathematical formulation of electromagnetic wave and acoustic scattering problems, see for examples, \cite{Bowman_et_al_1987, Dunster_2013}. Other application includes the study of Rogers-Ramanujan identities \cite{Ehrenpreis_1990, Ehrenpreis_1993}.
In this paper, we present a systematic study of generating functions of Bessel functions based on the Bessel module \eqref{E:Bessel_mod_intro} defined below. This is the first of a series of papers originates from our study of generating functions of classical special functions using $D$-module methodology. It is known that $D$-modules is an efficient language in exhibiting algebraic structures and in carrying out explicit computation of identities of special functions in general, see for examples \cite{Chyzak_2000, Chyzak_Salvy_1998, Ehrenpreis_1990, Ehrenpreis_1993, kashiwara, Kashiwara_Kawai_1981, Kashiwara_Schapira_1997, Mansour_Schork_2016, Paule_Schorn_1995, Wilf_Zeilberger_1992, Zeilberger_1990}. We demonstrate that this is indeed the case that the $D$-modules approach not only allows us to better understand some of the classical formulae about special functions but also to derive their new difference analogues. This work can be considered as a continuation of the previous works of \cite{Lommel, Lommel_1871, Nielson, Sonine, Schlfafli_1873, Truesdell_1947, Truesdell_1948} and to a less extent of \cite{Burchnall_1953, Ehrenpreis_1990, Ehrenpreis_1993} by algebraic methods. A noteworthy feature of this study is that the main results are first obtained in the $D$-modules framework before their $D$-modules structures are being manifested in complex domains differently. Thus the study unifies the classical Bessel functions and the recently discovered difference Bessel functions amongst other possible manifestations. The fact that the main results have been derived entirely within the appropriate $D$-modules that is subject to different manifestations in the complex plane is a vindication of the viewpoint from umbral calculus discovered centuries ago.
\subsection{Bessel\rq{}s modules} We extend Bessel\rq{}s classical generating function\footnote{It was known to Hansen in 1845 and earlier to Jacobi in the 1836 in a modified form \cite[\S 2.1]{Watson1944}.} \begin{equation}\label{E:bessel_classical_gf_0}
e^{\frac{x}{2}\big(t-\frac1t\big)}=\footnote{The series converges absolutely in each compact subset of $\mathbb{C}\times\mathbb{C}\backslash\{0\}$.} \sum_{n=-\infty}^\infty J_{n}(x)\, t^n
\end{equation}
for the Bessel coefficients $(J_n(x))$ which denotes the bilateral sequence of \textit{Bessel functions of the first kind} of integer orders, to the case of non-integer orders $(J_{\nu+n}(x))$ where $\nu\not=0$, amongst other results obtained, including their difference analogues, by studying
the quotient module
\begin{equation}\label{E:Bessel_mod_intro}
\mathcal{B}_\nu:= \frac{\mathcal{A}_2}
{\mathcal{A}_2(X_1\partial_1+(\nu+X_2\partial_2)-X_1X_2)+
\mathcal{A}_2(X_1\partial_1-(\nu+X_2\partial_2)+{X_1}/{X_2})},
\end{equation}be called the \textit{Bessel module of order} $\nu\in \mathbb{C}$ in this article, where the $\mathcal{A}_2$ denotes the $D$-modules of the Weyl-algebra generated by $\partial_i,\, X_i\ (j=1,\, 2)$ subject to the standard commutation relations
\begin{equation}\label{E:comm}
[\partial_j, X_k]=\delta_{j, k}, \qquad [\partial_j, \partial_k]=0,
\qquad [X_j, X_k]=0,\qquad \mbox{for } j,\, k=1,\, 2.
\end{equation} Here the two relations
\begin{equation}\label{E:2_elements}
X_1\partial_1+\nu+X_2\partial_2-X_1X_2,
\qquad
X_1\partial_1-(\nu+X_2\partial_2)+{X_1}/{X_2},
\end{equation} which define the Bessel module $\mathcal{B}_\nu$, are abstractions of the well-known recursions \footnote{Lommel \cite[1868]{Lommel}.}
\begin{equation}\label{E:2_PDE_d_delta}
\begin{split}
& xJ'_{\nu+n}(x)-(\nu+n) J_{\nu+n}(x)=-xJ_{\nu+n+1}(x),
\\
& xJ'_{\nu+n}(x)+(\nu+n) J_{\nu+n}(x)=xJ_{\nu+n-1}(x),
\end{split}
\end{equation}about Bessel functions, known as Gauss\rq{} contiguous relations\footnote{They and other generalised ones will be called PDEs in a board sense in this article. The list of all holonomic PDEs are listed in Appendix \S\ref{SS:holo_modules}.} in special functions, which hold for every $n\in \mathbb{Z}$, where the $\partial_1$ and $\partial_2$ in \eqref{E:2_elements} play the roles of differentiation with respect to $x$ and shift of subscripts $n$ respectively. As we shall see that the $\mathcal{B}_\nu$ is holonomic in the sense of Bernstein-Kashiwara-Sato's holonomic PDEs theory (see, e.g., \cite{coutinho, kashiwara}), and that solving $\mathcal{B}_0$ gives raise to the \eqref{E:bessel_classical_gf_0} as a special case of $\nu=0$. The two relations from \eqref{E:2_elements} forms a system of two PDEs from this viewpoint. Thus, we have derived
\begin{equation}\label{E:bessel_classical_gf_1}
t^{-\nu}e^{\frac{x}{2}\big(t-\frac1t\big)}\sim\footnote{The infinite sum is divergent, and the sign ``$\sim$" denotes \textit{Borel-resummation}; see the paragraph below the \eqref{gf-db-02}.} \sum_{n=-\infty}^\infty J_{\nu+n}(x)\, t^n,
\end{equation}
for general $\nu\not=0$, and its difference analogue
\begin{equation}\label{gf-db-02}
e^{i\pi\nu}
\frac{\sin(x-\nu)\pi}{\sin(\pi x)}\,
t^{-\nu}\big[\frac{1}{2}(t-\frac{1}{t})+1\big]^{x}
\sim\sum_{n=-\infty}^{\infty}
J^{\Delta}_{n+\nu}(x)\,t^n,
\end{equation}
for the \textit{difference Bessel functions} $(J_{\nu+n}^\Delta)$ discovered by Bohner and Cuchta \cite{BC_2017} to be discussed below. Here the ``$\sim$" signs used above denote the \textit{Borel-resummation} on the right-sides of the corresponding formulae since both the sequences $(J_{\nu+n})$ and the $(J^\Delta_{\nu+n}$) are $1$-Gervey\footnote{See Theorem \ref{T:Bessel_Gevrey} and Theorem \ref{T:delta_Bessel_Gevrey}.}\footnote{For an up-to-date discussion about the \textit{resurgence theory}, please see \cite[Part II]{Mitschi_Sauzin_2016}, \cite{Shawyer_Watson_1994} and \cite{Delabaere_Pham_1999}. } in the sense that in each compact set, $|J_{\nu+n}(x)|\le C^nn!$ and $|J^\Delta_{\nu+n}(x)|\le D^nn!$ for some positive constants $C,\ D$, see Example \ref{D:borel}.
Both sides of \eqref{E:2_PDE_d_delta} and \eqref{gf-db-02} satisfy the systems of PDEs\footnote{We use the terminology ``PDEs\rq\rq{} in a generalised sense that may include difference or shift operators or simply members in the $\mathcal{A}_2$ in this paper.}
\begin{equation}\label{E:PDE_gf_bessel}
\begin{split}
&y_x(x,t)+\big(1/t-t\big)/2\, y(x,t)=0,\\
&ty_t(x,t)+\nu y(x,t)-{x}/{2}\,\big(1/t+t\big)\, y(x,t)=0,
\end{split}
\end{equation} and
\begin{equation}\label{E:PDE_gf_dBessel}
\begin{split}
&y(x+1, t)-y(x, t)+(1/t-t)/2 y(x, t)=0,\\
&ty_t(x,t)+\nu y(x,t)-x(1/t+ t)/2y(x-1, t)=0
\end{split}
\end{equation} respectively.
The method of derivation of the above results and other results in this paper is based on construction of $D$-linear maps from the Bessel modules $\mathcal{B}_\nu$ (and other $D$-modules) to appropriately \textit{manifested} $D$-module analytic function spaces of two variables (e.g., $\mathcal{O}_{dd},\ \mathcal{O}_{\Delta d}$). Other quotient $D$-modules related to the half-Bessel module $\mathcal{B}_{1/2}$ studied in this paper includes \textit{Bessel polynomial modules} $\Theta, \mathcal{Y}$ and \textit{Glaisher modules} $\mathcal{G}_\pm$ which also are listed in Appendix \S\ref{SS:holo_modules}. This paper can be considered as a $D$-module study of Truesdell\rq{}s $F$-equation theory \cite{Truesdell_1948}, which summarised and extended research works about generating functions of predecessors from the nineteenth century, when applied to Bessel\rq{}s generating functions.\footnote{The authors only got to know Truesdell's work after obtaining the main results of this paper.} However, unlike the analytic approach used in \cite{Truesdell_1948}, the existence and uniqueness of the $F$-equation theory are replaced by the holonomicity of the $D$-modules concerned. The paper aims to show that $D$-modules is not only a natural language but also an efficient computational tool to study special functions. We have built up our applications of $D$-modules that represent some of the simplest holonomic equations which give raise to the exponential, trigonometric functions, etc from scratch in order to deal with the quotient modules studied in this paper. This is because we could reach our main results efficiently with our self-contained approach instead of utilising the full force of $D$-modules theory from, for examples, \cite{kashiwara, SST_2000}. This is especially the case when we derive difference analogues of the classical results. Closer to our approach but differ substantially in details were the pioneering works of Wilf and Zeilberger \cite{Zeilberger_1990, Wilf_Zeilberger_1992} in which the authors used holonomic $D$-modules methodology to derive hypergeometric type identities. Our approach is also different from the Lie algebraic approach of generating functions used by Weisner \cite{Weisner_1959}, see also \cite{Mcbride}.
The choice to start with Bessel functions is a natural one in regard to them being ubiquity in many research areas and wide range of applications, see for examples \cite{AAR, EAM2, Lommel, Wang_Guo1989, WW, Watson1944}.
For easy of computation of the relevant generating functions, we add and subtract the two relations \eqref{E:2_elements} to obtain, respectively, the two alternative relations
\begin{equation}\label{E:2_more_elements}
2X_1\partial_1-X_1(X_2-1/X_2),\qquad 2\nu+2X_2\partial_2-X_1(X_2+1/X_2)
\end{equation} for which we can, without loss of generality, eliminate the term ``$X_1$\rq\rq{} altogether from the first relation above. As an example with $\nu=0$, after a ``change of variables" \footnote{This change of variables corresponds to the ``characteristic method\rq\rq{}.},
the \eqref{E:2_more_elements} is rewritten as
\[
2\hat{X}_1\hat{\partial}_1-\hat{X}_1,\quad
2\hat{X}_2\hat{\partial}_2+\hat{X}_2,
\]where the $\{\hat{\partial}_1, \hat{X}_1, \hat{\partial}_2, \hat{X}_2\}$ satisfy similar commutation relations as those in the \eqref{E:comm}, and which when \textit{solved simultaneously}
leads us to the \textit{solution}
\begin{equation}\label{E:bessel_exp}
{\mathop{\rm E}} \Big[\frac{X_1}{2}(X_2-\frac{1}{X_2})\Big],
\end{equation} where the ${\mathop{\rm E}}$ denotes the \textit{Weyl exponential} (see Example \ref{Eg:exp}). Right multiplication by $e^{\frac{x}{2}(x-1/x)}$ gives the map $\mathcal{B}_0\to \mathcal{O}_{dd}$ where the $ \mathcal{O}_{dd}$ has the \textit{manifestation}, as it is called in this paper,
\begin{equation}\label{E:manifestation_1}
\begin{aligned}[l]
&(\partial_1 f)(x, t)=f_x(x,t),\\
&(X_1f)(x,t)=xf(x,t),
\end{aligned}
\qquad
\begin{aligned}[l]
&(\partial_2 f)(x, t)=f_t(x,t),\\
&(X_2f)(x,t)=tf(x,t)
\end{aligned}
\end{equation}
so that the $\mathcal{O}_{dd}$ becomes an $\mathcal{A}_2$-module, and the map itself is clearly well-defined. It turns out that the system of PDEs \eqref{E:2_more_elements} become, when $\nu=0$, the following system of PDEs
\begin{equation}\label{E:PDE_gf_bessel_0}
\begin{split}
&y_x(x,t)+\big(1/t-t\big)/2\, y(x,t)=0,\\
&ty_t(x,t)-x\,\big(1/t+t\big)/2\, y(x,t)=0,
\end{split}
\end{equation}via an $A_2$-linear map, and that both sides of the \eqref{E:bessel_classical_gf_0} satisfy. Indeed, the $e^{\frac{x}{2}\big(t-\frac1t\big)}$ is an image of \eqref{E:bessel_exp} via $\mathcal{A}_2/(\mathcal{A}_2\partial_1+\mathcal{A}_2\partial_2)\to \mathcal{O}_{dd}$.
On the other hand, we may equip the $\mathcal{O}^{\mathbb{Z}}$ consisting of sequences of analytic functions $(f_n)_{-\infty}^\infty$ all defined on an appropriate domain with a $D$-module structure by the \textit{manifestation} \begin{equation}
\begin{aligned}[l]\label{E:manifestation_2}
&(\partial_1 f)_n(x)=f^\prime_n(x),\\
&(X_1 f)_n(x)=xf_n(x),
\end{aligned}
\qquad
\begin{aligned}[l]
&(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),\\
&(X_2 f)_n(x)=f_{n-1}(x).
\end{aligned} \end{equation}It is easily verified that the $\mathcal{O}^{\mathbb{Z}}$ becomes an $\mathcal{A}_2$-module which is denoted by $\mathcal{O}^{\mathbb{Z}}_d$, so that right multiplication by $(J_n)$ shows that the map $\mathfrak{j}: \mathcal{B}_0\to \mathcal{O}^{\mathbb{Z}}_d$ is also $\mathcal{A}_2$-linear. Moreover, we have the systems\footnote{According to Watson \cite[p. 18, p. 45]{Watson1944} these formulae were discovered by Bessel in 1824 and later the \eqref{E:2_PDE_d_delta}by Lommel in 1868 \cite{Lommel} for general $\nu\not=0$.}
\begin{equation}\label{E:2_PDE_d_delta_0}
\begin{split}
& xJ'_{n}(x)-n J_{n}(x)=-xJ_{n+1}(x),
\\
& xJ'_{n}(x)+n J_{n}(x)=xJ_{n-1}(x),
\end{split}
\end{equation}that lie in the kernel of the $\mathcal{A}_2$-linear map $\mathfrak{j}: \mathcal{B}_0\to \mathcal{O}^{\mathbb{Z}}_d$, which are special cases of the \eqref{E:2_PDE_d_delta} when $\nu=0$. Then following diagram commutes
\begin{equation}\label{E:commute-0}
\begin{tikzcd} [row sep=large, column sep=huge]
\mathcal{B}_{0} \arrow{r}{\mathfrak{j}}
\arrow[swap]{d}{\times {\mathop{\rm E}} [\frac{X_1}{2}(X_2-\frac{1}{X_2})]}
& \mathcal{O}^{\mathbb{Z}}_d \arrow{d}{\mathfrak{z}}\\
\widetilde{\mathcal{A}}_2 \arrow{r}{\times 1} & \mathcal{O}_{dd},
\end{tikzcd}
\end{equation}where $\widetilde{\mathcal{A}}_2:= \overline{\mathcal{A}_2/\big[\mathcal{A}_2\partial_1+\mathcal{A}_2\partial_2\big]}$ is the completion of $\mathcal{A}_2/\big[\mathcal{A}_2\partial_1+\mathcal{A}_2\partial_2]$ and where the map $\mathfrak{z}$ is the $z$-transform, from which the classical formula \eqref{E:bessel_classical_gf_0} essentially follows because of the holonomicity of $\mathcal{B}_0$. A more general consideration gives the formula \eqref{E:bessel_classical_gf_1} for $\mathcal{B}_\nu$ (Theorem \ref{T:Bessel_gf}).
The above description illustrates how the $D$-module approach allows us to see each generating function is a solution for the different holonomic $D$-modules studied (see the table in Appendix \ref{SS:holo_modules}). They are obtained by solving the systems of holonomic PDEs from, for examples, the \eqref{E:PDE_gf_dBessel}, \eqref{E:PDE_gf_bessel_0}, \eqref{E:PDE_gf_dBessel_0}, \eqref{E:PDeltaE_neg_glaisher}, \eqref{E:delta_rev_bessel_pde} which are different manifestations inherited from the \textit{same} holonomic modules listed in Appendix \ref{SS:holo_modules}.
Before we discuss the system \eqref{E:PDE_gf_dBessel}, let us introduce the \textit{difference Bessel function} of order $\nu$ recently discovered by Bohner and Cuthta \cite{BC_2017}
\begin{equation}\label{E:dBessel}
J^{\Delta}_{\nu}(x):=\sum_{k=0}^{\infty} \frac{(-1)^k}{2^{\nu+2k}k!\Gamma(\nu+k+1)}(x)_{\nu+2k},
\end{equation}where $(x)_\nu=\Gamma(x+1)/\Gamma(x+1-\nu)$ for an arbitrary $\nu$ from our $D$-module perspective. Bohner and Cuthta \cite{BC_2017} shows that the Newton series solves the \textit{difference Bessel equation}
\begin{equation}\label{E:difference_bessel_eqn_0}
x(x-1)\triangle^2 y(x-2)+x\triangle y(x-1)+x(x-1)y(x-2)-\nu^2y(x)=0.
\end{equation} The first two authors and Tsang have shown in \cite{CCT3} that the function \eqref{E:dBessel} and the equation \eqref{E:difference_bessel_eqn_0} are manifestations of the Weyl-algebraic Bessel \eqref{E:pos_nu_map} and the Weyl-Bessel operator
\[
(X\partial)^2+ (X^2-\nu^2)
\] in the analytic function space $\mathcal{O}_\Delta$, where the subscript $\Delta$ indicates that the $\partial$ in $\mathbb{C}\langle\partial, X\rangle$ subjects to $[\partial, X]=\partial X-X\partial=1$, that is in $\mathcal{A}_1$, is interpreted as a forward difference operator instead, that is, $\partial f(x)=\Delta f(x)=f(x+1)-f(x)$ and $Xf(x)=xf(x-1)$ (see Example \ref{Eg:delta_Bessel_fn}.). Indeed, one recovers the classical Bessel equation
\begin{equation}\label{E:Bessel_eqn}
x^2y^{\prime\prime}(x)+xy^\prime(x)+(x^2-\nu^2)y(x)=0.
\end{equation}and classical Bessel function with the well-known manifestation $\partial f(x)= f^\prime (x),\ Xf(x)=xf(x)$.
Historically, both Sonine \cite{Sonine} and Nielsen \cite{Nielson} (see also \cite[\S3.9]{Watson1944}) amongst others from the second half of nineteenth century studied Bessel functions from the system of partial differential equations \eqref{E:2_PDE_d_delta} instead of \eqref{E:Bessel_eqn}. Indeed, Nielson distinguished the solutions to the system \eqref{E:2_PDE_d_delta} called \textit{cylindrical functions} against those from the \eqref{E:Bessel_eqn}. These earlier studies were in line with Truesdell's ``$F$-equation theory" \cite{Truesdell_1948} as noted by Truesdell himself, that we have already mentioned.
An advantage of our $D$-modules approach to the generating function \eqref{E:bessel_classical_gf_1} is that we could modify the $\mathcal{A}_2$-module structures \eqref{E:manifestation_1} and \eqref{E:manifestation_2}, which represent one such \textit{manifestation} that we have mentioned above, into different ones, say to
\begin{equation}\label{E:O_delta_d_0}
(\partial_1f)(x,t)=\Delta_x f(x, t)=f(x+1,t)-f(x,t),\quad
(X_1f)(x,t)=xf(x-1,t)
\end{equation}
and
\[
(\partial_1f)_n(x,t)=\Delta_x f_n(x, t)=f_n(x+1,t)-f_n(x,t),\quad
(X_1f)_n(x,t)=xf_n(x-1,t)
\]
on $\mathcal{O}_{\Delta d}$ and $\mathcal{O}^{\mathbb{Z}}_\Delta$ respectively,
while keeping the defining properties of the corresponding $\partial_2,\, X_2$ in \eqref{E:manifestation_1} and \eqref{E:manifestation_2} respectively unchanged. As a result, we arrive at the commutative diagram
\begin{equation*}
\begin{tikzcd} [row sep=huge, column sep=huge]
\mathcal{B}_0 \arrow{r}{\mathfrak{j}_\Delta}
\arrow[swap]{d}{\times{\mathop{\rm E}}\big[\frac{X_1}{2}\big(X_2-\frac{1}{X_2}\big)\big]}
& \mathcal{O}^{\mathbb{Z}}_\Delta
\arrow{d}{\mathfrak{z}_\Delta} \\
\widetilde{A}_2 \arrow{r}{\times 1}& \mathcal{O}_{\Delta d}
\end{tikzcd}
\end{equation*} similar to the \eqref{E:commute-0} for this new $\mathcal{A}_2$-module manifestation, which implies the new generating function
\begin{equation}\label{gf-db-01}
\Big[\frac{1}{2}(t-\frac{1}{t})+1\Big]^{x}=\sum_{n=-\infty}^{\infty}
J^{\vartriangle}_{n}(x)\, t^n
\end{equation}
when $\nu=0$. The full statements will be given in Theorem \ref{T:delta_bessel_gf} and Theorem \ref{T:delta_bessel_0_gf} respectively. Here, both sides of \eqref{gf-db-02} satisfy the system of PDEs
\begin{equation}\label{E:PDE_gf_dBessel_0}
\begin{split}
&y(x+1, t)-y(x, t)+(1/t-t)/2 y(x, t)=0,\\
&ty_t(x,t)-x(1/t+ t)/2y(x-1, t)=0
\end{split}
\end{equation}as images of the \eqref{E:2_more_elements}, with $\nu=0$, which are the difference analogues of those in \eqref{E:PDE_gf_bessel_0} in our $D$-modules interpretation. They are the special cases of \eqref{E:PDE_gf_dBessel} when $\nu=0$. The $(J^\triangle_{n}(x))$ in \eqref{gf-db-01} are precisely the difference Bessel functions $(J^\triangle_{\nu+n}(x))$ discovered by Bohner and Cuchta \cite{BC_2017}. The reason that we can derive the infinite sum representation of the generating function on the right-hand side of \eqref{gf-db-01} is because of the following formulae
\begin{align*}
x\Delta J^{\Delta}_{n}(x-1)+n J^{\Delta}_{n}(x)-
xJ^{\Delta}_{n-1}(x-1)&=0,
\\
x\Delta J^{\Delta}_{n}(x-1)-\nu J^{\Delta}_{n}(x)+
xJ^{\Delta}_{n+1}(x-1)&=0,
\end{align*}derived by brute-force verification in \cite{BC_2017} are manifestation of the \eqref{E:2_elements} in the kernel of the $\mathcal{A}_2$-linear map $\mathcal{B}_0\to \mathcal{O}^{\mathbb{Z}}_\Delta$ as similar to the case of the \eqref{E:2_PDE_d_delta}.
If we modify the the above commutator to arbitrary step size $h$, i.e., $\partial f(x)=\triangle_h f(x)=\frac1h\big(f(x+h)-f(x)\big)$ and $Xf(x)=xf(x-h)$, then the generating function \eqref{gf-db-01} is replaced by
\begin{equation}\label{E:gf-db_h-01}
\Big[\frac{h}{2}(t-\frac{1}{t})+1\Big]^{x/h}
=\sum_{n=-\infty}^{\infty}
J^{\vartriangle_h}_{n}(x)\, t^n.
\end{equation}Clearly, we recover the classical formula \eqref{E:bessel_classical_gf_0} from the \eqref{E:gf-db_h-01} after letting $h\to 0$ on both sides of the above formula. See Theorem \ref{T:h_limit} in the Appendix A for more details. By a similar consideration, the formulae \eqref{E:2_PDE_d_delta} become
\begin{equation}\label{P:delta_bessel_PDE}
\begin{split}
&x\triangle \mathscr{C}^{\Delta}_{\nu+n}(x-1)+(\nu+n) \mathscr{C} ^{\Delta}_{\nu+n}(x)-
x \mathscr{C} ^{\Delta}_{\nu+n-1}(x-1)=0,
\\
&x\triangle \mathscr{C} ^{\Delta}_{\nu+n}(x-1)-(\nu+n) \mathscr{C} ^{\Delta}_{\nu+n}(x)+
x \mathscr{C}^{\Delta}_{\nu+n+1}(x-1)=0. \\
\end{split}
\end{equation}we the difference manifestation of $\mathcal{A}_2$. If we choose $\mathscr{C}_\nu(x)=J^\Delta_\nu(x)$ in the above two formulae for $J^\Delta_\nu(x)$, then we recover the two corresponding formulae that Bohner and Cuthta derived in \cite{BC_2017} . Combining the PDEs \eqref{E:PDE_gf_dBessel} and \eqref{P:delta_bessel_PDE} and a commutative diagram similar to the last commutative diagram above implies the new generating function \eqref{gf-db-02}.
The difference Bessel functions are not discussed in Nikifornov \textit{et al}. We mention that the difference Bessel functions have not been discussed in recent major works on difference special functions such as \cite{Nikiforov_Suslov_1986} and \cite{Nikiforov_1991}, which mostly focus on orthogonal polynomials.
Before we discuss similar formulae that we have obtained for the new difference analogues of Bessel polynomials and Glaisher's trigonometric type generating functions below, let us first return to the discussion mentioned in the first paragraph about the aspect of our Weyl-algebraic methodology in \textit{solving}\footnote{See Definition \ref{D:soln}.} the two ``PDEs\rq\rq{} \eqref{E:2_elements} in the context of the Bessel module $\mathcal{B}_\nu$. In fact, the process of a priori computation for the ``generating functions" greatly simplifies the entire process of finding the generating function that are classically written within suitably defined analytic function spaces. This a priori computation is therefore \textit{no less significant} than the conventional approach done in a complex domain.
As a by-product of \eqref{gf-db-02} of ``extracting the residue" from the formula \eqref{gf-db-02}, we obtain the integral representation
\[
J^{\Delta}_{\nu}(x)=e^{i\pi\nu} \dfrac{\sin(x-\nu)\pi}{\sin(\pi x)}
\dfrac{1}{2\pi i}\int_\infty^{(0+)}
t^{-\nu-1} \Big[ \frac{1}{2}(t-\frac{1}{t})+1\Big]^x dt,\quad \mathop{\rm Re}(x)<\mathop{\rm Re}(\nu)
\]
where the integration path is a Hankel-type contour (Theorem \ref{T:integral rep of db}). This integral representation is the difference analogue of the classical Schl\"afli-Sonine integral of Corollary \ref{C:Sonine} \cite[p. 176]{Watson1944}. Thus, the classical Schl\"afli-Sonine integral can be considered as ``residue extraction" from the formula \eqref{E:bessel_classical_gf_1}.
The rest of this paper will focus on two main variants of the Bessel module of order half $\mathcal{B}_{1/2}$. They are related to classical Glaisher's trigonometric type generating functions and the classical Bessel polynomials. As a result we have also obtained their difference analogues. The solving of these variants of $\mathcal{B}_{1/2}$ (Theorem \ref{T:bessel_poly_gf_map}, Theorem \ref{T:bessel_poly_delta_gf}, Theorem \ref{T:neg_glaisher_cosine}, and Theorem \ref{T:pos_glaisher_sine}) are more complicated in the sense that certain \textit{quadratic extensions} on the base Weyl-algebra are needed.
\subsection{Glaisher's modules}
The treatment of Glaisher's Poisson type generating functions \cite[p. 140]{Watson1944} \[
\sqrt{\frac{2}{\pi x}} \cos\sqrt{x^2-2xt}
=\sum_{n=0}^\infty J_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!},
\] which is valid for $2|t|<|x|$, requires a change of characteristics variables
to \eqref{E:2_elements} with $\nu=-\frac12$ different from that for obtaining the \eqref{E:bessel_exp}, so that the alternative $\mathcal{A}_2$-module, called the negative Glaisher's module is given by
\[
\mathcal{G}_{-}=
\dfrac{\mathcal{A}_{2}}
{\mathcal{A}_{2}
\big(W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})- W_1\Xi_2^{-1}\big)
+\mathcal{A}_{2}\big(W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})+ W_1 \Xi_2\big)},
\]in which the $\Xi_1, W_1$ and $\Xi_2,\, W_2$ play the roles of $\partial_1,\, X_1$ and $\partial_2,\, X_2$ respectively, as defined in \eqref{E:comm}.
As an example, after a ``further change of variables" and solving the resulting PDEs finally results in the following difference analogue of Glaisher's generating functions for difference Bessel functions with half-integer orders (Theorem \ref{T:Delta_glaisher_gf}):
\[
\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{-\infty}^{(0+)} \frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\cos\sqrt{\lambda ^2-2\lambda t}}{\Gamma(-x)}\, d\lambda
=\sum_{n=0}^{\infty}J^{\Delta}_{n-\frac{1}{2}}(x)\frac{t^n}{n!},
\] valid on the whole $\mathbb{C}\times\mathbb{C}$. In particular, the generating function is a solution to the system of delay-differential equations\footnote{These equations are generally called PDEs in a generalised sense in this paper.} (Theorem \ref{T:difference_glaisher_gf})
\begin{equation}\label{E:PDeltaE_neg_glaisher}
\begin{split}
& tf_{tt}(x,\, t)+\big(x+1/2\big)f_t(x,\, t)-xf_t(x-1,\, t)-xf(x-1,\, t)=0,\\
& tf_{t}(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+1/2\big) f(x,\, t)=0.
\end{split}
\end{equation}
\subsection{Bessel polynomial modules} Next we consider half Bessel-module $\mathcal{B}_{1/2}$ which is more natural to treat the classical reverse polynomials $\theta_n(x)$ instead of the $y_n(x)=x^n\theta_n(1/x)$. If $X\partial$ is replaced by $X\partial-X-1/2$ in \eqref{E:2_elements}, then the $\mathcal{B}_{1/2}$ is replaced by the \textit{reverse Bessel polynomial module}
\[
\Theta=\dfrac{\mathcal{A}_2}{\mathcal{A}_2[\partial_1-1+X_1/\partial_2]+\mathcal{A}_2[X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2]}.
\]such that the two generators $\partial_1-1+X_1/\partial_2$ and $X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2$ would generate a third element
$X_1\partial_1^2-2(X_1+X_2\partial_2)\partial_1+2X_2\partial_2$ which is a Weyl-algebraic analogue of the classical Bessel polynomial equation. In particular, we have derived the difference equation (Theorem \ref{T:difference_Bessel_Poly_DD})
\begin{equation}
(x-2n)\,\theta_n^\Delta(x+1)-4(x-n)\,\theta_n^\Delta(x)+3x\, \theta_n^\Delta (x-1)=0,
\end{equation} which is satisfied by \textit{difference reverse Bessel (Newton) polynomial of degree} $n$
\[
\theta^\Delta_n(x) :=\sum_{k=0}^n \frac{(n+k)!}{2^k (n-k)!\, k!}\, (x)_{n-k}.
\]
In a similar fashion, the two generators $\partial_1-1+X_1/\partial_2$ and $X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2$ implies two delay-difference equations for difference reversed Bessel polynomials for each $n$ (Theorem \ref{T:difference_Bessel_Poly_PDE}):
\[
\theta^\Delta_n(x+1)-2\theta^\Delta_n(x)+x\theta^\Delta_{n-1}(x-1)=0,
\]
and
\[
\theta^\Delta_{n+1}(x)+(x-2n-1)\theta^\Delta_n(x)-2x\, \theta^\Delta_n(x-1)=0.
\] A ``characteristic change of variables" to the module $\Theta$ happens to be $\rho^2=1-2X_2$ and we formulate a ``quadratic extension of $\Theta$" in Theorem \ref{T:rev_bessel_poly_gf}
\[
\Theta(\rho):=\dfrac{\mathcal{A}_2(\rho)}{\mathcal{A}_2(\rho)[-\rho^2\partial_2^2
+3\partial_2+X_1^2]
+\mathcal{A}_2(\rho)[X_1\partial_1+\rho^2\partial_2-1-X_1]}.
\]so that the commutative diagram of $\mathcal{A}_2$-linear maps enable us to derive a generating function for the difference reverse Bessel polynomials
\[
\frac{e^{-i\pi x}}{2\pi i\sin\pi x\Gamma(-x)}\int_{-\infty}^{(0+)}e^\lambda(-\lambda)^{-x-1} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda=\sum_{n=0}^\infty
\theta_n^\Delta (x)\, \frac{t^n}{n!}
\]that is valid in $\mathbb{C}\times B(0,\, \frac12)$ in Theorem \ref{T:bessel_poly_delta_gf}. The generating function for the difference reverse Bessel polynomials satisfies the system of delay-differential equations (Theorem \ref{E:delta_bessel_poly_PDE})
\begin{equation}\label{E:delta_rev_bessel_pde}
\begin{split}
&f_t(x+1,\, t)-2f_t(x,\, t)+xf(x-1,\, t)=0,\\
&(1-2t) f_t(x,t)+(x-1)f(x, t)-2xf(x-1, t)=0.
\end{split}
\end{equation}
This paper is organised as follows. Section \ref{S:preliminaries} explains some examples of
$D$-modules structures on certain spaces of analytic functions and spaces of sequences of analytic functions $\mathcal{O}^{\mathbb{Z}}$, as well as $D$-linear maps between them. A list summarising these entities is given in Appendix \ref{SS:holo_modules}. The remaining parts of the \S\ref{S:preliminaries} gives a rudiment of holonomic $D$-modules theory needed in this article. A kind of generalised Borel resummation is also introduced. \S\ref{S:bessel} introduces the Bessel modules $\mathcal{B}_\nu$ for arbitrary $\nu$ and the method of characteristic in solving a system of PDEs. Before discussing about the Bessel modules, we also discuss analogues of ``elementary functions" such as the Weyl-exponentials, Weyl-trigonometry, and Weyl-Bessel.
This section also contains the statements and proofs of generating functions for the classical Bessel functions and difference Bessel functions, plus some immediate applications of these results to yield difference analogue of the classical Schl\"afli-Sonine integral representation from classical Bessel functions to difference Bessel functions. In section \S\ref{S:half_bessel_I} we modify the Bessel module $\mathcal{B}_{1/2}$ to the ``\textit{reverse Bessel module}" $\Theta$ by the introducing an integrating factor. We work out a characteristic for $\Theta$ as well as a variant $\mathcal{Y}$ corresponding to the usual Bessel polynomials $y_n$. These characteristics will be applied to find (Poisson type) generating functions in due course. As applications of the $D$-modules approach, we derive, in a \textit{uniform way}, the well-known three-term recursion formulae for Bessel polynomials as well as the corresponding ones for the difference reverse Bessel polynomials. Section \S\ref{S:Newton} introduces the \textit{Newton transform},
a kind of generalised Mellin transform but with very different purpose as a $D$-linear map from $\mathcal{O}_d$ to $\mathcal{O}_\Delta$, namely the $D$-modules of analytic functions endowed with a forward difference operator. The Newton transformation serves as an effective tool to obtain generating functions for difference Bessel functions. Utilising results obtained from \S\ref{S:half_bessel_I}, we
give detailed proofs of generating functions for the reverse Bessel polynomials as well as their difference analogues in \S\ref{SS:delta_reverse_bessel_poly}. Section \S\ref{S:half_bessel_II} take the half-Bessel modules from \S\ref{S:half_bessel_I} a step further to study trigonometric-type generating functions for the Bessel functions first obtained by Glaisher \cite{Watson1944}. Before the concluding section \S\ref{S:conclusion}, \S\ref{S:discussion}
discusses our main results against previous results of Truesdell amongst others with some historical notes. In particular, it focus the philosophy that Truesdell pursued, and similarly of Wilf and Zeilberger, to find general methods of discovery of what kinds of formulae should exist rather than by ad hoc manipulations of formulae, a viewpoint shared by this study. Appendix \ref{A:proofs} contains proofs of various Theorems and Propositions stated but not fully verified in all aspects in the main text. Appendix \ref{Append:arbitrary} extends the study of the manifestation of the Bessel modules $\mathcal{B}_\nu$ to $\mathcal{O}_{\Delta_h}$, that is finite difference with arbitrary step size $h$. In the case of $\mathcal{B}_0$, the generating function derived unifies the case when $h=1$ for the difference Bessel functions and the classical Bessel functions when $h\to 0$. This presents a further unification of the generating functions of the classical Bessel functions and that of difference Bessel functions of arbitrary step sizes. Appendix \S\ref{A:list} gives lists of $D$-modules used, transmutation formulae, the systems of PDEs encountered for various generating functions defined in their appropriate analytic functions space of two variables.
\section{Preliminaries}\label{S:preliminaries} \subsection{Linear maps between $D$-modules}
\begin{definition}[\cite{CCT3}] Let $n\in \mathbb{N}.$ The Weyl-algebra $\mathcal{A}_n$ is the $\mathbb{C}$-algebra with $2n$ generators
$X_1, \ldots, X_n, \partial_1, \ldots, \partial_n$
subject to the relations
\[
[X_i, X_j]=0;\quad[\partial_i, \partial_j]=0; \quad[\partial_i, X_j]=\delta_{ij}.
\] \end{definition} In this paper, we denote $\mathcal{A}_1=\mathbb{C} \langle X, \partial\rangle$ or by $\mathcal{A}$ when this is clear from the context.
We gather here preliminary material about $D$-module that we shall employ to exhibit a unified theory for the classical Bessel functions and generating functions for the recently discovered difference Bessel functions by Bohner and Cuchta \cite{BC_2017} in the next sections. More specifically, amongst the generating functions studied include
\begin{itemize}
\item extensions of generating functions of various kinds for the classical Bessel functions of arbitrary order $\nu$;
\item new generating functions for difference Bessel functions \cite{BC_2017};
\item new generating functions of Glaisher (Poisson)-type for difference Bessel functions;
\item new generating functions for new difference Bessel polynomials.
\end{itemize}
The primary concern of this paper is on ${D}$-linear maps defined on the Bessel modules and their induced generating functions to be introduced in the following two sections. Hence some generating functions found for difference Bessel functions may not converge in the same way as those of their classical counterparts.
The principle of obtaining these generating functions behind is based on the finite dimensionality of the space of $D$-linear maps from $\mathcal{B}_\nu$ to different $\mathcal{A}_2$-modules. The first step is the introduction of some examples of $\mathcal{A}_n$-modules and linear maps between them which are frequently used in this article.
\begin{definition} We denote by $\mathcal{A}(1/X)$ the Weyl-algebra $\mathbb{C}\langle\partial,X\rangle (1/X)= \mathbb{C}\langle\partial,X,{1}/{X}\rangle$ modulo the aforementioned relations. \end{definition}
\begin{example}\label{Eg:O_deleted_d} Let $\mathcal{O}$ be the space of analytic functions on a certain domain. Then the $\mathcal{O}$ becomes a left $\mathcal{A}$-module denoted by $\mathcal{O}_d$, if one endows it with the structure
\[
\begin{array}{l}
(\partial f)(t)=f^\prime (t),\\
(Xf)(t)=tf(t)\quad\mbox{for all }f\in\mathcal{O}\mbox{ and }t\mbox{ in this domain}.
\end{array} \]Similarly, let $\mathcal{O}_2$ be the space of analytic functions on a certain domain in $\mathbb{C}\times \mathbb{C}$. Then the $\mathcal{O}_2$ becomes a left $\mathcal{A}_2$-module denoted by $\mathcal{O}_{dd}$, if one endows it with the structure
\begin{equation}\label{E:O_dd_endow}
\begin{split}
(\partial_1 f)(x,\, t) &=f_x (x,\, t), \quad (X_1f)(x,\, t)=xf(x,\, t),\\
(\partial_2 f)(x,\, t) &=f_t(x,\, t), \quad (X_2f)(x,\, t)=tf(x,\, t).
\end{split}
\end{equation} \end{example}
\begin{example}\label{Eg:O_delta} Let $\mathcal{O}$ be the space of analytic functions. Then the $\mathcal{O}$ becomes a left $\mathcal{A}$-module, denoted by $\mathcal{O}_\Delta$, if one endows it with the structure
\[
\begin{split}
&(\partial f)(t) =f(t+1)-f(t), \\
&(Xf)(t)=tf(t-1).
\end{split}
\] \end{example}
\begin{example}\label{Eg:two-seq-numbers} Let $\mathbb{C}^\mathbb{Z}$ be the space of all functions $\mathbb{Z}\to\mathbb{C}$ (bilateral sequences). Let
\[
a=(a_n)=(\cdots, a_{-2},\, a_{-1},\, a_0,\, a_1,\, a_2,\, \cdots)
\]denote a bilateral sequence where $a_k\in\mathbb{C}$. Then the $\mathbb{C}^\mathbb{Z}$ becomes a left $\mathcal{A}$-module by defining \[ \begin{array}{l} (\partial a)_n=(n+1)a_{n+1}\\ (Xa)_n=a_{n-1}\quad\mbox{ for all bilateral sequences }(a_n). \end{array} \] Various growth conditions can be imposed to such bilateral sequences. \end{example}
\subsubsection{$z$-transform I} \begin{theorem}[\textbf{(Generating function of numeric sequences)}]\label{T:z-transform-1}
Suppose that $\mathbb{C}^\mathbb{Z}$ consists only of sequences $\{a_n\}$ satisfying that
\[
0\le\limsup_{n\to\infty}{|a_{-n}|^{1/n}}<\frac{1}{\limsup_{n\to\infty}{|a_n|^{1/n}}}\le+\infty,
\]
and let $\mathcal{O}$ be the space of analytic functions defined on a certain annulus centred at $0$. Then the \textit{z-transform}
\[
\begin{array}{rcl}
\mathfrak{z}:\mathbb{C}^\mathbb{Z}&\longrightarrow &\mathcal{O}_d,\\
(a_n)& \longmapsto & \displaystyle\sum_{n=-\infty}^\infty a_nt^n.
\end{array}
\]
is left $\mathcal{A}$-linear. \end{theorem}
\begin{proof} The fact that the map $\mathfrak{z}$ is $\mathbb{C}$-linear is obvious. The verification that both
\[
\mathfrak{z}\, \partial (a_n)=\partial\, \mathfrak{z} (a_n),
\quad \mbox{and}\quad
\mathfrak{z}\, X (a_n)=X\, \mathfrak{z} (a_n)
\] hold are routine, with the understanding that the maps $\partial$ and $X$ are interpreted appropriately in $\mathbb{C}^\mathbb{Z}$ and $\mathcal{O}_d$ respectively. Hence the map $\mathfrak{z}$ is $\mathcal{A}$-linear. Clearly, it is both injective and surjective. \end{proof}
In fact, Theorem~\ref{T:z-transform-1} can be extended to Theorem~\ref{T:z-transform-1}$'$ below, which handles a larger class of sequences $\{a_n\}$ so that $\sum_{n=-\infty}^\infty a_nt^n$ may not converge.
\begin{definition}[\textbf{(Borel resummation)}]\label{D:borel} Suppose that $\{a_n\}\in\mathbb{C}^{\mathbb{Z}}$ satisfies both
\begin{equation}\label{E:G1}
\limsup_{n\to\infty}{|a_n|^{1/n}}<+\infty \quad \mbox{and} \quad \limsup_{n\to\infty}{\left|\frac{a_{-n}}{n!}\right|^{1/n}}<+\infty.
\end{equation}
Then the \textit{Borel resummation} of the Laurent series $\sum_{n=-\infty}^\infty {a_nt^n}$ is defined by taking the Laplace transform of the Borel transform of its principal part, i.e.,
\begin{equation}\label{E:BL}
\int_0^{+\infty}{e^{-st}\left(\sum_{n=0}^{\infty}{\frac{a_{-n-1}}{n!}}s^n\right)\,ds} + \sum_{n=0}^\infty {a_nt^n}.
\end{equation} \end{definition}
\begin{remark}\label{R:gervey}
A power series or Laurent series whose sequence of coefficients $\{a_n\}$ satisfy that \[\limsup_{n\to\infty}{\left|\frac{a_n}{n!}\right|^{1/n}}<+\infty\] is said to be \textit{$1$-Gevrey}. Note that the Borel resummation of a $1$-Gevrey Laurent series always exists as an analytic function, and the Borel resummation of a convergent Laurent series is just its ordinary sum, which is again an analytic function. We refer the reader to \cite[Part II]{Mitschi_Sauzin_2016}, \cite{Shawyer_Watson_1994} and \cite{Delabaere_Pham_1999} for general discussions of $1$-Gevrey or $m$-Gevrey series and their Borel-resummation. \end{remark}
\subsubsection{$z$-transform II} \begin{theoremprime}{T:z-transform-1}[\textbf{(Generating function of numeric sequences)}]
Suppose that $\mathbb{C}^\mathbb{Z}$ consists only of sequences $\{a_n\}$ satisfying both
\[
\limsup_{n\to\infty}{|a_n|^{1/n}}<+\infty \quad \mbox{and} \quad \limsup_{n\to\infty}{\left|\frac{a_{-n}}{n!}\right|^{1/n}}<+\infty,
\]
and let $\mathcal{O}$ be the space of analytic functions defined on a certain region. Then the \textit{z-transform}
\[
\begin{array}{rcl}
\mathfrak{z}:\mathbb{C}^\mathbb{Z}&\longrightarrow &\mathcal{O}_d,\\
(a_n)& \longmapsto & \mathfrak{B} \displaystyle\sum_{n=-\infty}^\infty a _nt^n,
\end{array}
\]where the $\mathfrak{B} \sum_{n=-\infty}^\infty a _nt^n$ denotes the Borel-resummation of $\sum_{n=-\infty}^\infty a _nt^n$, is left $\mathcal{A}$-linear. \end{theoremprime}
\begin{proof} Since the Borel and Laplace transforms are both left $\mathcal{A}$-linear, the Borel resummation is left $\mathcal{A}$-linear also. \end{proof}
\begin{example}\label{Eg:O_delta_d} Let $\mathcal{O}_2$ be a space of analytic functions in two variables. Then $\mathcal{O}_2$ becomes a left $\mathcal{A}_2$-module by
\begin{equation}\label{E:O_delta_d}
\begin{array}{ll}
\partial_1f(x,\, t)=f(x+1,\, t)-f(x,\, t), & X_1f(x,\, t)=xf(x-1,\, t);\\
\partial_2f(x,\, t) =f_t(x,\, t), & X_2 f(x,\, t)=tf(x,\, t).
\end{array}
\end{equation} Such a left $\mathcal{A}_2$-module is denoted by $\mathcal{O}_{\Delta d}$. \end{example}
\begin{remark} In addition to the above notation $\mathcal{O}_{\Delta d}$, we shall liberally adopt notations such as $\mathcal{O}_{d d}$, $\mathcal{O}_{d \Delta}$, $\mathcal{O}_{\Delta\Delta}$ to represent different left $\mathcal{A}_2$-module structures endowed on $\mathcal{O}_2$ with respect to the corresponding operators. \end{remark}
We next extend the idea of space of bilateral sequences of numbers $\mathbb{C}^\mathbb{Z}$ in Example \ref{Eg:two-seq-numbers} to the space of bilateral sequences of analytic functions $\mathcal{O}^\mathbb{Z}$.
\begin{example}\label{Eg:two-seq-functions} Let $\mathcal{O}^\mathbb{Z}$ be the space of all bilateral sequences of analytic functions with appropriately imposed growth restriction. Let
\[
(f_n)=(\cdots, \, f_{-1},\, f_0,\, f_1,\, \cdots)
\]denote a bilateral infinite sequence of analytic functions. Then $\mathcal{O}^\mathbb{Z}$ becomes a left $\mathcal{A}_2$-module by
\[
\begin{array}{ll}
(\partial_1 f)_n(x)=f^\prime_n(x), & (X_1 f)_n(x)=xf_n(x)\\
(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),& (X_2 f)_n(x)=f_{n-1}(x),
\end{array}
\]for all $n$ and for all $x$. This left $\mathcal{A}_2$-module is denoted by $\mathcal{O}_d^\mathbb{Z}$. \end{example}
\begin{example}\label{Eg:two-seq-functions_2} Let $\mathcal{O}_\Delta^\mathbb{Z}$ be the space of all bilateral sequences of analytic functions with an appropriately imposed growth restriction. Let
\[
(f_n)=(\cdots, \, f_{-1},\, f_0,\, f_1,\, \cdots)
\]denote a bilateral infinite sequence of $f_k\in\mathcal{O}$. Then $\mathcal{O}^\mathbb{Z}$ becomes a left $\mathcal{A}_2$-module, by
\begin{equation}\label{E:two-seq-functions_2}
\begin{array}{ll}
(\partial_1 f)_n(x)=\Delta f_n(x)=f_n(x+1)-f_n(x), & (X_1 f)_n(x)=xf_n(x-1)\\
(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),& (X_2 f)_n(x)=f_{n-1}(x),
\end{array}
\end{equation}for all bilateral sequences of analytic functions $(f_n)$. This left $\mathcal{A}_2$-module is denoted by $\mathcal{O}_\Delta^\mathbb{Z}$. \end{example}
Similar to Theorem \ref{T:z-transform-1}, we have the following:
\begin{theorem}[\textbf{(Generating function of sequences in $\mathcal{O}_d$)}] \label{T:z-transform-2} The following \textit{z-transform}
\begin{equation}\label{E:z-transform-2}
\begin{array}{rcl}
\mathfrak{z}:\mathcal{O}^\mathbb{Z}_d &\longrightarrow &\mathcal{O}_{dd},\\
(f_n) &\longmapsto&\mathfrak{B}\displaystyle\sum_{n=-\infty}^\infty f_n(x)\, t^n
\end{array}
\end{equation}is left $\mathcal{A}_2$-linear. \end{theorem}
\begin{theorem}[\textbf{(Generating function of sequences in $\mathcal{O}_\Delta$)}]\label{T:z-transform-3} The following \textit{z-transform}
\begin{equation}\label{E:z-transform-3}
\begin{array}{rcl}
\mathfrak{z}_\triangle:\mathcal{O}^\mathbb{Z}_\Delta &\longrightarrow &\mathcal{O}_{\Delta d},\\
(f_n) &\longmapsto&\mathfrak{B}\displaystyle\sum_{n=-\infty}^\infty f_n(x)\, t^n
\end{array}
\end{equation}is left $\mathcal{A}_2$-linear. \end{theorem}
One needs to make a slight modification of defining rules for $\mathcal{A}$-module in order to study one-sided (ordinary) sequences. The motivation of this rule will be explained in Remark~\ref{R:poisson} after we introduce the Poisson transform.
\begin{example}\label{Eg:seq-functions_poisson} Let $\mathbb{C}^\mathbb{N}$ be the space of sequences
\[
a=(a_n)=(a_1,\, a_2,\, a_3,\, \cdots)
\] of complex numbers. It is equipped with the structure of a left $\mathcal{A}$-module by defining \[ \begin{array}{l} (Xa)_n=na_{n-1}\\ (\partial a)_n=a_{n+1}\quad\mbox{for all sequences }(a_n). \end{array} \] Similarly, let $\mathcal{O}^{\mathbb{N}_0}$ be the space of all sequences of analytic functions with appropriately imposed growth restriction. Let
\[
(f_n)=(f_{0},\, f_1,\, f_2,\, \cdots)
\]denote an infinite sequence of $f_k\in\mathcal{O}$. Then $\mathcal{O}^{\mathbb{N}_0}$ becomes a left $\mathcal{A}_2$-module by
\begin{equation}\label{E:seq-functions_poisson}
\begin{array}{ll}
(\partial_1 f)_n(x)=f^\prime_n(x), & (X_1 f)_n(x)=xf_n(x)\\
(\partial_2 f)_n(x)=f_{n+1}(x),& (X_2 f)_n(x)=nf_{n-1}(x),
\end{array}
\end{equation}for all sequences of analytic functions $(f_n)$. This left $\mathcal{A}_2$-module is usually denoted as $\mathcal{O}_d^{\mathbb{N}_0}$. \end{example}
\begin{example}\label{Eg:seq-functions_poisson_delta}
Let $\mathcal{O}^{\mathbb{N}_0}$ be the space of all sequences of analytic functions with appropriately imposed growth restriction. Let
\[
(f_n)=(f_0,\, f_1,\, f_2,\, \cdots)
\]denote an infinite sequence of $f_k\in\mathcal{O}$. Then $\mathcal{O}^{\mathbb{N}_0}$ becomes a left $\mathcal{A}_2$-module by
\begin{equation}\label{E:seq-functions_poisson_delta}
\begin{array}{ll}
(\partial_1 f)_n(x)=f_n(x+1)-f_n(x), & (X_1 f)_n(x)=xf_n(x-1)\\
(\partial_2 f)_n(x)=f_{n+1}(x),& (X_2 f)_n(x)=nf_{n-1}(x),
\end{array}
\end{equation}for all sequences of analytic functions $(f_n)$. This left $\mathcal{A}_2$-module is usually denoted as $\mathcal{O}_\Delta^{\mathbb{N}_0}$. \end{example}
We first demonstrate how we use the theory set up so far to derive a classical generating function for Bessel functions from the Bessel module $\mathcal{B}_\nu$ defined in Definition \ref{E:bessel_mod_gf} which is isomorphic to Definition
\ref{E:bessel_mod}.
A table summarising all the $D$-module structures relevant to this article is included in the Appendix \ref{SS:common_D_list}.
\subsection{Integrability criteria}\label{SS:integrable}
This section deals with the idea of holonomic modules, which can be found in \cite{coutinho}. A quick revision is included here so that this article is self-contained. For each multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{N}^n$, $X_1^{\alpha_1}\cdots X_n^{\alpha_n}$ will be abbreviated as $X^\alpha$. A similar abbreviation is adopted for the $\partial$'s. It is also a standard notation that $|\alpha|=|(\alpha_1,\ldots,\alpha_n)|=\sum_j\alpha_j$.
\begin{definition} The \textit{Bernstein filtration} of $\mathcal{A}_n$ is the filtration $F^0\mathcal{A}_n\subset F^1\mathcal{A}_1\subset\cdots\subset\mathcal{A}_n$ defined by \[
F^k\mathcal{A}_n=\mbox{the }\mathbb{C}\mbox{-vector space spanned by }\{X^\alpha\partial^\beta:|\alpha|+|\beta|\leq k\}. \] \end{definition}
\begin{remark} The Bernstein filtration enjoys the following properties. \begin{itemize} \item $F^i\mathcal{A}_n\cdot F^j\mathcal{A}_n\subset F^{i+j}\mathcal{A}_n$, \item $[F^i\mathcal{A}_n,F^j\mathcal{A}_n]\subset F^{i+j-1}\mathcal{A}_n$, \item the $k^{\footnotesize\mbox{th}}-$graded piece $\mathrm{Gr}^k\mathcal{A}_n=F^k\mathcal{A}_n/F^{k+1}\mathcal{A}_n$ has finite $\mathbb{C}$-dimension for all $k$. \end{itemize} In fact, in the following treatment, the Bernstein filtration can be replaced by any other filtration having the these properties. \end{remark}
\begin{lemma} The graded ring of the Bernstein filtration is a commutative $\mathbb{C}$-algebra. Indeed, \[ \mathrm{Gr}\,\mathcal{A}_n=\bigoplus_k\mathrm{Gr}^k\mathcal{A}_n\cong\mathbb{C}[X_1,\ldots,X_n,\partial_1,\ldots,\partial_n]. \] \end{lemma}
\begin{definition} Let $M$ be a left $\mathcal{A}_n$-module. A \textit{good filtration} of $M$ is a filtration $\Gamma$ which satisfies \begin{itemize} \item $F^i\mathcal{A}_n\cdot \Gamma^jM\subset\Gamma^{i+j}M$, \item there exists $J\in\mathbb{N}$ such that $F^i\mathcal{A}_n\cdot \Gamma^jM=\Gamma^{i+j}M$ for all $i$, $j>J$. \end{itemize} \end{definition}
\begin{lemma} Let $M$ be a finitely generated left $\mathcal{A}_n$-module and $\Gamma$ be a good filtration of $M$. Then the graded ring of $\Gamma$, that is, \[ \mathrm{Gr}\,M=\bigoplus_k{\Gamma^kM/\Gamma^{k+1}M}, \] is a commutative algebra over the commutative ring $\mathrm{Gr}\,\mathcal{A}_n$. \end{lemma}
\begin{theorem}[(Hilbert)] Let $M$ be a finitely generated left $\mathcal{A}_n$-module equipped with a good filtration $\Gamma$. Then there exists a polynomial $P$ such that
\[
P(k)=\dim_{\mathbb{C}}\Gamma^kM\quad\mbox{for every sufficiently large }k.
\] \end{theorem}
\begin{definition}\label{D:hilbert} The polynomial $P$ in the last theorem is called the \textit{Hilbert polynomial} of $M$. (It turns out that $P$ is independent of the choice of the good filtration chosen.). If $P$ is of degree $d$ and its leading coefficient is $m/d!$, then we call $d$ the \textit{dimension} of $M$ and $m$ the \textit{multiplicity} of $M$. \end{definition}
\begin{example} Consider $\mathcal{A}_n$ as a left $\mathcal{A}_n$-module and the Bernstein filtration is a good one. Then, \[ \dim_{\mathbb{C}}\Gamma^k\mathcal{A}_n=\binom{k+2n}{2n} \] so that the dimension of $\mathcal{A}_n$ is $2n$ and its multiplicity is $1$. \end{example}
\begin{example}\label{E:prime_integrable_eg} Consider \[ M=\mathcal{A}_n/\mathcal{A}_n\partial_1+\cdots+\mathcal{A}_n\partial_n, \] which inherits a good filtration $\Gamma$ from the Bernstein filtration. Then, \[ \dim_{\mathbb{C}}\Gamma^kM=\binom{k+n}{n} \] so that the dimension of $M$ is $n$ and its multiplicity is $1$. \end{example}
\begin{example}\label{E:prime_integrable_eg_2} Consider \[ M=\mathcal{A}_2/\mathcal{A}_2\partial_1^2+\mathcal{A}_2\partial_2, \] which inherits a good filtration $\Gamma$ from the Bernstein filtration. Then, \[ \dim_{\mathbb{C}}\Gamma^kM=(k+1)^2 \] so that the dimension of $M$ and its multiplicity are both equal to $2$. \end{example}
\begin{theorem}[(\textbf{Bernstein inequality})]\label{T:bernstein} For each non-trivial finitely generated left $\mathcal{A}_n$-module $M$, its dimension $d(M)$ satisfies \[ n\leq d(M)\leq 2n. \] \end{theorem}
\begin{definition} A finitely generated left $\mathcal{A}_n$-module is \textit{holonomic} if it is nontrivial and its dimension is $n$. \end{definition}
\begin{example} $\mathcal{A}_n$ is not a holonomic left $\mathcal{A}_n$-module. However, $M=\mathcal{A}_n/(\mathcal{A}_n\partial_1+\cdots+\mathcal{A}_n\partial_n)$ is a holonomic left $\mathcal{A}_n$-module. \end{example}
\begin{example}[(\textbf{Integrability})] \label{EG:integrable} Let $p(X_1,X_2)$, $q(X_1,X_2)\in\mathbb{C}[X_1,X_2]$ be polynomials in two variables. Denote their formal partial derivatives by subscripts. Consider the left $\mathcal{A}_2$-module \[ M=\dfrac{\mathcal{A}_2}{\mathcal{A}_2(\partial_1+p(X_1,X_2))+\mathcal{A}_2(\partial_2+q(X_1,X_2))}. \] \begin{itemize} \item If $p_2(X_1,X_2)\neq q_1(X_1,X_2)$, then $M$ is trivial and is therefore not holonomic. \item If $p_2(X_1,X_2)=q_1(X_1,X_2)$, then $M$ can be analyzed in a way similar to the previous examples, and it turns out that $M$ is holonomic with its (Hilbert) dimension two and multiplicity one in this case as described in Example \ref{E:prime_integrable_eg}. \end{itemize} The condition $p_2(X_1,X_2)=q_1(X_1,X_2)$ is known as an \textit{integrability condition}. Integrability conditions give rise to holonomic $\mathcal{A}_n$-modules, which are important because their ``solution space" (see below) has finite $\mathbb{C}$-dimension. Such a $D$-module thus ensembles the basic features of an ODE. The examples of $D$-modules which we study in this article are going to be holonomic. \end{example}
\subsection{Banach algebra and Poincar\'e-Perron theory} In order to study the convergence issue of Bessel's generating functions that are results of the $z$-transforms introduced in the last subsection, we extend the classical Poincar\'e-Perron theory from sequences of complex numbers to sequences in Banach algebra \cite{Yosida}. In our applications we only need to consider sequences in Banach algebra satisfying second order linear difference equations. However, our adaption can clearly be extended to higher order difference equations.
\begin{theorem}[(\bf{Poincar\'e-Perron theorem in Banach algebras})] \label{T:PP}
Let ($\mathbb{B},\|\cdot\|$) be a Banach algebra. Suppose that the $a$, $b\in\mathbb{B}$ and $(p_n)$, $(q_n)$ are sequences in $\mathbb{B}$ converging to zero as $n\to\infty$. If the characteristic equation of the associated difference equation
\begin{equation}\label{E:PP}
y_{n+2}+ay_{n+1}+by_n=0,
\end{equation} of the difference equation
\begin{equation}\label{E:PPP} y_{n+2}+(a+p_n)y_{n+1}+(b+q_n)y_n=0,
\end{equation} has roots $\lambda_1,\, \lambda_2$ distinct in norms, then for each solution $y_n$ to \eqref{E:PPP}, the limit
\[
\lim_{n\to\infty} \frac{\|y_{n+1}\|}{\| y_n\|}=\|\lambda_j\|,
\]holds, for some $j=1,\, 2$. \end{theorem}
Proofs of Poincar\'e's original statement can be found from either Gel'fond \cite[\S5.5.2]{Gelfond} or Milne-Thomson \cite[XVII]{MT}. The proof of the above theorem is a straightforward adaptation in the
proof of Poincar\'e's original statement with $\mathbb{B}$ in place of $\mathbb{C}$. Since the proof of Poincar\'e's theorem in full generality is long and the its adaption in Banach algebra $\mathbb{B}$ is almost verbatim except for the aforementioned replacement of ``$|\cdot |$\rq\rq{} by ``$\|\cdot \|$\rq\rq{} in all the computation and estimates,
one can verify that a shorter proof of Poincar\'e's theorem available for third order linear difference equations discussed in \cite{MT} can easily be adopted to the Banach algebra $\mathbb{B}$ setting.
Indeed, there are different adaptations of Poincar\'e-Perron theory in the literature. For example, the author in \cite{Schafke} considered objects that assume values in certain abelian groups in place of
complex numbers. Although the asymptotic of the classical Bessel function $J_{\nu+n}$ as $n\to\infty$ is well-known, a corresponding result for the difference Bessel function $J^\Delta_{\nu+n}$ is unknown and appears highly non-trivial. The adaption of Poincar\'e's theorem in a Banach algebra setting can easily produce the needed, though rough, asymptotic for the generating function of difference Bessel functions in to be stated in Theorem \ref{T:delta_Bessel_Gevrey}.
\section{Bessel modules and Bessel functions}\label{S:bessel}
In this section, we investigate the generating functions of the Bessel functions. We will construct an important holonomic left $\mathcal{A}_2$-module, called the Bessel module, from which we obtain a pair of PDEs that the generating function satisfy.
\begin{definition}[\cite{CCT3}]\label{D:soln} Let $M, N$ be left $\mathcal{A}_n$-modules. A \textit{solution} of $M$ in $N$ is a left $\mathcal{A}_n$-linear maps from $M$ to $N$. In particular, the set of \textit{all solutions} of $M$ in $N$ is denoted by
\[
\mathop{\rm Hom} (M, N).
\] \end{definition}
\begin{theorem}[\cite{CCT3}] Let $\mathcal{D}=\langle X\partial, X\rangle$ be the subalgebra of $\mathcal{A}$ generated by $X\partial$ and $X$. Then there exists an element $S=\sum_{k=0}^{\infty} a_kX^k$ in the $X$-adic completion $\mathbb{C}[[X]]$ so that the map
\[
\mathcal{D}/\mathcal{D}L
\stackrel{\times S}{\longrightarrow}
\mathcal{D}/\mathcal{D}(X\partial-\lambda)
\]is left $\mathcal{A}$-linear, i.e., it gives a solution of $\mathcal{D}/\mathcal{D}L$ in $\mathcal{D}/\mathcal{D}(X\partial-\lambda)$. \end{theorem}
The element $S$ in the above theorem is a Weyl-algebraic analogue of a \textit{Frobenius series expansion} of a solution of $L$. In fact, each left $\mathcal{A}$-linear map $\mathcal{A}/\mathcal{A}L \longrightarrow \mathcal{A}/\mathcal{A}K$ induces a map between solution sets
\[
\mathop{\rm Hom} (\mathcal{A}/\mathcal{A}K, N)\longrightarrow
\mathop{\rm Hom} (\mathcal{A}/\mathcal{A}L, N).
\] Thus finding solutions of a differential operator $L$ in terms of solutions of another differential operator $K$ is equivalent to identifying an $\mathcal{A}$-linear map
\[
\mathcal{A}/\mathcal{A}L
\stackrel{\times S }{\longrightarrow}
\mathcal{A}/\mathcal{A}K.
\]
For instance, finding ``power series solutions" of a differential operator $L$ is equivalent to a left $\mathcal{A}$-linear map $\mathcal{A}/\mathcal{A}L\to\mathcal{A}/\mathcal{A}\partial$; finding a ``Frobenius series solution" of $L$ is equivalent to a left $\mathcal{A}$-linear map $\mathcal{A}/\mathcal{A}L\to\mathcal{A}/\mathcal{A}(X\partial-\nu)$; and so on. We refer to \cite{CCT3} for the details. We illustrate the above discussion with the following examples that will be needed for the rest of this paper.
\subsection{Weyl series solutions}\label{SS:d-mod}
\begin{example}[(\textbf{Weyl-exponential})]\label{Eg:exp}
Let $a\in \mathbb{C}$. A solution of $L:=\partial-a$, i.e. a solution of $\mathcal{A}/\mathcal{A}(\partial-a)$ in any other $D$-module, is called a \textit{Weyl-exponential}. In this example, we consider solutions of $\mathcal{A}/\mathcal{A}(\partial-a)$ in $\mathcal{A}/\mathcal{A}\partial$, that is, we look for solutions as ``power series\rq\rq{} of $X$, which are called \textit{Weyl power series} to the Weyl exponential. For this purpose we compute the element $S=\sum_k c_kX^k$ which gives rise to a left $\mathcal{A}$-linear map
\[
\mathcal{A}/\mathcal{A}(\partial-a)
\stackrel{\times S }{\longrightarrow}
\overline{\mathcal{A}/\mathcal{A}\partial}
\] guaranteed by the study in \cite[\S4.1]{CCT3}. For each $k\ge 1$, since $\partial X^k=X^k \partial+k X^{k-1}$, we have that
\[
\begin{split}
0= (\partial-a )\left(\sum_{k=0}^\infty c_k X^k\right)
&= \sum_{k=0}^\infty c_k X^k \partial+\sum_{k=1}^\infty c_k k X^{k-1}-a\sum_{k=0}^\infty c_k X^k\\
&= \sum_{k=0}^\infty (c_{k+1}(k+1)-a c_k) X^k. \quad \qquad \mod \mathcal{A}\partial
\end{split}
\] The above formula gives recurrence of $c_k$ from the relation
\[
kc_k -ac_{k-1}=0,\qquad k\ge 1.
\] By choosing $c_0=1,$ we obtain $c_{k}=a^k/k!$. Thus we denote
\begin{equation}\label{E:exp}
S(X)={\mathop{\rm E}}(aX):=\sum_{k=0}^{\infty} \frac{a^k}{k!} X^k
\end{equation} to be the \textit{Weyl exponential series}. \end{example}
\begin{example}\label{Eg:trigo} (\textbf{Weyl Sine and Cosine}) The \textit{Weyl-sine} and the \textit{Weyl-cosine} are solutions of $L:=\partial^2+1$, i.e., solutions of the $D$-module $\mathcal{A}/\mathcal{A}(\partial^2+1)$. To express them as ``Weyl power series", i.e., in terms of powers of $X$, we again let $K=\partial$, and compute the element $S =\sum c_{k} X^k$ which yields the left $\mathcal{A}$
\begin{equation}\label{E:X^2+1}
\mathcal{A}/\mathcal{A}(\partial^2+1)
\stackrel{\times S }{\longrightarrow}
\overline{\mathcal{A}/\mathcal{A}\partial},
\end{equation} which are solutions of $\mathcal{A}/\mathcal{A}(\partial^2+1)$ in $\mathcal{A}/\mathcal{A}\partial$.
Since $\partial^2+1=(\partial-i)(\partial+i)$, we apply Example \ref{Eg:exp} with the choice $a=-i$ from the Example \ref{Eg:exp}. Then it follows immediately that
\[
(\partial-i)(\partial+i) {\mathop{\rm E}}(-iX)=0\qquad \mod \partial
\]
and since $\partial^2+1=(\partial+i)(\partial-i)$ so that
\[
(\partial+i)(\partial-i) {\mathop{\rm E}}(iX)=0\qquad \mod \partial,
\]where we have chosen $a=i$ from the Example \ref{Eg:exp}. Thus both ${\mathop{\rm E}}(iX)$ and ${\mathop{\rm E}}(-iX)$ in $\mathcal{A}$ are possible choices of $S$ in \eqref{E:X^2+1}. Hence both
\begin{equation}\label{E:sine-map}
\mathop{\rm Sin} (X):=\frac{1}{2i}\big({\mathop{\rm E}}(iX)-{\mathop{\rm E}}(-iX)\big)
\end{equation}
and
\begin{equation}\label{E:cosine-map}
\mathop{\rm Cos} (X):=\frac{1}{2}\big({\mathop{\rm E}}(iX)+{\mathop{\rm E}}(-iX)\big)
\end{equation}
are different choices of elements of $\mathcal{A}$, called the \textit{Weyl-Sine and Cosine series}, that can also serve for the $S$ in \eqref{E:X^2+1}. corresponding
Alternatively, we may compute the Weyl Sine and Cosine series $S =\sum c_{k} X^k$ \textit{directly} by
\[
\begin{split}
0&=(\partial-i)(\partial+i)\sum_{k=0}^\infty c_{k} X^k\\
&=\sum_{k=0}^\infty c_{k} X^k \partial^2+2\sum_{k=1}^\infty c_{k} kX^{k-1}\partial
+\sum_{k=2}^\infty c_{k}k(k-1) X^{k-2}+\sum_{k=0}^\infty c_{k} X^k\\
&=\sum_{k=0}^\infty c_{k+2}(k+1)(k+2) X^{k}+\sum_{k=0}^\infty c_{k} X^k.\qquad\mod\partial
\end{split}
\]That is,
\[
c_{k+2}=\frac{-c_{k}}{(k+1)(k+2)},\quad k\ge 0.
\]
Thus, if $c_0=0, c_{1}=1$, we obtain Weyl Sine series
$$\mathop{\rm Sin} X=X-\frac{X^3}{3!}+\frac{X^5}{5!}-\cdots+
\frac{(-1)^{k+1}}{(2k-1)!}X^{2k-1}+\cdots.$$
If $c_0=1, c_{1}=0$, we obtain Weyl (power) Cosine series
$$\mathop{\rm Cos} X=1-\frac{X^2}{2!}+\frac{X^4}{4!}-\cdots+
\frac{(-1)^{k}}{(2k)!}X^{2k}+\cdots.$$ Hence they serve as $S$ with ``appropriately imposed initial conditions". \end{example}
We now introduce one of the main subjects of study in this paper.
\begin{example}[(\textbf{Weyl Bessel of order $\nu$})]\label{Eg:bessel_map} Let $\nu\in\mathbb{C}$ such that $2\nu$ is not an integer, and let
\begin{equation}\label{E:bessel-eqn-map}
L=(X\partial)^2+X^2-\nu^2.
\end{equation} Let $\mathcal{D}=\langle X\partial, X\rangle$ be the subalgebra of $\mathcal{A}$ generated by $X\partial $ and $X$. Then the \textit{Weyl Bessel of order $\nu$} is a solution of $L$ in a left $\mathcal{D}$-module $N$ , i.e., a solution of $\mathcal{D}/\mathcal{D}L$ in $N$. To find the ``Frobenius series expansion" of this Weyl Bessel, we look for a solution of $\mathcal{D}/\mathcal{D}L$ in the $D$-module $\overline{\mathcal{D}/\mathcal{D}(X\partial-\nu)}$, i.e., we find the element $S=\sum_{k=0}^\infty c_kX^k$ that gives the left $\mathcal{A}$-linear map
\begin{equation}\label{E:bessel_map}
\mathcal{D}/\mathcal{D}L
\stackrel{\times S}{\longrightarrow}
\overline{\mathcal{D}/\mathcal{D}(X\partial-\nu)}.
\end{equation}
Since
\[
(X\partial+\nu)(X\partial-\nu)X^k
=X^k[(X\partial+\nu)+2k](X\partial-\nu)+(2\nu k+k^2)X^k,
\]holds for every $k\ge 0$ (as well as trivially when $k=0$), we have
\[
\begin{split}
0&=L\sum_{k=0}^\infty c_{k} X^k = \sum_{k=0}^\infty c_{k}[(X\partial+\nu)(X\partial-\nu)+X^2]X^k\\
&=\sum_{k=0}^\infty c_k X^k[(X\partial+\nu)+2k](X\partial-\nu) + \sum_{k=1}^\infty{c_kk(2\nu +k)X^k}+\sum_{k=0}^\infty c_k X^{k+2} \\
&=\sum _{k=1}^\infty c_k k(2\nu+k)X^k+\sum _{k=0}^\infty c_kX^{k+2}
\mod X\partial-\nu\\
&= c_1(1+2\nu)X+\sum _{k=2}^\infty (c_k k (2\nu+k)+ c_{k-2})X^{k},
\mod X\partial-\nu
\end{split}
\]which implies that $c_1=c_3=c_5=\cdots= 0$, and by choosing $c_0 \Gamma(\nu+1)=2^{-\nu},$ we must have
\[
c_{2k}=\frac{(-1)^k }{2^{\nu+2k} k!\Gamma(\nu+k+1)},\quad k\ge 0.
\]This shows that
\[
S=\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}
\] can be chosen as such a Weyl series $S$ in \eqref{E:bessel_map} that serves as a solution for \eqref{E:bessel_map}. We denote this specific Weyl Bessel series by $\mathcal{J}_\nu$ as solution in $\mathcal{D}/\mathcal{D}(X\partial-\nu)$, or more generally, in a left $\mathcal{A}$-module $N$, which can be considered as the Weyl algebraic version of the classical Bessel $J_\nu$. We caution that the reader not to confuse the series form of $\mathcal{J}_\nu$ in contrast to the original Bessel function of order $J_\nu$ since there is no factor ``$X^\nu$" that would appear there unless when the $\nu$ reduces to an integer.
\end{example}
\subsection{Transmutation formulae}\label{SS:transmutation_1}
After defining the Weyl Bessel as a solution of $L=L_\nu:=(X\partial)^2+X^2-\nu^2$ in the last subsection, we observe the following transmutation formulae for $L$. They can be verified directly, but they do not appear to be found immediately amongst the formulae in Derezi\'nski and Majewski \cite{Derezinski2014,Derezinski2020,Der_Maj}\footnote{Please see \S\ref{S:conclusion} Conclusion for discussion related to the work of Infeld and Hull \cite{Infeld_Hull_1951}.}. As we shall see that they are instrumental in the development of Bessel module to be defined in the next subsection.
\begin{proposition}\label{P:bessel_transmutation} For each $\nu\in\mathbb{C}$, we have \begin{align}
[(X\partial)^2+X^2-(\nu-1)^2]\left(\partial+\dfrac{\nu}{X}\right)&=\left(\partial+\dfrac{\nu-2}{X}\right)[(X\partial)^2+X^2-\nu^2],\label{E:bessel_transmutation_1}\\ [(X\partial)^2+X^2-(\nu+1)^2]\left(\partial-\dfrac{\nu}{X}\right)&=\left(\partial-\dfrac{\nu+2}{X}\right)[(X\partial)^2+X^2-\nu^2],\label{E:bessel_transmutation_2} \end{align}
which induce the following left $\mathcal{A}$-linear maps \[ \begin{array}{rcl}\mathcal{A}/\mathcal{A}((X\partial)^2+X^2-(\nu-1)^2)&\stackrel{\times(\partial+\frac{\nu}{X})}{\longrightarrow}&\mathcal{A}/\mathcal{A}((X\partial)^2+X^2-\nu^2),\\ \mathcal{A}/\mathcal{A}((X\partial)^2+X^2-(\nu+1)^2)&\stackrel{\times(\partial-\frac{\nu}{X})}{\longrightarrow}&\mathcal{A}/\mathcal{A}((X\partial)^2+X^2-\nu^2). \end{array} \] \end{proposition}
\begin{corollary}\label{C:bessel_formulae} Let $\nu\in\mathbb{C}$, the classical Bessel functions satisfy the following recurrence relations.
\begin{align}
J'_\nu(x)+\dfrac{\nu J_\nu(x)}{x}&=J_{\nu-1}(x),\label{E:bessel_formula_1}\\
J'_\nu(x)-\dfrac{\nu J_\nu(x)}{x}&=-J_{\nu+1}(x).\label{E:bessel_formula_2}
\end{align} \end{corollary}
\begin{proof} This proof serves to provide an alternative to the classical proof found in most literature. It is a direct consequence from the previous Proposition with the $D$-module structure on $\mathcal{O}$ introduced from the Example \ref{Eg:O_deleted_d}. Let $J_\nu$ be the classical Bessel function to $L_\nu$ with the asymptotic behaviour $\lim_{x\to 0^+}{J_\nu(x)}/{x^\nu}={1}/{2^\nu\Gamma(\nu+1)}$ \cite[p. 43]{Watson1944}. Then one applies the both sides of \eqref{E:bessel_transmutation_1}. It implies that $(\partial+\nu/X)J_\nu=CJ_{\nu-1}+DJ_{-\nu+1}$ holds for some constants $C, D$. One can easily verify that the $C=1,\ D=0$ from the asymptotic $J^\prime_\nu(x)/x^{\nu-1}\sim 1/2^{\nu-1}\Gamma(\nu)$ \cite[p. 43]{Watson1944}. This verifies the \eqref{E:bessel_formula_1}. The verification of \eqref{E:bessel_formula_2} is similar.
\end{proof}
\begin{corollary}\label{C:Bessel_trans} Let $\nu\in\mathbb{C}$ and a left $\mathcal{D}$-module $N$ be given. Then there exists a sequence $\mathrm{(}\mathfrak{J}_{\nu+n}\mathrm{)}$ in $N^\mathbb{Z}$ and for each $n$ the $\mathfrak{J}_{\nu+n}$ is a solution to $L_n=(X\partial)^2+X^2-(\nu+n)^2$ in $N$, such that we have
\begin{align}
X\partial\, \mathfrak{J}_{\nu+n} +(\nu+n) \mathfrak{J}_{\nu+n} - X\mathfrak{J}_{\nu+n-1}=0,\label{E:Bessel_trans_1}
\\
X\partial\, \mathfrak{J}_{\nu+n} -(\nu+n) \mathfrak{J}_{\nu+n} + X\mathfrak{J}_{\nu+n+1}=0.\label{E:Bessel_trans_2}
\end{align} \end{corollary}
\begin{remark} Reader are cautioned not to confuse the notation $\mathfrak{J}_\nu$ introduced in the last Corollary with notation $\mathcal{J}_\nu$ that denotes the Weyl-Bessel series introduced in Example \ref{Eg:bessel_map} that is defined as a solution with respect to the \eqref{E:bessel_map}.
\end{remark}
\begin{proof} Recall that $L_\nu=(X\partial)^2+X^2-\nu^2$. Let us replace $\nu$ by $\nu+n$ in \eqref{E:bessel_transmutation_1}. Then it is clear that the right-side of \eqref{E:bessel_transmutation_1} annihilates $\mathfrak{J}_{\nu+n}$. However the expression $(\partial+\nu/X)\mathfrak{J}_{\nu+n}=(X\partial+\nu)/X\, (X\mathfrak{J}_\nu$) on the left-side must be annihilated by $L_{\nu+n-1}$. Hence $X\partial\, \mathfrak{J}_{\nu+n} +(\nu+n) \mathfrak{J}_{\nu+n} = \pm X\mathfrak{J}_{\nu+n-1}$. This gives the \eqref{E:Bessel_trans_1} except for the ``$\mp$" in front of the third term $ X\mathfrak{J}_{\nu+n-1}$. We have chosen the minus sign over the plus sign in accordance with the corresponding formula for the classical Bessel function $J_\nu$. This establishes the \eqref{E:Bessel_trans_1}. The proof for \eqref{E:Bessel_trans_2} is similar. \end{proof}
\begin{corollary} [(\textbf{Three-term recurrence})] \label{C:Bessel_3term} Let $\nu\in\mathbb{C}$ and a left $\mathcal{D}$-module $N$ be given. Then there exists a sequence $\mathrm{(}\mathfrak{J}_{\nu+n}\mathrm{)}$ such that for each $n$ the $\mathfrak{J}_{\nu+n}$ is a solution to $L_n=(X\partial)^2+X^2-(\nu+n)^2$ in $N$, such that we have
\begin{equation}\label{E:Bessel_3term}
2(\nu+n)\,\mathfrak{J}_{\nu+n}-X\mathfrak{J}_{\nu+n-1}-X\mathfrak{J}_{\nu+n+1}=0
\end{equation} as an element in $N$. \end{corollary}
\subsection{Bessel modules}
Recall that the Weyl algebra $\mathcal{A}_2$ is the free $\mathbb{C}$-algebra $\mathbb{C}\langle \partial_1,\partial_2, X_1, X_2\rangle$ subject to \[ \partial_i\partial_j-\partial_j\partial_i=0,\quad X_iX_j-X_jX_i=0,\quad\partial_iX_j-X_j\partial_i=\delta_{ij}, \] for $i,j=1,2$. Then we define
\begin{definition}\label{D:A_2_mod} We denote the algebra obtained by adjoining the new elements $1/X_1,\, 1/X_2$ to the Weyl algebra $\mathcal{A}_2$ by
\[
\mathcal{A}_2(1/X_1,\, 1/X_2)
:=\mathbb{C}\langle\partial_1,\partial_2, X_1, \dfrac{1}{X_1},X_2,\dfrac{1} {X_2}\rangle.
\] \end{definition}
Let $\nu\in\mathbb{C}$. Motivated by Corollary \ref{C:Bessel_trans}, we now consider the left $\mathcal{A}_2$-module generanted by the following elements
\begin{equation}\label{E:bessel_PDE}
X_1\partial_1+(X_2\partial_2+\nu)-X_1X_2,\qquad
X_1\partial_1-(X_2\partial_2+\nu)+\frac{X_1}{X_2}.
\end{equation}
With the \eqref{E:bessel_PDE} in mind, we define
\begin{definition}[(\textbf{Bessel module of order $\nu$})]\label{D:bessel_mod}
Let $\nu\in\mathbb{C}$. The left $\mathcal {A}_{2}$-module\footnote{The $\mathcal{B}_\nu$ is in fact a quotient module. We have left out the word ``quotient" when no confusion can arise.}
\begin{equation}\label{E:bessel_mod}
\mathcal{B}_\nu=\frac{\mathcal{A}_2}
{\mathcal{A}_2(X_1\partial_1+(\nu+X_2\partial_2)-X_1X_2)+
\mathcal{A}_2(X_1\partial_1-(\nu+X_2\partial_2)+{X_1}/{X_2})}
\end{equation} is called the \textit{Bessel module of order} $\nu$. \end{definition}
First we check that the $\mathcal{B}_\nu$ is indeed holonomic.
\begin{proposition}\label{P:bessel_holonomic} Let $\nu\in\mathbb{C}$. The Bessel module $\mathcal{B}_\nu$ is a left $\mathcal{A}_2$-module \footnote{We omit the description that $\mathcal{B}_\nu$ is a left $\mathcal{A}_2$-quotient module for the sake of simplicity.} with dimension $2$ and multiplicity $1$. In particular, it is holonomic. \end{proposition}
\begin{proof} The two generators from Definition \ref{D:bessel_mod} that generate the Bessel module $\mathcal{B}_\nu$ satisfy the integrability criterion stated in Example \ref{E:prime_integrable_eg}. Hence the Bessel module has dimension $2$ and multiplicity $1$. \end{proof}
The following theorem shows that one can recover from $\mathcal{B}_\nu$ the ODE satisfied by Bessel functions.
\begin{proposition}[(\textbf{Bessel ODE module of order $\nu$})] \label{P:bessel_ode_mod}
The map \[
\dfrac{\mathcal{A}_2}{\mathcal{A}_2[(X_1\partial_1)^2+X_1^2-(\nu+X_2\partial_2)^2]}\longrightarrow\mathcal{B}_\nu \] is a well-defined left $\mathcal{A}_2$-linear surjection. \end{proposition}
\begin{proof} Recall that $\mathcal{B}_\nu$ is generated by the two elements \[ X_1\partial_1+X_2\partial_2-X_1X_2+\nu,\qquad X_1\partial_1-X_2\partial_2+\frac{X_1}{X_2}-\nu \] as in \eqref{E:bessel_PDE}, so as an element in $\mathcal{B}_\nu$, we have \begin{align*}
(X_1\partial_1)^2+X_1^2-(\nu+X_2\partial_2)^2 &= (X_1\partial_1)^2+X_1^2+(\nu+X_2\partial_2)(X_1\partial_1-X_1X_2) \\
&= (X_1\partial_1)(X_1\partial_1+\nu+X_2\partial_2)+(X_1X_2)(\frac{X_1}{X_2}-\nu-\partial_2X_2) \\
&= (X_1\partial_1)(X_1X_2)+(X_1X_2)(X_2\partial_2-X_1\partial_1-\partial_2X_2)\\
&= (X_1\partial_1)(X_1X_2)+(X_1X_2)(-\partial_1X_1)\\
&= 0. \end{align*} Hence the map is well defined. \end{proof}
We may rephrase the above result in the following form.
\begin{proposition} Let $\nu\in\mathbb{C}$ and $M$ be a left $\mathcal{A}$-module. If the sequence $(f_n)$ is a solution of $\mathcal{B}_\nu$ in $M^\mathbb{Z}$, then for each $n\in\mathbb{Z}$, $f_n$ is a solution of $(X\partial)^2+X^2-(\nu+n)^2$ in $M$. In other words, the element $(X_1\partial_1)^2+X_1^2-(\nu+X_2\partial_2)^2$ is equivalent to $0$ in $\mathcal{B}_\nu$. \end{proposition}
\subsection{Characteristics of Bessel modules}
We will see later that the Bessel module $\mathcal{B}_\nu$ gives rise to PDEs satisfied by the generating functions of the Bessel functions $(J_{\nu+n})$ in \S\ref{SS:classical_gf} as well as of the recently discovered difference Bessel functions $(J^\Delta_{\nu+n})$ in \S\ref{SS:delta_gf} below. The following theorem is an algebraic analogue of the \textit{method of characteristics}, which will help in solving such PDEs.
\begin{theorem}\label{T:bessel_gen_map_2} Let $\nu\in\mathbb{C}$ and $\mathcal{B}_\nu$ be defined above. Then there exists a left $\mathcal{A}_2(1/X_{j=1,2})$-linear map $S$
\begin{equation}\label{E:bessel_gen_map_1}
\mathcal{B}_\nu
\xrightarrow
{\times S}
\overline{
\mathcal{A}_2/\big[\mathcal{A}_2\partial_1+\mathcal{A}_2(X_2\partial_2+\nu)\big]}.
\end{equation} In fact, we can take
\begin{equation}\label{E:bessel_gen_map_2}
S={\mathop{\rm E}}\Big[\frac{X_1}{2}\Big(X_2-\frac{1}{X_2}\Big)\Big],
\end{equation}where ${\mathop{\rm E}}$ denotes the Weyl exponential series introduced in Example \ref{E:exp}. \end{theorem}
\begin{proof} Define new symbols $\Theta_1, \Theta_2, Y_1, Y_2$ by \[
Y_{1}=X_{1}X_{2},\qquad Y_{2}=\frac{X_{1}}{X_{2}},\qquad
2\Theta_{1}=\frac{\partial_1}{X_2}+\frac{\partial_{2}}{X_{1}},\qquad
2\Theta_{2}=X_{2}\partial_{1}-\frac{X^2_{2}\partial_{2}}{X_{1}}. \]
Direct verification yields
\begin{equation}\label{E:computation_0}
[\Theta_{1}, Y_{1}]=1, \quad [\Theta_{2}, Y_{2}]=1,
\quad
[\Theta_{1}, \Theta_{2}]=0,\quad [Y_1, Y_{2}]=0,
\quad [\Theta_i,\, Y_j]=0,\ i\not=j.
\end{equation}
The elements in \eqref{E:bessel_PDE} become
\[
2Y_1\Theta_{1}-Y_1+{\nu}\qquad \mbox{and}\qquad 2Y_2\Theta_{2}+Y_2-{\nu}.
\]
It follows from the Weyl exponential introduced in Example \ref{Eg:exp} for $a=\nu/2$ that
\[
\begin{split}
(2Y_1\Theta_1-Y_1+\nu)\, {\mathop{\rm E}}(Y_1/2)
&= 2Y_1{\mathop{\rm E}}(Y_1/2)\Theta_1 + \nu{\mathop{\rm E}}(Y_1/2)\\
&= {\mathop{\rm E}}(Y_1/2)(2Y_1\Theta_1+\nu)\\
&=0. \qquad\qquad\quad \mod \mathcal{A}_2 (Y_1\Theta_1+\nu/2)
\end{split}
\]
In general we have $S=F(Y_2)\cdot {\mathop{\rm E}}(Y_1/2)$ for some factor $F(Y_2)$ depending on $Y_2$ only. Next we require $F(Y_2)$ to satisfy that
\[
(2Y_2\Theta_2+Y_2-\nu)\,F(Y_2)\cdot {\mathop{\rm E}}(Y_1/2) =0.
\qquad\mod \mathcal{A}_2(Y_2\Theta_2-\nu/2)
\]
That is,
\[
(2Y_2\Theta_2+Y_2-\nu)\,F(Y_2)=0,\qquad\qquad \mod \mathcal{A}_2(Y_2\Theta_2-\nu/2).
\]
It follows from a similar consideration as above that $F(Y_2)=c{\mathop{\rm E}}(-Y_2/2)$ for some complex number $c$. Without loss of generality we may choose $c=1$ so that the above map becomes
\begin{equation}\label{E:bessel_gen_map_3}
S:={\mathop{\rm E}}(Y_1/2)\cdot {\mathop{\rm E}}(-Y_2/2)
=\sum_{k=0}^\infty \frac{X_1^kX_2^k}{2^kk!} \cdot
\sum_{\ell=0}^\infty \frac{(-1)^\ell X_1^\ell}{X_2^\ell 2^\ell \ell !}
={\mathop{\rm E}}\Big[\frac{X_1}{2}\Big(X_2-\frac{1}{X_2}\Big)\Big]
\end{equation} which is the map asserted in \eqref{E:bessel_gen_map_1}. \end{proof}
\begin{remark} We may rewrite the product of two series in $S$ in \eqref{E:bessel_gen_map_3} to get
\[
\sum_{k=0}^{\infty} \frac{X^k_{1}X^k_{2}}{2^k k!} \sum_{\ell=0}^{\infty}\frac{(-1)^l X^\ell_{1}}{X^\ell_{2}2^\ell !} =\sum_{n=-\infty}^{\infty}\left(\sum_{\ell=0}^{\infty}\frac{(-1)^\ell X^{n+2\ell}_{1}}{2^{n+2\ell} (n+\ell)! \ell!} \right) X_{2}^{n}.
\] The ``inner sum\rq\rq{}
\[
\mathcal{J}_{n} (X)=\sum_{\ell =0}^{\infty}
\frac{(-1)^\ell}{ (n+\ell)!l!}
\left(\frac{X}{2}\right)^{n+2\ell}
\] agrees with Example~\ref{Eg:bessel_map} after composing with the multiplication by $X^n$ from $\mathcal{A}/\mathcal{A}(X\partial-n)$ to $\mathcal{A}/\mathcal{A}\partial$. It gives a ``power series" of the Weyl Bessel of order $n$. Following the arguments used in Watson \cite[pp. 15-16]{Watson1944} that
\begin{equation}\label{E:neg_Bessel}
\mathcal{J}_{-n} (X)=(-1)^n \mathcal{J}_{n} (X)
\end{equation}
holds for each integer $n\ge 1$. \end{remark}
\subsection{Generating function of classical Bessel functions}\label{SS:classical_gf} As was explained by Watson \cite[Chap. 2., \S2,.1]{Watson1944} that the convergence of the infinite expansion of the generating function
\begin{equation}\label{E:bessel_classical_gf}
e^{\frac{x}{2}\big(t-\frac1t\big)}=\sum_{n=-\infty}^\infty J_n(x)\, t^n
\end{equation} of Bessel function is absolute in powers of $t$, and valid for all $x$ and $t\not=0$. We shall take this expansion as the basis of our study of various extensions of this generating function for various types of Bessel functions in this section.
\begin{example}[(\textbf{Frobenius series of classical Bessel functions of order $\nu$})]\label{Eg:Bessel} Let $\nu$ be a complex number such that $2\nu$ is not an integer, and consider the the Bessel operator $L=(X\partial)^2+X^2-\nu^2$ as in Example~\ref{Eg:bessel_map}. The classical Bessel function of order $\nu$, $J_\nu(x)$, is a solution of $\mathcal{D}/\mathcal{D}L$ in the $D$-module $\mathcal{O}_d$. To obtain a Frobenius series expansion of $J_\nu(x)$, we
recall from Example \ref{Eg:bessel_map} the element
\begin{equation}\label{E:pos_nu_map}
S=\mathcal{J}_\nu(X)=\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}
\end{equation} gives the composition map
of left $\mathcal{A}$-linear maps
\begin{equation}\label{E:pos_bessel_comp_map}
\mathcal{D}/\mathcal{D}L
\stackrel{\times S}{\longrightarrow}
\overline{\mathcal{D}/\mathcal{D}(X\partial-\nu)}
\stackrel{\times x^{\nu}}{\longrightarrow} \mathcal{O}_d,
\end{equation}
and so
\begin{equation}\label{E:pos_bessel-soln}
J_\nu(x)=x^\nu \sum_{k=0}^{\infty} \frac{(-1)^kx^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)},
\end{equation} gives raise to the classical Bessel function of order $\nu$.
On the other hand, if we replace the $\nu$ in \eqref{D:bessel_mod} by $-\nu$, and proceed in a similar manner for an analogous set up, then we would have obtained a ``linearly-independent" solution of $\mathcal{D}/\mathcal{D}L$ in $\mathcal{O}_d$ given by
\begin{equation}\label{E:neg_bessel_comp_map}
\mathcal{D}/\mathcal{D}L
\stackrel{\times S}{\longrightarrow}
\overline{\mathcal{D}/\mathcal{D}(X\partial+\nu)}
\stackrel{\times x^{-\nu}}{\longrightarrow} \mathcal{O}_d,
\end{equation}
where
\begin{equation}\label{E:neg_nu_map}
S=\mathcal{J}_{-\nu}(X)=\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}.
\end{equation} The \eqref{E:neg_bessel_comp_map} gives rise to the Frobenius series expansion
\begin{equation}\label{E:neg_bessel-soln}
J_{-\nu}(x)=x^{-\nu} \sum_{k=0}^{\infty} \frac{(-1)^kx^{2k}}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}.
\end{equation} \end{example}
The following example illustrates how sequences of analytic functions are realized from $D$-modules.
\begin{example}\label{Eg:bilateral_J} Let $\nu\in \mathbb{C}$ and consider the bilateral sequence of classical Bessel functions $(J_{\nu+n})_n$. Then the maps
\begin{equation}\label{E:j-map-bessel}
\begin{array}{rcl}
\mathfrak{j}_+:\mathcal{B}_\nu & \stackrel{\times (J_{\nu+n})}{\longrightarrow} &\mathcal{O}^\mathbb{Z}
\end{array}
\end{equation} and
\begin{equation}
\begin{array}{rcl}
\mathfrak{j}_-:\mathcal{B}_\nu & \stackrel{\times (J_{-\nu-n})}{\longrightarrow} &\mathcal{O}^\mathbb{Z}
\end{array}
\end{equation} are both left $\mathcal{A}_2$-linear.
More generally, a general solution of $\mathcal{B}_\nu$ in $\mathcal{O}^\mathbb{Z}$ will be denoted by $\times \mathrm{(}\mathscr{C}_{\nu+n}\mathrm{)}$. That is,
\begin{equation}\label{E:general-map-bessel}
\begin{array}{rcl}
\mathfrak{j}: \mathcal{B}_\nu & \stackrel{\times (\mathscr{C}_{\nu+n})}{\longrightarrow} &\mathcal{O}^\mathbb{Z}
\end{array}
\end{equation}
\end{example}
The following proposition and corollary are direct consequences of Corollary~\ref{C:Bessel_trans} with $M=\mathcal{O}_d$ and $\mathscr{C}_{\nu+n}$.
\begin{proposition}[(\textbf{Differential-difference formulae})]\label{P:PDE_cylinderical_bessel} Let $\mathcal{B}_\nu\ \substack{ \times (\mathcal{C}_{\nu+n})\\ \longrightarrow} \ \mathcal{O}^\mathbb{Z}_d$ $\mathrm{(}\mathscr{C}_{\nu+n}\mathrm{)}$ be a solution of $\mathcal{B_\nu}$ in $\mathcal{O}^\mathbb{Z}_d$. Then the $\mathrm{(}\mathscr{C}_{\nu+n}\mathrm{)}$
satisfies the following classical differential-difference formulae
\begin{equation}\label{E:any_bessel_recus_1}
x\mathscr{C}_{\nu}^\prime(x)+\nu\mathscr{C}_{\nu}(x)-
x\mathscr{C}_{\nu-1}(x)=0,
\end{equation} and
\begin{equation}\label{E:any_bessel_recus_2}
x\mathscr{C}^{\prime}_{\nu}(x)-\nu\mathscr{C}_{\nu+n}(x)+
x\mathscr{C}_{\nu+1}(x)=0.
\end{equation}
\end{proposition}
The usage of the notation $\mathscr{C}_{\nu+n}$ follows Sonine's notation (see \cite[p. 83]{Watson1944}) for functions that satisfy the system \eqref{E:any_bessel_recus_1} and \eqref{E:any_bessel_recus_2}.
Here is an application of the proposition to recover two well-known formulae about the classical Bessel functions. We shall see how this will lead to new results for difference Bessel functions and half-Bessel modules for difference reverse Bessel polynomials in \S\ref{SS:delta_reverse_bessel_poly}.
One recovers the well-known formulae \eqref{E:bessel_formula_1} and \eqref{E:bessel_formula_2} by choosing $\mathscr{C}_\nu=J_\nu$ in the above Corollary.
The following classical three-term recursion formula is a consequence of the above differential-difference formulae \eqref{E:any_bessel_recus_1} and \eqref{E:any_bessel_recus_2}.
\begin{corollary} [(\textbf{Three-term recurrence})]\label{C:any_bessel_3_term}
Let $\mathcal{B}_\nu\ \substack{ \times (\mathcal{C}_{\nu+n})\\ \longrightarrow} \ \mathcal{O}^\mathbb{Z}_d$ $\mathrm{(}\mathscr{C}_{\nu+n}\mathrm{)}$ be a solution of $\mathcal{B_\nu}$ in $\mathcal{O}^\mathbb{Z}_d$. Then
\begin{equation}\label{E:any_bessel_3_term}
2\nu\mathscr{C}_{\nu}(x)-x\mathscr{C}_{\nu-1}(x)-x\mathscr{C}_{\nu+1}(x)=0.
\end{equation} \end{corollary}
Now we come to the main result of this subsection, which is about the generating function of the classical Bessel functions $(J_{\nu+n}(x))$. It is a two-variable analytic solution of the holonomic system of PDEs obtained from the Bessel module $\mathcal{B}_\nu$ \eqref{E:bessel_mod}.
\begin{theorem}[(\bf{Asymptotic behaviour})]\label{T:Bessel_Gevrey} The sequence $(J_{n+\nu})$ is a uniformly 1-Gevrey compacta solution of $\mathcal{B}_\nu$ in $\mathcal{O}_d^\mathbb{Z}$. \end{theorem}
\begin{proof} It follows from the the Corollary \ref{C:any_bessel_3_term} that $J_{\nu+n}$ satisfies
\[
y_{n+2}-\dfrac{2(n+\nu+1)}{x}y_{n+1}+y_n=0.
\] As a result, the modified sequence ($Y_n$) where $Y_n:=J_{\nu+n}/n!$ satisfies the equation
\begin{equation}\label{E:modified_bessel_3_term}
Y_{n+2}-\dfrac{2(n+\nu+1)}{(n+2)x}Y_{n+1}+\dfrac{1}{(n+2)(n+1)}Y_n=0.
\end{equation}
Now one fixes a compact set $K\subset\mathbb{C}\backslash\{0\}$. The space of functions analytic on $K$ becomes a Banach algebra $\mathbb{B}_K$ equipped with the usual sup norm $\|\cdot\|_K$.Therefore, we may apply the generalized Poincar\'e-Perron theory stated in Theorem \ref{T:PP} on the Banach algebra $\mathbb{B}_K$ for the asymptotic behaviour of ($J_{\nu+n}$) on the compact set $K$. Since the characteristic equation of \eqref{E:modified_bessel_3_term} is given by
\[
\lambda^2-2\lambda/x=0,
\]so that its roots are $0$ and $2/x$ whose norms are obviously different. Consequently
\begin{equation}\label{E:uniform_gervey}
\|J_{n+\nu}/n!\|_K=O((2/d(K,0))^n).
\end{equation}That is, according to the Remark \ref{R:gervey}, the sequence $(J_{n+\nu})$ is $1$-Gevrey. \end{proof}
\begin{remark}
\begin{enumerate}
\item It is instructive to compare the estimate \eqref{E:uniform_gervey}
with the well-known asymptotic
\[
J_{\nu+n}(x)\sim \frac{e^{-\nu}}{\sqrt{2\pi n}}\Big(\frac{ex}{2n}\Big)^{\nu+n}
\]as given, see for example, in \cite[\S5]{Gautschi}. An application of Stirling formula to \eqref{E:uniform_gervey} shows that we have the same order of growths from the two asymptotic estimates. \footnote{In fact, this can be seen from the elementary growth estimate \cite[p. 225]{Watson1944}
\[
J_\nu(x)\approx \exp\big\{\nu+\nu\log(x/2) -(\nu+\frac12)\log \nu\big\}
\cdot \big[1/\sqrt{2\pi}+\frac{c_1}{\nu}+\frac{c_2}{\nu^2}+\cdots\big]
\]holds for a fixed $x$ as $\nu\to\infty$, so that although the ``analystic part\rq\rq{}, i.e., the ``$\sum_0^\infty$\rq\rq{} part of \eqref{E:gf_bessel} is convergent, while the ``principle part\rq\rq{}, or the ``$\sum_{-1}^{-\infty}$\rq\rq{} is divergent. }
\item Similarly, one sees that the sequence $(J_{-n+\nu})$ is ``uniformly 1-Gevrey compacta" also. The conclusion remains valid if the standard Bessel functions are replaced by an arbitrary sequence $(\mathscr{C}_{n+\nu})$ as formulated above.
\item It follows from the last remark item that the $z$-transform of the sequence $(J_{n+\nu})$, or the Borel resummation of $\sum_n J_{n+\nu}t^n$, is a well-defined analytic functions in two variables as defined in the Definition \ref{D:borel}.
\item We also note that the conclusion of Theorem \ref{T:Bessel_Gevrey} remains valid if the uniform compacta norm is replaced by any other norm.
\end{enumerate} \end{remark}
\begin{theorem}[(\textbf{Generating function of Bessel functions})]\label{T:Bessel_gf} Let $\nu\in\mathbb{C}$.
\begin{enumerate}
\item Let $(\mathscr{C}_{n+\nu})_n$ be a bilateral sequence of analytic functions which is a solution of the Bessel module $\mathcal{B}_\nu$ in $\mathcal{O}^\mathbb{Z}$. Then there exists a complex number $C_\nu$ such that
\begin{equation}\label{E:gf_bessel}
C_\nu\, t^{-\nu}\exp\Big[\dfrac{x}{2}\big(t-\dfrac{1}{t}\big)\Big]
\sim\sum_{n=-\infty}^\infty \mathscr{C}_{\nu+n}(x)\, t^n,
\end{equation}
\end{enumerate}where the symbol ``$\sim$\rq\rq{} above means that the left-side is the Borel resummation\footnote{Borel resummation is defined in Definition \ref{D:borel}.} of the right-hand side.
\item Moreover, the holonomic system of PDEs \eqref{E:bessel_PDE} when realized in $\mathcal{O}_{dd}$ in Example \ref{Eg:O_deleted_d} defined by \eqref{E:O_dd_endow} is given by
\begin{equation}\label{E:PDE_bessel}
y_x+(1/t-t)/2\, y=0,\quad
\nu y+ty_t-{x}/2\,(1/t+t)\, y=0\footnote{See also the Appendix \ref{SS:PDE_list}.}.
\end{equation} \end{theorem}
To prove Theorem~\ref{T:Bessel_gf}, we need the following two lemmas, which serve the purpose of simplifying and solving the holonomic system of PDEs \eqref{E:bessel_PDE} in $\mathcal{O}_{dd}$.
\begin{lemma}\label{T:bessel_gf_mod} The Bessel module $\mathcal{B}_\nu$ is isomorphic to
\begin{equation}\label{E:bessel_mod_gf}
\dfrac{\mathcal{A}_2}{\mathcal{A}_2[\partial_1+\frac{1}{2}({1}/{X_2}-X_2)]+\mathcal{A}_2[(\nu+X_2\partial_2)-\frac12 X_1(1/X_2+X_2)]}.
\end{equation} \end{lemma}
\begin{lemma} \label{C:classical_app} The map
\[
\begin{array}{rcl} \dfrac{\mathcal{A}_2}{\mathcal{A}_2[\partial_1+\frac{1}{2}({1}/{X_2}-X_2)]+\mathcal{A}_2[(\nu+X_2\partial_2)-\frac12 X_1(1/X_2+X_2)]}
&\stackrel{\times t^{-\nu}\exp[\frac{x}{2}(t-\frac{1}{t})]}{\longrightarrow}
&\mathcal{O}_{dd}
\end{array}
\]
is well-defined and left $\mathcal{A}_2$-linear. In fact, this linear map is unique up to a complex multiple. \end{lemma}
\begin{proof} It is straightforward to verify that the function $ t^{-\nu}\exp[\frac{x}{2}(t-\frac{1}{t})]$ satisfies the system of PDEs
\begin{equation}\label{E:bessel_PDE_fn}
\frac{\partial y}{\partial x} +\frac{1}{2}\big(\dfrac{1}{t}-t\big)y=0,\qquad
\nu y +t \frac{\partial y}{\partial t} -\frac{x}{2}\big(\dfrac{1}{t}+t\big)y=0.
\end{equation}
To see the uniqueness, for a general analytic function $y=y(x,\, t)$. The first equation yields $y(x,t)=g(t)\exp\big[(\dfrac{x}{2}(t-\dfrac{1}{t})\big]$ for some $g$. Substituting this $y$ into the second differential equation yields $g(t)=Ct^{-\nu}$ for some $C\in\mathbb{C}$. \end{proof}
\noindent\textit{Proof of Theorem~\ref{T:Bessel_gf}.}
Recall the left $\mathcal{A}_2$-linear maps $\mathfrak{j}$ and $\mathfrak{z}$ as defined in \eqref{E:general-map-bessel} and \eqref{E:z-transform-2} respectively.
We have the diagram
\begin{equation}\label{E:commute-1}
\begin{tikzcd} [row sep=large, column sep=huge]
\mathcal{B}_{\nu} \arrow{r}{\mathfrak{j}}
\arrow[swap]{d}{\times {\mathop{\rm E}} [\frac{X_1}{2}(X_2-\frac{1}{X_2})]}
& \mathcal{O}^{\mathbb{Z}} \arrow{d}{\mathfrak{z}}\\
\widetilde{\mathcal{A}}_2 \arrow{r}{\times t^{-\nu}} & \mathcal{O}_{d d}
\end{tikzcd}
\end{equation}
in which $\widetilde{\mathcal{A}}_2:=\mathcal{A}_2/\big[\mathcal{A}_2\partial_1+\mathcal{A}_2(X_2\partial_2+\nu)\big]$. Here the left vertical map $\mathcal{B}_\nu\longrightarrow \widetilde{\mathcal{A}}_2$ of \eqref{E:commute-1} is given by \eqref{E:bessel_gen_map_2} from Theorem \ref{T:bessel_gen_map_2}. Since the above diagram commutes up to a complex multiple, the sum $\sum_n \mathscr{C}_{\nu+n}(x)\, t^n$, being the image of $1$ in $\mathcal{O}_{dd}$ via the top-right path, is also a formal solution to the system of PDEs \eqref{E:bessel_PDE}. Now the bottom-left path of \eqref{E:commute-1} gives $t^{-\nu}\exp\big[\dfrac{x}{2}(t-\dfrac{1}{t})\big]$ which is a solution to \eqref{E:bessel_PDE} because of Lemma~\ref{T:bessel_gf_mod} and Lemma~\ref{C:classical_app}. Since $\mathcal{B}_\nu$ is holonomic, the $\mathbb{C}$-dimension of the local solution space of \eqref{E:bessel_PDE} equals the multiplicity of $\mathcal{B}_\nu$, which is one. So, in summary, we may regard \eqref{E:gf_bessel} holds up to a complex scalar multiple $C_\nu$ which can be identically zero.
\qed
The next corollary determines the constant $C_\nu$ that appears in \eqref{E:gf_bessel} for various classes of Bessel functions. Moreover, a connection between these generating functions and Sonine integrals, see e.g. Watson \cite[Chap. VI]{Watson1944} is established.
\begin{corollary}
\begin{enumerate}
\item Let $\nu\in\mathbb{C}$. Then
\begin{equation}\label{E:gf_J_nu}
t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\sim\sum_{n=-\infty}^\infty J_{\nu+n}(x)\, t^n.
\end{equation}
\item Let $\nu\in\mathbb{C}^\ast$. Then
\begin{equation}\label{E:gf_Y_nu}
e^{-i\pi/2}\, t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\sim \sum_{n=-\infty}^\infty Y_{\nu+n}(x)\, t^n.
\end{equation}
\item Let $\nu\in\mathbb{C}$. Then
\begin{equation}\label{E:gf_I_nu}
t^{-\nu}\exp\big[\dfrac{x}{2}\big(t+\dfrac{1}{t})\big]\sim \sum_{n=-\infty}^\infty I_{\nu+n}(x)\, t^n.
\end{equation}
\item Let $\nu\in\mathbb{C}^\ast$. Then
\begin{equation}\label{E:gf_K_nu}
i\pi \, t^{-\nu}\exp\big[-\dfrac{x}{2}\big(t+\dfrac{1}{t})\big]\sim \sum_{n=-\infty} ^\infty K_{\nu+n}(x)\, t^n.
\end{equation}
\end{enumerate}
where the notation ``$\sim$" denotes the Borel resummation as defined in Definition \ref{D:borel}. \end{corollary}
\begin{remark} We note that the \eqref{E:gf_J_nu} and \eqref{E:gf_I_nu} above become well-known equalities when $\nu$ is an integer. \end{remark}
\begin{proof} \begin{enumerate}
\item Suppose $\mathscr{C}_{\nu+n}=J_{\nu+n}$ in \eqref{E:gf_bessel}. We aim to show that $C_\nu=1$. In particular, we have
\begin{equation}\label{E:gf_J_nu_1}
C_\nu t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]-\frac{J_{0}(x)}{t}\sim\sum_{n=-\infty \atop n\not=0}^{+\infty} J_{\nu+n}(x)\, t^{n-1}.
\end{equation}
That is, there is a $F\in \mathcal{O}_{dd}$\footnote{
It is clear that
\[
\sum_{n=-\infty \atop n\not=0}^{+\infty} \frac{J_{\nu+n}(x)}{n}
\, t^{n}
\]is a formal primitive of the right-side of \eqref{E:gf_J_nu_1}.} such that
\begin{equation}\label{E:gf_J_nu_2}
C_\nu t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]-\frac{J_{0}(x)}{t}
=\mathfrak{z}\big( \partial_2(J_{\nu+n}/n)\big)=\partial_2 F
\end{equation}where the $z$-transform given by the Borel-resummation is defined in \eqref{E:z-transform-2}, and the $\partial_2$'s are defined in \eqref{E:two-seq-functions_2}, where $\lim_{t\to -\infty}F(x, t)=0$ if $\mathop{\rm Re}(x)>0$. According to Theorem \ref{T:z-transform-2}, the Borel transform $\mathfrak{B}$ representing the $\mathfrak{z}$-transform is $\mathcal{A}$-linear, it follows that the left-side of \eqref{E:gf_J_nu_1} also has a primitive, which can be written as
\[
\int \Big\{C_\nu t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]-\frac{J_{\nu}(x)}{t}\Big\}\, dt.
\]
Let $0<\delta< R$ be given and $|\arg x|<\pi/2$. Consider the contour
\[
\Gamma_{R, \delta}=(Re^{-i \pi},\, \delta e^{-i \pi}) \cup \mathcal{C}_\delta\cup (\delta e^{i \pi},\, Re^{i \pi}),
\]where $\mathcal{C}_\delta$ denotes the circle centred at the origin with radius $\delta$. Thus the contour $\Gamma_{R, \delta}$ can be considered a \textit{truncated Hankel\rq{}s contour} which emanates from $-\infty$ below the negative real-axis and then back to $-\infty$ above the negative real-axis after circulating the origin once in an anti-clockwise direction.
Hence
\[
C_\nu\int_{\Gamma_{R, \delta}} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt=\int_{\Gamma_{R, \delta}} \frac{J_{\nu}(x)}{t}\, dt
\]
We now take the Cauchy principal values on both sides of this equation under the limits that $R\to+\infty$ and $\delta\to 0$. The right-side yields
\[
C_\nu\int_{\Gamma_{\infty, 0}} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt
=2\pi i\, J_\nu (x).
\]However, since the integral on the left-side converges to a Hankel contour, so we obtain
\[
\int_{\Gamma_{R, \delta}} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt
\longrightarrow
\int_{-\infty}^{(0+)} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt=2\pi i\, J_\nu(x)
\]as $R\to+\infty$ and $\delta\to 0$, which actually equals to $2\pi i J_\nu(x)$ from the classical Schl\"afli-Sonine integrals which is valid for $|\arg x|<\frac{\pi}{2}$, see e.g., Watson \cite[Chap. VI, p. 176, (2)]{Watson1944}. This proves that $C_\nu=1$ after comparing the above calculations. Alternatively, one can verify directly that the Hankel-type contour as the limit of $R\to 0,\, \delta\to 0$ in \eqref{E:gf_J_nu_2} satisfies the Bessel equation \eqref{E:Bessel_eqn} and the following asymptotic
\[
\begin{split}
\int_{-\infty}^{(0+)} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt& \stackrel{t=\frac{2x}{u}}{
=} \frac{x^\nu}{2^\nu}
\int_{-\infty}^{(0+)} u^{-\nu-1}\exp\big[u-\dfrac{x^2}{4u}\big]\, du\\
&\approx \frac{x^\nu}{2^\nu}\frac{2^\nu}{\Gamma(\nu+1)}=\frac{2^\nu}{\Gamma(\nu+1)}
\end{split}
\]when $x\to 0$. See also Watson \cite[pp. 175-176]{Watson1944}. Hence
\[
\int_{-\infty}^{(0+)} t^{-\nu-1}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt=2\pi i\, J_\nu(x).
\]Therefore, we deduce $C_\nu=1$. This proves \eqref{E:gf_J_nu}.
\item Recall that
\[
Y_{\nu+n}(x)=\frac{\cos (\nu+n)\pi\, J_{n+\nu}-J_{-\nu-n}}{\sin \nu\pi}
=\cot\nu\pi\, J_{\nu+n}(x)-(-1)^n \csc\nu \pi\, J_{-\nu-n}(x).
\]Since $\big(J_{-\nu-n}(x)\big)_n$ corresponds to PDEs \eqref{E:bessel_PDE} with $\nu$ replaced by $-\nu$. Hence
\[
t^{\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\sim\sum_{n=-\infty}^\infty J_{-\nu-n}(x)\, t^n.
\]Thus
\[
\begin{split}
\sum_{n=-\infty}^\infty Y_{\nu+n}(x)
&\sim\sum_{n=-\infty}^\infty
\big[\cot \nu\pi J_{\nu+n} (x)-(-1)^n\csc\nu \pi J_{-\nu-n}(x)\big]\, t^n\\
&=\sum_{n=-\infty}^\infty \cot \nu\pi J_{\nu+n}(x)\, t^n
- \csc\nu\pi \sum_{n=-\infty}^\infty (-1)^nJ_{-\nu-n}(x) \, t^n\\
&=\cot \nu\pi\sum_{n=-\infty}^\infty J_{\nu+n}(x)\, t^n
- \csc\nu\pi \sum_{n=-\infty}^\infty J_{-\nu+n}(x) \, (-1/t)^n\\
&=\cot \nu\pi t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]
-\csc\nu\pi (-1/t)^\nu \exp\big[\dfrac{x}{2}\big(-\dfrac{1}{t}+t)\big]\\
&=\big(\cot \nu\pi-(-1)^\nu \csc\nu\pi\big) t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\\
&= e^{-i\pi/2} t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]
\end{split}
\]as asserted.
\item The proofs of (iii) and (iv) can be derived in a similar way as the proof of (ii) with the definitions $I_\nu(x)=J_\nu(ix)$ and $K_\nu(x)=\frac{\pi}{2}(I_{-\nu}(x)-I_\nu(x))/\sin\nu\pi$.
\end{enumerate} \end{proof}
By an argument using the Cauchy integral formula, we can now "extract" the $n$-th coefficient from the generating function \eqref{E:gf_J_nu} above to give a way to explain why the existence of the following integral representation of Bessel functions avoids using series expansions as in \cite[\S17.231]{WW}, \cite{Watson1944}, see also \cite[\S 10.10-11]{Derezinski2020}.
\begin{corollary}[(\textbf{Schl\"afli-Sonine integrals})]\label{C:Sonine}
Let $\nu\in\mathbb{C}$ be given and $|\arg x|<\displaystyle\frac{\pi}{2}$. Then we have
\begin{equation}\label{E:Sonine}
J_{\nu+n}(x)=\frac{1}{2\pi i}\int_{-\infty}^{(0+)} t^{-\nu-1-n}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\, dt
\end{equation}
holds for every $n\in \mathbb{Z}$. \end{corollary}
\subsection{Generating function of difference Bessel functions}\label{SS:delta_gf}
Let us recall the Example \ref{Eg:O_delta} that $\mathcal{O}_{\vartriangle}$ represents the space of analytic functions defined on a domain is a left $\mathcal{A}_1$-module with
\[
(\partial f)(x)=f(x+1)-f(x), \qquad
(Xf)(x)=xf(x-1),\quad f\in \mathcal{O}_{\vartriangle}.
\]
We will occasionally use the well-known \textit{Newton polynomials} written in falling factorial notations:
\[
(x)_0=1,\quad
(x)_n=
\begin{cases}
x(x-1)\cdots (x-n+1), & n\ge 1,\\
1/[(x+1)\cdots (x+n)] & n\le -1.
\end{cases}
\]For non-integer $\nu$, we adopt the notation
\begin{equation}\label{E:fractional_Newton}
(x)_\nu=\frac{\Gamma (x+1)}{\Gamma (x+1-\nu)}.
\end{equation}
\begin{example}[(\textbf{Difference Exponential function})]\label{Eg:delta_exp} Let $a\in\mathbb{C}$. We have seen in Example~\ref{Eg:exp} that the Weyl exponential is any solution of the $D$-module $\mathcal{A}/\mathcal{A}(\partial-a)$. Now in particular, the \textit{difference exponential function} $\exp_\Delta(\cdot;a)$ is defined as a solution of $\mathcal{A}/\mathcal{A}(\partial-a)$ in the $D$-module $\mathcal{O}_\Delta$, i.e., \[
\mathcal{A}/\mathcal{A}(\partial-a) \stackrel{\times\exp_\Delta(\cdot;a)}{\longrightarrow} \mathcal{O}_{\Delta}. \] To obtain a binomial (or ``Newton") series expansion of $\exp_\Delta(\cdot;a)$, we factorize the above left $\mathcal{A}$-linear map as \[
\mathcal{A}/\mathcal{A}(\partial-a) \stackrel{\times S}{\longrightarrow} \mathcal{A}/\mathcal{A}\partial \stackrel{\times 1}{\longrightarrow} \mathcal{O}_{\Delta}, \] in which $S=\sum{\frac{a_k}{k!}X^k}$ as in \eqref{E:exp}. We can illustrate the above by the commutative diagram \begin{equation}
\begin{tikzcd}
\mathcal{A}/\mathcal{A}(\partial-a) \arrow{r}{\times S}
\arrow[swap]{dr}{\times \exp_\Delta(\cdot;a)}
& \mathcal{A}/\mathcal{A}\partial
\arrow{d}{\times 1} \\
& \mathcal{O}_{\Delta}
\end{tikzcd}
\end{equation} As a result, we obtain
\[
\exp_\Delta(x;a)=\sum_{k=0}^{\infty} \frac{a^k}{k!} x(x-1)\cdots(x-k+1).
\]
It is well-known that this binomial series converges absolutely to $(a+1)^x$ for any complex $x$ and any $|a|<1$, when an appropriate branch of logarithm is chosen. \end{example}
\begin{example}[(\textbf{Difference trigonometric functions})]\label{Eg:dtrigo} We continue the list of producing difference versions of trigonometric functions from Example \ref{Eg:trigo}, where the Weyl sine and the Weyl cosine are solutions of the $D$-module $\mathcal{A}/\mathcal{A}(\partial^2+1)$ in a $D$-module $N$. Now in particular, we define the \textit{difference sine function} and the \textit{difference cosine function} to be solutions of $\mathcal{A}/\mathcal{A}(\partial^2+1)$ in the $D$-module $\mathcal{O}_\Delta$. To obtain binomial series expansions of these two functions, we use the factorization
\begin{equation}
\mathcal{A}/\mathcal{A}(\partial^2+1) \stackrel{\times S}{\longrightarrow} \mathcal{A}/\mathcal{A}\partial \stackrel{\times 1}{\longrightarrow} \mathcal{O}_{\Delta}
\end{equation} in which we respectively take $S=\sin X$ and $S=\cos X$ as in \eqref{E:sine-map} and \eqref{E:cosine-map}. This factorization can be illustrated by the commutative diagram
\begin{equation}
\begin{tikzcd}
\mathcal{A}/\mathcal{A}(\partial^2+1) \arrow{r}{\times S}
\arrow[swap]{dr}{\times \mathrm{Sin\, }X/\times \mathrm{Cos\, }X}
& \mathcal{A}/\mathcal{A}\partial
\arrow{d}{\times 1} \\
& \mathcal{O}_{\Delta}
\end{tikzcd}
\end{equation}
Thus we obtain
\[
\sin_\Delta x:=\frac{1}{2i}[(1+i)^x-(1-i)^x]=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{(2k-1)!}(x)_{2k-1}
\]
and
\[
\cos_\Delta x:=\frac{1}{2}[(1+i)^x+(1-i)^x]=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}(x)_{2k}.
\]
These two series converge absolutely if and only if either $\mathop{\rm Re}(x) > 0$ or $x = 0$, which are unlike the power series of the usual sine and cosine functions. \end{example}
\begin{example}[(\textbf{Difference Bessel functions})] \label{Eg:delta_Bessel_fn} Let $\mathcal{A}=\mathcal{A}_1$ as before. Recall that the Weyl Bessel of order $\nu$ \eqref{E:bessel_map} is any solution of the Bessel operator
\[
L=(X\partial)^2+X^2-\nu^2
\] in some $D$-module $N$, in particular, we define the \textit{difference Bessel function of order $\nu$}, denoted by $J^{\Delta}_{\nu}(x)$, to be a solution of $\mathcal{A}/\mathcal{A}L$ in the $D$-module $\mathcal{O}_\Delta$. Recall the factorization from \eqref{E:pos_bessel_comp_map}
\begin{equation}\label{E:composition}
\mathcal{A}/\mathcal{A}L\stackrel{\times S}{\longrightarrow} \mathcal{A}/\mathcal{A}(X\partial-\nu)\stackrel{\times (x)_\nu}{\longrightarrow}
\mathcal{O}_\Delta,
\end{equation}
in which
\[
S=\mathcal{J}_\nu=\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}.
\]
Thus we have obtained the Newton series expansion
\begin{equation}\label{E:difference_bessel_fn}
J^{\Delta}_{\nu}(x): =S((x)_\nu)
=\sum_{k=0}^{\infty} \frac{(-1)^k}{2^{\nu+2k}k!\Gamma(\nu+k+1)}(x)_{\nu+2k}.
\end{equation} $J^{\Delta}_{\nu}(x)$ solves the second order difference equation \eqref{E:difference_bessel_eqn_0} or in its equivalent form
\begin{equation}\label{E:difference_bessel_eqn}
[(x+2)^2-\nu^2] y(x+2)-(x+2)(2x+3)y(x+1)+2(x+2)(x+1)y(x)=0,
\end{equation}
which is obtained from the action of the element $L=(X\partial)^2+(X^2-\nu^2)$ in $\mathcal{O}_\Delta$, see Example \ref{Eg:O_delta}.
When $x\in \mathbb{Z}$, \eqref{E:difference_bessel_eqn} and \eqref{E:difference_bessel_fn} are precisely the difference Bessel equation and the difference Bessel function $ J^{\Delta}_{\nu}(x)$ that appear to be first studied by Bohner and Cuchta \cite{BC_2017}\footnote{Bohner and Cuchta \cite{BC_2017} called $J^\Delta_\nu(x)$ the ``Bessel difference function\rq\rq{} and the equation \eqref{E:difference_bessel_eqn} the ``Bessel difference equation\rq\rq{} instead.}. Either by replacing $\nu$ in \eqref{E:difference_bessel_fn} by $-\nu$ or by considering \eqref{E:neg_nu_map}, we easily obtain
\begin{equation}\label{E:difference_bessel_fn_2}
J^{\Delta}_{-\nu}(x):=\sum_{k=0}^{\infty} \frac{(-1)^k}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}(x)_{-\nu+2k},
\end{equation}
so that $\{J^{\Delta}_{\nu}, J^{\Delta}_{-\nu}\}$ are $\mathbb{C}$-linearly independent solutions of \eqref{E:difference_bessel_eqn} in $\mathcal{O}_\Delta$, provided that $2\nu\notin\mathbb{Z}$.
\end{example}
We now provide a statement about an half-plane convergence of the $ J^{\Delta}_{\nu}(x)$.
\begin{proposition}\label{P:bessel_convergence} Let $\nu\in \mathbb{C}$ be arbitrary. Then the difference Bessel function $ J^{\Delta}_{\nu}(x)$ \eqref{E:difference_bessel_fn} converges uniformly in each compact subset of $\mathop{\rm Re}(x)>0$. \end{proposition}
A justification of the above statement will be given in the Appendix \S\ref{SS:bessel_convergence}. However, it was demonstrated in \cite[p. 1569]{BC_2017} that although the series $ J^{\Delta}_{n}(x)$ diverges for $x<-1$, it converges at $x=-1$. Thus, a natural question is about the convergence in the strip $\{x: -1<\mathop{\rm Re} x<-\frac12\}$.
Similar to Example \ref{Eg:bilateral}, we have
\begin{example}\label{Eg:bilateral} Let $\nu\in \mathbb{C}$ and consider the bilateral sequence of classical Bessel functions $(J^\Delta_{\nu+n})_n$. Then the maps
\begin{equation}\label{E:j-delta-map-bessel}
\begin{array}{rcl}
\mathfrak{j}_\Delta^+:\mathcal{B}_\nu & \stackrel{\times (J^\Delta_{\nu+n})}{\longrightarrow} &\mathcal{O}^\mathbb{Z}
\end{array}
\end{equation} and
\begin{equation}
\begin{array}{rcl}
\mathfrak{j}_\Delta^-:\mathcal{B}_\nu & \stackrel{\times (J^\Delta_{-\nu-n})}{\longrightarrow} &\mathcal{O}^\mathbb{Z}
\end{array}
\end{equation} are both left $\mathcal{A}_2$-linear. More generally, a general solution of $\mathcal{B}_\nu$ in $\mathcal{O}_\Delta^\mathbb{Z}$ will be denoted by $\times \mathrm{(}\mathscr{C}^\Delta_{\nu+n}\mathrm{)}$. That is,
\begin{equation}\label{E:general-delta-map-bessel}
\begin{array}{rcl}
\mathfrak{j}_\Delta: \mathcal{B}_\nu & \stackrel{\times (\mathscr{C}^\Delta_{\nu+n})}{\longrightarrow} &\mathcal{O}_\Delta^\mathbb{Z}.
\end{array}
\end{equation}
\end{example}
The following proposition and corollary are direct consequences of Corollary~\ref{C:Bessel_trans} with $N=\mathcal{O}_\Delta$ and $\mathscr{C}^\Delta_{\nu+n}$.
\begin{proposition}[(\bf{Delay-difference formulae})]\label{P:bilateral_Delta_PDE} Let $\nu\in\mathbb{C}$ and $\mathrm{(}\mathscr{C}^\Delta_{\nu+n}\mathrm{)}$ be an arbitrary (bilateral sequence of) solution of $\mathcal{B}_\nu\to \mathcal{O}_\Delta^\mathbb{Z}$ as defined in \eqref{E:general-delta-map-bessel}. Then the $\mathrm{(}\mathscr{C}^\Delta_{\nu+n}\mathrm{)}$
satisfies
the following delay-difference formulae
\begin{equation}\label{E:delta_bessel_PDE_a}
x\Delta \mathscr{C}^{\Delta}_{\nu}(x-1)+\nu\mathscr{C}^{\Delta}_{\nu}(x)-
x\mathscr{C}^{\Delta}_{\nu-1}(x-1)=0
\end{equation} and
\begin{equation}\label{E:delta_bessel_PDE_b}
x\Delta \mathscr{C}^{\Delta}_{\nu}(x-1)-\nu\mathscr{C}^{\Delta}_{\nu}(x)+
x\mathscr{C}^{\Delta}_{\nu+1}(x-1)=0.
\end{equation} for each $n$.
\end{proposition}
The following three-term recurrence is a direct consequence of Corollary~\ref{C:Bessel_3term} with $N=\mathcal{O}_\Delta$, or directly from the above two corollary.
\begin{proposition}[(\textbf{Three-term recurrence})] For every $\nu\in\mathbb{C}$ and every $x\in\mathbb{C}$, we have
\begin{equation}\label{E:delta_bessel_3_term_a}
2\nu\mathscr{C}_{\nu}^\Delta(x)-x\mathscr{C}_{\nu-1}^\Delta(x-1)-x\mathscr{C}_{\nu+1}^\Delta(x-1)=0.
\end{equation} \end{proposition}
One deduces the following formulae originally derived by Bohner and and Cuchta \cite[Theorems 5-6, Corollary 7]{BC_2017} about $J_\nu^\Delta$ as consequences of the above when $\mathscr{C}_{\nu+n}^\Delta(x)=J_{\nu+n}^\Delta(x)$ for all $n$.
\begin{corollary}[({\cite[Theorems 5-6, Corollary 7]{BC_2017}})] For every $\nu\in\mathbb{C}$ and every $x\in\mathbb{C}$, we have
\begin{align}
x\Delta J^{\Delta}_{\nu}(x-1)+\nu J^{\Delta}_{\nu}(x)-
xJ^{\Delta}_{\nu-1}(x-1)&=0,\label{E:delta_bessel_PDE_1}
\\
x\Delta J^{\Delta}_{\nu}(x-1)-\nu J^{\Delta}_{\nu}(x)+
xJ^{\Delta}_{\nu+1}(x-1)&=0,\label{E:delta_bessel_PDE_2} \\
2\nu J_{\nu}^\Delta(x)-xJ_{\nu-1}^\Delta(x-1)-xJ_{\nu+1}^\Delta(x-1)&=0.\label{E:delta_bessel_3_term}
\end{align} \end{corollary}
\begin{proof} We provide an alternative proof based on the transmutation formulae from Proposition \ref{P:bessel_transmutation}. This more direct approach is similar to the derivation of the Corollary \ref{C:bessel_formulae}. Thus it suffices to prove the \eqref{E:delta_bessel_PDE_1}. Since the right-side of \eqref{E:bessel_transmutation_1} annihilates the $CJ^\Delta_\nu(x)$ for some constant $C$, and the fact that on the Newton series \eqref{E:difference_bessel_fn} gives
\[
\begin{split}
(\partial+\nu/X)J^\Delta_\nu(x)
&=
J_{\nu}^\Delta(x) -J_{\nu}^\Delta(x-1)+\nu J_{\nu}^\Delta(x+1)/(x+1)\\
&= \frac{\Gamma(x+1)}{2^\nu\Gamma(\nu+1)\Gamma(x+2-\nu)}
\Big( \frac2\nu+O\big(\frac1x\big)\Big)\\
&=\frac{\Gamma(x+1)}{2^{\nu-1}\Gamma(\nu)\Gamma(x+2-\nu)}
\Big( 1+O\big(\frac1x\big)
\Big)\\
\end{split}
\]as $x\approx 0$. On the other hand, according to the left-side of \eqref{E:bessel_transmutation_1}, the $(\partial+\nu/X)J^\Delta_\nu(x)$ must be equal to $CJ^\Delta_{\nu-1}(x)$ for some constant $C$. But
\[
J_{\nu-1}^\Delta(x)\approx \frac{\Gamma(x+1)}{2^{\nu-1}\Gamma(\nu)\Gamma(x+2-\nu)}
\]as $x\approx 0$. One immediately deduces that $C=1$ by comparing the above two asymptotes. This establishes the \eqref{E:delta_bessel_PDE_1}. The verification of \eqref{E:delta_bessel_PDE_2} is similar. One can derive the \eqref{E:delta_bessel_3_term} by subtracting the \eqref{E:delta_bessel_PDE_1} and \eqref{E:delta_bessel_PDE_2}.
\end{proof}
Now we establish that a difference analogue of the generating function $e^{\frac{x}{2}(t-\frac1t)}$ of the classical Bessel function for difference Bessel functions \eqref{E:bessel_classical_gf} is given by the function
\[
\big[\frac12\big(t-\frac1t\big)+1\big]^x
\]and to show that it serves as a generating function to integer order difference Bessel function to \eqref{E:difference_bessel_eqn} in $\mathcal{O}_\Delta$.
\begin{theorem}\label{T:delta_bessel_0_gf}
Let $ |\frac{1}{2}(t-\frac{1}{t})|<1$ and $x\in\mathbb{C}$. Then
\begin{equation}\label{gf-db-1}
\Big[\frac{1}{2}(t-\frac{1}{t})+1\Big]^{x}=\sum_{n=-\infty}^{\infty}
J^{\vartriangle}_{n}(x)\, t^n.
\end{equation} \end{theorem}
\begin{proof} Since $| \frac{1}{2}(t-\frac{1}{t})|<1$, so the series
\[
\Big[\frac{1}{2}(t-\frac{1}{t})+1\Big]^{x}=\sum_{k=0}^{\infty} \binom{x}{k}
\big[ \frac{1}{2}(t-\frac{1}{t})\big]^{k}
\]is absolutely convergent.
Writing
\[
\Big[ \frac{1}{2}(t-\frac{1}{t})\Big]^{k}= \sum_{s=0}^{k}\binom{k}{s}
(-\frac{1}{2t})^s (\frac{t}{2})^{k-s}.
\]
Then
\[
\Big[\frac{1}{2}(t-\frac{1}{t})+1\Big]^{x}=
\sum_{k=0}^{\infty}(x)_{k} \sum_{s=0}^{k}\frac{1}{(k-s)! s!}
(-\frac{1}{2t})^s (\frac{t}{2})^{k-s}.
\]
It is clear that the power of $t$ goes from $-\infty$ to $\infty$, as $k$ goes to infinity. Then we exchange the order of summations to arrive at
\[
\sum_{-\infty}^{\infty} t^n\sum_{s=0}^{\infty}(-1)^{s}
\frac{(x)_{n+2s}}{s!(n+s)!2^{n+2s}}
=\sum_{-\infty}^{\infty} J^{\Delta}_{n}(x)t^n.
\]
\end{proof}
The series manipulation technique used in the above proof is in the same spirit of that for the classical Bessel generating function \eqref{E:bessel_classical_gf} presented in \cite[\S2.1]{Watson1944}. In order to deal with the generating function when $\nu\not=0$, we first need to study the asymptotic behaviour of $J^\Delta_{\nu+n}$ as $n\to\infty$. We apply the same technique of Banach algebra to $(J^\Delta_{n+\nu})$ as in the proof of Theorem \ref{T:Bessel_Gevrey} with suitable modification.
\begin{theorem}[(\bf{Asymptotic behaviour})]\label{T:delta_Bessel_Gevrey} The sequence $(J^\Delta_{n+\nu})$ is a uniformly 1-Gevrey compacta solution of $\mathcal{B}_\nu$ in $\mathcal{O}_\Delta^\mathbb{Z}$. \end{theorem}
\begin{proof} The proof makes use of the Newton transform $\mathfrak{N}$ which is described in Theorem \ref{T:Newton_trans}. Fix a compact $K\subset\mathbb{C}$ and for each $f\in\mathcal{O}_d$, one defines a norm by \[
|\| f |\| =\|\mathfrak{N}(f)\|_{L^\infty,K}, \]
where the right-side is the sup norm on the compact set $K$. Then $(\mathcal{O}_d,|\|\cdot |\|)$ becomes a Banach algebra which can be verified easily.
The proof of Theorem \ref{T:Bessel_Gevrey} under this alternative norm shows that there exists $C>0$ such that \[
|\|J_{\nu+n} |\| \leq Cn!(\dfrac{2}{|\|x |\|})^n\quad\mbox{for large }n. \] Consequently, for large $n$, \[
\|J^\Delta_{\nu+n}\|_{L^\infty,K}=\|\mathfrak{N}(J_{\nu+n})\|_{L^\infty,K}=|\|J_{\nu+n}|\|\leq Cn!(\dfrac{2}{|\|x|\|})^n. \] \end{proof}
\begin{remark} Similarly, the sequence $(J^\Delta_{\nu-n})$ is also 1-Gevrey. The same is true if $J^\Delta$ is replaced by $\mathscr{C}^\Delta$. As a consequence, the $z$-transform of the sequence $(J^\Delta_{\nu+n})$ is a well-defined analytic function in two variables. \end{remark}
\begin{theorem}\label{T:difference_Bessel_gf} Let $\nu\in\mathbb{C}$.
\begin{enumerate}
\item Let $(\mathscr{C}^\Delta_{n+\nu})_n$ be a bilateral sequence of analytic functions which is a solution of the Bessel module $\mathcal{B}_\nu$ in $\mathcal{O}_{\Delta}^\mathbb{Z}$. Then there exists a $1$-periodic function $C_\nu$ such that
\begin{equation}\label{E:delta_bessel_gf}
C_\nu(x)\, t^{-\nu}\big[\dfrac{1}{2}(t-\dfrac{1}{t})+1\big]^x
\sim\sum_{n=-\infty}^\infty \mathscr{C}_{\nu+n}^\Delta (x)\, t^n,
\end{equation}
where the symbol $\sim$ means that the left-hand side is the Borel resummation \footnote{Definition \ref{D:borel}} of the right-hand side whenever it diverges.
\item Moreover, the manifestation of the holonomic system of PDEs \eqref{E:bessel_PDE} in $\mathcal{O}_{\Delta d}$ in Example \ref{Eg:O_delta_d} defined by \eqref{E:O_delta_d} is given by
\begin{equation}\label{E:PDE_dbessel}
y(x+1, t)-y(x, t)=
\frac{1}{2}(t-\frac{1}{t}) y(x, t),\quad
\nu y(x,t) +ty_t(x,t)=\frac{x}{2}\big(t+\frac{1}{t}\big)y(x-1, t).
\end{equation}
\end{enumerate} \end{theorem}
To prove the above theorem, we need the following lemma, which serves the purpose of solving the holonomic system of PDEs that arise from $\mathcal{B}_\nu$ in $\mathcal{O}_{\Delta d}$. \begin{lemma} \label{E:classical_app_2}
\[
\begin{array}{rcl} \dfrac{\mathcal{A}_2}{\mathcal{A}_2[\partial_1+\frac{1}{2}(1/{X_2}-X_2)]+\mathcal{A}_2[2(\nu+X_2\partial_2)-X_1(X_2+1/{X_2})]}&\stackrel{\times t^{-\nu}[\frac{1}{2}(t-\frac{1}{t})+1]^x }{\longrightarrow} &\mathcal{O}_{\Delta d}
\end{array}
\]is a left $\mathcal{A}_2$-linear map. \end{lemma}
\begin{proof} It follows from the two generators of the $\mathcal{B}_\nu$ defined above that we solve the system of PDEs
\[
\partial_1 y+\frac{1}{2}(\dfrac{1}{X_2}-X_2)y=0,\qquad
(\nu+ X_2\partial_2)y-\frac{X_1}{2}(\dfrac{1}{X_2}+X_2)y=0
\]for an analytic function $y=y(x,\, t)$ for the current choices of $\partial_j,\, X_j,\ j=1,\, 2$. That is,
\begin{equation}\label{E:PDE_dBessel_gf}
y(x+1, t)-y(x, t)=
\frac{1}{2}(t-\frac{1}{t}) y(x, t),\qquad
\nu y(x,t) +t\frac{\partial y}{dt}(x,t)=\frac{x}{2}( t+\frac{1}{t})y(x-1, t).
\end{equation}
The first equation yields $y(x,t)=g(t)\big[\dfrac{1}{2}(t-\dfrac{1}{t})+1\big]^x$ for some $g$. Substituting this $y$ into the second differential-difference equation yields $g(t)=C_\nu t^{-\nu}$ for some $C_\nu\not=0$. \end{proof}
\noindent\textit{Proof of Theorem~\ref{T:difference_Bessel_gf}.} Recall the maps $\mathfrak{j}_\Delta$ and $\mathfrak{z}_\Delta$ as defined in \eqref{E:general-delta-map-bessel} and \eqref{E:z-transform-3} respectively. We have the diagram
\begin{equation}
\begin{tikzcd} [row sep=huge, column sep=huge]
\mathcal{B}_\nu \arrow{r}{\mathfrak{j}_\Delta}
\arrow[swap]{d}{\times{\mathop{\rm E}}\big[\frac{X_1}{2}\big(X_2-\frac{1}{X_2}\big)\big]}
& \mathcal{O}^{\mathbb{Z}}_\Delta
\arrow{d}{\mathfrak{z}_\Delta} \\
\widetilde{A}_2 \arrow{r}{\times t^{-\nu}}& \mathcal{O}_{\Delta d}
\end{tikzcd}
\end{equation} in which $\widetilde{A}_2:=\overline{\mathcal{A}_2/\big[\mathcal{A}_2\partial_1+\mathcal{A}_2(X_2\partial_2+\nu)\big]}$. Since the above diagram commutes up to a periodic multiple, the sum $\sum_n \mathscr{C}_{\nu+n}^\Delta (x)\, t^n$, being the image of $1$ in $\mathcal{O}_{\Delta d}$ via the top right path, is also a formal solution to the system of PDEs \eqref{E:bessel_PDE}. Now $t^{-\nu}\big[\dfrac{1}{2}(t-\dfrac{1}{t})+1\big]^x $ is a solution to \eqref{E:bessel_PDE} because of Lemma~\ref{E:classical_app_2}. Since $\mathcal{B}_\nu$ is holonomic, the $\mathbb{C}$-dimension of the local solution space of \eqref{E:bessel_PDE} equals the multiplicity of $\mathcal{B}_\nu$, which is one. So \eqref{E:delta_bessel_gf} holds up to a complex $1$-periodic function $C_\nu(x)$.
\qed
\begin{theorem}\label{T:delta_bessel_gf}
\begin{enumerate}
\item Let $\mathop{\rm Re}(x)<\mathop{\rm Re}(\nu)$. Then \begin{equation}\label{gf-db-2} e^{i\pi\nu} \frac{\sin(x-\nu)\pi}{\sin(\pi x)}\, t^{-\nu}\big[\frac{1}{2}(t-\frac{1}{t})+1\big]^{x} {\color{blue}\sim} \sum_{n=-\infty}^{\infty} J^{\Delta}_{n+\nu}(x)\,t^n, \end{equation} where the notation ``$\sim$" means that the left-side is the Borel-resummation of the right-side.
\item Moreover, the the generating function from part \textrm{(i)} satisfies the system of PDE \eqref{E:PDE_dBessel_gf}\footnote{See also the Appendix \ref{SS:PDE_list}.}
\end{enumerate} \end{theorem}
\begin{proof} From Theorem~\ref{T:difference_Bessel_gf}, we have
\[
C_{\nu}(x)t^{-\nu}\big[\dfrac{1}{2}\big(t-\dfrac{1}{t})+1\big]^x\sim\sum_{n=-\infty}^\infty J^\Delta_{\nu+n}(x)\, t^n.
\]
Subtracting $J_\nu(x)$ from both sides and then dividing both sides by $t$, we obtain
\begin{equation}\label{Eq:integral3}
C_{\nu}(x)t^{-\nu-1}\big[\dfrac{1}{2}\big(t-\dfrac{1}{t})+1\big]^x-\frac{J_\nu(x)}{t}\sim\sum_{n=-\infty,n\ne 0}^\infty J^\Delta_{\nu+n}(x)\, t^{n-1}.
\end{equation}
Now note that the right-side of \eqref{Eq:integral3} has an anti-$\partial_2$ (e.g., $\partial_2(\sum_{n=-\infty,n\ne 0}^\infty \frac{J^\Delta_{\nu+n}(x)}{n}\, t^n)$). Since the Borel resummation is $\mathcal{A}$-linear, it follows that the LHS of \eqref{Eq:integral3} has an antiderivative.
Now let $0<\delta< R$ be given and $|\arg x|<\pi/2$, and consider the contour
\[
\Gamma_{R, \delta}=(Re^{-i \pi},\, \delta e^{-i \pi}) \cup \mathcal{C}_\delta\cup (\delta e^{i \pi},\, Re^{i \pi}),
\]where $\mathcal{C}_\delta$ denotes the circle centred at the origin with radius $\delta$. Thus the contour $\Gamma_{R, \delta}$ can be considered a truncated Hankel\rq{}s contour which emanates from $-\infty$ below the negative real-axis and then back to $-\infty$ above the negative real-axis after circulating the origin once in an anti-clockwise direction. By Cauchy integral theorem, the integral of the left-side of \eqref{Eq:integral3} along $\Gamma_{R, \delta}$ is zero, i.e.,
\begin{equation}\label{Eq:integral4}
C_{\nu}(x)\int_{\Gamma_{R, \delta}} t^{-\nu-1}\big[\dfrac{1}{2}\big(t-\dfrac{1}{t})+1\big]^x\, dt = \int_{\Gamma_{R, \delta}}\frac{J^\Delta_\nu(x)}{t} \,dt = 2\pi i J^\Delta_\nu(x).
\end{equation}
By \eqref{E:dBessel integral rep} which is an analytic result to be obtained in the next subsection, we have
\[
C_\nu(x)=e^{i\pi\nu}\frac{\sin(x-\nu)\pi}{\sin(\pi x)}.
\] \end{proof}
We remark that when $\nu=0$ in the above theorem, one can recover the infinite sum in \eqref{E:delta_bessel_gf} from the generating function $\big[\dfrac{1}{2}(t-\dfrac{1}{t})+1\big]^x$ by Theorem \ref{T:delta_bessel_0_gf}, thus showing that $C_0(x)\equiv 1$.
\subsection{Integral representations} Now we aim to find an integral representation of the difference Bessel functions that is analogous to the Sonine integral representation in Corollary~\ref{C:Sonine}.
\begin{theorem}\label{T:solutions of db equ}
Let $\nu\in \mathbb{C}$ and $\mathop{\rm Re} (x)<\mathop{\rm Re} (\nu)$. Let $U$ be a Hankel-type contour in the $t$-plane that starts from, $+\infty$ above the real-axis, circles around the origin in the counter-clockwise direction, and returns to $+\infty$ below the real-axis, and in particular
\[
t^{-\nu} \big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x}\bigg\vert_{\partial U}
=0.
\]
Then the integral
\[
y_\nu(x)=\int_{U} t^{-\nu-1} \big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x}\, dt
\]
\begin{enumerate}
\item satisfies the delay-difference equations \eqref{E:delta_bessel_PDE_1} and \eqref{E:delta_bessel_PDE_2} above,
\item and, in particular, the integral solves the Bessel difference equation of order $\nu$ \eqref{E:difference_bessel_eqn}:
\[
x(x-1)\triangle^2 y(x-2)+x\triangle y(x-1)+x(x-1)y(x-2)-\nu^2y(x)=0.
\]
\end{enumerate}
\end{theorem}
\begin{proof} We first note that because of $\mathop{\rm Re} (x)>\mathop{\rm Re} (-\nu)$, so that integral is a well-defined function of $x$.
\begin{enumerate}
\item We rewite the \eqref{E:delta_bessel_PDE_1} and \eqref{E:delta_bessel_PDE_2} in the equivalent forms
\[
\begin{split}
&2\triangle y_{\nu}(x)-y_{\nu-1}(x)+y_{\nu+1}(x)=0,\\
&2\nu y_{\nu}(x)-xy_{\nu-1}(x-1)-xy_{\nu+1}(x-1)=0.
\end{split}
\]
Then, it is easy to see that
\[
2\triangle y_{\nu}(x)
=\int_{U}( t^{-\nu} -t^{-\nu-2})
\big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x} dt
=y_{\nu-1}(x)-y_{\nu+1}(x)
\]holds. To verify the second equation, we substitute the $y_n$ into the equation and apply integration-by-parts once to get
\[
\begin{split}
xy_{\nu-1}(x-1)+xy_{\nu+1}(x-1)&=
x\int_{U}( t^{-\nu} +t^{-\nu-2})
\big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x-1} dt\\
&=2\int_{U} t^{-\nu}
d\big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x} \\
&=-2\int_{U}
\big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^{x} dt^{-\nu}\\
&=2\nu y_{\nu}(x),
\end{split}
\]as required
\item
The proof to this part actually follows directly from the part (i) and Proposition \ref{P:bessel_ode_mod}. However, we offer a direct verification. Applying $\triangle$ to the \eqref{E:delta_bessel_PDE_1} and replacing $\nu$ by $\nu-1$ in the \eqref{E:delta_bessel_PDE_2} respectively yield
\[
\begin{split}
&x\triangle^2 y_{\nu}(x-1)+\triangle y_{\nu}(x)-x\triangle y_{\nu-1}(x-1)
-y_{\nu-1}(x)+\nu \triangle y_{\nu}(x)=0,\\
&x\triangle y_{\nu-1}(x-1)+xy_{\nu}(x-1)-(\nu-1) y_{\nu-1}(x)=0.
\end{split}
\]
Adding the above two equations yields
\[
x\triangle^2 y_{\nu}(x-1)+\triangle y_{\nu}(x)+xy_{\nu}(x-1)-\nu y_{\nu-1}(x)+\nu \triangle y_{\nu}(x)=0.
\]
Replacing $x$ by $x-1$, and multiplying $x$ throughout both sides of the above equation yield
\[
x(x-1)\triangle^2 y_{\nu}(x-2)+x\triangle y_{\nu}(x-1)+x(x-1)y_{\nu}(x-2)-x\nu y_{\nu-1}(x-1)+x\nu \triangle y_{\nu}(x-1)=0.
\]
Multiplying by $\nu$ on both sides of \eqref{E:delta_bessel_PDE_1} and substitute the resulting equation into the last expression yield precisely the Bessel difference equation of order $\nu$ \eqref{E:difference_bessel_eqn}.
\end{enumerate}
\end{proof}
\begin{theorem}[(\textbf{Difference Schl\"afli-Sonine integrals})]\label{T:integral rep of db}
Let $\mathop{\rm Re}(x)<\mathop{\rm Re}(\nu)$. Let $U$ be an Hankel-type contour parametrised by $t$ that starts at $+\infty$, above the real-axis, circles around the origin in the counter-clockwise direction, and returns to $+\infty$ below the real-axis, and such that $|2t+t^2|\geq1$ on $U$. Then
\begin{equation}\label{E:dBessel integral rep}
J^{\Delta}_{\nu}(x)=e^{i\pi\nu} \dfrac{\sin(x-\nu)\pi}{\sin(\pi x)}
\dfrac{1}{2\pi i}\int_\infty^{(0+)}
t^{-\nu-1} \Big[ \frac{1}{2}(t-\frac{1}{t})+1\Big]^x dt
\end{equation} \end{theorem}
\begin{proof} The integrand has branch points $0,\, -1\pm\sqrt{2}$ and $ \infty $ when $\nu , x$ are not integers. We pick the branch cuts from $[-1-\sqrt{2},\, 0]$ and $[-1+\sqrt{2},\, +\infty]$ on the real-axis. Since $|2t+t^2|\geq1$ on $U$, the integrand allows an absolutely convergent expansion
\[
\begin{split}
t^{-\nu-1}\big[\frac{1}{2}(t-\frac{1}{t})+1\big]^{x}
&
=t^{-\nu-1}(1+\frac{t}{2})^x\big[1-\frac{1}{2t(1+t/2)}\big]^x\\
&=t^{-\nu-1}(1+\frac{t}{2})^x\sum_{k=0}^{\infty}\binom{x}{k}(-1)^k [\frac{1}{2t(1+t/2)}]^k.\\
\end{split}
\]
Observe that the \textit{lowest} term of \eqref{E:difference_bessel_fn} has
\[
\frac{2^{-\nu}\Gamma(x+1)}{\Gamma(\nu+1)\Gamma(x+1-\nu)}
\approx \frac{1}{2^\nu \Gamma(\nu+1)\Gamma(1-\nu)},
\]as $x\approx 0$.
Next we consider the lowest term of
\[
\begin{split}
&\dfrac{1}{2\pi i}\int_{\infty}^{(0+)}
t^{-\nu-1} \big[ \frac{1}{2}(t-\frac{1}{t})+1\big]^x dt\\
&=\dfrac{1}{2\pi i}\int_{\infty}^{(0+)}t^{-\nu-1}(1+\frac{t}{2})^x
\sum_{k=0}^{\infty}\binom{x}{k}(-1)^k [\frac{1}{2t(1+t/2)}]^k.\\
\end{split}
\] That is, \[ \dfrac{1}{2\pi i}\int_{\infty}^{(0+)}t^{-\nu-1}(1+\frac{t}{2})^xdt. \] Let $\frac{t}{2}=e^{i\pi} u$ in above integration. Then we have \[ \dfrac{1}{2\pi i}e^{-\nu\pi i} 2^{-\nu} \int_{-\infty}^{(0+)}u^{-\nu-1}(1-u)^{x}\, du. \] We recall a classical result about beta function, see for example \cite[p. 104 (12)]{Wang_Guo1989}, in an Hankel-type contour which happens to match exactly with the change of variable $t=2e^{i\pi} u$ above, namely that
\[
\dfrac{\Gamma(p+q+1)}{\Gamma(p+1)\Gamma(q+1)}
=\dfrac{1}{2\pi i} \int_{-\infty}^{(0+)}
t^{-p-1}(1-t)^{-q-1}dt,\quad \mathop{\rm Re}(p+q+1)>0
\]holds. Hence we obtain \[ \dfrac{1}{2\pi i}e^{-\nu\pi i} 2^{-\nu} \int_{-\infty}^{(0+)}u^{-\nu-1}(1-u)^{x}du =e^{-\nu\pi i} 2^{-\nu}\dfrac{\Gamma(\nu-x)}{\Gamma(\nu+1)\Gamma(-x)} \] which is valid for $\mathop{\rm Re}(x-\nu)<0$. Since \[ \dfrac{\Gamma(\nu-x)}{\Gamma(-x)} =\dfrac{\sin(-x)\pi\,\Gamma(x+1)}{\sin(\nu-x)\pi \,\Gamma(x+1-\nu)}. \] Hence we obtain \[ e^{i\pi\nu} \dfrac{\sin(x-\nu)\pi}{\sin(\pi x)} \dfrac{1}{2\pi i}\int_{\infty}^{(0+)} t^{-\nu-1} (1+\frac{t}{2})^xdt =\frac{2^{-\nu}\Gamma(x+1)}{\Gamma(\nu+1)\Gamma(x+1-\nu)}
\approx \frac{2^{-\nu} }{\Gamma(\nu+1)\Gamma(1-\nu)}, \]as $x\approx 0$ which matches the coefficient of the lowest term of the other side. This implies that both sides of \eqref{E:dBessel integral rep} are the same and \eqref{E:dBessel integral rep} holds. \end{proof}
We can modify the Hankel-type contour $U$ in the integral considered above to obtain a
second linearly independent solution to the difference Bessel equation \eqref{E:difference_bessel_eqn}.
\begin{theorem}\label{T:integral rep of db -nu}
Let $\mathop{\rm Re}(x)>\mathop{\rm Re}(\nu+1)$. Let $U^\prime$ be an Hankel-type contour parametrised by $s=-1/t$ where $t$ parametrised by the contour $U$ as is defined in Theorem \ref{T:integral rep of db}. Then
\begin{equation}\label{E:dBessel integral rep-nu}
J^{\Delta}_{-\nu}(x)=e^{i\pi} e^{-2i\pi\nu} \dfrac{\sin(x+\nu)\pi}{\sin\pi x}
\dfrac{1}{2\pi i}\int_{U^\prime}
s^{-\nu-1} \Big[ \frac{1}{2}(s-\frac{1}{s})+1\Big]^x ds.
\end{equation} \end{theorem}
\begin{proof}Notice that both ends of the contour $U^\prime$ would be at the origin which corresponds to $U$ at $+\infty$. The restriction $\mathop{\rm Re}(x)>\mathop{\rm Re}(\nu+1)$ ensures that the integral converges at the origin. Substituting $t=-1/s$ into the contour integral yields
\[
\frac{1}{2\pi i}\int_U (-t)^{\nu-1} \Big[ \frac{1}{2}(t-\frac{1}{t})+1\Big]^x dt
=\frac{-e^{i\nu\pi}}{2\pi i}\int_U t^{\nu-1} \Big[ \frac{1}{2}(t-\frac{1}{t})+1\Big]^x dt
\]where we need to be more restrictive under the extra temporary assumption that $\mathop{\rm Re}(x)<\mathop{\rm Re}(-\nu)$ holds in order to ensure that the integral converges at $+\infty$. Then the proof of the \eqref{E:dBessel integral rep-nu} follows closely that of the last theorem. One then considers the \textit{lowest term} from the integral and to compare it with that of the $J^\Delta_{-\nu}(x)$ in \eqref{E:difference_bessel_fn_2}. We can now remove the extra assumption $\mathop{\rm Re}(x)<\mathop{\rm Re}(-\nu)$. \end{proof}
\section{Half-Bessel modules I: Bessel polynomials}\label{S:half_bessel_I}
We recall the formula
\begin{equation}\label{E:K_bessel_poly}
K_{n+\frac12}(x)=\sqrt{\frac{\pi}{2}}\, e^{-x} x^{-n-\frac12} \theta_n(x)
\end{equation}that connects the modified Bessel function (or Macdonald function) of order $n+\frac12$ and the \textit{reversed Bessel polynomials} $\theta_n(x)$.
The modified Bessel functions $I_\nu(x)$ and $K_\nu(x)$ are essentially related to $J_\nu(ix)$ and $Y_\nu(ix)$ respectively. We refer to Watson \cite{Watson1944} for their precise definitions so that both $I_\nu(x)$ and $K_\nu(x)$ are real on the real axis. We first modify the adopt the transmutation formulae for the Bessel modules from subsection \S\ref{SS:transmutation_1} to half-Bessel modules that better suit our purpose of discussion in this section. Thus, it is more natural to discuss the reverse Bessel polynomials $\theta_n(x)$ than the Bessel polynomials $y_n(x)=x^{n}\theta_n(1/x)$ from our ${D}$-modules viewpoint. So the discussion of the $y_n(x)$ and its difference analogue will be postponed to the end of this section.
\subsection{Transmutation formulae}
Replacing $\partial$ by $-i\partial$ and $X$ by $iX$, we still have $[-i\partial,\, iX]=1$. So the transmutation formulae in Proposition \ref{P:bessel_transmutation} becomes
\begin{lemma}\label{L:Kbessel_transmutation} For each $\nu\in\mathbb{C}$, \begin{align}\label{E:Kbessel_transmutation} [(X\partial)^2-(X^2+(\nu+1)^2)]\left(\partial-\dfrac{\nu}{X}\right)&=\left(\partial-\dfrac{\nu+2}{X}\right)[(X\partial)^2-(X^2+\nu^2)],\\ [(X\partial)^2-(X^2+(\nu-1)^2)]\left(\partial+\dfrac{\nu}{X}\right)&=\left(\partial+\dfrac{\nu-2}{X}\right)[(X\partial)^2-(X^2+\nu^2)], \end{align} which induce the left $\mathcal{A}$-linear maps \[ \begin{array}{rcl} \mathcal{A}/\mathcal{A}((X\partial)^2-(X^2+(\nu+1)^2))&\stackrel{\times(\partial-\frac{\nu}{X})}{\longrightarrow}&\mathcal{A}/\mathcal{A}((X\partial)^2-(X^2+\nu^2)),\\ \mathcal{A}/\mathcal{A}((X\partial)^2-(X^2+(\nu-1)^2))&\stackrel{\times(\partial+\frac{\nu}{X})}{\longrightarrow}&\mathcal{A}/\mathcal{A}((X\partial)^2-(X^2+\nu^2)). \end{array} \] \end{lemma}
As a result we deduce from this corollary the following well-known formulae with the help of the asymptotic of $K_\nu(x)$ as $x\to 0$.
\begin{corollary} For each $\nu\in\mathbb{C}$, the modified Bessel functions $K_\nu(x)$ satisfies the following recurrence relations. \begin{align*}
K'_\nu(x)-\dfrac{\nu K_\nu(x)}{x}&=-K_{\nu+1}(x),\\
K'_\nu(x)+\dfrac{\nu K_\nu(x)}{x}&=-K_{\nu-1}(x). \end{align*} \end{corollary}
The formula \eqref{E:K_bessel_poly} suggests the following ``change of variable" of $X\partial $ to $X\partial-X-\nu$ and keep the $X$ unchanged that the transmutation formulae in Lemma \ref{L:Kbessel_transmutation} become
\begin{proposition}[(\textbf{Transmutation formulae})] \label{P:bessel_poly_transmutation_1} For each $\nu\in\mathbb{C}$, we have
\begin{align}
\Big(X\partial^2-2(\nu+\frac12+X)\partial & +2(\nu +\frac12)\Big)(X\partial-X-2\nu)\notag\\
&=\big(X\partial-X-2(\nu+\frac12)\big)\Big(X\partial^2-2(\nu-\frac12+X)\partial+2(\nu -\frac12)\Big),\label{E:bessel_poly_transmutation_1} \\
\Big(X\partial^2-2(\nu-\frac12+X)\partial &+2(\nu-\frac12)\Big)\frac1X(\partial-1)\notag\\
&=\frac1X (\partial-1-\frac1X)\Big(X\partial^2-2(\nu+\frac12+X)\partial+2(\nu+\frac12)\Big),\label{E:bessel_poly_transmutation_2}
\end{align}
which induce the left $\mathcal{A}$-linear maps \[ \begin{array}{rcl} \mathcal{A}/\mathcal{A}\Big(X\partial^2-2(\nu+\frac12+X)\partial +2(\nu +\frac12)\Big)&\stackrel{\times(X\partial-X-2\nu)}{\longrightarrow}&\mathcal{A}/\mathcal{A}\Big(X\partial^2-2(\nu-\frac12+X)\partial+2(\nu -\frac12)\Big),\\ \mathcal{A}/\mathcal{A}\Big(X\partial^2-2(\nu-\frac12+X)\partial +2(\nu -\frac12)\Big)&\stackrel{\times\frac{1}{X}(\partial-1)}{\longrightarrow}&\mathcal{A}/\mathcal{A}\Big(X\partial^2-2(\nu+\frac12+X)\partial+2(\nu +\frac12)\Big). \end{array} \] \end{proposition}
\subsection{Reverse Bessel polynomial modules} \label{SS:reverse_bessel_poly_mod}
\subsubsection{Reverse Bessel polynomial operator}\label{SSS:bessel_poly_oper}
Some of Burchnall\rq{}s formulae found in \cite{Burchnall_1953} are applications of the Weyl-algebraic formulae to be derived in this section. They are special cases in the manifestation of $D$-modules $\mathcal{O}_d$. We shall change the manifestation of $D$-modules to $\mathcal{O}_\Delta$ when we treat the difference Bessel polynomials later. We are ready to set $\nu=n+\frac12$ and $\nu=n-\frac12$ for $n\in \mathbb{Z}$ in \eqref{E:bessel_poly_transmutation_1} and \eqref{E:bessel_poly_transmutation_2} respectively to obtain
\begin{corollary}\label{C:bessel_poly_gauge_2} For each $n\in\mathbb{Z}$, the following are well-defined left $\mathcal{A}$-linear maps. \[ \begin{array}{rcl}
\mathcal{A}/\mathcal{A}\big(X\partial^2-2(n+1+X)\partial +2(n+1)\big)&\stackrel{\times(X\partial-X-2n-1)}{\longrightarrow}&\mathcal{A}/\mathcal{A} \big(X\partial^2-2(n+X)\partial +2n\big),\\
\mathcal{A}/\mathcal{A}\big(X\partial^2-2(n-1+X)\partial +2(n-1)\big) & \stackrel{\times\frac{1}{X}(\partial-1)}{\longrightarrow} & \mathcal{A}/\mathcal{A} \big(X\partial^2-2(n+X)\partial +2n\big). \end{array} \] \end{corollary}
Let $\Theta_0=1$ and consider for each $n\in\mathbb{Z}$ the gauge transformations defined by right multiplication by
\[
G_n :=-(X\partial-X-2n-1),\quad n\in\mathbb{Z}. \mod X\partial
\] Then we define
\begin{equation}\label{E:Theta_N}
\Theta_n(X)= G_{n-1}\, \cdots G_1\, G_0\, \Theta_0(X)\mod X\partial
\end{equation} to be the \textit{reversed Bessel polynomial of degree} $n$.
\begin{theorem}\label{T:theta_bessel_poly_map} For each $n\in\mathbb{N}\cup \{0\}$, the map
\[
\mathcal{A}/\mathcal{A}\big(X\partial^2-2(n+X)\partial +2n)\big)\stackrel{\times
\Theta_n(X)}{\longrightarrow}\mathcal{A}/\mathcal{A} \big(X\partial^2-2X\partial \big)
\]
where $\Theta_n(X)$ is defined in \eqref{E:Theta_N} is a well-defined left $\mathcal{A}$-linear map. \end{theorem}
In particular, we have
\begin{equation}
\begin{split}
\Theta_1(X)&=G_0\, \Theta_0 (X) = (-X\partial+X+1)=X+1\quad \mod X\partial\\
\Theta_2(X)&= G_1\, G_0\, \Theta_0(x) =
(-X\partial +X+3)\, (X+1)=X^2+3X+3,\quad \mod X\partial\\
\Theta_3(X)&=G_2\, G_1\, G_0 \, \Theta_0(x)
(-X\partial +x-5) X^2+3X+3=X^3+6X^2+15X+15,\quad \mod X\partial\\
\cdots & \cdots
\end{split}
\end{equation}
\begin{theorem}\label{T:theta_bessel_poly_map_2} Let $n\in\mathbb{N}_0$ and $\Theta_n(X)= G_{n-1} \cdots G_1 G_0 \Theta_0(X)$. Then we have
\begin{equation}
\Theta_n(X)=\sum_{k=0}^n \frac{(n+k)!}{2^k (n-k)!\, k!}X^{n-k},
\end{equation} and the map
\begin{equation}
\mathcal{A}/\mathcal{A}\big(X\partial^2-2(n+X)\partial +2n)\big) \stackrel{\times\Theta_n(X)} {\longrightarrow} \mathcal{A}/ \mathcal{A}\partial
\end{equation} is left $\mathcal{A}$-linear. \end{theorem}
On the other hand, let $\Theta_0=1$ and consider the gauge transformation $H$ defined by a left multiplication by the element $-\frac1X(\partial-1)$. Then we have
\begin{equation}
1=\Theta_0(X)= H^{n}(\Theta_n(X))=(H \, \cdots \, H)(\Theta_n(X)).
\end{equation}
That is,
\begin{equation}
\begin{split}
\Theta_0(X)&=H(\Theta_1 (X)) = 1/X(1-\partial) (X+1)\quad \mod \mathcal{A}\partial\\
\Theta_0(X)&=H^2(\Theta_2(X)) =\big[ 1/X(1-\partial)\big]^2( X^2+3X+3),\quad \mod \mathcal{A}\partial\\
\Theta_0(X)&=H^3(\Theta_3(X))=\big[ 1/X(1-\partial)\big]^3(X^3+6X^2+15X+15),\quad \mod \mathcal{A}\partial\\
\cdots & \cdots
\end{split}
\end{equation}
\begin{corollary}\label{C:Besselpoly_trans} For each $n$, let $\Theta_n$ be defined as above, which is a solution of $X\partial^2-2(n+X)\partial+2n$ in the left $\mathcal{A}$-module $\mathbb{C}[X]$. Then \begin{align*}
(X\partial-X-2n-1)\Theta_n + \Theta_{n+1}=0,\\
(\partial-1) \Theta_n + X\Theta_{n-1}=0. \end{align*} as elements in $M$. \end{corollary}
\begin{proof} We only proof the first expression. The second expression can be dealt with similarly. Let $\nu=n+\frac12$ in \eqref{E:bessel_poly_transmutation_1}. Then we note that the right-side of \eqref{E:bessel_poly_transmutation_1} annihilates the $\Theta_N(X)$, while the left-side of \eqref{E:bessel_poly_transmutation_1} therefore implies that
\[
(X\partial-X-(2n+1))\Theta_n(X)=C\Theta_{n+1}(X),\mod \mathcal{A}\partial
\]for some constant $C$. A simple comparison of the highest terms on both sides yields that $C=-1$. This proves the desired result. \end{proof}
We immediately deduce from the above Corollary the following recursion.
\begin{corollary} [(\textbf{Three-term recurrence})] \label{C:Besselpoly_3term} For each $n$, let $\Theta_n$ be defined as above, which is a solution of $X\partial^2-2(n+X)\partial+2n$ in the left $\mathcal{A}$-module $\mathbb{C}[X]$. Then
\[
\Theta_{n+1}-(2n+1)\Theta_n-X^2\Theta_{n-1}=0
\] as an element in $M$. \end{corollary}
We next show that the reverse Bessel polynomial equation \cite[p. 8]{Grosswald} or more generally the \textit{half-order Bessel polynomial operator}
\[
L_n=X\partial^2-2(n+X)\partial +2n
\]
can be studied as a special case of the \textit{half-Bessel module} below. The construction of this module is motivated by Corollary~\ref{C:Besselpoly_trans} and the $\mathcal{A}_2$-module structure endowed on $\mathcal{O}^{\mathbb{N}_0}$
as in Example~\ref{Eg:seq-functions_poisson}.
\begin{definition}\label{D:Theta} We define the \textit{reverse Bessel polynomial module} to be the modified half-Bessel module to be the left $\mathcal{A}_2$-module
\begin{equation}\label{E:Theta} \Theta=\dfrac{\mathcal{A}_2}{\mathcal{A}_2(\partial_1-1+{X_1}/{\partial_2})+\mathcal{A}_2(X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2)}.
\end{equation} \end{definition}
We first verify that the Bessel polynomial module $\Theta$ defined above is holomonic.
\begin{theorem} \label{T:B_poly_mod_holonomic} The left $\mathcal{A}_2$-module $\Theta$ has dimension two and multiplicity two. In particular, $\Theta$ is holonomic. \end{theorem}
\begin{proof} Multiply $\partial_2$ throughout the first generator from \eqref{E:Theta} yields \[
\partial_2\partial_1-\partial_2+X_1
\] before eliminating $\partial_1$ from the second generator from the \eqref{E:Theta}
\[
X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2.
\]This yields
\begin{equation}\label{E:B_poly_3th_generator}
\begin{split}
\partial_2\partial_1-\partial_2+{X_1}&=
\partial_2\big(\frac{2X_2}{X_1}\partial_2-\frac{1}{X_2}\partial_2+1+1/X_1\big)-\partial_2+X_1\\
&=((2X_1-1)\partial_2^2+3\partial_2+X_1^2)/X_1
\end{split}
\end{equation} Thus we can replace the original two generators by
\[
((2X_1-1)\partial_2^2+3\partial_2+X_1^2)/X_1,
\quad
\partial_1-(2X_2+1)/X_1\partial_2-1-1/X_1.
\]This renders the elements in the $\Gamma^k\Theta$ are spanned by $\{X_1^aX_2^b: a+b\le k\}$ and $\{X_1^{a\rq{}}X_2^{b\rq{}}\partial_2: a+b\le k-1\}$. Hence the $\Theta$ shares the same Hilbert polynomial with that studied in Example \ref{E:prime_integrable_eg_2} so that
\[ \dim_{\mathbb{C}}\mathrm{Gr}^kM=k^2. \]It follows that the $\Theta$ is holonomic, hence dimension two and with multiplicity two. \end{proof}
Eliminating $\partial_1$ again from the above two relations of $\mathcal{A}_2$ from the Definition $\Theta$ \eqref{D:Theta} yields the following element of $\mathcal{A}_2$-module.
\begin{proposition}[(\textbf{Three-term recursion})] \label{P:Weyl_mod_half_bessel_3term}
The natural map
\[ \dfrac{\mathcal{A}_2}{\mathcal{A}_2[(2X_1-1)\partial_2^2+3\partial_2+X_1^2]}\longrightarrow\Theta
\] is a well-defined left $\mathcal{A}_2$-linear surjection. \end{proposition}
The following records the relationship between the half-Bessel module $\Theta$ defined above and the second order half-Bessel operator $L_n$.
\begin{proposition}[(\textbf{Weyl-algebraic reverse Bessel-polynomial sequence of $\Theta$})]\label{P:Weyl_Bessel_Poly_Eqn} The natural map \[ \dfrac{\mathcal{A}_2}{\mathcal{A}_2[X_1\partial_1^2-2(X_1+X_2\partial_2)\partial_1+2X_2\partial_2]}\longrightarrow\Theta \] is a well-defined left $\mathcal{A}_2$-linear surjection. \end{proposition}
The derivation of this result is given in the Appendix \ref{A:Weyl_Bessel_Poly_Eqn}.
\begin{proof} See the Appendix. \end{proof}
\subsubsection{Classical reverse Bessel polynomials} To study Poisson generating functions, i.e., generating functions of the form $\displaystyle\sum_\mathbb{N} a_n {t^n}/{n!}$, one uses the $D$-module structure endowed on spaces $\mathbb{C}^{\mathbb{N}_0}$ and $\mathcal{O}^{\mathbb{N}_0}$ of (one-sided) sequences as in Example~\ref{Eg:seq-functions_poisson}. The purpose of the section is to illustrate that many classically known defining formulae about the reverse Bessel polynomials can be derived and presented in a coherent manner as consequences of manifestation of the $D$-module structures discussed in the last section. We shall show that the same $D$-module structure with a different manifestation would allow us to reach an analogous theory for difference reverse Bessel polynomials in the next subsection. Thus except for the transmutation formulae derived from above, the remaining formulae that we derive here are well-known in the classical Bessel polynomial literature, e.g., \cite{Grosswald}. Reader who are looking for new formulae about difference Bessel polynomials may proceed directly to the next section.
We deduce from Theorem \ref{T:theta_bessel_poly_map_2} the expected result:
\begin{proposition} Let $n\in \mathbb{N}_0$. Then the left $\mathcal{A}$-linear map
\[
\begin{array}{rclll}
\mathcal{A}/\mathcal{A}\big(X\partial^2-2(n+X)\partial +2n)\big)
&\stackrel{\times\Theta_n(X)}{\longrightarrow}
& \mathcal{A}/ \mathcal{A}\partial
&\stackrel{\times 1}{\longrightarrow}
&\mathcal{O}
\end{array}
\]
sends $1$ to the classical reversed Bessel polynomial
\[
\theta_n(x)=\sum_{k=0}^n \frac{(n+k)!}{2^k (n-k)!\, k!}\, x^{n-k}.
\] \end{proposition}
\begin{example}\label{Eg:bilateral_theta} Consider the sequence of classical Bessel polynomials $(\theta_n)$. Then the maps
\begin{equation}\label{E:j-map-bessel_poly}
\begin{array}{rcl}
\mathfrak{j}:\Theta & \stackrel{\times (\theta_n)}{\longrightarrow} &\mathcal{O}^{\mathbb{N}_0}
\end{array}
\end{equation} is well-defined and left $\mathcal{A}_2$-linear. \end{example}
The following differential-difference equations and three-term recurrence, as well as Corollaries~\ref{C:Besselpoly_trans} and \ref{C:Besselpoly_3term} about classical Bessel polynomials below present solution of the module $\Theta$ in $\mathcal{O}_d$. They follow from Example \ref{E:j-map-bessel_poly}, hence giving new proofs to these classical results. We shall see how solutions of the module $\Theta$ in $\mathcal{O}_\Delta$ gives their difference analogues below.
\begin{theorem}[(\textbf{differential-difference equations} \cite{Grosswald}, p. 19)]\label{T:differential-differential} The reverse classical Bessel polynomials $\{\theta_n(x)\}$ satisfy the following (well-known) formulae
\begin{equation}\label{E:bessel_poly_trans}
\begin{split}
&\theta_n^\prime(x)-\theta_n(x)+x\theta_{n-1}(x)=0,\\
&x\theta_n^\prime(x)-(x+2n+1)\theta_n(x)+\theta_{n+1}(x)=0
\end{split}
\end{equation} for all $n\ge 1$ and $x$.
\end{theorem}
\begin{proof} This follows from Corollary \ref{C:Besselpoly_trans} with the manifestation of analytic function space $\mathcal{O}_d$ as a $\mathcal{A}_1$-module given by the Example \ref{Eg:O_deleted_d}. \end{proof}
\begin{proposition}[(\textbf{Three-term recurrence} \cite{Grosswald}, p. 18)] The sequence $(\theta_n)$ satisfies the (well-known) three-term recurrence formula
\begin{equation}\label{E:bessel_poly_3term}
\theta_{n+2}(x)-(2n+3)\theta_{n+1}(x)-x^2\theta_n(x)=0
\end{equation}
for every $n\in\mathbb{N}_0$ and every $x$. \end{proposition}
\begin{proof} This follows from the last Corollary or directly from the Corollary \ref{C:Besselpoly_3term} with the manifestation of the $\mathcal{A}_1$-module $\mathcal{O}_d$ as given by the Example \ref{Eg:O_deleted_d}. \end{proof}
Recently, there are also complex analytic studies of the PDEs satisfied by generating functions of some classical special functions, including the Bessel polynomials \cite{Hu_Yang_2009, Hu_Li_2017}.
\subsubsection{Difference reverse Bessel polynomials} Now we consider the left $\mathcal{A}_2$-module structure of $\mathcal{O}_\Delta^{\mathbb{N}_0}$ as in Example~\ref{Eg:seq-functions_poisson_delta}. Recalling that $\Theta_n$ is a solution of $L_n:=X\partial^2-2(n+X)\partial+2n$, we immediately obtain the following
\begin{theorem}[(\textbf{Difference equation})]\label{T:difference_Bessel_Poly_DD} For each integer $n\ge 0$, let $\theta^\Delta_n$ be the \textit{difference reverse Bessel polynomial of degree} $n$ defined by
\begin{equation}\label{E:delta_Bessel_poly}
\theta^\Delta_n(x) :=\sum_{k=0}^n \frac{(n+k)!}{2^k (n-k)!\, k!}\, (x)_{n-k}.
\end{equation} Then for each integer $n\ge 0$ and each $x$, we have
\begin{equation}\label{E:delta_Bessel_poly_eqn}
(x-2n)\,\theta_n^\Delta(x+1)-4(x-n)\,\theta_n^\Delta(x)+3x\, \theta_n^\Delta (x-1)=0.
\end{equation} \end{theorem}
The following delay-differential relations and three-term recurrence are direct consequences of the module $\Theta$, as well as Corollaries~\ref{C:Besselpoly_trans} and \ref{C:Besselpoly_3term} with $N=\mathcal{O}_\Delta$.
\begin{theorem}[(\textbf{Delay-difference relations})]\label{T:difference_Bessel_Poly_PDE} The sequence of difference reverse Bessel polynomials $(\theta^\Delta_n)$ satisfy the following two delay-difference equations
\begin{equation}\label{E:delta_bessel_poly_trans}
\begin{split}
\theta^\Delta_n(x+1)-2\theta^\Delta_n(x)+x\theta^\Delta_{n-1}(x-1)=0, \\
\theta^\Delta_{n+1}(x)+(x-2n-1)\theta^\Delta_n(x)-2x\, \theta^\Delta_n(x-1)=0,
\end{split}
\end{equation}
for every $n\ge 1$ and every $x$. \end{theorem}
\begin{proof} This follows from Corollary \ref{C:Besselpoly_trans} with the manifestation of the analytic function space $\mathcal{O}_\Delta$ as a $\mathcal{A}_1$-module given by the Example \ref{Eg:O_delta}. \end{proof}
It follows from Proposition \ref{P:Weyl_mod_half_bessel_3term} and the structure of the above $\mathcal{A}_2$-module the new three-term recursion formula for the reverse difference Bessel polynomials.
\begin{theorem}[(\textbf{Three-term recurrence})] \label{T:3-term} The sequence of difference reversed Bessel polynomials $(\theta^\Delta_n)$ satisfies the following recurrence relation
\begin{equation}\label{E:delta_reverse_bessel_poly_3term}
\theta^\Delta_{n+2}(x)-(2n+3)\theta^\Delta_{n+1}(x)-x(x-1)\theta^\Delta_n(x-2)=0
\end{equation}
for every $n\in\mathbb{N}_0$ and every $x$. \end{theorem}
\begin{proof} This follows from the last Corollary or directly from the Corollary \ref{C:Besselpoly_3term} with the manifestation of the analytic function space $\mathcal{O}_d$ as a $\mathcal{A}_1$-module given by the Example \ref{Eg:O_delta}. \end{proof}
\subsection{Characteristics of modified half-Bessel modules}
We recall from \eqref{E:Theta} the definition of the half-Bessel module. We may change one of the two generators from the half-Bessel module \eqref{E:Theta} by $(2X_2-1)\partial_2^2+3\partial_2+X_1^2$, as suggested by Corollary \ref{C:Besselpoly_3term}.
\begin{theorem} The half-Bessel module $\Theta$ is isomorphic to the left $\mathcal{A}_2$-module \[ \dfrac{\mathcal{A}_2}{\mathcal{A}_2[(2X_2-1)\partial_2^2
+3\partial_2+X_1^2]
+\mathcal{A}_2[X_1\partial_1+(1-2X_2)\partial_2-1-X_1] }. \] \end{theorem}
\begin{theorem} The map \[ \dfrac{\mathcal{A}_2}{\mathcal{A}_2[(2X_2-1)\partial_2^2
+3\partial_2+X_1^2]
+\mathcal{A}_2[X_1\partial_1+(1-2X_2)\partial_2-1-X_1] } \longrightarrow \Theta \] is a left $\mathcal{A}_2$-linear isomorphism. \end{theorem}
\begin{theorem}\label{T:bessel_poly_gf_map} Let
\[
\mathcal{A}_{2}(\rho):=\frac{\mathcal{A}_2( \rho, {1}/{\rho})}{\langle\rho^2-1+2X_2\rangle}
\] and let $S$ be the element
\begin{equation}\label{E:bessel_poly_gf_map}
S = C_1\rho^{-1}{\mathop{\rm E}}[X_1(1-\rho)]+C_2\rho^{-1}{\mathop{\rm E}}[X_1(1+\rho)],
\end{equation}
for every choice of $C_1,\, C_2\in\mathbb{C}$, where the ${\mathop{\rm E}}$ is the Weyl exponential defined in Example \ref{Eg:exp}. Then the map
\begin{equation}
\begin{split}
\Theta(\rho):=\dfrac{\mathcal{A}_2(\rho)}{\mathcal{A}_2(\rho)[(2X_2-1)\partial_2^2
+3\partial_2+X_1^2]
+\mathcal{A}_2(\rho)[X_1\partial_1+(1-2X_2)\partial_2-1-X_1] }\\
\stackrel{\times S}{\longrightarrow}
\overline{ \dfrac{\mathcal{A}_2(\rho)}
{\mathcal{A}_2(\rho)\partial_1+\mathcal{A}_2(\rho)\partial_2}}
\end{split}
\end{equation}
is left $\mathcal{A}_{2}(\rho)$-linear. \end{theorem}
\begin{proof} Let
\begin{equation}\label{E:char05}
\Lambda_{1}=\partial_1
\quad \Lambda_{2}=\rho\partial_{2},\quad Z_{1}=X_{1},\quad Z_{2}=-\rho,
\end{equation}where $\rho^2=1-2X_2$. Since
\[
\partial_2\rho-\rho\partial_2=-\frac1\rho,
\] we deduce from direct computations that
\begin{equation}\label{E:computation_1}
[\Lambda_{1}, Z_{1}]=1, \quad [\Lambda_{2}, Z_{2}]=1,
\quad
[\Lambda_{1}, \Lambda_{2}]=0,\quad [Z_1, Z_{2}]=0,
\quad [\Lambda_i,\, Z_j]=0,\quad i\not=j
\end{equation}hold.
It is routine to check that
\begin{equation}\label{E:char_2}
(2X_2-1)\partial_2^2
+3\partial_2+X_1^2
=-\Lambda_2^2-\frac{2}{Z_2}\Lambda_2+Z_1^2.
\end{equation}
Let us make use of the characteristic change of elements by \eqref{E:char05} in the second generator $X_1\partial_1+(1-2X_2)\partial_2-1-X_1$ to obtain
\[
Z_1\Lambda_1-Z_2 \Lambda_2-1-Z_1
=Z_1(\Lambda_1-1)-\Lambda_2Z_2.
\]
Let $\widehat{\mathcal{A}_2}=\langle \Lambda_1,\, \Lambda_2,\, Z_1,\, Z_2\rangle$ and
\[
\widehat{\Theta}:= \frac{\widehat{\mathcal{A}_2}}
{\widehat{\mathcal{A}_2}[(Z_2\Lambda_2^2+2\Lambda_2-Z_2Z_1^2]+\widehat{\mathcal{A}_2}[(Z_1(\Lambda_1-1)-\Lambda_2Z_2)]}
\]Note that the above quotient $\widehat{\mathcal{A}_2}$-module $\widehat{\Theta}$ is holonomic with dimension and multiplicity both equal to $2$, see e.g., Example \ref{E:prime_integrable_eg_2}.
In order to find the map
\[
\widehat{\Theta}\
\stackrel{\times \widehat S}{\longrightarrow}\overline{
\frac{\widehat{\mathcal{A}_2}}
{{\widehat{\mathcal{A}_2}}\Lambda_1+{\widehat{\mathcal{A}_2}}\Lambda_2}},
\]we
left multiply the expression \eqref{E:char_2} by $Z_2$. Further simplification yield
\[
\begin{split}
(Z_2\Lambda_2)\Lambda_2+2\Lambda_2-Z_2Z_1^2
&=\Lambda_2^2 Z_2-Z_2Z_1^2\\
&=(\Lambda_2^2-Z_1^2)Z_2\\
&=(\Lambda_2+Z_1)(\Lambda_2-Z_1)Z_2.
\end{split}
\]
Since the two first-order operators above commute, so we deduce that
\[
\frac{1}{Z_2} F(Z_1) E(Z_1Z_2)+\frac{1}{Z_2} G (Z_1) E(-Z_1Z_2)
\]is the general solution to the above operator
for some $F(Z_1)$ and $G(Z_1)$. This is a sum of two solutions which also conform with our knowledge that the $\widehat{\Theta}$ has multiplicity $2$ discussed above.
We now make use of the second generator of $\widehat{\Theta}$ to determine the exact forms of the solutions $F$ and $G$. We first note that
\begin{align}
& \Lambda_1 {\mathop{\rm E}}(Z_1Z_2)=Z_{2}{\mathop{\rm E}}(Z_1Z_2)+{\mathop{\rm E}}(Z_1Z_2)\Lambda_1,\label{E:product_rule_1}\\
& \Lambda_2 {\mathop{\rm E}}(Z_1Z_2)=Z_1{\mathop{\rm E}}(Z_1Z_2)+{\mathop{\rm E}}(Z_1Z_2)\Lambda_2.\label{E:product_rule_2}
\end{align}It follows from \eqref{E:product_rule_1} and \eqref{E:product_rule_2} that
\[
\begin{split}
&\big(Z_1(\Lambda_1-1)-\Lambda_2Z_2\big)
\frac{1}{Z_2} F(Z_1) {\mathop{\rm E}}(Z_1Z_2)\\
&=Z_1(\Lambda_1-1)F(Z_1) {\mathop{\rm E}}(Z_1Z_2) - \Lambda_2 F(Z_1) {\mathop{\rm E}}(Z_1Z_2)\\
&=(Z_1/Z_2)(\Lambda_1-1)F(Z_1) {\mathop{\rm E}}(Z_1Z_2) - F(Z_1) Z_1 {\mathop{\rm E}}(Z_1Z_2)
-F(Z_1){\mathop{\rm E}}(Z_1Z_2)\Lambda_2\\
&=(Z_1/Z_2)(\Lambda_1-1)F(Z_1) {\mathop{\rm E}}(Z_1Z_2) - F(Z_1) Z_1 {\mathop{\rm E}}(Z_1Z_2)\quad \mod \widehat{\mathcal{A}}_2 \Lambda_2\\
&=(Z_1/Z_2)\big({\mathop{\rm E}}(Z_1Z_2)\Lambda_1F(Z_1)-F(Z_1)\Lambda_1{\mathop{\rm E}}(Z_1Z_2)\big)
-(Z_1/Z_2)F(Z_1)E(Z_1Z_2)\\
&\qquad- F(Z_1) Z_1 {\mathop{\rm E}}(Z_1Z_2)\\
&=(Z_1/Z_2){\mathop{\rm E}}(Z_1Z_2)\Lambda_1F(Z_1)-(Z_1/Z_2)F(Z_1)Z_2{\mathop{\rm E}}(Z_1Z_2)
-(Z_1/Z_2)F(Z_1)E(Z_1Z_2)\\
&\qquad - F(Z_1) Z_1 {\mathop{\rm E}}(Z_1Z_2)\\
&=(Z_1/Z_2){\mathop{\rm E}}(Z_1Z_2)\Lambda_1F(Z_1)-(Z_1/Z_2)F(Z_1)E(Z_1Z_2)\\
&=(Z_1/Z_2){\mathop{\rm E}}(Z_1Z_2)(\Lambda_1-1)F(Z_1)\\
&=0\quad \mod \widehat{\mathcal{A}}_2 \Lambda_1+\widehat{\mathcal{A}}_2 \Lambda_2,
\end{split}
\]where the last equal sign holds provided that
\[
(\Lambda_1 -1)F(Z_1)=0\quad \mod \widehat{\mathcal{A}}_2 \Lambda_1.
\]But this equation could hold when we have chosen $F(Z_1)=-{\mathop{\rm E}}(Z_1)$, see Example \ref{Eg:exp}, except perhaps for a constant multiple.
It remains to consider the following and a similar calculation yields
\[
\begin{split}
&\big(Z_1(\Lambda_1-1)-\Lambda_2Z_2\big)
\frac{1}{Z_2} G(Z_1) {\mathop{\rm E}}(-Z_1Z_2)\\
&={Z_1}/{Z_2}\Big({\mathop{\rm E}}(-Z_1Z_2)\Lambda_1G(Z_1)-Z_2{\mathop{\rm E}}(-Z_1Z_2)G(Z_1)\Big)
-{Z_1}/{Z_2}G(Z_1) {\mathop{\rm E}}(-Z_1Z_2)\\
&\qquad -\Big({\mathop{\rm E}}(-Z_1Z_2)\Lambda_2G(Z_1)-Z_1{\mathop{\rm E}}(-Z_1Z_2)G(Z_1)\Big)\qquad
\mod \widehat{\mathcal{A}}_2 \Lambda_1+\widehat{\mathcal{A}}_2 \Lambda_2\\
&={\mathop{\rm E}}(-Z_1Z_2)/Z_2\Big(Z_1(\Lambda_1-1)+Z_2\Lambda_2\Big)G(Z_1)
\qquad
\mod \widehat{\mathcal{A}}_2 \Lambda_1+\widehat{\mathcal{A}}_2 \Lambda_2\\
&=(1/Z_2){\mathop{\rm E}}(-Z_1Z_2) Z_1(\Lambda_1-1)G(Z_1)\\
&=0 \qquad \mod \widehat{\mathcal{A}}_2 \Lambda_1+\widehat{\mathcal{A}}_2 \Lambda_2,
\end{split}
\]where the last equal sign would hold if
\[
G(Z_1)={\mathop{\rm E}}(Z_1) \quad \mod \widehat{\mathcal{A}}_2\Lambda_1,
\]
which may differ by a constant multiple. This establishes the \eqref{E:bessel_poly_gf_map} and hence completes the proof. \end{proof}
\subsection{Bessel polynomials module}
In this subsection, we define the \textit{Weyl Bessel polynomial of degree} $n\ge 0$ by \begin{equation} Y_n(X)=X^n\Theta_n(1/X)=\sum_{k=0}^{n}\frac{(n+k)!}{(n-k)! k! 2^k}X^k. \end{equation} Right multiplication by $Y_n(X)$ gives a left $\mathcal{A}$-linear map \[ \mathcal{A}/\mathcal{A}(X^2\partial^2+(2X+2)\partial-n(n+1) \stackrel{\times Y_n(X)\ \ }{\longrightarrow}\overline{\mathcal{A}/\mathcal{A}\partial}. \]
We may further define when $n=-1$, then $Y_{-1}(X)=1$.
We deduce from Corollary \ref{C:Besselpoly_trans} the followings.
\begin{corollary}\label{C:Bpoly_trans} For each $n$, let $Y_n$ be defined as above, which is a solution of $X^2\partial^2+(2X+2)\partial-n(n+1)$ in a certain left $\mathcal{A}$-module $\mathbb{C}[X]$. Then \begin{align}
(X^2\partial+(n+1)X+1)Y_n - Y_{n+1}=0,\label{bessel_poly_recursion_1 in a A}\\
(X^2\partial-nX+1)Y_n - Y_{n-1}=0.\label{bessel_poly_recursion_2 in a A} \end{align} as elements in $N$. \end{corollary}
\begin{proof} Replacing $X$ by $1/X$ and $\partial$ by $-X^2\partial$ in Corollary~\ref{C:Besselpoly_trans}, we obtain \begin{align*}
(-X\partial-1/X-2n-1)\Theta_n(1/X) + \Theta_{n+1}(1/X)=0,\\
(-X^2\partial-1) \Theta_n(1/X) + (1/X)\Theta_{n-1}(1/X)=0. \end{align*} Left-multiplying both sides of the first equation by $X^{n+1}$ and both sides of the second equation by $X^n$, we obtain \begin{align*}
(-X^2\partial-1-(n+1)X)X^n\Theta_n(1/X) + X^{n+1}\Theta_{n+1}(1/X)=0,\\
(-X^2\partial-1+nX) X^n\Theta_n(1/X) + X^{n-1}\Theta_{n-1}(1/X)=0, \end{align*} and hence the result follows. \end{proof}
\begin{corollary} [(\textbf{Three-term recurrence})] \label{C:Bpoly_3term} For each $n$, let $Y_n$ be defined as above, which is a solution of $X^2\partial^2+(2X+2)\partial-n(n+1)$ in a certain left $\mathcal{A}$-module $C[x]$. Then
\begin{equation}\label{bessel_poly_recursion_3 in a A}
Y_{n+1}-(2n+1)XY_n-Y_{n-1}=0
\end{equation} as an element in $N$. \end{corollary}
We introduce $(\partial_2f)_n=f_{n+1}$ and $ (X_2 f)_n=nf_{n-1}$ to constrruct an $\mathcal{A}_2$-module, such that
\[
[\partial_1, X_1] =1, \quad [\partial_2, X_2] =1, \quad [X_1, X_2]=0, \quad [\partial_1, \partial_2]=0.
\]
Replacing $n$ by $n-1$ in \eqref{bessel_poly_recursion_1 in a A} and representing the resulting formula in terms of $X_1, \partial_i, i=1, 2$ yield
\begin{equation}\label{E:dBessel_1st_gen} X_{1}^2\partial_{1}+X_{1}X_{2}\partial_{2}+1-\partial_{2}.
\end{equation} Similarly, the equation \eqref{bessel_poly_recursion_2 in a A} can be rewritten as \[ X_{1}^2\partial_{1}-X_{1}X_{2}\partial_{2}+1-\partial_{2}^{-1}. \] Right multiplying $\partial_2$ to the above formula gives
\begin{equation}\label{E:dBessel_2nd_gen} X_{1}^2\partial_{1}\partial_{2}-X_{1}X_{2}\partial^2_{2}+\partial_{2}-1.
\end{equation}
Thus the \textit{Bessel polynomial module} is given by
\begin{definition}\label{D:y-1}
The \textit{Bessel polynomial module} is the left $\mathcal{A}_2$-module defined by \begin{equation}\label{E:y-1} \mathcal{Y}=\dfrac{\mathcal{A}_{2}} {\mathcal{A}_{2}
(X_{1}^2\partial_{1}\partial_{2}-X_{1}X_{2}\partial^2_{2}+\partial_{2}-1)
+\mathcal{A}_{2}
(X_{1}^2\partial_{1}+X_{1}X_{2}\partial_{2}+1-\partial_{2})}. \end{equation} \end{definition}
Similarly, we can represent \eqref{bessel_poly_recursion_3 in a A} in $\mathcal{A}_2$ by \begin{equation}\label{bessel_poly_recursion_3 in a A becomes} \partial_2^2-2X_1X_2\partial_2^2-X_1\partial_2-1. \end{equation}
\begin{proposition} The map \[ \dfrac{\mathcal{A}_2}{\mathcal{A}_2(X_1^2\partial_1^2+(2X_1+2)\partial_1-X^2_2\partial_2^2)} \longrightarrow \mathcal{Y} \] is a well-defined left $\mathcal{A}_2$-linear surjection. \end{proposition}
\subsubsection{Classical Bessel polynomials}
\begin{proposition} Let $n\in \mathbb{N}_0$. Then the classical Bessel polynomial
\[
y_{n}(x)=\sum_{k=0}^{n}\frac{(n+k)!}{2^k (n-k)!\, k!}\, x^{k},
\]for $n\ge 0$ and $y_{-1}(x)=1$,
is the image of $1$ via the left $\mathcal{A}$-linear map
\[
\begin{array}{rclll}
\mathcal{A}/\mathcal{A}\big(X^2\partial^2+2(X+2)\partial -n(n+1)\big)
&\stackrel{\times Y_{n}(X)}{\longrightarrow}
& \mathcal{A}/ \mathcal{A}\partial
&\stackrel{\times 1}{\longrightarrow}
&\mathcal{O}
\end{array}
\] \end{proposition}
The following PDEs (delay-differential relations) and three-term recurrence are direct consequences of the module $\mathcal{Y}$, as well as Corollaries~\ref{C:Bpoly_trans} and \ref{C:Bpoly_3term} with $N=\mathcal{O}_d$.
\begin{theorem}[(\textbf{Delay-differential equations})]\label{T:differential-differential-classical bessel poly}
The classical Bessel polynomials $\{y_n(x)\}$ \cite[p. 19]{Grosswald} satisfy the following (well-known) formulae
\begin{equation}\label{E:classical_bessel_poly_trans}
\begin{split}
&x^2y'_{n-1}(x)-[(n-1)x-1]y_{n-1}(x)-y_{n-2}(x)=0,\\
&x^2y'_{n-1}(x)-y_{n}(x)+(nx+1)y_{n-1}(x)=0.
\end{split}
\end{equation}
for all $n$ and all $x$. \end{theorem}
\begin{proof} This follows from the two generators \eqref{E:dBessel_1st_gen} and \eqref{E:dBessel_2nd_gen} of the Bessel polynomial module $\mathcal{Y}$ in the Definition \ref{D:y-1}, follows by the manifestation of the analytic function space $\mathcal{O}_d$ as a $\mathcal{A}_1$-module given by the Example \ref{Eg:O_deleted_d}. \end{proof}
\begin{proposition}[(\textbf{Three-term recurrence} \cite{Grosswald}, {p. 18})] The $(y_n)$ satisfies the (well-known) three-term recurrence formula
\begin{equation}\label{E:classical_bessel_poly_3term}
y_{n+1}(x)-(2n+1)xy_{\color{blue}n}(x)-y_{n-1}(x)=0.
\end{equation}
or every $n\in\mathbb{N}_0$ and every $x$. \end{proposition}
\begin{proof} This either follows from the last corollary or directly from the Corollary \ref{C:Bpoly_3term} with the manifestation of the analytic function space $\mathcal{O}_d$ as a $\mathcal{A}_1$-module given by the Example \ref{Eg:O_deleted_d}. \end{proof}
\subsubsection{Difference Bessel polynomials} \begin{proposition} Let $n\in \mathbb{N}_0$. Then the difference Bessel polynomial
\[
y^{\Delta}_{n}(x):=\sum_{k=0}^{n}\frac{(n+k)!}{2^k (n-k)!\, k!}\, (x)_k,
\]where we define $y^\Delta_{-1}(x):=1$, is the image of $1$ via the left $\mathcal{A}$-linear map
\[
\begin{array}{rclll}
\mathcal{A}/\mathcal{A}\big(X^2\partial^2+2(X+2)\partial -n(n+1)\big)
&\stackrel{\times Y_{n}(X)}{\longrightarrow}
& \mathcal{A}/ \mathcal{A}\partial
&\stackrel{\times 1}{\longrightarrow}
&\mathcal{O}_\Delta.
\end{array}
\] \end{proposition}
The following delay-differential relations and three-term recurrence are direct consequences of the module $\mathcal{Y}$, as well as Corollaries~\ref{C:Bpoly_trans} and \ref{C:Bpoly_3term} with $M=\mathcal{O}_\Delta$.
\begin{theorem}[(\textbf{Delay-difference equations})]\label{T:classical_Bessel_Poly_DD} The difference Bessel polynomials $(y_n)$ satisfy the following two delay-difference equations
\begin{equation}\label{E:delta_classical_bessel_poly_trans}
\begin{split} x(x-1) \Delta y^{\Delta}_{n-1}(x-2) -(nx-x)y^{\Delta}_{n-1}(x-1)+y^{\Delta}_{n-1}(x)- y^{\Delta}_{n-2}(x)=0, \\ x(x-1) \Delta y^{\Delta}_{n-1}(x-2)-y^{\Delta}_{n}(x)+ nxy^{\Delta}_{n-1}(x-1)+y^{\Delta}_{n-1}(x)=0
\end{split}
\end{equation}
for every $n$ and every $x$.
\end{theorem}
\begin{proof}
\end{proof}
\begin{proposition}[(\textbf{Three-term recurrence})] \label{P:3-term_bessel_poly} The difference Bessel polynomials $(y^\Delta_n)$ satisfy the three-term recurrence formula
\begin{equation}\label{E:delta_bessel_poly_3term}
y^{\Delta}_{n+1}(x)-x(2n+1)y^{\Delta}_{n}(x-1)-y^{\Delta}_{n-1}(x)=0
\end{equation} for every $n$ and $x$. \end{proposition}
\subsection{Characteristic of Bessel polynomial module} We change one of the two generators from the defining Bessel polynomial module \eqref{E:y-1} by the element from the three-term recursion \eqref{bessel_poly_recursion_3 in a A becomes} to obtain following theorem. \begin{theorem}\label{T: iso bp-n-1} The natural map \[ \dfrac{\mathcal{A}_{2}} {\mathcal{A}_{2}
((1-2X_1X_2)\partial_2^2-X_1\partial_2-1)
+\mathcal{A}_{2}
(X_{1}^2\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1)} \longrightarrow \mathcal{Y} \] is a left $\mathcal{A}_2$-linear isomorphism. \end{theorem}
\begin{theorem}\label{T:symbol gen yn-1}
Let
\[
\mathcal{A}_{2}(\eta):=\dfrac{\mathcal{A}_2\langle\eta\rangle}{\langle\eta^2-1+2X_1X_2\rangle},
\]
\[
\mathcal{Y}(\eta)=\dfrac{\mathcal{A}_{2}(\eta)}
{\mathcal{A}_{2}(\eta)(\eta^2\partial_2^2-X_1\partial_2-1)
+\mathcal{A}_{2}(\eta)
(X_{1}^2\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1)}
\]
and let
\[
S:=C_1\,{\mathop{\rm E}}(\frac{1-\eta}{X_1})+C_2\,{\mathop{\rm E}}(\frac{1+\eta}{X_1})
\]
for every choice of $C_1,\, C_2\in\mathbb{C}$, where ${\mathop{\rm E}}$ denote the Weyl exponential introduced in Example \ref{Eg:exp}.
Then $\mathcal{Y}(\eta)$ is a left $\mathcal{A}_{2}(\eta)$-module and the map
\begin{equation}\label{eta gen for bp}
\mathcal{Y}(\eta)
\xrightarrow[]{\times S}
\overline{\mathcal{A}_{2}(\eta)/(\mathcal{A}_{2}(\eta)\partial_1+
\mathcal{A}_{2}(\eta)\partial_2)},
\end{equation}
is left $\mathcal{A}_{2}(\eta)$-linear. \end{theorem}
Since the proof of Theorem \ref{T:symbol gen yn-1} is similar to that of Theorem \ref{T:bessel_poly_gf_map}, so we place it in the Appendix \ref{A:BP_chara}.
\section{Newton transformations} \label{S:Newton} To handle more sophisticated generating functions from Bessel modules in relation to difference calculus, we would like to introduce a transformation that allows us to ``transform\rq\rq{} entities with respect to differential operators to difference operators under different sittings. An origin of the transform can be traced to spectral analysis and integral forms of gamma functions, see e.g., \cite[\S 12.22]{WW}:
\[
\Gamma(x)=-\frac{1}{2i \sin\pi x}\int_{\infty}^{(0+)}
e^{-\lambda}(-\lambda)^{x-1} d\lambda\quad (|\arg(-\lambda)|<\pi)
\]which is valid on $\mathbb{C}$ except at negative integers including the origin, where the upper and lower limits denote the standard Hankel contour which starts from $+\infty$ above the real-axis, then circles around the origin in the counter-clockwise direction before returns to $+\infty$ below the real-axis. If we replace $x$ by $-x$, then the integral becomes
\begin{equation}\label{E: Gamma} \Gamma(-x)=\frac{e^{-i\pi x}}{2i \sin\pi x}\int_{-\infty}^{(0+)}
e^\lambda(-\lambda)^{-x-1} d\lambda\quad (|\arg(\lambda)|<\pi) \end{equation} where the integral contour starts at $-\infty$ before the negative real-axis, then circle around the origin in the counter-clockwise direction, before returns to $-\infty$ above negative real-axis.
\begin{theorem}[(\textbf{Newton transformations})]\label{T:Newton_trans} Let $\mathcal{O}_\Delta$ be the space of entire functions endowed with the structure of a left $\mathcal{A}$-module as in Example \ref{Eg:O_delta_d}, and $\mathcal{O}_d$ be the space of entire functions $f$ with growth rate \[
|f(x)\, e^{mx}|\to 0\quad\mbox{ as }\quad x\to-\infty \] for some $m<1$, endowed with the structure of a left $\mathcal{A}$-module as in Example \ref{Eg:O_deleted_d}. Then the map
\begin{equation}\label{E:newton_trans} \begin{array}{rcl}
\mathfrak{N}: \mathcal{O}_d&\longrightarrow&\mathcal{O}_\Delta\\ f&\mapsto&\dfrac{e^{-i\pi x}}{2i\sin(\pi x) \Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\, f(s)\, e^s(-s)^{-x-1}\,ds \end{array}
\end{equation} is left $\mathcal{A}$-linear. This map is called the Newton transformation. \end{theorem}
\begin{proof} We just need to verify that $\mathfrak{N}(\partial f) = \partial(\mathfrak{N}f)$ and $\mathfrak{N}(X f) = X(\mathfrak{N}f)$ as follows. \begin{align*}
\mathfrak{N}(\partial f)(x) &= \dfrac{e^{-i\pi x}}{2i\sin(\pi x) \Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\, f'(s)\, e^s(-s)^{-x-1}\,ds \\
&= -\dfrac{e^{-i\pi x}}{2i\sin(\pi x) \Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\, f(s)[e^s(-s)^{-x-1}+(x+1)e^s(-s)^{-x-2}]\,ds \\
&=(\mathfrak{N}f)(x+1) - (\mathfrak{N}f)(x) = \partial(\mathfrak{N}f)(x);\\
\mathfrak{N}(X f)(x) &= \dfrac{e^{-i\pi x}}{2i\sin(\pi x) \Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\, sf(s)\, e^s(-s)^{-x-1}\,ds \\
&= -\dfrac{e^{-i\pi x}}{2i\sin(\pi x) \Gamma(-x)}\displaystyle\int_{-\infty}^{(0+)}\, f(s)\, e^s(-s)^{-x}\,ds \\
&= x(\mathfrak{N}f)(x-1) = X(\mathfrak{N}f)(x). \end{align*} \end{proof}
\begin{theorem} The Newton transformation defined above is an injection. \end{theorem} \begin{proof} It follows from the injectivity of the Mellin transformation and the observation that the integral above is nothing but the Mellin transformation of $f$ times the exponential function. \end{proof}
\begin{example} Let us revisit the Weyl exponential ${\mathop{\rm E}}(aX)$ from Example \ref{Eg:exp} derived when finding a $\mathcal{A}$-linear map as a solution $S$ to
\[
\mathcal{A}/\mathcal{A}(\partial-a)
\stackrel{\times S }{\longrightarrow}
\mathcal{A}/\mathcal{A}\partial.
\] Since it is known that $\mathcal{A}/\mathcal{A}(\partial-a) \xrightarrow{\times {\mathop{\rm E}}(aX)} \mathcal{A}/\partial\mathcal{A} \xrightarrow{\times 1}\mathcal{O}_d$ can be given by the classical function $e^{ax}$ where $\mathcal{O}_d$ endowed with the $\mathcal{A}$-module structure described in Example \ref{Eg:O_deleted_d}. Then the Newton transformation above completes the following commutative diagram
\[
\begin{tikzcd}[row sep=large, column sep=large]
\mathcal{D}/\mathcal{D}(\partial-a) \arrow{r}{\times e^{ax}}
\arrow[swap]{dr}{\times \exp_\Delta(a;\,X)}
& \mathcal{O}_d
\arrow{d}{\mathfrak{N}} \\
& \mathcal{O}_{\Delta}
\end{tikzcd}
\] where the space of analytic functions $\mathcal{O}_\Delta$ is endowed with a $\mathcal{A}$-module structure defined in Example \ref{Eg:O_delta}. It is straightforward to verify that
\begin{equation}\label{E:newton_bases}
\mathfrak{N}: x^n\ \ \longmapsto\ \ x(x-1)\cdots (x-n+1),
\end{equation}for every integer $n$, so that the above commutative diagram gives the difference exponential function
\[
\exp_\Delta(a;\, x):=(a+1)^x=\sum_{k=0}^{\infty} \frac{a^k}{k!}\, x(x-1)\cdots(x-k+1).
\]
converges absolutely for any complex $x$, if $|a|<1$ directly. The $\exp_\Delta$ was already derived in Example \ref{Eg:delta_exp}. \end{example}
Moreover, it is easy to see that
\begin{equation}\label{E:newton_bases_2}
\mathfrak{N}: x^\nu\ \ \longmapsto\ \ \frac{\Gamma(x+1)}{\Gamma(x+1-\nu)}= (x)_\nu,
\end{equation}for every $\nu\in\mathbb{C}$.
We shall apply similar rationale as the above example to derive generating functions for certain half-Bessel
modules with respect to difference operator to be considered in subsequent sections. In fact, similar transformations already appeared in the literature in different applications spread over many decades. See for example \cite{BHPR_2019}, in integral form \cite[p. 5 (25)]{Truesdell_1948} for Laguerre polynomials, and again in \cite[p. 401]{Gessel_2003} in ad hoc manners.
\begin{remark} Variants of the Newton transformation are possible. For example, if $\mathcal{O}_d$ in Theorem \ref{T:Newton_trans} is replaced by the space of entire functions $f$ with growth rate such that \[
|f(x)\, e^{mx}|\to 0\mbox{ as }x\to\pm i\infty, \] holds for every $ m\in\mathbb{C}$, then one replaces the Hankel-type contour in the above Newton transformation by, for example,
Then the conclusion of Theorem \ref{T:Newton_trans} remains valid. \end{remark}
\section{Poisson-type generating functions for (difference) Bessel polynomials}
We are ready to apply the Theorem \ref{T:bessel_poly_gf_map} derived from the last section to recover the well-known (Poisson-type) generating function for classical Bessel polynomials $\theta_n(x)$ and to derive new \textit{difference Bessel polynomials} $\theta_n^\Delta(x)$ in this section.
\begin{theorem}[(\textbf{Poisson transformation I})] Let $\mathcal{O}_d^{\mathbb{N}_0}$ denote the space of sequences of analytic functions $(f_n(x))_0^\infty$ with appropriately restricted growth rate be endowed with the $\mathcal{A}_2$-module structure defined in Example \ref{Eg:seq-functions_poisson}. Let $\mathcal{O}_{dd}$ denote the space of analytic functions be endowed with the $\mathcal{A}_2$-module structure as in the Theorem \ref{T:Bessel_gf}. Then the map \begin{equation}\label{E:poisson} \begin{array}{rcl}
\mathfrak{p}:\mathcal{O}^{\mathbb{N}_0}&\to&\mathcal{O}_{dd} \\ (f_n)&\mapsto&\displaystyle \sum_{n=0}^\infty f_n\dfrac{t^n}{n!}, \end{array} \end{equation} called the ``Poisson transformation" is a left $\mathcal{A}_2$-linear map.
\end{theorem}
\begin{proof} This is a straightforward verification. \end{proof}
For finite difference operator $\Delta$, we have the following analogue.
\begin{theorem}[(\textbf{Poisson transformation II})] Let $\mathcal{O}_\Delta^{\mathbb{N}_0}$ denote the space of bilateral sequences of analytic functions $(f_n(x))$ with appropriately restricted growth rate be endowed with the $\mathcal{A}_2$-module structure defined in Example \ref{Eg:seq-functions_poisson_delta}. Let $\mathcal{O}_{\Delta d}$ denote the space of analytic functions be endowed with the $\mathcal{A}_2$-module structure as in the Theorem \ref{T:Bessel_gf}. Then the map \begin{equation}\label{E:poisson_2} \begin{array}{rcl}
\mathfrak{p}_\Delta:\mathcal{O}^{\mathbb{N}_0}&\to&\mathcal{O}_{\Delta d} \\ (f_n)&\mapsto&\displaystyle \sum_{n=0}^\infty f_n\dfrac{t^n}{n!}, \end{array} \end{equation} called the ``difference Poisson transformation" is a left $\mathcal{A}_2$-linear map.
\end{theorem}
\begin{remark}\label{R:poisson} The left $\mathcal{A}$-module structure of $\mathbb{C}^\mathbb{Z}$ as in Example~\ref{Eg:two-seq-numbers} is specifically designed so that the z-transform $\mathfrak{z}:\mathbb{C}^\mathbb{Z}\to\mathcal{O}_d$ in Theorem~\ref{T:z-transform-1} is left $\mathcal{A}$-linear. In a similar way, the left $\mathcal{A}$-module structure of $\mathbb{C}^\mathbb{N}$ as in Example~\ref{Eg:seq-functions_poisson} and Example~\ref{Eg:seq-functions_poisson_delta} is specifically designed so that the above Poisson transforms are left $\mathcal{A}$-linear. \end{remark}
\subsection{Classical reverse Bessel polynomials $\theta_n(x)$}
\begin{example}\label{Eg:Bessel_mod_no_root} Let $\mathcal{A}_{2}(\rho):=\dfrac{\mathcal{A}_2\langle \rho, {1}/{\rho}\rangle}{\langle\rho^2-1+2X_2\rangle}$ be defined as in Theorem \ref{T:bessel_poly_gf_map}, and
\[
\Theta (\rho):=\dfrac{\mathcal{A}_2(\rho)}{\mathcal{A}_2(\rho)(-\rho^2\partial_2^2
+3\partial_2+X_1^2)
+\mathcal{A}_2(\rho)(X_1\partial_1+\rho^2\partial_2-1-X_1)}.
\] Then $\Theta(\rho)$ is a left $\mathcal{A}_{2}(\rho)$-module. On the other hand, the space $\mathcal{O}_2$ of two-variable analytic functions with appropriate growth rate endowed with
\begin{equation}\label{E:O_dd_endow_1}
\begin{split}
(\partial_1 f)(x, t)) &=f_x (x, t), \quad (X_1f)(x, t)=xf(x, t),\\
(\partial_2 f)(x, t) &=f_t (x, t), \quad (X_2f)(x, t)=tf(x, t)
\end{split}
\end{equation}
and by spectra analysis, $\rho^2f$ can be defined so that \[
(\rho^2 f)(x,t) = (1-2t)f(x,t),
\] $\mathcal{O}_2$ is a left $\mathcal{A}_2(\rho)$-module again denoted by $\mathcal{O}_{dd}$, and the same space $\mathcal{O}_2$ endowed with \eqref{E:O_delta_d}
\begin{equation}\label{E:O_delta_d_2}
\begin{array}{ll}
\partial_1f(x,\, t)=f(x+1,\, t)-f(x,\, t), & X_1f(x,\, t)=xf(x-1,\, t);\\
\partial_2f(x,\, t) =f_t(x,\, t), & X_2 f(x,\, t)=tf(x,\, t),
\end{array}
\end{equation} and
\[
(\rho^2 f)(x, t)=(1-2t)f(x, t),
\] is a left $\mathcal{A}_2(\rho)$-module again denoted by $\mathcal{O}_{\Delta d}$; and the space $\mathcal{O}^{\mathbb{N}_0}$ of all sequences of analytic functions endowed with \eqref{E:seq-functions_poisson} and $\rho^2f$ can be defined so that
\begin{equation}\label{E:O_dd_endow_2}
(\rho^2 f)_n = f_n-2nf_{n-1}
\end{equation}
is also a left $\mathcal{A}_2(\rho)$-module. \end{example}
We are ready to state our first main result in this subsection for which the generating function of the classical reverse Bessel polynomials derived by Burchnall \cite{Burchnall_1953} is a special case.
\begin{theorem}\label{T:general_rev_bessel_poly_gf}
\begin{enumerate}
\item Let $(\vartheta_n)$ be a sequence of analytic functions which is a solution of the reverse Bessel polynomial module $\Theta$ in $\mathcal{O}_d^{\mathbb{N}_0}$. Then there exist complex constants $C_1,\, C_2$ such that
\begin{equation}\label{E:gen_rev_bessel_poly_gf}
\frac{1}{\sqrt{1-2t}}\Big\{ C_1\exp\big[x(1-\sqrt{1-2t})\big]
+C_2 \exp\big[x(1+\sqrt{1-2t})\big]\Big\}
=\sum_{n=0}^\infty
\vartheta_n(x)\, \frac{t^n}{n!}
\end{equation}
which converges uniformly in each compact subset of $\mathbb{C}\times B(0,\, \frac12)$.
\item Moreover, the generating function in the part $\mathrm{(i)}$ above satisfies the system of partial differential equations
\begin{equation}\label{E:bessel_poly_PDE}
\begin{split}
&f_{xt}(x,\, t)-f_t(x,\, t)+xf(x, \, t)=0,\\
&xf_x (x,\, t)+(1-2t)f_t(x,\, t)-(1+x)f(x,\, t)=0
\end{split}
\end{equation} for all $x$ and $t$.\footnote{See also the Appendix \ref{SS:PDE_list}.}.
\end{enumerate} \end{theorem}
We skip the proof to this theorem since it is very similar to that of the following special case.
\begin{theorem}[(\cite{Grosswald}, p. 42)]\label{T:rev_bessel_poly_gf}
\begin{enumerate}
\item The reverse Bessel polynomials $(\theta_n)$ have the generating function
\begin{equation}\label{E:rev_bessel_poly_gf}
\frac{1}{\sqrt{1-2t}} \exp\big[x(1-\sqrt{1-2t})\big]=\sum_{n=0}^\infty
\theta_n(x)\, \frac{t^n}{n!}
\end{equation}
which converges uniformly in each compact subset of $\mathbb{C}\times B(0,\, \frac12)$.
\item Moreover, the generating function in the part $\mathrm{(i)}$ above satisfies the system of partial differential equations \eqref{E:bessel_poly_PDE}\footnote{See also the Appendix \ref{SS:PDE_list}.}.
\end{enumerate} \end{theorem}
\begin{proof} Recall the map $\mathfrak{p}$ as defined in \eqref{E:poisson} and the Example \ref{Eg:Bessel_mod_no_root} just discussed above that $\Theta(\rho)$ is a $\mathcal{A}_2(\rho)$-modules above. Then we have the diagram
\begin{equation}\label{E:commute-2}
\begin{tikzcd} [row sep=huge, column sep=huge]
\Theta(\rho) \arrow{r}{\times(\theta_n(x))}
\arrow[swap]{d}{\times \{C_1\rho^{-1}{\mathop{\rm E}}[X_1(1-\rho)]+C_2\rho^{-1}{\mathop{\rm E}}[X_1(1+\rho)]\}}
& \mathcal{O}^{\mathbb{N}_0}
\arrow{d}{\mathfrak{p}} \\
\widetilde{A}_2(\rho) \arrow{r}{\times 1}& \mathcal{O}_{d d}
\end{tikzcd}
\end{equation} in which $\widetilde{A}_2(\rho):=\overline{\mathcal{A}_2(\rho)/\big[\mathcal{A}_2(\rho)\partial_1+\mathcal{A}_2(\rho)\partial_2\big]}$. Since the above diagram commutes, the sum $\sum_{n=0}^\infty\theta_n (x)\, t^n/n!$ being the image of $1$ in $\mathcal{O}_{dd}$ via the top-right path, is also a solution to the system of differential-difference equations
\begin{equation}\label{E:bessel-poly-PDEs}
\begin{split}
&\theta_n^\prime(x)-\theta_n(x)+x\theta_{n-1}(x)=0,\\
&x\theta_n^\prime(x)-(x+2n+1)\theta_n(x)+\theta_{n-1}(x)=0.
\end{split}
\end{equation}
mentioned in the Theorem \ref{T:differential-differential}. Now the function $\frac{1}{\sqrt{1-2t}} \exp\big[x(1-\sqrt{1-2t})\big]$ is a solution to the system of PDEs \eqref{E:bessel_poly_PDE}
\begin{align}
&f_{xt}(x,\, t)-f_t(x,\, t)+xf(x, \, t)=0,\label{E:rev_bessel_poly_PDE_1}
\\
&xf_x (x,\, t)+ (1-2t)f_t(x,\, t)-(1+x)f(x,\, t)=0.\label{E:rev_bessel_poly_PDE_2}
\end{align} Since the PDEs \eqref{E:rev_bessel_poly_PDE_1} and \eqref{E:rev_bessel_poly_PDE_2} are images of the two generators of the Bessel module $\Theta$ in Definition \ref{D:Theta}, so the dimension of the local solution space of the PDEs coincides with the multiplicity two as stated in Theorem \ref{T:B_poly_mod_holonomic}.The Theorem \ref{T:bessel_poly_gf_map} directly verifies that the solution space of the PDEs has $\mathbb{C}$-dimension two by solving the explicit solution as a linear combination of $\rho^{-1}{\mathop{\rm E}}[X_1(1-\rho)]$ and $\rho^{-1}{\mathop{\rm E}}[X_1(1+\rho)]$ of $\Theta(\rho)$ in $\tilde{A}_2(\rho)$ that effects the bottom-left path in the commutative diagram \eqref{E:commute-2}. So
\[
\frac{1}{\sqrt{1-2t}}\big\{C_1 \exp\big[x(1-\sqrt{1-2t})\big]+C_2\exp\big[x(1+\sqrt{1-2t})\big]\big\}=\sum_{n=0}^\infty
\theta_n(x)\, \frac{t^n}{n!}
\]
for complex scalars $C_1,\, C_2$. Substituting $t=0$ into the above equation yields $C_1=1$ and $C_2=0$. One way to verify the convergence of \eqref{E:rev_bessel_poly_gf} is to combine \eqref{E:rev_bessel_poly_PDE_1} and \eqref{E:rev_bessel_poly_PDE_2} into a single ODE. Subtracting the equation \eqref{E:rev_bessel_poly_PDE_1} from \eqref{E:rev_bessel_poly_PDE_2} after it being differentiated partially with respect to $t$ yields the ODE
\[
(1-2t)f_{tt}(x,\, t)-3f_t(x,\, t)-x^2f(x,\, t)=0.
\]For each fixed, but otherwise arbitrary, $x$ the ODE has a regular singularity at $t=\frac12$. Hence the infinite sum in \eqref{E:rev_bessel_poly_gf} converges uniformly in each compact subset of $\mathbb{C}\times B(0,\, \frac12)$. This completes the proof. \end{proof}
\subsection{Difference reverse Bessel polynomials $\theta^\Delta_n(x)$} \label{SS:delta_reverse_bessel_poly} \begin{example}\label{Eg:BP_delta_J} Let $\Theta(\rho)$ and
$(\theta^\Delta_n(x))$ to denote the Bessel polynomial module defined in Example \ref{Eg:Bessel_mod_no_root} and infinite sequence of difference Bessel polynomials respectively. Suppose that $\mathcal{O}^{\mathbb{N}_0}=\mathcal{O}^{\mathbb{N}_0}_\Delta$ is endowed with the $\mathcal{A}_2$-module structure as defined in Example \ref{Eg:seq-functions_poisson_delta}.
Then the map
\begin{equation}\label{E:BP_delta_J}
\begin{array}{rcl}
\mathfrak{j}_\Delta:\Theta(\rho) & \stackrel{\times (\theta^\Delta_{n})}{\longrightarrow} &\mathcal{O}^{\mathbb{N}_0}_\Delta
\end{array}
\end{equation} is left $\mathcal{A}_2$-linear. \end{example}
We are now ready to make use of the Example \ref{Eg:Bessel_mod_no_root} and to apply Newton's transform $\mathfrak{N}$ to ``map" the classical generating for the reverse Bessel polynomials derived in the last subsection to the difference reverse Bessel polynomials.
\begin{theorem}\label{T:bessel_poly_delta_gf}
\begin{enumerate}
\item The difference reverse Bessel polynomials have the generating function
\begin{equation}\label{E:bessel_difference_poly_gf}
\frac{e^{-i\pi x}}{2i\sin\pi x\Gamma(-x)}\int_{-\infty}^{(0+)}e^\lambda(-\lambda)^{-x-1} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda=\sum_{n=0}^\infty
\theta_n^\Delta (x)\, \frac{t^n}{n!},
\end{equation}
where the infinite series converges uniformly in each compact subset of $\mathbb{C}\times B(0,\, \frac12)$.
\item Moreover, the generating function in part $\mathrm{(i)}$ above satisfies the delay-differential equations\footnote{See also the Appendix \ref{SS:PDE_list}.}
\begin{equation}\label{E:delta_bessel_poly_PDE}
\begin{split}
&f_t(x+1,\, t)-2f_t(x,\, t)+xf(x-1,\, t)=0,\\
&(1-2t)f_t(x,\, t)+(x-1)f(x,\, t)-2xf(x-1,\, t)=0.
\end{split}
\end{equation}
\end{enumerate} \end{theorem}
\begin{proof} We first draw on the simple fact from the Example \ref{Eg:Bessel_mod_no_root} that $\Theta (\rho)$ is a $\mathcal{A}_2(\rho)$-module and from the Example \ref{Eg:BP_delta_J} that the map $\mathfrak{j}_\Delta:\Theta(\rho) \stackrel{\times (\theta^\Delta_{n})}{\longrightarrow} \mathcal{O}^{\mathbb{N}_0}_\Delta $ is $\mathcal{A}_2(\rho)$-linear. Then the map $\mathfrak{p}_\Delta: \mathcal{O}^{\mathbb{N}_0} \longrightarrow\mathcal{O}_{\Delta d}$ as defined in \eqref{E:poisson_2} and their composition is also an $\mathcal{A}_2(\rho)$-linear. Thus $\mathfrak{p}_\Delta\circ \mathfrak{j}_\Delta$ acting on the identity yields the right-side of \eqref{E:bessel_difference_poly_gf}. Moreover, we have the commutative diagram \begin{equation}\label{E:commute-dbessel-2}
\begin{tikzcd} [row sep=large, column sep=large]
& \Theta (\rho) \arrow[swap]{dl}{\times\rho^{-1}{\mathop{\rm E}}[X_1(1-\rho)]} \arrow{d}{\mathfrak{j}} \arrow{dr}{\mathfrak{j}_\Delta} \\
\widetilde{\mathcal{A}}_2(\rho) \arrow{dr}{\times 1} & \mathcal{O}_d^{\mathbb{N}_0} \arrow{d} {\mathfrak{p}} & \mathcal{O}_\Delta^{\mathbb{N}_0} \arrow{d}{\mathfrak{p}_\Delta}\\
& \mathcal{O}_{d d} \arrow{r}{\mathfrak{N}} & \mathcal{O}_{\Delta d}
\end{tikzcd}
\end{equation}
in which $\widetilde{\mathcal{A}}_2(\rho):=\overline{\mathcal{A}_2(\rho)/\big[\mathcal{A}_2(\rho)\partial_1+\mathcal{A}_2(\rho)\partial_2\big]}$, and the growth of the analytic functions in $\mathcal{O}_{d d}$ and $\mathcal{O}_{\Delta d}$ are suitably restricted. Let
\[
y(x,t)=\int_{-\infty}^{(0+)}e^\lambda(-\lambda)^{-x-1} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda.
\]
Since $\frac{1}{\sqrt{1-2t}} \exp\big[x(1-\sqrt{1-2t})\big]$ is a solution to the system of PDEs \eqref{E:bessel_poly_PDE} according to the leftmost path of the above diagram, this $y(x,\, t)$ satisfies
\[
\begin{split}
&[\partial_2\partial_1-\partial_2+X_1]y(x,\, t)=0,\\
&\big[X_1\partial_2+(1-2X_2)\partial_2-1-X_1\big] y(x,\, t)=0,
\end{split}
\]by Theorem~\ref{T:Newton_trans} in the analytic function space $\mathcal{O}_{\Delta d}$ endowed with a $\mathcal{A}_2$-module structure given by the Example \ref{Eg:O_delta_d}. Hence the $y(x,\, t)$ satisfies the system of delay-differential equations \eqref{E:delta_bessel_poly_PDE}. We instead offer an alternative approach by direct verification that the $y(x,\, t)$ to satisfies the equivalent system\footnote{This would make the verification shorter.}
\[
\begin{split}
&[(2X_1-1)\partial_2^2+3\partial_2+X_1^2]y(x,\, t)=0,\\
&\big[ X_1\partial_1+(1-2X_2)\partial_2-1-X_1\big]y(x,\, t)=0,
\end{split}
\]where the first expression above can be found in Proposition \ref{P:Weyl_mod_half_bessel_3term}. It is straightforward to note that
\[
\partial_2 y=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x-1}
\big[ (1-2t)^{-1}+\lambda(1-2t)^{-\frac{1}{2}}\big]
\frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda
\] and \[ \begin{split} \partial^2_2y &=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x-1} \big[ 3(1-2t)^{-2} +3\lambda(1-2t)^{-\frac{3}{2}} +\lambda^{2}(1-2t)^{-1}\big]\\ &\quad\times \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda\\ \end{split} \] so that \[ \begin{split} (2X_2-1)\partial_2^2y
=&\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x-1}
\big[ -3(1-2t)^{-1}-3\lambda(1-2t)^{-\frac{1}{2}} -\lambda^{2}\big]\\
&\quad \times\frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda\\ &=(-3\partial_2-X_1^2) y. \end{split} \] Hence
\[
\big[(2X_2-1)\partial_2^2+3\partial_2+X_1^2\big]y=0.
\] On the other hand, we have \[ X_1\partial_1y=\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x-1} (x-\lambda) \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda\\ \]and
\[ \begin{split} \big[(1-2X_2)\partial_2-1-X_1\big] y& =\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x} (1-\sqrt{1-2t}) \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda\\ =&\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x} \frac{1}{\sqrt{1-2t}} d\exp\big[\lambda(1-\sqrt{1-2t})\big]. \end{split} \] Integration-by-parts of the above improper integral over an Hankel-type contour yields\[ \frac{1}{\Gamma(-x)}e^\lambda(-\lambda)^{-x} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\Big\vert_{-\infty}^{(0+)} =0 \]so that \[ \begin{split} \big[(1-2X_2)\partial_2-1-X_1\big] y=&-\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d(e^\lambda(-\lambda)^{-x})\\ =&-X_1\partial_1\frac{1}{\Gamma(-x)}\int_{-\infty}^{(+0)}e^\lambda(-\lambda)^{-x-1} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda. \end{split} \]Hence
\[
\big[ X_1\partial_1+(1-2X_2)\partial_2-1-X_1\big]y=0
\]as desired.
That is, $y(x,\, t)$ satisfies
\begin{align}
&(2t-1)f_{tt}(x,\, t)+3f_t(x, t)+x(x-1)f(x-2, t)=0, \label{E:delta_rev_bessel_PDE_1}\\
&(1-2t) f_t(x,t)+(x-1)f(x, t)-2xf(x-1, t)=0.\label{E:delta_rev_bessel_PDE_2}
\end{align}
Hence $y(x,\, t)$ satisfies the original system of delay-differential equations \eqref{E:delta_bessel_poly_PDE}.
Let us return to the right-half of the commutative diagram above. We notice that the rightmost path of the commutative diagram \eqref{E:commute-dbessel-2}, the sum $\sum_{n=0}^\infty\theta_n^\Delta (x)\, t^n/n!$ is a (formal sum) solution to the system of delay-difference equations
\begin{align*}
\theta^\Delta_n(x+1)-2\theta^\Delta_n(x)+x\theta^\Delta_{n-1}(x-1)&=0, \\
\theta^\Delta_{n+1}(x)+(x-2n-3)\theta^\Delta_n(x)-2x\, \theta^\Delta_n(x-1)&=0
\end{align*}
for each $n$, which were
derived in the Theorem \ref{T:difference_Bessel_Poly_DD}.
Let us determine the region of convergence of \eqref{E:bessel_difference_poly_gf}. First we differentiate the equation \eqref{E:delta_rev_bessel_PDE_2} with respect to $t$. This yields
\begin{equation}
\label{E:delta_rev_bessel_PDE_4}
(1-2t)f_{tt}(x,\, t)+(x-3)f_t(x,\, t)-2xf_t(x-1,\, t)=0.
\end{equation} Replace $x$ by $x-1$ in equation \eqref{E:delta_rev_bessel_PDE_2} and multiply the resulting equation by $x$ throughout yield
\begin{equation}\label{E:delta_rev_bessel_PDE_5}
x(1-2t)f_{t}(x-1,t) +x(x-2)f(x-1, t)-2x(x-1)f(x-2,\, t)=0.
\end{equation} Multiply the equation \eqref{E:delta_rev_bessel_PDE_1} throughout by $2$ and added to the \eqref{E:delta_rev_bessel_PDE_5} yield
\begin{equation}\label{E:delta_rev_bessel_PDE_6}
2(2t-1)f_{tt}(x,t) +6 f_t(x,t) +x(1-2t) f_t(x-1, t)+x(x-2) f(x-1, t)=0.
\end{equation} Multiply the equation \eqref{E:delta_rev_bessel_PDE_4} throughout by $1-2t$ yields
\begin{equation}\label{E:delta_rev_bessel_PDE_8}
(1-2t)^2f_{tt}(x,t)+(1-2t)(x-3)f_t(x,t)-2(1-2t)x f_t(x-1,t)=0.
\end{equation} Multiply the equation \eqref{E:delta_rev_bessel_PDE_6} by $2$ and add the resulting equation to \eqref{E:delta_rev_bessel_PDE_8} yield
\begin{equation}\label{E:delta_rev_bessel_PDE_10}
[(1-2t)^2+4(2t-1)]f_{tt}(x,t)+[(1-2t)(x-3)+12]f_t(x,t)+2x(x-2)f(x-1,t)=0.
\end{equation}
Now multiply the equation \eqref{E:delta_rev_bessel_PDE_2} throughout by the factor $x-2$ yields
\begin{equation}\label{E:delta_rev_bessel_PDE_12}
(x-2)(1-2t)f_t(x,t)+(x-2)(x-1)f(x,t)-2x(x-2)f(x-1,t)=0.
\end{equation}
It is now clear that the result of adding \eqref{E:delta_rev_bessel_PDE_10} and \eqref{E:delta_rev_bessel_PDE_12} yields, after simplification, the equation
\begin{equation}
\label{E:delta_rev_bessel_PDE_14}
(2t-1)(2t+3) f_{tt}(x,t)+[(1-2t)(2x-5)+12] f_t(x,t)+(x-1)(x-2)f(x,t)=0.
\end{equation}
For a fixed $x$, the above equation \eqref{E:delta_rev_bessel_PDE_14} is a second order linear ODE with two finite regular singularities and $t=\frac12$ is the one closest to the origin $t=0$ which is an ordinary point. Hence we conclude that the expansion in \eqref{E:bessel_difference_poly_gf} converges uniformly in each compact subset of $\mathbb{C}\times B(0,\, \frac12)$ as asserted.
This completes the proof of the theorem. \end{proof}
\subsection{Classical Bessel polynomials $y_n(x)$} \begin{example} Let $\mathcal{A}_{2}(\eta):=\dfrac{\mathcal{A}_2\langle\eta\rangle}{\langle\eta^2-1+2X_1X_2\rangle}$ be as defined in Theorem \ref{T:symbol gen yn-1}. Then $\mathcal{Y}(\eta)$ as defined in Theorem \ref{T:symbol gen yn-1} is a left $\mathcal{A}_{2}(\eta)$-module. Other examples of $\mathcal{A}_{2}(\eta)$-modules include $\mathcal{O}_{dd}$, $\mathcal{O}_{\Delta d}$, $\mathcal{O}^{\mathbb{N}_0}$ and so on. \end{example}
In much the same spirit of the derivation of the generating function for the classical reverse Bessel polynomials in Theorem \ref{T:rev_bessel_poly_gf}, we have the corresponding theorems for the generating function of the classical Bessel polynomials $(y_n(x))$ \cite{Krall_Frink_1949, Grosswald}.
\begin{theorem}\label{T:general_bessel_poly_gf}
\begin{enumerate}
\item Let $(\mathscr{Y}_n)$ be a sequence of analytic functions which is a solution of the Bessel polynomial module $\mathcal{Y}$ in $\mathcal{O}_d^{\mathbb{N}_0}$. Then there exist complex constants $C_1,\, C_2$ such that
\begin{equation}\label{E:gen_bessel_poly_gf}
C_1\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)
+C_2 \exp\left(\frac{1+(1-2xt)^{1/2}}{x}\right)
=\sum_{n=0}^\infty
\mathscr{Y}_{n-1}(x)\, \frac{t^n}{n!},
\end{equation}
which holds for $|t|<\left|\frac{1}{2x}\right|$, with an appropriate choice of $\mathscr{Y}_{-1}$.
\item Moreover, the holonomic system of PDEs of $\mathcal{Y}$ from the Definition \ref{D:y-1} when manifested in $\mathcal{O}_{dd}$ is given by
\begin{equation}\label{E:classical_bessel_poly_PDE}
\begin{split}
&x^2f_{xt}(x,\, t) -xtf_{tt}(x,\, t)+f_t(x,\,t)-f(x,\,t)=0,\\
&x^2f_x(x,\, t)-f_t(x,\, t)+xtf_t(x,\, t)+f(x,\,t)=0,
\end{split}
\end{equation}for which both sides of \eqref{E:gen_bessel_poly_gf} satisfy.
\end{enumerate} \end{theorem}
\begin{theorem}\label{T:Bessel p_gf_1}
\begin{enumerate}
\item The classical Bessel polynomials have the generating function
\begin{equation}\label{E:bessel p_gf-1}
\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)
=\sum_{n=0}^\infty y_{n-1}(x)\, \frac{t^n}{n!}
\end{equation}
which holds for $|t|<\left|\frac{1}{2x}\right|$.
\item Moreover, the generating function from part \textrm{(i)} satisfies the PDE system \eqref{E:classical_bessel_poly_PDE}\footnote{See also the Appendix \ref{SS:PDE_list}.}.
\end{enumerate} \end{theorem}
\begin{proof} Recall the Poisson transform $\mathfrak{p}$ as defined in \eqref{E:poisson}. Then we have the commutative diagram
\begin{equation}\label{E:commute-diff bess pol}
\begin{tikzcd} [row sep=huge, column sep=huge]
\mathcal{Y}(\eta) \arrow{r}{\times(y_{n-1}(x))}
\arrow[swap]{d}{{\mathop{\rm E}}(\frac{1-\eta}{X_1})}
& \mathcal{O}^{\mathbb{N}_0}_d
\arrow{d}{\mathfrak{p}} \\
\widetilde{A}_2(\eta) \arrow{r}{\times 1}& \mathcal{O}_{d d}
\end{tikzcd}
\end{equation} in which $\widetilde{\mathcal{A}_2}(\eta):=\overline{
\mathcal{A}_2(\eta)/\big[\mathcal{A}_2(\eta)\partial_1+\mathcal{A}_2(\eta)\partial_2\big] }$ and the analytic functions in $\mathcal{O}_{d d}$ are suitably restricted. Due to the similarity of the proof of Theorem \ref{T:rev_bessel_poly_gf}, the remaining of the proof will be completed in the Appendix \S\ref{A:BP_gf_1}.
\end{proof}
\subsection{Difference Bessel polynomials $y^\Delta_n(x)$} \label{S:detal_BP_gf_1}
\begin{theorem}\label{T:delta Bessel_gf_1}
\begin{enumerate}
\item The difference Bessel polynomials $(y^\Delta_n(x))$ have the Poisson-type generating function
\begin{equation}\label{E:bessel dp_gf-1}
\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}d\lambda
=\sum_{n=0}^\infty y^{\Delta}_{n-1}(x)\, \frac{t^n}{n!},
\end{equation}
which converges uniformly in each compact subset of $\mathbb{C}\times B(0,\frac12)$.
\item Moreover, the generating function from part $\mathrm{(i)}$ above satisfies the PDEs (delay-differential equations)\footnote{See also the Appendix \ref{SS:PDE_list}.}
\begin{equation}\label{E:delta_classical_bessel_poly_PDE}
\begin{split}
&x(x-1)[f_t(x-1,\, t)-f_t(x-2, \, t)]-xtf_{tt}(x-1,\, t)+f_t(x,\, t)-f(x, \, t)=0,\\
&x(x-1)[f(x-1,\, t)-f(x-2, \, t)]-f_{t}(x,\, t)+xtf_t(x-1,\, t)+f(x, \, t)=0.
\end{split}
\end{equation}
\end{enumerate} \end{theorem}
\begin{proof} Recall the maps $\mathfrak{p}$ and $\mathfrak{p}_\Delta$ as defined in \eqref{E:poisson} and \eqref{E:poisson_2} respectively. Then we have the diagram
\begin{equation}\label{E:commute-dbp}
\begin{tikzcd} [row sep=large, column sep=large]
& \mathcal{Y}(\eta) \arrow[swap]{dl}{{\mathop{\rm E}}(\frac{1-\eta}{X_1})} \arrow{d}{\times(y_{n-1}(x))} \arrow{dr}{\times(y^\Delta_{n-1}(x))} \\
\widetilde{\mathcal{A}}_2(\eta) \arrow{dr}{\times 1} & \mathcal{O}^{\mathbb{N}_0} \arrow{d} {\mathfrak{p}} & \mathcal{O}^{\mathbb{N}_0} \arrow{d}{\mathfrak{p}_\Delta} \\
& \mathcal{O}_{d d} \arrow{r}{\times\mathfrak{N}} & \mathcal{O}_{\Delta d}
\end{tikzcd}
\end{equation}
in which $\widetilde{\mathcal{A}}_2(\eta):=\overline{\mathcal{A}_2/\big[\mathcal{A}_2(\eta)\partial_1+\mathcal{A}_2(\eta)\partial_2\big]}$ and the analytic functions in $\mathcal{O}_{\Delta d}$ have suitably restricted growth. The remaining of the proof, being similar to that of the Theorem \ref{T:bessel_poly_delta_gf}, will be completed in the Appendix \S \ref{A:detal_BP_gf_1}. \end{proof}
\section{Half-Bessel modules II: Glaisher's generating functions}\label{S:half_bessel_II}
In a number of publications between the years 1873-1881 (see e.g., \cite{Glaisher1881}), the derivations of trigonometric-type generating functions for Bessel functions
\[
\begin{split}
&\sqrt{\frac{2}{\pi x}} \cos\sqrt{x^2-2xt}
=\sum_{n=0}^\infty J_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!},\\
&\sqrt{\frac{2}{\pi x}} \sin\sqrt{x^2+2xt}
=\sum_{n=0}^\infty J_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}
\end{split}
\] obtained by Glaisher are known to be ``formidable" to compute as noted by Watson \cite[p. 144]{Watson1944}. We shall show in this section that how a suitable ``characteristic change of variables" from the generators of the Bessel module of order $n+\frac12$ in Definition \ref{D:bessel_mod} can simplify Glaisher\rq{}s computation \cite[pp. 774-781]{Glaisher1881}. Moreover, in addition to recovering and extending Glaisher's results for classical Bessel functions, we also derive analogous generating functions for the difference Bessel functions in this section. Analogous generating functions for $q$-Bessel functions will be investigated in a separate publication. On the other hand, Glaisher's results have been superseded by Lommel's results from a different perspective, see also \cite[p. 144]{Watson1944}. Weisner \cite{Weisner_1959} re-derived Lommel's results with Lie-algebraic method, whose idea can also be found in \cite{Mcbride}. We comment that no explicit Lie algebraic method has been employed in our approach.
\subsection{Characteristics of Glaisher modules} \subsubsection{Negative Glaisher module}
The characteristic change of variables we shall apply are
\begin{equation}\label{E:glaisher_char_1}
\begin{array}{ll}
\Xi_1=\partial_1, & W_1=X_1\\
\Xi_2={1}/{X_2}, & W_2=X_2\partial_2 X_2.
\end{array}
\end{equation}
Then one can easily verify that
\[
[\pm \Xi_{1}, \pm W_{1}]=1, \quad [\Xi_{2}, W_{2}]=1,
\quad
[\Xi_{1}, \Xi_{2}]=0,\quad [W_1, W_{2}]=0,
\quad [\Xi_i,\, W_j]=0,\ i\not=j
\]hold.
Substitute the above characteristic change of variables into \eqref{E:bessel_PDE} and choose
$\nu=-\frac{1}{2}$ yield
\begin{align}
&W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})-W_1\Xi_2^{-1},\label{E:PDEs_neg_glaisher_1}\\
&W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})+W_1 \Xi_2.\label{E:PDEs_neg_glaisher_2}
\end{align}
\begin{definition}\label{D:neg_glaisher_module}
Let $W_i$ and $\Xi_i$ be defined as in \eqref{E:glaisher_char_1}. The
\begin{equation}\label{E:neg_glaisher_module}
\mathcal{G}_{-}=
\dfrac{\mathcal{A}_{2}}
{\mathcal{A}_{2}
\big(W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})- W_1\Xi_2^{-1}\big)
+\mathcal{A}_{2}\big(W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})+ W_1 \Xi_2\big)}
\end{equation}
is called the \textit{negative Glaisher module}. \end{definition}
\begin{theorem}[\textbf{(Integrability)}]\label{T:neg_glaisher_holonomic} The negative Glaisher module $\mathcal{G}_{-}$ is a left $\mathcal{A}_2$-module that has dimension two and multiplicity two. In particular, it is holonomic.
\end{theorem}
\begin{proof} Let us effect the following ``change of variables"
\[
\widetilde{W}_2=\Xi_2,\quad
\widetilde{\Xi}_2=-W_2
\]
in the two generators of the negative Glaisher module \eqref{E:neg_glaisher_module}. Note that $[\widetilde{\Xi}_2,\, \widetilde{W}_2]=1$.
This yields
\[
\begin{split}
&W_1 \Xi_1+(\widetilde{W}_2\widetilde{\Xi}_2-\frac{1}{2})-W_1\widetilde{W}_2^{-1},\\
&W_1 \Xi_1-(\widetilde{W}_2\widetilde{\Xi}_2-\frac{1}{2})+W_1 \widetilde{W}_2.
\end{split}
\] Subtracting and adding the above two relations yield
\[
\begin{split}
& \widetilde{\Xi}_2+\frac12\Big(\frac{1}{\widetilde{W}_2}+W_1+\frac{W_1}{\widetilde{W}_2^2}\Big)\\
& \Xi_1+\frac12\Big(\widetilde{W}_2-\frac{1}{\widetilde{W}_2}\Big).
\end{split}
\]One immediately verifies that the two relations satisfy the \textit{integrability condition} discussed from Example \ref{EG:integrable}.
A disadvantage of the above argument is although the ``change of variables" method preserves holonomicity, there is no guarantee that the multiplicity of the negative Glaisher's module remains unchanged. Indeed, we could follow the argument used in the proof of Theorem \ref{T:B_poly_mod_holonomic} to conclude directly that the dimension and multiplicity of the module are both two. We skip the details. \end{proof}
We immediately obtain from the negative Glaisher module \eqref{E:neg_glaisher_module} the following surjection.
\begin{proposition}
The natural map
\[
\dfrac{\mathcal{A}_2}
{\mathcal{A}_2
[(W_1\Xi_1)^2+W_1^2-(W_2 \Xi_2
-\frac{1}{2})^2]}
\rightarrow \mathcal{B}_{-\frac{1}{2}}
\]
is a well-defined $\mathcal{A}_2$-linear surjection.
\end{proposition}
We omit its routine verification.
\begin{theorem}\label{T:neg_glaisher_cosine}
Let $\mathcal{G}_{-}$ be the negative Glaisher module. Set $\omega_-^2=W_1^2- 2W_1W_2$ and
\[
\mathcal{A}_{2}(\omega_-)=\langle \Xi_1, \Xi_2,
W_1, W_2, \omega_-^{\pm 1}\rangle.
\]
Then the
\[
S=C_1\mathop{\rm Cos} \omega_-+ C_2\mathop{\rm Sin} \omega_-
\] for each choice of complex constants $C_1, C_2$, where $\mathop{\rm Sin}$ and $\mathop{\rm Cos}$ are the Weyl sine and the Weyl cosine defined in \eqref{E:sine-map} and \eqref{E:cosine-map}, is an element of $\mathcal{A}_2(\omega_-)$, such that the map
\[
\mathcal{G}_{-}
\xrightarrow
{\times S}
\overline{\mathcal{A}_{2}(\omega_-)/[\mathcal{A}_{2}(\omega_-)
(W_1\Xi_1+\frac{1}{2})
+\mathcal{A}_{2}(\omega_-)\Xi_2]}
\] is well-defined and left $\mathcal{A}_2$-linear.
\end{theorem}
\begin{proof} Suppose
\[
\delta_1=\Xi_1\, \omega_--\omega_-\,\Xi_1,
\qquad
\delta_2=\Xi_2\, \omega_--\omega_-\, \Xi_2
\]
and
\begin{equation}\label{E:glaisher_char_3}
\delta_1\, \omega=\omega_-\,\delta_1,\quad \omega_-\delta_2=\delta_2\omega_-,
\quad [\omega_-,\, W_1]=0,\quad [\omega_-,\, W_2]=0.
\end{equation}
Then it is easy to compute
\[
\Xi_1\, \omega_-^2-\omega_-^2\,\Xi_1=2(W_1-W_2), \quad
\Xi_2\, \omega_-^2-\omega_-^2\, \Xi_2=-2W_1.
\] Hence
\begin{equation}\label{E:glaisher_char_4}
\delta_1= \Xi_1\, \omega_- -\omega_-\, \Xi_1=\frac{W_1-W_2}{\omega_-},
\quad
\delta_2=\Xi_2\, \omega_--\omega_-\, \Xi_2=-\frac{W_1}{\omega_-}.
\end{equation}
Subtracting the two elements \eqref{E:PDEs_neg_glaisher_1} and \eqref{E:PDEs_neg_glaisher_2} yields the expression
\[
-2(W_2\Xi_2-\frac12\big)+W_1\Xi_2+W_2/\Xi_2.
\]Left multiplication to the above expression throughout by $\Xi_2$ on the left yields
\[
\Xi_2[(W_1-2W_2)\Xi_2+1+W_1/\Xi_2]=(W_1-2W_2)\Xi_2^2-\Xi_2+W_1.
\]
Right multiplication of the above expression by $W_1$ throughout yields
\[
W_1(W_1-2W_2)\Xi_2^2-W_1\Xi_2+W_1^2=\omega^2\, \Xi_2^2-W_1\Xi_2+W_1^2.
\]
Substituting the second expression from \eqref{E:glaisher_char_4} into the above expression and utilising the \eqref{E:glaisher_char_3} lead to
\[
(\omega_-\, \Xi_2)^2+W_1^2
=W_1^2 \big[\big(\frac{\omega_-\, \Xi_2}{W_1}\big)^2+1\big]=W_1^2 (\Xi^2+1).
\]Let us denote
\begin{equation}\label{E:PDEs_neg_glaisher_3}
\Xi=-\frac{\omega_-\, \Xi_2}{W_1}.
\end{equation}It follows from the second expression of \eqref{E:glaisher_char_4} that
\[
[\Xi,\, \omega_-]=1.
\] Therefore, it follows from the above commutator and \eqref{E:PDEs_neg_glaisher_2} that it is appropriate to set
\begin{align}
\aleph: &= \Xi_1+\Big(1-\frac{W_2}{W_1}\Big)\Xi_2,\label{E:PDEs_neg_glaisher_4} \\
W: &= W_1.\label{E:PDEs_neg_glaisher_5}
\end{align} Then it is easy to check that
\[
[\aleph, W]=1,\quad
[\Xi,\, \omega_-]=1,\quad
[\aleph, \Xi]=0,\quad
[\aleph, \omega_-]=0,\quad
[\Xi, W]=0,\quad
[\omega_-, W]=0
\]hold. Hence we may consider
\[
\widehat{\mathcal{A}}_2=\mathbb{C}\langle \aleph , \Xi, W, \omega_-\rangle,
\] and to rewrite the negative Glaisher module \eqref{E:neg_glaisher_module} in the form with the above ``change of variables\rq\rq{}:
\[
\widehat{\mathcal{G}}_- :=\frac{\widehat{\mathcal{A}}_2}
{\widehat{\mathcal{A}}_2(W\aleph+\frac12)+\widehat{\mathcal{A}}_2 (\Xi^2+1)}.
\]
We observe from Example \ref{E:prime_integrable_eg_2} that the $\widehat{\mathcal{G}}_-$ is a holonomic module. In particular, its dimension and multiplicity are both equal to $2$. On the other hand, it follows from Example \ref{Eg:trigo} that there exists a left $\widehat{\mathcal{A}_2}$-linear map
\[
\widehat{\mathcal{A}}_2/
\widehat{\mathcal{A}}_2 (\Xi^2+1) \ \xrightarrow {\times S}\
\overline{ \widehat{\mathcal{A}}_2/
\widehat{\mathcal{A}}_2 \Xi}
\]
of the form
\[
S= C_1\mathop{\rm Cos} (\omega_-)+C_2\mathop{\rm Sin} (\omega_-)
\]for some complex numbers $C_1,\, C_2$.
Hence a solution map $S$ for
\[
\widehat{\mathcal{G}}_-
\ \xrightarrow {\times S}\
\overline{\widehat{\mathcal{A}}_2/\big[
\widehat{\mathcal{A}}_2(W\aleph+\frac12) +\widehat{\mathcal{A}}_2\Xi\big]}
\]must assume the form
\[
S=f_1(W)\mathop{\rm Cos} (\omega_-)+f_2(W)\mathop{\rm Sin} (\omega_-),
\]where the maps $f_1, f_2$ depends on $W$ only.
Without loss of generality, we may consider, $f_1$ only. As a result,
\[
\begin{split}
0=&\big(W\aleph+\dfrac12)f_1(W)\,\mathrm{Cos} \,\omega_-
\ \mod \widehat{\mathcal{A}_2}(W\aleph+\dfrac12)\\
=&\mathrm{Cos} \,\omega_-(W\aleph+\dfrac12\big)f_1(W)
\ \mod \widehat{\mathcal{A}_2}(W\aleph+\dfrac12)\\
=&W\aleph f_1(W),
\end{split}
\]
which implies $\aleph f_1(W)=0$, Hence $f_1(W)$ reduces to a constant. Similarly $f_2(W)$ also reduces to a constant.
Hence it follows from \eqref{E:PDEs_neg_glaisher_3}, \eqref{E:PDEs_neg_glaisher_4} and \eqref{E:PDEs_neg_glaisher_5} that we have derived the asserted map $S$
\[
\mathcal{G}_-(\omega_-)
\xrightarrow
{\times (C_1\,\mathrm{Cos} \,\omega_-+ C_2\,\mathrm{Sin} \,\omega_-)}
\overline{\mathcal{A}_2(\omega_-)/
[\mathcal{A}_2(\omega_-)(W_1\Xi_1+1/2)+
\mathcal{A}_2(\omega_-)\Xi_2}].
\]This completes the proof.
\end{proof}
\subsubsection{Positive Glaisher module}
The characteristic change of variables we shall apply are specific sign changes of those of \eqref{E:glaisher_char_1}
\begin{equation}\label{E:glaisher_char_2}
\begin{array}{ll}
\Xi_1=-\partial_1, & W_1=-X_1,\\
\Xi_2={1}/{X_2}, & W_2=X_2\partial_2 X_2
\end{array}
\end{equation}with $\nu=\frac12$
in \eqref{E:bessel_PDE} and choose
\begin{equation}\label{E:PDEs+half_bessel}
\begin{split}
&W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})+W_1\Xi_2^{-1},\\
&W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})-W_1 \Xi_2.
\end{split}
\end{equation}
\begin{definition}\label{D:pos_glaisher_module} Let $W_i$ and $\Xi_i$ be defined as in \eqref{E:glaisher_char_2}. Then
\begin{equation}\label{E:pos_glaisher_module}
\mathcal{G}_{+}=
\dfrac{\mathcal{A}_{2}}
{\mathcal{A}_{2}
\big(W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})+W_1\Xi_2^{-1}\big)
+\mathcal{A}_{2}\big(W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})-W_1 \Xi_2\big)}
\end{equation}
a \textit{positive Glaisher module}. \end{definition}
We obviously have the following companion theorem for the negative Glaisher module.
\begin{theorem}\label{T:pos_glaisher_holonomic} The positive Glaisher module $\mathcal{G}_{+}$ is a left $\mathcal{A}_2$-module that has dimension two and multiplicity two. In particular, it is holonomic. \end{theorem}
We state without giving details the theorem
\begin{theorem}\label{T:pos_glaisher_sine}
Let $\mathcal{G}_{+}$ be defined in Definition
\ref{D:pos_glaisher_module}. Set $\omega_+^2=W_1^2+2W_1W_2$ and
\[
\mathcal{A}_{2}(\omega_+)=\langle -\Xi_1, \Xi_2,
-W_1, W_2, \omega_+^{\pm 1}\rangle.
\]
Then the expression
\[
S=C_1\mathop{\rm Cos} \omega_++ C_2\mathop{\rm Sin} \omega_+,
\]
where $C_1, C_2$ are constants, $\mathop{\rm Sin}$ and $\mathop{\rm Cos}$ are the Weyl sine and the Weyl cosine defined in \eqref{E:sine-map} and \eqref{E:cosine-map} respectively, is an element of $\mathcal{A}_2(\omega_+)$, such that the map
\[
\overline{\mathcal{G}_{+}}
\xrightarrow
{\times S}
\overline{\mathcal{A}_{2}(\omega_+)/[\mathcal{A}_{2}(\omega_+)
(W_1\Xi_1+\frac{1}{2})
+\mathcal{A}_{2}(\omega_+)\Xi_2]}
\]
is left $\mathcal{A}_{2}(\omega_+)$-linear.
\end{theorem}
\subsection{Glaisher's generating functions}
Combining the discussion from the previous subsections concerning the positive and negative Glaisher-modules, which are modified half-Bessel modules, yields the followings.
\begin{theorem}\label{T:classical_glaisher_gf}
\begin{enumerate}
\item Let $(\mathscr{C}_{n+\nu})_n$ be a bilateral sequence of analytic functions which is a solution of the Glaisher modules $\mathcal{G}_\mp$ in $\mathcal{O}^\mathbb{Z}$. Then there exist complex numbers $C_1,\, C_2$ such that
\begin{equation}\label{E: general half bessel_gf}
x^{-\frac12}
\big[C_1\cos\sqrt{x^2\pm 2xt}+C_2\sin \sqrt{x^2\pm 2xt} \big]
=\sum_{n=0}^\infty \mathscr{C}_{\mp n\pm \frac{1}{2}}(x)\, \frac{t^n}{n!},
\end{equation}where the convergence of the infinite sum in \eqref{E: general half bessel_gf} is uniform in each compact subset of $\mathbb{G}:=\{(x,\, t)\in \mathbb{C}^\dagger\times\mathbb{C}: 2|t|<|x|\}$, $\mathbb{C}^\dagger:=\mathbb{C}\backslash\{x:x\le 0\}$.
\item Moreover, the holonomic systems of PDEs that generates the Glaisher's modules $\mathcal{G}_{-}$ \eqref{E:neg_glaisher_module} and $\mathcal{G}_{+}$ \eqref{E:pos_glaisher_module} with manifestation in $\mathcal{O}_{dd}$ as defined in \eqref{E:O_dd_endow} from Example \ref{Eg:O_deleted_d} are given, respectively, by\footnote{See also the Appendix \ref{SS:PDE_list} for the list of holonomic PDE systems.}
\begin{align}
&xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)-xf(x, t)=0,\label{E:glaisher_PDE_1}\\
&xf_x(x,\, t)+(x-t)f_t(x,\, t)+ \frac12 f(x,\, t)=0\label{E:glaisher_PDE_2}
\end{align}
and
\begin{align}
&xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)+xf(x, t)=0,\label{E:glaisher_PDE_6}
\\
&xf_x(x,\, t)-(x+t)f_t(x,\, t)+\frac12 f(x,\, t)=0.\label{E:glaisher_PDE_7}
\end{align}
\end{enumerate} \end{theorem}
\begin{proof} Recall the map $\mathfrak{p}$ as defined in \eqref{E:poisson}. Then we have the diagram
\begin{equation*}
\begin{tikzcd} [row sep=large, column sep=large]
\mathcal{G}_{\pm} \arrow{r}{\times(\mathscr{C}_{\pm n\mp\frac12})}
\arrow[swap]{d}{\mathop{\rm Cos}\omega_\pm\slash\mathop{\rm Sin}\omega_\pm}
& \mathcal{O}^{\mathbb{N}_0} \arrow{d}{\mathfrak{p}}\\
\widetilde{\mathcal{A}_2}(\omega_\pm) \arrow{r}{\times x^{-\frac12}} & \mathcal{O}_{d d}
\end{tikzcd}
\end{equation*} in which
\[
\widetilde{\mathcal{A}_2}(\omega_\pm):=\overline{\displaystyle
{\mathcal{A}_{2}(\omega_\pm)}/{\big[\mathcal{A}_{2}(\omega_\pm)(W_1\Xi_1+\frac{1}{2})
+\mathcal{A}_{2}(\omega_\pm)\Xi_2\big]}},
\]where the left-vertical map given in terms of
Weyl cosine and sine $\mathop{\rm Cos} \omega_\pm\slash \mathop{\rm Sin} \omega_\pm$ which are defined as in Example \ref{Eg:trigo}, is guaranteed by the Theorem \ref{T:neg_glaisher_cosine}. We also note that the analytic functions in $\mathcal{O}_{dd}$ are suitably restricted.
We first consider the case of negative Glaisher's module $\mathcal{G}_-$. Since $\omega_-^2 =W_1^2-2W_1W_2$, the lower left path of the above commutative diagram shows that
\begin{equation}\label{E:glaisher_fn}
x^{-\frac12}
\big(C_1\cos\sqrt{x^2- 2xt}+C_2\sin \sqrt{x^2- 2xt} \big)
\end{equation}solves the PDEs \eqref{E:PDEs_neg_glaisher_1} and \eqref{E:PDEs_neg_glaisher_2} when manifested in the $\mathcal{A}_2$-module $\mathcal{O}_{dd}$ endowed with the interpretation in the Example \ref{Eg:O_deleted_d}, i.e.,
\begin{align*}
&xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)-xf(x, t)=0,
\\
&xf_x(x,\, t)+(x-t)f_t(x,\, t)+\frac12 f(x,\, t)=0
\end{align*}
which are precisely the PDEs \eqref{E:glaisher_PDE_1} and \eqref{E:glaisher_PDE_2}. On the other hand, via the top-right path of the diagram, the sum
\[
\sum_{n=0}^\infty J_{n-\frac{1}{2}}(x)\,\frac{ t^n}{n!}
\]is a solution to the system of PDEs \eqref{E:PDEs_neg_glaisher_1} and \eqref{E:PDEs_neg_glaisher_2}.
Since we know from Theorem \ref{T:neg_glaisher_holonomic} that the $\mathcal{G}_-$ is holonomic with both dimension and multiplicity equal to two, so the dimension of the local solution space about $t=0$ for the PDEs \eqref{E:PDEs_neg_glaisher_1} and \eqref{E:PDEs_neg_glaisher_2} is therefore two. We conclude that the formula \eqref{E: general half bessel_gf} holds. We next investigate the convergence region of the series \eqref{E: general half bessel_gf}.
Differentiating the equation \eqref{E:glaisher_PDE_2} with respect to $t$ yields
\begin{equation}\label{E:glaisher_PDE_3}
xf_{xt}(x,\, t)+(x-t)f_{tt}(x,\, t)-\frac12f_t(x,\, t)=0.
\end{equation}
Substituting the equation \eqref{E:glaisher_PDE_1} into equation \eqref{E:glaisher_PDE_3} yields the new equation
\begin{equation}\label{E:glaisher_PDE_4}
(x-2t)f_{tt}(x,\, t) -f_t(x, t)+xf(x,\, t)=0.
\end{equation}The equation is a second order linear ODE for a fixed $x$ and it has an ordinary point at $t=0$ and a regular singularity at $t={x}/{2}$. Hence the expansion \eqref{E: general half bessel_gf} converges uniformly in each compact subset of $\mathbb{C}\times \mathbb{C}$ where $2|t|<|x|$.
The handling of the second case when $\omega_+=W_1^2+2W_1W_2$ when $\nu=\frac12$ was dealt with in the last subsection which is similar to that of the case $\mathcal{G}_-$ and is therefore omitted. We only note that the PDEs \eqref{E:glaisher_PDE_1} and \eqref{E:glaisher_PDE_2} are replaced by
\begin{align*}
&xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)+xf(x, t)=0,
\\
&xf_x(x,\, t)-(x+t)f_t(x,\, t)+\frac12 f(x,\, t)=0
\end{align*}which are precisely the \eqref{E:glaisher_PDE_6} and \eqref{E:glaisher_PDE_7}
A similar procedure that has led to the equation \eqref{E:glaisher_PDE_4}
applied to the above equations \eqref{E:glaisher_PDE_6} and \eqref{E:glaisher_PDE_7} yields
\begin{equation}\label{E:glaisher_PDE_8}
(x+2t)f_{tt}(x,\, t) + f_t(x,\, t)+xf(x,\, t)=0.
\end{equation}This equation is similar to the equation \eqref{E:glaisher_PDE_4} which also has an ordinary point at $t=0$ and a regular singular point at $t=-x/2$. So we omit the details. \footnote{It is unclear to the authors of the reason why Watson indicated in \cite[p. 140]{Watson1944} that the restriction $2|t|<|x|$ in the above theorem only applies to the case $\nu=\frac12$, that is to $\mathcal{G}_+$ instead to both since $t=x/2$ is a common regular singularity for both of the differential equations \eqref{E:glaisher_PDE_4} and \eqref{E:glaisher_PDE_8}. }
The above analysis shows that the commutative diagram \eqref{E:commute-glaisher} holds. This completes the proof. \end{proof}
One can deduce the following Glaisher-type generating functions for other classical Bessel functions from the above Theorem.
\begin{corollary} \label{C:Graisher_gf}
Let $\mathbb{G}:=\{2|t|<|x|:\ (x,\, t)\in \mathbb{C}^\dagger\times\mathbb{C}\}$ where $\mathbb{C}^\dagger:=\mathbb{C}\backslash\{x:x\le 0\}$. Then the following series converge uniformly in any compact subset of $\mathbb{G}$.\footnote{Modified forms of the (i) and (ii) can be found from \cite[p. 439]{AS_1964} for $j_n(x)=\sqrt{\pi/2x}J_{n+\frac12}(x)$ and $y_n(x)=\sqrt{\pi/2x}Y_{n+\frac12}(x)$.}
\begin{enumerate}
\item
\begin{equation}\label{E: first half bessel_gf}
\begin{split}
&\sqrt{\frac{2}{\pi x}} \cos(x^2-2xt)^{\frac{1}{2}}
=\sum_{n=0}^\infty J_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!},\\
&\sqrt{\frac{2}{\pi x}} \sin(x^2+2xt)^{\frac{1}{2}}
=\sum_{n=0}^\infty J_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!};
\end{split} \end{equation}
\item
\begin{equation}\label{E: second half bessel gf}
\begin{split}
&\sqrt{\frac{2}{\pi x}}\sin(x^2-2xt)^{\frac{1}{2}}
=\sum_{n=0}^{\infty} Y_{n-\frac{1}{2}}(x)\,\frac{t^n}{n!},\\
-&\sqrt{\frac{2}{\pi x}} \cos(x^2+2xt)^{\frac{1}{2}}
=\sum_{n=0}^{\infty} Y_{-n+\frac{1}{2}}(x)\,\frac{t^n}{n!};
\end{split}
\end{equation}
\item
\begin{equation}\label{E: first modify half bessel gf}
\begin{split}
&\sqrt{\frac{2}{\pi x}}\cos(-x^2-2xt)^{\frac{1}{2}}=
\sum_{n=0}^{\infty} I_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!},\\
-&i\sqrt{\frac{2}{\pi x}}\sin(-x^2-2xt)^{\frac{1}{2}}=
\sum_{n=0}^{\infty} I_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!};
\end{split}
\end{equation}
\item \begin{equation}\label{E: second modify half bessel gf} \sqrt{\frac{\pi}{2 x}}e^{i (-x^2+2xt)^{1/2}}= \sum_{n=0}^{\infty} K_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!}= \sum_{n=0}^{\infty} K_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}. \end{equation}
\end{enumerate}
\end{corollary}
\begin{proof}
\begin{enumerate}
\item By choosing $t=0$ in equation \eqref{E: general half bessel_gf}, we immediately have
\[
x^{-\frac{1}{2}} \big[ C_1\cos x +C_2\sin x\big]=
\mathscr{C}_{-\frac{1}{2}}(x)
\]
and
\[
x^{-\frac{1}{2}} \big[ C_1\sin x +C_2\cos x\big]
=\mathscr{C}_{\frac{1}{2}}(x).
\]
Since
\[
J_{-\frac{1}{2}}(x)= \sqrt{2/\pi} x^{-\frac{1}{2}} \cos x,
\qquad
J_{\frac{1}{2}}(x)= \sqrt{2/\pi} x^{-\frac{1}{2}} \sin x.
\]
Hence $C_1=C_2=\sqrt{2/\pi}$ and we obtain \eqref{E: first half bessel_gf}.
\item
Applying the well-known formulae
\[
Y_{-\frac{1}{2}}(x)= \sqrt{2/\pi} x^{-\frac{1}{2}} \sin x,
\qquad
Y_{\frac{1}{2}}(x)= -\sqrt{2/\pi} x^{-\frac{1}{2}} \cos x
\]
to $\mathscr{C}_{-\frac{1}{2}}(x)$ and
$\mathscr{C}_{\frac{1}{2}}(x)$,
we obtain \eqref{E: second half bessel gf} immediately.
\item We have
\begin{align*}
\sum_{n=0}^{\infty} I_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!} &=
\begin{cases}
\sum_{n=0}^{\infty}(e^{-\frac{ i\pi}{2}(n-\frac{1}{2}) }
J_{n-\frac{1}{2}}(xe^{\frac{i\pi}{2}}))\, \frac{t^n}{n!},
& -\pi <\arg x\leqslant \frac{1}{2}\pi\\
\sum_{n=0}^{\infty}(e^{\frac{3}{2}i \pi(n-\frac{1}{2})} J_{n-\frac{1}{2}}(xe^{\frac{-3i\pi}{2}}))\, \frac{t^n}{n!},
& \frac{\pi }{2}<\arg x\leqslant \pi,
\end{cases} \\
&=
\begin{cases}
e^{\frac{ i\pi}{4} }\sum_{n=0}^{\infty}
J_{n-\frac{1}{2}}(xe^{\frac{i\pi}{2}}) \, \frac{(te^{-\frac{i\pi}{2}})^n}{n!},
& -\pi <\arg x\leqslant \frac{1}{2}\pi,\\
e^{-\frac{3}{4}i \pi} \sum_{n=0}^{\infty}
J_{n-\frac{1}{2}}(xe^{\frac{-3i\pi}{2}})\, \frac{(te^{\frac{3i\pi}{2}})^n}{n!},
& \frac{\pi }{2}<\arg x\leqslant \pi.
\end{cases} \\
&=
\begin{cases}
\sqrt{2/\pi}x^{-\frac{1}{2}}
\cos(-x^2-2xt)^{\frac{1}{2}} ,
& -\pi <\arg x\leqslant \frac{1}{2}\pi,\\
\sqrt{2/\pi}x^{-\frac{1}{2}}
\cos(-x^2-2xt)^{\frac{1}{2}}
& \frac{\pi }{2}<\arg x\leqslant \pi.
\end{cases}
\end{align*}
from (i). Similarly, we have
\[
\sum_{n=0}^{\infty} I_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}=
\begin{cases}
e^{-\frac{ i\pi}{4} }\sum_{n=0}^{\infty}
J_{-n+\frac{1}{2}}(xe^{\frac{i\pi}{2}})
\, \frac{(te^{\frac{i\pi}{2}})^n}{n!},
& -\pi <\arg x\leqslant \frac{1}{2}\pi,\\
e^{\frac{3}{4}i \pi} \sum_{n=0}^{\infty}
J_{-n+\frac{1}{2}}(xe^{\frac{-3i\pi}{2}})
\, \frac{(te^{\frac{-3i\pi}{2}})^n}{n!},
& \frac{\pi }{2}<\arg x\leqslant \pi.
\end{cases}
\] We can now substitute the generating function for $J_{-n+\frac{1}{2}}(x)$ derived from (i) above to obtain the desired result.
\item
Recall the relationships
\[
K_{n-\frac{1}{2}}(x)=\frac{\pi}{2} (-1)^n
\big[ I_{n-\frac{1}{2}}(x) -I_{-n+\frac{1}{2}}(x) \big],
\]
and
\[
K_{-n+\frac{1}{2}}(x)=\frac{\pi}{2} (-1)^n
\big[ I_{n-\frac{1}{2}}(x) -I_{-n+\frac{1}{2}}(x) \big]
\]
and we obtain the corresponding generating functions.
\end{enumerate}
\end{proof}
\subsection{Difference Glaisher's generating functions}
\begin{theorem}\label{T:difference_glaisher_gf}
\begin{enumerate}
\item
The difference Bessel functions have the Glaisher's generating function
\begin{equation}\label{E:delta_glaisher_gf}
\begin{split}
\dfrac{e^{-i\pi x}}{2i\sin(\pi x)} \int_{-\infty}^{(0+)}
& \lambda^{-\frac12} \frac{\big[C_1(x)\sin \sqrt{\lambda^2\mp 2\lambda t} +C_2(x)\cos \sqrt{\lambda^2\mp 2\lambda t}\big]}{\Gamma(-x)} e^\lambda (-\lambda)^{-x-1}\, d\lambda\\
&=\sum_{n=0}^\infty \mathscr{C}^\Delta_{\pm n\mp \frac{1}{2}}(x)\, \frac{t^n}{n!},
\end{split}
\end{equation}where the infinite sum converges uniformly in each compact subset of $\mathbb{C}\times\mathbb{C}$.
\item The holonomic systems of PDEs (delay-differential equations) that generates Glaisher's modules $\mathcal{G}_{-}$ \eqref{E:neg_glaisher_module} and $\mathcal{G}_{+}$ \eqref{E:pos_glaisher_module} with manifestation in $\mathcal{O}_{\Delta d}$ as defined by \eqref{E:O_delta_d} from Example \ref{Eg:O_delta_d} are given, respectively, by\footnote{See also the Appendix \ref{SS:PDE_list}.}
\begin{align}
& tf_{tt}(x,\, t)+\big(x+\frac12\big)f_t(x,\, t)-xf_t(x-1,\, t)-xf(x-1,\, t)=0,\label{E:PDeltaE_neg_glaisher_-1}\\
& tf_{t}(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+\frac12\big) f(x,\, t)=0\label{E:PDeltaE_neg_glaisher_-2}
\end{align}
and
\begin{align}
& tf_{tt}(x,\, t)+\big(x+\frac12\big)f_t(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)=0,\label{E:PDeltaE_pos_glaisher_+1}\\
& tf_{t}(x,\, t)+xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+\frac12\big) f(x,\, t)=0.\label{E:PDeltaE_pos_glaisher_+2}
\end{align}
\end{enumerate} \end{theorem}
\begin{proof} Recall the maps $\mathfrak{p}$, $\mathfrak{p}_\Delta$ and $\mathfrak{N}$ as defined in \eqref{E:poisson}, \eqref{E:poisson_2} and \eqref{E:newton_trans} respectively. Then we have the diagram
\begin{equation}\label{E:commute-half besssel-2}
\begin{tikzcd} [row sep=large, column sep=large]
& \mathcal{G}_{\pm}(\omega_\pm) \arrow[swap]{dl}{\mathop{\rm Cos}\omega\slash\mathop{\rm Sin}\omega} \arrow{d}{\times(\mathscr{C}_{\pm n\mp \frac12})} \arrow{dr}{\times(\mathscr{C}^\Delta_{\pm n\mp \frac12})} \\
\widetilde{\mathcal{A}_2}(\omega_\pm) \arrow{dr}{\times x^{-\frac12}} & \mathcal{O}^{\mathbb{N}_0} \arrow{d}{\mathfrak{p}} & \mathcal{O}^{\mathbb{N}_0} \arrow{d}{\mathfrak{p}_\Delta} \\
& \mathcal{O}_{d d} \arrow{r}{\times\mathfrak{N}} & \mathcal{O}_{\Delta d}
\end{tikzcd} \end{equation} in which $\widetilde{\mathcal{A}_2}(\omega_\pm):=\overline{\displaystyle\frac{\mathcal{A}_{2}(\omega_\pm)}{\big[\mathcal{A}_{2}(\omega_\pm)(W_1\Xi_1+\frac{1}{2})
+\mathcal{A}_{2}(\omega_\pm)\Xi_2\big]}}$, the Weyl cosine and sine $\mathop{\rm Cos} \omega_\pm\slash \mathop{\rm Sin} \omega_\pm$ are as defined in Example \ref{Eg:trigo}, and the analytic functions in $\mathcal{O}_{\Delta d}$ are suitably restricted.
Again, it suffices to consider the case for the negative Glaisher module $\mathcal{G}_-$ for the proof of $\mathcal{G}_+$ being similar. Since $\omega_-^2 =W_1^2-2W_1W_2$, the leftmost path in the above commutative diagram asserts that the analytic function from \eqref{E:glaisher_fn}
\begin{equation}\label{E:glaisher_fn_2}
x^{-\frac12}
\big(C_1\cos\sqrt{x^2- 2xt}+C_2\sin \sqrt{x^2- 2xt} \big)
\end{equation}
satisfies the system of PDEs \eqref{E:glaisher_PDE_1} and \eqref{E:glaisher_PDE_2}. A further composition of this analytic function under the Newton transformation \eqref{E:newton_trans} gives
\begin{equation}\label{E:Glaisher_dintegral_soln}
f_-(x,\, t)= \int_{-\infty}^{(0+)}
\lambda^{-\frac12} \frac{\big[C_1(x)\sin \sqrt{\lambda^2- 2\lambda t}+C_2(x)\cos \sqrt{\lambda^2-2\lambda t}\big]}{\Gamma(-x)} e^\lambda (-\lambda)^{-x-1}\, d\lambda,
\end{equation}
which satisfies the system of delay-differential equations \eqref{E:PDEs_neg_glaisher_1} and \eqref{E:PDEs_neg_glaisher_2} by Theorem~\ref{T:Newton_trans}, i.e.,
\begin{align}
& tf_{tt}(x,\, t)+\big(x+\frac12\big)f_t(x,\, t)-xf_t(x-1,\, t)-xf(x-1,\, t)=0,\label{E:PDeltaE_neg_glaisher_1}\\
& tf_{t}(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+\frac12\big) f(x,\, t)=0.\label{E:PDeltaE_neg_glaisher_2}
\end{align}Without loss of generality, we may assume that the $1$-periodic factor ${e^{-i\pi x}}/{2i\sin(\pi x)}$ is included in the unknown $1$-periodic functions $C_1(x),\, C_2(x)$ in the following computation. The handling of the second case when $\omega_+=W_1^2+2W_1W_2$ when $\nu=\frac12$ is similar and is therefore omitted.
On the other hand, the rightmost path in the above commutative diagram shows that the sum $\sum_{n=0}^\infty \mathscr{C}^\Delta_{n-\frac{1}{2}}(x)\, t^n/{n!}$ is a solution to the system of PDEs \eqref{E:PDeltaE_neg_glaisher_1} and \eqref{E:PDeltaE_neg_glaisher_2}. It is clear that the $\mathcal{G}_\pm(\omega_\pm)$ defined above is holonomic with multiplicity two. Hence \eqref{E:delta_glaisher_gf} holds apart from two $1$-periodic functions in $x$ which can easily be incorporated into the constants $C_1,\, C_2$.
We now derive a third order differential equation from the equations \eqref{E:PDeltaE_neg_glaisher_1} and \eqref{E:PDeltaE_neg_glaisher_2} in order to determine the range of convergence of the \eqref{E:delta_glaisher_gf}. Adding and subtracting the equations \eqref{E:PDeltaE_neg_glaisher_1} and \eqref{E:PDeltaE_neg_glaisher_2} yields the equations
\begin{equation}
\label{E:PDeltaE_neg_glaisher_3}
tf_{tt}(x,\, t)+\big(x+t+\frac12\big)f_t(x,\, t)-2x f_t(x-1,\, t)-\big(x+\frac12\big)f(x,\, t)=0
\end{equation}
and
\begin{equation}
\label{E:PDeltaE_neg_glaisher_4}
tf_{tt}(x,\, t)+\big(x-t+\frac12\big)f_t(x,\, t)-2xf(x-1,\, t)+\big(x+\frac12\big)f(x,\, t)=0
\end{equation}
respectively. Differentiating \eqref{E:PDeltaE_neg_glaisher_4} with respect to $t$ yields
\begin{equation}
\label{E:PDeltaE_neg_glaisher_5}
tf_{ttt}(x,\, t)+\big(x-t+\frac32\big) f_{tt}(x,\, t)+\big(x-\frac12\big)f_t(x,\, t) -2xf_t(x-1,\, t)=0.
\end{equation}Subtracting \eqref{E:PDeltaE_neg_glaisher_3} from \eqref{E:PDeltaE_neg_glaisher_5} yields the equation
\begin{equation}
\label{E:PDeltaE_neg_glaisher_6}
tf_{ttt}(x,\, t)+\big(x-2t+\frac32\big) f_{tt}(x,\, t)-(t+1)f_t(x,\, t)+\big(x+\frac12\big)f(x,\, t)=0,
\end{equation}
which is a third order linear ordinary differential equation in $t$ for each fixed $x$. Clearly, the equation \eqref{E:PDeltaE_neg_glaisher_6} has only one finite regular singularity at $t=0$, so that any power series solution of it must converge unformly in each compact subset of $\mathbb{C}\times\mathbb{C}$. Substituting a Frobenius solution $f(x,\cdot)=\sum_{k=0}^\infty a_kx^{\sigma+k}$ into the third-order equation \eqref{E:PDeltaE_neg_glaisher_6} further confirming that $\sigma=0,\, 1$ to be the only integer indicial roots. Hence the \eqref{E:delta_glaisher_gf} follows.
The handling of the $\mathcal{G}_\frac12$ is similar to that for $\mathcal{G}_{-\frac12}$ and is therefore omitted. This completes the proof. \end{proof}
\begin{remark} We offer a direct verification that the \eqref{E:Glaisher_dintegral_soln} solves the delay-differential equations \eqref{E:PDeltaE_neg_glaisher_1} and \eqref{E:PDeltaE_neg_glaisher_2}. Since the computation of the sine and cosine functions are similar, so it suffice to consider only the special case with cosine kernel. Moreover, the $1$-periodic function $C_1(x)$ can be factored out in the computation below since it is independent of $t$ and so it is also omitted. Let
\[
y(x, t)=\displaystyle\int_{-\infty}^{(0+)} e^{\lambda}
\dfrac{\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}} d\lambda.
\]We now show that the $y(x,\, t)$ is a solution to the delay-differential equation \eqref{E:PDeltaE_neg_glaisher_2}. We split the computation of this equation into two parts. We substitute this $y$ into the third and the fourth terms from \eqref{E:PDeltaE_neg_glaisher_2}. This yields
\begin{equation}\label{E:three-terms}
xy(x, t)-xy(x-1, t)+\frac{1}{2}y(x, t)\\
=\displaystyle\frac{1}{\Gamma(-x)}
\int_{-\infty}^{(0+)} (x-\lambda+\frac{1}{2})e^{\lambda}
(-\lambda)^{-x-1}
\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}
\lambda^{-\frac12}d\lambda.
\end{equation}We now substitute the $y(x,\, t)$ into the first two terms in \eqref{E:PDeltaE_neg_glaisher_2} before performing an integration-by-parts once yield
\[
\begin{split}
&x\frac{dy(x-1, t)}{dt}-t\frac{dy(x, t)}{dt}
=\displaystyle\frac{1}{\Gamma(-x)}
\int_{-\infty}^{(0+)} e^{\lambda}
(-\lambda)^{-x-1} \lambda^{\frac{1}{2}} \,
d[-\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}]\\
&=-e^{\lambda}
\frac{(-\lambda)^{-x-1} }{\Gamma(-x)}\lambda^{\frac{1}{2}}
\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}
\Big|_{-\infty}^{(0+)}
+\displaystyle\frac{1}{\Gamma(-x)}
\int_{-\infty}^{(0+)}
\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}
d(e^{\lambda}
(-\lambda)^{-x-1} \lambda^{\frac{1}{2}} ).
\end{split} \]Since $\mathop{\rm Re}(x)<-\frac{1}{2}$, then the limit of the first term above vanishes. That is, \[ -e^{\lambda} \frac{(-\lambda)^{-x-1} }{\Gamma(-x)}\lambda^{\frac{1}{2}}
\cos(\lambda^2-2\lambda t)^{\frac{1}{2}} \Big|_{-\infty}^{(0+)}=0. \]Thus the above expression becomes, after this integration-by-parts,
\[
\begin{split}
&x\frac{dy(x-1, t)}{dt}-t\frac{dy(x, t)}{dt}\\
&=\displaystyle\frac{1}{\Gamma(-x)}
\int_{-\infty}^{(0+)}
\cos(\lambda^2-2\lambda t)^{\frac{1}{2}}
e^{\lambda}(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
(\lambda-x-\frac{1}{2}) d\lambda\\
\end{split}
\]which is precisely the negative of the expression above \eqref{E:three-terms}. This verifies that $y(x,\, t)$ solves the first equation from \eqref{E:PDeltaE_neg_glaisher_2}. The verification of $y(x, t)$ to the \eqref{E:PDeltaE_neg_glaisher_1} is similar and is therefore omitted. \end{remark}
The following difference analogues for the Glaisher-type generating functions \eqref{E: first half bessel_gf} for difference Bessel functions $J_{n+\frac12}^\Delta(x)$ are amongst the special cases described in the above theorem.
\begin{theorem}\label{T:Delta_glaisher_gf} The series
\begin{enumerate}
\item
\begin{equation}\label{E: gf-Delta_glaisher_cos}
\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{-\infty}^{(0+)}
\frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\cos\sqrt{\lambda ^2-2\lambda t}}{\Gamma(-x)}\, d\lambda
=\sum_{n=0}^{\infty}J^{\Delta}_{n-\frac{1}{2}}(x)\frac{t^n}{n!},
\end{equation}
\item and
\begin{equation}\label{E: gf-Delta_glaisher_sin}
\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{-\infty}^{(0+)}
\frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\sin\sqrt{\lambda ^2+2\lambda t}}{\Gamma(-x)}\, d\lambda
=\sum_{n=0}^{\infty}J^{\Delta}_{-n+\frac{1}{2}}(x)\frac{t^n}{n!}
\end{equation}
\end{enumerate} converge uniformly in each compact subset of $ \mathbb{C}\times\mathbb{C}$.
\end{theorem}
It may be possible to derive the two generating functions of the above theorem from the Theorem \ref{T:difference_glaisher_gf}, but we choose a more direct method by applying the Newton transform Theorem \ref{T:Newton_trans} to Glaisher\rq{}s formulae directly.
\begin{proof} Due to the similarity of the two relations \eqref{E: gf-Delta_glaisher_cos} and \eqref{E: gf-Delta_glaisher_sin}, so it suffices to only establish the \eqref{E: gf-Delta_glaisher_cos}.
Let
\[
L=(X\partial)^2+(X^2-\nu^2),
\] and to recall that a $\mathcal{D}-$linear map \eqref{E:neg_nu_map}
\[
S_{-\nu}=\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}
\]
in the following diagram \begin{equation}\label{E:commute-glaisher}
\begin{tikzcd} [row sep=large, column sep=large]
\displaystyle\frac{\mathbb{C}[[X\partial, X]]}{\mathbb{C}[[X\partial, X]]L} \arrow{r}{\times S_{-\nu}}
& \displaystyle\frac{\mathbb{C}[[X\partial, X]]}{\mathbb{C}[[X\partial, X]](X\partial+\nu)}
\arrow{d}{\times 1}
\arrow{dr}{\times 1}\\
& \mathcal{O}_{d } \arrow{r}{\times\mathfrak{N}} & \mathcal{O}_{\Delta}
\end{tikzcd}
\end{equation}where the $\mathfrak{N}$ again denote the Newton transformation. Since \eqref{E:newton_bases_2}. Let us apply a truncated Newton transformation $\mathfrak{N}_R$ of \eqref{E:newton_trans} which has the truncated Hankel-type contour $\Gamma_R\ (R>0)$ starts from $-R$ below the negative real-axis, goes around the origin in the counter-clockwise direction before returning to $x=-R$
\[
\begin{split}
\mathfrak{N}_R(J_{-\nu}(x)) &= \dfrac{e^{-i\pi x}} {2i \sin(\pi x)\Gamma(-x)}
\int_{Ne^{-i\pi}}^{Ne^{i\pi}} \sum_{k=0}^{\infty} \frac{(-1)^k \lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^\lambda(-\lambda)^{-x-1}\, d\lambda
\\
&=\sum_{k=0}^{\infty}
\dfrac{e^{-i\pi x}} {2i \sin(\pi x)\Gamma(-x)}\int_{Ne^{-i\pi}}^{Ne^{i\pi}}
\frac{(-1)^k \lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^\lambda(-\lambda)^{-x-1}\, d\lambda\\
&\to \sum_{k=0}^{\infty}
\dfrac{e^{-i\pi x}} {2i \sin(\pi x)\Gamma(-x)}\int_{-\infty}^{(0+)}
\frac{(-1)^k \lambda^{-\nu+2k}}{2^{\nu+2k}k!\Gamma(-\nu+k+1)}e^\lambda(-\lambda)^{-x-1}\, d\lambda\\
&=\sum_{k=0}^{\infty} \frac{(-1)^k}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}(x)_{-\nu+2k}
=J^\Delta_{-\nu}(x),
\end{split}
\]as $R\to\infty$, where the interchange of the integral and summation signs above is guaranteed by the uniform convergence of the classical Bessel function $J_{\nu}(x)$ in any compact subset of $\mathbb{C}^\dagger=\mathbb{C}\backslash\{x:\, x\le 0\}$.
Let $\nu=n-\frac12$. Apply a truncated Newton transformation \eqref{E:newton_trans} over the truncated Hankel-type contour $\Gamma_R$ as defined above to both sides of \eqref{E: first half bessel_gf} with respect to $\lambda$. This yields \begin{equation}\label{E: gf-Delta_glaisher_cos_2}
\begin{split}
\sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
&\int_{Ne^{-i\pi}}^{Ne^{i\pi}}
\frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\cos\sqrt{\lambda ^2-2\lambda t}}{\Gamma(-x)}\, d\lambda\\
& =\sum_{n=0}^{\infty} \Big(\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{Ne^{-i\pi}}^{Ne^{i\pi}}
\frac{e^{\lambda}
(-\lambda)^{-x-1}J_{n-\frac12}(\lambda)}{\Gamma(-x)}\, d\lambda\Big)\frac{t^n}{n!},\\
\end{split}
\end{equation}where the interchange of the integral and summation signs above is allowed because of the uniform convergence of the \eqref{E: first half bessel_gf} in each compact subset of $\mathbb{G}:=\{2|t|<|x|:\ (x,\, t)\in \mathbb{C}^\dagger\times\mathbb{C}\}$ where $\mathbb{C}^\dagger:=\mathbb{C}\backslash\{x:x\le 0\}$ as shown there. We finally obtain the formula \eqref{E: gf-Delta_glaisher_cos} after letting $R\to\infty$ in the truncated contour $\Gamma_R$. We may analytic continue the $x$ to the whole of $\mathbb{C}$ by deforming the Hankel contour when necessary. The proof of the \eqref{E: gf-Delta_glaisher_sin} is similar. This completes the proof. \end{proof}
We give a direct derivation of the integral representations of $J^\Delta_{\pm\frac12}$ from the \eqref{E: gf-Delta_glaisher_cos} which are of different type from those given in \eqref{E:dBessel integral rep} and \eqref{E:dBessel integral rep-nu}.
\begin{corollary} Let $x\in \mathbb{C}$. Then
\begin{enumerate}
\item
\[
J^\Delta_{-\frac12}(x)=\sqrt{\frac{2}{\pi}}
\frac{e^{-i\pi x}}{2i \sin\pi x}
\int_{-\infty}^{(0+)}
\dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}\cos \lambda\, d\lambda
\] and
\item
\[
J^\Delta_{\frac12}(x)=\sqrt{\frac{2}{\pi}}
\frac{e^{-i\pi x}}{2i \sin\pi x}
\int_{-\infty}^{(0+)}
\dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}\sin \lambda\, d\lambda.
\]
\end{enumerate} \end{corollary}
\begin{proof}Expanding the $\cos\lambda$ into series in
\[
\begin{split}
&\frac{e^{-i\pi x}}{2i \sin\pi x}
\int_{-\infty}^{(0+)}
\dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}\cos \lambda\, d\lambda\\
&=\frac{e^{-i\pi x}}{2i \sin\pi x}\int_{-\infty}^{(0+)} \dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}
\sum_{k=0}^{\infty} \frac{(-1)^k\lambda^{2k}}{(2k)!}\, d\lambda\\
&=\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}\, G_k(x)
\end{split}
\]where, with applications of the generalised Gamma function and Euler's reflection formula,
\[
\begin{split}
G_k(x)&:=\frac{e^{-i\pi x}}{2i \sin\pi x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}
\int_{-\infty}^{(0+)}e^{\lambda}(-\lambda)^{-x-\frac{1}{2}+2k-1}d\lambda\\
&=\frac{e^{-i\pi x}}{2i \sin\pi x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}
\frac{e^{-i\pi(x+\frac12-2k)}}{e^{-i\pi(x+\frac12-2k)}}
\frac{\sin\pi(-x-\frac12+2k)}{\sin\pi(-x-\frac12+2k)}
\frac{\Gamma(-x-\frac12+2k)}{\Gamma(-x-\frac12+2k)}
\int_{-\infty}^{(0+)} e^\lambda(-\lambda)^{-x-\frac12+2k-1}\, d\lambda
\\
&=\frac{e^{-i\pi x}}{\sin\pi x}\frac{e^{-i\frac{\pi}{2}}}{\Gamma(-x)}
\frac{\sin\pi(-x-\frac12+2k)\Gamma(-x-\frac12+2k)}{e^{-i\pi(x+\frac12-2k)}}\,
\\
&=\frac{\Gamma(-x-\frac{1}{2}+2k)}{\Gamma(-x)}
\frac{\sin(x+\frac{1}{2}-2k)\pi}{\sin (-\pi x)}\\
&=\frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}.
\end{split}
\]
Hence
\begin{equation}\label{E:glaisher_cos_expansion}
\frac{e^{-i\pi x}}{2i \sin\pi x}
\int_{-\infty}^{(0+)}
\dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}\cos \lambda\, d\lambda\
=\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!}\, \frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}=\sqrt{\frac{\pi}{2}}\, J^\Delta_{-\frac12}(x)
\end{equation}
since, with applications of the well-known identity $\Gamma(k+\frac12)k!\, 2^{2k}=\sqrt{\pi}(2k)!$, see for example \cite[p. 211]{Copson_1935}, that
\begin{equation}\label{E:difference_bessel_1/2}
\begin{split}
J^{\Delta}_{-\frac12}(x)&:=\sum_{k=0}^{\infty} \frac{(-1)^k}{2^{-\frac12+2k}k!\, \Gamma(\frac12+k)}(x)_{-\frac12+2k}
=\sum_{k=0}^{\infty}
\frac{(-1)^k}{2^{-\frac12}\sqrt{\pi}(2k)!}(x)_{-\frac12+2k}\\
&=\sqrt{\frac{2}{\pi}}\sum_{k=0}^{\infty}
\frac{(-1)^k}{(2k)!} \frac{\Gamma(x+1)}{\Gamma(x+\frac{3}{2}-2k)}.
\end{split}
\end{equation}
Similarly, we have
\begin{equation}\label{E:glaisher_sin_expansion}
\frac{e^{-i\pi x}}{2i \sin\pi x}
\int_{-\infty}^{(0+)}
\dfrac{e^{\lambda}}
{\Gamma(-x) (-\lambda)^{x+1}\lambda^\frac{1}{2}}\sin \lambda\, d\lambda\
=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}\, \frac{\Gamma(x+1)}{\Gamma(x+\frac{1}{2}-2k)}=\sqrt{\frac{\pi}{2}}\, J^\Delta_{\frac12}(x).
\end{equation} \end{proof}
\section{Discussion}\label{S:discussion}
\subsection*{Truesdell\rq{}s $F$-equation theory}
It had been observed by Lommel \cite{Lommel_1871}, Nielson \cite{Nielson} and Sonine \cite{Sonine},
all from the nineteenth century, to characterize Bessel functions from the pair of recurrence formulae, which we call PDEs in this article (Proposition \ref{P:PDE_cylinderical_bessel}),
\begin{equation}\label{E:any_bessel_recus_0}
\begin{split}
& x\mathscr{C}_{\nu}^\prime(x)+\nu\mathscr{C}_{\nu}(x)-
x\mathscr{C}_{\nu-1}(x)=0,\\
&x\mathscr{C}^{\prime}_{\nu}(x)-\nu\mathscr{C}_{\nu+n}(x)+
x\mathscr{C}_{\nu+1}(x)=0,
\end{split}
\end{equation} as analytic functions of two complex variables. They called the functions that satisfy both of the equations \textit{cylinder functions}, see also \cite[pp. 82-84]{Watson1944}. In fact, according to Watson \cite{Watson1944}, Sonine refrained from using Bessel's differential equation \eqref{E:Bessel_eqn} in his study. Truesdell published the book \cite{Truesdell_1948} in 1948 perfecting the viewpoint started by the aforementioned nineteenth century scholars as an analytic theory of differential-difference equation that can be written in the forms
\begin{equation}\label{E:Truesdell}
\frac{\partial}{\partial x}f(x,\alpha)=A(x,\alpha)f(x,\alpha)+B(x,\alpha)f(x,\alpha+1).
\end{equation}If the $f(x,\alpha)$, assumes a specific form that after a suitable change of variables, can be written in an equivalent form
\begin{equation}\label{E:F_eqn}
\frac{\partial}{\partial x}F(x,\alpha)=F(x, \alpha+1),
\end{equation}
then he demonstrated that derived many known classical special function identities\footnote{Thirty five formulae have been listed in \cite{Truesdell_1947}.}
as their generating functions and to discover new ones for different $f$ for all sorts of classical special functions. Truesdell called the equation \eqref{E:F_eqn} an $F$-equation. We quote from \cite[p. 7] {Truesdell_1948}
\begin{quote}
``The aim of this essay is to provide a general theory which motivates, discovers, and coordinates such seemingly unconnected relations among familiar special functions as the formulas (1) through (35).\rq\rq{}
\end{quote}
Truesdell's aimed to find ``rational methods of discovery" \footnote{Truesdell, p. 7.} of the numerous classical special functions formulae found in the literature that often left beginners, and even experienced researchers, wondering about their origins and the methods of their derivation. For example, Truesdell commented that if a textbook has the following question:
\begin{quote}
``Find a formula which gives Laguerre polynomials in terms of Bessel functions.\rq\rq{}\footnote{Truesdell, p. 7.}
\end{quote} then, it would not be straightforward to any student unless he/she had a priori knowledge of such formulae. However, once the student sees such a formula, then it would be a typical textbook exercise for a student to give a rigorous proof. Thus Truesdell would like to ``coordinate", ``to bring order into the part of collection of known relations concerning special functions by showing that they are simple special cases of about a dozen general formulas and by adding to their number some of the missing analogues which do not seem to have been discovered thus far." \footnote{\cite{Truesdell_1948}.}
Truesdell further commented that it is not with the particular special function formulae (and some of them were claimed to be new) that was his focus, since
\begin{quote}
``it is of no great task to construct ad hoc rigorous proofs and ... to any of the formulas we have just listed,"
\end{quote} but rather a general property, although it had been observed by earlier researchers in various literature over a period of several decades in the second half of the nineteenth century, that was largely neglected \footnote{\cite[p. 8]{Truesdell_1948}.} is the result of applying his $F$-equation theory that he wanted to bring about. Indeed Truesdell was not alone that Ehrenpreis, without explicit mention, essentially also took the $F$-equation viewpoint in his study of classical analogues of the Rogers-Ramanujan identities in \cite{Ehrenpreis_1990, Ehrenpreis_1993} in which a generalised system of \eqref{E:PDE_gf_bessel_0} that is related to multi-variable Bessel functions has been proposed. From the $D$-modules viewpoint employed in this article, although Truesdell and his predecessors certainly had the insight to study the classical Bessel functions $J_{\nu}$ via the recursions \eqref{E:any_bessel_recus_0}, the ``general and neglected property" is replaced by the algebraic property that the corresponding modules, such as the Bessel module studied in this paper, are holonomic in the sense of Bernstein-Kashiwara-Sato holonomic modules theory. Moreover, the essence of the $F$-equation theory which is about the existence and uniqueness of the \eqref{E:F_eqn} having analytic solutions is being replaced by the specific algebraic properties of the $D$-modules determined by their generators. In the case of the Bessel module $\mathcal{B}_\nu$ considered in this paper, the generators are given by \eqref{E:bessel_PDE}. What is definitely new are the difference analogues of \eqref{E:any_bessel_recus_0} given in \eqref{P:bilateral_Delta_PDE} and other formulae obtained in this article.
In addition to the well-known works of \cite{Paule_Schorn_1995, Chyzak_Salvy_1998, Chyzak_2000, Zeilberger_1990, Wilf_Zeilberger_1992} of applying $D$-modules to compute for ($q$-)hypergeometric type identities and the standard works on computations with $D$-modules \cite{SST_2000, IMST_2020}, in fact, rudiments of $D$-modules ideas had been employed by Boole in his seminar work \cite{Boole}\footnote{Boole was awarded the first gold medal in mathematics by the Royal Society in 1844 for this paper.} in which a very general commutation relation on operational method has been applied to solve a wide variety of linear differential equations including a variant of confluent hypergeometric equation. \footnote{See \cite{CCC_1} for a more detailed discussion.} Of course, in Boole's time, he could not possibly have the knowledge of the modern $D$-modules theory. However, the use of his generalised commutator in solving for series solutions of several linear differential equations is a vindication of his enormous insights into the nature.\footnote{Indeed, the paper together with other earlier works on differential and difference equations of Boole could be considered as preludes to his celebrated work on mathematical logic \cite{Boole_1847}, see \cite[\S 3]{Laita}.} In addition to the difference in notations used and problems tackled, a major differences between our work and those of Wilf and Zeilberger and his successors \cite{Zeilberger_1990, Wilf_Zeilberger_1992, Paule_Schorn_1995, Chyzak_Salvy_1998, Chyzak_2000} is that the way that the generators used to define our $D$-modules which is based on an algebraization of Truesdell's $F$-equation theory. For example, the two generators \eqref{E:2_elements} of the Bessel module $\mathcal{B}_\nu$ are abstractions of the consequences of the gauge transformations on the Bessel operators parametrised by integers derived from the the corresponding transmutation formulae in Proposition \ref{P:bessel_transmutation}. The $\partial_1,\partial_2, X_1, X_2$ in the $\mathcal{B}_\nu$ carry different interpretations (manifestations) after appropriate $\mathcal{A}_2$-linear maps in the construction of Bessel's generating functions. Lemma 4.1 in \cite{Zeilberger_1990}, although not needed in this paper, provides a theoretical basis for the existence of characteristic change of variables made in the proof of Theorem \ref{T:bessel_gen_map_2}, Theorem \ref{T:bessel_poly_gf_map}, Theorem \ref{T:symbol gen yn-1}, Theorem \ref{T:neg_glaisher_cosine} and Theorem \ref{T:pos_glaisher_sine} in order to find (well-defined) $\mathcal{A}_2$-linear maps. The well-defined $\mathcal{A}_2$-linear maps found from these Theorems represent the generating functions derived (in $\mathcal{O}_{dd}$ and $\mathcal{O}_{d\Delta}$) in the main theorems of this papers.
\subsection*{Umbral calculus} We also note that the Weyl-exponential ${\mathop{\rm E}}(X)$, Weyl-sine $\mathop{\rm Sin} (X)$ and Weyl-cosine $\mathop{\rm Cos} (X)$ derived as ``solutions" to the holomorphic
$D$-modules $\mathcal{A}_1/\mathcal{A}_1(\partial-a)$ and $\mathcal{A}_1/\mathcal{A}_1(\partial^2+1)$ relative to $\mathcal{A}_1/\mathcal{A}_1\partial$ in Example \ref{Eg:exp} and Example \ref{Eg:trigo}. The manifestation of these ``solutions" in $\mathcal{O}_\Delta$ result in
the difference-exponential, difference-sine and cosine introduced in Example \ref{Eg:delta_exp} and Example \ref{Eg:dtrigo} respectively, can be found in various literature under ``umbral calculus", such as \cite{Curtright_Zachos_2013, Gessel_2003,LNOS_2008,Roman_1984}. In this spirit, it is interesting to ask if the
\[
{\mathop{\rm E}}[X_1(X_2-1/X_2)/2]
\] constructed in the Theorem \ref{T:bessel_gen_map_2} for the generating function of Bessel module $\mathcal{B}_0$, the
\[
\rho^{-1}{\mathop{\rm E}}[X_1(1-\rho)],\qquad \rho^2=1-2X_2
\] constructed in Theorem \ref{T:bessel_poly_gf_map} for the Bessel polynomials module $\Theta$, and the
\[
C_1\mathop{\rm Cos}\omega_\mp+C_2\mathop{\rm Sin}\omega_\mp,
\qquad \omega_\mp^2=W_1^2\mp 2W_1W_2
\]constructed in Theorem \ref{T:neg_glaisher_cosine} for the Glaisher modules $\mathcal{G}_\mp$ can be interpreted from umbral calculus's viewpoint.
\section{Conclusion}\label{S:conclusion} This paper revisits system of PDEs satisfied by the generating function of the classical Bessel functions of the first kind $J_n(x)$ from an holonomic $D$-modules viewpoint. The approach unifies the generating functions of the classical Bessel functions and the recently found difference Bessel functions $J^\Delta_n(x)$. One can view this as an algebraization of the $F$-equation theory, which is analytic in nature, proposed by Truesdell in 1948. The key ingredients of our arguments involve new transmutation formulae for the Bessel functions and Bessel polynomials in certain $\mathcal{A}_1$-modules, the derivation of well-defined $\mathcal{A}_2$-linear maps of the $\mathcal{A}_2$-modules, which includes the Bessel module $\mathcal{B}_\nu$, reverse Bessel polynomial module $\Theta$, Bessel polynomial module $\mathcal{Y}$, and the Glaisher modules $\mathcal{G}_\pm$. Although the transmutation formulae can be viewed as abstractions of the relatedf factorization method proposed by Infeld and Hull in \cite{Infeld_Hull_1951}, the scope of the study here and details involved are very different.
We have studied ``solutions" in each $\mathcal{A}_2$-module above (see also Appendix \ref{SS:holo_modules}) in the form of well-defined $\mathcal{A}_2$-linear maps that amount to solving systems of linear PDEs defining the holonomic $D$-modules in ``closed-forms". We have thus extended the classical Bessel's generating function to include $J_\nu(x)$ for arbitrary $\nu\in\mathbb{C}$ and its difference analogue by constructing $\mathcal{A}_2$-linear maps from the Bessel module $\mathcal{B}_\nu$ to analytic function spaces of two variables $\mathcal{O}_{dd}$ and $\mathcal{O}_{\Delta d}$ which are also $D$-modules with different ``manifestations" as indicated by their respective subscripts. The difference Bessel functions $J^{\Delta}_\nu(x)$ that appear in the aforementioned difference manifestation of Bessel's generating function coincide with those found by Bohner and Cuthta in 2017. These new generating functions are divergent (1-Gevrey) series unless $\nu=0$ which are Borel-resummable. These new Bessel's generating functions for general $\nu\not=0$ of asymptotic type explains the classical Schl\"afli-Sonine integral representation \eqref{E:Sonine} of the Bessel function $J_\nu(x)$ that can be obtained via the usual procedure of ``residue extraction" of $J_{\nu+n}$ for any $n$ from the asymptotic formula \eqref{E:gf_J_nu} instead of series manipulation technique usually used, see \cite[\S 6.2]{Watson1944}. In fact, no knowledge of the series expansion of the Bessel function $J_\nu(x)$ is needed with the ``residue extraction" from the \eqref{E:gf_J_nu}. This viewpoint conforms with the way of studying the sequence of Bessel functions $(J_{\nu+n})$ as \textit{coefficients} of generating functions discussed in \cite[Chapter 2]{Watson1944}. The ``residue extraction" method has been applied to the generating function formula \eqref{gf-db-2} to obtain a \textit{difference analogue of Schl\"afli-Sonine integral representation} for the difference Bessel functions $J^\Delta_\nu(x)$ in \eqref{E:dBessel integral rep}. The results obtained thus far (\S 3) in this article suggests that one can approach the classical Bessel functions as (Weyl-)algebraic entities without necessarily recourse to Bessel's series expansions as has been done in most modern literature.
Before we discuss about modifications of the half Bessel module $\mathcal{B}_{1/2}$, we would like to emphasize that although we have devoted this entire paper to Bessel functions, the choice of topics studied and results presented in this paper, are by no means exhaustive, only serves to illustrate our objectives and methodology of studying special functions, with the Bessel function as an example, by solving the holonomic system PDEs in the generalised sense represented by the respective $D$-modules. In fact, it is clear to us that the $D$-modules approach could apply to many other topics about Bessel functions found, for example, in Watson's book \cite{Watson1944}, and to other type of special functions (see below). Thus the choice of topics studied in this paper is not limited by the range of applicability on the methodology, but is limited rather by confining the paper to within a reasonable scope.
We next turn to the modifications of the Bessel module $\mathcal{B}_\nu$ when $\nu=\frac12$, that would lead to a new generating functions to the new difference Bessel polynomials and new difference Glaisher trigonometric type generating functions in addition to their classical counterparts, which can be found from \S\ref{S:half_bessel_I} and \S\ref{S:half_bessel_II} respectively. Each modification of $\mathcal{B}_{1/2}$ requires certain ``quadratic extensions" in addition to their respective ``change of variables". One important feature of $\mathcal{B}_{1/2}$ comes from the fact that the half Bessel operators can be factorised (because it is \textit{solvable} \cite{Morales_Ruiz_1999}).
In the case about the Weyl-Bessel polynomials studied in \S\ref{S:half_bessel_I}, the focus on $\mathcal{B}_{1/2}$ is that after a suitable ``change of variables" the reverse polynomial operator possesses the transmutation formula
\[
X\partial^2-(2+X)\partial+2n=(X\partial-2n)(\partial-1)=(\partial-1)(X\partial-2n-1)
\] that can be further derived from the transmutation formula \eqref{C:bessel_poly_gauge_2}. This factorization of the reverse polynomial operator shows the reason that one can find \textit{closed-form} solutions which can always be transformed to a terminated Frobenius solution or a polynomial solution if needed. This is the case only when $\nu=1/2$ in $\mathcal{B}_\nu$. Indeed, most of the algebraic results derived for the Weyl-Bessel polynomials in \S\ref{SSS:bessel_poly_oper}, to some extent, can be found in \cite{Burchnall_1953, Burchnall_Chaundy_1931, Grosswald} without the benefit of $D$-modules language. However, the different manifestations of the reverse Bessel polynomial operator \S\ref{SSS:bessel_poly_oper} in the analytic function spaces $\mathcal{O}_{d}$ and $\mathcal{O}_{\Delta}$ give rise to the classical reverse Bessel polynomials $\theta_n(x)$ and the new difference reverse Bessel polynomials $\theta^\Delta_n(x)$ and their respectively recursive formulae. The introduction of the reverse Bessel polynomial module $\Theta$ in the same spirit as the Bessel module $\mathcal{B}_\nu$ allows us to solve the corresponding PDEs that leads to generating functions of the reverse classical Bessel polynomials and their difference analogue respectively. The same has been done for the Bessel polynomial module $\mathcal{Y}$ where the new difference Bessel polynomials which are denoted by ($y^\Delta_n(x)$).
The consideration of Glaisher modules $\mathcal{G}_\pm$ in \S\ref{S:half_bessel_II} is achieved after the ``change of variables"
\[
\begin{array}{ll}
\Xi_1=\partial_1, & W_1=X_1\\
\Xi_2={1}/{X_2}, & W_2=X_2\partial_2 X_2.
\end{array}
\]
in \eqref{E:glaisher_char_1} to $\mathcal{B}_{1/2}$ can be considered a kind of `` Borel transform" on $\mathcal{B}_{1/2}$, which is instrumental in solving the PDEs in the Glaisher modules $\mathcal{G}_\pm$ leading to the trigonometric type generating functions in Corollary \ref{C:Graisher_gf}, Theorem \ref{T:difference_glaisher_gf} and Theorem \ref{T:Delta_glaisher_gf}.
To summarise, it is clear from within the limited scope of study made in this article that there is a close relationships amongst the systems of PDEs (in the generalised sense) for which the generating functions of various type of Bessel functions or polynomials satisfy are holonomic, and related integral representations of Bessel functions. This relationship enables us to derive the first integral representation for the recently discovered difference Bessel functions. It also follows from our study that these difference Bessel functions are on ``equal footing" with the classical Bessel functions. Similar relationship also applies to the newly discovered difference Bessel polynomials in this article and their classical counterparts. This investigation show that the space of differential operators annihilating
some generating functions of Bessel functions have rich algebraic structures and that both the classical results and the new ones for the difference Bessel functions and polynomials can be derived from the same Bessel module $\mathcal{B}_\nu$, Bessel polynomial modules $\Theta, \mathcal{Y}$ and the Glaisher modules $\mathcal{G}_\pm$.
\section{Outlook} The findings from this article have brought up more questions than the problems solved. For example, it is very likely that our method could also derive the more sophisicated generating function
\[
(x+h)^{-\nu/2}J_\nu (2\sqrt{x+h})=\sum_{k=0}^\infty \frac{(-h)^k}{k!}
x^{-(\nu+k)/2}J_{\nu+k}(2\sqrt{x}),\quad |h|<|x|,
\] found by Lommel \cite[\S5.22]{Watson1944}\footnote{Bessel actually discovered the formula when $\nu=0$ in 1826.}, its difference analogue and the likes. What are mathematical basis behind those systems of PDEs that allows for such generating functions? Since generating functions can be viewed as a natural way to possess certain combinatorial structures that re-organise the various symbols $\partial_1,\partial_2, X_1, X_2$, and the generating functions are solutions to the specific systems of PDEs studied in this article, so a question is how these combinatorial structures are described and understood against the algebraic structures hidden in these systems of holonomic PDEs. Another question is about the ``orthogonality" of the new difference Bessel polynomials found in this paper. The orthogonality issue of the new difference Bessel polynomials will be dealt with in a separate article \cite{CCL_2023_2}. It turns out that the classical orthogonal polynomial theory \cite{Ismail_2005} does not seem to apply to these new polynomials immediately which are naturally organised as Newton polynomials. Thus a new approach that treats ``orthogonality" as residue pairings has been developed in the forthcoming article \cite{CCL_2023_2}. An advantage of such an algebraic approach is that it could treat the orthogonality for both the classical Bessel polynomials, i.e., the classical theory, and the difference Bessel polynomials simultaneously. Finally, in spite of the lack of understanding of deeper algebraic structures encoded in these systems of holonomic PDEs ($D$-modules) that lead to the Bessel's generating functions studied in this article, similar strategy has been applied to other types of $D$-modules of special functions in another forthcoming article \cite{CCC_2}.
\appendix
\section{Proofs of some theorems/propositions}\label{A:proofs} \subsection{Proof of Proposition \ref{P:bessel_convergence}} \label{SS:bessel_convergence}
\begin{proof}
We rewrite the series of $J^{\Delta}_{\nu}(x)$ \eqref{E:difference_bessel_fn} as
\begin{equation}\label{E:difference_bessel_fn_2_a}
J^{\Delta}_{\nu}(x)=\sum_{k=0}^{\infty}
\dfrac{(-1)^k \Gamma(x+1)}
{k! 2^{\nu+2k}\Gamma(\nu+k+1)\Gamma(x+1-\nu-2k)}.
\end{equation}
An asymptotic formula of Gamma function \cite[Theorem 1.4.2]{AAR}, \cite{WW}
for a complex number $x$ not equal to zero or a negative real number, can be derived from the formula
\[
\log \Gamma(x)=\frac{1}{2} \log 2\pi +(x-\frac{1}{2})\log x-x+\sum_{j=1}^{m}
\frac{B_{2j}(0)}{(2j-1)2j}\frac{1}{x^{2j-1}}-\frac{1}{2m}\int_{0}^{\infty}
\frac{B_{2m}(t-[t])}{(x+t)^{2m}}dt,
\]where the $B_{2m}(t)$ is the Bernoulli polynomial of degree $2m$.
The value of $\log \Gamma(x)$ is the branch with $\log \Gamma(x)$ being real when $x$ is real and positive.
Set
\[
g(x)=\sum_{j=1}^{m}
\frac{B_{2j}(0)}{(2j-1)2j}\frac{1}{x^{2j-1}}-\frac{1}{2m}\int_{0}^{\infty}
\frac{B_{2m}(t-[t])}{(x+t)^{2m}}dt.
\]
Then
\[
\Gamma(x)=\sqrt{2\pi}x^{x-1/2} e^{-x+g(x)}.
\] We rewrite those Gamma functions that appear in the denominator of $ J^{\Delta}_{\nu}(x)$ as
\[
\begin{split}
&\Gamma(k+1)=\sqrt{2\pi}\,k^{k+1/2}(1/k+1)^{k+1/2} e^{-k-1+g(k+1)},\\
&\Gamma(\nu+k+1)=\sqrt{2\pi}\,k^{\nu+k+1/2}(\nu/k+1/k+1)^{\nu+k+1/2} e^{-\nu-k-1+g(\nu+k+1)},\\
&\Gamma(x-\nu-2k+1)=\sqrt{2\pi}
\dfrac{ (x/2k-\nu/2k+1/2k-1)^{x-\nu-2k+1/2} e^{g(x-\nu-2k+1)}}
{(2k)^{-x+\nu+2k-1/2}e^{x-\nu-2k+1}}.
\end{split}
\]
Set
\[
u_k=\dfrac{2^{-\nu-2k}}{k!\Gamma(\nu+k+1)\Gamma(x-\nu-2k+1)}.
\] Since $g(x)\to 0$ as $x\to\infty$ with $\mathop{\rm Re}(x)>0$, so that it is routine to verify that
\[
u_k\approx \frac{ (-1)^{2k+\nu-\frac12-x}2^{-\nu-2k}}
{\sqrt{2}k^{\frac32} (2k)^x2^{-\nu-2k}}
=\frac{(\textrm{Const.}) (-1)^x}{k^{x+\frac32}}
=\frac{(\textrm{Const.}) e^{\pi \mathop{\rm Im} x}}{k^{x+\frac32}}
\]
holds as $k\to\infty$. Hence
\[
\sum_{k=1}^\infty u_k
\]converges uniformly in each compact subset of $\mathbb{C}$ where $\mathop{\rm Re} (x)>-\frac12$. Therefore, the same holds for the \eqref{E:difference_bessel_fn_2_a}.
\end{proof}
\subsection{Proof of Proposition \ref{P:Weyl_Bessel_Poly_Eqn}}\label{A:Weyl_Bessel_Poly_Eqn} \begin{proof} Multiplying the expression
\[
\partial_1-1+\dfrac{X_1}{\partial_2}
\]
by $\partial_1\partial_2$ yields
\[
\partial_2\partial_1^2 -\partial_1\partial_2+\partial_1X_1.
\] Left multiplying the expression $X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2$ by $\partial_1$ and taking into account of the last two relations yield
\[
\begin{split}
\partial_1 X_1 \partial_1-2X_2\partial_1\partial_2-\partial_1-\partial_1X_1+\partial_1\partial_2
&= X_1\partial_1^2-2X_2\partial_1\partial_2+\partial_2\partial_1^2\\
&= X_1\partial_1^2-2X_2\partial_1\partial_2 +\partial_1\partial_2-\partial_1X_1\\
&=X_1\partial_1^2-(X_1+2X_2\partial_2)\partial_1+\partial_1\partial_2-1\\
&=X_1\partial_1^2-2(X_1+X_2\partial_2)\partial_1+X_1\partial_1+\partial_1\partial_2-1\\
&=X_1\partial_1^2-2(X_1+X_2\partial_2)\partial_1+X_1\partial_1+\partial_2-X_1-1\\
&=X_1\partial_1^2-2(X_1+X_2\partial_2)\partial_1+
1+2X_2\partial_2-1\\
\end{split}
\]as desired.
\end{proof}
\subsection{Proof of Theorem \ref{T:symbol gen yn-1}}\label{A:BP_chara}
\begin{proof}
It is easy to verify that
\[
\partial_{1}\eta^{2}-\eta^{2}\partial_1=-2X_2,\qquad
\partial_{2}\eta^{2}-\eta^{2}\partial_2=-2X_1
\]hold. Suppose that
\[
\delta_1=\partial_1\eta-\eta\partial_1,
\qquad
\delta_2=\partial_2\eta-\eta\partial_2,
\]
and $\eta$ commutes with $\delta_i, i=1, 2$, i.e.,
\[
\delta_1\eta=\eta \delta_1, \qquad \delta_2\eta=\eta \delta_2.
\]
Then we obtain
\[
\partial_1 \eta-\eta\partial_1=-\frac{X_2}{\eta},\qquad
\partial_2\eta-\eta\partial_{2}=-\frac{X_1}{\eta}.
\]
Let
\[
\Theta_1=\partial_1-\dfrac{1-X_1X_2}{X_1^2}\partial_2,\qquad Y_1=X_1,\qquad
\Theta_2=\eta\partial_2,\qquad Y_2=-\dfrac{\eta}{X_1}.
\]
We deduce from direct computation that
\[
[\Theta_1, Y_1]=1,\quad [\Theta_2,Y_2]=1, \quad [Y_1, Y_2]=0,
\quad [\Theta_1, Y_2]=0,\quad [\Theta_2, Y_1]=0, \quad [\Theta_1,\Theta_2]=0.
\]
Then $\Theta_i, Y_i, i=1, 2$ generate the following module
\[
\widehat{\mathcal{A}_{2}}=\mathbb{C}\langle \Theta_1, \,\Theta_2, \,Y_1, \,Y_2\rangle.
\] The two generators of $\mathcal{Y}(\eta)$ in Theorem \ref{T: iso bp-n-1} become, when written in terms of $\Theta_i, Y_i, i=1, 2$,
\[
(\Theta_2-1)(\Theta_2+1), \qquad \Theta_1+ \dfrac{1}{Y^2_1}.
\]
Thus we define a modified $\widehat{\mathcal{A}}_2$-module
\[
\widehat{\mathcal{Y}}=\dfrac{\widehat{\mathcal{A}_{2}}}
{\widehat{\mathcal{A}_{2}}((\Theta_2-1)(\Theta_2+1))
+\widehat{\mathcal{A}_{2}}(\Theta_1+ \dfrac{1}{Y^2_1})}.
\]
It follows from Example \ref{E:prime_integrable_eg_2} that
$\widehat{\mathcal{Y}}$ is a holonomic module with dimension and multiplicity both equal to two.
We apply Example \ref{Eg:exp} to derive that the map
\[
\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}(\Theta^2_2-1)
\xrightarrow[]{\times [{\mathop{\rm E}}(\pm Y_2)f(Y_1)]}
\overline{\widehat{\mathcal{A}}_{2}/\widehat{\mathcal{A}}_{2}\Theta_2}
\]
is well-defined left $\widehat{\mathcal{A}}_{2}$-linear if the unknown relations $f(Y_1), g(Y_1)$ depending only on $Y_1$ can be worked out. So we require
\[
\begin{split}
(\Theta_1+ \dfrac{1}{Y^2_1}){\mathop{\rm E}}(Y_2)f(Y_1)
=&{\mathop{\rm E}}(Y_2)\Theta_1f(Y_1)+\dfrac{1}{Y^2_1}{\mathop{\rm E}}(Y_2)f(Y_1)\\
=&{\mathop{\rm E}}(Y_2)(\Theta_1+\dfrac{1}{Y^2_1})f(Y_1)\mod \widehat{\mathcal{A}_{2}}\Theta_1\\
=&0.
\end{split}
\]
This would hold if $f(X_1)={\mathop{\rm E}}({1}/{Y_1})$. Similarly, we can obtain $g(X_1)={\mathop{\rm E}}({1}/{Y_1})$.
Hence, for each choice of constants $C_1, C_2$, we have
\[
\widehat{\mathcal{Y}}
\xrightarrow[]{\times [C_1{\mathop{\rm E}}(Y_2+\frac{1}{Y_1})+C_2{\mathop{\rm E}}(-Y_2+\frac{1}{Y_1})]}
\overline{\widehat{\mathcal{A}}_{2}/(\widehat{\mathcal{A}}_{2}\Theta_1+\widehat{\mathcal{A}}_{2}\Theta_2)},
\]
which gives rise to the $\mathcal{A}_2(\eta)$-linear map
\[
\mathcal{Y}(\eta)
\xrightarrow[]{\times [C_1{\mathop{\rm E}}(\frac{1-\eta}{X_1})+C_2{\mathop{\rm E}}(\frac{1+\eta}{X_1})]}
\overline{\mathcal{A}_{2}(\eta)/(\mathcal{A}_{2}(\eta)\partial_1+
\mathcal{A}_{2}(\eta)\partial_2)}.
\] \end{proof}
\subsection{Completion of proof of Theorem \ref{T:Bessel p_gf_1}}\label{A:BP_gf_1}
\begin{proof} We continue the proof that is left unfinished below the Theorem \ref{T:Bessel p_gf_1}.
\begin{enumerate}
\item The $\mathcal{O}^{\mathbb{N}_0}$ being an $\mathcal{A}_2(\eta)$-module is essentially verified from Example \ref{Eg:seq-functions_poisson}. The composition of the horizontal map and the Poisson transform $\mathfrak{p}$ of the top-right path from the commutative diagram \eqref{E:commute-diff bess pol}
\[
\begin{array}{rlcll}
\mathcal{Y}(\eta) & \xrightarrow[]{\times (y_{n-1})}
& \mathcal{O}^{\mathbb{N}_0}
&\xrightarrow[]{\times 1} & \mathcal{O}_{dd},\\
&&&\\
\end{array}
\]is a left $\mathcal{A}_{2}(\eta)$-linear map. Hence the composition map yields
\[
1 \longmapsto
\displaystyle\sum_{n=0}^\infty y_{n-1}(x)\, \dfrac{t^n}{n!},
\]which implies that the sum $\sum_n y_{n-1}(x)\, {t^n}/{n!}$ is a solution to the following system of PDEs \eqref{pde-1} and \eqref{pde-2} below.
\item The first map from the bottom-left path of the commutative diagram \eqref{E:commute-diff bess pol} has already been constructed in Theorem \ref{T:symbol gen yn-1}. On the other hand, the following maps
\[
\begin{array}{rlcll}
\mathfrak{g}_p: \mathcal{Y}(\eta) &\xrightarrow{ C_1{\mathop{\rm E}}(\frac{1-\eta}{X_1})+C_2{\mathop{\rm E}}(\frac{1+\eta}{X_1})} & \overline{\mathcal{A}_{2}(\eta)/\big[\mathcal{A}_{2}(\eta)\partial_1
+\mathcal{A}_{2}(\eta)\partial_2\big] }
&\xrightarrow{ \times 1} \mathcal{O}_{d d}
\end{array}
\]
and \eqref{E:commute-diff bess pol} show that the
\begin{equation}\label{E:gen_bessel_poly_gf_1}
C_1\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)+C_2
\exp\left(\frac{1+(1-2xt)^{1/2}}{x}\right)
=\sum_{k=0}^\infty \frac{y_k(x)}{k!}t^k
\end{equation}holds for some $C_1,\, C_2\in\mathbb{C}$ and that both sides satisfy the system of PDEs
\begin{equation}\label{pde-1}
(1-2xt)f_{tt}(x, \,t)-f(x, \,t)-xf_t(x, \, t)=0,
\end{equation}
\begin{equation}\label{pde-2}
x^2f_x(x,\, t)-f_{t}(x,\,t)+xtf_t(x, \, t)+f(x,\, t)=0.
\end{equation}
Putting $t=0$ in both sides of \eqref{E:gen_bessel_poly_gf_1} implies that $C_1=1$ and $C_2=0$.
\end{enumerate} \end{proof}
We recall the well-known theorem that describes which half-plane does a Newton series and its associated series, called \textit{faculty series} by N\"orlund, would converge.
\subsection{Proof of Theorem \ref{T:delta Bessel_gf_1} }\label{A:detal_BP_gf_1} Here we give full details of the proof to Theorem \ref{T:delta Bessel_gf_1} in addition to the commutative diagram \eqref{E:commute-dbp}.
\begin{proof}
\begin{enumerate}
\item The composition of the following maps
\[
\mathcal{Y}(\eta) \xrightarrow[]{\times (y^\Delta_{n-1})}
\mathcal{O}_\Delta^{\mathbb{N}_0}
\xrightarrow[]{\times 1} \mathcal{O}_{\Delta d}
\]
is left $\mathcal{A}(\eta)$-linear. Hence the infinite sum from
\[
1 \longmapsto
\displaystyle\sum_{n=0}^\infty y_{n-1}^\Delta (x)\, \frac{t^n}{n!}
\]
is a solution to the following delay-differential equations
\begin{align}
&x(x-1)[f(x-1, t)-f(x-2, t)]-f_t(x, t)+xtf_t(x-1, t)+f(x, t)=0,\label{diff-Bessel yn-1 eq1}\\
&f_{tt}(x, t)-2xtf_{tt}(x-1, t)-f(x, t)-xf_t(x-1, t)=0.\label{diff-Bessel yn-1 eq2}
\end{align}
\item To deal with the left-side of the equation \eqref{E:bessel dp_gf-1}, we make use of the commutative diagram \eqref{E:commute-diff bess pol} from the proof of Theorem \ref{T:Bessel p_gf_1} from the last subsection. Thus to complete the left-half the commutative diagram \eqref{E:commute-dbp}, the final step of the composition map
\[
\mathcal{Y}(\eta) \xrightarrow[]{\times {\mathop{\rm E}}[\frac{1-\eta}{X_1}]} {\tilde{\mathcal{A}}_2(\eta)}
\xrightarrow[]{\times 1}\mathcal{O}_{dd}
\xrightarrow[]{\times \mathfrak{N}} \mathcal{O}_{\Delta d}
\]that is given by the Newton transformation \eqref{E:newton_trans} yields
\[
\mathfrak{N}\circ {\mathop{\rm E}}(\frac{1-\eta}{X_1})\cdot 1=
\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}d\lambda
\]as asserted.
Although the $\mathcal{A}_2(\eta)$-linearity consideration of the above composition map is sufficient to complete the proof, we offer a direct verification that the above $ \mathfrak{N}\circ {\mathop{\rm E}}(\frac{1-\eta}{X_1})\cdot 1$ is indeed a solution to the system of PDEs \eqref{diff-Bessel yn-1 eq1} and \eqref{diff-Bessel yn-1 eq2} here.
Let
\[
f(x, t)=\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }.
(-\lambda)^{-x-1}d\lambda.
\]
Then we verify
\[
\begin{split}
&(X_{1}^{2}\partial_1-\partial_2+X_1X_2\partial_2+1) f(x, t)=0,\\
&(1-2X_1X_2)\partial_2^2-X_1\partial_2-1) f(x, t)=0.
\end{split}
\]
We first compute
\[
\begin{split}
&\partial_{2}^2\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}d\lambda\\
&=\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}
\big[(1-2\lambda t)^{-1}+
\lambda(1-2\lambda t)^{-3/2}\big]
d\lambda,
\end{split}
\]
and
\[
\begin{split}
&(1-2X_1X_2)\partial_{2}^2
\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}d\lambda\\
&=\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}
\big[1+\lambda(1-2\lambda t)^{-1/2}\big]
d\lambda.
\end{split}
\]
Combine the above computations yields
\[
((1-2X_1X_2)\partial_2^2-X_1\partial_2-1)
\displaystyle\frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1} d\lambda=0.
\]
Similarly, we can compute
\[
\begin{split}
&X_1^2\partial_{1}\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1} d\lambda\\
&=\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x+1}
\big[\dfrac{x-1}{\lambda}-1\big]d\lambda\\
&=-\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
d(e^{\lambda} (-\lambda)^{-x+1} ).
\end{split}
\]
Integration-by-part yields
\[
\begin{split}
&(X_1^2\partial_{1}+1)\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1} d\lambda\\
&=(1-X_1X_2)\partial_2
\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1} d\lambda,
\end{split}
\]
which implies
\[
(X_1^2\partial_{1}-(1-X_1X_2)\partial_2+1)
\displaystyle\frac{1}{\Gamma(-x)}\int_{-\infty}^{0}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1} d\lambda=0.
\]
Hence $f(x, t)$ satisfies the system of differential-difference equations
\eqref{diff-Bessel yn-1 eq1} and
\eqref{diff-Bessel yn-1 eq2}.
Similar to the argument used in the proof of Theorem \ref{T:Bessel p_gf_1} that
the solution space of the PDEs \eqref{diff-Bessel yn-1 eq1} and \eqref{diff-Bessel yn-1 eq2} is two-dimensional. Hence \eqref{E:gen_bessel_poly_gf} from Theorem \ref{T:general_bessel_poly_gf} holds for some constants $C_1, \, C_2$ with $\mathscr{Y}_{n-1}$ replaced by $y_{n-1}$.
Substitute $t=0$ into both sides of \eqref{E:bessel dp_gf-1} yields $C_1=1,\, C_2=0$.
Then we determine the region of convergence of \eqref{E:bessel dp_gf-1}.
The sum of the two generators yields
\[
X_1\partial_1\partial_2+X_1\partial_1-X_2\partial^2_2+X_2\partial_2
\]
which gives rise to the equation
\[
x[f_t(x,\, t)-f_t(x-1, \,t)]+x[f(x, \, t)-f(x-1,\, t)]-tf_{tt}(x,\, t)+tf_t(x,\, t)=0.
\]
We differentiate above equation with respect to $t$ to obtain
\begin{equation}\label{diff-Bessel yn-1 eq3}
x[f_{tt}(x,\, t)-f_{tt}(x-1, \,t)]+x[f_t(x, \, t)-f_t(x-1,\, t)]-tf_{ttt}(x,\, t)+(t-1)f_{tt}(x,\, t)+f_t(x,\,t)=0.
\end{equation}
Since the manifestation of equation \eqref{bessel_poly_recursion_3 in a A becomes}
in $\mathcal{O}_{\Delta d}$ is given by
\begin{equation}\label{diff-Bessel yn-1 eq4}
f_{tt}(x,\,t)-2xtf_{tt}(x-1,\, t)-xf_t(x-1,\, t)-f(x, \,t)=0.
\end{equation}
Multiplying the equation \eqref{diff-Bessel yn-1 eq3} by $2t$ and subtract the resulting equation from \eqref{diff-Bessel yn-1 eq4} yield the first equation in \eqref{diff-Bessel yn-1 eq5}. Subtracting the \eqref{diff-Bessel yn-1 eq3} and \eqref{diff-Bessel yn-1 eq4} yields the second equation in \eqref{diff-Bessel yn-1 eq5}:
\begin{equation}\label{diff-Bessel yn-1 eq5}
\begin{split}
&xf_t(x-1,\, t)=\dfrac{1}{1-2t}[(1-2tx+2t-2t^2)f_{tt}(x, t)+2t^2f_{ttt}-2t(x+1)f_t(x, t)-f(x, t)],\\
&xf_{tt}(x-1,\, t)=\dfrac{1}{1-2t}[(x-2+t)f_{tt}(x, t)-tf_{ttt}(x, t)+(x+1)f_t(x, t)+f(x, t)].
\end{split}
\end{equation}
We replace $x$ by $x-1$ in the equation \eqref{diff-Bessel yn-1 eq4} and differentiate the resulting equation with respect to $t$ yield
\begin{equation}\label{diff-Bessel yn-1 eq100}
f_{ttt}(x-1,\,t)-2(x-1)tf_{ttt}(x-2,\, t)-3(x-1)f_{tt}(x-2,\, t)-f_t(x-1, \,t)=0.
\end{equation}
We multiply $x$ to the equation \eqref{diff-Bessel yn-1 eq100} and subtract the resulting equation to the equation \eqref{diff-Bessel yn-1 eq1} after it being differentiated three times and multiply by the $2t$ throughout the whole equation would eliminate the term $f_{ttt}(x-2,\, t)$ from the equation \eqref{diff-Bessel yn-1 eq100}. Multiply the equation \eqref{diff-Bessel yn-1 eq100} by $x$ and subtract 3 times the \eqref{diff-Bessel yn-1 eq1} after it being differentiated twice would eliminate the term $f_{tt}(x-2,\, t)$ from the equation \eqref{diff-Bessel yn-1 eq100}. This results in the following fourth order equation
\begin{equation}\label{diff-Bessel yn-1 eq6}
\begin{split}
&(x-2tx^2-7xt)f_{ttt}(x-1, \, t)-2xt^2f_{tttt}(x-1,\, t)+(-3x^2-3x)f_{tt}(x-1,\, t)-xf_t(x-1,\, t)\\
&+2tf_{tttt}(x,\, t)+(3-2t)f_{ttt}(x,\, t)-3f_{tt}(x,\, t)=0.
\end{split}
\end{equation} We then eliminate the terms $f_{tttt}(x-1,\, t)$ and $f_{ttt}(x-1,\, t)$ from the \eqref{diff-Bessel yn-1 eq6} utilizing the equation \eqref{diff-Bessel yn-1 eq4} after it being differentiated several times. The substitutions of the terms $f_{tttt}(x-1,\, t)$ and
$f_{ttt}(x-1,\, t)$ from the equation \eqref{diff-Bessel yn-1 eq6} yield
\[
\begin{split}
\dfrac{-3x}{2t}f_{tt}(x-1, t)-xf_t(x-1, t)+&tf_{tttt}(x, t)+(\dfrac{4t-4t^2-2xt+1}{2t})
f_{ttt}(x, t)\\
&+(t-3)f_{tt}-\dfrac{1-2tx-2t}{2t}f_t(x, t)=0.
\end{split}
\]
Substituting \eqref{diff-Bessel yn-1 eq5} into above equation yields
\begin{equation}\label{diff-Bessel yn-1 eq7}
\begin{split}
&tf_{tttt}(x, \,t)+\big(\dfrac{-4t^3+12t^2-5t-4xt^2+2xt-1}{2t(2t-1)}\big)f_{ttt}(x, \,t)\\
&+\big(\dfrac{-10t^2-4t^2x+11t-6+3x}{2t(2t-1)}\big)f_{tt}(x,\, t)
+\big(\dfrac{3x+4-4t-2xt}{2t(2t-1)}\big)f_t(x, \,t)\\
&+\dfrac{3-2t}{2t(2t-1)}f(x, \,t)=0,
\end{split}
\end{equation}
which is a fourth order linear ordinary differential equation in $t$ for each fixed $x$. Since the equation \eqref{diff-Bessel yn-1 eq5} has a regular points at $t=0,\, 1/2$, so that every analytic solution \eqref{diff-Bessel yn-1 eq5} has a radius of convergence $1/2$ in the neighbourhood of $x=0$. Hence the \eqref{E:bessel dp_gf-1} converges uniformly in each compact subset of $\mathbb{C}\times B(0; \frac12)$. Substituting a Frobenius solution $f(x,\cdot)=\sum_{k=0}^\infty a_kx^{\sigma+k}$ into the equation \eqref{diff-Bessel yn-1 eq5} further confirming that $\sigma=0,\, 1,\, 2$ are the only integer indicial roots. Hence the \eqref{E:bessel dp_gf-1} holds.
\end{enumerate} \end{proof}
\section{Difference operators with arbitrary step sizes}\label{Append:arbitrary} In practical applications, one would probably like to consider those $D$-modules like $\mathcal{O}_\Delta$ from Example~\ref{Eg:O_deleted_d} or $\mathcal{O}_{\Delta d}$ from Example~\ref{Eg:O_delta_d} with any step size $h\in\mathbb{C}\setminus\{0\}$. So we gather some formulae about these $D$-modules and the corresponding realizations in various function spaces.
\begin{example}\label{Eg:O_delta_h} Let $\mathcal{O}$ be the space of analytic functions. Then $\mathcal{O}$ becomes a left $\mathcal{A}$-module with
\[
\begin{split}
&(\partial f)(x) =\frac1h \Delta_h f(x)=
\frac1h \big(f(x+h)-f(x)\big), \\
&(Xf)(x)=xf(x-h).
\end{split}
\]
This left $\mathcal{A}$-module is denoted by $\mathcal{O}_{\Delta_h}$. \end{example}
Clearly $\mathcal{O}_{\Delta_h}$ is the same as $\mathcal{O}_\Delta$ when $h=1$. Similarly in $\mathcal{O}_{\Delta_h}$, if we denote
\[
(x)_{n,h}:=x(x-h)\cdots \big(x-(n-1)h\big)=h^n\dfrac{\Gamma(x/h+1)}{\Gamma(x/h-n+1)},
\] then we have
\[
X^n\cdot 1 =x(x-h)\cdots \big(x-(n-1)h\big)= (x)_{n, h}.
\]
As a result, the element $L=(X\partial)^2+X^2-\nu^2$ gives the difference Bessel equation
\begin{equation}\label{E:Bessel_eqn_delta_h}
\begin{split}
\Big(\frac{x+2h)^2}{h^2}-\nu^2\Big)f(x+2h) &-\frac{(x+2h)(2x+3h)}{h^2}f(x+h)\\
&+2\Big(1+\frac{1}{h^2}\Big)(x+2h)(x+h)f(x)=0,
\end{split}
\end{equation}which converges to Bohner and Cuchta\rq{}s difference Bessel equation \eqref{E:difference_bessel_eqn} with step $h\to 1$. One can also recover the classical Bessel equation \eqref{E:Bessel_eqn} when the step $h\to 0$ with a more careful analysis. Classical solutions of \eqref{E:Bessel_eqn_delta_h} are solutions of $\mathcal{D}/\mathcal{D}L$ in $\mathcal{O}_{\Delta_h}$. One of them is
\begin{equation}
\mathcal{D}/\mathcal{D}L\stackrel{\times S_\nu}{\longrightarrow} \mathcal{D}/\mathcal{D}(X\partial-\nu)\stackrel{\times (x)_{\nu,h}}{\longrightarrow}
\mathcal{O}_{\Delta_h},
\end{equation}
in which $S_\nu=\displaystyle\sum_{k=0}^{\infty} \frac{(-1)^kX^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}$, and gives rise to the Newton series
\begin{equation}\label{E:difference_bessel_fn_h}
J^{\Delta_h}_{\nu}(x):=\sum_{k=0}^{\infty} \frac{(-1)^kh^{2k}}{2^{\nu+2k}k!\Gamma(\nu+k+1)}(x)_{\nu+2k,h}.
\end{equation}
Another solution in $\mathcal{O}_{\Delta_h}$ corresponding to $S_{-\nu}$ yields the Newton series
\begin{equation}
J^{\Delta_h}_{-\nu}(x):=\sum_{k=0}^{\infty} \frac{(-1)^kh^{2k}}{2^{-\nu+2k}k!\Gamma(-\nu+k+1)}(x)_{-\nu+2k,h},
\end{equation}
which is ``linearly independent" to the first one in case that $2\nu$ is not an integer.
\begin{example}\label{Eg:O_delta_h_d} Let $\mathcal{O}_2$ be the space of analytic functions in two variables. It becomes a left $\mathcal{A}_2$-module when endowed with the structure
\begin{equation}\label{E:O_delta_h_d}
\begin{array}{ll}
\partial_1f(x,\, t)=\frac1h\big(f(x+h,\, t)-f(x,\, t)\big), & X_1f(x,\, t)=xf(x-h,\, t);\\
\partial_2f(x,\, t) =f_t(x,\, t), & X_2 f(x,\, t)=tf(x,\, t).
\end{array}
\end{equation}
Such a left $\mathcal{A}_2$-module is denoted by $\mathcal{O}_{\Delta_h d}$. \end{example}
\begin{example}\label{Eg:two-seq-functions_2_h} Let $\mathcal{O}^\mathbb{Z}$ be the space of all bilateral sequences of analytic functions with an appropriately imposed growth restriction. Let
\[
(f_n)=(\cdots, \, f_{-1},\, f_0,\, f_1,\, \cdots)
\]denote a bilateral infinite sequence of $f_k\in\mathcal{O}$. Then $\mathcal{O}^\mathbb{Z}$ becomes a left $\mathcal{A}_2$-module by
\begin{equation}
\begin{array}{ll}
(\partial_1 f)_n(x)=\frac1h\big(f_n(x+h)-f_n(x)\big), & (X_1 f)_n(x)=xf_n(x-h),\\
(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),& (X_2 f)_n(x)=f_{n-1}(x),
\end{array}
\end{equation}for all bilateral sequences of analytic functions $(f_n)$. Such a left $\mathcal{A}_2$-module is sometimes denoted by $\mathcal{O}_{\Delta_h}^\mathbb{Z}$. \end{example}
We state without proof the following extensions of the Theorem \ref{T:difference_Bessel_gf} with an arbitrary step size $h$:
\begin{theorem}\label{T:difference_h_Bessel_gf} Let $\nu\in\mathbb{C}$.
\begin{enumerate}
\item Let $(\mathscr{C}^\Delta_{n+\nu})_n$ be a bilateral sequence of analytic functions which is a solution of the Bessel module $\mathcal{B}_\nu$ in $\mathcal{O}_{\Delta}^\mathbb{Z}$. Then there exists a $1$-periodic function $C_\nu$ such that
\begin{equation}\label{gf-db_h-2}
e^{i\pi\nu}
\frac{\sin(x/h-\nu)\pi}{\sin(\pi x/h)}\,
t^{-\nu}\big[\frac{h}{2}(t-\frac{1}{t})+1\big]^{x/h}
\sim \sum_{n=-\infty}^{\infty} J^{\Delta_h}_{n+\nu}(x)\,t^n, \end{equation} where the symbol $\sim$ means that the left-hand side is the Borel resummation \footnote{Definition \ref{D:borel}} of the right-hand side whenever it diverges.
\item Moreover, the manifestation of the holonomic system of PDEs (delay-differential equations) \eqref{E:bessel_PDE} in $\mathcal{O}_{\Delta d}$ in Example \ref{Eg:O_delta_h_d} defined by \eqref{E:O_delta_d} is given by
\begin{equation}\label{E:PDE_delta_h_d}
y(x, t)-y(x-h, t)=
\frac{h}{2}(t-\frac{1}{t}) y(x-h, t),\quad
\nu y(x,t) +ty_t(x,t)=\frac{x}{2}\big(t+\frac{1}{t}\big)y(x-h, t).
\end{equation}
\end{enumerate} \end{theorem}
We note that although the above result is valid for each fixed $h\not=0$, one cannot take the limit $h\to 0$ so that the \eqref{E:PDE_delta_h_d} becomes the generating function \eqref{E:bessel_classical_gf_1} for the classical Bessel functions, unless when $\nu=0$, as in the following theorem.
\begin{theorem}\label{T:h_limit}
\begin{enumerate}
\item Let $(J^{\Delta_h}_n(x))$ and $(J_n(x))$ be sequences in $\mathcal{O}_{\Delta_h}^{\mathbb{Z}}$ and ${\mathcal{O}_d}^\mathbb{Z}$. Then we have the following limits
\begin{equation}\label{E:h_limit}
\begin{tikzcd} [row sep=large, column sep=large]
\displaystyle\big[\frac{h}{2}(t-\frac{1}{t})+1\big]^{x/h}\qquad =
\arrow[swap]{d}{h\to 0}
& \displaystyle \sum_{n=-\infty}^{\infty} J^{\Delta_h}_{n}(x)\,t^n \arrow{d}{h\to 0}\\
\displaystyle\exp\big[\frac{x}{2}(t-\frac{1}{t})\big]
\qquad= & \displaystyle\sum_{n =-\infty}^{\infty} J_{n}(x)\,t^n
\end{tikzcd}
\end{equation}
\item Moreover, the system of PDEs \eqref{E:O_delta_h_d} with $\nu=0$, that is
\begin{equation}\label{E:O_delta_h_d_0}
y(x, t)-y(x-h, t)=
\frac{h}{2}(t-\frac{1}{t}) y(x-h, t),\quad
ty_t(x,t)=\frac{x}{2}\big(t+\frac{1}{t}\big)y(x-h, t).
\end{equation}
becomes the system of PDEs \eqref{E:PDE_gf_bessel_0} as $h\to 0$.
\end{enumerate} \end{theorem}
The case of the above limits when $\nu\not=0$, while likely to hold, is more delicate that would be dealt with in a separate publication.
\eject
\section{List of tables} \label{A:list}
\subsection{List of commonly used $D$-modules}\label{SS:common_D_list}
\begin{table}[h!]\label{T:common_D_list}
{\small
\begin{tabular}[h!]{|c|c|c|l|} \hline $D$-module & \begin{tabular}{c}``ground"\\ Weyl algebra\end{tabular} & \begin{tabular}{c}underlying\\ space\end{tabular} & Manifestation \\ \hline $\mathcal{A}/\mathcal{A}L$ & $\mathcal{A}$ & $\mathcal{A}/\mathcal{A}L$ & same as multiplication in $\mathcal{A}$ \\ \hline $\dfrac{\mathcal{A}_2}{\mathcal{A}_2L_1+\mathcal{A}_2L_2}$ & $\mathcal{A}_2$ & $\dfrac{\mathcal{A}_2}{\mathcal{A}_2L_1+\mathcal{A}_2L_2}$ & same as multiplication in $\mathcal{A}_2$ \\ \hline $\mathbb{C}^{\mathbb{N}_0}$ & $\mathcal{A}$ & $\mathbb{C}^{\mathbb{N}_0}$ & \begin{tabular}{l}$(\partial a)_n=a_{n+1}$\\ $(Xa)_n=na_{n-1}$\end{tabular}\\ \hline $\mathbb{C}^\mathbb{Z}$ & $\mathcal{A}$ & $\mathbb{C}^\mathbb{Z}$ & \begin{tabular}{l}$(\partial a)_n=(n+1)a_{n+1},$\\ $(Xa)_n=a_{n-1}$\end{tabular}\\ \hline $\mathcal{O}_d$ & $\mathcal{A}$ & $\mathcal{O}$ & \begin{tabular}{l}$\partial f(x)=f'(x),$\\ $Xf(x)=xf(x)$\end{tabular}\\ \hline $\mathcal{O}_\Delta$ & $\mathcal{A}$ & $\mathcal{O}$ & \begin{tabular}{l}$\partial f(x)=f(x+1)-f(x),$\\ $Xf(x)=xf(x-1)$\end{tabular}\\ \hline $\mathcal{O}_{dd}$ & $\mathcal{A}_2$ & $\mathcal{O}_2$ & \begin{tabular}{l}$\partial_1 f(x,t)=f_x(x,t),$\\ $X_1 f(x,t)=xf(x,t)$\\ $\partial_2 f(x,t)=f_t(x,t),$\\ $X_2 f(x,t)=tf(x,t)$\end{tabular}\\ \hline $\mathcal{O}_{\Delta d}$ & $\mathcal{A}_2$ & $\mathcal{O}_2$ & \begin{tabular}{l}$\partial_1 f(x,t)=f(x+1,t)-f(x,t),$\\ $X_1 f(x,t)=xf(x-1,t)$\\ $\partial_2 f(x,t)=f_t(x,t),$\\ $X_2 f(x,t)=tf(x,t)$\end{tabular}\\ \hline $\mathcal{O}_d^{\mathbb{N}_0}$ & $\mathcal{A}_2$ & $\mathcal{O}^{\mathbb{N}_0}$ & \begin{tabular}{l}$(\partial_1 f)_n(x)=f_n'(x),$\\ $(X_1 f)_n(x)=xf_n(x)$\\ $(\partial_2 f)_n(x)=f_{n+1}(x),$\\ $(X_2 f)_n(x)=nf_{n-1}(x)$\end{tabular}\\ \hline
$\mathcal{O}_d^\mathbb{Z}$ & $\mathcal{A}_2$ & $\mathcal{O}^\mathbb{Z}$
& \begin{tabular}{l}$(\partial_1 f)_n(x)=f_n'(x),$\\ $(X_1 f)_n(x)=xf_n(x)$\\
$(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),$\\
$(X_2 f)_n(x)=f_{n-1}(x)$
\end{tabular}
\\ \hline
$\mathcal{O}_\Delta^\mathbb{Z}$ & $\mathcal{A}_2$ & $\mathcal{O}^\mathbb{Z}$ & \begin{tabular}{l} $(\partial_1 f)_n(x)=f_n(x+1)-f_n(x),$\\ $(X_1 f)_n(x)=xf_n(x-1)$;\\ $(\partial_2 f)_n(x)=(n+1)f_{n+1}(x),$\\ $(X_2 f)_n(x)=f_{n-1}(x)$\end{tabular}\\ \hline
$\mathcal{O}_\Delta^{\mathbb{N}_0}$ & $\mathcal{A}_2$ & $\mathcal{O}^{\mathbb{N}_0}$ &
\begin{tabular}{l} $(\partial_1 f)_n(x)=f_n(x+1)-f_n(x),$\\ $(X_1 f)_n(x)=xf_n(x-1)$;\\ $(\partial_2 f)_n(x)=f_{n+1}(x),$\\ $(X_2 f)_n(x)=nf_{n-1}(x)$\end{tabular}\\ \hline \end{tabular}\\ \begin{center} \caption{\label{demo-table}List of $D$-modules used.\newline
}\end{center} } \end{table}
Each ground Weyl algebra above is allowed to be slightly modified, e.g. $\mathcal{A}$ can be modified into $\mathcal{A}(1/X)$, and $\mathcal{A}_2$ can be modified into $\mathcal{A}_2(1/X)$, etc.
\eject
\subsection{List of transmutation formulae} \begin{table}[h!]
\scriptsize{
\begin{tabular}[h!]{|c|c|c|} \hline
$D$-modules & Transmutation formulae & No.\\
\hline
$\mathcal{B}_\nu$ &
\begin{tabular}{l}
$[(X\partial)^2+X^2-(\nu-1)^2]\big(\partial+{\nu}/{X}\big)$\\
\hskip4cm$=\big(\partial+(\nu-2)/{X}\big)[(X\partial)^2+X^2-\nu^2]$,
\\
$[(X\partial)^2+X^2-(\nu+1)^2]\big(\partial-\nu/X\big)$\\
\hskip4cm $=\big(\partial-(\nu+2)/X\big)[(X\partial)^2+X^2-\nu^2]$
\end{tabular}
&
\begin{tabular}{c} \eqref{E:bessel_transmutation_1}\\
-\eqref{E:bessel_transmutation_2}
\end{tabular}\\
\hline
$\Theta$ &
\begin{tabular}{l}
$\Big(X\partial^2-2(\nu+\frac12+X)\partial +2(\nu +\frac12)\Big)(X\partial-X-2\nu)$\\
$=\big(X\partial-X-2(\nu+1)\big)\Big(X\partial^2-2(\nu-\frac12+X)\partial+2(\nu -\frac12)\Big)$, \\
$\Big(X\partial^2-2(\nu-\frac12+X)\partial +2(\nu-\frac12)\Big)\frac1X(\partial-1)$\\
$=\frac1X (\partial-1+\frac1X)\Big(X\partial^2-2(\nu+\frac12+X)\partial+2(\nu+\frac12)\Big)$
\end{tabular}
&
\begin{tabular}{c}
\eqref{E:bessel_poly_transmutation_1}\\
-\eqref{E:bessel_poly_transmutation_2}
\end{tabular}
\\ \hline
\end{tabular} } \caption{\label{demo-table}List of transmutation formulae.} \end{table}
\subsection{List of holonomic modules}\label{SS:holo_modules}
\scriptsize{
\begin{tabular}[h!]{|c|c|c|c|} \hline \begin{tabular}{l} Quotient modules\\
$\mathcal{A}_2/(\mathcal{A}_2L_1+\mathcal{A}_2L_2)$
\end{tabular} & \begin{tabular}{l}$D$-module\\
terminologies
\end{tabular} & Generators & Definitions no. \\ \hline
$\mathcal{B}_\nu$ & Bessel modules &
\begin{tabular}{c}
$L_1=X_1\partial_1+(\nu+X_2\partial_2)-X_1X_2$, \\
$L_2=X_1\partial_1-(\nu+X_2\partial_2)+{X_1}/{X_2}$
\end{tabular} &
Definition \ref{E:bessel_mod}
\\ \hline
$\Theta$ &
\begin{tabular}{l}
Half-Bessel module:\\
reverse Bessel\\
polynomial module
\end{tabular}
&
\begin{tabular}{l}
$L_1=\partial_1-1+{X_1}/{\partial_2}$,\\
$L_2=X_1\partial_1-2X_2\partial_2-1-X_1+\partial_2$
\end{tabular} &
Definition \ref{D:Theta}\\ \hline
$\mathcal{Y}$ &
\begin{tabular}{l}
Half-Bessel module:\\
Bessel polynomial\\
module
\end{tabular}
&
\begin{tabular}{l}
$L_1=X_{1}^2\partial_{1}\partial_{2}-X_{1}X_{2}\partial^2_{2}+\partial_{2}-1$,\\
$L_2=X_{1}^2\partial_{1}-\partial_{2}+X_{1}X_{2}\partial_{2}+1$
\end{tabular} &
Definition \ref{D:y-1}\\ \hline
$\mathcal{G}_{-}$ &
\begin{tabular}{l}
Half-Bessel module:\\
Negative Glaisher\\
module
\end{tabular}
&
\begin{tabular}{l}
$L_1=W_1 \Xi_1+(W_2\Xi_2-\frac{1}{2})-W_1/\Xi_2$,\\
$L_2=W_1 \Xi_1-(W_2 \Xi_2-\frac{1}{2})+W_1 \Xi_2$.
\end{tabular} &
Definition \ref{D:neg_glaisher_module}\\ \hline
$\mathcal{G}_{+}$ &
\begin{tabular}{l}
Half-Bessel module:\\
Positive Glaisher\\
module
\end{tabular}
&
\begin{tabular}{l}
$L_1=W_1 \Xi_1+(W_2\Xi_2+\frac{1}{2})+W_1/\Xi_2$,\\
$L_2=W_1 \Xi_1-(W_2 \Xi_2+\frac{1}{2})-W_1 \Xi_2$.
\end{tabular} &
Definition \ref{D:pos_glaisher_module}\\ \hline \end{tabular} }
\subsection{List of formulae in $\mathcal{O}_d$, $\mathcal{O}_\Delta$}
\small{
\begin{tabular}[h!]{|c|c|c|} \hline $D$-modules & PDEs. & No. \\ \hline $\mathcal{B}_\nu$ in $\mathcal{O}_d$ &
\begin{tabular}{l}
$x\mathscr{C}_{\nu}^\prime(x)+\nu\mathscr{C}_{\nu}(x)-x\mathscr{C}_{\nu-1}(x)=0$,\\
$x\mathscr{C}^{\prime}_{\nu}(x)-\nu\mathscr{C}_{\nu}(x)+
x\mathscr{C}_{\nu+1}(x)=0$,\\
$2\nu\mathscr{C}_{\nu}(x)-x\mathscr{C}_{\nu-1}(x)-x\mathscr{C}_{\nu+1}(x)=0$
\end{tabular}
& \eqref {E:any_bessel_recus_1}, \eqref {E:any_bessel_recus_2}, \eqref {E:any_bessel_3_term}\\ \hline $\mathcal{B}_\nu$ in $\mathcal{O}_\Delta$ &
\begin{tabular}{l}
$x\Delta \mathscr{C}^{\Delta}_{\nu}(x-1)+\nu\mathscr{C}^{\Delta}_{\nu}(x)-
x\mathscr{C}^{\Delta}_{\nu-1}(x-1)=0$,\\
$x\Delta \mathscr{C}^{\Delta}_{\nu}(x-1)-\nu\mathscr{C}^{\Delta}_{\nu}(x)+
x\mathscr{C}^{\Delta}_{\nu+1}(x-1)=0$,\\
$2\nu\mathscr{C}_{\nu}^\Delta(x)-x\mathscr{C}_{\nu-1}^\Delta(x-1)-x\mathscr{C}_{\nu+1}^\Delta(x-1)=0$
\end{tabular}
& \eqref {E:delta_bessel_PDE_a}, \eqref {E:delta_bessel_PDE_b}, \eqref{E:delta_bessel_3_term_a}
\\ \hline $\Theta$ in $\mathcal{O}_d$
&
\begin{tabular}{l}
$\theta_n^\prime(x)-\theta_n(x)+x\theta_{n-1}(x)=0$,\\
$x\theta_n^\prime(x)-(x+2n+1)\theta_n(x)+\theta_{n+1}(x)=0$,\\
$\theta_{n+2}(x)-(2n+3)\theta_{n+1}(x)-x^2\theta_n(x)=0$.
\end{tabular}
&\eqref{E:bessel_poly_trans},
\eqref{E:bessel_poly_3term} \\ \hline
$\Theta$ in $\mathcal{O}_\Delta$ &
\begin{tabular}{l}
$\theta^\Delta_n(x+1)-2\theta^\Delta_n(x)+x\theta^\Delta_{n-1}(x-1)=0$, \\
$ \theta^\Delta_{n+1}(x)+(x-2n-1)\theta^\Delta_n(x)-2x\, \theta^\Delta_n(x-1)=0$,\\
$\theta^\Delta_{n+2}(x)-(2n+3)\theta^\Delta_{n+1}(x)-x(x-1)\theta^\Delta_n(x-2)=0$
\end{tabular}
& \eqref{E:delta_bessel_poly_trans},
\eqref{E:delta_reverse_bessel_poly_3term}
\\ \hline
$\mathcal{Y}$ in $\mathcal{O}_d$
&
\begin{tabular}{l}
$x^2y'_{n-1}(x)-[(n-1)x-1]y_{n-1}(x)-y_{n-2}(x)=0$,\\
$x^2y'_{n-1}(x)-y_{n}(x)+(nx+1)y_{n-1}(x)=0$,\\
$y_{n+1}(x)-(2n+1)xy_{\color{blue}n}(x)-y_{n-1}(x)=0$.
\end{tabular}
& \eqref{E:classical_bessel_poly_trans}, \eqref{E:classical_bessel_poly_3term}
\\ \hline
$\mathcal{Y}$ in $\mathcal{O}_\Delta$
&
\begin{tabular}{l}
$x(x-1) \Delta y^{\Delta}_{n-1}(x-2)-(nx-x)y^{\Delta}_{n-1}(x-1)+y^{\Delta}_{n-1}(x)- y^{\Delta}_{n-2}(x)=0$, \\
$x(x-1) \Delta y^{\Delta}_{n-1}(x-2)-y^{\Delta}_{n}(x)+nxy^{\Delta}_{n-1}(x-1)+y^\Delta_{n-1}(x)=0$,\\
$y^{\Delta}_{n+1}(x)-x(2n+1)y^{\Delta}_{n}(x-1)-y^{\Delta}_{n-1}(x)=0$.
\end{tabular}
& \eqref{E:delta_classical_bessel_poly_trans}, \eqref{E:delta_bessel_poly_3term} \\ \hline \end{tabular} }
\subsection{List of PDE systems in complex domains}\label{SS:PDE_list}
\vskip-.8cm
\begin{sidewaystable} {\small
\begin{tabular}[h!]{|c|c|c|c|c|} \hline $D$-modules & \begin{tabular}{c} ``target"\\ Weyl algebras\end{tabular} & PDEs & Gen. fns & Eqn no. \\ \hline $\mathcal{B}_\nu$ & $\mathcal{O}_{dd}$ & \begin{tabular}{l}
$y_x(x,t)+(1/t-t)/2\, y(x,t)=0$,\\
$\nu y(x,t)+ty_t(x,t)-{x}/2\,(1/t+t)\, y(x,t)=0$.
\end{tabular}
& $t^{-\nu}e^{\frac{x}{2}(t-\frac1t)}$
& \eqref{E:PDE_bessel}
\\ \hline $\mathcal{B}_\nu$ & $\mathcal{O}_{\Delta d}$ & \begin{tabular}{l}
$y(x+1, t)-y(x, t)=
\frac{1}{2}(t-\frac{1}{t}) y(x, t)$,\\
$ \nu y(x,t) +ty_t(x,t)={x}/2(t+{1}/{t})y(x-1, t)$.
\end{tabular} & $t^{-\nu} \big[\frac12(t-\frac1t)+1\big]^x$ & \eqref{gf-db-2}
\\ \hline $\Theta$ & $\mathcal{O}_{dd}$ &
\begin{tabular}{l}
$f_{xt}(x,\, t)-f_t(x,\, t)+xf(x, \, t)=0$,\\
$xf_x (x,\, t)+ (1-2t)f_t(x,\, t)-(1+x)f(x,\, t)=0$.
\end{tabular}
& $(1-2t)^{-\frac12}\exp[x(1-\sqrt{1-2t)}]$
& \eqref{E:bessel_poly_PDE}\\ \hline $\Theta$ & $\mathcal{O}_{\Delta d}$ &
\begin{tabular}{l}
$f_t(x+1,\, t)-2f_t(x,\, t)+xf(x-1,\, t)=0$,\\
$(1-2t)f_t(x,\, t)+(x-1)f(x,\, t)-2xf(x-1,\, t)=0$
\end{tabular}
& $\mathfrak{N} (1-2t)^{-\frac12}\exp[x(1-\sqrt{1-2t)}]$
& \eqref{E:delta_bessel_poly_PDE}\\ \hline $\mathcal{Y}$ & $\mathcal{O}_{d d}$ &
\begin{tabular}{l}
$x^2f_{xt}(x,\, t) -xtf_{tt}(x,\, t)+f_t(x,\,t)-f(x,\,t)=0$,\\
$x^2f_x(x,\, t)-f_t(x,\, t)+xtf_t(x,\, t)+f(x,\,t)=0$
\end{tabular}
& $\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)$
& \eqref{E:classical_bessel_poly_PDE}\\ \hline $\mathcal{Y}$ & $\mathcal{O}_{\Delta d}$ &
\begin{tabular}{l}
$x(x-1)[f_t(x-1,\, t)-f_t(x-2, \, t)]-xtf_{tt}(x-1,\, t)+f_t(x,\, t)-f(x, \, t)=0$,\\
$x(x-1)[f(x-1,\, t)-f(x-2, \, t)]-f_{t}(x,\, t)+xtf_t(x-1,\, t)+f(x, \, t)=0.$
\end{tabular}
& $\mathfrak{N} \exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)$
& \eqref{E:delta_classical_bessel_poly_PDE}\\ \hline $\mathcal{G}_-$ & $\mathcal{O}_{d d}$ &
\begin{tabular}{l}
$xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)-xf(x, t)=0$,\\
$xf_x(x,\, t)+(x-t)f_t(x,\, t)+\frac12 f(x,\, t)=0$
\end{tabular}
& $\sqrt{2/\pi x}\cos\sqrt{x^2-2xt}$
&
\begin{tabular}{c}
\eqref{E:glaisher_PDE_1},\\
\eqref{E:glaisher_PDE_2}
\end{tabular}
\\ \hline $\mathcal{G}_+$ & $\mathcal{O}_{d d}$ &
\begin{tabular}{l}
$xf_{xt}(x,\, t)+tf_{tt}(x,\, t)+\frac12 f_t(x,\, t)+xf(x, t)=0$,\\
$xf_x(x,\, t)-(x+t)f_t(x,\, t)+\frac12 f(x,\, t)=0$
\end{tabular}
& $\sqrt{2/\pi x}\sin\sqrt{x^2+2xt}$
&
\begin{tabular}{c}
\eqref{E:glaisher_PDE_6},\\
\eqref{E:glaisher_PDE_7}
\end{tabular}
\\ \hline $\mathcal{G}_-$ & $\mathcal{O}_{\Delta d}$ &
\begin{tabular}{l}
$tf_{tt}(x,\, t)+\big(x+\frac12\big)f_t(x,\, t)-xf_t(x-1,\, t)-xf(x-1,\, t)=0$,\\
$tf_{t}(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+\frac12\big) f(x,\, t)=0$
\end{tabular}
& $\mathfrak{N}\sqrt{2/\pi x}\cos\sqrt{x^2-2xt}$
&
\begin{tabular}{c}
\eqref{E:PDeltaE_neg_glaisher_-1},\\
\eqref{E:PDeltaE_neg_glaisher_-2}
\end{tabular}
\\ \hline
$\mathcal{G}_+$ & $\mathcal{O}_{\Delta d}$ &
\begin{tabular}{l}
$tf_{tt}(x,\, t)+\big(x+\frac12\big)f_t(x,\, t)-xf_t(x-1,\, t)+xf(x-1,\, t)=0$,\\
$ tf_{t}(x,\, t)+xf_t(x-1,\, t)+xf(x-1,\, t)-\big(x+\frac12\big) f(x,\, t)=0$
\end{tabular}
& $\mathfrak{N}\sqrt{2/\pi x}\sin\sqrt{x^2+2xt}$
&
\begin{tabular}{c}
\eqref{E:PDeltaE_pos_glaisher_+1},\\
\eqref{E:PDeltaE_pos_glaisher_+2}
\end{tabular}
\\ \hline \end{tabular} }\caption{\label{demo-table}List of systems of holonomic PDEs.} \end{sidewaystable}
\eject
\subsection{List of generating functions}\label{SS:gf_list}
{\footnotesize
\begin{tabular}[h!]{|l|c|c|} \hline Generating functions (${}^\ast$ denote likely new formulae)& $D$-mod. & Eqn no. \\ \hline
$\displaystyle
t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\sim\sum_{n=-\infty}^\infty J_{\nu+n}(x)\, t^n$& $\mathcal{B}_\nu$ & $\eqref{E:gf_J_nu}^\ast$\\ \hline
$\displaystyle
\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]=\sum_{n=-\infty}^\infty J_{n}(x)\, t^n$& $\mathcal{B}_0$ & \eqref{E:bessel_classical_gf}\\ \hline
$\displaystyle e^{-i\pi/2}\, t^{-\nu}\exp\big[\dfrac{x}{2}\big(t-\dfrac{1}{t})\big]\sim \sum_{n=-\infty}^\infty Y_{\nu+n}(x)\, t^n$
& $\mathcal{B}_\nu$ & $\eqref{E:gf_Y_nu}^\ast$\\ \hline
$\displaystyle t^{-\nu}\exp\big[\dfrac{x}{2}\big(t+\dfrac{1}{t})\big]\sim \sum_{n=-\infty}^\infty I_{\nu+n}(x)\, t^n$
& $\mathcal{B}_\nu$ & $\eqref{E:gf_I_nu}^\ast$\\ \hline
$\displaystyle i\pi \, t^{-\nu}\exp\big[-\dfrac{x}{2}\big(t+\dfrac{1}{t})\big]\sim \sum_{n=-\infty}^\infty K_{\nu+n}(x)\, t^n$
& $\mathcal{B}_\nu$ & $\eqref{E:gf_K_nu}^\ast$\\ \hline
$\displaystyle e^{i\pi\nu} \frac{\sin(x-\nu)\pi}{\sin(\pi x)}\, t^{-\nu}\big[\frac{1}{2}(t-\frac{1}{t})+1\big]^{x} \sim \sum_{n=-\infty}^{\infty} J^{\Delta}_{n+\nu}(x)\,t^n$
& $\mathcal{B}_\nu$ & $\eqref{gf-db-2}^\ast$\\ \hline
$\displaystyle \big[\frac{1}{2}(t-\frac{1}{t})+1\big]^{x} = \sum_{n=-\infty}^{\infty} J^{\Delta}_{n}(x)\,t^n$, \quad $|\frac{1}{2}(t-\frac{1}{t})|<1, \ x\in \mathbb{C}$
& $\mathcal{B}_0$ & $\eqref{gf-db-1}^\ast$\\ \hline
$\displaystyle \frac{1}{\sqrt{1-2t}}\Big\{ C_1\exp\big[x(1-\sqrt{1-2t})\big]
+C_2 \exp\big[x(1+\sqrt{1-2t})\big]\Big\}
=\sum_{n=0}^\infty \vartheta_n(x)\, \frac{t^n}{n!},
\ (\mathbb{C}\times B(0,1/2))$
& $\Theta$ & $\eqref{E:gen_rev_bessel_poly_gf}^\ast$\\ \hline
$\displaystyle \frac{1}{\sqrt{1-2t}} \exp\big[x(1-\sqrt{1-2t})\big]=\sum_{n=0}^\infty
\theta_n(x)\, \frac{t^n}{n!}, \ (\mathbb{C}\times B(0,1/2))$
& $\Theta$ & \eqref{E:rev_bessel_poly_gf}\\ \hline
$\displaystyle \frac{e^{-i\pi x}}{2i\sin\pi x\Gamma(-x)}\int_{-\infty}^{(0+)}e^\lambda(-\lambda)^{-x-1} \frac{1}{\sqrt{1-2t}} \exp\big[\lambda(1-\sqrt{1-2t})\big]\,d\lambda=\sum_{n=0}^\infty \theta_n^\Delta (x)\, \frac{t^n}{n!}, \ (\mathbb{C}\times B(0,1/2))$
& $\Theta$ & $\eqref{E:bessel_difference_poly_gf}^\ast$\\ \hline
$\displaystyle C_1\exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)
+C_2 \exp\left(\frac{1+(1-2xt)^{1/2}}{x}\right)
=\sum_{n=0}^\infty
\mathscr{Y}_{n-1}(x)\, \frac{t^n}{n!}, \ (|t|<1/|2x|)$
& $\mathcal{Y}$ & $\eqref{E:gen_bessel_poly_gf}^\ast$\\ \hline
$\displaystyle \exp\left(\frac{1-(1-2xt)^{1/2}}{x}\right)
=\sum_{n=0}^\infty y_{n-1}(x)\, \frac{t^n}{n!},\ (|t|<1/|2x|)$
& $\mathcal{Y}$ & \eqref{E:bessel p_gf-1}\\ \hline
$\displaystyle \frac{e^{-i\pi x}}{2i \sin \pi x\Gamma(-x)}\int_{-\infty}^{(0+)}
e^{\lambda+\frac{1-\sqrt{1-2\lambda t}}{\lambda} }
(-\lambda)^{-x-1}d\lambda
=\sum_{n=0}^\infty y^{\Delta}_{n-1}(x)\, \frac{t^n}{n!}, \ (\mathbb{C}\times B(0,1/2))$
& $\mathcal{Y}$ & $\eqref{E:bessel dp_gf-1}^\ast$\\ \hline
$\displaystyle x^{-\frac12}
\big[C_1\cos\sqrt{x^2\pm 2xt}+C_2\sin \sqrt{x^2\pm 2xt} \big]=\sum_{n=0}^\infty \mathscr{C}_{\mp n\pm \frac{1}{2}}(x)\, \frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_\pm$ & $\eqref{E: general half bessel_gf}^\ast$\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi x}} \cos(x^2-2xt)^{\frac{1}{2}}
=\sum_{n=0}^\infty J_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!}$,\
($|t|<1/|2x|$) & $\mathcal{G}_-$ & \eqref{E: first half bessel_gf}\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi x}} \sin(x^2+2xt)^{\frac{1}{2}}
=\sum_{n=0}^\infty J_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_+$ &\eqref{E: first half bessel_gf}\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi x}}\sin(x^2-2xt)^{\frac{1}{2}}
=\sum_{n=0}^{\infty} Y_{n-\frac{1}{2}}(x)\,\frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_-$ & \eqref{E: second half bessel gf}\\ \hline
$\displaystyle -\sqrt{\frac{2}{\pi x}} \cos(x^2+2xt)^{\frac{1}{2}}
=\sum_{n=0}^{\infty} Y_{-n+\frac{1}{2}}(x)\,\frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_+$ &\eqref{E: second half bessel gf}\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi x}}\cos(-x^2-2xt)^{\frac{1}{2}}=
\sum_{n=0}^{\infty} I_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_-$ &\eqref{E: first modify half bessel gf}\\ \hline
$\displaystyle -i\sqrt{\frac{2}{\pi x}}\sin(-x^2-2xt)^{\frac{1}{2}}=
\sum_{n=0}^{\infty} I_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_+$ &\eqref{E: second half bessel gf}\\ \hline
$\displaystyle \sqrt{\frac{\pi}{2 x}}e^{i (-x^2+2xt)^{1/2}}= \sum_{n=0}^{\infty} K_{n-\frac{1}{2}}(x)\, \frac{t^n}{n!}= \sum_{n=0}^{\infty} K_{-n+\frac{1}{2}}(x)\, \frac{t^n}{n!}$,
\ ($|t|<1/|2x|$) & $\mathcal{G}_\mp$ &\eqref{E: second modify half bessel gf}\\ \hline
$\displaystyle {}\int_{-\infty}^{(0+)}
\lambda^{-\frac12} \frac{\big[C_1(x)\sin \sqrt{\lambda^2\mp 2\lambda t} +C_2(x)\cos \sqrt{\lambda^2\mp 2\lambda t}\big]}{2i\sin(\pi x)\Gamma(-x)} e^{-i\pi x+\lambda} (-\lambda)^{-x-1}\, d\lambda
=\sum_{n=0}^\infty \mathscr{C}^\Delta_{\pm n\mp \frac{1}{2}}(x)\, \frac{t^n}{n!}$,
($\mathbb{C}^2$)
& $\mathcal{G}_\mp$ &$\eqref{E:delta_glaisher_gf}^\ast$\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{-\infty}^{(0+)}
\frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\cos\sqrt{\lambda ^2-2\lambda t}}{\Gamma(-x)}\, d\lambda
=\sum_{n=0}^{\infty}J^{\Delta}_{n-\frac{1}{2}}(x)\frac{t^n}{n!}$,
($\mathbb{C}\times\mathbb{C}$) & $\mathcal{G}_-$ & $\eqref{E: gf-Delta_glaisher_cos}^\ast$\\ \hline
$\displaystyle \sqrt{\frac{2}{\pi}}\dfrac{e^{-i\pi x}} {2i \sin(\pi x)}
\int_{-\infty}^{(0+)}
\frac{e^{\lambda}
(-\lambda)^{-x-1}\lambda^{-\frac{1}{2}}
\sin\sqrt{\lambda ^2+2\lambda t}}{\Gamma(-x)}\, d\lambda
=\sum_{n=0}^{\infty}J^{\Delta}_{-n+\frac{1}{2}}(x)\frac{t^n}{n!}$,
($\mathbb{C}\times\mathbb{C}$) & $\mathcal{G}_+$ & $\eqref{E: gf-Delta_glaisher_sin}^\ast$\\ \hline \end{tabular} }
\noindent{\textbf{Acknowledgement.}} The authors would like to thank Aimo Hinkkanen for pointing out an error found in an earlier draft to the statement of the Theorem \ref{T:Bessel_gf}. We thank Chun-Kong Law for his suggestion to include tables of results obtained. The authors benefit from the critical and constructive comments from Henry Cheng that improve the readability of the original manuscript. Special thanks also go to Yum Tong Siu who spent time to explain to the first author of the importance of $D$-modules in relation to special functions.
\end{document}
\end{document} | arXiv |
channel bandwidth in wireless communication
x ( While bandwidth is officially measured as a frequency (Hz), it … | = The frequency hopping rate is usually chosen to be equal to or faster than the symbol rate. 1 These standards only included voice communications. The use of various MIMO schemes is essential in providing high spectral efficiencies (by enabling SU-MIMO/MU-MIMO) and greater coverage (via beam-forming). 1 ( For example, a signal-to-noise ratio of 30 dB corresponds to a linear power ratio of + N Typically only FM, and no other form of wireless communication is conducted in this range. ( The off-chip electrical channel density currently faces hard packaging limits that are the largest impediment toward increasing the off-chip bandwidth of processors. X Time localization Time localization is important to efficiently enable (dynamic) TDD and support low applications. h C Y 2 1 , When the sensors are scattered throughout a person's body, WPANs are transformed to WBANs. , 1 having an input alphabet A WLAN is a ubiquitous and broadband wireless resource that uses low-bandwidth channels that meet the requirements for reliable and robust communication with speeds of up to 54 Mb/s. Uplink transmission from MS to BS. {\displaystyle p_{X}(x)} 1 However, due to the relaying feature the coverage of LTE can be further extended. 0 An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: C is measured in bits per second if the logarithm is taken in base 2, or nats per second if the natural logarithm is used, assuming B is in hertz; the signal and noise powers S and N are expressed in a linear power unit (like watts or volts2). X 1 , BW = the receiver noise bandwidth, in Hz, for the communications channel being evaluated m i = the peak modulation of the light source by each RF signal As an example, if a source has an RIN of −160 dB/Hz and is modulated 3% (typical for a 77-channel system) by an NTSC video channel whose noise is measured in a 4-MHz bandwidth, the C/N will be approximately 60.5 dB. The UG channel bandwidth and capacity are vital parameters in wireless underground communication system design. ) H The local oscillating signal at the receiving end is also an FH signal with the hopping law that is the same with the FH signal in the transmitting end, but a frequency difference fi exists between them that is just right to equal IF at the receiving end. Y R If Q is a uniform quantizer of step Δ, then |x – Q(x) | ≤ Δ/2; and if |x| < Δ/2, then Q(x) = 0. be a random variable corresponding to the output of ) , {\displaystyle \lambda } 1 ( , 2 ∑ The application of FPGA allows the user to vary the number of real-time channel response averages, channel sampling interval, and duration of measurement. Y X 1 and ( ( ¯ y advertisement. , 1 p p Y Let I 2 Index TermsŠ Channel, Coding, communication, Diversity, Multi Antenna, Receiver, transmitter, Wireless,. ( + 2 ) − WBANs (along with WPANs) will be more thoroughly covered later. 1 Robustness to channel frequency selectivity Typically, wideband wireless channels are strongly frequency selective and robustness to frequency selectivity is fundamental to support high throughput communication over wideband channels. 1 Hussein T. Mouftah, Melike Erol-Kantarci, in Handbook of Green Information and Communication Systems, 2013. {\displaystyle X_{2}} 1 h These transmission categories are briefly described in the next sections. The peak data rates for LTE are around 300 Mbps at the downlink and 80 Mbps at the uplink with 20 MHz channel bandwidth and 4 × 4 MIMO antennas. p 2 ( ¯ p Chapter 10 describes the implementation of audio transform codes. P p ) The more bandwidth there is available, the more bits per second can be sent on the channel. By the introduction of LTE and LTE-A, cellular networks will be capable of carrying the high volume of smart meter data, in addition to the data of phasor measurement units (PMUs). 2 | I Y {\displaystyle H(Y_{1},Y_{2}|X_{1},X_{2})=H(Y_{1}|X_{1})+H(Y_{2}|X_{2})} Y as: H {\displaystyle Y_{1}} 1 1 X p For high mobility applications, peak data rates will be around 100 Mbps in LTE-A. Time-division multiplexing (TDM) operates by giving the entire channel bandwidth to a stream of characters or bits from each of the low-speed channels for a small segment of time, as illustrated in Fig. P 1 y 2 Y For example, in home applications the idea of a smart home is beginning to unfold, where domestic electric appliances, like vacuum cleaners, lights, or microwave ovens have sensors that communicate one with the other internally, but also remotely from the Internet, and allow the user to gain control over them. {\displaystyle n} ( . Communication outages are known to occur during natural disasters or terrorist attacks even without the additional data load of the smart grid. = 2 1 1 as X 2 2 Y I Y x , Jason Orcutt, ... Vladimir Stojanović, in Optical Fiber Telecommunications (Sixth Edition), 2013. is linear in power but insensitive to bandwidth. ( ) 0 , 10 Compression is thus a sparse approximation problem. I {\displaystyle \epsilon } In particular, Chapter 3 is centered on the point-to-point communication scenario and there the focus is on diversity as a way to mitigate the adverse effect of fading. Some of the above frequency bands will be used only in certain countries and regions while some of them are allocated for global use as indicated above. R {\displaystyle {\mathcal {Y}}_{1}} | , y {\displaystyle p_{X,Y}(x,y)} For health monitoring, there are applications that provide an easy-to-use interface for disabled persons or diagnostics and drug administration in hospitals. X X 2 ( I Copyright © 2021 Elsevier B.V. or its licensors or contributors. 2 [4] It means that using two independent channels in a combined manner provides the same theoretical capacity as using them independently. 2 = X By summing this equality over all Flexibility and scalability Flexibility and scalability of the waveform is important to enable diverse use cases and deployment scenarios. This loss may be due to several reasons, but let's leave that for now. , for N The key result states that the capacity of the channel, as defined above, is given by the maximum of the mutual information between the input and output of the channel, where the maximization is with respect to the input distribution. W 1 1 × N {\displaystyle {\mathcal {Y}}_{1}} ( The best candidate technologies for WWANs are considered to be Worldwide Interoperability for Microwave Access (WiMAX) and fourth generation (4G) and beyond 4G cellular networks. ( To address this problem, many technological solutions have been proposed. H | Due to the low-bandwidth requirements of WPAN applications, the connections used in WPANs have low bandwidth and are cost-effective and flexible. x 2 2 2 1 x y {\displaystyle p_{2}} , The relaying concept is specifically useful for low density deployments. y | , {\displaystyle {\bar {P}}} The dominant packaging technology for CPUs, flip-chip bonding, will enable scaling-limited electrical connections at a pitch of approximately 100 μm in a 2D array. Finally, the transmission signal will be recovered by passing through the demodulation for the IF signal. ) X is logarithmic in power and approximately linear in bandwidth. ( given The data can be forwarded through point-to-point links between the source and the destination or can be relayed through a mesh network via multiple hops to a sink or gateway node for processing and (further) redistribution to other nodes on the network or even outside of it, eg, Internet. ) 1 be the conditional probability distribution function of 2 X Assuming a current limit of 100 mA per pin in a fine pin-pitch package [6], the 100 A required for a 100 W processor running at 1 V consumes 2000 pins. ( + + Note that these data rates are expected to be valid for low mobility. ; Channel Bandwidth – the range of signal bandwidths allowed by a communication channel without significant loss of energy (attenuation). 2 , p ( , depends on the random channel gain In any signaling interval, the transmitted signal occupies one or more of the available frequency slots. , ( When the SNR is small (SNR << 0 dB), the capacity x t 1 FHSS signals are primarily used in AJ and CDMA systems. , = ) , {\displaystyle {\mathcal {Y}}_{2}} | 2 P Abstract: The relatively unused millimeter-wave (mmWave) spectrum offers excellent opportunities to increase mobile capacity due to the enormous amount of available raw bandwidth. Provided the pseudorandom codes between the transmitting and receiving ends are synchronized to each other, the hopping frequencies generated by the frequency synthesizer at both ends will also be synchronized, thus the output from the mixer will be IF signal. X ( {\displaystyle Y_{1}} ( It is developed between sensors and gateways that are scattered at a distance of 30–50 m … ) 1 ) through the channel ( p Y ¯ Durability: Ambient, body, and environmental sensors are subject to various weather challenges and hazards but need to be able to maintain their operation and functionality for long periods of time, regardless of the conditions that they are facing. {\displaystyle C(p_{1})} ) ) Chapter 4 looks at cellular wireless networks as a whole and introduces several multiple access and interference management techniques. {\displaystyle Y_{2}} 2 Channel capacity in wireless communications, AWGN Channel Capacity with various constraints on the channel input (interactive demonstration), Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Channel_capacity&oldid=995602672, Articles needing additional references from January 2008, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 December 2020, at 23:10. Together, these sub-carriers overlap to fully utilize the 16.25 MHz channel bandwidth dedicated per channel. ( 2 Frequency localization is not a major performance indicator at high frequencies where large channel bandwidths are available. The high speed scenarios are relevant in large cell deployments. The capacity of the frequency-selective channel is given by so-called water filling power allocation. X H 1 Depending on the occasion, multiple sinks could also be used for the processing procedures. ( 2 WPAN is a dedicated network that allows transfer of information between various electronic devices within a very small area, including home electrical appliances, smart meters and health monitoring systems, energy management devices, and smart wearables. X be two independent random variables. X What is the maximum allowable path loss? 2 1 , 1 X {\displaystyle I(X_{1},X_{2}:Y_{1},Y_{2})\geq I(X_{1}:Y_{1})+I(X_{2}:Y_{2})} ( Because of the difficulty of maintaining phase references as the frequency hops, FSK modulation and non-coherent demodulation are normally used in FHSS, rather than PSK and coherent demodulation. : , two probability distributions for C ) X , x This value is known as the Robustness to Power Amplifier Non-linearity The impact of the nonlinear PA increases as the signal bandwidth increases (see Chapter 4). 2 This means that the "information" about a sparse representation is mostly geometric. More formally, let 2 Y P The highest-frequency channel (165) operates on 5.825 GHz. 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Winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.
Not to be confused with Map winding number.
Mathematical analysis → Complex analysis
Complex analysis
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Basic theory
• Zeros and poles
• Cauchy's integral theorem
• Local primitive
• Cauchy's integral formula
• Winding number
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Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory).
Intuitive description
Suppose we are given a closed, oriented curve in the xy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin.
When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three.
Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be any integer. The following pictures show curves with winding numbers between −2 and 3:
$\cdots $
−2 −1 0
$\cdots $
1 2 3
Formal definition
Let $\gamma :[0,1]\to \mathbb {C} \setminus \{a\}$ :[0,1]\to \mathbb {C} \setminus \{a\}} be a continuous closed path on the plane minus one point. The winding number of $\gamma $ around $a$ is the integer
${\text{wind}}(\gamma ,a)=s(1)-s(0),$
where $(\rho ,s)$ is the path written in polar coordinates, i.e. the lifted path through the covering map
$p:\mathbb {R} _{>0}\times \mathbb {R} \to \mathbb {C} \setminus \{a\}:(\rho _{0},s_{0})\mapsto a+\rho _{0}e^{i2\pi s_{0}}.$
The winding number is well defined because of the existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of $p$ are of the form $\rho _{0}\times (s_{0}+\mathbb {Z} )$ (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed.
Alternative definitions
Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above:
Alexander numbering
A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865[1] and again independently by James Waddell Alexander II in 1928.[2] Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve).
Differential geometry
In differential geometry, parametric equations are usually assumed to be differentiable (or at least piecewise differentiable). In this case, the polar coordinate θ is related to the rectangular coordinates x and y by the equation:
$d\theta ={\frac {1}{r^{2}}}\left(x\,dy-y\,dx\right)\quad {\text{where }}r^{2}=x^{2}+y^{2}.$
Which is found by differentiating the following definition for θ:
$\theta (t)=\arctan {\bigg (}{\frac {y(t)}{x(t)}}{\bigg )}$
By the fundamental theorem of calculus, the total change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral:
${\text{wind}}(\gamma ,0)={\frac {1}{2\pi }}\oint _{\gamma }\,\left({\frac {x}{r^{2}}}\,dy-{\frac {y}{r^{2}}}\,dx\right).$
The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first de Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number.
Complex analysis
Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem). In the context of complex analysis, the winding number of a closed curve $\gamma $ in the complex plane can be expressed in terms of the complex coordinate z = x + iy. Specifically, if we write z = reiθ, then
$dz=e^{i\theta }dr+ire^{i\theta }d\theta $
and therefore
${\frac {dz}{z}}={\frac {dr}{r}}+i\,d\theta =d[\ln r]+i\,d\theta .$
As $\gamma $ is a closed curve, the total change in $\ln(r)$ is zero, and thus the integral of $ {\frac {dz}{z}}$ is equal to $i$ multiplied by the total change in $\theta $. Therefore, the winding number of closed path $\gamma $ about the origin is given by the expression[3]
${\frac {1}{2\pi i}}\oint _{\gamma }{\frac {dz}{z}}\,.$
More generally, if $\gamma $ is a closed curve parameterized by $t\in [\alpha ,\beta ]$, the winding number of $\gamma $ about $z_{0}$, also known as the index of $z_{0}$ with respect to $\gamma $, is defined for complex $z_{0}\notin \gamma ([\alpha ,\beta ])$ as[4]
$\mathrm {Ind} _{\gamma }(z_{0})={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z_{0}}}={\frac {1}{2\pi i}}\int _{\alpha }^{\beta }{\frac {\gamma '(t)}{\gamma (t)-z_{0}}}dt.$
This is a special case of the famous Cauchy integral formula.
Some of the basic properties of the winding number in the complex plane are given by the following theorem:[5]
Theorem. Let $\gamma :[\alpha ,\beta ]\to \mathbb {C} $ :[\alpha ,\beta ]\to \mathbb {C} } be a closed path and let $\Omega $ be the set complement of the image of $\gamma $, that is, $\Omega :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])$ :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])} . Then the index of $z$ with respect to $\gamma $,
$\mathrm {Ind} _{\gamma }:\Omega \to \mathbb {C} ,\ \ z\mapsto {\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z}},$
is (i) integer-valued, i.e., $\mathrm {Ind} _{\gamma }(z)\in \mathbb {Z} $ for all $z\in \Omega $; (ii) constant over each component (i.e., maximal connected subset) of $\Omega $; and (iii) zero if $z$ is in the unbounded component of $\Omega $.
As an immediate corollary, this theorem gives the winding number of a circular path $\gamma $ about a point $z$. As expected, the winding number counts the number of (counterclockwise) loops $\gamma $ makes around $z$:
Corollary. If $\gamma $ is the path defined by $\gamma (t)=a+re^{int},\ \ 0\leq t\leq 2\pi ,\ \ n\in \mathbb {Z} $, then $\mathrm {Ind} _{\gamma }(z)={\begin{cases}n,&|z-a|<r;\\0,&|z-a|>r.\end{cases}}$
Topology
In topology, the winding number is an alternate term for the degree of a continuous mapping. In physics, winding numbers are frequently called topological quantum numbers. In both cases, the same concept applies.
The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps $S^{1}\to S^{1}:s\mapsto s^{n}$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the group of the integers, Z; and the winding number of a complex curve is just its homotopy class.
Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.
Turning number
One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated at the beginning of this article has a winding number of 3, because the small loop is counted.
This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map.
This is called the turning number, rotation number,[6] rotation index[7] or index of the curve, and can be computed as the total curvature divided by 2π.
Polygons
Further information: Density (polytope) § Polygons
In polygons, the turning number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q.
Space curves
Turning number cannot be defined for space curves as degree requires matching dimensions. However, for locally convex, closed space curves, one can define tangent turning sign as $(-1)^{d}$, where $d$ is the turning number of the stereographic projection of its tangent indicatrix. Its two values correspond to the two non-degenerate homotopy classes of locally convex curves.[8] [9]
Winding number and Heisenberg ferromagnet equations
The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc. Solutions of the last equations are classified by the winding number or topological charge (topological invariant and/or topological quantum number).
Applications
Point in polygon
Further information: Point in polygon § Winding number algorithm
A point's winding number with respect to a polygon can be used to solve the point in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not.
Generally, the ray casting algorithm is a better alternative to the PIP problem as it does not require trigonometric functions, contrary to the winding number algorithm. Nevertheless, the winding number algorithm can be sped up so that it too, does not require calculations involving trigonometric functions.[10] The sped-up version of the algorithm, also known as Sunday's algorithm, is recommendable in cases where non-simple polygons should also be accounted for.
See also
• Argument principle
• Coin rotation paradox
• Linking coefficient
• Nonzero-rule
• Polygon density
• Residue theorem
• Schläfli symbol
• Topological degree theory
• Topological quantum number
• Twist (mathematics)
• Wilson loop
• Writhe
References
1. Möbius, August (1865). "Über die Bestimmung des Inhaltes eines Polyëders". Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Klasse. 17: 31–68.
2. Alexander, J. W. (April 1928). "Topological Invariants of Knots and Links". Transactions of the American Mathematical Society. 30 (2): 275–306. doi:10.2307/1989123. JSTOR 1989123.
3. Weisstein, Eric W. "Contour Winding Number". MathWorld. Retrieved 7 July 2022.
4. Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. p. 201. ISBN 0-07-054235-X.
5. Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill. p. 203. ISBN 0-07-054234-1.
6. Abelson, Harold (1981). Turtle Graphics: The Computer as a Medium for Exploring Mathematics. MIT Press. p. 24.
7. Do Carmo, Manfredo P. (1976). "5. Global Differential Geometry". Differential Geometry of Curves and Surfaces. Prentice-Hall. p. 393. ISBN 0-13-212589-7.
8. Feldman, E. A. (1968). "Deformations of closed space curves". Journal of Differential Geometry. 2 (1): 67–75. doi:10.4310/jdg/1214501138. S2CID 116999463.
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| Wikipedia |
Topology-based representative datasets to reduce neural network training resources
Rocio Gonzalez-Diaz1,
Miguel A. Gutiérrez-Naranjo2 &
Eduardo Paluzo-Hidalgo ORCID: orcid.org/0000-0002-4280-59451
Neural Computing and Applications volume 34, pages 14397–14413 (2022)Cite this article
One of the main drawbacks of the practical use of neural networks is the long time required in the training process. Such a training process consists of an iterative change of parameters trying to minimize a loss function. These changes are driven by a dataset, which can be seen as a set of labeled points in an n-dimensional space. In this paper, we explore the concept of a representative dataset which is a dataset smaller than the original one, satisfying a nearness condition independent of isometric transformations. Representativeness is measured using persistence diagrams (a computational topology tool) due to its computational efficiency. We theoretically prove that the accuracy of a perceptron evaluated on the original dataset coincides with the accuracy of the neural network evaluated on the representative dataset when the neural network architecture is a perceptron, the loss function is the mean squared error, and certain conditions on the representativeness of the dataset are imposed. These theoretical results accompanied by experimentation open a door to reducing the size of the dataset to gain time in the training process of any neural network.
The success of the different architectures used in the framework of neural networks is doubtless [1]. The achievements made in areas such as video imaging [2], recognition [3], or language models [4] show the surprising potential of such architectures. In spite of such success, they still have some shortcomings. One of their main drawbacks is the long time needed in the training process. Such a long training time is usually associated with two factors: first, the large amount of weights to be adjusted in the current architectures and second, the huge datasets used to train neural networks. In general, the time needed to train a complex neural network from the scratch is so long that many researchers use pretrained neural networks as, for example, Oxford VGG models [5], Google Inception Model [6], or Microsoft ResNet Model [7]. Other attempts to reduce the training time are, for example, to partition the training task in multiple training subtasks with submodels, which can be performed independently and in parallel [8], to use asynchronous averaged stochastic gradient descent [9], and to reduce data transmission through a sampling-based approach [10]. Besides, in [11], the authors studied how the elimination of "unfavorable" samples improve generalization accuracy for convolutional neural networks. Finally, in [12], an unweighted influence data subsampling method is proposed.
Roughly speaking, a training process consists of searching for a local minimum of a loss function in an abstract space where the states are sets of weights. Each of the training sample batches provides an extremely small change in weights according to the training rules. The aim of such changes is to find the "best" set of weights that minimizes the loss function. If we consider a "geometrical" interpretation of the learning process, such changes can be seen as tiny steps in a multi-dimensional metric space of parameters that follow the direction settled by the gradient of the loss function. In some sense, one may think that two "close" points of the dataset with the same label provide "similar" information to the learning process since the gradient of the loss function on such points is similar.
Such a viewpoint leads us to look for a representative dataset, "close" to and with a smaller number of points than the original dataset but keeping its "topological information," allowing the neural network to perform the learning process taking less time without losing accuracy. Proving such a general result for any neural network architecture is out of the scope of this paper and, probably, it is not possible since the definition of neural network is continuously evolving over time. Due to such difficulties, we will begin in this paper by proving the usefulness of representative datasets in the perceptron case; that is, we formally prove, for the perceptron case, that the accuracy of a neural network evaluated on the representative dataset is similar to the accuracy of the neural network evaluated on the original dataset under some constraints on the representativeness of the dataset. Besides, experimental evidence indicates that neural networks trained on representative datasets perform similar to neural networks trained on the original datasets in the case of multi-layer neural networks.
Moreover, in order to "keep the shape" of the original dataset, the concept of representative datasets is associated with a notion of nearness independent of isometric transformations. As a first approach, the Gromov–Hausdorff distance is used to measure the representativeness of the dataset. Nonetheless, as the Gromov–Hausdorff distance complexity is an open problemFootnote 1, the bottleneck distance between persistence diagrams [14] is used instead as a lower bound to the Gromov–Hausdorff distance since its time complexity is cubic on the size of the dataset (see [15]).
The paper is organized as follows. In Sect. 2, basic definitions and results from neural networks and computational topology are given. The notion of representative datasets is introduced in Sect. 3. Persistence diagrams are used in Sect. 4 to measure the representativeness of a dataset. In Sect. 5, the perceptron architecture is studied to theoretically prove that the accuracy of a perceptron evaluated on original and representative datasets coincide under some constraint. In Sect. 6.1, experimental results are provided for the perceptron case showing the good performance of representative datasets, compared to random datasets. In Sect. 6.2, we illustrate experimentally the same fact for multi-layer neural networks. Finally, some conclusions and future work are provided in Sect. 7.
Next, we recall some basic definitions and notations used throughout the paper.
The research field of neural networks is extremely vivid and new architectures are continuously being presented (see, e.g., CapsNets [7], Bidirectional Feature Pyramid Networks [16] or new variants of the Gated Recurrent Units [17, 18]), so the current notion of neural network is far from the classic multi-layer perceptron or radial basis function networks [19].
As a general setting, a neural network is a mapping \({\mathcal {N}}_{w,\Phi }: {\mathbb {R}}^n \rightarrow \llbracket 0,k \rrbracket\) (where \(\llbracket 0,k \rrbracket =\{0,1,\dots ,k\}\)) that depends on a set of weights w and a set of parameters \(\Phi\) which involve the description of the synapses between neurons, layers, activation functions and whatever consideration in its architecture. To train the neural network \({\mathcal {N}}_{w,\Phi }\), we use a dataset which is a finite set of pairs \({\mathcal {D}}= \big \{(x,c_x)\) where point x lies in \(X\subset {\mathbb {R}}^n\) and label \(c_x\) lies in \({\llbracket 0,k \rrbracket }\big \}\), for a fixed integer \(k\in \mathbb {N}\). Observe that it should be satisfied that a point cannot have different labels. The sets X and \({\llbracket 0,k \rrbracket }\) are called, respectively, the set of points and the set of labels in \({\mathcal {D}}\).
To perform the learning process, we use: (1) a loss function which measures the difference between the output of the network (obtained with the current weights) and the desired output; and (2) a loss-driven training method to iteratively update the weights.
Finally, let us introduce the concept of accuracy as a measure to evaluate the performance of a neural network.
Definition 1
The accuracy of a neural network \({\mathcal {N}}_{w,\Phi }\) evaluated on a dataset \({\mathcal {D}}= \big \{(x,c_x): x\in X\subset {\mathbb {R}}^n\) and \(c_x\in \llbracket 0,k \rrbracket \big \}\), is defined as:
$$\begin{aligned} {\mathbb {A}}({\mathcal {D}},{\mathcal {N}}_w)=\frac{1}{|X|}\sum _{x\in X}I_w(x), \end{aligned}$$
where, for any \(x\in X\),
$$\begin{aligned} {\mathcal {I}}_w(x) = \left\{ \begin{array}{cl} 1 &{} if\ c_x = {\mathcal {N}}_{w,\Phi }(x), \\ 0 &{} otherwise. \\ \end{array} \right. \end{aligned}$$
Persistent homology
In this paper, the representativeness of a dataset will be measured using methods from the recent developed area called computational topology whose main tool is persistent homology. A detailed presentation of this field can be found in [14].
Homology provides mathematical formalism to count holes where holes refer to connected components, tunnels, cavities, and so on, being the q-dimensional homology group the mathematical representation for the q-dimensional holes in a given space. Persistent homology is usually computed when the homology cannot be determined. An example of the latter appears when a surface is sampled by a point cloud.
Persistent homology is based on the concept of filtration, which is an increasing sequence of simplicial complexes. The building blocks of a simplicial complex are q-simplices, being a 0-simplex a point, a 1-simplex a line segment, a 2-simplex a triangle, a 3-simplex a tetrahedron, and so on. An example of filtration is the Vietoris–Rips filtration (see [20]) that is built by successively increasing the radius of the balls centered at the points of a given set in an n-dimensional space and joining those vertices (points) whose balls intersect forming new simplices.
We say that a q-dimensional hole is born when it appears at a specific time along the filtration and it dies when it merges with another q-dimensional hole at a specific time along the filtration. One of the common graphical representations of births and deaths of the q-dimensional holes over time is the so-called (q-dimensional) persistence diagram which consists of a set of points on the Cartesian plane. This way, a point of a persistence diagram represents the birth and the death of a hole. Since deaths happen only after births, all the points in a persistence diagram lie above the diagonal axis. Furthermore, those points in a persistence diagram that are far from the diagonal axis are candidates to be "topologically significant" since they represent holes that survive for a long time. The so-called bottleneck distance can be used to compare two persistence diagrams.
The (q-dimensional) bottleneck distance between two (q-dimensional) persistence diagrams \(\hbox {Dgm}\) and \(\widetilde{\hbox {Dgm}}\) is:
$$\begin{aligned} d_B(\hbox {Dgm},\widetilde{\hbox {Dgm}})= \inf _{\phi } \;\sup _{\alpha } \Vert \alpha -\phi (\alpha ) \Vert _{\infty } \end{aligned}$$
where \(\alpha \in \hbox {Dgm}\) and \(\phi\) is any possible bijection between \(\hbox {Dgm}\cup \Delta\) and \(\widetilde{\hbox {Dgm}}\cup \Delta\), being \(\Delta\) the set of points in the diagonal axis.
An useful result used in this paper is the following one that connects the Gromov–Hausdorff distance between two metric spaces and the bottleneck distance between the persistence diagrams obtained from their corresponding Vietoris–Rips filtrations. For the sake of brevity, the (q-dimensional) persistence diagram obtained from the Vietoris–Rips filtration computed from a subset X of \({\mathbb {R}}^n\), with \(q\le n\), will be simply called the (q-dimensional) persistence diagram of X and denoted by \(\hbox {Dgm}_q(X)\).
Theorem 1
[21, Theorem 5.2] For any two subsets X and Y of \({\mathbb {R}}^n\), and for any dimension \(q\le n\), the bottleneck distance between the persistence diagrams of X and Y is bounded by the Gromov–Hausdorff distance of X and Y:
$$\begin{aligned} d_B(\hbox {Dgm}_q(X),\hbox {Dgm}_q(Y)) \le 2d_{GH}(X,Y). \end{aligned}$$
Let us recall that \(d_{GH}(X,Y) =\frac{1}{2} \inf _{f,g}\big \{\,d_H (f(X), g(Y))\, \big \},\) where \(d_H (X,Y) = \max \big \{ \,\sup _{x\in X} \;\inf _{y\in Y} \;\Vert x-y\Vert ,\) \(\sup _{y\in Y} \;\inf _{x\in X}\; \Vert x-y\Vert \, \big \},\) and \(f:X\rightarrow Z\) (resp. \(g:Y\rightarrow Z\)) denotes an isometric transformation of X (resp. Y) into some metric space Z. Summing up, we can conclude that
$$\begin{aligned} \frac{1}{2}d_B(\hbox {Dgm}_q(X),\hbox {Dgm}_q(Y))\le d_{GH}(X,Y)\le d_{H}(X,Y). \end{aligned}$$
Representative datasets
In this section, we provide the definition of representative datasets which is independent of the neural network architecture considered. The intuition behind this definition is to keep the "shape" of the original dataset while reducing its number of points.
Firstly, let us introduce the notion of \(\varepsilon\)-representative point.
A labeled point \((\tilde{x},c_{\tilde{x}})\in {\mathbb {R}}^n\times {\llbracket 0,k \rrbracket }\) is \(\varepsilon\)-representative of \((x,c_x)\in {\mathbb {R}}^n\times {\llbracket 0,k \rrbracket }\) if \(c_x=c_{\tilde{x}}\) and \(\Vert x-\tilde{x}\Vert \le \varepsilon\), where \(\varepsilon \in {\mathbb {R}}\) is the representation error. We denote \(\tilde{x}\approx _\varepsilon x\).
The next step is to define the concept of \(\varepsilon\)-representative dataset. Notice that if a dataset can be correctly classified by a neural network, any isometric transformation of such dataset can also be correctly classified by the neural network (after adjusting the weights). Therefore, the definition of \(\varepsilon\)-representative dataset should be independent of such transformations. The concept of \(\lambda\)-balanced \(\varepsilon\)-representative datasets is also introduced and it will be used in Sect. 5 to ensure that similar accuracy results are obtained when a trained perceptron is evaluated on a representative dataset rather than on the original dataset.
A dataset \(\tilde{{\mathcal {D}}}= \big \{(\tilde{x},c_{\tilde{x}}) : \tilde{x}\in \tilde{X}\subset {\mathbb {R}}^n\) and \(c_{\tilde{x}}\in {\llbracket 0,k \rrbracket }\big \}\) is \(\varepsilon\)-representative of \({\mathcal {D}} = \big \{(x,c_x): x\in X\subset {\mathbb {R}}^n\) and \(c_x\in {\llbracket 0,k \rrbracket }\big \}\) if there exists an isometric transformation \(f:\tilde{X}\rightarrow {\mathbb {R}}^n\), such that for any \((x,c_x)\in {\mathcal {D}}\) there exists \((\tilde{x},c_{\tilde{x}})\in \tilde{{\mathcal {D}}}\) satisfying that \(f(\tilde{x})\approx _\varepsilon x\). The minimum of all those possible \(\varepsilon\) for which an isometric transformation exists is called to be optimal. Finally, the dataset \(\tilde{{\mathcal {D}}}\) is said to be \(\lambda\)-balanced if for each \((\tilde{x},c_{\tilde{x}})\in \tilde{\mathcal {D}}\), the set \(\{(x,c_x) : f(\tilde{x})\approx _\varepsilon x \}\) contains \(\lambda\) points and for each \((x,c_x)\in {\mathcal {D}}\) there exists only one \((\tilde{x},c_{\tilde{x}})\in \tilde{{\mathcal {D}}}\) such that \(f(\tilde{x})\approx _\varepsilon x\).
Let us point out that \(\lambda\)-balanced datasets cannot be computed for most datasets when they are required to be subsets of the datasets. Even if it is too restrictive, in Sect. 5, we will provide several theoretical results using that assumption. Nevertheless, we will prove experimentally in Sect. 6 that, without that assumption, \(\varepsilon\)-representative datasets still perform well.
Proposition 1
Let \(\tilde{\mathcal {D}}\) be an \(\varepsilon\)-representative dataset (with set of points \(\tilde{X}\subset {\mathbb {R}}^n\)) of a dataset \({\mathcal {D}}\) (with set of points \(X\subset {\mathbb {R}}^n\)). Then,
$$\begin{aligned}d_{GH}(X,\tilde{X})\le \varepsilon .\end{aligned}$$
By definition of \(\varepsilon\)-representative datasets, there exists an isometric transformation from \(\tilde{X}\) to \({\mathbb {R}}^n\) where for all \(x\in X\) there exists \(\tilde{x}\in \tilde{X}\) such that \(\Vert x-f(\tilde{x})\Vert \le \varepsilon\). Therefore, \(d_H(X,f(\tilde{X}))\le \varepsilon\). Then, by the definition of the Gromov–Hausdorff distance, \(d_{GH}(X,\tilde{X})\le d_H(X,f(\tilde{X}))\le \varepsilon\). \(\square\)
Notice that the definition of \(\varepsilon\)-representative datasets is not useful when \(\varepsilon\) is "big." The following result, which is a consequence of Proposition 1, provides the optimal value for \(\varepsilon\).
Corollary 1
The parameter \(\varepsilon\) is optimal if and only if \(\varepsilon =d_{GH}(X,\tilde{X})\).
Therefore, one way to discern if a dataset \(\tilde{\mathcal {D}}\) is "representative enough" of \({\mathcal {D}}\) is to compute the Gromov–Hausdorff distance between X and \(\tilde{X}\). If the Gromov–Hausdorff distance is "big," we could say that the dataset \(\tilde{{\mathcal {D}}}\) is not representative of \({\mathcal {D}}\). However, the Gromov–Hausdorff distance is not useful in practice because of its high computational cost. An alternative approach to this problem is given in Sect. 4.
Proximity graph algorithm
In this section, for a given \(\varepsilon >0\), we propose a variant of the proximity graph algorithm [22] to compute an \(\varepsilon\)-representative dataset \(\tilde{\mathcal {D}}\) of a dataset \({\mathcal {D}}=\big \{(x,c_x):x\in X\subset {\mathbb {R}}^n \hbox { and } c_x\in {\llbracket 0,k \rrbracket }\big \}\).
Firstly, a proximity graph is built over X, establishing adjacency relations between the points of X, represented by edges.
Given \(\varepsilon >0\), an \(\varepsilon\)-proximity graph of X is a graph \(G_\varepsilon (X)=(X,E)\) such that if \(x,y\in X\) and \(\Vert x-y\Vert \le \varepsilon\) then \((x,y)\in E\).
See Fig. 1 in which the proximity graph of one of the two interlaced solid torus is drawn for a fixed \(\varepsilon\).
A point cloud sampling two interlaced solid torus and the \(\varepsilon\)-proximity graph of one of them for a fixed \(\varepsilon\).
Secondly, from an \(\varepsilon\)-proximity graph of X, a dominating dataset (also known as a vertex cover) \(\tilde{X}\subseteq X\) is computed satisfying that if \(x\in X\) then \(x\in \tilde{X}\) or there exists \(y\in \tilde{X}\) adjacent to x. We then obtain an \(\varepsilon\)-representative dataset \(\tilde{\mathcal {D}} = \{(\tilde{x},c_{\tilde{x}}):\tilde{x}\in \tilde{X}\) and \((\tilde{x},c_{\tilde{x}})\in {\mathcal {D}}\}\) also called dominating dataset of \({\mathcal {D}}\). Algorithm 1 shows the pseudo-code used in this paper to compute a dominating dataset of \({\mathcal {D}}\).
Here, DominatingSet\((G_\varepsilon (X_c))\) refers to a dominating set obtained from the proximity graph \(G_\varepsilon (X_c)\). Among the existing algorithms in the literature to obtain a dominating set, we will use, in our experiments in Sect. 6, the algorithm proposed in [23] that runs in \(O(|X |\cdot |E|)\). Therefore, the complexity of Algorithm 1 is \(O(|X |^2+|X |\cdot |E|)\) because of the size of the matrix of distances between points and the complexity of the algorithm to obtain the dominating set. Let us observe that the algorithm proposed in this paper is just an example to show how we can compute representative datasets. Other more efficient algorithms to compute representative datasets are left for future work in Sect. 7.
Lemma 1
The dominating dataset \(\tilde{\mathcal {D}}\) obtained by running Algorithm 1 is an \(\varepsilon\)-representative dataset of \({\mathcal {D}}\).
Let us prove that for any \((x,c_x)\in {\mathcal {D}}\) there exists \((\tilde{x},c_{x})\in \tilde{{\mathcal {D}}}\) such that \(x \approx _{\varepsilon } \tilde{x}\). Two possibilities arise:
If \((x,c_x)\in \tilde{{\mathcal {D}}}\), it is done.
If \((x,c_x)\not \in \tilde{{\mathcal {D}}}\), since \(\tilde{X}_{c_x}\) is a dominating dataset of \(G_\varepsilon (X_{c_x})\), then there exists \(\tilde{x}\in \tilde{X}_{c_x}\) such that \(\tilde{x}\) is adjacent to x in \(G_\varepsilon (X_{c_x})\). Therefore, \((\tilde{x},c_x)\in \tilde{{\mathcal {D}}}\) and \(x\approx _{\varepsilon }\tilde{x}\).
\(\square\)
Persistent homology to Infer the representativeness of a dataset
In this section, we show the role of persistent homology as a tool to infer the representativeness of a dataset.
Firstly, from Theorem 1 in page 5, we will establish that the bottleneck distance between persistence diagrams is a lower bound of the representativeness of the dataset.
Let \(\tilde{\mathcal {D}}\) be an \(\varepsilon\)-representative dataset (with set of points \(\tilde{X}\subset {\mathbb {R}}^n\)) of a dataset \({\mathcal {D}}\) (with set of points \(X\subset {\mathbb {R}}^n\)). Let \(\hbox {Dgm}_q(X)\) and \(\hbox {Dgm}_q(\tilde{X})\) be the q-dimensional persistence diagrams of X and \(\tilde{X}\), respectively. Then, for \(q\le n\),
$$\begin{aligned} \frac{1}{2} d_{B}\big (\hbox {Dgm}_q(X),\hbox {Dgm}_q(\tilde{X})\big )\le \varepsilon .\end{aligned}$$
Since \(\tilde{\mathcal {D}}\) is an \(\varepsilon\)-representative dataset of \({\mathcal {D}}\) then \(d_{GH}(X\), \(\tilde{X})\le \varepsilon\) by Proposition 1. Now, by Theorem 1, \(\frac{1}{2}d_B(\hbox {Dgm}_q(X),\hbox {Dgm}_q(Y)) \le d_{GH}(X,Y)\le \varepsilon\).\(\square\)
As a direct consequence of Lemma 2 and the fact that the Hausdorff distance is an upper bound of the Gromov–Hausdorff distance, we have the following result.
Let \(\tilde{\mathcal {D}}\) be an \(\varepsilon\)-representative dataset (with set of points \(\tilde{X}\subset {\mathbb {R}}^n\)) of a dataset \({\mathcal {D}}\) (with set of points \(X\subset {\mathbb {R}}^n\)) where the parameter \(\varepsilon\) is optimal. Let \(\hbox {Dgm}_q(X)\) and \(\hbox {Dgm}_q(\tilde{X})\) be the q-dimensional persistence diagrams of X and \(\tilde{X}\), respectively. Then,
$$\begin{aligned} \frac{1}{2} d_B\big (\hbox {Dgm}_q(X),\hbox {Dgm}_q(\tilde{X}))\big )\le \varepsilon \le d_H\big (X,\tilde{X}\big ).\end{aligned}$$
In order to illustrate the usefulness of this last result, we will discuss a simple example. In Fig. 2a, we can see a subsample of a circumference (the original dataset) together with two classes corresponding, respectively, to the upper and lower part of the circumference. In Fig. 2c, we can see a subset of the original dataset and a decision boundary "very" different to the one given in Fig. 2a. Then, we could say that the dataset shown in Fig. 2c does not "represent" the same classification problem than the original dataset. However, the dataset shown in Fig. 2b could be considered a representative dataset of the original one since both decision boundaries are "similar." This can be determined by computing the Hausdorff distance between the original and the other datasets, and the bottleneck distance between the persistence diagrams of the corresponding datasets (see the values shown in Table 1). Using Corollary 2, we can infer that \(0.08\le \varepsilon _1\le 0.18\) for the dataset given in Fig. 2b and \(0.13\le \varepsilon _2\le 0.3\) for the dataset given in Fig. 2c. Therefore, the dataset given in Fig. 2b can be considered "more" representative of the dataset shown in Fig. 2a than the dataset given in Fig. 2c, as expected.
Illustration of a binary classification problem and the representative dataset concept
Table 1 The 0-dimensional bottleneck distance (\(d_{B0}\)), the 1-dimensional bottleneck distance (\(d_{B1}\)), and the Hausdorff distance (\(d_H\)) between the persistence diagrams of the dataset given in Fig. 2a and the datasets given in Fig. 2b and c, respectively
In Table 2, we show the output of Algorithm 1 applied to the dataset pictured in Fig. 2a and different values of \(\varepsilon\), in order to obtain different dominating datasets. Let us observe that, depending on the value of the parameter \(\varepsilon\), the size of the resulting dominating dataset is different. Finally, let us remark that the values of the parameter used in Algorithm 1 does not correspond, in general, to the optimal \(\varepsilon\) (see Definition 4).
Table 2 The 0-dimensional bottleneck distance (\(d_{B_0}\)) and the Hausdorff distance (\(d_H\)) between the persistence diagrams of the dataset (\({\mathcal {D}}\)) given in Fig. 2a, composed of 22 points, and the dominating datasets (\(\tilde{\mathcal {D}}\)) obtained applying Algorithm 1 for different values of \(\varepsilon\), and a random dataset (\({\mathcal {R}}\)) of the same size than the corresponding dominating dataset
Theoretical results on the perceptron case
One of the simplest neural network architecture is the perceptron. Our goal in this section is to formally prove that the accuracy of a perceptron evaluated on the original dataset and on its representative dataset are equivalent when we impose certain conditions on the representativeness.
For the sake of simplicity, in this section, we will restrict our interest to a binary classification problem, although our approach is valid for any classification problem. Therefore, our input is a binary dataset \({\mathcal {D}} = \big \{(x,c_x): x\in X\subset {\mathbb {R}}^n\) and \(c_x\in \{0,1\}\big \}\). Besides, we will assume in this section that the training process tries to minimize the following error function:
$$\begin{aligned}{\mathbb {E}}(w,{{\mathcal {D}}}) = \frac{2}{|X |} \sum _{x\in X}E_x(w),\end{aligned}$$
where, for \((x,c_x)\in {\mathcal {D}}\) and \(w\in {\mathbb {R}}^{n+1}\),
$$\begin{aligned} E_x(w) = \frac{1}{2}(c_x - y_{w}(x))^2\end{aligned}$$
is the loss function considered, called the mean squared error (MSE). An example of such a training process is the gradient descent training algorithm.
First, let us introduce the definition of a perceptron.
A perceptron \({\mathcal {N}}_w:{\mathbb {R}}^n\rightarrow \{0,1\}\), with weights \(w =(w_0,w_1,\dots ,w_n)\in {\mathbb {R}}^{n+1}\), is defined as:
$$\begin{aligned} {\mathcal {N}}_w(x) = \left\{ \begin{array}{cc} 1 &{} if\ y_w(x) \ge \frac{1}{2}, \\ 0 &{} otherwise; \\ \end{array} \right. \end{aligned}$$
being \(y_{w}: {\mathbb {R}}^{n} \rightarrow \, (0,1)\) defined as
$$\begin{aligned} y_{w}(x)=\sigma (wx) \end{aligned}$$
where, for \(x=(x_1,\dots ,x_n) \in {\mathbb {R}}^n\),
$$\begin{aligned} wx=w_0+ w_1x_1+\dots + w_n x_n, \end{aligned}$$
and \(\sigma : {\mathbb {R}} \rightarrow (0,1)\), defined as
$$\begin{aligned} \sigma (z) =\frac{1}{1+ e^{-z}}, \end{aligned}$$
is the sigmoid function.
In the previous definition, let us point out that the condition \(y_w(x)\ge \frac{1}{2}\) is the same as the condition \(wx\ge 0\) that usually appears in the definition of perceptron.
A useful property of the sigmoid function is the easy expression of its derivative. Let \(\sigma ^m\) denote the composition \(\sigma ^m=\sigma {\mathop {\cdots }\limits ^{{m\hbox {-times}}}}\sigma\).
If \(m\in \mathbb {N}\) and \(z\in {\mathbb {R}}\) then
$$\begin{aligned}0<(\sigma ^m)'(z)= m\sigma ^{m}(z)(1-\sigma (z))\le \left( \frac{m}{m+1}\right) ^{m+1}. \end{aligned}$$
Firstly, let us observe that \((\sigma ^m)'(z)= m\sigma ^{m}(z)(1-\sigma (z))>0\) since \(0<\sigma (z)<1\) for all \(z\in {\mathbb {R}}\). Secondly, let us find the local extrema of \((\sigma ^m)'\) by computing the roots of its derivative:
$$\begin{aligned} (\sigma ^m)''(z) =m\sigma ^{m}(z)(1-\sigma (z))(m-(m+1)\sigma (z)). \end{aligned}$$
Now, \((\sigma ^m)''(z)=0\) if and only if \(m-(m+1)\sigma (z)=0\). The last expression vanishes at \(z=\log (m)\). Besides, \((\sigma ^m)''(z)>0\) if and only if \(m-(m+1)\sigma (z)>0\) which is true for all \(z\in (-\infty ,\log (m))\). Analogously, \((\sigma ^m)''(z)<0\) for all \(z\in (\log (m),+\infty )\), concluding that \(z= \log (m)\) is a global maximum. Finally, \((\sigma ^m)'(\log (m))=\left( \frac{m}{m+1}\right) ^{m+1}\) concluding the proof. \(\square\)
From now on, we will consider that the associated isometric transformation f by which the dataset is \(\varepsilon\)-representative is applied to the representative dataset. Therefore, by abuse of notation, \(\tilde{x}\) will mean \(f(\tilde{x})\). Analogously, \(\tilde{X}\) will mean \(f(\tilde{X})\) and \(\tilde{\mathcal {D}}\) will mean \(f(\tilde{\mathcal {D}})\).
In the following lemma, we prove that the difference between the outputs of the function \(y^m_w\) evaluated at a point x and at its \(\varepsilon\)-representative point \(\tilde{x}\) depends on the weights w and the parameter \(\varepsilon\).
Let \(w\in {\mathbb {R}}^{n+1}\) and x, \(\tilde{x}\in {\mathbb {R}}^n\) with \(\tilde{x}\approx _{\varepsilon } x\). Then,
$$\begin{aligned} \Vert y^m_{w}(\tilde{x})-y^m_{w}(x)\Vert \le \rho _{m} \Vert w\Vert _*\varepsilon , \end{aligned}$$
$$\begin{aligned} \rho _{m} =\rho _{(wx,w\tilde{x},m)} = {\left\{ \begin{array}{ll} (\sigma ^m)'(z), &{} \quad \hbox {if } \log (m)< z,\\ (\sigma ^m)'(\tilde{z}), &{}\quad \hbox {if } \tilde{z} < \log (m),\\ (\sigma ^m)'(\log (m)), &{}\quad \hbox {otherwise.} \end{array}\right. } \end{aligned}$$
with \(z= \min \{wx,w\tilde{x}\}\) and \(\tilde{z}=\max \{wx,w\tilde{x}\}\).
Let us assume, without loss of generality, that \(wx\le w\tilde{x}\). Then, using the mean value theorem, there exists \(\beta \in (wx,w\tilde{x})\) such that
$$\begin{aligned} y^m_{w}(\tilde{x})-y^m_{w}(x) = (\sigma ^m)'(\beta )(w\tilde{x}-wx). \end{aligned}$$
By Lemma 3, the maximum of \((\sigma ^m)'\) in the interval \([z,\tilde{z}]\) is reached at \(\log (m)\) if \(z<\log (m)<\tilde{z}\), at z if \(\log (m)\le z\), and at \(\tilde{z}\) if \(\tilde{z}\le \log (m)\), with \(z= \min \{wx,w\tilde{x}\}\) and \(\tilde{z}=\max \{wx,w\tilde{x}\}\). Consequently,
$$\begin{aligned} \Vert y^m_{w}(x)-y^m_{w}(x)\Vert \le \rho _m \Vert w(\tilde{x}-x)\Vert . \end{aligned}$$
Applying now the Hölder inequality we obtain:
$$\begin{aligned} \Vert w(\tilde{x}-x)\Vert \le \Vert w\Vert _*\Vert \tilde{x}-x\Vert \le \Vert w\Vert _*\varepsilon . \end{aligned}$$
Replacing \(\Vert w(\tilde{x}-x)\Vert\) by \(\Vert w\Vert _*\varepsilon\) in Eq. (1), we obtain the desired result. \(\square\)
The following result is a direct consequence of Lemma 3 and Lemma 4.
$$\begin{aligned}\Vert y^m_{w}(\tilde{x})-y^m_{w}(x)\Vert \le \left( \frac{m}{m+1}\right) ^{m+1}\Vert w\Vert _*\varepsilon .\end{aligned}$$
The next result establishes under which conditions \(\varepsilon\)-representative points are classified under the same label as the points they represent.
Let \({\tilde{\mathcal {D}}}\) be an \(\varepsilon\)-representative dataset of the binary dataset \({\mathcal {D}}\). Let \({\mathcal {N}}_w\) be a perceptron with weights \(w\in {\mathbb {R}}^{n+1}\). Let \((x,c)\in {\mathcal {D}}\) and \((\tilde{x},c)\in \tilde{\mathcal {D}}\) with \(\tilde{x}\approx _\varepsilon x\). If \(\varepsilon \le \frac{\Vert wx\Vert }{\Vert w\Vert }\) then
$$\begin{aligned}{\mathcal {N}}_w(x)={\mathcal {N}}_w(\tilde{x}).\end{aligned}$$
First, if \(wx=0\), then \(\varepsilon =0\), therefore \(x=\tilde{x}\) and then \({\mathcal {N}}_w(x)={\mathcal {N}}_w(\tilde{x})\). Now, let us suppose that \(wx<0\). Then, \(y_w(x)<\frac{1}{2}\) and \({\mathcal {N}}_w(x)=0\) by the definition of perceptron. Since \(\varepsilon < \frac{\Vert wx\Vert }{\Vert w\Vert }\) then x and \(\tilde{x}\) belong to the same semispace in which the space is divided by the hyperplane \(wx=0\). Therefore, \(w\tilde{x}<0\), then \(y_w(\tilde{x})<\frac{1}{2}\) and finally \({\mathcal {N}}_w(\tilde{x})=0\). Similarly, if \(wx>0\), then \({\mathcal {N}}_w(x)=1={\mathcal {N}}_w(\tilde{x})\), concluding the proof. \(\square\)
By Lemma 5, we can state that if \(\varepsilon\) is "small enough" then the perceptron \({\mathcal {N}}_w\) evaluated on \({\mathcal {D}}\) and \(\tilde{\mathcal {D}}\) will coincide.
In the following results, we will restrict ourselves to \(\lambda\)-balance datasets as a theoretical convenience.
The next result relies on the accuracy (see Definition 1) of a perceptron evaluated on the original dataset and its representative dataset.
Let \(\tilde{\mathcal {D}}\) be a \(\lambda\)-balanced \(\varepsilon\)-representative dataset of the binary dataset \({\mathcal {D}}\). Let \({\mathcal {N}}_w\) be a perceptron with weights \(w\in {\mathbb {R}}^{n+1}\). If \(\varepsilon \le \min \Big \{ \frac{\Vert wx\Vert }{\Vert w\Vert }:\) \((x,c_x)\in {\mathcal {D}}\Big \}\) then
$$\begin{aligned} {{\mathbb {A}}}({\mathcal {D}},{\mathcal {N}}_w)={{\mathbb {A}}}(\tilde{\mathcal {D}},{\mathcal {N}}_w). \end{aligned}$$
Since \(\tilde{\mathcal {D}}\) is \(\lambda\)-balance \(\varepsilon\)-representative of \({\mathcal {D}}\), then \({|X|=\lambda \cdot |\tilde{X}|}\) and we have:
$$\begin{aligned} \begin{aligned} {{\mathbb {A}}}({\mathcal {D}},{\mathcal {N}}_w)-{{\mathbb {A}}}(\tilde{\mathcal {D}},{\mathcal {N}}_w)&= \frac{1}{|{X} |}\sum _{ {x\in X}} I_w(x)-\frac{1}{|{\tilde{X}} |}\sum _{ {\tilde{x}\in \tilde{X}}} I_w(\tilde{x}) \\&=\frac{1}{|{X} |}\sum _{ {x\in X} }\left( I_w(x)-\lambda \cdot I_w(\tilde{x})\right) \\ {}&=\frac{1}{|{X} |}\sum _{ {\tilde{x}\in \tilde{X}}} \;\sum _{x\approx _\varepsilon \tilde{x}}\left( I_w(x)-I_w(\tilde{x})\right) . \end{aligned} \end{aligned}$$
Finally, \(I_w(x)=I_w(\tilde{x})\) for all \(x\approx _{\varepsilon } \tilde{x}\) and \((\tilde{x},c_{\tilde{x}})\in \tilde{\mathcal {D}}\) by Lemma 5 since \(\varepsilon <\frac{\Vert wx\Vert }{\Vert w\Vert }\) for all \((x,c_x)\in {\mathcal {D}}\). \(\square\)
Next, let us compare the two errors \({\mathbb {E}}(w,{{\mathcal {D}}})\) and \({\mathbb {E}}(w,{\tilde{\mathcal {D}}})\) obtained when considering the binary dataset \({\mathcal {D}}\) and its \(\lambda\)-balanced \(\varepsilon\)-representative dataset \(\tilde{\mathcal {D}}\).
Let \(\tilde{\mathcal {D}}\) be a \(\lambda\)-balanced \(\varepsilon\)-representative dataset of the binary dataset \({\mathcal {D}}\). Then:
$$\begin{aligned}\Vert {\mathbb {E}}(w, {{\mathcal {D}}})-{\mathbb {E}}(w, {\tilde{\mathcal {D}}})\Vert \le \frac{1}{|X |}\sum _{x\in X} \big ( 2c_{x}\rho _1+\rho _2\big ) \Vert w (x-\tilde{x})\Vert \end{aligned}$$
where \(\rho _m\) (being \(m=1,2\)) was defined in Lemma 4, and for each addend, \(x\approx _{\varepsilon }\tilde{x}\).
First, let us observe that:
$$\begin{aligned} \begin{aligned}&{\mathbb {E}}(w, {{\mathcal {D}}} )-{\mathbb {E}}(w, {\tilde{\mathcal {D}}} ) \\ =&\frac{1}{|X|}\sum _{x\in X} (c_x-y_{w}(x))^2 -\frac{1}{|\tilde{X}|}\sum _{\tilde{x}\in \tilde{X}} (c_{\tilde{x}}-y_{w}(\tilde{x}))^2\\ =&\frac{1}{|X|\cdot |\tilde{X}|}\Big (|\tilde{X}|\sum _{x\in X}(c_x-y_{w}(x))^2 -|X|\sum _{\tilde{x}\in \tilde{X}}(c_{\tilde{x}}-y_{w}(\tilde{x}))^2 \Big ). \end{aligned} \end{aligned}$$
Now, since \(\tilde{\mathcal {D}}\) is \(\lambda\)-balanced \(\varepsilon\)-representative of \({\mathcal {D}}\) then \(|X |=\lambda \cdot {|\tilde{X} |}\). Therefore,
$$\begin{aligned} \begin{aligned}&\Vert {\mathbb {E}}(w, {{\mathcal {D}}} )-{\mathbb {E}}(w, {\mathcal{\tilde{D}}} )\Vert \\&=\frac{1}{|X|}\; \Vert \sum _{x\in X} 2c_x \big ( y_{\tilde{w}}(\tilde{x})-y_{w}(x)\big ) +y_{w}^2(x)-y_{\tilde{w}}^2(\tilde{x})\big )\Vert \\&\le \frac{1}{|X|} \sum _{x\in X} 2c_x \Vert y_{\tilde{w}}(\tilde{x})-y_{w}(x)\Vert +\Vert y_{w}^2(x)-y_{\tilde{w}}^2(\tilde{x})\Vert , \end{aligned} \end{aligned}$$
where, for each addend, \({{\tilde{x}}}\approx _{\varepsilon } x\). Applying Lemma 4 for \(m=1,2\) to the last expression, we get:
$$\begin{aligned} \Vert {\mathbb {E}}(w, {{\mathcal {D}}} )-{\mathbb {E}}(w, \tilde{\mathcal {D}})\Vert \le \frac{1}{|X|}\sum _{x\in X} \big ( 2c_{x}\rho _1+\rho _2\big ) \Vert w (x-\tilde{x})\Vert . \end{aligned}$$
From this last result, we can infer the following: We can always fix the parameter \(\varepsilon\) "small enough" so that the difference between the error obtained when considering the dataset \({\mathcal {D}}\) and its \(\varepsilon\)-representative dataset is "close" to zero. Fig. 3 aims to provide intuition for this result.
Let \(\delta >0\). Let \(\tilde{\mathcal {D}}\) be a \(\lambda\)-balanced \(\varepsilon\)-representative dataset of the binary dataset \({\mathcal {D}}\). Let \({\mathcal {N}}_w\) be a perceptron with weights \(w\in {\mathbb {R}}^{n+1}\). If \(\varepsilon \le \frac{54}{43\Vert w\Vert _*}\delta\), then
$$\begin{aligned}\Vert {\mathbb {E}}(w, {{\mathcal {D}}})-{\mathbb {E}}(w, {\tilde{\mathcal {D}}})\Vert {\le }\delta .\end{aligned}$$
First, \(\rho _1\le \frac{1}{4}\) and \(\rho _2\le \frac{8}{27}\) by Corollary 3. Second, since \(c_x\in \{0,1\}\), we have:
$$\begin{aligned} \begin{aligned}&\frac{1}{|X |}\sum _{x\in X} \big ( 2c_{x}\rho _1+\rho _2\big ) \Vert w (x-\tilde{x})\Vert \\&\le \frac{1}{|X |} \sum _{x\in X} \Big ( \frac{1}{2}+\frac{8}{27}\Big )\Vert w (x-\tilde{x})\Vert \\&=\frac{43}{54} \,\frac{1}{|X |} \sum _{x\in X}\Vert w (x-\tilde{x})\Vert . \end{aligned} \end{aligned}$$
Applying Hölder inequality to the last expression, we get:
$$\begin{aligned} \frac{1}{|X |}\sum _{x\in X} \big ( 2c_{x}\rho _1+\rho _2\big ) \Vert w (x-\tilde{x})\Vert \le \frac{43}{54}\Vert w\Vert _*\varepsilon . \end{aligned}$$
Therefore, by Theorem 3, if \(\varepsilon \le \frac{54}{43\Vert w\Vert _*}\delta\), then \(\Vert {\mathbb {E}}(w,X)-{\mathbb {E}}(w,\tilde{X})\Vert <\delta\) as stated. \(\square\)
Intuition for Theorem 4. The error function can be understood as an error surface. For a fixed set of weights w, the difference between the error computed on the original dataset, \(E={\mathbb {E}}(w,{\mathcal {D}})\), and on its \(\lambda\)-balanced \(\varepsilon\)-representative dataset, \(\tilde{E}={\mathbb {E}}(w,\tilde{\mathcal {D}})\), is bounded
Summing up, we have proved that the accuracy and error of a perceptron evaluated on the binary dataset \({\mathcal {D}}\) or on its \(\lambda\)-balanced \(\varepsilon\)-representative dataset, are equivalent. This fact will be highlighted in Sect. 6.1 for the perceptron case and in Sect. 6.2 for neural networks with more complex architectures.
Experimental results
In this section, we experimentally prove that the accuracy of a neural network trained on the original dataset and on a dominating dataset is correlated with the parameter \(\varepsilon\). Besides, we also show that the accuracy is worse if we train the neural network on a random dataset. Evaluation metrics will be computed to show the performance of the trained neural network using different datasets. Specifically, MSE is the mean squared error, Recall is the ratio of positive identifications correctly classified over all positive identifications. Precision is the ratio of positive identifications correctly classified over all those classified positive. AUC is the area under the ROC curve. The ROC curve plots true positive rates vs. false positive rates at different classification thresholds.
The perceptron case
In this section, two experiments are provided to support our theoretical results for the perceptron case and to illustrate the usefulness of our method.
In the first experiment (Sect. 6.1.1), several synthetic datasets are presented showing different distributions. Random weight initialization is considered, and the holdout procedure is applied (i.e., the datasets were split into training dataset and test set) to test the generalization capabilities.
In the second experiment (Sect. 6.1.2), the Iris dataset is considered. The perceptron is initiated with random weights and trained on three different datasets: the original dataset, a representative dataset (being the output of Algorithm 1) and a random dataset of the same size as the size of the representative dataset. Now, the trained perceptron is evaluated on the original dataset. This experiment supports that a perceptron trained on representative datasets get similar accuracy to a perceptron trained on the original dataset.
Besides, we show that the training time, in the case of the gradient descent training, is lower when using a representative dataset and that representative datasets ensure good performance, while the random dataset provides no guarantees.
Synthetic datasets
In this experiment, different datasets were generated using a Scikit-learn Python package implementation.Footnote 2 Roughly speaking, it creates clusters of normally distributed points in an hypercube and adds some noise. Specifically, we considered three different situations: (1) distribution without overlapping; (2) distribution with overlapping; and (3) a dataset with a "thin" class and a high \(\varepsilon\). In the last experiment, we wanted to show that the choice of \(\varepsilon\) is important, and that there are cases where representative datasets are not so useful. In all three cases, the perceptron was trained using the stochastic gradient descent algorithm and the mean squared error as the loss function.
The methodology followed in the experiments performed in this section is outlined in Fig. 4 and summarized in the following steps.
Input: A dataset \({\mathcal {D}} = \big \{(x,c_x): x\in X\subset {\mathbb {R}}^2\) and \(c_x\in \{0,1\}\big \}\) and a parameter \(\varepsilon >0\).
Divide the dataset \({\mathcal {D}}\) in a training dataset \({\mathcal {S}}\) and a test set \({\mathcal {T}}\).
Compute a dominating dataset \(\tilde{\mathcal {S}}\) of \({\mathcal {S}}\) using Algorithm 1.
Compute a random dataset \({\mathcal {R}}\) of \({\mathcal {S}}\).
Train the perceptron \({\mathcal {N}}_{w}\) on \({\mathcal {X}}\), for \({\mathcal {X}}\in \{{\mathcal {S}}, \tilde{\mathcal {S}}, {\mathcal {R}}\}\).
Evaluate the trained perceptron on \({\mathcal {T}}\).
Methodology followed in the experiments carried out in Sects. 6.1.1 and 6.2.1
The aim of the second step in the methodology is to compute an \(\varepsilon\)-representative dataset using Algorithm 1 (see Lemma 1 of Sect. 3.1). Nevertheless, using Algorithm 1 is not mandatory, and we could replace it by any other process to compute an \(\varepsilon\)-representative dataset. Observe that the reduction is done in the fifth step since the idea is to train the neural network on the \(\varepsilon\)-representative dataset instead of on the training dataset.
In the first case (see Fig. 5a), 5000 points were taken with the two clusters well differentiated, i.e., without overlapping between classes. The \(20\%\) of the points were selected to belong to the test set, and the rest of the points constituted the training dataset. Then, an \(\varepsilon\)-representative dataset of the training dataset was computed using Algorithm 1 with \(\varepsilon =0.8\), obtaining a dominating dataset with just 17 points. Similarly, 17 random points were chosen from the training dataset (see Fig. 5c). Later, a perceptron was trained on each dataset for 20 epochs and evaluated on the test set. The mean accuracy results after 5 repetitions were: 0.96 for the dominating dataset, 0.82 for the random dataset, and 0.98 for the training dataset. Besides, the random dataset reached very low accuracy in general. In the second case (see Fig. 5d), a dataset composed of 5000 points with overlapping classes was generated. As in the first case, the dataset was split into a training dataset and a test set. Then, the \(\varepsilon\)-representative dataset of the training dataset was computed using Algorithm 1 with \(\varepsilon =0.5\), resulting in a dominating dataset of size 22. After training a perceptron for 20 epochs and repeating the experiments 5 times, the mean accuracy values were: 0.73 for the dominating dataset; 0.67 for the random dataset; and 0.86 for the training dataset. Finally, in the third case (see Fig. 5g), one of the classes was very "thin," in the sense that the points were very close to each other displaying a thin line. Therefore, if a "big" \(\varepsilon\) were chosen, that class would be represented by a pointed line as shown in Fig. 5h where \(\varepsilon =0.8\), reducing the dominating dataset to 15 points. With this example, we wanted to show a case where representative datasets were not so useful. The perceptron was trained for 20 epochs, and the mean accuracy of 5 repetitions was: 0.72 for the dominating dataset; 0.76 for the random dataset; and 0.99 for the training dataset. In terms of time, the training for 20 epochs on the training dataset took around 20 seconds, and the training on the dominating dataset took half a second. The computation of the dominating dataset took around 7 seconds. In Table 3, some evaluation metrics are provided on the test set.
Different synthetic datasets generated using the Scikit-learn python package implementation. The first column corresponds to original datasets, the second column corresponds to dominating datasets of the training datasets, and the third column corresponds to random subsets of the training datasets of the same size as the corresponding dominating set
Table 3 Evaluation metrics on the test set for a perceptron trained on the training datasets, the dominating datasets and the random datasets computed from the synthetic datasets shown in Fig. 5
The iris dataset
In this experiment, we used the Iris DatasetFootnote 3 which corresponds to a classification problem with three classes. It is composed by 150 4-dimensional instances. We limited our experiment to two of the three classes, keeping a balanced dataset of 100 points that will be our original dataset.
Input: The original dataset \({\mathcal {D}} = \big \{(x,c_x): x\in X\subset {\mathbb {R}}^4\) and \(c_x\in \{0,1\}\big \}\) and a parameter \(\varepsilon >0\).
Compute a dominating dataset \(\tilde{\mathcal {D}}\) of \({\mathcal {D}}\) using Algorithm 1.
Compute a random dataset \({\mathcal {R}}\) of \({\mathcal {D}}\).
Train a perceptron \({\mathcal {N}}_{w}\) on \({\mathcal {X}}\), for \({\mathcal {X}}\in \{{\mathcal {D}}, \tilde{\mathcal {D}}, {\mathcal {R}}\}\).
Evaluate the trained perceptron on \({\mathcal {D}}\).
Methodology followed in the experiments performed in Sects. 6.1.2 and 6.2.2.
Algorithm 1 was applied to the original dataset to obtain an \(\varepsilon\)-representative dataset of 16 points with \(\varepsilon \le 0.5\). A random dataset extracted from the original dataset with the same number of points than the dominating dataset was also computed. These datasets are represented in \({\mathbb {R}}^3\) in Fig. 7a, b and c, respectively. Besides, the associated persistence diagrams are shown in Fig. 8a, b and c. The Hausdorff and the 0-dimensional bottleneck distances between the original dataset, and the dominating and random datasets are given in Table 4.
Visualization of the Iris dataset: the original dataset is composed of 100 points, and the dominating dataset and the random dataset are composed of 16 points
Persistence diagram of the original, dominating and random datasets obtained from the Iris dataset and Digits dataset, respectively
Table 4 The Hausdorff distance (\(d_H\)) and the 0-dimensional bottleneck distance (\(d_B\)) of the dominating and random datasets with respect to the original datasets obtained from the Iris dataset and the Digits dataset, respectively
We trained the perceptron with different initial weights and observed that the perceptron trained on the dominating and the original datasets converged to similar errors. In Table 5, the difference between the errors using a fixed set of weights for the dominating and the random dataset is provided. In Table 6, different metrics were evaluated on the original dataset when training on the original, the dominating and the random dataset, respectively. The table shows that the dominating dataset provides better metrics than the random dataset (Table 6). In Table 7, the computation time in seconds when using the different datasets is shown.
Table 5 Comparison between the exact error differences computed over the random and the dominating datasets for the Iris classification problem
Table 6 Different metrics for the Iris and the Digits dataset experiments calculated as the mean values of 5 repetitions and evaluated on the original dataset
Table 7 Time (in seconds) required to compute the dominating datasets using Algorithm 1 and time (in seconds) required for the training process on the Iris and Digits datasets
The multi-layer neural network case
In this section, we will experimentally check the usefulness of representative datasets for more complex neural network architectures than a perceptron. Two different experiments were made, one using synthetic datasets and the other using the Digits datasetFootnote 4.
This experiment consists of two different binary classification problems on synthetic datasets with 5000 points.
Select a multi-layer neural network architecture \({\mathcal {N}}_{w,\Phi }\) to classify \({\mathcal {D}}\).
Train the multi-layer neural network \({\mathcal {N}}_{w,\Phi }\) on \({\mathcal {X}}\) for \({\mathcal {X}}\in \{{{\mathcal {S}},\tilde{S}},{\mathcal {R}}\}\).
Evaluate the trained neural network on \({\mathcal {T}}\).
Each synthetic dataset (case A and case B) was split into a training dataset and a test set with proportions of \(80\%\) and \(20\%\), respectively. Then, a dominating dataset of the training dataset, and a random subset of the training dataset with the same size as the dominating dataset, were computed and used for training a \(3\times 12 \times 6 \times 1\) neural network. It used ReLU activation function in the inner layers and sigmoid function in the output layer and was trained using stochastic gradient descent and mean squared error as the loss function for 20 epochs.
In the first case (see Fig. 9a), an unbalanced dataset with overlapping was considered, and an \(\varepsilon\)-dominating dataset was computed with \(\varepsilon =0.8\) composed of 67 points (see Fig. 9b). Then, a random dataset with the same size as the dominating dataset was considered. The multi-layer neural network was trained, and the mean accuracy values after 5 repetitions were: 0.85 for the dominating dataset; 0.74 for the random dataset; and 0.86 for the training dataset. In the second case, a balanced dataset with overlapping was considered (see Fig. 9d), and the same process as in the first case was carried out but with \(\varepsilon =0.3\), obtaining a dominating dataset of size 319. The mean accuracy values after 5 repetitions were: 0.92 for the dominating dataset; 0.91 for the random dataset; and 0.93 for the training dataset. In Table 8, different evaluation metrics on the test set for the two cases are shown.
The first column shows the training datasets obtained as subsets of two different synthetic datasets generated using the Scikit-learn python package implementation. The second column shows dominating datasets computed from the training datasets, and the third column shows random subsets of the training datasets with the same size as the corresponding dominating dataset
Table 8 Evaluation metrics on the test set for the training of a multi-layer neural network on the training dataset, on the dominating dataset and on the random dataset obtained from the synthetic dataset experiment
The digits dataset
The Digits datasetFootnote 5 used in this experiment consists of images classified in 10 different classes corresponding to digits from 0 to 9. An example of an image of each class is shown in Fig. 10. The Digits dataset is composed by 1797 64-dimensional instances.
(Digits) Example of an image of each class of the digits dataset. They are \(32\times 32\) arrays in gray scale
Input: A dataset \({\mathcal {D}} = \big \{(x,c_x): x\in X\subset {\mathbb {R}}^{64}\) and \(c_x\in \{0,1,...,9\}\big \}\) and a parameter \(\varepsilon >0\).
Train the multi-layer neural network \({\mathcal {N}}_{w,\Phi }\) on \({\mathcal {X}}\) for \({\mathcal {X}}\in \{{{\mathcal {D}},\tilde{D}},{\mathcal {R}}\}\).
Evaluate the trained neural network on \({\mathcal {D}}\).
In this experiment, Algorithm 1 was applied with \(\varepsilon =0.2\) to obtain a dominating dataset of size 173. The corresponding persistence diagrams can be seen in Fig. 8d, e and f. The Hausdorff and the bottleneck distances are shown in Table 4. In this case, we used a multi-layer neural network with \(64\times 400 \times 300 \times 800 \times 300 \times 10\) neurons with sigmoid activation function in the hidden layers and softmax activation function in the output layer. The neural network was trained using Adam algorithm and categorical cross-entropy as the loss function for 1000 epochs. It was launched 5 times for the dominating dataset and the random dataset. The mean accuracy values of different metrics for the 5 repetitions when training the neural network on the three different datasets and evaluated on the original dataset are shown in Table 6. Finally, in Table 9, different values for \(\varepsilon\) were used to compute the size of the dominating dataset and the accuracy of the neural network trained with the dominating dataset both on the dominating dataset and on the original dataset.
Table 9 Mean accuracy values after 5 repetitions of the neural network trained on the dominating dataset and evaluated on both the dominating dataset and the original dataset for the Digits classification problem
Conclusions and future work
The success of practical applications and the availability of new hardware (e.g., GPUs [24] and TPUs [25]) have led to focus neural network research on the development of new architectures rather than on theoretical issues. Nevertheless, a deeper understanding of the data structure is also necessary for field development, such as new discoveries on adversarial examples [26] have shown, or the one given in [27], where the redundancy of several datasets is empirically shown.
In this paper, we propose the use of representative datasets as a new approach to reduce learning time in neural networks based on the topological structure of the input dataset. Specifically, we have defined representative datasets using a notion of nearness that has the Gromov–Hausdorff distance as the lower bound. Nevertheless, the bottleneck distance of persistence diagrams (which is a lower bound of the Gromov–Hausdorff distance) is used to measure the representativeness of the dataset since it is computationally less expensive. Besides, we have theoretically proved that the accuracy of a perceptron evaluated on the original dataset coincides with the accuracy of the neural network evaluated on its representative dataset when the neural network architecture is a perceptron, the loss function is the mean square error and certain conditions on the representativeness of the dataset are imposed. Furthermore, the agreement between the provided theoretical results and the experiments supports that representative datasets can be a good approach to reach an efficient "summarization" of a dataset to train a neural network.
Planned future work is to provide more experiments using high-dimensional real data and different reduction algorithms. Furthermore, we plan to formally prove that the proposed approach can be extended to other neural network architectures and training algorithms using milder constraints. Let us observe that in the case that the dataset is already small, and then, it will not make sense to compute a representative dataset. The sparsity of the dataset may be a feature to be considered as a future work. Finally, we plan to investigate more efficient ways of computing dominating datasets in particular and representative datasets in general, since the algorithm proposed is just an example and, in general, will not compute the smallest possible representative dataset nor is it the fastest possible.
It seems to be intractable in practice [13].
It can be found in https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_classification.html.
https://archive.ics.uci.edu/ml/datasets/iris.
https://scikit-learn.org/stable/auto_examples/datasets/plot_digits_last_image.html.
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This work was partly supported by the Agencia Estatal de Investigación/10.13039/501100011033 under grant PID2019-107339GB-100 and the Agencia Andaluza del Conocimiento under grant P20-01145.
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Dept. of Applied Mathematics I, Universidad de Sevilla, 41012, Sevilla, Spain
Rocio Gonzalez-Diaz & Eduardo Paluzo-Hidalgo
Dept. of Computer Science and Artificial Intelligence, Universidad de Sevilla, 41012, Sevilla, Spain
Miguel A. Gutiérrez-Naranjo
Rocio Gonzalez-Diaz
Eduardo Paluzo-Hidalgo
Correspondence to Eduardo Paluzo-Hidalgo.
The authors declare that there is no conflict of interest.
The implementation of the methodology followed and the experimentation carried out can be consulted online in https://github.com/Cimagroup/Experiments-Representative-datasets.
R. Gonzalez-Diaz, M. A. Gutiérrez-Naranjo and E. Paluzo-Hidalgo have contributed equally to this work.
Gonzalez-Diaz, R., Gutiérrez-Naranjo, M.A. & Paluzo-Hidalgo, E. Topology-based representative datasets to reduce neural network training resources. Neural Comput & Applic 34, 14397–14413 (2022). https://doi.org/10.1007/s00521-022-07252-y
Data reduction
Computational topology | CommonCrawl |
RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
Vodopyanov, Sergei Konstantinovich
Statistics Math-Net.Ru
Total publications: 79
Scientific articles: 71
Presentations: 3
This page: 7275
Abstract pages: 20297
Full texts: 6098
References: 1543
Doctor of physico-mathematical sciences (1993)
Birth date: 9.12.1946
Keywords: Lebesgue spaces of differential forms, distortion of mappings, quasiconformal mapping, cogomology of Riemannian manifolds.
http://www.mathnet.ru/eng/person13262
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/230666
Publications in Math-Net.Ru
1. S. K. Vodopyanov, "On the Analytic and Geometric Properties of Mappings in the Theory of $\mathscr Q_{q,p}$-Homeomorphisms", Mat. Zametki, 108:6 (2020), 925–929 ; Math. Notes, 108:6 (2020), 889–894
2. S. K. Vodopyanov, A. I. Tyulenev, "Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$", Mat. Sb., 211:6 (2020), 40–94 ; Sb. Math., 211:6 (2020), 786–837
3. S. K. Vodopyanov, "О регулярности отображений, обратных к соболевским, и теория $\mathscr{Q}_{q,p}$-гомеоморфизмов", Sibirsk. Mat. Zh., 61:6 (2020), 1257–1299
4. S. K. Vodopyanov, "Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds", Mat. Sb., 210:1 (2019), 63–112 ; Sb. Math., 210:1 (2019), 59–104
5. S. K. Vodopyanov, "Isomorphisms of Sobolev spaces on Riemannian manifolds and quasiconformal mappings", Sibirsk. Mat. Zh., 60:5 (2019), 996–1034 ; Siberian Math. J., 60:5 (2019), 774–804
6. N. A. Kudryavtseva, S. K. Vodopyanov, "On the convergence of mappings with $k$-finite distortion", Probl. Anal. Issues Anal., 7(25):special issue (2018), 88–100
7. S. K. Vodopyanov, "Differentiability of mappings of the Sobolev space $W^1_{n-1}$ with conditions on the distortion function", Sibirsk. Mat. Zh., 59:6 (2018), 1240–1267 ; Siberian Math. J., 59:6 (2018), 983–1005
8. S. K. Vodopyanov, "Basics of the quasiconformal analysis of a two-index scale of spatial mappings", Sibirsk. Mat. Zh., 59:5 (2018), 1020–1056 ; Siberian Math. J., 59:5 (2018), 805–834
9. A. I. Tyulenev, S. K. Vodop'yanov, "On the Whitney Problem for Weighted Sobolev Spaces", Dokl. Akad. Nauk, 472:6 (2017), 634–638 ; Dokl. Math., 95:1 (2017), 79–83
10. S. K. Vodop'yanov, N. A. Kudryavtseva, "On the Convergence of Mappings with $k$-Finite Distortion", Mat. Zametki, 102:6 (2017), 943–948 ; Math. Notes, 102:6 (2017), 878–883
11. S. K. Vodop'yanov, A. O. Molchanova, "Lower semicontinuity of mappings with bounded $(\theta,1)$-weighted $(p,q)$-distortion", Sibirsk. Mat. Zh., 57:5 (2016), 999–1011 ; Siberian Math. J., 57:5 (2016), 778–787
12. S. K. Vodop'yanov, N. A. Evseev, "Isomorphisms of Sobolev spaces on Carnot groups and quasiconformal mappings", Sibirsk. Mat. Zh., 56:5 (2015), 989–1029 ; Siberian Math. J., 56:5 (2015), 789–821
13. A. N. Baykin, S. K. Vodop'yanov, "Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion", Sibirsk. Mat. Zh., 56:2 (2015), 290–321 ; Siberian Math. J., 56:2 (2015), 237–261
14. S. K. Vodop'yanov, N. A. Evseev, "Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings", Sibirsk. Mat. Zh., 55:5 (2014), 1001–1039 ; Siberian Math. J., 55:5 (2014), 817–848
15. S. G. Basalaev, S. K. Vodopyanov, "Approximate differentiability of mappings of Carnot–Carathéodory spaces", Eurasian Math. J., 4:2 (2013), 10–48
16. S. K. Vodopyanov, "Regularity of mappings inverse to Sobolev mappings", Mat. Sb., 203:10 (2012), 3–32 ; Sb. Math., 203:10 (2012), 1383–1410
17. D. V. Isangulova, S. K. Vodopyanov, "Coercive estimates and integral representation formulas on Carnot groups", Eurasian Math. J., 1:3 (2010), 58–96
18. S. K. Vodop'yanov, "Spaces of differential forms and maps with controlled distortion", Izv. RAN. Ser. Mat., 74:4 (2010), 5–32 ; Izv. Math., 74:4 (2010), 663–689
19. S. K. Vodop'yanov, "Spaces of differential forms and mappings with controlled distortion", Dokl. Akad. Nauk, 424:6 (2009), 727–731 ; Dokl. Math., 79:1 (2009), 105–109
20. S. K. Vodop'yanov, N. A. Kudryavtseva, "Nonlinear potential theory for Sobolev spaces on Carnot groups", Sibirsk. Mat. Zh., 50:5 (2009), 1016–1036 ; Siberian Math. J., 50:5 (2009), 803–819
21. S. K. Vodop'yanov, N. N. Romanovskii, "Sobolev classes of mappings on a Carnot–Carathéodory space: Various norms and variational problems", Sibirsk. Mat. Zh., 49:5 (2008), 1028–1045 ; Siberian Math. J., 49:5 (2008), 814–828
22. S. K. Vodop'yanov, I. M. Pupyshev, "The Traces of Bessel Potentials on Regular Subsets of Carnot Groups", Mat. Tr., 10:2 (2007), 19–61 ; Siberian Adv. Math., 18:1 (2008), 44–75
23. S. K. Vodop'yanov, I. M. Pupyshev, "Traces of Sobolev functions on the Ahlfors sets of Carnot groups", Sibirsk. Mat. Zh., 48:6 (2007), 1201–1221 ; Siberian Math. J., 48:6 (2007), 961–978
24. S. K. Vodop'yanov, "Differentiability of mappings in the geometry of Carnot manifolds", Sibirsk. Mat. Zh., 48:2 (2007), 251–271 ; Siberian Math. J., 48:2 (2007), 197–213
25. S. K. Vodop'yanov, D. V. Isangulova, "Differentiability of the mappings of Carnot–Caratheodory spaces in the Sobolev and $BV$-topologies", Sibirsk. Mat. Zh., 48:1 (2007), 46–67 ; Siberian Math. J., 48:1 (2007), 37–55
26. S. K. Vodop'yanov, I. M. Pupyshev, "Boundary Values of Differentiable Functions Defined on an Arbitrary Domain of a Carnot Group", Mat. Tr., 9:2 (2006), 23–46 ; Siberian Adv. Math., 17:1 (2007), 62–78
27. S. K. Vodop'yanov, I. M. Pupyshev, "Whitney-type theorems on extension of functions on Carnot groups", Sibirsk. Mat. Zh., 47:4 (2006), 731–752 ; Siberian Math. J., 47:4 (2006), 601–620
28. S. K. Vodop'yanov, "Foundations of the theory of mappings with bounded distortion on Carnot groups", Dokl. Akad. Nauk, 405:1 (2005), 7–12
29. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. II", Mat. Tr., 7:1 (2004), 13–49 ; Siberian Adv. Math., 15:1 (2005), 91–125
30. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I", Mat. Tr., 6:2 (2003), 14–65 ; Siberian Adv. Math., 14:4 (2004), 78–125
31. S. K. Vodop'yanov, "Differentiability of maps of Carnot groups of Sobolev classes", Mat. Sb., 194:6 (2003), 67–86 ; Sb. Math., 194:6 (2003), 857–877
32. S. K. Vodop'yanov, "The geometry of Carnot-Carathéodory spaces, quasiconformal analysis, and geometric measure theory", Vladikavkaz. Mat. Zh., 5:1 (2003), 14–34
33. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Superposition operators in Sobolev spaces", Izv. Vyssh. Uchebn. Zaved. Mat., 2002, 10, 11–33 ; Russian Math. (Iz. VUZ), 46:10 (2002), 9–31
34. S. K. Vodop'yanov, "Closure of Classes of Mappings with Bounded Distortion on Carnot Groups", Mat. Tr., 5:2 (2002), 92–137 ; Siberian Adv. Math., 14:1 (2004), 84–125
35. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Superposition operators in Lebesgue spaces and the differentiability of quasi-additive set functions", Vladikavkaz. Mat. Zh., 4:1 (2002), 11–33
36. S. K. Vodop'yanov, "Topological and geometrical properties of mappings with summable Jacobian in Sobolev classes. I", Sibirsk. Mat. Zh., 41:1 (2000), 23–48 ; Siberian Math. J., 41:1 (2000), 19–39
37. S. K. Vodop'yanov, "Mappings with bounded distortion and with finite distortion on Carnot groups", Sibirsk. Mat. Zh., 40:4 (1999), 764–804 ; Siberian Math. J., 40:4 (1999), 644–677
38. S. K. Vodop'yanov, I. G. Markina, "Classification of sub-Riemannian manifolds", Sibirsk. Mat. Zh., 39:6 (1998), 1271–1289 ; Siberian Math. J., 39:6 (1998), 1096–1111
39. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Sobolev spaces and $(P,Q)$-quasiconformal mappings of Carnot groups", Sibirsk. Mat. Zh., 39:4 (1998), 776–795 ; Siberian Math. J., 39:4 (1998), 665–682
40. S. K. Vodopyanov, A. V. Greshnov, "Continuation of differentiable functions, and quasiconformal mappings on Carnot groups", Dokl. Akad. Nauk, 348:1 (1996), 15–18
41. S. K. Vodopyanov, "Quasiconformal mappings on Carnot groups and their applications", Dokl. Akad. Nauk, 347:4 (1996), 439–442
42. S. K. Vodop'yanov, "Monotone functions and quasiconformal mappings on Carnot groups", Sibirsk. Mat. Zh., 37:6 (1996), 1269–1295 ; Siberian Math. J., 37:6 (1996), 1113–1136
43. S. K. Vodop'yanov, N. A. Kudryavtseva, "Normal families of mappings on Carnot groups", Sibirsk. Mat. Zh., 37:2 (1996), 273–286 ; Siberian Math. J., 37:2 (1996), 232–244
44. S. K. Vodop'yanov, A. D.-O. Ukhlov, "Approximately differentiable transformations and change of variables on nilpotent groups", Sibirsk. Mat. Zh., 37:1 (1996), 70–89 ; Siberian Math. J., 37:1 (1996), 62–78
45. S. K. Vodopyanov, A. D.-O. Ukhlov, "Weakly contact transformations and change of variables on nilpotent groups", Dokl. Akad. Nauk, 341:4 (1995), 439–441
46. S. K. Vodop'yanov, A. V. Greshnov, "Analytic properties of quasiconformal mappings on Carnot groups", Sibirsk. Mat. Zh., 36:6 (1995), 1317–1327 ; Siberian Math. J., 36:6 (1995), 1142–1151
47. S. K. Vodop'yanov, A. V. Greshnov, "On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric", Sibirsk. Mat. Zh., 36:5 (1995), 1015–1048 ; Siberian Math. J., 36:5 (1995), 873–901
48. S. K. Vodop'yanov, I. G. Markina, "Exceptional sets for solutions to subelliptic equations", Sibirsk. Mat. Zh., 36:4 (1995), 805–818 ; Siberian Math. J., 36:4 (1995), 694–706
49. S. K. Vodop'yanov, "Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations", Sibirsk. Mat. Zh., 36:2 (1995), 278–300 ; Siberian Math. J., 36:2 (1995), 246–264
50. S. K. Vodop'yanov, "Thin sets in weighted potential theory and degenerate elliptic equations", Sibirsk. Mat. Zh., 36:1 (1995), 28–36 ; Siberian Math. J., 36:1 (1995), 24–32
51. M. S. Alborova, S. K. Vodopyanov, "Removable singularities of bounded solutions of quasi-elliptic equations", Sibirsk. Mat. Zh., 33:4 (1992), 3–14 ; Siberian Math. J., 33:4 (1992), 543–556
52. S. K. Vodopyanov, "Weighted $L_p$ potential theory on homogeneous groups", Sibirsk. Mat. Zh., 33:2 (1992), 29–48 ; Siberian Math. J., 33:2 (1992), 201–218
53. Yu. I. Boyarskii, S. K. Vodopyanov, "Intrinsic metrics and boundary values of functions of Zygmund classes", Sibirsk. Mat. Zh., 32:2 (1991), 3–12 ; Siberian Math. J., 32:2 (1991), 177–185
54. S. K. Vodopyanov, "Weighted $L_p$ potential theory on homogeneous groups", Dokl. Akad. Nauk SSSR, 314:1 (1990), 37–41 ; Dokl. Math., 42:2 (1991), 265–270
55. S. K. Vodop'yanov, "The $L_p$ potential theory for generalized kernels", Mat. Zametki, 47:5 (1990), 146–148
56. S. K. Vodop'yanov, "Potential theory on homogeneous groups", Mat. Sb., 180:1 (1989), 57–77 ; Math. USSR-Sb., 66:1 (1990), 59–81
57. S. K. Vodopyanov, "Mappings of homogeneous groups and imbeddings of functional spaces", Sibirsk. Mat. Zh., 30:5 (1989), 25–41 ; Siberian Math. J., 30:5 (1989), 685–698
58. S. K. Vodopyanov, "Intrinsic geometries and boundary values of differentiable functions. I", Sibirsk. Mat. Zh., 30:2 (1989), 29–42 ; Siberian Math. J., 30:2 (1989), 191–202
59. S. K. Vodopyanov, "Potential theory on homogeneous groups", Dokl. Akad. Nauk SSSR, 303:1 (1988), 11–15 ; Dokl. Math., 38:3 (1989), 447–451
60. S. K. Vodopyanov, "Equivalent normings of spaces of differentiable functions in domains and their applications", Dokl. Akad. Nauk SSSR, 300:4 (1988), 777–781 ; Dokl. Math., 37:3 (1988), 708–713
61. S. K. Vodopyanov, "Quasielliptic $L_p$-theory of potential and its applications", Dokl. Akad. Nauk SSSR, 298:4 (1988), 780–784 ; Dokl. Math., 37:1 (1988), 133–137
62. S. K. Vodopyanov, "The maximum principle in potential theory and embedding theorems for anisotropic spaces of differentiable functions", Sibirsk. Mat. Zh., 29:2 (1988), 17–33 ; Siberian Math. J., 29:2 (1988), 176–189
63. S. K. Vodopyanov, "Geometric properties of domains and estimates for the norm of an extension operator", Dokl. Akad. Nauk SSSR, 292:4 (1987), 791–795
64. S. K. Vodopyanov, "Isoperimetric relations and conditions for the extension of differentiable functions", Dokl. Akad. Nauk SSSR, 292:1 (1987), 11–15
65. S. K. Vodop'yanov, V. M. Gol'dstein, Yu. G. Reshetnyak, "On geometric properties of functions with generalized first derivatives", Uspekhi Mat. Nauk, 34:1(205) (1979), 17–65 ; Russian Math. Surveys, 34:1 (1979), 19–74
66. S. K. Vodopyanov, V. M. Gol'dstein, T. G. Latfullin, "A criterion for the extension of functions of the class $L_2^1$ from unbounded plane domains", Sibirsk. Mat. Zh., 20:2 (1979), 416–419 ; Siberian Math. J., 20:2 (1979), 298–301
67. V. M. Gol'dstein, S. K. Vodop'yanov, "Metric completion of a domain by using a conformal capacity invariant under quasi-conformal mappings", Dokl. Akad. Nauk SSSR, 238:5 (1978), 1040–1042
68. S. K. Vodopyanov, V. M. Gol'dstein, "A test of the removability of sets for $L_p^1$ spaces of quasiconformal and quasi-isomorphic mappings", Sibirsk. Mat. Zh., 18:1 (1977), 48–68 ; Siberian Math. J., 18:1 (1977), 35–50
69. S. K. Vodopyanov, V. M. Gol'dstein, "Functional characterizations of quasi-isometric mappings", Sibirsk. Mat. Zh., 17:4 (1976), 768–773 ; Siberian Math. J., 17:4 (1976), 580–584
70. S. K. Vodopyanov, V. M. Gol'dstein, "Quasiconformal mappings, and spaces of functions with first generalized derivatives", Sibirsk. Mat. Zh., 17:3 (1976), 515–531 ; Siberian Math. J., 17:3 (1976), 399–411
71. S. K. Vodop'yanov, V. M. Gol'dstein, "A criterion for the possibility of eliminating sets for the spaces $W_p^1$ of quasiconformal and quasi-isometric mappings", Dokl. Akad. Nauk SSSR, 220:4 (1975), 769–771
72. S. K. Vodopyanov, "The boundary correspondence for quasiconformal mappings of $n$-dimensional domains", Sibirsk. Mat. Zh., 16:3 (1975), 630–633 ; Siberian Math. J., 16:3 (1975), 487–490
73. S. K. Vodopyanov, V. M. Gol'dstein, "Lattice isomorphisms of the spaces $W_n^1$ and quasiconformal mappings", Sibirsk. Mat. Zh., 16:2 (1975), 224–246 ; Siberian Math. J., 16:2 (1975), 174–189
74. S. K. Vodop'yanov, V. M. Gol'dstein, "Lattice isomorphisms of the spaces $W_n^1$, and quasiconformal mappings", Dokl. Akad. Nauk SSSR, 215:1 (1974), 24–26
75. S. K. Vodopyanov, "Estimates of the deviation of quasi-umbilical surfaces from a sphere", Sibirsk. Mat. Zh., 11:5 (1970), 971–987 ; Siberian Math. J., 11:5 (1970), 724–735
76. A. V. Abanin, E. K. Basaeva, A. O. Vatulyan, S. K. Vodop'yanov, A. E. Gutman, V. A. Koibaev, Yu. F. Korobeinik, S. B. Klimentov, S. S. Kutateladze, A. A. Makhnev, B. B. Tasoev, S. M. Umarkhadzhiev, M. Z. Hudalov, "To the 65-th anniversary of prof. A. G. Kusraev", Vladikavkaz. Mat. Zh., 20:2 (2018), 111–119
77. F. G. Avkhadiev, V. A. Botvinnik, S. K. Vodop'yanov, M. Vuorinen, V. M. Gol'dstein, V. V. Goryainov, A. A. Grigor'yan, V. N. Dubinin, I. V. Zhuravlev, V. A. Zorich, V. M. Kesel'man, A. A. Klyachin, V. A. Klyachin, T. G. Latfullin, A. V. Loboda, A. G. Losev, O. Martio, V. I. Pelikh, S. I. Pinchuk, Yu. G. Reshetnyak, A. S. Romanov, A. G. Sergeev, V. G. Tkachev, E. M. Chirka, "Vladimir Mikhailovich Miklyukov (obituary)", Uspekhi Mat. Nauk, 69:3(417) (2014), 173–176 ; Russian Math. Surveys, 69:3 (2014), 565–568
78. D. A. Archangelsky, B. S. Baizhanov, O. V. Belegradek, V. Ya. Belyaev, L. A. Bokut, M. K. Valiev, S. K. Vodopyanov, M. Gitik, Yu. S. Gurevich, D. O. Daderkin, A. M. Dekhtyar, M. I. Dekhtyar, A. Ya. Dikovsky, S. M. Dudakov, E. I. Zelmanov, B. I. Zilber, S. L. Krushkal, S. S. Kutateladze, Yu. V. Matiyasevich, G. E. Mints, I. Kh. Musikaev, A. K. Rebrov, Yu. G. Reshetnyak, A. L. Semenov, A. P. Stolboushkin, I. A. Taimanov, B. A. Trakhtenbrot, "Mikhail Abramovich Taitslin (1936–2013)", Sib. Èlektron. Mat. Izv., 10 (2013), 54–65
79. S. K. Vodop'yanov, E. I. Gordon, A. E. Gutman, A. V. Koptev, S. S. Kutateladze, S. A. Malyugin, Yu. G. Reshetnyak, "Anatoly Georgievich Kusraev is 60", Sib. Èlektron. Mat. Izv., 10 (2013), 13–29
80. S. K. Vodop'yanov, E. I. Gordon, A. E. Gutman, A. V. Koptev, S. S. Kutateladze, S. A. Malyugin, Yu. G. Reshetnyak, "Anatolii Georgievich Kusraev (on the occasion of his 60th anniversary)", Vladikavkaz. Mat. Zh., 15:1 (2013), 90–97
81. A. Yu. Vesnin, S. K. Vodop'yanov, S. S. Kutateladze, S. P. Novikov, I. A. Taimanov, "Yurii Grigor'evich Reshetnyak (on his 80th birthday)", Uspekhi Mat. Nauk, 64:5(389) (2009), 185–188 ; Russian Math. Surveys, 64:5 (2009), 961–965
82. A. Yu. Vesnin, S. K. Vodop'yanov, S. S. Kutateladze, I. A. Taimanov, "Yurii Grigor'evich Reshetnyak (on the occasion of his 80th birthday)", Sibirsk. Mat. Zh., 50:5 (2009), 959–962
83. Yu. G. Reshetnyak, S. K. Vodop'yanov, V. I. Kuz'minov, S. S. Kutateladze, I. A. Taimanov, "Yuri Fedorovich Borisov (1925–2007)", Sib. Èlektron. Mat. Izv., 4 (2007), 28–30
Presentations in Math-Net.Ru
1. КВАЗИКОНФОРМНЫЙ АНАЛИЗ И ЗАДАЧИ НЕЛИНЕЙНОЙ ТЕОРИИ УПРУГОСТИ
S. K. Vodopyanov
Seminar on Theory of Functions of Several Real Variables and Its Applications to Problems of Mathematical Physics
2. Properties of a new class of spatial mappings and its applications
S. K. Vodop'yanov
International Conference "Geometric Analysis and Control Theory"
3. Sobolev Spaces, geometric function theory and its application
International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Peoples' Friendship University of Russia, Moscow
Novosibirsk State University, Mechanics and Mathematics Department
Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2021 | CommonCrawl |
\begin{definition}[Definition:Category of Relations]
The '''category of (binary) relations''', denoted $\mathbf{Rel}$, is the metacategory with:
{{DefineCategory
|ob = sets
|mor = binary relations $\RR \subseteq A \times B$.
|comp = composition of relations
|id = identity mappings
}}
\end{definition} | ProofWiki |
\begin{document}
\begin{frontmatter}
\title{A new bijection relating $q$-Eulerian polynomials}
\author{Ange Bigeni\fnref{myfootnote}} \address{Institut Camille Jordan\\ Université Claude Bernard Lyon 1\\ 43 boulevard du 11 novembre 1918\\ 69622 Villeurbanne cedex\\ France } \ead{[email protected]}
\begin{abstract} On the set of permutations of a finite set, we construct a bijection which maps the 3-vector of statistics $(\maj-\text{exc},\text{des},\text{exc})$ to a 3-vector $(\maj_2,\widetilde{\text{des}_2},\text{inv}_2)$ associated with the $q$-Eulerian polynomials introduced by Shareshian and Wachs in \textit{Chromatic quasisymmetric functions, arXiv:1405.4269(2014).} \end{abstract}
\begin{keyword} $q$-Eulerian polynomials\sep descents\sep ascents \sep major index \sep exceedances\sep inversions. \end{keyword}
\end{frontmatter}
\linenumbers
\section*{Notations}
For all pair of integers $(n,m)$ such that $n<m$, the set $\{n,n+1,\hdots,m\}$ is indifferently denoted by $[n,m]$,$]n-1,m]$,$[n,m+1[$ or $]n-1,m+1[$.
The set of positive integers $\{1,2,3,\hdots\}$ is denoted by $\mathbb{N}_{>0}$.
For all integer $n \in \mathbb{N}_{>0}$, we denote by $[n]$ the set $[1,n]$ and by $\mathfrak{S}_n$ the set of the permutations of $[n]$. By abuse of notation, we assimilate every $\sigma \in \mathfrak{S}_n$ with the word $\sigma(1) \sigma(2) \hdots \sigma(n)$.
If a set $S = \{n_1,n_2,\hdots,n_k\}$ of integers is such that $n_1 < n_2 < \hdots < n_k$, we sometimes use the notation $S = \{n_1 < n_2 < \hdots < n_k \}$.
\section{Introduction} \label{sec:intro} Let $n$ be a positive integer and $\sigma \in \mathfrak{S}_n$. A \textit{descent} (respectively \textit{exceedance point}) of $\sigma$ is an integer $i \in [n-1]$ such that $\sigma(i) > \sigma(i+1)$ (resp. $\sigma(i) > i$). The set of descents (resp. exceedance points) of $\sigma$ is denoted by $\text{DES}(\sigma)$ (resp. $\EXC(\sigma)$) and its cardinal by $\text{des}(\sigma)$ (resp. $\text{exc}(\sigma)$). The integers $\sigma(i)$ with $i \in \EXC(\sigma)$ are called exceedance values of $\sigma$.
It is due to MacMahon~\cite{macmahon} and Riordan~\cite{riordan} that $$\sum\limits_{\sigma \in \mathfrak{S}_n} t^{\text{des}(\sigma)} = \sum\limits_{\sigma \in \mathfrak{S}_n} t^{\text{exc}(\sigma)} = A_n(t)$$ where $A_n(t)$ is the $n$-th Eulerian polynomial~\cite{euler}. A statistic equidistributed with des or exc is said to be \textit{Eulerian}. The statistic ides defined by $\text{ides}(\sigma) = \text{des}(\sigma^{-1})$ obviously is Eulerian.
The \textit{major index} of a permutation $\sigma \in \mathfrak{S}_n$ is defined as $$\maj(\sigma) = \sum\limits_{i \in \text{DES}(\sigma)} i.$$ It is also due to MacMahon that $$\sum\limits_{\sigma \in \mathfrak{S}_n} q^{\maj(\sigma)} = \prod\limits_{i=1}^n \dfrac{1-q^i}{1-q}.$$
A statistic equidistributed with maj is said to be \textit{Mahonian}. Among Mahonian statistics is the statistic inv, defined by $\text{inv}(\sigma) = |\text{INV}(\sigma)|$ where $\text{INV}(\sigma)$ is the set of \textit{inversions} of a permutation $\sigma \in \mathfrak{S}_n$, \textit{i.e.} the pairs of integers $(i,j) \in [n]^2$ such that $i < j$ and $\sigma(i) > \sigma(j)$.
In \cite{shareshianwachs}, the authors consider analogous versions of the above statistics : let $\sigma \in \mathfrak{S}_n$, the set of \textit{2-descents} (respectively \textit{2-inversions}) of $\sigma$ is defined as $$\text{DES}_2(\sigma) = \{i \in [n-1], \sigma(i) > \sigma(i+1)+1\}$$ (resp. $$\text{INV}_2(\sigma) = \{1 \leq i < j \leq n, \sigma(i) = \sigma(j)+1\})$$ and its cardinal is denoted by $\text{des}_2(\sigma)$ (resp. $\text{inv}_2(\sigma)$).
It is easy to see that $\text{inv}_2(\sigma) = \text{ides}(\sigma)$. The \textit{2-major index} of $\sigma$ is defined as $$\maj_2(\sigma) = \sum\limits_{i \in \text{DES}_2(\sigma)} i.$$
By using quasisymmetric function techniques, the authors of \cite{shareshianwachs} proved the equality \begin{equation} \label{eq:shareshianwachs} \sum_{\sigma \in \mathfrak{S}_n} x^{\maj_{2}(\sigma)} y^{\text{inv}_{2}(\sigma)} = \sum_{\sigma \in \mathfrak{S}_n} x^{\maj(\sigma)-\text{exc}(\sigma)} y^{\text{exc}(\sigma)}. \end{equation}
Similarly, by using the same quasisymmetric function method as in \cite{shareshianwachs}, the authors of \cite{hanceli} proved the equality \begin{equation}
\label{eq:hanceli}
\sum\limits_{\sigma \in \mathfrak{S}_n} x^{\text{amaj}_2(\sigma)} y^{\widetilde{\text{asc}_2}(\sigma)} z^{\text{ides}(\sigma)} = \sum\limits_{\sigma \in \mathfrak{S}_n} x^{\maj(\sigma)-\text{exc}(\sigma)} y^{\text{des}(\sigma)} z^{\text{exc}(\sigma)} \end{equation} where $\text{asc}_2(\sigma)$ is the number of \textit{2-ascents} of a permutation $\sigma \in \mathfrak{S}_n$, \textit{i.e.} the elements of the set $\text{ASC}_2(\sigma) = \{i \in [n-1], \sigma(i) < \sigma(i+1)+1\}$, which rises the statistic $\text{amaj}_2$ defined by $$\text{amaj}_2(\sigma) = \sum\limits_{i \in \text{ASC}_2(\sigma)} i,$$ and where $$\widetilde{\text{asc}_2}(\sigma) = \begin{cases} \text{asc}_2(\sigma) &\text{if $\sigma(1) = 1$,} \\ \text{asc}_2(\sigma)+1 &\text{if $\sigma(1) \neq 1$.} \end{cases}$$
\begin{defi} \label{def:des2tilde} Let $\sigma \in \mathfrak{S}_n$. We consider the smallest $2$-descent $d_2(\sigma)$ of $\sigma$ such that $\sigma(i) = i$ for all $i \in [d_2(\sigma)-1]$ (if there is no such $2$-descent, we define $d_2(\sigma)$ as $0$ and $\sigma(0)$ as $n+1$).
Now, let $d_2'(\sigma) > d_2(\sigma)$ be the smallest $2$-descent of $\sigma$ greater than $d_2(\sigma)$ (if there is no such $2$-descent, we define $d_2'(\sigma)$ as $n$).
We define an inductive property $\mathcal{P}(d_2(\sigma))$ by : \begin{enumerate} \item $\sigma(d_2(\sigma)) < \sigma(i)$ for all $(i,j) \in \text{INV}_2(\sigma)$ such that $d_2(\sigma) < i < d_2'(\sigma)$; \item if $(d_2'(\sigma),j) \in \text{INV}_2(\sigma)$ for some $j$, then either $\sigma(d_2(\sigma)) < \sigma(d_2'(\sigma))$, or $d_2'(\sigma)$ has the property $\mathcal{P}(d_2'(\sigma))$ (where the role of $d_2(\sigma)$ is played by $d_2'(\sigma)$ and that of $d_2'(\sigma)$ by $d_2''(\sigma)$ where $d_2''(\sigma) > d_2'(\sigma)$ is the smallest $2$-descent of $\sigma$ greater than $d_2'(\sigma)$, defined as $n$ if there is no such $2$-descent). \end{enumerate} This property is well-defined because $(n,j) \not\in \text{INV}_2(\sigma)$ for all $j \in [n]$.
Finally, we define a statistic $\widetilde{\text{des}_2}$ by : $$\widetilde{\text{des}_2}(\sigma) = \begin{cases} \text{des}_2(\sigma) &\text{if the property $\mathcal{P}(d_2(\sigma))$ is true,} \\ \text{des}_2(\sigma)+1 &\text{otherwise.} \end{cases}$$ \end{defi}
In the present paper, we prove the following theorem.
\begin{theo} \label{theo:existsbijection} There exists a bijection $\varphi : \mathfrak{S}_n \rightarrow \mathfrak{S}_n$ such that $$(\maj_{2}(\sigma),\widetilde{\text{des}_2}(\sigma),\text{inv}_{2}(\sigma)) = (\maj(\varphi(\sigma))-\text{exc}(\varphi(\sigma)),\text{des}(\varphi(\sigma)),\text{exc}(\varphi(\sigma))).$$ \end{theo}
As a straight corollary of Theorem \ref{theo:existsbijection}, we obtain the equality
\begin{equation} \label{eq:bigeni} \sum\limits_{\sigma \in \mathfrak{S}_n} x^{\maj_2(\sigma)} y^{\widetilde{\text{des}_2}(\sigma)} z^{\text{inv}_2(\sigma)} = \sum\limits_{\sigma \in \mathfrak{S}_n} x^{\maj(\sigma)-\text{exc}(\sigma)} y^{\text{des}(\sigma)} z^{\text{exc}(\sigma)} \end{equation}
which implies Equality (\ref{eq:shareshianwachs}).
The rest of this paper is organised as follows.
In Section \ref{sec:graphic}, we introduce two graphical representations of a given permutation so as to highlight either the statistic $(\maj-\text{exc},\text{des},\text{exc})$ or $(\maj_2,\widetilde{\text{des}_2},\text{inv}_2)$. Practically speaking, the bijection $\varphi$ of Theorem \ref{theo:existsbijection} will be defined by constructing one of the two graphical representations of $\varphi(\sigma)$ for a given permutation $\sigma \in \mathfrak{S}_n$.
We define $\varphi$ in Section \ref{sec:varphi}.
In Section \ref{sec:varphim1}, we prove that $\varphi$ is bijective by constructing $\varphi^{-1}$.
\section{Graphical representations} \label{sec:graphic}
\subsection{Linear graph}
Let $\sigma \in \mathfrak{S}_n$. The linear graph of $\sigma$ is a graph whose vertices are (from left to right) the integers $\sigma(1), \sigma(2), \hdots, \sigma(n)$ aligned in a row, where every $\sigma(k)$ (for $k \in \text{DES}_2(\sigma)$) is boxed, and where an arc of circle is drawn from $\sigma(i)$ to $\sigma(j)$ for every $(i,j) \in \text{INV}_2(\sigma)$.
For example, the permutation $\sigma = 34251 \in \mathfrak{S}_5$ (such that \linebreak[4]$(\maj_2(\sigma),\widetilde{\text{des}_2}(\sigma),\text{inv}_2(\sigma)) =~(6,3,2) $) has the linear graph depicted in Figure \ref{fig:FIGm1}.
\begin{figure}
\caption{Linear graph of $\sigma = 34251 \in \mathfrak{S}_5$.}
\label{fig:FIGm1}
\end{figure}
\subsection{Planar graph}
Let $\tau \in \mathfrak{S}_n$. The planar graph of $\tau$ is a graph whose vertices are the integers $1,2,...,n$, organized in ascending and descending slopes (the height of each vertex doesn't matter) such that the $i$-th vertex (from left to right) is the integer $\tau(i)$, and where every vertex $\tau(i)$ with $i \in \EXC(\tau)$ is encircled.
For example, the permutation $\tau = 32541 \in \mathfrak{S}_5$ (such that \linebreak[4]$(\maj(\tau)-\text{exc}(\tau),\text{des}(\tau),\text{exc}(\tau)) = (6,3,2))$ has the planar graph depicted in Figure \ref{fig:FIG0}.
\begin{figure}
\caption{Planar graph of $\tau = 32541 \in \mathfrak{S}_5$.}
\label{fig:FIG0}
\end{figure}
\section{Definition of the map $\varphi$ of Theorem \ref{theo:existsbijection}} \label{sec:varphi}
Let $\sigma \in \mathfrak{S}_n$. We set $(r,s) = (\text{des}_2(\sigma),\text{inv}_2(\sigma))$, and \begin{align*} \text{DES}_2(\sigma) &= \left \{d_2^k(\sigma), k \in [r] \right \},\\ \text{INV}_2(\sigma) &= \{(i_l(\sigma),j_l(\sigma)),l \in [s]\} \end{align*} with $d_2^k(\sigma) < d_2^{k+1}(\sigma)$ for all $k$ and $i_l(\sigma) < i_{l+1}(\sigma)$ for all $l$.
We intend to define $\varphi(\sigma)$ by constructing its planar graph. To do so, we first construct (in Subsection \ref{subsec:graph}) a graph $\mathcal{G}(\sigma)$ made of $n$ circles or dots organized in ascending or descending slopes such that two consecutive vertices are necessarily in a same descending slope if the first vertex is a circle and the second vertex is a dot. Then, in Subsection \ref{subsec:labelling}, we label the vertices of this graph with the integers $1,2,\hdots,n$ in such a way that, if $y_i$ is the label of the $i$-th vertex $v_i(\sigma)$ (from left to right) of $\mathcal{G}(\sigma)$ for all $i \in [n]$, then :
\begin{enumerate} \item $y_i <y_{i+1}$ if and only if $v_i$ and $v_{i+1}$ are in a same ascending slope; \item $y_i > i$ if and only if $v_i$ is a circle. \end{enumerate}
The permutation $\tau = \varphi(\sigma)$ will then be defined as $y_1 y_2\hdots y_n$, \textit{i.e.} the permutation whose planar graph is the labelled graph $\mathcal{G}(\sigma)$.
With precision, we will obtain $$\tau\left(\EXC(\tau)\right)=\{j_k(\sigma),k\in [s]\}$$ (in particular $\text{exc}(\tau) = s = \text{inv}_2(\sigma)$), and $$\text{DES}(\tau) = \begin{cases} \{d^k(\sigma),k \in [1,r]\} \text{ if $\widetilde{\text{des}_2}(\sigma) = r$},\\ \{d^k(\sigma),k \in [0,r]\} \text{ if $\widetilde{\text{des}_2}(\sigma) = r+1$} \end{cases}$$
for integers $0 \leq d^0(\sigma) < d^1(\sigma) < \hdots < d^{r}(\sigma) \leq n$ (with $d^0(\sigma) = 0 \Leftrightarrow \widetilde{\text{des}_2}(\sigma) = \text{des}_2(\sigma)$) defined by $$d^k(\sigma) = d_2^k(\sigma) + c_k(\sigma)$$ (with $d_2^0(\sigma) :=0$) where $(c_{k}(\sigma))_{k \in [0,r]}$ is a sequence defined in Subsection \ref{subsec:graph} such that $\sum_k c_k(\sigma) = \text{inv}_2(\sigma) = \text{exc}(\tau)$. Thus, we will obtain $\text{des}(\tau) = \widetilde{\text{des}_2}(\sigma)$ and $\maj(\tau) = \maj_2(\sigma) + \text{exc}(\tau)$.
\subsection{Construction of the unlabelled graph $\mathcal{G}(\sigma)$} \label{subsec:graph}
We set $\left( d_2^0(\sigma),\sigma(d_2^0(\sigma)) \right) =(0,n+1)$ and $\left( d_2^{r+1}(\sigma),\sigma(n+1) \right) = (n,0)$.
For all $k \in [r]$, we define the top $t_k(\sigma)$ of the 2-descent $d_2^k(\sigma)$ as
\begin{equation} \label{eq:definitionbk} t_k(\sigma) = \min\{d_2^l(\sigma),1\leq l \leq k,d_2^l(\sigma) = d_2^k(\sigma) - (k-l)\}, \end{equation}
in other words $t_k(\sigma)$ is the smallest 2-descent $d_2^l(\sigma)$ such that the 2-descents $d_2^l(\sigma),d_2^{l+1}(\sigma),\hdots,d_2^k(\sigma)$ are consecutive integers.
The following algorithm provides a sequence $(c^0_{k}(\sigma))_{k \in [0,r]}$ of nonnegative integers.
\begin{algo} \label{algo:suiteck0} Let $I_r(\sigma) = \text{INV}_2(\sigma)$. For $k$ from $r = \text{des}_2(\sigma)$ down to $0$, we consider the set $S_k(\sigma)$ of sequences $(i_{k_1}(\sigma),i_{k_2}(\sigma), \hdots, i_{k_m}(\sigma))$ such that : \begin{enumerate} \label{algo} \item $(i_{k_p}(\sigma),j_{k_p}(\sigma)) \in I_{k}(\sigma)$ for all $p \in [m]$; \item $t_k(\sigma) \leq i_{k_1}(\sigma) < i_{k_2}(\sigma) < \hdots < i_{k_m}(\sigma)$; \item $\sigma(i_{k_1}(\sigma)) < \sigma(i_{k_2}(\sigma)) < \hdots < \sigma(i_{k_m}(\sigma))$. \end{enumerate} The \textit{length} of such a sequence is defined as $l = \sum_{p=1}^m n_p$ where $n_p$ is the number of \textit{consecutive} 2-inversions whose beginning is $i_{k_p}$, \textit{i.e.} the maximal number $n_p$ of 2-inversions $(i_{k_p^1}(\sigma),j_{k_p^1}(\sigma)),(i_{k_p^2}(\sigma),j_{k_p^2}(\sigma)),\hdots,(i_{k^{n_p}_p}(\sigma),j_{k^{n_p}_p}(\sigma))$ such that $k^1_p = k_p$ and $j_{k^i_p}(\sigma) = i_{k^{i+1}_p}(\sigma)$ for all $i$. If $I_k(\sigma) \neq \emptyset$, we consider the sequence $(i_{k^{max}_1}(\sigma),i_{k^{max}_2}(\sigma), \hdots, i_{k^{max}_m}(\sigma)) \in I_k(\sigma)$ whose length $l^{max} = \sum_{p=1}^m n_p^{max}$ is maximal and whose elements $i_{k^{max}_1}(\sigma),i_{k^{max}_2}(\sigma),\hdots,i_{k^{max}_m}(\sigma)$ also are maximal (as integers). Then, \begin{itemize} \item if $I_k(\sigma) \neq \emptyset$, we set $c^0_k(\sigma) = l^{max}$ and $$I_{k-1}(\sigma) = I_k(\sigma) \backslash \left( \cup_{p=1}^m \{(i_{k^{max}_i}(\sigma),j_{k^{max}_i}(\sigma)),i\in [n_p^{max}]\} \right);$$ \item else we set $c_k^0(\sigma) = 0$ and $I_{k-1}(\sigma) = I_k(\sigma)$. \end{itemize} \end{algo}
\begin{ex} Consider the permutation $\sigma = 549321867 \in \mathfrak{S}_9$, with $\text{DES}_2(\sigma) = \{3,7\}$ and $I_2(\sigma) = \text{INV}_2(\sigma) = \{(1,2),(2,4),(3,7),(4,5),(5,6),(7,9)\}$. In Figure \ref{fig:FIG1} are depicted the $\text{des}_2(\sigma)+1 = 3$ steps $k \in \{2,1,0\}$ (at each step, the 2-inversions of the maximal sequence are drawed in red then erased at the following step) : \begin{figure}\label{fig:FIG1}
\end{figure} \begin{itemize} \item $k=2$ : there is only one legit sequence $(i_{k_1}(\sigma)) = (7)$, whose length is $l = n_1 = 1$. We set $c^0_2(\sigma) = 1$ and $I_1(\sigma) = I_2(\sigma) \backslash \{(7,9)\}$. \item $k=1$ : there are three legit sequences $(i_{k_1}(\sigma)) = (3)$ (whose length is $l = n_1 =1$) then $(i_{k_1}(\sigma)) = (4)$ (whose length is $l=n_1 =2$) and $(i_{k_1}(\sigma)) = (5)$ (whose length is $l =n_1=1$). The maximal sequence is the second one, hence we set $c^0_1(\sigma) = 2$ and $I_0(\sigma) = I_1(\sigma) \backslash \{(4,5),(5,6)\}$. \item $k=0$ : there are three legit sequences $(i_{k_1}(\sigma),i_{k_2}(\sigma)) = (1,3)$ (whose length is $l = n_1 + n_2 = 2+1=3$) then $(i_{k_1}(\sigma),i_{k_2}(\sigma)) = (2,3)$ (whose length is $l =n_1+n_2 = 1+1=2$) and $(i_{k_1}(\sigma)) = (3)$ (whose length is $l =n_1 = 1$). The maximal sequence is the first one, hence we set $c^0_0(\sigma) = 3$ and $I_{-1}(\sigma) = I_0(\sigma) \backslash \{(1,2),(2,4),(3,7)\} = \emptyset$. \end{itemize} \end{ex}
\begin{lem} \label{lem:lemme1} The sum $\sum_k c^0_k(\sigma)$ equals $\text{inv}_2(\sigma)$ (\textit{i.e.} $I_{-1}(\sigma) = \emptyset$) and, for all $k \in [0,r] = [0,\text{des}_2(\sigma)]$, we have $c^0_k(\sigma) \leq d_2^{k+1}(\sigma) - d_2^k(\sigma)$ with equality only if $c^0_{k+1}(\sigma) >~0$ (where $c^0_{r+1}(\sigma)$ is defined as $0$). \end{lem} \begin{proof} With precision, we show by induction that, for all $k \in \{\text{des}_2(\sigma),\hdots,1,0\}$, the set $I_{k-1}(\sigma)$ contains no 2-inversion $(i,j)$ such that $d_2^k(\sigma) < i$. For $k=0$, it will mean $I_{-1}(\sigma) = \emptyset$ (recall that $d_2^0(\sigma)$ has been defined as $0$).
$\star$ If $k = \text{des}_2(\sigma) = r$, the goal is to prove that $c^0_r(\sigma) < n - d_2^r(\sigma)$. Suppose there exists a sequence $(i_{k_1}(\sigma),i_{k_2}(\sigma), \hdots, i_{k_m}(\sigma))$ of length $c^0_{r}(\sigma) \geq n-d_2^{r}(\sigma)$ with $t_{r}(\sigma) \leq i_{k_1}(\sigma) < i_{k_2}(\sigma) < \hdots < i_{k_m}(\sigma)$. In particular, there exist $c^0_{r}(\sigma)~\geq~n-d_2^{r}(\sigma)$ 2-inversions $(i,j)$ such that $d_2^{r}(\sigma) < j$, which forces $c^0_{r}(\sigma)$ to equal $n-d_2^{r}(\sigma)$ and every $j > d_2^{r}(\sigma)$ to be the arrival of a 2-inversion $(i,j)$ such that $t_{r}(\sigma) \leq i$. In particular, this is true for $j = d_2^{r}(\sigma)+1$, which is absurd because $\sigma(i) \geq \sigma \left( d_2^{r}(\sigma) \right) > \sigma \left( d_2^{r}(\sigma) + 1 \right) +1$ for all $i \in [t_{r}(\sigma),d_2^{r}(\sigma)]$. Therefore $c^0_{r}(\sigma) < n-d_2^{r}(\sigma)$. Also, it is easy to see that every $i > d_2^{r}(\sigma)$ that is the beginning of a 2-inversion $(i,j)$ necessarily appears in the maximal sequence $\left( i_{k^{max}_1}(\sigma),i_{k^{max}_2}(\sigma), \hdots, i_{k^{max}_m}(\sigma) \right)$ whose length defines $c^0_{r}(\sigma)$, hence $(i,j) \not\in I_{r-1}(\sigma)$.
$\star$ Now, suppose that $c^{0}_{k}(\sigma) \leq d_2^{k+1}(\sigma) - d_2^k(\sigma)$ for some $k \in [\text{des}_2(\sigma)]$ with equality only if $c^0_{k+1}(\sigma) > 0$, and that no 2-inversion $(i,j)$ with $d_2^{k}(\sigma) < i$ belongs to $I_{k-1}(\sigma)$.
If $t_{k-1}(\sigma) = t_k(\sigma)$ (\textit{i.e.}, if $d_2^{k-1}(\sigma) = d_2^k(\sigma) - 1$), since $I_{k-1}(\sigma)$ does not contain any 2-inversion $(i,j)$ with $d_2^{k}(\sigma) <i$, then $c^0_{k-1}(\sigma) \leq 1 = d_2^k(\sigma) - d_2^{k-1}(\sigma)$. Moreover, if $c^0_{k-1}(\sigma) = 1$, then there exists a 2-inversion $(i,j) \in I_{k-1}(\sigma) \subset I_k(\sigma)$ such that $i \in [t_{k-1}(\sigma),d_2^k(\sigma)]$. Consequently $(i)$ was a legit sequence for the computation of $c^0_k(\sigma)$ at the previous step (because $t_k(\sigma) = t_{k-1}(\sigma)$), which implies $c^0_k(\sigma)$ equals at least the length of $(i)$. In particular $c^0_k(\sigma) > 0$.
Else, consider a sequence $(i_{k_1}(\sigma),i_{k_2}(\sigma), \hdots, i_{k_m}(\sigma))$ that fits the three conditions of Algorithm \ref{algo:suiteck0} at the step $k-1$. In particular $t_{k-1}(\sigma) \leq i_{k_1}(\sigma)$. Also $i_{k_m}(\sigma) \leq d_2^k(\sigma)$ by hypothesis. Since $\sigma(i_{k_p}(\sigma)) < \sigma(i_{k_{p+1}}(\sigma))$ for all $p$, and since $\sigma(t_{k-1}(\sigma)) > \sigma(t_{k-1}(\sigma)+1) > \hdots > \sigma \left( d_2^{k-1}(\sigma) \right) > \sigma \left( d_2^{k-1}(\sigma)+1 \right)$, then only one element of the set $[t_{k-1}(\sigma),d_2^{k-1}(\sigma)+1]$ may equal $i_{k_p}(\sigma)$ for some $p \in [m]$. Thus, the length $l$ of the sequence verifies $l \leq d_2^k(\sigma)-d_2^{k-1}(\sigma)$, with equality only if $i_{k_m}(\sigma) = d_2^k(\sigma)$ (which implies $c^0_k(\sigma) > 0$ as in the previous paragraph). In particular, this is true for $l = c^0_{k-1}(\sigma)$.
Finally, as for $k = \text{des}_2(\sigma)$, every $i \in [d_2^{k-1}(\sigma)+1,d_2^k(\sigma)]$ that is the beginning of a 2-inversion $(i,j)$ necessarily appears in the maximal sequence $\left( i_{k^{max}_1}(\sigma),i_{k^{max}_2}(\sigma), \hdots, i_{k^{max}_m}(\sigma) \right)$ whose length defines $c^0_{k-1}(\sigma)$, hence $(i,j) \not\in I_{k-2}(\sigma)$.
So the lemma is true by induction. \end{proof}
\begin{defi} \label{def:gzerosigma} We define a graph $\mathcal{G}^0(\sigma)$ made of circles and dots organised in ascending or descending slopes, by plotting : \begin{itemize} \item for all $k \in [0,r]$, an ascending slope of $c_k^0(\sigma)$ circles such that the first circle has abscissa $d_2^k(\sigma) + 1$ and the last circle has abscissa $d_2^k(\sigma) + c_k^0(\sigma)$ (if $c_k^0(\sigma) = 0$, we plot nothing). All the abscissas are distinct because $$d_2^0(\sigma) + c_0 < d_2^1(\sigma) + c_1 < \hdots < d_2^r(\sigma) + c_r$$ in view of Lemma \ref{lem:lemme1}; \item dots at the remaining $n-s = n-\text{inv}_2(\sigma)$ abscissas from $1$ to $n$, in ascending and descending slopes with respect to the descents and ascents of the word $\omega(\sigma)$ defined by \begin{equation} \label{eq:defomega} \omega(\sigma) = \sigma(u_1(\sigma)) \sigma(u_2(\sigma)) \hdots \sigma(u_{n-s}(\sigma)) \end{equation} where $$\{u_1(\sigma) < u_2(\sigma) < \hdots < u_{n-s}(\sigma) \} := \mathfrak{S}_n \backslash \{i_1(\sigma) < i_2(\sigma) < \hdots < i_{s}(\sigma)\}.$$ \end{itemize} \end{defi}
\begin{ex} The permutation $\sigma_0 = 425736981 \in \mathfrak{S}_9$ (with $\text{DES}_2(\sigma_0) = \{1,4,8\}$ and $\text{INV}_2(\sigma_0) = \{(1,5),(2,9),(4,6),(7,8)\}$), which yields the sequence $(c^0_k(\sigma_0))_{k \in [0,3]} = (1,1,2,0)$ (see Figure \ref{fig:FIG2} where all the 2-inversions involved in the computation of a same $c^0_k(\sigma_0)$ are drawed in a same color) and the word $\omega(\sigma_0) = 53681$, provides the unlabelled graph $\mathcal{G}^0(\sigma_0)$ depicted in Figure \ref{fig:FIG4}.
\begin{figure}
\caption{$(c^0_k(\sigma_0))_{k \in [0,3]} = (\textcolor{red}{1},\textcolor{green}{1},\textcolor{blue}{2},0).$}
\label{fig:FIG2}
\end{figure}
\begin{figure}
\caption{Graph $\mathcal{G}^0(\sigma_0)$.}
\label{fig:FIG4}
\end{figure} \end{ex}
The following lemma is easy.
\begin{lem} \label{lem:modifsequence} For all $i \in [n]$, if the $i$-th vertex (from left to right) $v_i^0(\sigma)$ of $\mathcal{G}^0(\sigma)$ is a dot and if $i$ is a \textit{descent} of $\mathcal{G}^0(\sigma)$ (i.e., if $v_i^{0}(\sigma)$ and $v_{i+1}^{0}(\sigma)$ are two dots in a same descending slope) whereas $i \not\in \text{DES}_2(\sigma)$, let $k_i$ such that $$d_2^{k_i}(\sigma)~+~c^{0}_{k_i}(\sigma)~<~i <~d_2^{k_i+1}(\sigma)$$ and let $p \in [n-s]$ such that $v_i^0(\sigma)$ is the $p$-th dot (from left to right) of $\mathcal{G}^{0}(\sigma)$. Then : \begin{enumerate} \item $u_p(\sigma)$ is the greatest integer $u < d_2^{k_i+1}(\sigma)$ that is not the beginning of a 2-inversion of $\sigma$; \item $u_{p+1}(\sigma)$ is the smallest integer $u > d_2^{k_i+1}(\sigma)$ that is not a 2-descent or the beginning of a 2-inversion of $\sigma$; \item $c^{0}_k(\sigma) >0$ for all $k$ such that $d_2^{k_i+1}(\sigma) \leq d_2^k(\sigma) \leq u_{p+1}(\sigma)$. \end{enumerate} In particular $c^{0}_{k_i+1}(\sigma) > 0$. \end{lem}
Lemma \ref{lem:modifsequence} motivates the following definition.
\begin{defi} \label{def:ckgsigma} For $i$ from $1$ to $n-1$, let $k_i \in [0,r]$ such that $$d_2^{k_i}(\sigma)~+~c^{0}_{k_i}(\sigma)~<~i <~d_2^{k_i+1}(\sigma).$$ If $i$ fits the conditions of Lemma \ref{lem:modifsequence}, then we define a sequence $(c^i_k(\sigma))_{k \in [0,r]}$ by \begin{align*} c^i_{k_i}(\sigma) &= c^{i-1}_{k_i}(\sigma)+1,\\ c^i_{k_i+1}(\sigma) &= c^{i-1}_{k_i+1}(\sigma)-1,\\ c^i_k(\sigma) &= c^{i-1}_k(\sigma) \text{ for all $k \not\in \{k_i,k_i+1\}$.} \end{align*} Else, we define $(c^i_k(\sigma))_{k \in [0,r]}$ as $(c^{i-1}_k(\sigma))_{k \in [0,r]}$.
The final sequence $(c^n_k(\sigma))_{k \in [0,r]}$ is denoted by $$(c_k(\sigma))_{k \in [0,r]}.$$ \end{defi}
By construction, and from Lemma \ref{lem:lemme1}, the sequence $(c_k(\sigma))_{k \in [0,r]}$ has the same properties as $(c^0_k(\sigma))_{k \in [0,r]}$ detailed in Lemma \ref{lem:lemme1}.
Consequently, we may define an unlabelled graph $$\mathcal{G}(\sigma)$$ by replacing $(c^0_k(\sigma))_{k \in [0,r]}$ with $(c_k(\sigma))_{k \in [0,r]}$ in Definition \ref{def:gzerosigma}.
\begin{ex} In the graph $\mathcal{G}^0(\sigma)$ depicted in Figure \ref{fig:FIG4} where $\sigma_0 = 425736981 \in \mathfrak{S}_9$, we can see that the dot $v_3^0(\sigma_0)$ is a descent whereas $3 \not\in \text{DES}_2(\sigma_0)$, hence, from the sequence $(c^0_k(\sigma_0))_{k \in [0,3]} = (1,1,2,0)$, we compute $(c_k(\sigma_0))_{k \in [0,3]} = (1,2,1,0)$ and we obtain the graph $\mathcal{G}(\sigma_0)$ depicted in Figure \ref{fig:FIG5}. \begin{figure}
\caption{Graph $\mathcal{G}(\sigma_0)$.}
\label{fig:FIG5}
\end{figure} \end{ex}
Let $v_1(\sigma),v_2(\sigma),\hdots,v_n(\sigma)$ be the $n$ vertices of $\mathcal{G}(\sigma)$ from left to right.
By construction, the descents of the unlabelled graph $\mathcal{G}(\sigma)$ (\textit{i.e.}, the integers $i \in [n-1]$ such that $v_i(\sigma)$ and $v_{i+1}(\sigma)$ are in a same descending slope) are the integers $$d^k(\sigma) = d_2^k(\sigma) + c_k(\sigma)$$ for all $k \in [0,r]$.
\subsection{Labelling of the graph $\mathcal{G}(\sigma)$} \label{subsec:labelling}
\subsubsection{Labelling of the circles} We intend to label the circles of $\mathcal{G}(\sigma)$ with the integers $$j_1(\sigma),j_2(\sigma),\hdots,j_{s}(\sigma).$$
\begin{algo} \label{algo:labelcircles} For all $i \in [n]$, if the vertex $v_i(\sigma)$ is a circle (hence $i < n$), we label it first with the set $$[i+1,n] \cap \{j_1(\sigma),j_2(\sigma),\hdots,j_{s}(\sigma)\}.$$
Afterwards, if a circle $v_i(\sigma)$ is found in a descending slope such that there exists a quantity of $a$ circles above $v_i(\sigma)$, and in an ascending slope such that there exists a quantity of $b$ circles above $v_i(\sigma)$, then we remove the $a+b$ greatest integers from the current label of $v_i(\sigma)$ (this set necessarily had at least $a+b+1$ elements) and the smallest integer from every of the $a+b$ labels of the $a+b$ circles above $v_i(\sigma)$ in the two related slopes. At the end of this step, if an integer $j_k(\sigma)$ appears in only one label of a circle $v_i(\sigma)$, then we replace the label of $v_i(\sigma)$ with $j_k(\sigma)$.
Finally, we replace every label that is still a set by the unique integer it may contain with respect to the order of the elements in the sequence $$(j_1(\sigma),j_2(\sigma), \hdots,j_{s}(\sigma))$$ (from left to right). \end{algo}
\begin{ex} For $\sigma_0 = 425736981$ (see Figure \ref{fig:FIG2}) whose graph $\mathcal{G}(\sigma_0)$ is depicted in Figure \ref{fig:FIG5}, we have $s = \text{inv}_2(\sigma) = 4$ and $\{j_1(\sigma_0),j_2(\sigma_0),j_3(\sigma_0),j_4(\sigma_0)\} = \{5,6,8,9\}$, which provides first the graph labelled by sets depicted in Figure \ref{fig:FIG6}. Afterwards, since the circle $v_2(\sigma_0)$ is in a descending slope with $a=1$ circle above it (the vertex $v_1(\sigma_0)$) and in an ascending slope with also $b=1$ circle above it (the vertex $v_3(\sigma_0)$), then we remove the $a+b=2$ integers $8$ and $9$ from its label, which becomes $\{5,6\}$, and we remove $5$ from the labels of $v_1(\sigma_0)$ and $v_3(\sigma_0)$. Also, since the label of $v_2(\sigma_0)$ is the only set that contains $5$, then we label $v_2(\sigma_0)$ with $5$ (see Figure \ref{fig:FIG7}). Finally, the sequence $(j_1(\sigma_0),j_2(\sigma_0),j_3(\sigma_0),j_4(\sigma_0)) = (5,9,6,8)$ gives the order (from left to right) of apparition of the remaining integers $6,8,9$ (see Figure \ref{fig:FIG8}).
\begin{figure}\end{figure} \end{ex}
\subsubsection{Labelling of the dots}
Let $$\{p_1(\sigma) < p_2(\sigma) < \hdots < p_{n-s}(\sigma)\} = [n] \backslash \bigsqcup_{k=0}^r ]d_2^k(\sigma),d^k(\sigma)].$$
We intend to label the dots $\{v_{p_i(\sigma)}(\sigma), i \in [n-s]\}$ of $\mathcal{G}(\sigma)$ with the elements of $$\{1 = e_1(\sigma) < e_2(\sigma) < \hdots < e_{n-s}(\sigma) \} = [n] \backslash \{j_1(\sigma),j_2(\sigma),\hdots,j_{s}(\sigma)\}.$$
\begin{algo} \label{algo:labeldots} \begin{enumerate} \item For all $k \in [n-s]$, we label first the dot $v_{p_k(\sigma)}(\sigma)$ with the set $$[\min(p_k(\sigma),u_k(\sigma))] \cap ([n] \backslash \{j_1(\sigma),j_2(\sigma),\hdots,j_{s}(\sigma)\})$$ where $u_1(\sigma),u_2(\sigma),\hdots,u_{n-s}(\sigma)$ are the integers introduced in $(\ref{eq:defomega})$. \item Afterwards, similarly as for the labelling of the circles, if a dot $v_i(\sigma)$ is found in a descending slope such that $a$ dots are above $v_i(\sigma)$, and in an ascending slope such that $b$ dots are above $v_i(\sigma)$, then we remove the $a+b$ greatest integers from the current label of $v_i(\sigma)$ and the smallest integer from every of the $a+b$ labels of the dots above $v_i(\sigma)$ in the two related slopes. At the end of this step, if an integer $l$ appears in only one label of a dot $v_i(\sigma)$, then we replace the label of $v_i(\sigma)$ with $l$. \item Finally, for $k$ from $1$ to $n-s$, let \begin{equation} \label{eq:defwik} w_1^k(\sigma) < w_2^k(\sigma) < \hdots < w_{q_k(\sigma)}^k(\sigma) \end{equation} such that $$\{p_{w_i^k(\sigma)}(\sigma),i\} = \left \{p_i(\sigma), \text{ $e_k(\sigma)$ appear in the label of $p_i(\sigma)$}\right \},$$ and let $i(k) \in [q_k(\sigma)]$ such that $$\sigma\left(u_{w_{i(k)}^k(\sigma)}(\sigma)\right) = \min \{ \sigma\left(u_{w_i^k(\sigma)}(\sigma)\right),i \in [q_k(\sigma)]\}.$$ Then, we replace the label of the dot $p_{w_{i(k)}^k(\sigma)}(\sigma)$ with the integer $e_k(\sigma)$ and we erase $e_k(\sigma)$ from any other label (and if an integer $l$ appears in only one label of a dot $v_i(\sigma)$, then we replace the label of $v_i(\sigma)$ with $l$). \end{enumerate} \end{algo}
\begin{ex} For $\sigma_0 = 425736981$ whose graph $\mathcal{G}(\sigma_0)$ has its circles labelled in Figure \ref{fig:FIG8}, the sequence $(u_1(\sigma_0),u_2(\sigma_0),u_3(\sigma_0),u_4(\sigma_0),u_5(\sigma_0)) = (3,5,6,8,9)$ provides first the graph labelled by sets depicted in Figure \ref{fig:FIG9}. \begin{figure}\end{figure} \begin{figure}
\caption{Labelled graph $\mathcal{G}(\sigma_0)$.}
\label{fig:FIG10}
\end{figure} The rest of the algorithm goes from $k=1$ to $n-s=9-4=5$. \begin{itemize} \item $k=1$ : in Figure \ref{fig:FIG9}, the integer $e_1(\sigma_0) = 1$ appears in the labels of the dots $v_{p_1(\sigma_0)}(\sigma_0) = v_4(\sigma_0)$, $v_{p_2(\sigma_0)}(\sigma_0) = v_6(\sigma_0)$ and $v_{p_5(\sigma_0)}(\sigma_0) = v_9(\sigma_0)$, so, from $$(\sigma_0(u_1(\sigma_0)),\sigma_0(u_2(\sigma_0)),\sigma_0(u_5(\sigma_0)) = (5,3,1),$$ we label the dot $v_{p_5(\sigma_0)}(\sigma_0) = v_9(\sigma_0)$ with the integer $e_1(\sigma_0) = 1$ and we erase $1$ from any other label, and since the integer $4$ now only appears in the label of the dot $v_7(\sigma_0)$, then we label $v_7(\sigma_0)$ with $4$ (see Figure \ref{fig:FIG91}). \item $k=2$: in Figure \ref{fig:FIG91}, the integer $e_2(\sigma_0) = 2$ appears in the labels of the dots $v_{p_1(\sigma_0)}(\sigma_0) = v_4(\sigma_0)$ and $v_{p_2(\sigma_0)}(\sigma_0) = v_6(\sigma_0)$ so, from $$(\sigma_0(u_1(\sigma_0)),\sigma_0(u_2(\sigma_0))) = (5,3),$$ we label the dot $v_{p_2(\sigma_0)}(\sigma_0) = v_6(\sigma_0)$ with the integer $e_2(\sigma_0) = 2$ and we erase $2$ from any other label, which provides the graph labelled by integers depicted in Figure \ref{fig:FIG10}. \item The three steps $k = 3,4,5$ change nothing because every dot of $\mathcal{G}(\sigma_0)$ is already labelled by an integer at the end of the previous step. \end{itemize} So the final version of the labelled graph $\mathcal{G}(\sigma_0)$ is the one depicted in Figure \ref{fig:FIG10}. \end{ex}
\subsection{Definition of $\varphi(\sigma)$}
By construction of the labelled graph $\mathcal{G}(\sigma)$, the word $y_1 y_2 \hdots y_n$ (where the integer $y_i$ is the label of the vertex $v_i(\sigma)$ for all $i$) obviously is a permutation of the set $[n]$, whose planar graph is $\mathcal{G}(\sigma)$.
We define $\varphi(\sigma) \in \mathfrak{S}_n$ as this permutation.
For the example $\sigma_0 = 425736981 \in \mathfrak{S}_9$ whose labelled graph $\mathcal{G}(\sigma_0)$ is depicted in Figure \ref{fig:FIG10}, we obtain $\varphi(\sigma_0) = 956382471 \in \mathfrak{S}_9$.
In general, by construction of $\tau = \varphi(\sigma) \in \mathfrak{S}_n$, we have \begin{equation} \label{eq:exceedancevalues} \tau \left( \EXC(\tau) \right) =\{j_k(\sigma),k\in [\text{inv}_2(\sigma)]\} \end{equation} and \begin{equation} \label{eq:descents} \text{DES}(\tau) = \begin{cases} \{d^k(\sigma),k \in [1,\text{des}_2(\sigma)]\} & \text{if $c_0(\sigma) = 0 (\Leftrightarrow d^0(\sigma) =0)$}, \\ \{d^k(\sigma),k \in [0,\text{des}_2(\sigma)]\} & \text{otherwise}. \end{cases} \end{equation} Equality (\ref{eq:exceedancevalues}) provides $$\text{exc}(\tau) = \text{inv}_2(\sigma).$$ By $d^k(\sigma) = d_2^k(\sigma) + c_k(\sigma)$ for all $k$, Equality (\ref{eq:descents}) provides $$\maj(\tau) = \maj_2(\sigma) + \sum_{k \geq 0} c_k(\sigma),$$ and by definition of $(c_k(\sigma))_k$ and Lemma \ref{lem:lemme1} we have $\sum_{k \geq 0} c_k(\sigma) = \sum_{k \geq 0} c^0_k(\sigma) = \text{inv}_2(\sigma) = \text{exc}(\tau)$ hence $$\maj(\tau) - \text{exc}(\tau) = \maj_2(\sigma).$$ Finally, it is easy to see that $\widetilde{\text{des}_2}(\sigma) = \text{des}_2(\sigma)$ if and only if $c_0(\sigma) = 0$, so Equality (\ref{eq:descents}) also provides $$\text{des}(\tau) = \widetilde{\text{des}_2}(\sigma).$$ As a conclusion, we obtain $$(\maj(\tau)-\text{exc}(\tau),\text{des}(\tau),\text{exc}(\tau)) = (\maj_2(\sigma),\widetilde{\text{des}_2}(\sigma),\text{inv}_2(\sigma))$$ as required by Theorem \ref{theo:existsbijection}.
\section{Construction of $\varphi^{-1}$} \label{sec:varphim1}
To end the proof of Theorem \ref{theo:existsbijection}, it remains to show that $\varphi : \mathfrak{S}_n \rightarrow \mathfrak{S}_n$ is surjective. Let $\tau \in \mathfrak{S}_n$. We introduce integers $r \geq 0$, $s = \text{exc}(\tau)$, and $$0 \leq d^{0,\tau} < d^{1,\tau} < \hdots < d^{r,\tau} < n$$ such that \begin{align*} \text{DES}(\tau) &= \{d^{k,\tau},k \in [0,r]\} \cap \mathbb{N}_{>0},\\ d^{0,\tau} &= 0 \Leftrightarrow \tau(1) = 1. \end{align*} In particular $\text{des}(\tau) = \begin{cases} r &\text{ if $\tau(1) = 1$,}\\ r+1 &\text{ otherwise}. \end{cases}$
For all $k \in [0,r]$, we define \begin{align*} c_k^{\tau} &= \EXC(\tau) \cap ]d^{k-1,\tau},d^{k,\tau}] \text{ (with $d^{-1,\tau} := 0$),}\\ d_2^{k,\tau} &= d^{k,\tau} - c_k^{\tau}. \end{align*}
We have $$0 = d_2^{0,\tau} < d_2^{1,\tau} < \hdots < d_2^{r,\tau} < n$$ and similarly as Formula \ref{eq:definitionbk}, we define \begin{equation} \label{eq:definitionbkbis} t_k^{\tau} = \min \{d_2^{l,\tau}, 1 \leq l \leq k, d_2^{l,\tau} = d_2^{k,\tau} - (k-l)\} \end{equation} for all $k\in [r]$.
We intend to construct a graph $\mathcal{H}(\tau)$ which is the linear graph of permutation $\sigma \in \mathfrak{S}_n$ such that $\varphi(\sigma) = \tau$.
\subsection{Skeleton of the graph $\mathcal{H}(\tau)$}
We consider a graph $\mathcal{H}(\tau)$ whose vertices $v_1^{\tau},v_2^{\tau},\hdots,v_n^{\tau}$ (from left to right) are $n$ dots, aligned in a row, among which we box the $d_2^{k,\tau}$-th vertex $v_{d_2^{k,\tau}}^{\tau}$ for all $k \in [r]$. We also draw the end of an arc of circle above every vertex $v_j^{\tau}$ such that $j = \tau(i)$ for some $i \in \EXC(\tau)$.
For the example $\tau_0 = 956382471 \in \mathfrak{S}_9$ (whose planar graph is depicted in Figure \ref{fig:FIG10}), we have $r = \text{des}(\tau_0)-1 = 3$ and \begin{align*} (c_k^{\tau_0})_{k \in [0,3]} &= (1,2,1,0),\\ (d_2^{k,\tau_0})_{k \in [0,3]} &= (1-1,3-2,5-1,8-0) = (0,1,4,8),\\ \tau_0(\EXC(\tau_0)) &= \{5,6,8,9\}, \end{align*}
and we obtain the graph $\mathcal{H}(\tau_0)$ depicted in Figure \ref{fig:FIG11}. \begin{figure}
\caption{Incomplete graph $\mathcal{H}(\tau_0)$.}
\label{fig:FIG11}
\end{figure}
In general, by definition of $\varphi(\sigma)$ for all $\sigma \in \mathfrak{S}_n$, if $\varphi(\sigma) = \tau$, then $r = \text{des}_2(\sigma)$ and $d_2^k(\sigma)$ (respectively $c_k(\sigma),d^k(\sigma),t_k(\sigma)$) equals $d_2^{k,\tau}$ (resp. $c_k^{\tau},d^{k,\tau},t_k^{\tau}$) for all $k \in [0,r]$ and $\{j_l(\sigma),l \in [\text{inv}_2(\sigma)]\} = \tau(\EXC(\tau))$. Consequently, the linear graph of $\sigma$ necessarily have the same skeleton as that of $\mathcal{H}(\tau)$.
The following lemma is easy.
\begin{lem} \label{lem:3facts} If $\tau = \varphi(\sigma)$ for some $\sigma \in \mathfrak{S}_n$, then : \begin{enumerate} \label{enum:abc} \item If $j = \tau(l)$ with $l \in \EXC(\tau)$ such that $l \in ]d_2^{k,\tau},d^{k,\tau}]$, and if $(i,j) \in \text{INV}_2(\sigma)$, then $t_{k}^{\tau} \leq i$. \item A pair $(i,i+1)$ cannot be a $2$-inversion of $\sigma$ if $i \in \text{DES}_2(\sigma)$ ($\Leftrightarrow$ if the vertex $v_i^{\tau}$ of $\mathcal{H}(\tau)$ is boxed). \item For all pair $(l,l') \in \EXC(\tau)^2$, if the labels of the two circles $v_l(\sigma)$ and $v_{l'}(\sigma)$ can be exchanged without modifying the skeleton of $\mathcal{G}(\sigma)$, let $i$ and $i'$ such that $(i,l) \in \text{INV}_2(\sigma)$ and $(i',l') \in \text{INV}_2(\sigma)$, then $i < i' \Leftrightarrow l < l'$. \end{enumerate} \end{lem}
Consequently, in order to construct the linear graph of a permutation $\sigma \in \mathfrak{S}_n$ such that $\tau = \varphi(\sigma)$ from $\mathcal{H}(\tau)$, it is necessary to extend the arcs of circles of $\mathcal{H}(\tau)$ to reflect the three facts of Lemma \ref{lem:3facts}. When a vertex is necessarily the beginning of an arc of circle, we draw the beginning of an arc of circle above it. When there is only one vertex $v_i^{\tau}$ that can be the beginning of an arc of circle, we complete the latter by making it start from $v_i^{\tau}$.
\begin{ex} For $\tau_0 = 956382471 \in \mathfrak{S}_9$, the graph $\mathcal{H}(\tau_0)$ becomes as depicted in Figure \ref{fig:FIG12}. \begin{figure}
\caption{Incomplete graph $\mathcal{H}(\tau_0)$.}
\label{fig:FIG12}
\end{figure} Note that the arc of circle ending at $v_6^{\tau_0}$ cannot begin at $v_5^{\tau_0}$ because otherwise, from the third point of Lemma \ref{lem:3facts}, and since $(6,8) = (\tau_0(l),\tau_0(l'))$ with $3 = l < l' = 5$, it would force the arc of circle ending at $v_8^{\tau_0}$ to begin at $v_{i'}^{\tau_0}$ with $6 \leq i'$, which is absurd because a permutation $\sigma \in \mathfrak{S}_9$ whose linear graph would be of the kind $\mathcal{H}(\tau_0)$ would have $c_2(\sigma) = 2 \neq 1 = c_2^{\tau_0}$. Also, still in view of the third point of Lemma \ref{lem:3facts}, and since $\tau_0^{-1}(9) < \tau^{-1}(6)$, the arc of circle ending at $v_9^{\tau_0}$ must start before the arc of circle ending at $v_6^{\tau_0}$, hence the configuration of $\mathcal{H}(\tau_0)$ in Figure \ref{fig:FIG12}. \end{ex}
The following two facts are obvious.
\begin{fact} \label{fact:ascents} If $\tau = \varphi(\sigma)$ for some $\sigma \in \mathfrak{S}_n$, then : \begin{enumerate} \item A vertex $v_i^{\tau}$ of $\mathcal{H}(\tau)$ is boxed if and only if $i \in \text{DES}_2(\sigma)$. In that case, in particular $i$ is a descent of $\sigma$. \item If a pair $(i,i+1)$ is not a $2$-descent of $\sigma$ and if $v_i^{\tau}$ is not boxed, then $i$ is an ascent of $\sigma$, \textit{i.e.} $\sigma(i) < \sigma(i+1)$. \end{enumerate} \end{fact}
To reflect Facts \ref{fact:ascents}, we draw an ascending arrow (respectively a descending arrow) between the vertices $v_i^{\tau}$ and $v_{i+1}^{\tau}$ of $\mathcal{H}(\tau)$ whenever it is known that $\sigma(i) < \sigma(i+1)$ (resp. $\sigma(i) > \sigma(i+1)$) for all $\sigma \in \mathfrak{S}_n$ such that $\varphi(\sigma) = \tau$.
For the example $\tau_0 = 956382471 \in \mathfrak{S}_9$, the graph $\mathcal{H}(\tau_0)$ becomes as depicted in Figure \ref{fig:FIG13}. Note that it is not known yet if there is an ascending or descending arrow between $v_7^{\tau_0}$ and $v_8^{\tau_0}$.
\begin{figure}
\caption{Incomplete graph $\mathcal{H}(\tau_0)$.}
\label{fig:FIG13}
\end{figure}
\subsection{Completion and labelling of $\mathcal{H}(\tau)$}
The following lemma is analogous to the third point of Lemma \ref{lem:3facts} for the dots instead of the circles and follows straightly from the definition of $\varphi(\sigma)$ for all $\sigma \in \mathfrak{S}_n$.
\begin{lem} \label{lem:equivdots} Let $\sigma \in \mathfrak{S}_n$ such that $\varphi(\sigma) = \tau$. For all pair $(l,l') \in ([n]\backslash \EXC(\tau))^2$, if the labels of the two dots $v_l(\sigma)$ and $v_{l'}(\sigma)$ can be exchanged without modifying the skeleton of $\mathcal{G}(\sigma)$, let $k$ and $k'$ such that $l = p_k(\sigma)$ and $l' = p_{k'}(\sigma)$, then $\tau(l) < \tau(l') \Leftrightarrow \sigma(u_k(\sigma)) < \sigma(u_{k'}(\sigma))$. \end{lem}
Now, the ascending and descending arrows between the vertices of $\mathcal{H}(\tau)$ introduced earlier, and Lemma \ref{lem:equivdots}, induce a partial order on the set $\{v_i^{\tau},i \in [n]\}$:
\begin{defi} \label{def:partialorder} We define a partial order $\succ$ on $\{v_i^{\tau},i \in [n]\}$ by : \begin{itemize} \item $v_i^{\tau } \prec v_{i+1}^{\tau}$ (resp. $v_i^{\tau} \succ v_{i+1}^{\tau}$) if there exists an ascending (resp. descending) arrow between $v_i^{\tau}$ and $v_{i+1}^{\tau}$; \item $v_i^{\tau} \succ v_j^{\tau}$ (with $i<j$) if there exists an arc of circle from $v_i^{\tau}$ to $v_j^{\tau}$; \item if two vertices $v_i^{\tau}$ and $v_j^{\tau}$ are known to be respectively the $k$-th and $k'$-th vertices of $\mathcal{H}(\tau)$ that cannot be the beginning of a complete arc of circle, let $l$ and $l'$ be respectively the $k$-th and $k'$-th non-exceedance point of $\tau$ (from left to right), if $(l,l')$ fits the conditions of Lemma \ref{lem:equivdots}, then we set $v_i^{\tau} \prec v_j^{\tau}$ (resp. $v_i^{\tau} \succ v_j^{\tau}$) if $\tau(l) < \tau(l')$ (resp. $\tau(l) > \tau(l')$). \end{itemize} \end{defi}
\begin{ex} \label{ex:partialorder} For the example $\tau_0 = 956382471$, according to the first point of Definition \ref{def:partialorder}, the arrows of Figure \ref{fig:FIG13} provide $$v_1^{\tau_0} \succ v_2^{\tau_0} \prec v_3^{\tau_0} \prec v_4^{\tau_0} \succ v_5^{\tau_0} \prec v_6^{\tau_0} \prec v_7^{\tau_0}$$ and $$v_8^{\tau_0} \succ v_9^{\tau_0}.$$ \end{ex}
\begin{defi} \label{def:minimalvertices} A vertex $v_i^{\tau}$ of $\mathcal{H}(\tau)$ is said to be \textit{minimal} on a subset $S \subset [n]$ if $v_i^{\tau} \not \succ v_j^{\tau}$ for all $j \in S$. \end{defi}
Let $$1 = e_1^{\tau} < e_2^{\tau} < \hdots < e_{n-s}^{\tau}$$ be the non-exceedance values of $\tau$ (\textit{i.e.}, the labels of the dots of the planar graph of $\tau$).
\begin{algo} \label{algo:varphim1} Let $S = [n]$ and $l = 1$. While the vertices $\{v_i^{\tau},i \in [n]\}$ have not all been labelled with the elements of $[n]$, apply the following algorithm. \begin{enumerate} \item If there exists a unique minimal vertex $v_i^{\tau}$ of $\tau$ on $S$, we label it with $l$, then we set $l := l+1$ and $S := S \backslash \{v_i^{\tau}\}$. Afterwards, \begin{enumerate} \item If $v_i^{\tau}$ is the ending of an arc of circle starting from a vertex $v_j^{\tau}$, then we label $v^{\tau}_{j}$ with the integer $l$ and we set $l := l+1$ and $S := S \backslash \{v_j^{\tau}\}$. \item If $v^{\tau}_{i}$ is the arrival of an incomplete arc of circle (in particular $i = \tau(l)$ for some $l \in \EXC(\tau)$), we intend to complete the arc by making it start from a vertex $v^{\tau}_{j}$ for some integer $j \in [t_k^{\tau},j[$ (where $l \in ]d_2^{k,\tau},d^{k,\tau}]$) in view of the first point of Lemma \ref{lem:3facts}. We choose $v_j^{\tau}$ as the rightest minimal vertex on $[t_k^{\tau},j[ \cap S$ from which it may start in view of the third point of Lemma \ref{lem:3facts}, and we label this vertex $v^{\tau}_{j}$ with the integer $l$. Then we set $l := l+1$ and $S := S \backslash \{v_j^{\tau}\}$. \end{enumerate} Now, if there exists an arc of circle from $v_j^{\tau}$ (for some $j$) to $v_i^{\tau}$, we apply steps (a),(b) and (c) to the vertex $v_j^{\tau}$ in place of $v_i^{\tau}$. \item Otherwise, let $k \geq 0$ be the number of vertices $v_i^{\tau}$ that have already been labelled and that are not the beginning of arcs of circles. Let $$l_1 < l_2 < \hdots < l_q$$ be the integers $l \in [n]$ such that $l \geq \tau(l) \geq e_{k+1}^{\tau}$ and such that we can exchange the labels of dots $\tau(l)$ and $e_{k+1}^{\tau}$ in the planar graph of $\tau$ without modifying the skeleton of the graph. It is easy to see that $q$ is precisely the number of minimal vertices of $\tau$ on $S$. Let $l_{i_{k+1}} = \tau^{-1}(e_{k+1}^{\tau})$ and let $v_j^{\tau}$ be the $i_{k+1}$-th minimal vertex (from left to right) on $S$. We label $v_j^{\tau}$ with $l$, then we set $l := l+1$ and $S := S \backslash \{v_j^{\tau}\}$, and we apply steps 1.(a), (b) and (c) to $v_j^{\tau}$ instead of $v_i^{\tau}$. \end{enumerate} \end{algo}
By construction, the labelled graph $\mathcal{H}(\tau)$ is the linear graph of a permutation $\sigma \in \mathfrak{S}_n$ such that $$\text{DES}_2(\sigma) = \{d_2^{k,\tau},k \in [r]\}$$ and $$\{j_l(\sigma),l \in [\text{inv}_2(\sigma)]\} = \tau(\EXC(\tau)).$$
\begin{ex} Consider $\tau_0 = 956382471 \in \mathfrak{S}_9$ whose unlabelled and incomplete graph $\mathcal{H}(\tau_0)$ is depicted in Figure \ref{fig:FIG13}. \begin{itemize} \item As stated in Example \ref{ex:partialorder}, the minimal vertices of $\tau_0$ on $S = [9]$ are $(v_2^{\tau_0},v_5^{\tau_0},v_9^{\tau_0})$. Following step 2 of Algorithm \ref{algo:varphim1}, $k=0$ and the integers $ l \in [9]$ such that $\tau_0(l) \geq e_{k+1}^{\tau_0} = 1$ and such that the labels of dots $\tau_0(l)$ can be exchanged with $1$ in the planar graph of $\tau_0$ (see Figure \ref{fig:FIG10}) are $(l_1,l_2,l_3) = (4,6,9)$. By $\tau_0^{-1}(1) = 9 = l_3$, we label the third minimal vertex on $[9]$, \textit{i.e.} the vertex $v_9^{\tau_0}$, with the integer $l=1$.
Afterwards, following step 1.(b), since $v_9^{\tau_0}$ is the arrival of an incomplete arc of circle starting from a vertex $v_j^{\tau_0}$ with $1 = t_1^{\tau_0} \leq j$, and with $j <5$ because that arc of circle must begin before the arc of circle ending at $v_6^{\tau_0}$ in view of Fact 3 of Lemma \ref{lem:3facts}, we complete that arc of circle by making it start from the unique minimal vertex $v_j^{\tau_0}$ on $[1,5[$, \textit{i.e.} $j=2$, and we label $v_2^{\tau_0}$ with the integer $l = 2$ (see Figure \ref{fig:FIG14}). Note that as from now we know that the arc of circle ending at $v_5^{\tau_0}$ necessarily begins at $v_1^{\tau_0}$, because otherwise $v_1^{\tau_0}$, being the beginning of an arc of circle, would be the beginning of the arc of circle ending at $v_6^{\tau_0}$, which is absurd in view of Fact 3 of Lemma \ref{lem:3facts} because $\tau_0^{-1}(9) < \tau^{-1}(6)$, so we complete that arc of circle by making it start from $v_1^{\tau_0}$, which has been depicted in Figure \ref{fig:FIG14}.
\begin{figure}
\caption{Beginning of the labelling of $\mathcal{H}(\tau_0)$.}
\label{fig:FIG14}
\end{figure}
We now have $S = [9] \backslash \{2,9\}$ and $l=3$.
\item From Figure \ref{fig:FIG14}, the minimal vertices on $S = [9] \backslash \{2,9\}$ are $(v_3^{\tau_0},v_5^{\tau_0})$. Following step 2 of Algorithm \ref{algo:varphim1}, $k=1$ and the integers $l \in [9]$ such that $l \geq \tau_0(l) \geq e_{k+1}^{\tau_0} = 2$ and such that the labels of dots $\tau_0(l)$ can be exchanged with $2$ in the planar graph of $\tau_0$ (see Figure \ref{fig:FIG10}) are $(l_1,l_2) = (4,6)$. By $\tau_0^{-1}(2) = 6 = l_2$, we label the second minimal vertex on $S$, \textit{i.e.} the vertex $v_5^{\tau_0}$, with the integer $l=3$.
Afterwards, following step 1.(a), since $v_5^{\tau_0}$ is the arrival of the arc of circle starting from the vertex $v_1^{\tau_0}$, we label $v_1^{\tau_0}$ with the integer $l = 4$ (see Figure \ref{fig:FIG15}).
\begin{figure}
\caption{Beginning of the labelling of $\mathcal{H}(\tau_0)$.}
\label{fig:FIG15}
\end{figure}
We now have $S = [9] \backslash \{1,2,5,9\}$ and $l=5$.
\item From Figure \ref{fig:FIG15}, the minimal vertices on $S = \{3,4,6,7,8\}$ are $(v_3^{\tau_0},v_6^{\tau_0})$. Following step 2 of Algorithm \ref{algo:varphim1}, $k=2$ and the integers $l \in [9]$ such that $l \geq \tau_0(l) \geq e_{k+1}^{\tau_0} = 3$ and such that the labels of dots $\tau_0(l)$ can be exchanged with $3$ in the planar graph of $\tau_0$ (see Figure \ref{fig:FIG10}) are $(l_1,l_2) = (4,7)$. By $\tau_0^{-1}(3) = 4 = l_1$, we label the first minimal vertex on $S$, \textit{i.e.} the vertex $v_3^{\tau_0}$, with the integer $l=5$ (see Figure \ref{fig:FIG16}). Note that as from now we know that the arc of circle ending at $v_6^{\tau_0}$ necessarily begins at $v_4^{\tau_0}$ since it it is the only vertex left it may start from. Consequently, the arc of circle ending at $v_8^{\tau_0}$ necessarily starts from $v_7^{\tau_0}$ (otherwise it would start from $v_6^{\tau_0}$, which is prevented by Definition \ref{def:partialorder} because we cannot have $v_8^{\tau_0} \prec v_6^{\tau_0} \prec v_7^{\tau_0} \prec v_8^{\tau_0}$). The two latter remarks are taken into account in Figure \ref{fig:FIG16}.
\begin{figure}
\caption{Beginning of the labelling of $\mathcal{H}(\tau_0)$.}
\label{fig:FIG16}
\end{figure}
We now have $S = \{4,6,7,8\}$ and $l=6$.
\item From Figure \ref{fig:FIG16}, there is only one minimal vertex on $S = \{4,6,7,8\}$, \textit{i.e.} the vertex $v_6^{\tau_0}$. Following step 1 of Algorithm \ref{algo:varphim1}, we label $v_6^{\tau_0}$ with $l=6$.
Afterwards, following step 1.(a), since $v_6^{\tau_0}$ is the arrival of the arc of circle starting from the vertex $v_4^{\tau_0}$, we label $v_4^{\tau_0}$ with the integer $l = 7$ (see Figure \ref{fig:FIG17}).
\begin{figure}
\caption{Beginning of the labelling of $\mathcal{H}(\tau_0)$.}
\label{fig:FIG17}
\end{figure}
We now have $S = \{7,8\}$ and $l=8$.
\item From Figure \ref{fig:FIG17}, there is only one minimal vertex on $S = \{7,8\}$, \textit{i.e.} the vertex $v_6^{\tau_0}$. Following step 1 of Algorithm \ref{algo:varphim1}, we label $v_6^{\tau_0}$ with $l=8$.
Afterwards, following step 1.(a), since $v_8^{\tau_0}$ is the arrival of the arc of circle starting from the vertex $v_7^{\tau_0}$, we label $v_7^{\tau_0}$ with the integer $l = 9$ (see Figure \ref{fig:FIG18}).
\begin{figure}
\caption{Labelled graph $\mathcal{H}(\tau_0)$.}
\label{fig:FIG18}
\end{figure} \end{itemize}
As a conclusion, the graph $\mathcal{H}(\tau_0)$ is the linear graph of the permutation $\sigma_0 = 425736981 \in \mathfrak{S}_9$, which is mapped to $\tau_0$ by $\varphi$. \end{ex}
\begin{prop} \label{prop:bijection} We have $\varphi(\sigma) = \tau$, hence $\varphi$ is bijective. \end{prop}
\begin{proof} By construction , for all $k \in [0,\text{des}_2(\sigma)] = [0,r]$, \begin{align*} d_2^k(\sigma) &= d^{k,\tau} - c_k^{\tau},\\ c_k(\sigma) &= c_k^{\tau},\\ d^k(\sigma) &= d_2^k(\sigma) + c_k(\sigma) = d_2^{k,\tau}+c_k^{\tau} = d^{k,\tau}, \end{align*} so $\mathcal{G}(\sigma)$ has the same skeleton as the planar graph of $\tau$, \textit{i.e.} $\text{DES}(\varphi(\sigma)) = \text{DES}(\tau)$ and $\EXC(\varphi(\sigma)) = \EXC(\tau)$.
The labels of the circles of $\mathcal{G}(\sigma)$ are the elements of $$\{j_l(\sigma),l \in [s]\} = \tau(\EXC(\tau)),$$ and by construction of $\sigma$, every pair $(l,l') \in \EXC(\tau)^2$ such that we can exchange the labels $\tau(l)$ and $\tau(l')$ in the planar graph of $\tau$ is such that $$i < i' \Leftrightarrow l < l'$$ where $(i,\tau(l))$ and $(i,\tau(l'))$ are the two corresponding $2$-inversions of $\sigma$. Consequently, by definition of $\varphi(\sigma)$, the labels of the circles of $\mathcal{G}(\sigma)$ appear in the same order as in the planar graph of $\tau$ (\textit{i.e.} $\varphi(\sigma)(i) = \tau(i)$ for all $i \in \EXC(\varphi(\sigma)) = \EXC(\tau)$).
As a consequence, the dots of $\mathcal{G}(\sigma)$ and the planar graph of $\tau$ are labelled by the elements $$1 = e_1(\sigma) = e_1^{\tau} < e_2(\sigma) = e_2^{\tau} < \hdots < e_{n-s}(\sigma) = e_{n-s}^{\tau}.$$ As for the labels of the circles, to show that the above integers appear in the same order among the labels of $\mathcal{G}(\sigma)$ and the planar graph of $\tau$, it suffices to prove that $$\varphi(\sigma)^{-1}(e_i^{\tau}) < \varphi(\sigma)^{-1}(e_j^{\tau}) \Leftrightarrow \tau^{-1}(e_i^{\tau}) < \tau^{-1}(e_j^{\tau})$$ for all pair $(i,j)$ such that we can exchange the labels $e_i^{\tau}$ and $e_j^{\tau}$ in the planar graph of $\tau$ (hence in $\mathcal{G}(\sigma)$ since the two graphs have the same skeleton). This is guaranteed by Definition \ref{def:partialorder} because the vertices $v_i^{\tau}$ that are not the beginning of an arc of circle correspond with the labels of the dots of the planar graph of $\tau$.
As a conclusion, the planar graph of $\tau$ is in fact $\mathcal{G}(\sigma)$, \textit{i.e.} $\tau = \varphi(\sigma)$. \end{proof}
\section{Open problem} In view of Formula (\ref{eq:hanceli}) and Theorem \ref{theo:existsbijection}, it is natural to look for a bijection $\mathfrak{S}_n \rightarrow \mathfrak{S}_n$ that maps $(\maj_2,\widetilde{\text{des}_2},\text{inv}_2)$ to $(\text{amaj}_2,\widetilde{\text{asc}_2},\text{ides})$.
Recall that $\text{ides} = \text{des}_2$ and that for a permutation $\sigma \in \mathfrak{S}_n$, the equality $\widetilde{\text{des}_2}(\sigma) = \text{des}_2(\sigma)$ is equivalent to $\varphi(\sigma)(1) = 1$, which is similar to the equivalence $\widetilde{\text{asc}_2}(\tau) = \text{asc}_2(\tau) \Leftrightarrow \tau(1) = 1$ for all $\tau \in \mathfrak{S}_n$.
Note that if $\text{DES}_2(\sigma) = \bigsqcup_{p = 1}^r [i_p,j_p]$ with $j_p+1 < i_{p+1}$ for all $p$, the permutation $\pi = \rho_1 \circ \rho_2 \circ \hdots \circ \rho_r \circ \sigma$, where $\rho_p$ is the $(j_p-i_p+2)$-cycle $$\begin{pmatrix}
i_p & i_p+1 & i_p+2 & \hdots & j_p & j_p+1\\
\sigma(j_p+1) & \sigma(j_p) & \sigma(j_p-1) & \hdots & \sigma(i_p+1) & \sigma(i_p)
\end{pmatrix}$$
for all $p$, is such that $\text{DES}_2(\sigma) \subset \text{ASC}_2(\pi)$ and $\text{INV}_2(\sigma) = \text{INV}_2(\pi)$. One can try to get rid of the eventual unwanted $2$-ascents $i \in \text{ASC}_2(\pi) \backslash \text{DES}_2(\sigma)$ by composing $\pi$ with adequate permutations.
\nocite{*}
\section*{References}
\label{sec:biblio}
\end{document} | arXiv |
A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures
KRM Home
February 2019, 12(1): 243-267. doi: 10.3934/krm.2019011
Time-splitting methods to solve the Hall-MHD systems with Lévy noises
Zhong Tan 1, , Huaqiao Wang 1,2, and Yucong Wang 1,2,,
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
* Corresponding author: Yucong Wang
Received March 2017 Revised March 2018 Published July 2018
Fund Project: Z. Tan and Y.C. Wang is supported by the National Natural Science Foundation of China No. 11271305, 11531010. H. Wang is supported by National Postdoctoral Program for Innovative Talents No. BX201600020
In this paper, we establish the existence of a martingale solution to the stochastic incompressible Hall-MHD systems with Lévy noises in a bounded domain. The proof is based on a new method, i.e., the time splitting method and the stochastic compactness method.
Keywords: Hall-MHD systems, Lévy noises, martingale solutions, stochastic compactness method, time splitting method.
Mathematics Subject Classification: Primary: 35Q35, 60H15, 76N10.
Citation: Zhong Tan, Huaqiao Wang, Yucong Wang. Time-splitting methods to solve the Hall-MHD systems with Lévy noises. Kinetic & Related Models, 2019, 12 (1) : 243-267. doi: 10.3934/krm.2019011
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2018 Impact Factor: 1.38
Zhong Tan Huaqiao Wang Yucong Wang | CommonCrawl |
Solids with weak and strong electron correlations [PDF]
Prof. Dr. Peter Fulde
Abstract: A number of methods are discussed which may serve for a treatment of electron correlations in solids. When the electron correlations are relatively weak like in semiconductors or a number of ionic crystals one may start from a self-consistent field calculation and include correlations by quantum chemical methods. An incremental computational scheme enables us to obtain results of high quality for the ground state of those systems. A number of examples demonstrates that explicitely. Solids with strongly correlated electrons require the use of model Hamiltonians. With their help one can tackle the problem of determining spectral densities for those systems. The projection technique is a useful tool here. In strongly correlated $f$ electron systems electron or holes can crystallize with quite different physical consequences as in the case of a Wigner crystal or Verwey transition. Finally, different routes to heavy-fermion behavior are discussed, another hallmark of strongly correlated electrons.
Correlations in optical phonon spectra of complex solids [PDF]
G. Fagas,Vladimir I. Fal'ko,C. J. Lambert,Yuval Gefen
Physics , 1999, DOI: 10.1103/PhysRevB.61.9851
Abstract: Spectral correlations in the optical phonon spectrum of a solid with a complex unit cell are analysed using the Wigner-Dyson statistical approach. Despite the fact that all force constants are real, we find that the statistics are predominantly of the GUE type depending on the location within the Brillouin zone of a crystal and the unit cell symmetry. Analytic and numerical results for the crossover from GOE to GUE statistics are presented.
Optical signatures of electron correlations in the cuprates [PDF]
D. van der Marel
Abstract: The f-sum rule is introduced and its applications to electronic and vibrational modes are discussed. A related integral over the intra-band part of sigma(omega) which is also valid for correlated electrons, becomes just the kinetic energy if the only hopping os between nearest-neighbor sites. A summary is given of additional sum-rule expressions for the optical conductivity and the dielectric function, including expressions for the first and second moment of the optical conductivity, and a relation between the Coulomb correlation energy and the energy loss function. It is shown from various examples, that the optical spectra of high Tc materials along the c-axis and in the ab-plane direction can be used to study the kinetic energy change due to the appearance of superconductivity. The results show, that the pairing mechanism is highly unconventional, and mostly associated with a lowering of kinetic energy parallel to the planes when pairs are formed. Keywords: Optical conductivity, spectral weight, sum rules, reflectivity, dielectric function, inelastic scattering, energy loss function, inelastic electron scattering, Josephson plasmon, multi-layers, interlayer tunneling, transverse optical plasmon, specific heat, pair-correlation, kinetic energy, correlation energy, internal energy.
Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices [PDF]
E. V. Gorelik,D. Rost,T. Paiva,R. Scalettar,A. Klümper,N. Blümer
Physics , 2012, DOI: 10.1103/PhysRevA.85.061602
Abstract: We determine antiferromagnetic (AF) signatures in the half-filled Hubbard model at strong coupling on a cubic lattice and in lower dimensions. Upon cooling, the transition from the charge-excitation regime to the AF Heisenberg regime is signaled by a universal minimum of the double occupancy at entropy s=S/(N k_B)=s*=ln(2) per particle and a linear increase of the next-nearest neighbor (NNN) spin correlation function for s
A Coherent Nonlinear Optical Signal Induced by Electron Correlations [PDF]
Shaul Mukamel,Rafal Oszwaldowski,Lijun Yang
Physics , 2007, DOI: 10.1063/1.2820379
Abstract: The correlated behavior of electrons determines the structure and optical properties of molecules, semiconductor and other systems. Valuable information on these correlations is provided by measuring the response to femtosecond laser pulses, which probe the very short time period during which the excited particles remain correlated. The interpretation of four-wave-mixing techniques, commonly used to study the energy levels and dynamics of many-electron systems, is complicated by many competing effects and overlapping resonances. Here we propose a coherent optical technique, specifically designed to provide a background-free probe for electronic correlations in many-electron systems. The proposed signal pulse is generated only when the electrons are correlated, which gives rise to an extraordinary sensitivity. The peak pattern in two-dimensional plots, obtained by displaying the signal vs. two frequencies conjugated to two pulse delays, provides a direct visualization and specific signatures of the many-electron wavefunctions.
Strong electron correlations in FeSb2: An optical investigation and comparison with RuSb2 [PDF]
A. Herzog,M. Marutzky,J. Sichelschmidt,F. Steglich,S. Kimura,S. Johnsen,B. B. Iversen
Physics , 2010, DOI: 10.1103/PhysRevB.82.245205
Abstract: We report investigations of the optical properties of the narrow gap semiconductor FeSb2 in comparison with the structural homolog RuSb2. In the infrared region the latter shows insulating behavior in whole investigated temperature range (10 - 300 K) whereas the optical reflectivity of FeSb2 shows typical semiconductor behavior upon decreasing the temperature. The conduction electron contribution to the reflectivity is suppressed and the opening of a direct and an indirect charge excitation gap in the far-infrared energy region is observed. Those gap openings are characterized by a redistribution of spectral weight of the optical conductivity in an energy region much larger than the gap energies indicating that strong electron-electron correlations are involved in the formation of the charge gap. Calculations of the optical conductivity from the band structure also provided evidence for the presence of strong electronic correlations. Analyzing the spectra with a fundamental absorption across the gap of parabolic bands yields a direct gap at 130 meV and two indirect gaps at 6 and 31 meV. The strong reduction in the free-carrier concentration at low energies and low temperatures is also reflected in a change in the asymmetry of the phonon absorption which indicates a change in the phonon-conduction-electron interaction.
Nanodiamond Landmarks for Subcellular Multimodal Optical and Electron Imaging [PDF]
Mark A. Zurbuchen,Michael P. Lake,Sirus A. Kohan,Belinda Leung,Louis-S. Bouchard
Physics , 2015, DOI: 10.1038/srep02668
Abstract: There is a growing need for biolabels that can be used in both optical and electron microscopies, are non-cytotoxic, and do not photobleach. Such biolabels could enable targeted nanoscale imaging of sub-cellular structures, and help to establish correlations between conjugation-delivered biomolecules and function. Here we demonstrate a subcellular multi-modal imaging methodology that enables localization of inert particulate probes, consisting of nanodiamonds having fluorescent nitrogen-vacancy centers. These are functionalized to target specific structures, and are observable by both optical and electron microscopies. Nanodiamonds targeted to the nuclear pore complex are rapidly localized in electron-microscopy diffraction mode to enable "zooming-in" to regions of interest for detailed structural investigations. Optical microscopies reveal nanodiamonds for in-vitro tracking or uptake-confirmation. The approach is general, works down to the single nanodiamond level, and can leverage the unique capabilities of nanodiamonds, such as biocompatibility, sensitive magnetometry, and gene and drug delivery.
Investigation on Dynamic Calibration for an Optical-Fiber Solids Concentration Probe in Gas-Solid Two-Phase Flows [PDF]
Guiling Xu,Cai Liang,Xiaoping Chen,Daoyin Liu,Pan Xu,Liu Shen,Changsui Zhao
Sensors , 2013, DOI: 10.3390/s130709201
Abstract: This paper presents a review and analysis of the research that has been carried out on dynamic calibration for optical-fiber solids concentration probes. An introduction to the optical-fiber solids concentration probe was given. Different calibration methods of optical-fiber solids concentration probes reported in the literature were reviewed. In addition, a reflection-type optical-fiber solids concentration probe was uniquely calibrated at nearly full range of the solids concentration from 0 to packed bed concentration. The effects of particle properties (particle size, sphericity and color) on the calibration results were comprehensively investigated. The results show that the output voltage has a tendency to increase with the decreasing particle size, and the effect of particle color on calibration result is more predominant than that of sphericity.
Signatures of electron-electron correlations in the optical spectra of $α$-(BEDT-TTF)$_2M$Hg(SCN)$_4$ ($M$=NH$_4$ and Rb) [PDF]
N. Drichko,M. Dressel,A. Greco,J. Merino,J. Schlueter
Physics , 2005, DOI: 10.1007/BF02679519
Abstract: We interpret the optical spectra of $\alpha$-(BEDT-TTF)$_2M$Hg(SCN)$_4$ (M=NH$_4$ and Rb) in terms of a 1/4 filled metallic system close to charge ordering and show that in the conductivity spectra of these compounds a fraction of the spectral weight is shifted from the Drude-peak to higher frequencies due to strong electronic correlations. Analyzing the temperature dependence of the electronic parameters, we distinguish between different aspects of the influence of electronic correlations on optical properties. We conclude, that the correlation effects are slightly weaker in the NH$_4$ compound compared to the Rb one.
Electron and orbital correlations in Ca_{2-x}Sr_{x}RuO_{4} probed by optical spectroscopy [PDF]
J. S. Lee,Y. S. Lee,T. W. Noh,S. -J. Oh,Jaejun Yu,S. Nakatsuji,H. Fukazawa,Y. Maeno
Physics , 2002, DOI: 10.1103/PhysRevLett.89.257402
Abstract: The doping and temperature dependent optical conductivity spectra of the quasi-two-dimensional Ca_{2-x}Sr_xRuO_4 (0.0= | CommonCrawl |
\begin{document}
\title{The splitting number can be smaller than the matrix chaos number}
\author{Heike Mildenberger and Saharon Shelah}
\thanks{The first author was supported by a Minerva fellowship.}
\thanks{The second author's research was partially supported by the ``Israel Science Foundation'', founded by the Israel Academy of Science and Humanities. This is the second author's work number 753}
\address{Heike Mildenberger, Saharon Shelah, Institute of Mathematics, The Hebrew University of Jerusalem,
Givat Ram, 91904 Jerusalem, Israel }
\email{[email protected]}
\email{[email protected]}
\begin{abstract}
Let $\chi$ be the minimum cardinal of a subset of $2^\omega$ that cannot be made convergent
by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that $\gs < \chi$ is consistent. We thus answer a question by \Vo. We give two kinds of models for the strict inequality. The first is the combination of an
$\aleph_2$-iteration of some proper forcing with adding $\aleph_1$ random reals. The second kind of models is got by adding $\delta$ random reals to a model of $\MA_{<\kappa}$ for some $\delta \in [\aleph_1,\kappa)$. It was a conjecture of Blass that $\gs=\aleph_1 < \chi = \kappa$ holds in such a model. For the analysis of the second model we again use the creature forcing from the first model. \end{abstract}
\subjclass{03E15, 03E17, 03E35, 03D65}
\maketitle
\newcommand{\LOC}{{\mathbb L}}
\setcounter{section}{-1} \section{Introduction} We consider products of $\omega\times \omega$ matrixes $A=(a_{i,j})_{i,j\in\omega}$ and functions from $\omega$ to 2 or to some bounded interval of the reals.
The product $A \cdot f$ is defined
as usual in linear algebra, i.e., $(A \cdot f)(i) = \sum_ {j \in \omega} a_{i,j} \cdot f(j)$. We define \[ A\lim f := \lim_{i\to\infty} \sum_{j=0}^{\infty} (a_{i,j}\cdot f(j)). \]
Toeplitz (cf.\ \cite{Cooke}) showed: $A\lim$ is an extension of the ordinary limit iff $A$ is a regular matrix\index{regular matrix}, i.e.\ iff $\exists m \; \forall i\;
\sum_{j=0}^{\infty} |a_{i,j}| < m$ and $\lim_{i \to \infty} \sum_{j=0}^{\infty} a_{i,j} = 1$ and $\forall j \; \lim_{i \to \infty} a_{i,j} = 0$. Regular matrices are also called Toeplitz\index{Toeplitz matrix} matrices.
We are interested whether for many $f$'s simultaneously there is one $A$ such that all $A\lim f$ exist, and formulate our question in terms of cardinal characteristics.
Let $\ell^\infty$ denote the set of bounded real sequences, and let $\mathbb M$ denote the set of all Toeplitz matrices. \Vo\ \cite{Vojtas88} defined for ${\mathbb A} \subseteq {\mathbb M}$ the chaos relations $\chi_{{\mathbb A}, \infty}$ and their norms $\Vert \chi_{{\mathbb A}, \infty} \Vert$ \begin{eqnarray*} \chi_{{\mathbb A},\infty} &=& \{(A,f) \such A \in {\mathbb A} \;\wedge\; f \in \ell^\infty \;\wedge\; A \lim f \mbox{ does not exist}\},\\
\Vert \chi_{{\mathbb A},\infty} \Vert &=& \min \{ |{\mathcal F}| \such {\mathcal F} \subseteq \ell^\infty \: \wedge \\ && \makebox[2cm]{} (\forall A \in {\mathbb A}) \; (\exists f \in {\mathcal F}) \; A \lim f \mbox{ does not exist}\}. \end{eqnarray*} By replacing $\ell^\infty$ by ${}^\omega 2$, the set of $\omega$-sequences with values in 2, we get the variations $\chi_{{\mathbb A},2}$. In \cite{Mi6} we showed that for the cardinals we are interested in, ${}^\omega 2$ and $\ell^\infty$ give the same result. From now on we shall work with ${}^\omega 2$.
\Vo\ (cf.\ \cite{Vo2}) also gave some bounds valid for any ${\mathbb A}$ that contains at least all matrices which have exactly one non-zero entry in each line: \[ \gs \leq \Vert\chi_{{\mathbb A},2}\Vert \leq \gb \cdot \gs.\] We write $\chi$ for $\Vert\chi_{{\mathbb M},2}\Vert$.
In \cite{Mi6} we showed that $\chi < \gb \cdot \gs$ is consistent relative to \zfc. Here, we show the complementary consistency result, that
$\gs < \chi$ is consistent. We get the convergence with positive matrices.
Now we recall here the definitions of the cardinal characteristics $\gb$ and $\gs$ involved: The order of eventual dominance $\leq^\ast$ is defined as follows: For $f,g \in \omega^\omega$ we say $f \leq^\ast g$ if there is $k \in \omega$ such that for all $n \geq k$ we have $f(n) \leq g(n)$.
The unbounding number $\gb$ is the smallest size of a subset ${\mathcal B} \subseteq {}^\omega \omega$ such that for each $f \in {}^\omega \omega$ there is some $b \in {\mathcal B}$ such that $b \not\leq^\ast f$. The splitting number $\gs$ is the smallest size of a subset ${\mathcal S} \subseteq [\omega]^\omega$ such that for each $X \in \potinf$ there is some $S \in {\mathcal S}$ such that $X \cap S$ and $X \setminus S$ are both infinite. The latter is expressed as ``$S$ splits $X$'', and $\mathcal S$ is called a splitting family. For more information on these cardinal characteristics, we refer the reader to the survey articles \cite{Blasshandbook, vanDouwen, Vaughan}.
If $A \lim f$ exists, then also $A' \lim f$ exists for any $A'$ that is gotten from $A$ be erasing rows and moving the remaining (infinitely many) rows together. We may further change $A'$ by keeping only finitely many non-zero entries in each row, such that the neglected ones have a negligible absolute sum, and then possibly multiplying the remaining ones such that they again sum up to 1. Hence, after possibly further deleting of lines we may restrict the set of Toeplitz matrices to linear Toeplitz matrices. A matrix is linear iff each column $j$ has at most one entry $a_{i,j} \neq 0$ and for $j < j'$ the $i$ with $a_{i,j} \neq 0$ is smaller or equal to the $i$ with $a_{i,j'} \neq 0$ if both exist, in picture \begin{small} \begin{equation*} \begin{pmatrix} c_0(0) & \dots c_0(\mup(c_0) -1) & 0 & \dots & 0 & 0 & \dots\\ 0 & \dots 0 & c_1(\mdn(c_1)) & \dots & c_1(\mup(c_1) -1) & 0 & \dots\\ 0 & \dots 0 & 0 &\dots& 0 & c_2(\mdn(c_2)) &\dots \\ \vdots \end{pmatrix} \end{equation*} \end{small}
Linear matrices can be naturally (as in the picture)
read as $(c_n \such n \in \omega)$ where $c_n \colon [\mdn(c_n), \mup(c_n)) \to [0,1]$, $c_n(j) = a_{n,j}$, give the finitely many non-zero entries in row $n$, and $\mup(c_{n-1}) = \mdn(c_n)$. The $c_n$ are special instances of the weak creatures in the sense of \cite{RoSh:470}. In the next two sections we shall show: The $c_n$'s coming from the trunks of the conditions in the generic filter of our forcing $Q$
give matrices that make, after multiplication, members of ${}^\omega 2$ from the ground model and members of ${}^\omega 2$ of any random extension convergent.
\section{A creature forcing}\label{S1}
In this section, we give a self-contained description of the creature forcing $Q$ which is the main tool for building the two kinds of models in the next section.
Moreover, we explain the connections and give the references to \cite{RoSh:470}, so that the reader can identify it as a special case of an extensive framework.
\begin{definition}\label{1.1} a) We define a notion of forcing $Q$. Its members $p$ are of the form $p= (n,c_0,c_1, \dots )= (n^p, c_0^p, c_1^p, \dots )$ such that \begin{myrules} \item[(1)] $n^p \in \omega$. \item[(2)] For each $i \in \omega$ there are $\mdn(c_i) < \mup(c_i) < \omega$ such that\\ $c_i \colon [\mdn(c_i),\mup(c_i)) \to [0,1]$, such that $(\forall k \in \dom(c_i))(c_i(k) \cdot k! \in {\mathbb Z})$. \item[(3)] $w(c_i) = \{ k \in [\mdn(c_i),\mup(c_i)) \such c_i(k) \neq 0 \}$, and $\sum_{k \in w(c_i)} c_i(k) = 1$. \item We let $\norm(c_i) = \mdn(c_i)$. We denote by $K$ the set of those $c_i$. \item[(4)] $\mup(c_i) = \mdn(c_{i+1})$. \end{myrules}
We let $p \leq q$ (``$q$ is stronger than $p$'', we follow the Jerusalem convention) if \begin{myrules} \item[(5)] $n^p \leq n^q$. \item[(6)] $c_0^p = c_0^q, \dots , c^p_{n^p-1} = c^q_{n^p-1}$. \item[(7)] there are $n^p \leq k_{n^p} < k_{n^p+1} < \dots $ and there are non-empty sets $u \subseteq [k_n,k_{n+1})$ and rationals $d_\ell >0$ for $\ell \in u$ such that $c^q_n = \sum\{d_\ell \cdot c_\ell^p \such \ell \in u\}$ and $\sum_{\ell \in u } d_\ell = 1$. We let $ \Sigma(\langle c_\ell \such \ell \in [k_n, k_{n+1}) \rangle$ denote the collection of all $c_n$ gotten with any $u \subseteq [k_n,k_{n+1})$ and any weights $d_\ell$ for $\ell \in u$. Thus $\mdn(c^q_n) = \mdn(c^p_{k_n})$ and $\mup(c^q_n)= \mup(c^p_{k_{n+1} -1})$. \end{myrules}
b) We write $p \leq_i q$ iff $n^p = n^q$ and $c_j^q = c_j^p$ for $j < n^p + i$.
\end{definition}
\begin{remark} The notation we used in \ref{1.1} is natural to describe our forcing in a compact manner. However, it does not coincide with the notation given for the general framework in \cite{RoSh:470}. Here is a translation: We write \\ $((c_0^p,c_1^p, \dots c_{n-1}^p), c_n^p, c_{n+1}^p, \dots)$ instead of $(n^p, c_0^p, c_1^p, \dots)$, which contains the same information. Then we write \begin{equation}\tag{$\ast$}\label{ast} ((c_0^p,c_1^p, \dots c_{n-1}^p), c_n^p, c_{n+1}^p, \dots) = (w^p, t_0^p, t_1^p \dots). \end{equation} Then the $t_i^p$ are (simple cases) of components of weak creatures in the sense of \cite[1.1.1 to 1.1.10]{RoSh:470}. If we write $\bf t = (\norm({\bf t}), \val({\bf t}), \dis({\bf t}))$ for a weak creature in the sense of \cite{RoSh:470}, then we have that $\dis$ is the empty function, and $t_i$ is part of such a ${\bf t}$ in the following sense: $\norm({\bf t}) = \mdn(t_i)$, $\rge(\val({\bf t})) = \{ t_i\}$. We set ${\bf H}(i) = \left\{0,\frac{1}{i!},\frac{2}{i!}, \dots , \frac{i! - 1}{i!}, 1\right\}$ and $t_i \in \prod_{m \in [\mdn(t_i), \mup(t_i))} {\bf H}(i)$. $K$ is a collection of weak creatures, and $\Sigma$ from \ref{1.1}(7) is a composition operation. Thus our $Q$ is $Q^*_{s\infty}(K,\Sigma)$ in Ros\l anowski's and Shelah's framework and is finitary and nice and satisfies some norm-conditions. We do not give the definitions of these properties, because we are working with our specific case. The interested reader should consult \cite{RoSh:470}. We use $w$, $w^p$, $w^q$ for the trunks in the representation as in \eqref{ast}.
\end{remark}
In order to make our work self-contained, we write a proof that $Q$ allows continuous reading of names and hence is proper. In this section, we use the notation as in \eqref{ast}, because it is more suitable.
\begin{definition} \label{1.3} $q=(w^q, t_0^q, \dots)$ approximates $\mathunderaccent\tilde-3 {\tau}$ at $t_n^q$ iff for all $r$ (if $q \leq r$ and $r$ forces a value to $\mathunderaccent\tilde-3 {\tau}$, then $r^{q,n}$ forces this, where $t_i^{r^{q,n}} = t_i^r$ for $i < n$ and $\{ t_i^{r^{q,n}} \such i \geq n \} = \{ t_i^q \such i < \omega, \mdn(t_i^q) \geq \mup(t^{r}_{n-1}) \}.$) \end{definition}
\begin{definition}\label{1.4} For $w\in\bigcup\limits_{m<\omega}\prod\limits_{i<m}{\bf H}(i)$ and ${\mathcal S}\in [K]^{\textstyle {\leq}\omega}$ we define the set $\pos(w,{\mathcal S})$ of possible extensions of $w$ from the point of view of ${\mathcal S}$ (with respect to $(K,\Sigma)$) as: \begin{eqnarray*} \pos^*(w,{\mathcal S}) & =& \Sigma({\mathcal S})\\ (&=& \{u: (\exists s\in\Sigma({\mathcal S}))(\langle w,u\rangle\in\val[s])\} \\ &&\mbox{for a general creature forcing}), \end{eqnarray*}
\[\hspace{-1cm} \begin{array}{ll} \pos(w,{\mathcal S})=\{u:&\!\!\!\!\mbox{there are disjoint sets }{\mathcal S}_i\mbox{ (for $i<m<\omega$) with }\bigcup\limits_{i<m}{\mathcal S}_i={\mathcal S}\\ \ &\mbox{and a sequence }0<\ell_0<\ldots<\ell_{m-1}<\lh(u)\mbox{ such that}\\ \ & u{\restriction} \ell_0\in\pos^*(w,{\mathcal S}_0)\ \&\ \\ \ & u{\restriction} \ell_1\in\pos^*(u{\restriction}\ell_0,{\mathcal S}_1)\ \&\ \ldots\ \&\ u\in\pos^*(u{\restriction} \ell_{m-1},{\mathcal S}_{m-1})\}.\\ \end{array} \] \end{definition}
\begin{lemma}\label{1.5} \label{deciding}
(The case $\ell=0$ of \cite[Theorem 2.1.4]{RoSh:470}) $Q$ has continuous reading of names, i.e. if $p \Vdash \mathunderaccent\tilde-3 {\tau} \colon \omega \to V \mbox{ (old universe) }$ there is $q=(w^q,s_0, s_1 \dots )$ such that \begin{myrules} \item[$(\alpha)$] $p \leq_0 q \in Q$, \item[$(\beta)$] if $n < \omega$ and $m \leq \mup(s_{n-1})$ then the condition $q$ approximates $\mathunderaccent\tilde-3 {\tau}(m)$ at $s_n$. \nothing{ d
$q \leq r$ and $r$ forces a value to $\mathunderaccent\tilde-3 {\tau}(i)$ and $i < \mup(t^r_{n_r -1})$, then $r^q$ forces this, where $n^{r^q} = n^r$, $t_n^{r^q} = t_n^r$ for $n < n^r$ and $\{ t_n^{r^q} \such n \geq n^r \} = \{ t_n^q \such n < \omega, \mdn(t_n^q) > \mup(t^{r^q}_{n^{r^q}-1} \}$. } \end{myrules} \end{lemma}
\proof Let $p = (w^p,t_0^p,t_1^p, \dots )$. Let $w^q = w^p$. Now, by induction on $n \geq 0$ we define $q_n, s_n, t^n_{n+1}, t^n_{n+2}, \dots$ such that: \begin{myrules} \item[(i)] $q_0 = p$, \item[(ii)] $q_{n+1} = ( w^p,s_0,\dots,s_n,t^n_{n+1},t^n_{n+2},\dots) \in Q$, \item[(iii)] $q_n \leq_n q_{n+1}$, \item[(iv)] if $w_1 \in {\rm pos}(w^p,s_0,\dots s_{n-1})$, and $m \leq \mup(s_{n-1})$ and there is a condition $r \in Q$, $r \geq_0 (w_1,s_n,t^n_{n+1},t^n_{n+2}, \dots )$ which decides the value of $\mathunderaccent\tilde-3 {\tau}(m)$ then the condition $(w_1,s_n,t^n_{n+1},t^n_{n+2}, \dots )$ already does it. \end{myrules}
Arriving at stage $n \geq 0$ we have defined $$q_n=(w^p,s_0,s_1,\dots,s_{n-1},t^{n-1}_n,t^{n-1}_{n+1},\dots).$$ Let $\langle (w^n_i, m^n_i): i< K_n\rangle$ be an enumeration of \[{\rm pos}(w^p,s_0,\ldots,s_{n-1})\times(\mup(s_{n-1})+1)\] (since each ${\bf H}(m)$ is finite, $K_n$ is finite). Next choose by induction on $k\leq K_n$ conditions $q_{n,k}\in Q$ such that: \begin{myrules} \item[$(\alpha)$] $q_{n, 0}=q_n$. \item[$(\beta)$] $q_{n,k}$ is of the form $(w^p,s_0,\dots,s_{n-1},t_n^{n,k}, t^{n, k}_{n+1}, t^{n, k}_{n + 2},\ldots)$. We set $w^n_k = (w^p,s_0, \dots s_{n-1})$. \item[$(\gamma)$] $q_{n,k}\leq_n q_{n,k+1}$. \item[$(\delta)$] If, in $Q$, there is a condition $r\geq_0 (w^n_k,t^{n,k}_n,t^{n,k}_{n+1},t^{n,k}_{n+2},\ldots)$ which decides (in $Q$) the value of $\mathunderaccent\tilde-3 {\tau}(m^n_k)$, then \[(w^n_k,t^{n,k+1}_n,t^{n,k+1}_{n+1},t^{n,k+1}_{n+2},\ldots)\in Q\] is a condition which forces a value to $\mathunderaccent\tilde-3 {\tau}(m^n_k)$.
\end{myrules} For this part of the construction we need our standard assumption that we may iterate the process in \ref{1.1}(7). Note, that choosing $(w^n_k,t^{n,k+1}_n,t^{n,k+1}_{n+1},t^{n,k+1}_{n+2},\ldots)$ we want to be sure that \[(w^p,s_0,\ldots,s_{n-1},t^{n,k+1}_n,t^{n,k+1}_{n+1},t^{n,k+1}_{n+2},\ldots) \in Q.\] Next, the condition $q_{n+1}\stackrel{\rm def}{=} q_{n,K_n}\in Q$ satisfies (iv): the keys are the clause ($\delta$) and the fact that \[(w^n_k,t^{n,k+1}_n,t^{n,k+1}_{n+1},t^{n,k+1}_{n+2},\ldots)\leq (w^n_k,t^{n,K_n}_n, t^{n,K_n}_{n+1},t^{n,K_n}_{n+2},\ldots)\in Q.\] Thus $s_n\stackrel{\rm def}{=} t^{n,K_n}_n$, $t^{n+1}_{n+k}\stackrel{\rm def}{=} t^{n,K_n}_{n+k}$ and $q_{n+1}= (w^p,s_0, \dots, s_n,t^{n+1}_n, \dots )$
are as required.
Now, by a fusion argument \[q\stackrel{\rm def}{=}(w^p,s_0,s_1,\ldots,s_l,s_{l+1},\ldots)=\lim_n q_n\in Q.\] It is easily seen that $q$ satisfies the assertions of the theorem. \proofend \nothing{ \begin{definition}\label{1.6} For $p,q \in Q$ we write $q \leq_{apr} q$ if there is some $n$, $w^q$ such that $q=(w^q,t_n^p,t_{n+1}^p,\dots)$ and $p \leq q$. That is, $q$ is only in the trunk stronger than $p$. \end{definition} }
\begin{lemma}(\cite[Corollary 2.1.6]{RoSh:470}) \begin{myrules} \item[(a)] Suppose that $\mathunderaccent\tilde-3 {\tau}_n$ are $Q$-names for ordinals and $q\in Q$ is a condition satisfying $(\beta)$ of \ref{deciding}. Further assume that $q\leq r\in Q$ and $r\Vdash $``$\mathunderaccent\tilde-3 {\tau}_m=\alpha$'' (for some ordinal $\alpha$).\\ Then $q'= r^{q,m}$ forces this.
\item[(b)] The forcing notion $Q$ is proper.
\end{myrules} \end{lemma}
\proof (a) is a special case of the previous lemma.
For (b), we use the equivalent definition of properness given in \cite[III.2.13]{Sh:h}, and the fact that $\{q'\!\in Q: (\exists r \geq q)(\exists n) q'=r^{q,n}\}$ is countable provided $\bigcup\limits_{i<\omega}{\bf H}(i)$ is countable.
\section{The effect of $Q$ on random reals}\label{S2}
Let $G$ be $Q$-generic over $V$. We set $c^G_n = c_n^q$ for $q \in G$ and $n^q >n$. This is well defined.
Let $\mathunderaccent\tilde-3 {c_n}$ be a name for it. Our aim is to show that multiplication by the matrix whose $n$-th row is $c_n$ makes any real from the ground model and even any real from a random extension of the ground model convergent. For background information about random reals we refer the reader to \cite[\S 42]{Jech}. The Lebesgue measure ist denoted by $\Leb$. With ``adding $\kappa$ random reals'' we mean forcing with the measure algebra $R_\kappa$ on $2^{\omega \times\kappa}$, that is adding $\kappa$ random reals at once or ``side-by-side'' and not successively.
\begin{definition}\label{2.1} \begin{myrules} \item[(1)] Let $\may_k(p) = \{ c_n^r \such p \leq_k r, n \geq n^p + k\}$. \item[(2)] For a creature $c$ and $\eta \in {}^\omega 2$ let $\aver(\eta,c) = \sum_{k \in w(c)} c(k) \eta(k)$. \end{myrules} \end{definition}
\begin{mainlemma}\label{2.2} Assume that \begin{myrules} \item[(A)] $\mathunderaccent\tilde-3 {\eta}$ is a random name of a member of ${}^\omega 2$, $ \mathunderaccent\tilde-3 {\eta} = f(\mathunderaccent\tilde-3 {r})$ where $f$ is Borel and $\mathunderaccent\tilde-3 {r}$ is as name of the random generic real, \item[(B)] $p \in Q$, \item[(C)] $k^* < \omega$. \end{myrules} Then for every $k \geq k^*$ there is some $q(k) \in Q$ such that \begin{myrules} \item[($\alpha$)]$p \leq_{k^*} q(k)$, \item[$(\beta)$] for all $\ell$, if $k^* \leq k < \ell < \omega$ and $c_1, c_2 \in \may_\ell(q(k))$ then
$$ \frac{1}{\ell!} >
\Leb\left\{r \such \frac{3}{2^k} \leq \left| \aver(f(r), c_1) -
\aver(f(r), c_0) \right| \right\}.$$
\end{myrules} \end{mainlemma}
\proof For $q \in Q$ and $k, \ell \in \omega$, $i \in \{0,1, \dots, 2^k\}$ we set \begin{eqnarray*}
\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c) &=& \Expect\left(\left|\aver(\mathunderaccent\tilde-3 {\eta},c) -
\frac{i}{2^k}\right|\right)\\ & = & \int_0^1
\left|\aver(f(r),c)
- \frac{i}{2^k} \right| \; d \Leb(r),\\ \eee^\ell_{k,i}(\mathunderaccent\tilde-3 {\eta},q) &=& \inf \{ \err_{k,i}(\mathunderaccent\tilde-3 {\eta},c) \such c \in \may_\ell(q)\}. \end{eqnarray*}
Note that $\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c)$ is a real and no longer a random name. So the infimum is well-defined.
Now, if $\ell_1 < \ell_2$ then $\may_{\ell_1}(q) \supseteq \may_{\ell_2}(q)$ and hence \begin{equation*}\label{mono} \eee^{\ell_1}_{k,i}(\mathunderaccent\tilde-3 {\eta},q) \leq \eee^{\ell_2}_{k,i}(\mathunderaccent\tilde-3 {\eta},q). \end{equation*}
So $\langle \eee^\ell_{k,i}(\mathunderaccent\tilde-3 {\eta},q) \such \ell \in \omega \rangle$ is an increasing bounded sequence and
\begin{equation*} \eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta},q) = \lim \langle \eee^\ell_{k,i}(\mathunderaccent\tilde-3 {\eta},q) \such \ell \in \omega \rangle \end{equation*}
is well-defined.
We fix $i \leq 2^k$, until Subclaim 4, when we start looking at all $i$ together. \nothing{, and later we shall vary $k$ as well.}
Subclaim 1: There is some $q^{k,i}_1=q_1 \geq_{k^\ast} p$ such that for $\ell \geq k^\ast$ $$\eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta}, p) - \frac{1}{\ell} \leq \err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c^{q_1}_\ell) \leq \eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta}, p) + \frac{1}{\ell}.$$ Moreover, if $\mdn(c_{\ell'}^{q_1}) = \mdn(c^p_{\ell})$ then $\eee^{\ell'}_{k,i}(\mathunderaccent\tilde-3 {\eta},q_1) \geq \eee_{k,i}^*(\mathunderaccent\tilde-3 {\eta},p)-\frac{1}{\ell}$.
Why? We choose $c_\ell^{q_1}$ by induction on $\ell$: For $\ell \leq n^p + k^\ast$, we take $c_\ell^{q_1} = c_\ell^p$. Suppose that we have chosen $c_m^{q_1}$ for $m < \ell$ and that we are to choose $c_\ell^{q_1}$, $\ell > n^p + k^\ast$. We set $\eps = \frac{1}{\ell}$. By possibly end-extending $c_{\ell -1}^{q_1}$ by zeroes we may assume that $\mup(c_{\ell -1}^{q_1}) = \mup(c_{\ell'}^p)$ for such a large $\ell' \geq \ell$ such that for all $\ell'' \geq \ell'$, $\eee_{k,i}^{\ell''}(\mathunderaccent\tilde-3 {\eta},p) \geq \eee_{k,i}^\ast(\mathunderaccent\tilde-3 {\eta},p) - \eps$. Then we take $c_\ell = c_\ell^{q_1} \in \may_{\ell''}(p)$ such that $\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c_\ell^{q_1}) \leq \eee_{k,i}^{\ell''}(\mathunderaccent\tilde-3 {\eta},p) + \eps \leq \eee_{k,i}^\ast(\mathunderaccent\tilde-3 {\eta},p) + \eps$. On the other side we have that $\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c_\ell^{q_1}) \geq \eee_{k,i}^{\ell''}(\mathunderaccent\tilde-3 {\eta},p) \geq \eee_{k,i}^\ast(\mathunderaccent\tilde-3 {\eta},p) - \eps$. The fact that
this holds also for $\ell' \leq \ell$ if $\mdn(c_{\ell'}^{q_1}) = \mdn(c^p_{\ell})$ yields the ``moreover'' part.
Subclaim 2: In Claim 1, if $\ell \geq k^*$ and $q^{k,i}_1 \leq_\ell q_2$ then $$\eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta},q) - \frac{1}{\ell} \leq \err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c_\ell^{q_2}) \leq \eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta},q) + \frac{1}{\ell}.$$.
Why? By the definition if suffices to show: \begin{equation}\tag{$\otimes$}\label{otimes} \begin{split} & \mbox{if } \ell_1 < \cdots < \ell_t < \omega \mbox{ and } d_1 , \dots d_t \geq 0 \mbox{ and } d_1 + \cdots + d_t =1,\\ & \mbox{and } c_\ell^{q_2} =d_1 c_1^{q_1} + \cdots + d_t c_t^{q_1},\\ &\mbox{then } \eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta},q_1) - \frac{1}{\ell} \leq \err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c_\ell^{q_2}) \leq \eee^*_{k,i}(\mathunderaccent\tilde-3 {\eta},q_1) + \frac{1}{\ell}. \end{split} \end{equation}
The first inequality holds by the ``moreover'' after the first inequality in the previous claim. For the second inequality it suffices to show that $$\err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c) \leq \sum_{s=1}^t d_s \err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c_s^{q_1}).$$ For this is suffices to show that
$$\Expect\left(\left|\aver(\mathunderaccent\tilde-3 {\eta},c) - \frac{i}{2^k} \right|\right) \leq \sum_{s=1}^t d_s
\Expect\left(\left|\aver(\mathunderaccent\tilde-3 {\eta},c_{s}^{q_1}) -
\frac{i}{2^k} \right|\right), $$ and writing this explicitly noting that $\Expect$ is actually a Lebesgue integral and that $d_s \geq 0$ and that $\sum_{s} d_s = 1$ we finish by the triangular inequality.
Subclaim 3: Let $q^{k,i}$ be as in Subclaim 2. For all $\ell$, if $c_0, c_1 \in \may_\ell(q^{k,i}_1)$, then $$ \frac{2^{k+1}}{\ell} \geq \Leb\left\{ r \such \aver(f(r), c_0) \geq \frac{i+1}{2^k} \wedge \aver(f(r),c_1) \leq \frac{i-1}{2^k} \right\}.$$
Why? Consider $c = \frac{1}{2} c_0 + \frac{1}{2} c_1 \in \may_\ell(q_1)$ . Write \\ $A=\left\{ r \such \aver(f(r), c_0)
\geq \frac{i+1}{2^k} \wedge \aver(f(r),c_1) \leq \frac{i-1}{2^k} \right\}$.
\begin{eqnarray*} \frac{2}{\ell} & \geq & \frac{1}{2} \err_{k,i}(\mathunderaccent\tilde-3 {\eta},c_0) + \frac{1}{2} \err_{k,i}(\mathunderaccent\tilde-3 {\eta},c_1) -\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c)\\ & = & \int_0^1
\left(\frac{1}{2}\left|\aver(f(r),c_0)
- \frac{i}{2^k} \right| +\right.
\frac{1}{2}\left|\aver(f(r),c_1)
- \frac{i}{2^k} \right| -\\ &&
\left.\left|\aver(f(r)),c)
- \frac{i}{2^k} \right|\right) d \Leb(r)\\ &\geq& \int_A
\left(\frac{1}{2}\left|\aver(f(r),c_0)
- \frac{i}{2^k} \right| +\right.
\frac{1}{2}\left|\aver(f(r),c_1)
- \frac{i}{2^k} \right| -\\ &&
\left.\left|\aver(f(r)),c)
- \frac{i}{2^k} \right|\right) d \Leb(r)\\ &\geq& \frac{1}{2^{k}} \Leb(A). \end{eqnarray*}
\nothing{ We divide the truth value in the defintion of $\err_{k,i}(\mathunderaccent\tilde-3 {\eta},c)$ by the events above. Outside it is $\leq \frac{1}{2} \err_{k,i}(\mathunderaccent\tilde-3 {\eta}, c_1) + \frac{1}{2} \err_{k,i}(\mathunderaccent\tilde-3 {\eta},c_2) \leq \err^*_{k,i} + \frac{1}{\ell}$. Inside the result drops by $\Leb$(event above) $\times \frac{1}{2^{k-1}}$
as $d_1 \leq -\frac{1}{2^k}$ and $d_2 \geq \frac{1}{2^k}$ implies that $d_1 + d_2| \leq |d_1| +|d_2| - \frac{1}{2^{k_1}}$. }
Subclaim 4: For every $q \in Q$ and $k^*$ we can find $q^{k}$ such that \begin{myrules} \item[$\alpha$)] $q \leq_{k^*} q^{k}$,
\item[$\beta$)] if $\ell \in [k,\omega)$ and $c_0,c_1 \in \may_\ell(q^{k})$ and $i \in \{1,2, \dots, 2^k-1\}$ then $\frac{2^{k+1}}{\ell} > \Leb\left\{ r \such \aver(f(r), c_0) \geq \frac{i+1}{2^k} \wedge \aver(f(r),c_1) \leq \frac{i-1}{2^k} \right\}.$
\item[$\gamma$)] This holds also for every $q^{*} \geq q^{k}$. \end{myrules} Why? Repeat Subclaims 1 and 2 and 3 choosing $q^{k,i}$, $i = 0,1, \dots, 2^k$. We let $q_0 = q$ and choose $q^{k,i+1} $ such that it
relates to $q^{k,i}$ like $q_1$ to $q$.
Now $q^{k}=q^{k,2^k}$ is o.k. Note that according to \eqref{otimes} thinning and averaging can only help.
Subclaim 5: Let $q^k$ be as in Subclaim 4. For $\ell \geq k$ there is $q(k,\ell)\geq_{\ell-1}q^k$ such that for $c_0,c_1 \in \may_\ell(q(k,\ell))$, $$ \frac{1}{\ell!} >
\Leb\left\{r \such \frac{3}{2^k} \leq \left| \aver(f(r), c_1) -
\aver(f(r), c_0) \right| \right\}.$$
Why? The event $\frac{3}{2^k} \leq \left| \aver(f(r), c_1) -
\aver(f(r), c_0) \right|$ implies that for some $i \in \{1,2,\dots,2^k-1\}$ we have $\aver(f(r), c_1) \geq \frac{i+1}{2^k} \wedge \aver(f(r), c_2) \leq \frac{i-1}{2^k}$ or vice versa. So it is incuded in the union of $2 \times (2^k -1)$ events, each of measure $\leq \frac{2^{k+1}}{\ell}$.
Hence it itself has measure $\leq \frac{2^{2k+2}}{\ell}$. By thinning out $q^k$ (by moving the former $\ell$ far out by putting in a lot of zeroes and thus having as new $c_\ell$'s weak creatures that were formerly labelled with a much larger $\ell$ and thus giving a much smaller quotient according to Subclaim 4) we replace $\frac{2^{2k+2}}{\ell}$ by $\frac{1}{\ell!}$.
Subclaim 6: Finally we come to the $q(k)$ from part $(\beta)$ of the lemma:
For any $k$ there is $q(k)$ such that $q \leq_{k^*} q(k)$ and for any $\ell\geq k$ and any
$c_1, c_2 \in \may_\ell(q^*)$ then
$$ \frac{1}{\ell!} >
\Leb\left\{r \such \frac{3}{2^k} \leq \left| \aver(f(r), c_1) -
\aver(f(r), c_0) \right| \right\}.$$
Why? Like in the previous claim we choose inductively $q(k,\ell)$ such that $q_0 =p$ and $q(k,\ell+1) \geq_\ell q(k,\ell)$ and $(q(k,\ell+1),q(k,\ell), \ell)$ are like $(q(k,\ell), q, \ell)$ from Subclaim 5, but for larger and larger $\ell$. Now $$q(k) =(n^p + k, c_0^p, \dots, c_{n^p + k}^p, c_{n^p + k +1}^{q(k,n^p+k+1)}, c_{n^p +k+1}^{q(k,n^p+k+2)}, \dots) $$ is as required in $(\alpha)$ and $(\beta)$ of the conclusion; we have even $q(k) \geq_k p$. \proofend
\begin{conclusion}\label{2.3} $\Vdash_Q $ ``if $\mathunderaccent\tilde-3 {\eta} \in V$ is a random name of a
member in $2^\omega$ (i.e.\ a name for a real in $V^{R_\omega}$) then ``$\Vdash_{R_\omega} \langle \aver(\mathunderaccent\tilde-3 {\eta}, \mathunderaccent\tilde-3 {c}_n) \such n \in \omega \rangle$ converges'' '' \end{conclusion}
\proof Let $q \in Q$ and $\eps > 0$ be given. Let $\mathunderaccent\tilde-3 {\eta} = f(\mathunderaccent\tilde-3 {r})$, $f \in V$, be a random name for a real. We take $k_0$ such that $\frac{3}{2^k} < \eps$. Then we take for $q(k)\geq q$ as in the Main Lemma. We set $$A_{k,c_0,c_1} = \left\{ r \such
\frac{3}{2^k} > \left| \aver(f(r), c_1) -
\aver(f(r), c_0) \right| \right\}.$$ Since $\sum_{\ell \geq 1} \frac{1}{\ell !} < \infty$, we can apply the Borell Cantelli lemma and get:
For any sequence $\langle c_\ell \such \ell \in \omega \rangle$ such that $c_\ell \in \may_\ell(q^*(k))$ we have that $$\Leb \left(\bigcup_{K \in [k,\omega)} \bigcap_{\ell \geq K} A_{k_0,c_\ell,c_{\ell+1}} \right) =1.$$ So $r \in \bigcap_{\ell \geq K} A_{k,c_\ell,c_{\ell+1}}$ for some $K \geq k$. So $q(k)$ forces that $\langle c_\ell \such \ell \in \omega \rangle$ describes a matrix whose product with $\eta$ lies eventually within an $\eps$ interval. Now we take smaller and smaller $\eps$'s and a density argument. \proofend
\begin{conclusion} \label{2.4} Let $P_{\omega_2}=\langle P_i, \mathunderaccent\tilde-3 {Q}_j \such i \leq \omega_2, j < \omega_2 \rangle$ be a countable support iteration of $\mathunderaccent\tilde-3 {Q}_i$, where $Q_i$ is $Q$ defined in $V^{P_i}$, and let $\mathunderaccent\tilde-3 {R}_{\omega_1}$ be a $P_{\omega_2}$ name of the $\aleph_1$-random algebra. Then in $V^{P_{\omega_2} \ast \mathunderaccent\tilde-3 {R}_{\omega_1}}$ we have $ \gs = \aleph_1$ and $\chi > \aleph_1$. \end{conclusion}
\proof Dow proves in \cite[Lemma 2.3]{Dow} that
$s = \aleph_1$ after adding $\aleph_1$ or more random reals, over any ground model. In order to show $\chi > \aleph_1$, let $\eta_i$ , $i < \omega_1$ be reals in $V^{P_{\omega_2} \ast \mathunderaccent\tilde-3 {R}_{\omega_1}}$. Over $V^{P_{\omega_2}}$, each $\eta_i$ has a $R_{\omega_1}$-name $\mathunderaccent\tilde-3 {\eta_i}$.
Since the random algebra is c.c.c, there are w.l.o.g.\ only countably many of the $\aleph_1$ random reals mentioned in $\mathunderaccent\tilde-3 {\eta_i}$. Let $\mathunderaccent\tilde-3 {\eta'_i}$ be got from $\mathunderaccent\tilde-3 {\eta_i}$ by replacing these countably many by the first $\omega$ ones and then doing as if it were just one random real. This is possible because $R_1$ and $R_\omega$ are equivalent forcings. Since the random algebra is c.c.c., the name $\mathunderaccent\tilde-3 {\eta'_i}$ can be coded as a single real $r_i$
in $V^{P_{\omega_2}}$. Now, by \cite[V.4.4.]{Properforcing} and by the properness of the $ \mathunderaccent\tilde-3 {Q_j}$, this name $r_i$ appears at some stage $\alpha(\eta_i) <\aleph_2$ in the iteration $P_{\omega_2}$. We take the supremum $\alpha$ of all the $\alpha(\eta_i)$, $i < \omega_1$. We apply the Main Lemma to the $\mathunderaccent\tilde-3 {\eta'_i}$. Thus $Q_\alpha$ adds a Toeplitz matrix, that makes after multiplication all the $\eta'_i$ convergent. Since the Main Lemma applies to all random algebras simultaneously, this matrix makes also the $\eta_i$ convergent. \proofend
\begin{definition} \label{2.5} \begin{myrules} \item[(1)]
$Q_{pr}= \{p \in Q \such n^p = 0\}$ is called the pure part of $Q$. \item[(2)] We write $p \leq^* q$ if there are some $w$, $n$ such that $p \leq (w,t_n^q,t_{n+1}^q \dots )$. So, it is up to a finite ``mistake'' $p \leq q$. \end{myrules} \end{definition}
\begin{fact}\label{2.6} If $\langle p_i \such i < \gamma \rangle$ is $\leq^*$-increasing in $Q$ and
$\MA_{|\gamma|}$ holds, then there is $p \in Q_{pr}$ such that for all $i < \delta$, $p_i \leq^* p$. \end{fact}
\proof We apply $\MA_{|\gamma|}$ to the following partial order $P$: Conditions are $(s,F)$ where $s=( t_0^p, \dots, t_n^p)$ is an initial segment of a condition in $Q_{pr}$ and $F \subset \gamma$ is a finite set. We let $(s,F) \leq_P (t,G)$ iff $s \trianglelefteq t$ and $F \subseteq G$ and $(\forall n \in \lgg(t) - \lgg(s))(\forall \alpha \in F) (n > $ (all mistakes between the $p_\alpha) \rightarrow t_n \in \Sigma(c_i^{p_\alpha} \such i \in {\mathcal S}(\alpha,n)$ for suitable ${\mathcal S}(\alpha,n)))$. This forcing is c.c.c., because conditions with the same first component are compatible and because there are only countably many possibilities for the first component. It is easy to see that for $\alpha < \delta$ the sets $D_\alpha = \{ (s,F) \such \alpha \in F \}$ is dense and that for $n \in \omega$ the sets $D^n =\{ (s,F) \such \lgg(s) \geq n \}$ are dense. Hence if $G$ is generic, then $p= \bigcup\{s \such \exists F (s,F) \in G \} \geq^* p_\alpha$ for all $\alpha$. \proofend
\begin{conclusion}\label{2.7} If $V \models \MA_\kappa$ and $\kappa > \delta > \aleph_0$, then in $V^{R_\delta}$ then matrix number is $\geq \kappa$ and the splitting number is $\aleph_1$. \end{conclusion}
\proof As mentioned, \cite{Dow} shows the the result on the splitting number. For the matrix number, let random names
$\mathunderaccent\tilde-3 {\eta_i}$, $i < \gamma$ be given in $V$, $\gamma < \kappa$.
We fix $\eps > 0$ and $K$ as in the proof of \ref{2.3}.
We choose for $i < \gamma$, $p^i = \langle c^i_k \such k \in \omega \rangle$
as in the end of the proof of \ref{2.3} for $\mathunderaccent\tilde-3 {\eta_i}$ and use and Fact 2.6. $\gamma+1$ times iteratively and find a pure condition $p = \langle c_k \such k \in \omega\rangle \geq^\ast p^i$ for all $i < \gamma$, that gives the lines of a matrix which brings everything into an $\eps$-range. We denote these $c_k$ by
$c_k = c_k(\eps)$. Now by induction we choose $c_k$: $c_0 = c_0(1)$, and $c_k= c_{k'}(\frac{1}{k'+1}) $ if $k' > k$ is the first $k''$ such that $\mdn(c_{k''}(\frac{1}{k''+1})) > \mdn(c_{k-1})$. The matrix with $c_k$ in the $k$th line
acts as desired. (Now $\mup(c_k) > \mdn(c_{k+1})$ is possible but this does not do any harm.) \proofend
{\bf Acknowledgement:} The first author would like to thank Andreas Blass for discussions on the subject and for reading and commenting.
\end{document} | arXiv |
\begin{document}
\title{Homology of $E_{n}$ Ring Spectra and Iterated $THH$} \author{Maria Basterra} \address{Department of Mathematics, University of New Hampshire, Durham, NH} \email{[email protected]} \author{Michael A. Mandell} \address{Department of Mathematics, Indiana University, Bloomington, IN} \email{[email protected]} \thanks{The second author was supported in part by NSF grant DMS-0804272}
\date{December 31, 2010}
\subjclass{Primary 55P43; Secondary 55P48}
\begin{abstract} We describe an iterable construction of $THH$ for an $E_{n}$ ring spectrum. The reduced version is an iterable bar construction and its $n$-th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented $E_{n}$ algebra. \end{abstract}
\maketitle
\section{Introduction}
Over the past two decades, topological Hochschild homology ($THH$) and its refinement topological cyclic homology ($TC$) have become standard tools in algebraic topology and algebraic $K$-theory. Waldhausen originally conjectured the theory $THH$ and used an ad hoc version of it to split the algebraic $K$-theory of spaces into stable homotopy theory and stable pseudo-isotopy theory \cite{WaldII}. The theory $TC$ provides the key tool in the proof of the $K$-theoretic Novikov conjecture by B\"okstedt, Hsiang, and Madsen \cite{BHM}, and in the algebraic $K$-theory computations pioneered by Hesselholt and Madsen \cite{HM1,HM2,HM3}. $K$-theory computations in $TC$ also led to Ausoni and Rognes' mysterious chromatic red shift phenomenon \cite{AusoniRognes}. Partly because of this, recently interest has grown in iterating $THH$ and $TC$ and in higher versions of $THH$ and $TC$ studied by Pirashvili \cite{Pirashvili} and by Brun, Carlsson, Douglas, and Dundas \cite{HigherTC1,HigherTC2}.
Although $THH$ makes sense for any ring or $A_{\infty}$ ring spectrum $A$, the applications to $K$-theory computations typically take advantage of the multiplicative structure on $THH(A)$ and in \cite{AusoniRognes} the extended power operation structure on $THH(A)$ that arises only when $A$ has more structure, for example that of an $E_{\infty}$ ring spectrum. Likewise, to iterate $THH$, $THH(A)$ must have at least an $A_{\infty}$ multiplication. In \cite{MultTHH}, Brun, Fiedorowicz, and Vogt showed that when $A$ is an $E_{n}$ ring spectrum $THH(A)$ is an $E_{n-1}$ ring spectrum; in this case $THH$ can be iterated up to $n$ times, and $THH(A)$ admits certain power operations. The construction in \cite{MultTHH} requires replacing $A$ by an equivalent ring spectrum over a different operad. As a consequence, to iterate the construction requires re-approximation at each stage.
This paper describes a construction of $THH$ of $E_{n}$ ring spectra which is iterable without re-approximation. Working in one of the modern categories of spectra, such as the category of EKMM $S$-modules \cite{ekmm} or the category of symmetric spectra of topological spaces \cite{hss,MMSS}, we study algebras over the little $n$-cubes operad $\LC$ of Boardman and Vogt \cite{BV}. In fact, because the output of our $THH$ construction is not quite a $\LC[n-1]$ algebra, we work with a mild generalization called a ``partial $\LC$-algebra'', which we review in Section~\ref{secpart}. A partial $\LC$-algebra is a partial $\LC[1]$-algebra by neglect of structure; we construct a cyclic bar construction version of $THH$ for partial $\LC[1]$-algebras and prove the following theorem.
\begin{thm}\label{mainthh} For a partial $\LC$-algebra $A$ satisfying mild technical hypotheses, the cyclic bar construction $THH(A)$ is naturally a partial $\LC[n-1]$-algebra. \end{thm}
The mild technical hypotheses amount to the partial algebra generalization of the usual hypothesis on algebras that the inclusion of the unit is a cofibration of the underlying $S$-modules (or symmetric or orthogonal spectra). We write out the hypotheses explicitly as Hypothesis~\ref{hypprop} below and show in Proposition~\ref{propitprop} that it is inherited on $THH(A)$, allowing iteration.
When working in the context of symmetric spectra, there are two different constructions of $THH$ of an $S$-algebra: A cyclic bar construction as in the previous theorem and the original construction of B\"okstedt. Because only B\"okstedt's construction is known to be suitable for constructing $TC$, the previous theorem still leaves open the problem of directly constructing an iterable version of $TC$ from an $E_{n}$ ring structure; the authors plan to return to this question in a future paper. On the other hand, the cyclic bar construction does admit a relative variant for partial $\LCR$-algebras (see Definition~\ref{defpart}), where we use a different base commutative $S$-algebra $R$ in place of the sphere spectrum $S$.
\begin{thm}\label{mainrel} For $R$ a commutative $S$-algebra, and $A$ a partial $\LCR$-algebra satisfying Hypothesis~\ref{hypprop}, $THH^{R}(A)$ is a partial $\LCR[n-1]$-algebra. \end{thm}
The relative construction also admits a reduced version for augmented algebras, which amounts to taking coefficients in $R$. This is the analogue of the bar construction of an augmented algebra, so we write $B^{R}(A)$ or $BA$ rather than $THH^{R}(A;R)$. For a partial augmented $\LCR$-algebra $A$, $BA$ is a partial augmented $\LCR[n-1]$-algebra, and so we can iterate the bar construction up to $n$ times. Up to a shift, the fiber of the augmentation $B^{n}A\to R$ is the $R$-module of $\LCR$-algebra derived indecomposables representing reduced topological Quillen homology (see Section~\ref{sectqh} for a review of this theory).
\begin{thm}\label{maintq} For an augmented $\LCR$-algebra $A$, there is a natural isomorphism in the derived category of $R$-modules \[ R\vee \Sigma^{n}Q^{\ld}_{\LC}(A)\cong B^{n}A, \] where $Q^{\ld}_{\LC}$ denotes the $\LCR$-algebra cotangent complex at the augmentation (the derived $\LCR$-algebra indecomposables). \end{thm}
We regard the previous result as the main theorem in this paper: It allows an inductive approach to obstruction theory for connective $E_{n}$ ring spectra. We explain this idea and apply it in a future paper to prove that $BP$ is an $E_{4}$ ring spectrum. Other non-iterative versions of Theorem~\ref{maintq} can be found in~\cite{FrancisThesis} and~\cite{DAGVI}. An algebraic version can be found in \cite{FresseItBar}.
\subsection*{Terminology}
Because partial ${\opsymbfont{C}}$-algebras (for various ${\opsymbfont{C}}$) play a fundamental role in the structure and results of the paper, we will sometimes use the terminology ``true ${\opsymbfont{C}}$-algebra'' (for ${\opsymbfont{C}}$-algebras in the usual sense) when necessary for emphasis, for contrast, or to avoid confusion with the terminology ``partial ${\opsymbfont{C}}$-algebra''.
\subsection*{Outline}
Section~\ref{secpart} reviews the definition of a partial algebra over an operad and the Kriz-May rectification theorem, which gives an equivalence between the homotopy category of partial algebras and the homotopy category of true algebras over an operad. Sections~\ref{secmoore} and~\ref{secncyc} construct the cyclic nerve of a partial $\LCR[1]$-algebra, breaking the construction into two steps. For a $\LCR[1]$-algebra $A$, we construct in Section~\ref{secmoore} a closely related partial associative $R$-algebra, called the ``Moore'' partial algebra, which has the same relationship to $A$ as the Moore loop space has to the loop space. In Section~\ref{secncyc}, we construct the cyclic nerve of a partial associative $R$-algebra and study its multiplicative structure when applied to the Moore algebra of a $\LCR[n]$-algebra, proving Theorems~\ref{mainthh} and ~\ref{mainrel} above. We produce the iterated bar construction for augmented partial $\LCR$-algebras in Section~\ref{secbar} and for non-unital $\LCR$-algebras in Section~\ref{secbar2}. Section~\ref{sectqh} proves Theorem~\ref{maintq}. Finally, Section~\ref{secopn} proves two compatibility results for the $E_{n-1}$-structure on the bar construction: it shows that the usual diagonal map on the bar construction preserves the multiplication (Theorem~\ref{thmhopf}) and that the bar construction has the expected behavior with respect to power operations (Theorem~\ref{thmopn}).
\section{Partial Operadic Algebras} \label{secpart}
In this section, we give a brief review of the definition and basic homotopy theory of partial algebras over an operad. We review the Kriz-May rectification theorem, which shows that any partial algebra over an appropriate operad is naturally weakly equivalent to a true algebra over that same operad. This in particular shows how to recover an operadic algebra from a partial operadic algebra, gives an equivalence of the homotopy theory of partial operadic algebras and of true operadic algebras, and justifies the perspective of the statements of the main theorems in Section~1.
In this section and throughout this paper, we work in one of the modern categories of spectra of \cite{MMSS} or \cite{ekmm}. We write ${\catsymbfont{M}}_{S}$ for any of these categories, and refer to an object in it as an ``$S$-module''; we write $\wedge_{S}$ for the smash product, reserving $\wedge$ for the smash product with a based space. For a commutative $S$-algebra $R$, we have the category of $R$-modules ${\catsymbfont{M}}_{R}$ that has a symmetric monoidal product $\wedge_{R}$. We avoid needless redundancy by typically working in ${\catsymbfont{M}}_{R}$; the case of $S$-modules being precisely the special case $R=S$.
In our cases of interest, we work with the little $n$-cubes operads $\LC$. With this in mind, we fix an operad ${\opsymbfont{C}}$ in spaces such that each ${\opsymbfont{C}}(m)$ is a free $\Sigma_{m}$-CW complex. Then in our context, a ${\opsymbfont{C}}$-algebra (or $\alg{{\opsymbfont{C}}}{R}$-algebra when $R$ needs to be made explicit) consists of an $R$-module $A$ and maps \[ {\opsymbfont{C}}(m)_{+}\wedge_{\Sigma_{m}} (A\wedge_{R}\dotsb \wedge_{R}A)\to A \] satisfying certain associativity and unit properties. A partial ${\opsymbfont{C}}$-algebra replaces the smash powers with an equivalent system of $R$-modules.
\begin{defn}\label{defpps} An \term{op-lax power system} of $R$-modules consists of \begin{enumerate} \item A sequence of $R$-modules $X_{1}, X_{2},\dotsc$, \item A $\Sigma_{m}$ action on $X_{m}$, and \item A $\Sigma_{m}\times \Sigma_{n}$-equivariant map $\lambda_{m,n}\colon X_{m+n}\to X_{m}\wedge_{R}X_{n}$ for each $m,n$ \end{enumerate} such that the following diagrams commute for all $m,n,p$, where $\tau_{m,n}\in \Sigma_{m+n}$ denotes the permutation that switches the first block of $m$ past the last block of $n$. \[ \xymatrix{ X_{m+n}\ar[r]^-{\lambda_{m,n}}\ar[d]_{\tau_{m,n}} &X_{m}\wedge_{R}X_{n}\ar[d]^{\tau}&
X_{m+n+p}\ar[r]^-{\lambda_{m,n+p}}\ar[d]_{\lambda_{m+n,p}} &X_{m}\wedge_{R}X_{n+p}\ar[d]^{\lambda_{n,p}}\\
X_{n+m}\ar[r]_-{\lambda_{n,m}}& X_{n}\wedge_{R}X_{m}&
X_{m+n}\wedge_{R}X_{p}\ar[r]_-{\lambda_{m,n}} &X_{m}\wedge_{R}X_{n}\wedge_{R}X_{p} } \] We write $\lambda_{m,n,p}$ for the common composite in the righthand diagram, and more generally, $\lambda_{m_{1},\dotsc,m_{r}}$ for the iterated composites \[ X_{m_{1}+\dotsb+m_{r}}\to X_{m_{1}}\wedge_{R}\dotsb \wedge_{R}X_{m_{r}}. \] We write $X_{0}=R$ and take $\lambda_{0,n}$ and $\lambda_{n,0}$ to be the (inverse) unit isomorphisms. A map of op-lax power systems $A\to B$ consists of equivariant maps $X_{n}\to Y_{n}$, which make the evident diagrams in the structure maps $\lambda_{m,n}$ commute. An op-lax power system is a \term{partial power system} when the maps $\lambda_{m_{1},\dotsc,m_{r}}$ are weak equivalences for all $m_{1},\dotsc,m_{r}$. \end{defn}
\begin{example} For an $R$-module $X$, we get a partial power system $X_{m}=X^{(m)}$ ($m$-th smash power over $R$ for some fixed association) with the usual symmetric group actions and the maps $\lambda_{m,n}$ the associativity isomorphism. We call such a partial power system a \term{true power system}. \end{example}
This definition depends strongly on the underlying symmetric monoidal category ${\catsymbfont{M}}_{R}$: An op-lax power system in $R$-modules does not have an underlying op-lax power system in $S$-modules. Definition~\ref{defpps} provides the appropriate framework for the cyclic bar construction of $THH$. (A version suitable for the B\"okstedt construction of $THH$ requires an ``external'' smash product formulation that is significantly more complex.)
We define a \term{weak equivalence} of partial power systems as a map $X\to Y$ that is a weak equivalence $X_{1}\to Y_{1}$. This compensates for the awkward fact that the smash product of $R$-modules does not preserve all weak equivalences. To help alleviate this difficulty, we introduce the following additional terminology.
\begin{defn}\label{deftidy} We say that a partial power system $X$ is \term{tidy} when the canonical maps \[ X_{1}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}X_{1}\to X_{1}\wedge_{R}\dotsb\wedge_{R}X_{1} \] from the derived smash powers to the point-set smash powers of $X_{1}$ are isomorphisms in the derived category ${\catsymbfont{D}}_{R}$. \end{defn}
We note that a partial power system $X$ is tidy if and only if the natural maps \[ X_{j_{1}}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}X_{j_{r}}\to X_{j_{1}}\wedge_{R}\dotsb\wedge_{R}X_{j_{r}} \] are all isomorphisms in ${\catsymbfont{D}}_{R}$. To see this, consider the following commutative diagram in ${\catsymbfont{D}}_{R}$, where $j=\sum j_{i}$. \[ \xymatrix@R-1pc@C-2pc{ X_{j_{1}}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}X_{j_{r}}\ar[rr] \ar[dd]_{\lambda_{1,\dotsc,1}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}\lambda_{1,\dotsc,1}}^{\sim} &&X_{j_{1}}\wedge_{R}\dotsb\wedge_{R}X_{j_{r}} \ar[dd]^{\lambda_{1,\dotsc,1}\wedge_{R}\dotsb \wedge_{R}\lambda_{1,\dotsc,1}}\\ &X_{j}\ar[ur]^(.3){\lambda_{j_{1},\dotsc j_{r}}}_-{\sim} \ar[dr]_(.3){\lambda_{1,\dotsc,1}}^-{\sim}\\ (X_{1}\wedge_{R}\dotsb \wedge_{R}X_{1})\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}} (X_{1}\wedge_{R}\dotsb \wedge_{R}X_{1})\ar[rr] &&X_{1}\wedge_{R}\dotsb \wedge_{R} X_{1} } \] The maps labelled $\sim$ are isomorphisms in ${\catsymbfont{D}}_{R}$ by the definition of partial power system, and so we see that the vertical maps are isomorphisms in ${\catsymbfont{D}}_{R}$. It follows that the top horizontal map is an isomorphism in ${\catsymbfont{D}}_{R}$ for all $j_{1},\dotsc,j_{r}$ exactly when the bottom horizontal map is an isomorphism in ${\catsymbfont{D}}_{R}$ for all $j_{1},\dotsc,j_{r}$. Looking at the diagram \[ \xymatrix{ (X_{1}\wedge_{R}\dotsb \wedge_{R}X_{1})\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}} (X_{1}\wedge_{R}\dotsb \wedge_{R}X_{1})\ar[r] &X_{1}\wedge_{R}\dotsb \wedge_{R} X_{1}\\ (X_{1}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}X_{1}) \wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}} (X_{1}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}}X_{1})\ar[u]\ar[r]_-{\cong} &X_{1}\wedge_{R}^{\mathbf{L}}\dotsb \wedge_{R}^{\mathbf{L}} X_{1}\ar[u] } \] we see that this happens exactly when $X$ is tidy.
Intuitively, tidy means that the $m$-th partial power $X_{m}$ is equivalent to the derived $m$-th smash power of $X_{1}$. True power systems need not be tidy in general, but if $X$ is cofibrant or if we are working in the category of symmetric spectra, orthogonal spectra, or EKMM $S$-modules and $X$ is cofibrant in the category of ${\opsymbfont{C}}$-algebras (for ${\opsymbfont{C}}$ as above) or even cofibrant in the category of commutative $R$-algebras, then the true power system $X_{m}=X^{m}$ is tidy.
To define a partial ${\opsymbfont{C}}$-algebra structure on a partial power system $A$, we need slightly more than the sequence of maps \[ {\opsymbfont{C}}(m)_{+}\wedge_{\Sigma_{m}}A_{m}\to A_{1}, \] that suffices for a true ${\opsymbfont{C}}$-algebra; rather, we need maps of the form \[ ({\opsymbfont{C}}(j_{1})\times\dotsb \times {\opsymbfont{C}}(j_{r}))_{+} \wedge_{\Sigma_{j_{1}}\times \dotsb \times \Sigma_{j_{r}}} A_{m}\to A_{r} \] for $j_{1}+\dotsb+j_{r}=m$. The coherence is most easily expressed by generalizing the monadic description of operadic algebras.
\begin{cons}\label{consmonad} For an op-lax power system $X$, let \[ (\bgsO^{\sharp} X)_{m}=\bigvee_{j_{1},\dotsc,j_{m}} ({\opsymbfont{C}}(j_{1})\times \dotsb \times {\opsymbfont{C}}(j_{m}))_{+} \wedge_{\Sigma_{j_{1}}\times \dotsb \times \Sigma_{j_{m}}} X_{j_{1}+\dotsb+j_{m}}. \] This obtains a $\Sigma_{m}$-action by permuting the $j_{i}$'s and performing the corresponding block permutation on $X_{j_{1}+\dotsb+j_{m}}$. We make $\bgsO^{\sharp} X$ an op-lax power system using the structure maps $\lambda$ of $X$ and the associativity isomorphism \begin{multline*} \bigvee_{j_{1},\dotsc,j_{m+n}} ({\opsymbfont{C}}(j_{1})\times \dotsb \times {\opsymbfont{C}}(j_{m+n}))_{+} \wedge_{\Sigma_{j_{1}}\times \dotsb \times \Sigma_{j_{m+n}}} (X_{j_{1}+\dotsb+j_{m}} \wedge_{R} X_{j_{m+1}+\dotsb +j_{m+n}}) \\ \cong (\bgsO^{\sharp} X)_{m}\wedge_{R}(\bgsO^{\sharp} X)_{n}. \end{multline*} \end{cons}
We have a natural map of op-lax power systems $X\to \bgsO^{\sharp} X$ induced by the inclusion of the identity element in ${\opsymbfont{C}}(1)$. We have a natural transformation of op-lax power systems \[ \bgsO^{\sharp} \bgsO^{\sharp} X \to \bgsO^{\sharp} X \] induced by the operadic multiplication on ${\opsymbfont{C}}$. An easy check of diagrams then proves the following proposition.
\begin{prop} The structure above makes $\bgsO^{\sharp}$ a monad in the category of op-lax power systems. \end{prop}
Because we have assumed that each ${\opsymbfont{C}}(m)$ is a free $\Sigma_{m}$-CW complex, the smash products \[ ({\opsymbfont{C}}(j_{1})\times \dotsb \times {\opsymbfont{C}}(j_{m}))_{+} \wedge_{\Sigma_{j_{1}}\times \dotsb \times \Sigma_{j_{m}}}(-) \] preserve weak equivalences. In particular, we see that when $X$ is a partial power system, so is $\bgsO^{\sharp} X$. Thus, we obtain the following proposition.
\begin{prop}\label{propmonad} Under the assumptions on ${\opsymbfont{C}}$ above, $\bgsO^{\sharp}$ is a monad in the category of partial power systems. \end{prop}
Note that for a true power system $X_{m}=X^{(m)}$, we have \[ (\bgsO^{\sharp} X)_{m}\cong (\bgsO^{\sharp} X)_{1}^{(m)}=({\mathbb{C}} X)^{(m)}, \] where ${\mathbb{C}}$ is the usual monad in $R$-modules associated to the operad ${\opsymbfont{C}}$. The monad $\bgsO^{\sharp}$ therefore generalizes to partial power systems the monad of $R$-modules associated to the operad ${\opsymbfont{C}}$. We now have the following monadic definition of a partial ${\opsymbfont{C}}$-algebra, generalizing the monadic definition of a true ${\opsymbfont{C}}$-algebra.
\begin{defn}\label{defpart} A \term{partial ${\opsymbfont{C}}$-algebra} (or \term{partial $\alg{{\opsymbfont{C}}}{R}$-algebra} when $R$ needs to be made explicit) is an algebra over the monad $\bgsO^{\sharp}$ in partial power systems. \end{defn}
In particular, a true $\alg{{\opsymbfont{C}}}{R}$-algebra structure on $A$ is precisely a partial $\alg{{\opsymbfont{C}}}{R}$-algebra structure on the true power system $A_{m}=A^{(m)}$. We could define a more general notion of op-lax ${\opsymbfont{C}}$-algebra in terms of the op-lax power systems; we learned from Leinster that such an algebra is precisely a strictly unital op-lax symmetric monoidal functor from an appropriate category of operators associated to ${\opsymbfont{C}}$. A formulation of the theory of partial ${\opsymbfont{C}}$-algebras along these lines as well as generalizations and further examples may be found in Leinster's preprint \cite{LeinsterPartial}.
The remainder of the section discusses and reviews the proof of the following theorem, the Kriz-May rectification theorem. In it, we write $T$ for the functor from $R$-modules to partial power systems that takes an $R$-module $X$ to the true power system $(TX)_{m}=X^{(m)}$.
\begin{thm}[May \cite{MayMult}, Kriz-May \cite{kmbook}] \label{thmrect} For an operad ${\opsymbfont{C}}$ as above, there exists a functor $R$ from partial ${\opsymbfont{C}}$-algebras to true ${\opsymbfont{C}}$-algebras, an endofunctor $E^{\sharp}$ on partial ${\opsymbfont{C}}$-algebras, and natural weak equivalences of partial ${\opsymbfont{C}}$-algebras \[ \Id \from E^{\sharp}\to TR. \] Moreover, there exists an endofunctor $E$ on ${\opsymbfont{C}}$-algebras and natural weak equivalences of ${\opsymbfont{C}}$-algebras \[ \Id\from E \to RT. \] The functors and natural transformations are natural also in the operad ${\opsymbfont{C}}$. \end{thm}
As a consequence, the categories of partial ${\opsymbfont{C}}$-algebras and of true ${\opsymbfont{C}}$-algebras become equivalent after formally inverting the weak equivalences.
The proof of theorem is an application of the two-sided monadic bar construction. To avoid confusion, write $UX$ for $X_{1}$ for a partial power system $X$. We note that for any partial power system $X$, the structure maps $\lambda$ induce a natural map of partial power systems $X\to TUX$ (in fact, a weak equivalence), which together with the identity $UTY=Y$, identify $U$ and $T$ as adjoints. In this notation, the monad ${\mathbb{C}}$ in $R$-modules is $U\bgsO^{\sharp} T$.
We let $RA=B({\mathbb{C}} U,\bgsO^{\sharp},A)$ be the geometric realization of the simplicial $R$-module \[ B_{\ssdot}({\mathbb{C}} U,\bgsO^{\sharp},A)= {\mathbb{C}} U\, \underbrace{\bgsO^{\sharp}\dotsb \bgsO^{\sharp}}_{\text{$\bullet$ iterates}}\, A. \] Here the face maps are induced by the $\bgsO^{\sharp}$-action on $A$, the monadic multiplication on $\bgsO^{\sharp}$, and the multiplication \[ {\mathbb{C}} U\bgsO^{\sharp} X \to {\mathbb{C}} {\mathbb{C}} UX\to {\mathbb{C}} UX. \] The degeneracy maps are induced by the unit of the monad $\bgsO^{\sharp}$. Because the monad ${\mathbb{C}}$ commutes with geometric realization, $RA$ is naturally a ${\opsymbfont{C}}$-algebra.
We let $E^{\sharp}A=B(\bgsO^{\sharp},\bgsO^{\sharp},A)$ be the geometric realization of the ``standard resolution'' \[ B_{\ssdot}(\bgsO^{\sharp},\bgsO^{\sharp},A)=\bgsO^{\sharp} \, \underbrace{\bgsO^{\sharp}\dotsb \bgsO^{\sharp}}_{\text{$\bullet$ iterates}}\, A \] (where we understand the geometric realization of a partial power system to be performed objectwise). The augmentation $B_{\ssdot}(\bgsO^{\sharp},\bgsO^{\sharp},A)\to A$ is a map of simplicial partial ${\opsymbfont{C}}$-algebras (where we regard $A$ as a constant simplicial object) and a simplicial homotopy equivalence of partial power systems. Thus, the geometric realization is a natural map of partial ${\opsymbfont{C}}$-algebras and a weak equivalence (in fact, a homotopy equivalence of $\Sigma_{m}$-equivariant $R$-modules in each partial power).
The natural map $\bgsO^{\sharp} X\to TU\bgsO^{\sharp} TU X = T{\mathbb{C}} UX$ is a map of partial ${\opsymbfont{C}}$-algebras; it is a weak equivalence since it is induced by the maps $\lambda_{m_{1},\dotsc,m_{r}}$. Moreover, the map is compatible with the left action of the monad $\bgsO^{\sharp}$. Thus, we get a map of simplicial partial ${\opsymbfont{C}}$-algebras \[ B_{\ssdot}(\bgsO^{\sharp},\bgsO^{\sharp},A)\to B_{\ssdot}(T{\mathbb{C}} U,\bgsO^{\sharp},A) \] which is a weak equivalence in each simplicial degree. The geometric realization is a weak equivalence and induces the natural weak equivalence of partial ${\opsymbfont{C}}$-algebras $E^{\sharp}\to TR$ in the statement.
On the ${\opsymbfont{C}}$-algebra side, we take $E$ to be the geometric realization of the standard resolution $B_{\ssdot}({\mathbb{C}},{\mathbb{C}},-)$. The construction and study of the natural transformations are analogous to the partial ${\opsymbfont{C}}$-algebra case described in detail above.
\section[The Moore Algebra] {The Moore Partial Algebra of a Partial $\LC[1]$-Algebra} \label{secmoore}
Moore constructed a variant of the loop space of a topological space where the concatenation of loops is strictly associative and unital and not just associative and unital up to homotopy. This construction easily extends to any $\LC[1]$ space, and in fact quite generally to $\LC[1]$-algebras in topological categories. In this section, we generalize Moore's construction to the partial context, constructing a partial associative $R$-algebra from a partial $\LCR[1]$-algebra. In the next section, we use this to construct the cyclic bar construction of a partial $\LCR[1]$-algebra.
We begin by reviewing the construction of the Moore algebra of a (true) $\LC[1]$-algebra, before treating the slightly more complicated partial case. Recall that an element of $\LC[1](m)$ consists of an ordered list of almost disjoint subintervals of the unit interval, not necessarily in their natural order. The operadic multiplication $a\circ_{i}b$ replaces the $i$-th subinterval in $a$ with a scaled version of the subintervals in $b$. The element \[ \gamma=([0,1/2],[1/2,1]) \in \LC[1](2) \] represents the loop multiplication for the action of $\LC[1]$ on a loop space; the two composites \[ \gamma\circ_{1}\gamma=([0,1/4],[1/4,1/2],[1/2,1]) \quad\text{and}\quad \gamma\circ_{2}\gamma=([0,1/2],[1/2,3/4],[3/4,1]) \] in $\LC[1](3)$ represent the two associations of the multiplication of three loops in a loop space. The basic idea of Moore's construction is to add a length parameter to build an associative multiplication from a $\LC[1]$-multiplication. Specifically, given lengths $r$ and $s$, we get an element $\gamma_{r,s}$ in $\LC[1](2)$ that models the concatenation of a loop of length $r$ with a loop of length $s$, rescaled to the unit interval. \[ \underbrace{ \subseg{r}{10em} \subseg{s}{6em}}_{r+s} \] Namely, $\gamma_{r,s}$ is the element \[ \gamma_{r,s}=([0,r/(r+s)],[r/(r+s),1])\in \LC[1](2). \] Writing $P=(0,\infty)\subset {\mathbb{R}}$ for the space of positive real numbers, the ``concatenation formula'' \[ \Gamma \colon P\times P\to P\times \LC[1](2) \] sends $(r,s)$ to $(r+s,\gamma_{r,s})$.
For a $\LC[1]$-algebra $A$, we get an associative multiplication on $P_{+}\wedge A$ using $\Gamma$ and the $\LC[1](2)$-action, \begin{multline*} (P_{+}\wedge A)\wedge_{R}(P_{+}\wedge A)\cong (P\times P)_{+}\wedge (A\wedge_{R}A)\to\\ (P\times \LC[1](2))_{+}\wedge (A\wedge_{R}A)\cong P_{+}\wedge (\LC[1](2)_{+}\wedge (A\wedge_{R}A)) \to P_{+}\wedge A. \end{multline*} To make this unital, let $\bar P=[0,\infty)\subset {\mathbb{R}}$ denote the non-negative real numbers, and define the $R$-module $\MA$ by the following pushout diagram. \begin{equation}\label{eqmoore} \begin{gathered} \xymatrix@-1pc{ P_{+}\wedge R\ar[r]\ar[d]&P_{+}\wedge A\ar[d]\\ \bar P_{+}\wedge R\ar[r]&\MA } \end{gathered} \end{equation} The multiplication above then extends to an associative multiplication on $\MA$, and is now unital with the unit $R\to \MA$ induced by the inclusion of $\{0\}_{+}\wedge R$ into $\bar P_{+}\wedge R$.
\begin{prop} For a $\LC[1]$-algebra $A$, $\MA$ is naturally an associative $R$-algebra. \end{prop}
To relate $A$ and $\MA$, note that forgetting the lengths (collapsing $P$ and $\bar P$ to a point), we obtain a map $\MA\to A$. This map is a homotopy equivalence of $R$-modules: The map $A\to \MA$ induced by the inclusion of $1$ in $P$ provides the homotopy inverse. The composite map on $A$ is the identity and compatible null homotopies on $P$ and $\bar P$ induce a homotopy from the identity to the composite on $\MA$. (See \cite[\S6]{E1E2E3E4} for a comparison of $A$ and $\MA$ as $\LC[1]$-algebras.)
To extend this to the case of partial algebras, we need to construct an appropriate partial power system with the $m$-th partial power analogous to the $m$-th smash power of the pushout in~\eqref{eqmoore}. Let ${\opsymbfont{T}}$ be the diagram with objects $a,b,c$ and arrows $\alpha,\beta$ as pictured \[ \xymatrix@-1pc{ &c\ar[dl]_{\beta}\ar[dr]^{\alpha}\\b&&a } \] so that a pushout is a colimit indexed on ${\opsymbfont{T}}$.
\begin{cons} Let $A$ be a partial $\LC[1]$-algebra. We construct the op-lax power system $\MA$ as follows. Let $\MA[1]$ be the pushout \[ \MA[1]=(P_{+} \wedge A_{1})\cup_{P_{+}\wedge R} (\bar P_{+}\wedge R), \] where the map $R\to A_{1}$ is induced by the unique element of $\LC[1](0)$. Inductively, having defined $\MA[1],\dotsc,\MA[m-1]$, we define $\MA[m]$ as a colimit over the following diagram $D_{m}$ indexed on ${\opsymbfont{T}}^{m}$. At a spot indexed by $(x_{1},\dotsc,x_{m})$, we put a copy of \[ (P_{x_{1}}\times \dotsb \times P_{x_{m}})_{+}\wedge A_{\#a} \] where $\#a=\#a(x_{1},\dotsc,x_{m})$ is the number of occurrences of $a$ in the $x_{i}$'s, and \[ P_{x_{i}}=\begin{cases} P&x_{i}=a\text{ or }x_{i}=c\\ \bar P&x_{i}=b. \end{cases} \] The map \[ (x_{1},\dotsc,x_{i-1},c,x_{i+1},\dotsc,x_{m}) \to (x_{1},\dotsc,x_{i-1},y,x_{i+1},\dotsc,x_{m}) \] induced by $\alpha$ (when $y=a$) is induced by the map $A_{\#a-1}\to A_{\#a}$ induced by the element \[ (\id,\dotsc,\id,1,\id,\dotsc,\id)\in \LC[1](1)^{\#a(x_{1},\dotsc,x_{i-1})}\times \LC[1](0)\times \LC[1](1)^{\#a(x_{i+1},\dotsc,x_{n})}, \] where $\id$ is the identity element in $\LC[1](1)$ and $1$ is the unique element in $\LC[1](0)$. The map induced by $\beta$ (when $y=b$) is induced by the inclusion of $P$ in $\bar P$ in the appropriate factor. We give $\MA[m]$ the $\Sigma_{m}$-action induced by permuting the factors of ${\opsymbfont{T}}^{m}$, together with the appropriate re-arrangement of the $P_{x_{i}}$ factors and the $\Sigma_{\#a}$-action on $A_{\#a}$ (corresponding to restricting a permutation in $\Sigma_{m}$ to the rearrangement of the $a$'s in the object of ${\opsymbfont{T}}^{m}$). The structure maps of $A$ induce the structure maps $\MA[m+n]\to \MA[m]\wedge_{R}\MA[n]$ for $\MA$. \end{cons}
Collapsing the $P_{x_{i}}$ factors to a point defines a map of op-lax power systems $\MA\to A$. An argument like the one above shows that each map $\MA[m]\to A_{m}$ is a $\Sigma_{m}$-equivariant homotopy equivalence of $R$-modules.
\begin{prop} Each map $\MA[m]\to A_{m}$ is a $\Sigma_{m}$-equivariant homotopy equivalence of $R$-modules. In particular, $\MA$ is a partial power system and is tidy exactly when $A$ is. \end{prop}
A partial associative $R$-algebra structure is a partial ${\opsymbfont{A}}$-algebra structure for ${\opsymbfont{A}}$ the associative operad. We write $\mu_{j}\in {\opsymbfont{A}}(j)$ for the canonical element, representing the $j$-fold multiplication (if $j>1$), the operadic identity (if $j=1$), or the unit (if $j=0$). To define a partial $R$-algebra structure on the partial power system $X$, we need to specify maps \[ (\mu_{j_{1}},\dotsc,\mu_{j_{m}})\colon X_{j_{1}+\dotsb +j_{m}}\to X_{m} \] for all $j_{1},\dotsc,j_{m}\geq 0$, satisfying the associativity and identity diagrams implicit in Definition~\ref{defpart}. These conditions become easier to verify by thinking of the sequence $j_{1},\dotsc,j_{m}$ as specifying a map of totally ordered sets \[ \{1,\dotsc,j\}\to \{1,\dotsc,m\} \] where $j=j_{1}+\dotsb+j_{m}$: It specifies the unique weakly increasing map $\phi$ such that the cardinality of $\phi^{-1}(i)$ is $j_{i}$. The following well-known fact is a consequence of an easy check of the diagrams, q.v.~\cite[VII\S5]{maclane}. In it, we write $\otos[m]$ for the totally ordered set $\{1,\dotsc,m\}$, and $\underline\DDelta$ for the category whose objects are the totally ordered sets $\otos[0],\otos[1],\dotsc $ and whose morphisms are the weakly increasing maps.
\begin{prop}\label{propmaclane} Let $X$ be a partial power system in the category of $R$-modules. A partial $R$-algebra structure on $X$ consists of a map $X_{\phi}\colon X_{j}\to X_{m}$ for each $\phi \colon \otos[j]\to\otos[m]$ in $\underline\DDelta$, making $X$ a functor from $\underline\DDelta$ to $R$-modules, such that the diagram \[ \xymatrix{ X_{i+j}\ar[r]^-{\lambda_{j,k}}\ar[d]_{X_{\theta+\phi }} &X_{i}\wedge_{R}X_{j}\ar[d]^{X_{\theta}\wedge_{R}X_{\phi}}\\ X_{m+n}\ar[r]_-{\lambda_{m,n}}&X_{m}\wedge_{R}X_{n} } \] commutes for all $\theta \colon \otos[i]\to\otos[m]$, $\phi\colon \otos[j]\to\otos[n]$. \end{prop}
More intrinsically, the previous proposition states that a partial associative $R$-algebra is a strictly unital op-lax monoidal functor $(X,\lambda)$ from $\underline\DDelta$ to $R$-modules together with a $\Sigma_{m}$-action on $X_{m}=X_{\otos[m]}$ making $X$ into a partial power system, cf.~\cite[1.6(a),2.2.1]{LeinsterPartial}.
To construct the partial associative $R$-algebra structure on $\MA$, first we must generalize the elements $\gamma_{r,s}$ above. For $\vec x=(x_{1},\dotsc,x_{m})$ in ${\opsymbfont{T}}^{m}$ and $\vec r=(r_{1},\dotsc,r_{m})$ in $P_{x_{1}}\times \dotsb \times P_{x_{m}}$, let $\gamma^{\vec x}_{\vec r}$ be the following element of $\LC[1](\#a)$, where $\#a=\#a(x_{1},\dotsc,x_{n})$, the number of $a$'s among the $x_{i}$. If $\#a=0$, then $\gamma^{\vec x}_{\vec r}$ is the unique element of $\LC[1](0)$. Otherwise, define $k_{1} < k_{2} < \dotsb < k_{\#a}$ by $x_{k_{i}}=a$, and note that $r_{k_{i}}>0$ for all $i$. Let $\gamma^{\vec x}_{\vec r}$ consist of the subintervals \[ \left[\frac{r_{1}+\dotsb+r_{k_{i}-1}}{r_{1}+\dotsb+r_{m}}, \frac{r_{1}+\dotsb+r_{k_{i}}}{r_{1}+\dotsb+r_{m}}\right] \] (in their natural order) for $i=1,\dotsc,\#a$. For example, if $m=\#a$ (i.e., all the $x_{i}'s$ are $a$'s), then $\gamma^{\vec x}_{\vec r}$ is the element the of $\LC[1](m)$ that subdivides the unit interval into $m$ subintervals of lengths the given proportions $r_{1},\dotsc,r_{m}$; when one of the $x_{i}$'s is $c$ or $b$ with $r_{i}>0$, then $\gamma^{\vec x}_{\vec r}$ contains a gap proportional to $r_{i}$ where the subinterval would have been if that $x_{i}$ were $a$. An $x_{i}$ that is $b$ with $r_{i}=0$, has neither a subinterval nor a gap corresponding to it; in the multiplication below, it will behave like a unit factor in an $m$-th smash power.
Next, for fixed $\phi \colon \otos[j]\to\otos[m]$ in $\underline\DDelta$, let $F_{\phi}$ be the functor from ${\opsymbfont{T}}^{j}$ to ${\opsymbfont{T}}^{m}$ that sends $(x_{1},\dotsc,x_{j})$ to $(y_{1},\dotsc,y_{m})$ where \[ y_{i}=\begin{cases} a&\text{if at least one $k\in \phi^{-1}(i)$ satisfies $x_{k}=a$}\\ b&\text{if every $k\in \phi^{-1}(i)$ satisfies $x_{k}=b$ (or $\phi^{-1}(i)$ is empty)}\\ c&\text{if at least one $k\in \phi^{-1}(i)$ satisfies $x_{k}=c$ and none satisfy $x_{k}=a$}\\ \end{cases} \] and does the only possible thing on maps.
We defined $\MA[m]$ as the colimit of a diagram $D_{m}$ indexed on ${\opsymbfont{T}}^{m}$; we now define $\MA[\phi]\colon \MA[j]\to\MA[m]$ to be the map on colimits induced by a natural transformation $D_{\phi}\colon D_{j}\to F_{\phi}^{*}D_{m}$ as follows. For $(x_{1},\dotsc,x_{j})$ in ${\opsymbfont{T}}^{j}$, define $J_{i}$ and $j_{i}$ by $\phi^{-1}(i)=\{J_{i}+1,\dotsc,J_{i}+j_{i}\}$. (In other words, let
$j_{i}=|\phi^{-1}(i)|$ and $J_{i}=j_{1}+\dotsb+j_{i-1}$.) write $\vec x_{\phi^{-1}(i)}$ for $(x_{J_{i}+1},\dotsc,x_{J_{i}+j_{i}})$, and for $(r_{1},\dotsc,r_{j})\in P_{x_{1}}\times \dotsb \times P_{x_{j}}$, write $\vec r_{\phi^{-1}(i)}$ for $(r_{J_{i}+1},\dotsc,r_{J_{i}+j_{i}})$. Then let $D_{\phi}$ be the natural transformation \[ (P_{x_{1}}\times \dotsb \times P_{x_{j}})_{+}\wedge A_{\#a(\vec x)} \to (P_{y_{1}}\times \dotsb \times P_{y_{m}})_{+}\wedge A_{\#a(\vec y)} \] (for $(y_{1},\dotsc,y_{m})=F_{\phi}(x_{1},\dotsc,x_{j})$ as above), sending $(r_{1},\dotsc,r_{j})\in P_{x_{1}}\times \dotsb \times P_{x_{j}}$ to \[ \bigg(\sum \vec r_{\phi^{-1}(1)},\dotsc, \sum \vec r_{\phi^{-1}(m)}\bigg) \in P_{y_{1}}\times \dotsb \times P_{y_{m}} \] (where we understand the sum to be zero when $\phi^{-1}(i)$ is empty), and sending $A_{\#a(\vec x)}\to A_{\#a(\vec y)}$ by the map induced by \[ \bigg(\gamma^{\vec x_{\phi^{-1}(k_1)}}_{\vec r_{\phi^{-1}(k_1)}},\dotsc \gamma^{\vec x_{\phi^{-1}(k_\ell)}}_{\vec r_{\phi^{-1}(k_\ell)}}\bigg) \in \LC[1](\#a(\vec x_{\phi^{-1}(k_1)} ))\times \dotsb \times \LC[1](\#a(\vec x_{\phi^{-1}(k_\ell)} )). \] where $k_1,...,k_\ell$ are the position of the $a$'s in $\vec y = F_{\phi}(\vec x)$ and $\ell = \#a(\vec y).$
The formula in the $P_{x_{k}}$ factors lands in the appropriate $P_{y_{i}}$ since $\sum \vec r_{\phi^{-1}(i)}$ can only be $0$ when every object in the list $\vec x_{\phi^{-1}(i)}$ is $b$.
To see that $D_{\phi}$ are natural in ${\opsymbfont{T}}^{j}$, it suffices to check the maps $\alpha$ and $\beta$ separately. For the maps $\beta$, the formulas in the $P_{x_{k}}$ factors are clearly natural in ${\opsymbfont{T}}^{n}$. For the maps $\alpha$, we note that $\alpha$ is induced by $1\in \LC[1](0)$ and the composition $\gamma^{\vec x}_{\vec r}\circ_{i}1$ drops the $i$-th subinterval.
This completes the construction of the natural transformation $D_{\phi}$ and of the map $\MA[\phi]$. A straightforward check of the diagrams now proves the following theorem.
\begin{thm}\label{thmmoore} The maps $\MA[\phi]$ above make $\MA$ into a partial associative $R$-algebra. \end{thm}
\section{$THH$ of a Partial $\LC[n]$-Algebra} \label{secncyc}
In this section, we study the multiplicative structure on $THH^{R}(A)$ for a partial $\LC$-algebra $A$. Starting with the cyclic bar construction of a partial associative $R$-algebra, we define $THH^{R}(A)$ as the cyclic bar construction of the Moore partial algebra $\MA$. This depends only on the underlying $\LC[1]$-structure; we use ``interchange'' (see Definition~\ref{definterchange}) of the $\LC[1]$-structure with a $\LC[n-1]$-structure to make $THH^{R}_{\ssdot}(\MA)$ into a partial $\LC[n-1]$-algebra, proving Theorems~\ref{mainthh} and~\ref{mainrel} from the introduction. Finally, in Construction~\ref{consitthh} below, we give a closed description of iterated $THH^{R}$.
We begin by reviewing the cyclic bar construction of a partial associative $R$-algebra.
\begin{cons}[The cyclic bar construction]\label{consncyalg} For a partial associative $R$-algebra $A$, let $THH^{R}(A)$ be the geometric realization of the cyclic $R$-module \[ THH^{R}_{p}(A)=A_{p+1} \] with action of the cyclic group $C_{p+1}$ inherited from the action of the symmetric group $\Sigma_{p+1}$, with face maps $d_{i}$ induced by \[ (\id,\dotsc,\id,\mu,\id,\dotsc,\id)\in {\opsymbfont{A}}(1)^{i}\times {\opsymbfont{A}}(2)\times {\opsymbfont{A}}(1)^{p-i-1} \] for $i=0,\dotsc,p-1$ (with $d_{p}$ induced by the cyclic permutation followed by $d_{0}$), and degeneracy maps $s_{i}$ induced by \[ (\id,\dotsc,\id,1,\id,\dotsc,\id)\in {\opsymbfont{A}}(1)^{i}\times {\opsymbfont{A}}(0)\times {\opsymbfont{A}}(1)^{p-i}. \] \end{cons}
In the case of a partial $\LCR[1]$-algebra, we apply this construction to the Moore partial algebra.
\begin{defn}\label{defncyen} For a partial $\LCR[1]$-algebra $A$, we define $THH^{R}(A)=THH^{R}(\MA)$. \end{defn}
We remark that this point-set construction does not generally represent the correct homotopy type without additional assumptions on $A$. When $A$ is tidy (Definition~\ref{deftidy}), then $\MA$ is tidy, and the simplicial object $THH^{R}_{\ssdot}(A)$ has the correct homotopy type. For the geometric realization to have the correct homotopy type, it suffices for the simplicial object to be ``proper'' (for the inclusion of the union of the degeneracies at each stage to be a Hurewicz cofibration). The following proposition usually suffices for most purposes. We provide an iterable generalization in Proposition~\ref{propitprop} at the end of this section.
\begin{prop}\label{proptrueprop} Working in the context of $R$-modules of symmetric spectra, orthogonal spectra, or EKMM $S$-modules, assume that $A$ is a true $\LCR[1]$-algebra such that the map $R\to A$ is a cofibration of the underlying $R$-modules. Then $\MA$ is tidy and $THH^{R}_{\ssdot}(\MA)$ is a proper simplicial object. \end{prop}
\begin{proof} Under the hypotheses above, each map $A^{m-1}\to A^{m}$ induced by $\LC[1](0)$ is a cofibration of the underlying $R$-modules, and in the case of symmetric spectra or orthogonal spectra, it follows that (after passing to a retraction if necessary), each degeneracy map from $THH^{R}_{p-1}(\MA)$ is the inclusion of a subcomplex in a fixed relative ${\opsymbfont{I}}'$-cell complex \cite[5.4]{MMSS} structure on $THH^{R}_{p}(\MA)$ (relative to the inclusion of $R$), where ${\opsymbfont{I}}'={\opsymbfont{I}}$ is the set of generating cofibrations in the model structure \cite[6.2]{MMSS}. In the context of EKMM $S$-modules, the same holds, but for ${\opsymbfont{I}}'$ the set of generating cofibrations together with the maps $R\wedge S^{j-1}_{+}\to R\wedge B^{j-1}_{+}$ (for $j\geq 0$). It follows that the union (colimit) of these subcomplexes is a subcomplex and its inclusion is a Hurewicz cofibration. \end{proof}
This completes the generalization of the cyclic bar construction to a partial $\LC[1]$-algebra $A$. The remainder of the section studies the multiplicative structure when $A$ is a $\LC$-algebra. In this case, we understand $A$ to be a $\LC[1]$-algebra via the \term{first-coordinate embedding} of $\LC[1]$ in $\LC[n]$: This embedding takes the sequence of subintervals \[ ( [x_{1},y_{1}],\dotsc,[x_{m},y_{m}] ) \in \LC[1](m) \] to the sequence of subrectangles \[ ( [x_{1},y_{1}]\times [0,1]^{n-1},\dotsc , [x_{m},y_{m}] \times [0,1]^{n-1}) \in \LC(m). \] We also have a \term{last coordinates embedding} of $\LC[n-1]$ in $\LC[n]$, taking the sequence of subrectangles \[ ( [x^{1}_{1},y^{1}_{1}]\times \dotsb \times [x^{n-1}_{1},y^{n-1}_{1}], \dotsc, [x^{1}_{m},y^{1}_{m}]\times \dotsb \times [x^{n-1}_{m},y^{n-1}_{m}]) \in \LC[n-1](m) \] to the sequence of subrectangles \[ ( [0,1]\times [x^{1}_{1},y^{1}_{1}]\times \dotsb \times [x^{n-1}_{1},y^{n-1}_{1}], \dotsc, [0,1]\times [x^{1}_{m},y^{1}_{m}]\times \dotsb \times [x^{n-1}_{m},y^{n-1}_{m}]) \in \LC(m). \] Both of these embeddings are special cases of the following ``interchange'' map.
\begin{defn}\label{definterchange} For $\ell,m\geq 0$, the \term{interchange map} is the map \[ \rho \colon \LC[1](\ell)\times \LC[n-1](m)\to \LC(\ell m) \] that takes the pair \[ ([a^{i},b^{i}]\mid 1\leq i\leq \ell), ([x^{j}_{1},y^{j}_{1}]\times \cdots \times [x^{j}_{n-1},y^{j}_{n-1}] \mid 1\leq j\leq m) \] to the sequence of subrectangles of $[0,1]^{n}$, \[ [a^{i},b^{i}]\times [x^{j}_{1},y^{j}_{1}]\times \cdots \times [x^{j}_{n-1},y^{j}_{n-1}], \] (for $1\leq i\leq \ell$, $1\leq j\leq m$), ordered by lexicographical order in $(i,j)$. \end{defn}
The first coordinate embedding is then \[ \rho_{\mathrm{first}}(-)=\rho (-,([0,1]^{n-1}))\colon \LC[1]\to \LC[n] \] and the last coordinate embedding is \[ \rho_{\mathrm{last}}(-)=\rho(([0,1]),-)\colon \LC[n-1]\to \LC[n]. \] Using $\rho_{\mathrm{last}}$, for any element $c$ in $\LC[n-1](m)$, we get an element $\rho_{\mathrm{last}}(c)$ in $\LC(m)$ and hence a map $A_{m}\to A$.
We call $\rho$ the interchange map because for a (true) $\LC$-algebra $A$, for any $a$ in $\LC[1](\ell)$ and $c$ in $\LC[n-1](m)$, both composites in the diagram \begin{equation}\label{eqint} \begin{gathered} \xymatrix@C+2pc{ (A^{(m)})^{(\ell)}\ar[r]^-{(\rho_{\mathrm{last}}(c))^{(\ell)}} \ar[d]_{a} &A^{(\ell)}\ar[d]^{a}\\ A^{(m)}\ar[r]_-{\rho_{\mathrm{last}}(c)}&A } \end{gathered} \end{equation} are the induced map of $\rho(a,c)\colon A^{(\ell m)}\to A$ under the isomorphism $A^{(\ell m)}\cong (A^{(m)})^{(\ell)}$ induced by lexicographical order. In the diagram, the left vertical arrow is the action of $a$ in the ``diagonal'' $\LC$-algebra structure on $A^{(m)}$: Its has the structure map \begin{equation}\label{eqdiagact} \LC(\ell)_{+}\wedge (A^{(m)})^{(\ell)} \to (\LC(\ell)_{+})^{(m)}\wedge (A^{(m)})^{(\ell)} \cong (\LC(\ell)_{+}\wedge A^{(\ell)})^{(m)} \to A^{(m)} \end{equation} where the first map is induced by the diagonal map on $\LC(\ell)$, the last map is the action map for $A$ on each of the $m$-factors, and the isomorphism in the middle is the permutation $\sigma_{m,\ell}$ that rearranges the $\ell$ blocks of $m$ factors of $A$ into $m$ blocks of $\ell$ factors. The following proposition is an immediate consequence of the commutative diagram.
\begin{prop}\label{proptrueint} Let $A$ be a true $\LC$-algebra. For any $c$ in $\LC[n-1](m)$, the map $A^{(m)}\to A$ induced by $\rho_{\mathrm{last}}(c)$ is a map of $\LC[1]$-algebras. \end{prop}
The only obstacle to extending the previous proposition to partial $\LC$-algebras is understanding $A_{m}$ as a partial $\LC$-algebra, and the only obstacle here is understanding $A_{m}$ as a partial power system. We overcome these obstacles in the following definition.
\begin{defn}\label{defppspower} For $X$ an op-lax power system and $m>0$, let $X^{[m]}$ be the op-lax power system with $X^{[m]}_{p}=X_{pm}$, symmetric group action induced by block permutation (with blocks of size $m$), and structure maps \[ \lambda_{p,q}\colon X^{[m]}_{p+q}\to X^{[m]}_{p}\wedge_{R}X^{[m]}_{q} \] induced from the structure map $\lambda_{pm,qm}$ for $X$. Then $X^{[1]}=X$, and we let $X^{[0]}=R$. \end{defn}
We note that when $X$ is a partial power system, $X^{[m]}$ is as well. For a partial $\LC$-algebra $A$, the partial power system $A^{[m]}$ obtains a ``diagonal'' partial $\LC$-algebra structure from the $\LC$-action on $A$: The element \[ (c_{1},\dotsc,c_{j})\in \LC(\ell_{1})\times \dotsb \times \LC(\ell_{j}) \] induces the map $A^{[m]}_{\ell}\to A^{[m]}_{j}$ (for $\ell=\ell_{1}+\dotsb +\ell_{j}$) given by \begin{equation}\label{eqperm} \sigma_{m,j}^{-1}\circ (c_{1},\dotsc,c_{j},\dotsc,c_{1},\dotsc,c_{j})\circ \sigma_{m,\ell}\colon A_{\ell m}\to A_{jm} \end{equation} (with $c_{1},\dotsc,c_{j}$ repeated $m$ times) where $\sigma_{m,k}$ denotes the permutation in $\Sigma_{km}$ that rearranges the $k$ blocks of $m$ into $m$ blocks of $k$, as in~\eqref{eqdiagact}. Given an element $c$ in $\LC(m)$, we can also make $c$ into a map of partial power systems $A^{[m]}\to A$, by defining the map $c_{\ell}$ on the $\ell$-th partial power level to be $(c,\dotsc,c)$, i.e., $c$ repeated $\ell$ times, without permutations. Then regarding \[ (a_{1},\dotsc,a_{j})\in \LC[1](\ell_{1})\times \dotsb \times \LC[1](\ell_{j}) \] as a partial $\LC[1]$-algebra structure map and using the map of partial power systems $\rho_{\mathrm{last}}(c)$ for $c\in \LC[n-1](m)$, the following interchange diagram commutes. \[ \xymatrix@C+2pc{ A^{[m]}_{\ell}\ar[d]_{(a_{1},\dotsc,a_{j})} \ar[r]^{\rho_{\mathrm{last}}(c)_{\ell}} &A_{\ell}\ar[d]^{(a_{1},\dotsc,a_{j})}\\ A^{[m]}_{j}\ar[r]_{\rho_{\mathrm{last}}(c)_{j}} &A_{j} } \] This is the partial analogue of~\eqref{eqint} and proves the following partial analogue of Proposition~\ref{proptrueint}.
\begin{prop}\label{propint} Let $A$ be a partial $\LC$-algebra. For any $c$ in $\LC[n-1](m)$, the map $A^{[m]}\to A$ induced by $\rho_{\mathrm{last}}(c)$ is a map of $\LC[1]$-algebras. \end{prop}
Returning to the cyclic bar construction, we now have the terminology and notation to describe the multiplicative structure and prove Theorems~\ref{mainthh} and~\ref{mainrel}. Even in the case when we start with a true $\LC$-algebra $A$, $THH^{R}(A)$ will only have a partial multiplicative structure. We construct the op-lax power system as follows.
\begin{cons}\label{consthhps} For a partial $\LCR[1]$-algebra $A$, we define the op-lax power system $THH^{R}(A)$ by $(THH^{R}(A))_{m}=THH^{R}(\MoAlg{A^{[m]}})$. \end{cons}
When each of the simplicial objects $THH^{R}_{\ssdot}(\MoAlg{A^{[m]}})$ is proper, this defines a partial power system, since geometric realization then preserves the degreewise weak equivalences. With just this mild hypotheses, we can now prove the following version of Theorem~\ref{mainrel}.
\begin{thm}\label{thmmainrel} For $R$ a commutative $S$-algebra, let $A$ be a partial $\LCR$-algebra such that the op-lax power system $THH^{R}(A)$ is a partial power system. Then $THH^{R}(A)$ is a partial $\LCR[n-1]$-algebra, naturally in maps of $\LCR$-algebra maps in $A$. \end{thm}
\begin{proof} Applying Proposition~\ref{propint}, we see that every element $c$ in $\LC[n-1](m)$ induces a map of partial associative $R$-algebras \[ \MoAlg{A^{[m]}}\to \MA. \] More generally, the argument for Propositions~\ref{proptrueint} and~\ref{propint} implies that elements $c_{1},\dotsc,c_{r}$ of $\LC[n-1](j_{i})$ induce a map of partial $\LC[1]$-algebras \[ A^{[j_{1}+\dotsb +j_{r}]}\to A^{[r]} \] and hence a map of partial associative $R$-algebras \begin{equation}\label{eqactmoore} \MoAlg{A^{[j_{1}+\dotsb+j_{r}]}}\to \MoAlg{A^{[r]}}. \end{equation} Restricting to the $p$ power and putting these maps together for all elements of $\LC[n-1]$ at once, we get maps of $R$-modules \begin{equation}\label{eqlevel} (\LC[n-1](j_{1})\times \dotsb \times \LC[n-1](j_{r}))_{+} \wedge_{\Sigma_{j_{1}}\times \dotsb\times \Sigma_{j_{r}}} \MoAlg{A^{[j_{1}+\dotsb+j_{r}]}}_{p}\to \MoAlg{A^{[r]}}_{p}. \end{equation} Because the maps~\eqref{eqactmoore} are maps of partial power systems, the maps~\eqref{eqlevel} commute with the $\Sigma_{p}$-action. Because the maps~\eqref{eqactmoore} are maps of partial associative $R$-algebras, the maps~\eqref{eqlevel} commute with the simplicial face maps when we view $\MoAlg{A^{[m]}}_{p}$ as $(THH^{R}_{p-1}(A))_{m}$ (simplicial degree $p-1$ in the $m$-th power). Likewise, the maps~\eqref{eqlevel} commute with the degeneracy operations. Commuting the smash product and geometric realization, we therefore get a map of partial power systems \begin{equation}\label{eqact} \bLC[n-1] THH^{R}(A) \to THH^{R}(A) . \end{equation} Using the associativity and unity of the $\LC$-action on $A$ and the fact that $\rho_{\mathrm{last}}$ is a homomorphism of operads, a straightforward check shows that \eqref{eqact} defines a partial $\LC[n-1]$-algebra structure on $THH^{R}(A)$. \end{proof}
As a special case, when $n=2$, $THH^{R}(A)$ is a $\LC[1]$-algebra. Iterating $THH^{R}$ would mean applying $THH^{R}$ to the Moore partial algebra $\MoAlg{THH^{R}(A)}$. This partial associative $R$-algebra admits a closed description in terms of the colimit diagrams that construct the Moore partial algebra. The simplification in terms of the ``main'' submodules $P^{m}_{+}\wedge A_{m}\subset \MA[m]$ captures the main ideas while avoiding the complications. Since geometric realization commutes with the Moore partial algebra construction, it is enough to describe the construction on each simplicial level. In these terms, the main submodule of $\MoAlg{THH^{R}_{p}(A)}_{m}$ (simplicial degree $p$, partial power $m$) is \[ P^{(p+1)+m}_{+}\wedge A_{(p+1)m}. \] Instead of having a length for each $A$ power, this rather has a length for each column and each row, where we think of $A_{(p+1)m}$ as organized into $m$ rows of $p+1$ columns. The main submodule of $THH^{R}(THH^{R}(A))_{1}$ in bisimplicial degree $p,q$ then is \[ P^{(p+1)+(q+1)}_{+}\wedge A_{(p+1)(q+1)}. \] The face maps in the $p$-direction add the column lengths and concatenate squares (in $\LC[2]$) in the horizontal direction, while the face maps in the $q$-direction add the row length and concatenate squares in the vertical direction.
More generally and precisely, for a partial $\LC$-algebra, we give the following closed construction of the $n$-th iterate of $THH^{R}$
\begin{cons}\label{consitthh} Let $A$ be a partial $\LC$-algebra. The $n$-th iterate of $THH^{R}$ is isomorphic to the geometric realization of the following $n$-fold simplicial set $T^{n}(A)$. In multisimplicial degree $p_{1},\dotsc,p_{n}$, $T^{n}(A)$ is the colimit of the diagram indexed on ${\opsymbfont{T}}^{(p_{1}+1)+\dotsb+(p_{n}+1)}$ that at the object \[ \vec x=(x_{1},\dotsc,x_{q})= (x^{1}_{0},\dotsc,x^{1}_{p_{1}},x^{2}_{0},\dotsc,x^{2}_{p_{2}}, \dotsc,x^{n}_{0},\dotsc,x^{n}_{p_{n}}) \] is the $R$-module \[ (P_{x_{1}}\times \dotsb \times P_{x_{q}})_{+}\wedge A_{m_{1}\dotsb m_{n}}, \] where $m_{i}=\#a(x^{i}_{0},\dotsc,x^{i}_{p_{i}})$ (the number of $a$'s among the $x^{i}_{j}$'s). The degeneracies are induced by inserting $b$ in the appropriate spot in $\vec x$ with zero as the associated length in $P_{b}=\bar P$. In the $i$-th simplicial direction, the $j$-th face map (for $j<p_{i}$) is induced as follows. The lengths $(r,s)\in P_{x^{i}_{j}}\times P_{x^{i}_{j+1}}$ add. If $x^{i}_{j}=x^{i}_{j+1}=a$, we use the element \[ c=[0,1]^{i-1}\times \gamma_{r,s}\times [0,1]^{n-i} \] of $\LC(2)$, applied $m_{1}\dotsb \hat m_{i}\dotsb m_{n}$ times, with the appropriate permutations (as in~\eqref{eqperm}) to produce the map \[ A_{m_{1}\dotsb m_{n}}\to A_{m_{1}\dotsb (m_{i}-1)\dotsb m_{n}}. \] If one of $x^{i}_{j},x^{i}_{j+1}$ is $a$, we use the appropriate element \begin{align*} &([0,1]^{i-1} \times [0,r/(r+s)]\times [0,1]^{n-i}), \quad \text{or}\\ &([0,1]^{i-1} \times [r/(r+s),1]\times [0,1]^{n-i}) \end{align*} of $\LC(1)$. We use the identity on the $A$'s factor if neither of $x^{i}_{j},x^{i}_{j+1}$ is $a$. The $p_{i}$-th face map is the appropriate permutation followed by the zeroth face map. \end{cons}
Using the concrete construction above, we can see that the multisimplicial object $T^{n}_{\ssdot}(A)$ has the correct homotopy type when $A$ is tidy, and hence in this case a thickened realization will capture the correct homotopy type for iterated $THH^{R}$. Moreover, we can see that $T^{n}_{\ssdot}(A)$ is proper under reasonable hypotheses on $A$ like the one discussed above or its iterable generalization, which we now discuss.
For the iterable generalization of Proposition~\ref{proptrueprop}, we use the following technical hypothesis. In it, ${\opsymbfont{I}}'$ is as in the proof of Proposition~\ref{proptrueprop}: In the context of symmetric spectra or orthogonal spectra, ${\opsymbfont{I}}'$ is the collection of generating cofibrations in the model structure; in the context of EKMM $S$-modules, ${\opsymbfont{I}}'$ also includes the maps $R\wedge S^{j-1}_{+}\to R\wedge B^{j}_{+}$.
\begin{hyp}[Technical hypothesis on a partial {$\LC[1]$}-algebra $A$] \label{hypprop} For all $m$, the maps $R\to A_{m}$ are relative ${\opsymbfont{I}}'$-cell complexes (for ${\opsymbfont{I}}'$ as above) such that each of the $m$ maps $A_{m-1}\to A_{m}$ (induced by the action of $\LC[1](0)$) is the inclusion of a relative subcomplex. \end{hyp}
\begin{prop}\label{propitprop} Working in the category of $R$-modules of symmetric spectra, orthogonal spectra, or EKMM $S$-modules, let $A$ be a partial $\LCR[n]$-algebra satisfying Hypothesis~\ref{hypprop}. Then $\MA$ is tidy, $THH^{R}_{\ssdot}(\MoAlg{A^{[m]}})$ is a proper simplicial object for all $m$, and the op-lax power system $THH^{R}(A)$ is a partial power system. Moreover, if $n\geq 2$, then the partial $\LCR[n-1]$-algebra $THH^{R}(A)$ also satisfies Hypothesis~\ref{hypprop}. \end{prop}
\begin{proof} Since $\LC(0)$ is a point, the maps $A_{m-1}\to A_{m}$ induced by $\LC[1](0)$ (using the first coordinate embedding) coincide with the maps $A_{m-1}\to A_{m}$ induced by $\LC[n-1](0)$ (using the last coordinates embedding). The proof is now a straightforward cell argument in terms of the construction of $\MA$ and $THH^{R}$. \end{proof}
We now have Theorem~\ref{mainrel} as an immediate corollary of the previous proposition and Theorem~\ref{thmmainrel}. Theorem~\ref{mainthh} is the special case $R=S$.
\section{The Bar Construction for Augmented $\LC$-Algebras} \label{secbar}
In this section we study the reduced version of $THH$, usually called the bar construction, defined for an augmented partial $\LC$-algebra. We begin with the definition of augmented partial $\LC$-algebras and augmented partial associative $R$-algebras.
\begin{defn} An \term{augmented} partial $\LCR$-algebra consists of a partial $\LCR$-algebra $A$ together with a map of partial $\LCR$-algebras $\epsilon \colon A\to R$ called the \term{augmentation}. Likewise, an augmented partial associative $R$-algebra consists of a partial associative $R$-algebra $A$ and a map of partial associative $R$-algebras $A\to R$. \end{defn}
In general for an op-lax power system $X$, given a map $X\to R$, we get a pair of maps $X_{m}\to X_{m-1}$ as composites \[ \begin{gathered} e_{1}\colon X_{m}\to X_{1}\wedge_{R}X_{m-1}\to R\wedge_{R}X_{m-1}\cong X_{m-1}\\ e_{m}\colon X_{m}\to X_{m-1}\wedge_{R}X_{1}\to X_{m-1}\wedge_{R}R\cong X_{m-1} \end{gathered} \] using $\lambda_{1,m-1}$ and $\lambda_{m-1,1}$. We think of $e_{1}$ and $e_{m}$ as applying the augmentation to the first and last spots in $X_{m}$. Note that $e_{m}$ can be obtained from $e_{1}$ and the permutation actions on $X_{m}$ and $X_{m-1}$, and using permutations like this, we can define analogous maps $e_{2},\dotsc,e_{m-1}$ that apply the augmentation to an arbitrary spot in $X_{m}$.
In the case of an augmented partial associative $R$-algebra $A$, the maps $e_{j}$ make the following diagram commute. \[ \xymatrix{ A_{m}\ar[r]^{e_{1}}\ar[d]_{(\mu,\id,\dotsc,\id)}&A_{m-1}\ar[d]^{e_{1}} &&A_{m}\ar[r]^{e_{m}}\ar[d]_{(\id,\dotsc,\id,\mu)}&A_{m-1}\ar[d]^{e_{m-1}}\\ A_{m-1}\ar[r]_{e_{1}}&A_{m-2}&&A_{m-1}\ar[r]_{e_{m-1}}&A_{m-2} } \] We use these maps in the following construction.
\begin{cons}\label{consbar} For an augmented partial associative $R$-algebra $A$, let $BA$ be the geometric realization of the simplicial $R$-module $B_{\ssdot} A$ that is $A_{m}$ in simplicial degree $m$, with degeneracy maps $s_{i}$ induced by the action of ${\opsymbfont{A}}(0)$, with face maps $d_{i}$ for $1\leq i\leq m-1$ induced by the action of $\mu \in {\opsymbfont{A}}(2)$, and with face maps $d_{0}=e_{1}$ and $d_{m}=e_{m}$ as defined above. \end{cons}
As in the previous section, the construction may not have the correct homotopy type without the additional assumption that $A$ is tidy and an additional assumption ensuring that the simplicial object is proper.
For $A$ a partial augmented $\LC[n]$-algebra, we set $BA=B\MA$, and we extend $BA$ to a partial power system by setting $BA_{m}=B(A^{[m]})=B(\MoAlg{A^{[m]}})$. The trick used in the proof of Theorems~\ref{mainthh} and~\ref{mainrel} now constructs the $\LC[n-1]$-structure in the following theorem.
\begin{thm}\label{thmbarmult} Let $A$ be an augmented partial $\LC$-algebra $A$ such that $B_{\ssdot} A^{[m]}$ is a proper simplicial object for all $m$. Then the bar construction $BA$ is naturally an augmented partial $\LC[n-1]$-algebra. \end{thm}
The properness hypothesis holds in particular when $A$ satisfies Hypothesis~\ref{hypprop} or the hypothesis of Proposition~\ref{proptrueprop}. The augmentation in the theorem is the map $BA\to R$ induced by the augmentations $\MoAlg{A^{[m]}}_{p}\to R$. We now give a detailed description of the iterated bar construction along the lines of Construction~\ref{consitthh}.
\begin{cons} Let $A$ be an augmented partial $\LC$-algebra. The $n$-th iterate of the bar construction is isomorphic to the geometric realization of the following $n$-fold simplicial set $B^{n}(A)$. In multisimplicial degree $p_{1},\dotsc,p_{n}$, $B^{n}(A)$ is the colimit of the diagram indexed on ${\opsymbfont{T}}^{p_{1}+\dotsb+p_{n}}$ that at the object \[ \vec x=(x_{1},\dotsc,x_{q})= (x^{1}_{1},\dotsc,x^{1}_{p_{1}},x^{2}_{1},\dotsc,x^{2}_{p_{2}}, \dotsc,x^{n}_{1},\dotsc,x^{n}_{p_{n}}) \] is the $R$-module \[ (P_{x_{1}}\times \dotsb \times P_{x_{q}})_{+}\wedge A_{m_{1}\dotsb m_{n}}, \] where $m_{i}=\#a(x^{i}_{1},\dotsc,x^{i}_{p_{i}})$ (the number of $a$'s among the $x^{i}_{j}$'s). The degeneracies are induced by inserting $b$ in the appropriate spot in $\vec x$ with zero as the associated length in $P_{b}=\bar P$. In the $i$-th simplicial direction, the $j$-th face map (for $0<j<p_{i}$) is induced as follows. The lengths $(r,s)\in P_{x^{i}_{j}}\times P_{x^{i}_{j+1}}$ add. If $x^{i}_{j}=x^{i}_{j+1}=a$, we use the element \[ c=[0,1]^{i-1}\times \gamma_{r,s}\times [0,1]^{n-i} \] of $\LC(2)$, applied $m_{1}\dotsb \hat m_{i}\dotsb m_{n}$ times, with the appropriate permutations (as in~\eqref{eqperm}) to produce the map \[ A_{m_{1}\dotsb m_{n}}\to A_{m_{1}\dotsb (m_{i}-1)\dotsb m_{n}}. \] If one of $x^{i}_{j},x^{i}_{j+1}$ is $a$, we use the appropriate element \begin{align*} &([0,1]^{i-1} \times [0,r/(r+s)]\times [0,1]^{n-i}), \quad \text{or}\\ &([0,1]^{i-1} \times [r/(r+s),1]\times [0,1]^{n-i}) \end{align*} of $\LC(1)$. We use the identity on the $A$'s factor if neither of $x^{i}_{j},x^{i}_{j+1}$ is $a$. The $0$-th and $p_{i}$-th face maps are obtained by application of the appropriate maps $e_{j}$. \end{cons}
The category of $R$-modules under and over $R$ has $R$ as both an initial and final object. As a consequence, this category admits a tensor with based spaces: For $M$ an $R$-module under and over $R$ and $X$ a based space, the tensor of $M$ with $X$, $M\mathbin{\widehat\otimes} X$, is formed as a pushout \[ \xymatrix{ R\wedge X_{+}\cup_{R}M \ar[r]\ar[d]&M\wedge X_{+}\ar[d]\\ R\ar[r]&M\mathbin{\widehat\otimes} X. } \] In particular, we have a suspension in the category of $R$-modules under and over $R$ that we denote as $\Sigma_{R}$. For an augmented partial $R$-algebra $A$, the first partial power $A_{1}$ is an $R$-module under and over $R$, as is the first partial power $\MA[1]$ of the Moore partial algebra. The unit maps $\MA[1]\to \MA[m]$ induce maps \[ \underbrace{\MA[1]\cup_{R}\dotsb \cup_{R}\MA[1]}_{m\text{ factors}} \to \MA[m], \] which together induce a map of simplicial objects \[ \MA[1]\mathbin{\widehat\otimes} S^{1}_{\ssdot} \to B_{\ssdot} A, \] which on geometric realization induces a map $\Sigma_{R}\MA[1]\to BA$, natural in $A$. Using the explicit description of the multisimplicial object $B^{n}A$ above, the simplicial map above generalizes to a multi-simplicial map \[ \MA[1]\mathbin{\widehat\otimes} (S^{1}_{\ssdot}\wedge \dotsb \wedge S^{1}_{\ssdot})\to B^{n}_{\bullet,\dotsc,\bullet}A \] as follows. Thinking of an element of $S^{1}_{p}$ as an element of the based set $\{0,\dotsc,p\}$, a non-basepoint element of $S^{n}_{p_{1},\dotsc,p_{n}}$ is an $n$-tuple $\vec j=(j_{1},\dotsc,j_{n})$ with $1\leq j_{i}\leq p_{i}$. We have one copy of $\MA[1]$ in $\MA[1]\mathbin{\widehat\otimes} S^{n}_{p_{1},\dotsc,p_{n}}$ for each $\vec j$, which we map into $B^{n}_{p_{1},\dotsc,p_{n}}A$ by a map induced by a map of diagrams. For fixed $\vec j$, we have the functor ${\opsymbfont{T}}\to {\opsymbfont{T}}^{p_{1}+\dotsb +p_{n}}$ sending $x$ to the object $\vec x$ where $x^{i}_{j_{i}}=x$ and $x^{i}_{k}=b$ for $k\neq j_{i}$. We use the map of diagrams compatible with this functor sending $P_{+}\wedge A_{1}$ (or $\bar P_{+}\wedge R$ or $P_{+}\wedge R$, for $x=a$, $b$, or $c$, respectively) into $(P_{x_{1}}\times\dotsb \times P_{x_{q}})_{+}\wedge A_{1}$ (or $(P_{x_{1}}\times\dotsb \times P_{x_{q}})_{+}\wedge R$) induced by the identity on $A_{1}$ (or $R$) and sending $r$ in $P$ to $(r^{1}_{1},\dotsc,r^{n}_{p_{n}})$ in $P_{x^{1}_{1}}\times\dotsb \times P_{x^{n}_{p_{n}}}$ with $r^{i}_{j_{i}}=r$ and $r^{i}_{k}=0$ for $k\neq j_{i}$. This then describes a map from $\MA[1]\mathbin{\widehat\otimes} S^{n}_{p_{1},\dotsc,p_{n}}$ to $B^{n}_{p_{1},\dotsc,p_{n}}A$ that respects the face and degeneracy maps in all simplicial direction. On geometric realization, we get a map \begin{equation}\label{eqsusp} \Sigma^{n}_{R}\MA[1]\to B^{n}A, \end{equation} natural in $A$. The map of $R$-modules under and over $R$ from $\MA[1]$ to $A_{1}$ (obtained by forgetting length coordinate) induces a map \[ \Sigma^{n}_{R}\MA[1]\to \Sigma^{n}_{R}A_{1} \] which is a weak equivalence when the unit $R\to A_{1}$ is a Hurewicz cofibration, so in particular, when the hypothesis of Theorem~\ref{thmbarmult} holds.
\section{The Bar Construction for Non-Unital Partial $\LC$-algebras} \label{secbar2}
In the next section, we study the reduced Andr\'e--Quillen cohomology of a $\LCR$-algebra; as we will see there, this is easiest when we work with ``non-unital'' $\LCR$-algebras in place of augmented $\LCR$-algebras. For this reason, we take this section to redo the work of the previous section in the non-unital context. We begin with the definition of a non-unital partial $\LCR$-algebra.
\begin{defn}\label{defnonunital} Let $\NC$ be the operad with $\NC(0)$ empty and $\NC(m)=\LC(m)$ for $m>0$. A \term{non-unital} partial $\LC$-algebra is a partial $\NC$-algebra. \end{defn}
If $N$ is a non-unital partial $\LC$-algebra, then we can form an associated augmented partial $\LC$-algebra by formally adding a unit. Noting that for an $R$-module $X$, \[ (X\vee R)^{m}=\bigvee_{s\subset \otos[m]} X^{(s)}, \] (smash power indexed on the set $s$) where $\otos[m]=\{1,\dotsc,m\}$, the $R$-modules \[
KN_{m}=\bigvee_{s\subset \otos[m]} N_{|s|} \] naturally form a partial power system with $KN_{1}=N_{1}\vee R$. The partial $\NC$-algebra structure on $N$ extends to a partial $\LC$-algebra structure on $KN$, with the missing elements in the subsets acting like place-holders for the unit $1\in \LC(0)$ and with the action of $\LC(0)$ manipulating the sets $\otos[m]=\{1,\dotsc,m\}$ and their subsets. Specifically, on the summand corresponding to $s\subset \otos[m]$, the element \[ (c_{1},\dotsc,c_{j})\in \LC(m_{1})\times \dotsb \times \LC(m_{j}) \] (for $m_{1}+\dotsb +m_{j}=m$) induces the map \begin{equation}\label{eqnuact}
(c'_{1},\dotsc,c'_{j'})\colon N_{|s|}\to N_{|t|} \end{equation} into the $t\subset \otos[j]=\{1,\dotsc,j\}$ summand, as follows. Noting that $(c_{1},\dotsc,c_{j})$ has $m$ total inputs, we form \[ (c'_{1},\dotsc,c'_{j'})\in \LC(m'_{1})\times \dotsb \times \LC(m'_{j'}) \]
(with $m'_{1}+\dotsb+m'_{j'}=|s|$) by plugging $1\in \LC(0)$ into each input that corresponds to an element not in $s$, and then dropping any $c_{i}$ whose inputs all become plugged. The subset $t$ then consists of those elements $i$ in $\otos[j]$ where $c_{i}$ was not dropped.
In the case of true algebras, we can also go from augmented $\LC$-algebras to non-unital $\LC$-algebras: For a true $\LC$-algebra $A$, we can form the true non-unital $\LC$-algebra $N$ as the homotopy pullback of the augmentation, $N=R^{I}\times_{R}A$. In the next section, we will see that for true $\LC$-algebras, up to homotopy, working with non-unital $\LC$-algebras is equivalent to working with augmented $\LC$-algebras (Theorem~\ref{thmaugqe}).
We have a corresponding notion of non-unital partial associative $R$-algebra, and the construction $K$ above also defines a functor from non-unital partial associative $R$-algebras to augmented partial associative $R$-algebras. We can generalize the Moore algebra to this context. For a non-unital partial $\LC[1]$-algebra $N$, the power system \[ \MN[m]=P^{m}_{+}\wedge N_{m} \] forms a non-unital partial associative $R$-algebra using the length concatenation construction in the Moore algebra. To compare this with $\MoAlg{KN}$, note that \[ K\MN[m]=\bigvee_{s\subset \otos[m]}
P^{s}_{+}\wedge N_{|s|}, \] (the cartesian product of copies of $P$ indexed on $s$), while \[ \MoAlg{KN}_{m}=\bigvee_{s\subset \otos[m]}
(P^{m}_{s})_{+}\wedge N_{|s|}, \] where $P^{m}_{s}$ is the subset of $\bar P^{m}=[0,\infty)^{m}$ of points $(r_{1},\dotsc, r_{m})$ with $r_{i}>0$ if $i\in s$. The inclusion of $P^{s}$ in $P^{m}_{s}$ as the subset where $r_{i}=0$ for $i\not\in s$ induces a map $K\MN$ to $\MoAlg{KN}$ that is a map of partial associative $R$-algebras and a $\Sigma_{m}$-equivariant homotopy equivalence of $R$-modules in each partial power.
We can construct a version of the bar construction for a non-unital $\LC$-algebra, taking \[ B_{m} N = \MN[m] \] for the non-unital Moore algebra construction above. Since $N$ and $\MN$ do not have units, this collection of $R$-modules does not admit degeneracy maps, but the face maps in Construction~\ref{consbar} still make sense (using the trivial map for the zeroth and last face map in place of the augmentation). Then $B_{\ssdot} N$ forms a $\Delta$-object (a simplicial object without degeneracies), and we can form $BN$ as the geometric realization (by gluing $B_{m}\wedge \Delta[m]_{+}$ along the face maps). We compare $BN$ and $BKN$ in the following proposition.
\begin{prop} The $R$-modules $BN$ and $B(K\MN)$ are canonically isomorphic and the map of $R$-modules $B(K\MN)\to B(\MoAlg{KN})=BKN$ is a homotopy equivalence. \end{prop}
\begin{proof} Associated to any $\Delta$-object we obtain a simplicial object by formally attaching degeneracies; the geometric realization of the $\Delta$-object is canonically isomorphic to the geometric realization of the associated simplicial object. In this case, the simplicial object associated to $BN$ is in simplicial degree $m$ the $R$-module \[ \bigvee_{j\leq m}\ \bigvee_{f\colon \tos[m]\to\tos[j]} \MN[j], \] where the maps $f\colon \tos[m]\to\tos[j]$ in the inner wedge are the iterated degeneracies in the category ${\mathbf \varDelta}$, i.e., the weakly increasing epimorphisms of totally ordered sets $\{0,\dotsc,m\}\to\{0,\dotsc,j\}$. We have a one-to-one correspondence between the set of such epimorphisms $f$ and the set of $j$ element subsets $s$ of $\{1,\dotsc,m\}$ (where the elements in $s$ are the first elements of $\{0,\dotsc,m\}$ that $f$ sends to each of the elements $1,\dotsc,j$ of $\{0,\dotsc,j\}$), and this defines an isomorphism between the $R$-module above and $K\MN[m]$. As $m$ varies, these isomorphisms preserve the face and degeneracy maps in the simplicial objects. This proves the first statement. For the second statement, we note that the homotopy equivalences $K\MN[p]\to \MoAlg{KN}_{p}$ (using linear homotopies) commute with the face and degeneracy maps in the bar construction and so extend to homotopy equivalences on the geometric realizations. \end{proof}
We note that the properness issues that plague the discussion of the cyclic bar construction and the bar construction for augmented algebras disappear for non-unital algebras: For the simplicial version of the construction $BN$, each degeneracy is the inclusion of a wedge summand and the inclusion of the union of degeneracies likewise is the inclusion of a wedge summand. For the construction $BKN$, each degeneracy is induced by smashing with the inclusion of $\{0\}_{+}$ in $\bar P_{+}=[0,\infty)_{+}$ followed by the inclusion of a wedge summand, and the inclusion of the union of the degeneracies admits a similar description on each of its wedge summands. Thus, in particular, we have the following version of Theorem~\ref{thmbarmult}.
\begin{thm}\label{thmbarmultnu} For a non-unital $\LC$-algebra $N$, $BKN$ is a naturally an augmented partial $\LC[n-1]$-algebra. \end{thm}
We have a variant of $BN$ where we use the trivial $R$-module in place of $R$ at the zero level. We denote this as $\tilde B N$, and we identify the partial power system $B(K\MN)$ as $K\tilde B N$. Specifically, $(\tilde B N)_{m}$ is the geometric realization of the simplicial object which in degree $p$ is \[ (\tilde B_{p}N)_{m}= \bigvee_{s_{1},\dotsc,s_{m}}
P^{|s_{1}\cup\dotsb \cup s_{m}|}_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{m}|}, \] with the wedge over the $m$-tuples of non-empty subsets of $\otos[p]=\{1,\dotsc,p\}$. We can identify $(\tilde B_{p}N)_{m}$ as a submodule of \[ (B_{p}KN)_{m}=\bigvee_{s_{1},\dotsc,s_{m}} (P^p_{s_{1}\cup \dotsb \cup s_{m}})_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{m}|} \] with the wedge over the $m$-tuples of all subsets of $\otos[p]$. From the work above we have a partial $\LC[n-1]$-algebra structure on the partial power system $B_{p}KN$; this restricts to a non-unital partial $\LC[n-1]$-algebra structure on the partial power system $\tilde B_{p}N$ as follows.
The action becomes easier to describe when we re-index the summands by writing \[ (B_{p}KN)_{m}=\bigvee_{s_{1},\dotsc,s_{p}} (P(s_{1})\times \dotsb \times P(s_{p}))_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{p}|} \] where the sum is over the $p$-tuples of subsets of $\otos[m]=\{1,\dotsc,m\}$ and $P(s)=\bar P$ if $s$ is empty and $P(s)=P$ if $s$ is non-empty; the relationship between these two indexings of the wedge sum corresponds to arranging $\{1,\dotsc ,pm\}$ into the $p$ blocks of $m$ elements $(1,\dotsc,m),(m+1,\dotsc, 2m),\dotsc, ((p-1)m+1,\dotsc,pm)$ in the first description or the $m$ blocks of $p$ elements $(1,m+1,2m+1,\dotsc,(p-1)m+1),\dotsc, (m,2m,\dotsc ,pm)$ in the new description. Letting $P'(s)=\{0\}$ if $s$ is empty and $P'(s)=P(s)=P$ if $s$ is non-empty, then in this formulation, \[ (\tilde B_{p}N)_{m}= \bigvee_{s_{1},\dotsc,s_{p}} (P'(s_{1})\times \dotsb \times P'(s_{p}))_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{p}|}, \] where the sum is over the $p$-tuples of subsets of $\{1,\dotsc,m\}$ such that $s_{1}\cup\dotsb \cup s_{p}=\{1,\dotsc,m\}$. In the $\LC[n-1]$-action, for $m_{1}+\dotsb +m_{j}=m$, $m_{i}>0$, an element \[ (c_{1},\dotsc,c_{j})\in \LC[n-1](m_{1})\times \dotsb \times \LC[n-1](m_{j}) \] acts on the $s_{1},\dotsc,s_{p}$ summand by the identity map on the $P$ factors and by the map \[ (c^{1}_{1},\dotsc,c^{1}_{k_{1}},\dotsc,c^{p}_{1},\dotsc,c^{p}_{k_{p}}) \in \LC[n-1](m^{1}_{1})\times \dotsb \times \LC[n-1](m^{p}_{k_{p}}) \]
from $N_{|s_{1}|+\dotsb +|s_{p}|}$ to $N_{|t_{1}|+\dotsb +|t_{p}|}$ (using the last coordinates embedding of $\LC[n-1]$ in $\LC[n]$), where $t_{i}$ and $(c^{i}_{1},\dotsc,c^{i}_{k_{i}})$ are as in~\eqref{eqnuact} above: $(c^{i}_{1},\dotsc,c^{i}_{k_{i}})$ is formed by plugging $1\in \LC[n-1](0)$ into the input corresponding to elements of $\{1,\dotsc,m\}$ not in $s_{i}$, dropping any $c_{r}$ where all inputs are plugged; $t_{i}$ consists of the indexes $r$ where $c_{r}$ is not dropped when forming $(c^{i}_{1},\dotsc,c^{i}_{k_{i}})$. To see that this action restricts to $\tilde B_{p}N$, we just need to observe that when $s_{1}\cup\dotsb \cup s_{p}=\{1,\dotsc ,m\}$, then $t_{1}\cup\dotsb \cup t_{p}=\{1,\dotsc ,j\}$.
Just as for $B_{p}KN$, the $\LC[n-1]$-action on $\tilde B_{p}N$ is compatible with the face and degeneracy maps in the simplicial object $\tilde B N$. This makes $\tilde B N$ into a non-unital partial $\LC[n-1]$-algebra with the map $K\tilde B N\to BKN$ a map of partial $\LC[n-1]$-algebras.
\begin{thm}\label{thmbarnu} For a non-unital partial $\LC$-algebra $N$, the bar construction $\tilde B N$ is naturally a non-unital $\LC[n-1]$-algebra and the weak equivalence $K\tilde B N\to B(KN)$ is a natural map of partial $\LC[n-1]$-algebras. \end{thm}
Next we describe the iterated bar construction $\tilde B^{n}N$. Using the first description of the partial power system $\tilde B_{p}N$ above, we see that $\tilde B^{2}N$ is the $\Delta$-object in simplicial $R$-modules which in degree $p,q$ is \[ \MoAlg{\tilde B_{p}N}_{q}= P^{q}_{+}\wedge \bigl(\bigvee_{s_{1},\dotsc,s_{q}}
P^{|s_{1}\cup\dotsb \cup s_{q}|}_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{q}|}\bigr), \] with the wedge over the $q$-tuples of non-empty subsets of $\otos[p]=\{1,\dotsc,p\}$. In bidegree $p,q$, the associated bisimplicial $R$-module is then \[ \tilde B^{2}_{p,q}N= \bigvee_{\putatop{t\subset \{1,\dotsc,q\}}{t\neq\{\}}}\
\bigvee_{s_{1},\dotsc,s_{|t|}}
P^{|s_{1}\cup\dotsb \cup s_{|t|}|+|t|}_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{|t|}|}, \]
with the inside wedge over the $|t|$-tuples of non-empty subsets of $\otos[p]$. If we re-index the subsets $s_{i}$ by the elements of $t$ and set $s_{i}=\{\}$ for $i\not\in t$, then we get \[ \tilde B^{2}_{p,q}N= \bigvee_{\putatop{s_{1},\dotsc,s_{q}\subset \otos[p]} {s_{1}\cup\dotsb \cup s_{q}\neq \{\}}}
P^{|s_{1}\cup\dotsb \cup s_{q}|+n(s_{1},\dotsc,s_{q})}_{+}\wedge
N_{|s_{1}|+\dotsb +|s_{q}|}, \] where $n(s_{1},\dotsc,s_{q})$ denotes the number of $s_{1},\dotsc,s_{q}$ that are non-empty. Finally, the collections $s_{1},\dotsc,s_{q}\subset \{1,\dotsc p\}$ satisfying $s_{1}\cup\dotsb \cup s_{q}\neq \{\}$ are in one to one correspondence with the non-empty subsets of \[ \otos[p]\times \otos[q]=\{1,\dotsc,p\}\times \{1,\dotsc,q\}. \] For $s\subset \otos[p]\times \otos[q]$, write $\n^{p,q}_{1}(s)$ for the subset of $\otos[p]$ of elements $i$ such that $s\cap (\{i\}\times \otos[q])$ is non-empty and likewise write $\n^{p,q}_{2}(s)$ for the subset of $\otos[q]$ of elements $i$ such that $s\cap (\otos[p]\times \{i\})$ is non-empty. In other words, $\n^{p,q}_{1}$ and $\n^{p,q}_{2}$ are the images of the projection maps from $\otos[p]\times \otos[q]$ to $\otos[p]$ and $\otos[q]$, respectively. Now we can identify $\tilde B^{2}_{p,q}N$ as \[ \tilde B^{2}_{p,q}N= \bigvee_{\putatop{s\subset \otos[p]\times \otos[q]}{s\neq\{\}}}
(P^{\n^{p,q}_{2}(s)}\times P^{\n^{p,q}_{1}(s)})_{+}\wedge N_{|s|}. \] Working inductively, we see that in multisimplicial degree $p_{1},\dotsc,p_{n}$, \begin{equation}\label{eqconsitbar} \tilde B^{n}_{p_{1},\dotsc,p_{n}}N= \bigvee_{\putatop{s\subset \otos[p]_{1}\times\dotsb \times \otos[p]_{n}}{s\neq\{\}}} (P^{\n^{p_{1},\dotsc,p_{n}}_{n}(s)}\times \dotsb \times
P^{\n^{p_{1},\dotsc,p_{n}}_{1}(s)})_{+}\wedge N_{|s|}, \end{equation} where $\n^{p_{1},\dotsc,p_{n}}_{i}(s)$ is the subset of elements $j$ in $\otos[p]_{i}$ such that \[ s\cap (\otos[p]_{1}\times \dotsb \times \otos[p]_{i-1} \times \{j\}\times \otos[p]_{i+1}\times \dotsb \times \otos[p]_{n}) \] is non-empty.
For $B^{n}KN$, we obtain a completely analogous description, but with $s=\{\}$ in the wedge sum and with different length coordinates. Recall that for $u\subset \otos[p]$, $P^{p}_{u}$ denotes the subset of $\bar P^{p}=[0,\infty)^{p}$ of elements $(r_{1},\dotsc,r_{p})$ where $r_{j}>0$ for all $j\in u$. Then the length coordinates on the summand indexed by $s$ is $P^{p_{i}}_{\n^{p_{1},\dotsc,p_{n}}_{i}(s)}$.
In other words, \[ B^{n}_{p_{1},\dotsc,p_{n}}KN= \bigvee_{s\subset \otos[p]_{1}\times\dotsb \times \otos[p]_{n}} (P^{p_{n}}_{\n^{p_{1},\dotsc,p_{n}}_{n}(s)}\times \dotsb \times
P^{p_{1}}_{\n^{p_{1},\dotsc,p_{n}}_{1}(s)})_{+}\wedge N_{|s|}. \]
To complete the closed description of the iterated bar construction, we describe the face and degeneracy maps. The degeneracy map $s_{j}$ in the $i$-th simplicial direction is induced by the map $\otos[p]_{i}$ to $\{1,\dotsc,p_{i}+1\}$ sending $1,\dotsc,j-1$ by the identity and $j,\dotsc,p_{i}$ by $m\mapsto m+1$.
For $0<j<p_{i}$ the face map $d_{j}$ adds the appropriate pair $r,s\mapsto r+s$ in the $P$ factors (corresponding to $j,j+1\in \otos[p]_{i}$), and performs the action \[ c=\id^{i-1}\times \gamma_{r,s}\times \id^{n-i}\in \LC(2) \] on the corresponding spots in
$N_{|s|}$: We first use a permutation to rearrange from lexicographical order on $s$ to the lexicographical order where the $i$-th index is least significant. Then for fixed $a_{k}\in \otos[p]_{k}, k\neq i$, in this order, the elements \[ (a_{1},\dotsc,a_{i-1},j,a_{i+1},\dotsc,a_{n}) \qquad \text{and}\qquad (a_{1},\dotsc,a_{i-1},j+1,a_{i+1},\dotsc,a_{n}) \]
are adjacent when they both appear in $s$. For such elements, we apply $c$ on the appropriate spot on $N_{|s|}$, but when one is missing, we plug $1\in \LC(0)$ in that input of $c$ and apply that element of $\LC(1)$ to the spot in $N_{|s|}$. (When neither element is in $s$, no action needs to be taken for that pair.) We then use the permutation to rearrange back to the natural lexicographical order.
The zeroth face map $d_{0}$ is the trivial map on the summands indexed by those subsets $s$ where $1\in \n^{p_{1},\dotsc,p_{n}}_{i}(s)$. Fixing $s$ with $1\not\in \n^{p_{1},\dotsc,p_{n}}_{i}(s)$, the action of $d_{0}$ sends this summand to summand indexed by $s'$, where $s'$ is obtained by subtracting $1$ from the $i$-th coordinate of each element of $s$. On the length factors, the first factor gets dropped and for $j\ge 1$ the $j+1$ coordinate becomes the new $j$
coordinate. On the $N$ factor, $|s|=|s'|$ and $N_{|s|}$ maps by the identity to $N_{|s'|}$. The last face map $d_{p_{i}}$ has a similar description: it is the trivial map on the summands indexed by those subsets $s$ where $p_{i}\in \n^{p_{1},\dotsc,p_{n}}_{i}(s)$ and on those summands where $p_{i}\not\in \n^{p_{1},\dotsc,p_{n}}_{i}(s)$, it lands in $s\subset \otos[p]_{1}\times \dotsb \times \otos[p]'_{i}\times \dotsb \times \otos[p]_{n}$ (for
$p_{i}'=p_{i}-1$), dropping the last length coordinate and acting by the identity on $N_{|s|}$.
We note from the description above that we have a canonical inclusion of $\Sigma^{n}N_{1}$ into $\tilde B^{n}N$: Using the singleton subsets of $\otos[p]_{1}\times \dotsb \times \otos[p]_{n}$ and the element $1\in P$, we get a map of multisimplicial $R$-modules $N_{1}\wedge S^{1}_{\ssdot} \wedge \dotsb \wedge S^{1}_{\ssdot}\to \tilde B^{n}_{\bullet,\dotsc,\bullet}N$, which on geometric realization induces a map \[ \Sigma^{n}N_{1}\to \tilde B^{n}N. \] We have an inclusion of $KN_{1}=R\vee N_{1}$ in $M(KN)_{1}$ where we send the $R$ factor in as length zero and the $N_{1}$ factor in as length one. This defines a map in the category of $R$-modules under and over $R$, splitting the usual map $M(KN)_{1}\to KN_{1}$ (induced by forgetting lengths). Noting that $\Sigma^{n}_{R}KN_{1}=R\vee \Sigma^{n}N_{1}$, we have the following commutative diagram, relating the map above to the map~\eqref{eqsusp}. \begin{equation}\label{eqsuspsplit} \begin{gathered} \xymatrix{ &R\vee \Sigma^{n}N_{1}\ar[r]\ar[d]\ar[ld]_{=}&R\vee \tilde B^{n}N\ar[d]\\ \Sigma^{n}_{R}KN_{1}&\Sigma^{n}_{R}MKN_{1}\ar[l]^{\simeq}\ar[r]&B^{n}KN } \end{gathered} \end{equation}
\section {Reduced Topological Quillen Homology and the Iterated Bar Construction} \label{sectqh}
In this section we relate the iterated bar construction for augmented $\LC$-algebras of the previous section to reduced topological Quillen homology, proving Theorem~\ref{maintq}. Quillen homology theories are defined in terms of derived indecomposables, and (as shown in~\cite{mbthesis}) work best in the context of non-unital algebras. We begin by reviewing the Quillen equivalence of augmented and non-unital algebras.
Recall from the previous section the functor $K$ from (true) non-unital $\LC$-algebras to (true) augmented $\LC$-algebras that wedges on a unit, $KN=R\vee N$. This functor is left adjoint to the functor $I$ from augmented $\LC$-algebras to non-unital $\LC$-algebras that takes the (point-set) fiber of the augmentation map, $IA=*\times_{R}A$. We have a Quillen closed model structure on each of these categories where the fibrations and weak equivalences are the underlying fibrations and weak equivalences of $R$-modules; the cofibrations are the retracts of $\btNC {\opsymbfont{I}}$-cell complexes (in the non-unital case) and of $\btLC {\opsymbfont{I}}$-cell complexes (in the augmented case) where ${\opsymbfont{I}}$ is the set of generating cofibrations. The functor $I$ preserves fibrations and acyclic fibrations, and so the adjoint pair $K,I$ forms a Quillen adjunction. Since we can calculate the effect on homotopy groups of $K$ on arbitrary non-unital $\LC$-algebras and of $I$ on fibrant augmented $\LC$-algebras, we see that when $A$ is fibrant, a map of augmented $\LC$-algebras $KN\to A$ is a weak equivalence if and only if the adjoint map $N\to IA$ is a weak equivalence; it follows that $K,I$ is a Quillen equivalence.
\begin{thm}\label{thmaugqe} The functors $K$ and $I$ form a Quillen equivalence between the category of non-unital $\LC$-algebras and the category of augmented $\LC$-algebras. \end{thm}
Given an $R$-module $M$, we can make $M$ into a non-unital $\LC$-algebra by giving it the trivial $\LC$-action, letting \[ \LC(m)_{+}\wedge_{\Sigma_{m}}M^{(m)}\to M \] be the trivial map for $m>1$ and the composite $\LC(1)_{+}\wedge M\to *_{+}\wedge M\cong M$ for $m=1$. This defines a functor $Z$ (the ``zero multiplication'' functor) from $R$-modules to non-unital $R$-algebras. The functor $Z$ has a left adjoint ``indecomposables'' functor $Q$, which can be constructed as the coequalizer \[ \xymatrix@C-1pc{ \displaystyle \bigvee_{m>0}\LC(m)_{+}\wedge_{\Sigma_m}N^{(m)}\mathstrut \ar[r]<-.5ex>\ar[r]<.5ex> &N\ar[r]&QN, } \] where one map is the action map for $N$ and the other map is the zero multiplication action map. Since the functor $Z$ preserves fibrations and weak equivalences, we see that $Q,Z$ forms a Quillen adjunction.
\begin{thm}\label{thmzq} The functors $Q$ and $Z$ form a Quillen adjunction between the category of non-unital $\LC$-algebras and the category of $R$-modules. \end{thm}
For an augmented $\LC$-algebra $A$ and an $R$-module $M$, one can define the reduced topological Quillen cohomology groups in terms of derivations of $A$ with coefficients in $M$, i.e., as maps in the homotopy category of augmented $\LC$-algebras from $A$ to $KZM$ (or $KZ\Sigma^{*}M$). Applying the Quillen equivalence of Theorem~\ref{thmaugqe} and the Quillen adjunction of Theorem~\ref{thmzq}, we can identify this as maps in the derived category of $R$-modules from $QN$ to $M$ (or $\Sigma^{*}M$), where $N$ is a cofibrant non-unital $\LC$-algebra with $KN$ equivalent to $A$. In other words, the left derived functor $Q^{\ld}$ of $Q$ produces an object representing topological Quillen homology. We write $Q^{\ld}_{\LC}$ for the composite of $Q^{\ld}$ with the right derived functor of $I$.
\begin{defn}\label{deftqh} For an augmented $\LC$-algebra $A$, the $\LC$-algebra cotangent complex at the augmentation is the $R$-module of derived indecomposables $Q^{\ld}_{\LC}$. \end{defn}
We have a canonical natural map $\tilde B^{n}N\to \Sigma^{n}QN$ defined as follows. Thinking of $\Sigma^{n}QN$ as the geometric realization of the multisimplicial $R$-module $QN \wedge (S^{1}_{\ssdot})^{n}$, we send the summand indexed by $s\subset \otos[p]_{1}\times \dotsb \times \otos[p]_{n}$ in~\eqref{eqconsitbar} by the counit of the $Q,Z$ adjunction $N\to QN$ (and dropping the $P$
factors) when $|s|=1$ and by the trivial map when $|s|>1$. This clearly commutes with the degeneracy maps and commutes with the face maps since every face map that changes the cardinality of the indexing set is either the trivial map or lands in the decomposables.
Using the constructions above, Theorem~\ref{maintq} becomes the following theorem stated in terms of non-unital $\LC$-algebras.
\begin{thm}\label{thmtq} The natural map $\tilde B^{n}N\to \Sigma^{n}QN$ is a weak equivalence when $N$ is a cofibrant non-unital $\LC$-algebra. \end{thm}
To prove this theorem, we use the monadic bar construction trick from~\cite[\S5]{mbthesis}. The key observation is that both the functors $\tilde B^{n}$ and $Q$ commute with geometric realization. This is clear from the construction for $\tilde B^{n}$, but follows for $Q$ because $Q$ is a topological left adjoint and because geometric realization of simplicial operadic algebras can be formed as a topological colimit in the category of algebras; see \cite[VII\S3]{ekmm}. In particular, applied to the monadic bar construction $B(\btNC,\btNC,N)$, we get isomorphisms \[ \tilde B^{n}B(\btNC,\btNC,N)\cong B(\tilde B^{n}\btNC,\btNC,N) \quad \text{and}\quad QB(\btNC,\btNC,N)\cong B(Q\btNC,\btNC,N). \] The argument now simplifies from \cite{mbthesis} since in our context a cofibrant non-unital $\LC$-algebra is cofibrant as an $R$-module. Regarding $B(\btNC,\btNC,N)$ as a topological colimit in non-unital $\LC$-algebras, it follows that $B(\btNC,\btNC,N)$ is cofibrant as a non-unital $\LC$-algebra when $N$ is. Since the simplicial objects \[ B_{\ssdot}(\tilde B^{n}\btNC,\btNC,N) \qquad \text{and}\qquad B_{\ssdot}(Q\btNC,\btNC,N). \] are always proper (the inclusion of the union of the degeneracies in each simplicial degree is induced by the inclusion of $\id$ in $\LC(1)$ and the inclusion of a wedge summand), Theorem~\ref{thmtq} now reduces to the following lemma.
\begin{lem}\label{lemqt} For any cofibrant $R$-module $X$, the natural map $\tilde B^{n}\btNC X\to \Sigma^{n}Q\btNC X$ is a weak equivalence. \end{lem}
The functor $\Sigma^{n}Q\btNC X$ is canonically isomorphic to the $n$-th suspension functor $\Sigma^{n}$; moreover, the natural transformation $\tilde B^{n}\btNC \to \Sigma^{n}$ is split by the natural transformation \[ \Sigma^{n}\to \Sigma^{n}\btNC \to \tilde B^{n}\btNC, \] induced by the inclusion of the singleton subsets in~\eqref{eqconsitbar}. Thus, to prove Lemma~\ref{lemqt}, it suffices to prove the following lemma.
\begin{lem}\label{lemreverse} For any cofibrant $R$-module $X$, the natural map $\Sigma^{n}X\to \tilde B^{n}\btNC X$ is a weak equivalence. \end{lem}
We can make a further reduction by identifying $\tilde B^{n}\btNC X$ as the functor associated to a symmetric sequence. Applying the explicit description of $\tilde B^{n}$ in the previous section, we note that the face and degeneracies in $\tilde B^{n}\btNC X$ preserve homogeneous degrees in $X$. We can therefore decompose $\tilde B^{n}\btNC$ naturally into a wedge sum of its homogeneous pieces, \[ \tilde B^{n}\btNC X= \bigvee_{m> 0} {\opsymbfont{B}}(m)\wedge_{\Sigma_{m}}X^{(m)}, \] where each ${\opsymbfont{B}}(m)$ is a based $\Sigma_{m}$-space; in fact, each ${\opsymbfont{B}}(m)$ is a free based $\Sigma_{m}$-cell complex since it is the geometric realization of a multisimplicial $\Sigma_{m}$-space that in each multisimplicial degree is a wedge of pieces of the form \[ \left( (\bar P^{i} \times P^{j})\times (\LC(m_{1})\times\dotsb \times \LC(m_{r})) \times_{\Sigma_{m_{1}}\times \dotsb \times \Sigma_{m_{r}}}\Sigma_{m} \right)_{+}. \] The natural map $\Sigma^{n}X\to \tilde B^{n}\btNC X$ is induced by the inclusion of $S^{n}$ in ${\opsymbfont{B}}(1)$. Thus, it suffices to show that the map $S^{n}\to {\opsymbfont{B}}(1)$ induces a weak equivalence on $R$-homology and that each ${\opsymbfont{B}}(m)$ has trivial $R$-homology.
Although it is not hard to show directly that the map $S^{n}\to {\opsymbfont{B}}(1)$ is a weak equivalence, analysis of the construction of ${\opsymbfont{B}}(m)$ is rather complicated for a direct argument (cf.~\cite[\S 8]{FresseItBar}) and we take a shorter oblique approach in terms of the functor $\tilde B^{n}\btNC$. Let $R^{0}_{c}$ be a cofibrant $R$-module equivalent to $R$, and consider the set with $m$ elements, $\otos[m]=\{1,\dotsc,m\}$. We note that ${\opsymbfont{B}}(m)\wedge_{\Sigma_{m}}(\otos[m]_{+})^{(m)}$ contains ${\opsymbfont{B}}(m)$ as a wedge summand, and so \[ \pi_{*}(\tilde B^{n}\btNC(R^{0}_{c}\wedge \otos[m]_{+})) \] contains the $R$-homology $R_{*}{\opsymbfont{B}}(m)$ as a direct summand. Rewriting in terms of the natural transformation $\Sigma^{n}\to \tilde B^{n}\btNC$, we have reduced Lemma~\ref{lemreverse} to the following lemma.
\begin{lem}\label{lemfs} The natural map $\Sigma^{n}(R^{0}_{c}\wedge X_{+})\to \tilde B^{n}\btNC (R^{0}_{c}\wedge X_{+})$ is a weak equivalence for every finite set $X$. \end{lem}
Lemma~\ref{lemfs} follows from the analogous statement in terms of augmented algebras, that the map \[ R\vee \Sigma^{n}(R^{0}_{c}\wedge X_{+})\to B^{n}\btLC (R^{0}_{c}\wedge X_{+}) \] is a weak equivalence for all finite sets $X$. We have an isomorphism of $\LC$-algebras \[ \btLC (R \wedge X_{+})\cong R\wedge (\btLC X)_{+} \] where $\btLC X$ is the free $\LC$-space on $X$. The weak equivalence $R^{0}_{c}\to R$ induces a weak equivalence of $\LC$-algebras \[ \btLC(R^{0}_{c}\wedge X_{+})\to \btLC(R\wedge X_{+})\cong R\wedge (\btLC X)_{+}, \] but this is not a map of augmented $\LC$-algebras: The augmentation on the left is induced by the trivial map $X_{+}\to *$, but the augmentation on the right is induced by the trivial map $\btLC X\to *$. In terms of $\btLC(R\wedge X_{+})$, the right-hand augmentation is induced by $X_{+}\to S^{0}$ and the $\LC$-action $\btLC R\to R$. Since we are free to choose any cofibrant model $R^{0}_{c}$, choosing one that is a suspension, we can construct a map \[ \alpha \colon R^{0}_{c}\wedge X_{+}\to R^{0}_{c} \vee R^{0}_{c}\wedge X_{+} \] that represents the sum of the map $X_{+}\to S^{0}$ and the identity on $X_{+}$ smashed with $R^{0}_{c}$. Write $\epsilon$ for the composite map \[ R^{0}_{c}\wedge X_{+}\to R^{0}_{c}\to R, \] which is homotopic to (but probably not equal to) the map induced by $X_{+}\to S^{0}$, and choose a homotopy $h$. Using $\alpha$, we get a map of $\LC$-algebras \[ \bar \alpha \colon \btLC(R^{0}_{c}\wedge X_{+})\to \btLC(R^{0}_{c}\wedge X_{+}), \] which respects augmentations when we give the copy on the left the augmentation induced by $\epsilon$; an easy filtration argument shows that this map is a weak equivalence. Now we get a diagram of weak equivalences of augmented $\LC$-algebras \[ \xymatrix{ \btLC(R^{0}_{c}\wedge X_{+}\wedge\{0\}_{+})\ar[r]\ar[d]_{\bar \alpha} &\btLC(R^{0}_{c}\wedge X_{+}\wedge I_{+}) &\btLC(R^{0}_{c}\wedge X_{+}\wedge \{1\}_{+})\ar[l]\ar[d]\\ \btLC(R^{0}_{c}\wedge X_{+}) &&R\wedge (\btLC X)_{+} } \] which respects augmentations when we use the augmentation induced by $X_{+}\to S^{0}$ on the right and the augmentation induced by $h$ in the middle (on $\btLC(R^{0}_{c}\wedge X_{+}\wedge I_{+})$). Since applying $B^{n}$ preserves these weak equivalences, we get that \[ B^{n}\btLC(R^{c}_{0}\wedge X_{+}) \qquad\text{and}\qquad B^{n}(R \wedge (\btLC X)_{+}). \] are weakly equivalent.
The map in Lemma~\ref{lemfs} is induced by the inclusion of $R^{0}_{c}\wedge X_{+}$ in $\btNC (R^{0}_{c}\wedge X_{+})$ and the section~\eqref{eqsuspsplit} of the natural map~\eqref{eqsusp} \[ \Sigma^{n}_{R}M(\btLC (R^{0}_{c}\wedge X_{+})) \to B^{n}\btLC (R^{0}_{c}\wedge X_{+}). \] We can follow the natural map~\eqref{eqsusp} along the diagram of weak equivalences between $B^{n}\btLC(R^{c}_{0}\wedge X_{+})$ and $B^{n}(R \wedge (\btLC X)_{+})$ above, and lift the map from \[ R\vee \Sigma^{n}(R^{0}_{c}\wedge X_{+}) = \Sigma^{n}_{R}(R^{0}_{c}\wedge X_{+}) \] up to homotopy all the way around to a map \[ R\vee \Sigma^{n}(R^{0}_{c}\wedge X_{+})\to \Sigma^{n}_{R}M(R \wedge (\btLC X)_{+})\to B^{n}(R \wedge (\btLC X)_{+}). \] Specifically, the map from $R\vee \Sigma^{n}(R^{0}_{c}\wedge X_{+})$ to each of \[ \begin{small} \xymatrix@C-1pc@R-1pc{ \Sigma^{n}_{R}M\btLC(R^{0}_{c}\wedge X_{+}\wedge\{0\}_{+})\ar[d]\ar[r] &\Sigma^{n}_{R}M\btLC(R^{0}_{c}\wedge X_{+}\wedge I_{+}) &\Sigma^{n}_{R}M\btLC(R^{0}_{c}\wedge X_{+}\wedge \{1\}_{+})\ar[l]\ar[d]\\ \Sigma^{n}_{R}M\btLC(R^{0}_{c}\wedge X_{+}), &&\Sigma^{n}_{R}M(R\wedge (\btLC X)_{+}) } \end{small} \] induces a weak equivalence on the submodules in homogeneous filtration one (and below). In particular, on the bottom right, looking at the map \[ R\vee \Sigma^{n}(R^{0}_{c}\wedge X_{+})\to \Sigma^{n}_{R}M(R \wedge (\btLC X)_{+}) \simeq \Sigma^{n}_{R}(R \wedge (\btLC X)_{+})= R\wedge (\Sigma^{n}(\btLC X))_{+}, \] we now see that the map in Lemma~\ref{lemfs} is a weak equivalence if and only if the map \[ R\wedge (\Sigma^{n}X_{+})_{+}\to B^{n}(R \wedge (\btLC X)_{+}) \] induced by the inclusion of $X$ in $\btLC X$ is a weak equivalence. This reduces Lemma~\ref{lemfs} to the following lemma.
\begin{lem}\label{lemswitch} The natural map $R\wedge \Sigma^{n}(X_{+})_{+}\to B^{n}(R \wedge (\btLC X)_{+})$ is a weak equivalence for every finite set $X$. \end{lem}
The point of this lemma is that it lets us compare with the classical bar construction on spaces. The construction $B^{n}$ in the previous section used little about the category of $R$-modules and generalizes to an iterated bar construction on the category of partial $\LC$-spaces (where we use the cartesian product to define partial power systems). In fact, in spaces, it is much easier to describe because we can talk in terms of elements. For a $\LC$-space $A$, each element of the Moore construction $\MA$ has a length, and we make the bar construction \[ B_{\ssdot} A=B_{\ssdot}(\MA)=\MA\times \dotsb \times \MA \] into a partial power system by insisting that the lengths match up: We take the $m$-th partial power $(B_{p} A)_{m}$ to be the subset of $(B_{p}A)^{m}=(\MA^{p})^{m}$ where the length vectors for each of the $m$ copies of $\MA^{p}$ all agree. We have an entirely similar description when $A$ is a partial $\LC$-space (noting that elements of $\MA[\bullet]$ also have sequences of lengths). We also note that when $A=\Omega Z$ with its $\LC[1]$-structure coming from the standard $\LC[1]$-structure on the loop space, then by construction, $\MA$ is the Moore loop space $\Omega_{M}Z$.
The functor $R\wedge(-)_{+}$ from unbased spaces to $R$-modules takes partial power systems to partial power systems. By inspection, for any partial $\LC$-space $Z$, we have an isomorphism of partial $\LC[n-1]$-algebras \[ B(R\wedge Z_{+})\cong R\wedge (BZ)_{+}, \] and iterating, an isomorphism $B^{n}(R\wedge Z_{+})\cong R\wedge (B^{n}Z)_{+}$. Finally, Lemma~\ref{lemswitch} is a consequence of the following proposition.
\begin{prop}\label{propdeloop} For any finite set $X$, the map $\Sigma^{n} X_{+}\to B^{n}\btLC X$ is a weak equivalence. \end{prop}
To prove the proposition, we inductively analyze $B^{i}\btLC X$. The inclusion $X\to \Omega^{n} \Sigma^{n} X_{+}$ induces a map of $\LC$-spaces $\btLC X\to \Omega^{n} \Sigma^{n} X_{+}$. By the group completion theorem, this map induces a weak equivalence \begin{equation}\label{eqbase} B\btLC X\to B\Omega^{n} \Sigma^{n}X_{+}\simeq \Omega^{n-1}\Sigma^{n}X_{+}. \end{equation} Since up to homotopy this map is compatible with the inclusion of $\Sigma X_{+}$ in $B\btLC X$, this completes the argument in the case $n=1$. For $n>1$, we need the reduced free $\LC[j]$-space functor $C_{j}$ from \cite[2.4]{MayGILS}; its fundamental formal property is that it gives an adjunction between \emph{based} maps from a based spaced $Y$ to a $\LC[j]$-space $Z$ and maps of $\LC[j]$-spaces from $C_{j}Y$ to $Z$. Its fundamental homotopical property is that when $Y$ is connected and nondegenerately based, the universal map $C_{j}Y\to \Omega^{j}\Sigma^{j}Y$ is a weak equivalence. The fundamental formal property also holds when the target $Z$ is a partial $\LC[j]$-space: For a based space $Y$, based maps of partial power systems $Y\to Z$ are in bijective correspondence with maps of partial $\LC[j]$-spaces $C_{j}Y\to Z$.
To be specific about the weak equivalence $B\Omega Z\simeq Z$ in~\eqref{eqbase}, we use the zigzag in~\cite[14.3]{MayClass} \begin{equation}\label{eqmayclass} B(\Omega Z)=B(\Omega_{M}Z) \xleftarrow{\,\simeq\,} B(P_{M}Z,\Omega_{M}Z,*)\xrightarrow{\,\simeq\,} Z, \end{equation} which is a weak equivalence whenever $Z$ is connected. Here $P_{M}Z$ denotes the Moore based path space (the space of positive length paths ending at the base point), the middle term the classical two-sided bar construction for the action of the Moore loop space on the Moore path space, and the maps are induced by the trivial map $P_{M}Z\to *$ (on the left) and the start point projection $P_{M}Z\to Z$ on the right. When $Z$ is a $\LC[j]$-space, we can make \[ B_{\ssdot}(P_{M}Z,\Omega_{M}Z,*)= P_{M}Z\times \Omega_{M}Z\times \dotsb \times \Omega_{M}Z\times * \] a partial power system by taking the $m$-the partial power to be the subset of the $m$-the power where the lengths match up, as for $B$ above. Then both maps in the zigzag become maps of partial $\LC[j]$-spaces (with the true power system for $Z$).
Returning to~\eqref{eqbase}, we have a zigzag of weak equivalences of partial $\LC[n-1]$-spaces \[ B\btLC X\to B\Omega^{n} \Sigma^{n}X_{+} \from B(P_{M}\Omega^{n-1}\Sigma^{n}X_{+},\Omega_{M}\Omega^{n-1}\Sigma^{n}X_{+},*) \to \Omega^{n-1}\Sigma^{n}X_{+}. \] We can now see that the inclusion of $\Sigma X_{+}$ into $B\btLC X$ induces a weak equivalence of partial $\LC[n-1]$-spaces \[ C_{n-1}\Sigma X_{+}\to B\btLC X \] (for the true $\LC[n-1]$-space $C_{n-1}\Sigma X_{+}$). We use this as the base case of an inductive argument: Assume by induction that the natural map $\Sigma^{i}X_{+}\to B^{i}\btLC X$ induces a weak equivalence of partial $\LC[n-i]$-spaces \[ C_{n-i}\Sigma^{i}X_{+}\to B^{i}\btLC X. \] Applying $B$, the weak equivalence of $\LC[n-i]$-spaces $C_{n-i}\Sigma^{i}X_{+}\to \Omega^{n-i}\Sigma^{n}X_{+}$ and the zigzag~\eqref{eqmayclass} give us a zigzag of weak equivalences of partial $\LC[n-(i+1)]$-spaces \begin{multline*} \Omega^{n-(i+1)}\Sigma^{n}X_{+}\from B(P_{M}\Omega^{n-(i+1)}\Sigma^{n}X_{+},\Omega_{M}\Omega^{n-(i+1)}\Sigma^{n}X_{+},*)\\ \to B\Omega^{n-i}\Sigma^{n}X_{+}\from BC_{n-i}\Sigma^{i}X_{+}\to B^{i+1}\btLC X. \end{multline*} The inclusions of $\Sigma^{i+1}X_{+}$ into each of these spaces agree up to homotopy under these maps, and so the induced map of partial $\LC[n-(i+1)]$-spaces $C_{n-(i+1)}\Sigma^{i+1}X_{+}\to B^{i+1}\btLC X$ is a weak equivalence. This completes the proof of Proposition~\ref{propdeloop}, which completes the proof of Theorem~\ref{thmtq}.
\begin{rem} For $Z=\Omega^{j}Y$ for a $(j-1)$-connected space $Y$, the identification of~\eqref{eqmayclass} as a zigzag of weak equivalences of partial $\LC[j]$-spaces implies by induction that $B^{n}$ is an ``$n$-fold de-looping machine''. As an alternative argument, it should be possible to deduce Proposition~\ref{propdeloop} from a uniqueness theorem for $n$-fold de-looping machines such as \cite{dunndeloop}; however, translating the problem to the context in which such a theorem applies is more complicated than the direct argument above. \end{rem}
\section{Further Structure on the Bar Construction}\label{secopn}
With an eye to using Theorem~\ref{maintq} for computations, we take this final section to verify two of the expected properties of the multiplication on the bar construction. We begin by studying the diagonal map on the bar construction, and we show that it commutes with the $\LC[n-1]$-multiplication constructed in Section~\ref{secbar}. We then study power operations, showing that the (dimension shifting) map on homotopy groups from a non-unital $\LC$-algebra $N$ to its bar construction $BN$ preserves power operations in the expected way. In this section, we work in the context of true algebras since that is where these remarks are of primary interest.
Given an augmented $R$-algebra $A$, it is well-known that the bar construction $BA$ admits a diagonal map \[ BA\to BA\wedge_{R} BA \] that is associative up to homotopy, even up to coherent homotopy. The best construction of this map uses ``edgewise subdivision'' \cite[\S1]{BHM}. For a simplicial object, $X_{\ssdot}$, the edgewise subdivision is the object $\sd X_{\ssdot}$ where $\sd X_{n}=X_{2n+1}$. The argument for \cite[1.1]{BHM} shows that just as in the context of simplicial sets or simplicial spaces, in the context of simplicial $R$-modules, we have a natural isomorphism between the geometric realization of $X_{\ssdot}$ and the geometric realization of the edgewise subdivision $\sd X_{\ssdot}$. We get the diagonal map on the bar construction as the composite \[
BA \cong |\sd B_{\ssdot} A| \to |B_{\ssdot} A\wedge_{R} B_{\ssdot} A|\cong BA\wedge_{R}BA \] for a particular simplicial map $\sd B_{\ssdot} A\to B_{\ssdot} A\wedge_{R}B_{\ssdot} A$. This map in degree $m$ is the map \[ (\sd B_{\ssdot} A)_{m} = A^{(2m+1)}\to A^{(m)}\wedge_{R}A^{(m)}= B_{m}A\wedge_{R}B_{m}A \] that performs the augmentation $A\to R$ on the $(m+1)$-st factor of $A$. We prove the following theorem.
\begin{thm}\label{thmhopf} Let $N$ be a non-unital $\LC$-algebra. The diagonal map $BKN\to BKN\wedge_{R}BKN$ above is a map of $\LC[n-1]$-algebras. \end{thm}
\begin{proof} The edgewise subdivision functor and isomorphism on geometric realization preserve smash products of $R$-modules in the sense that the diagram of natural isomorphisms \[ \xymatrix{
|X_{\ssdot} \wedge_{R} Y_{\ssdot}|\ar[rr]\ar[d]
&&|X_{\ssdot}|\wedge_{R}|Y_{\ssdot}|\ar[d]\\
|\sd(X_{\ssdot}\wedge_{R} Y_{\ssdot})|\ar[r]
&|(\sd X_{\ssdot})\wedge_{R} (\sd Y_{\ssdot})|\ar[r]
&|\sd X_{\ssdot}|\wedge_{R}|\sd Y_{\ssdot}| } \] commutes. It therefore suffices to check that the map from $\sd B_{\ssdot} A$ to $B_{\ssdot} A\wedge_{R}B_{\ssdot} A$ is a map of simplicial $\LC[n-1]$-algebras (for $A=MKN$), and this is clear from the construction of the $\LC[n-1]$-structure. \end{proof}
We close with a remark on power operations. For technical reasons about homotopy groups, we restrict to the context of $R$-modules of orthogonal spectra or EKMM $S$-modules for this discussion. Consider a non-unital true $\LC$-algebra $N$ and choose a representative map of $R$-modules $R^{q}_{c}\to N$ where $R^{q}_{c}$ is some cofibrant version of the $q$-sphere $R$-module. We then get a map of non-unital true $\LC$-algebras $\btNC R^{q}_{c}\to N$, where (as above) $\btNC$ denotes the free non-unital $\LC$-algebra functor in $R$-modules. The induced map \[ \bigoplus_{m>0} R_{*}(\LC(m)_{+}\wedge_{\Sigma_{m}}S^{(mq)}) \cong \pi_{*}(\btNC R^{q}_{c})\to \pi_{*}N \] depends only on the original $x\in \pi_{q}N$ and not on the choice of representative. Restricting to the $m$-th homogeneous piece, we get the map \[ {\opsymbfont{P}}^{m}(x)\colon R_{*}(\LC(m)_{+}\wedge_{\Sigma_{m}}S^{(mq)})\to \pi_{*}N, \] which we think of as the total $m$-ary $\LC$-algebra power operation of $x$; we think of $R_{*}(\LC(m)_{+}\wedge_{\Sigma_{m}}S^{(mq)})$ as parametrizing the $m$-ary power operations on $\pi_{q}N$. We relate the power operations on $\pi_{q}$ to the power operations on $\pi_{q+1}$ using the suspension sequence \[ \btNC R^{q}_{c}\to \btNC CR^{q}_{c}\to \btNC R^{q+1}_{c}. \] The composite map is the trivial map and the middle term has a canonical contraction; this then defines a map \[ \pi_{*}(\btNC R^{q}_{c})\to \pi_{*+1}(\btNC R^{q+1}_{c}) \] and in particular a map \[ \sigma \colon R_{*}(\LC(m)_{+}\wedge_{\Sigma_{m}}S^{(mq)})\to R_{*+1}(\LC(m)_{+}\wedge_{\Sigma_{m}}S^{(m(q+1))}) \] In terms of our work above, we have the following result.
\begin{thm}\label{thmopn} For a non-unital true $\LC$-algebra $N$, the canonical map \[ \sigma \colon \pi_{*}N\to \pi_{*+1}\tilde B N \] preserves $m$-ary $\LC[n-1]$-algebra power operations for all $m$, meaning that the diagrams \[ \xymatrix{ R_{*}(\LC[n-1](m)_{+}\wedge_{\Sigma_{m}}S^{(mq)}) \ar[r]^{\sigma}\ar[d]_{{\opsymbfont{P}}^{m}(x)}& R_{*+1}(\LC[n-1](m)_{+}\wedge_{\Sigma_{m}}S^{(m(q+1))}) \ar[d]^{{\opsymbfont{P}}^{m}(\sigma x)}\\ \pi_{*}N\ar[r]_{\sigma}&\pi_{*+1}\tilde B N } \] commute for all $x\in \pi_{q}N$. Here we regard $N$ as a non-unital $\LC[n-1]$-algebra via the last coordinates embedding of $\LC[n-1]$ in $\LC[n]$. \end{thm}
\begin{proof} We have an ``$E$'' version of the bar construction where $E_{0}N=K\MN$ and \[ E_{\ssdot} N = \underbrace{K\MN \wedge_{R} \dotsb \wedge_{R}K\MN} _{\bullet\ \text{factors}} \wedge_{R} K\MN \] for $\bullet >0$. Likewise, we have a $\tilde E$ version such that $EN=K\tilde E N$. Constructions analogous to those above make these into partial non-unital $\LC[n-1]$-algebras, and the inclusion of $N$ in $\MN$ in $\tilde E_{0}N$ (as, say, $\{1\}_{+}\wedge N\subset P_{+}\wedge N$) induces a map of partial $\LC[n-1]$-algebras $N\to \tilde E N$. The trivial map $\MN\to *$ induces a map of partial $\LC[n-1]$-algebras $\tilde E N\to \tilde B N$, giving us a sequence of maps of partial $\LC[n-1]$-algebras \[ N \to \tilde E N \to \tilde B N \] with the composite map $N\to \tilde B N$ the trivial map. The usual simplicial contraction argument shows that $\tilde E$ is contractible. Choosing a contraction, any map of $R$-modules $R^{q}_{c}\to N$ gives us a map of partial power systems $CR^{q}_{c}\to \tilde E N$ and hence a map of partial power systems $\Sigma R^{q}_{c}\to \tilde B N$. This correspondence lifts the map $\pi_{*}N\to \pi_{*+1}\tilde B N$ to representatives. Applying the free functor $\bNC[n-1]$ and unwinding the definition of \[ \sigma\colon R_{*}(\LC[n-1](m)_{+}\wedge_{\Sigma_{m}}S^{(mq)}) \to R_{*+1}(\LC[n-1](m)_{+}\wedge_{\Sigma_{m}}S^{(m(q+1))}) \] above, the result follows. \end{proof}
\def\noopsort#1{}\def\MR#1{}
\end{document} | arXiv |
\begin{document}
{\obeylines \small \vspace*{0.2cm} \hspace*{6.0cm}Ils ne savent pas ce qu'ils perdent \hspace*{6.0cm}Tous ces sacr\'es cabotins. \hspace*{6.0cm}Sans le latin, sans le latin, \hspace*{6.0cm}La messe nous emmerde. \footnote{Georges Brassens} \vspace*{0.8cm} \hspace*{7.5cm}\indent{\it To Seraina and Theres} \vspace*{1.5cm}
} \title[Catalan-Fermat] {A Cyclotomic Investigation of the Catalan -- Fermat Conjecture. DRAFT} \author{Preda Mih\u{a}ilescu} \address[P. Mih\u{a}ilescu]{Gesamthochschule Paderborn} \email[P. Mih\u{a}ilescu]{[email protected]} \date{Version 1.0 \today}
\begin{abstract} With give some new, simple results on the equation $x^{p}+y^{p} = z^{q}$ using classical methods of cyclotomy. \end{abstract} \tableofcontents \maketitle \vspace*{1.0cm} \section{Introduction} Consider the equation \begin{eqnarray} \label{FC} x^{p} + y^{p} = z^{q}, \quad \hbox{with} \quad x, y, z \in \mathbb{Z}, \ (x, y, z) = 1 \quad \hbox{and $p, q$ odd primes.} \end{eqnarray} This is the natural generalization the equations $x^{p} + y^{p} + z^{p} = 0$ of Fermat and $x^{p} - y^{q} = 1$ of Catalan, and we shall denote it by \textit{Fermat - Catalan} equation. This name is used by some authors for the more general \begin{eqnarray} \label{SF} x^{p} + y^{q} = z^{r}, \end{eqnarray} while others \cite{Za} use the term of \textit{super - Fermat} equation, for \rf{SF}. We shall also adopt this terminology in this paper. There has been an increasing literature on the subject in the last decade, yet general results are still scarce. Thus F. Buekers enumerates all the solutions of \rf{SF} for $\chi = 1/p + 1/q + 1/r \geq 1$ , while for $\chi < 1$, Darmon and Granville \cite{DG} have proved that there are at most finitely many solutions for a fixed set of exponents. Specific curves are used for fixed triples of exponents by Darmon and coauthors \cite{Da1}, \cite{DM}, Ellenberg \cite{El}, Bruin \cite{Br}, Poonen et. al. \cite{PSS}. We refer the reader to \cite{Be} for a nice survey of the topic and a comprehensive overview of the cases known up to fall 2004.
A program generalizing the Wiles proof of Fermat's Last Theorem was proposed by Darmon in \cite{Da}; it suggests replacing elliptic curves by hyperelliptic curves or even surfaces, in the general case. A particular case we shall consider is the \textit{rational} Catalan equation: \begin{eqnarray} \label{ratcat} X^{p} + Y^{q} = 1, \quad \hbox{with odd primes $p, q$ and } \quad X, Y \in \mathbb{Q}; \end{eqnarray} it is easily shown (see below) that this is equivalent to the special case $x^{p}+y^{q}= z^{pq}$ of \rf{FC}. Tijdeman and Shorey emitted in \cite{ST}, Chapter XII, the conjecture, that \rf{ratcat} has at most finitly many rational solutions. We shall find conditions on $(p, q)$ for which that equation has no non-trivial rational solutions at all.
Our purpose in this paper is to investigate \rf{FC} separately, using classical cyclotomic approaches. Unsurprisingly, the results obtained are partial, but the set of exponents $p, q$ for which they are valid are unbounded. The results are ordered in increasing order of conditions on the (odd prime) exponents $p, q$. Based on a simple relative class number divisibility condition we reduce first \rf{FC} to a Fermat - type equation (with only one exponent) over a totally real field:
\begin{theorem} \label{genfer} Let $p, q$ be odd primes with $p >3, q \not \hspace{0.25em} \mid h_{p}$ - with $h_{p}$ the class number of the \nth{p} cyclotomic extension - and for which the Fermat - Catalan equation \rf{FC} has a non trivial solution. Then the equation \begin{eqnarray} \label{fereq} a X^{q} + b Y^{q} + c Z^{q} = 0 \quad & & X \in \mathbb{Z}; \quad Y, Z \in \mathbb{Z}[\zeta+\overline \zeta] \end{eqnarray} has a non - trivial solution. Here $b, c$ are units, while $a$ is either unit or the principal ideal $(a)$ is a power of the ramified prime ideal above $p$. \end{theorem} After deriving the distinction of the first and second case in \rf{FC}, and the analogs of the Barlow - Abel formulae \cite{Ri}, we prove a theorem, which allows a useful additional case distinction:
\begin{theorem} \label{main} If the equation \rf{FC} has a solution $(x,y,z; p, q)$ and the exponents are such that $\max\{p,\frac{p(p-20)}{16}\} > q$ and $q \not \hspace{0.25em} \mid h_{p}^{-}$, with $h_{p}^{-}$ the relative class number of the \nth{p} cyclotomic extension, then \begin{eqnarray} \label{qpot2} x+f \cdot y & \equiv & 0 \mod q^{2} \quad \hbox{ with } \quad f \in \{-1,0,1\}. \end{eqnarray} \end{theorem}
This leads to a delicate case by case analysis (six cases in total). For five of the six cases we are able to find some simple algebraic conditions, while one of the cases ($z \not \equiv 0 \mod p$ and $x \equiv 0 \mod q^{2}$) remains unsolved. The main results of this paper are the following:
\begin{theorem} \label{main1} Let $p, q$ be odd primes such that $q \not \hspace{0.25em} \mid h(p,q); \ p \not \equiv 1 \mod q$ and $\max\{p,\frac{p(p-20)}{16}\} > q$. Then the equation \[ x^{p} + y^{p} = z^{q} , \quad (x, y, z) = 1 \] has no non - trivial solutions for \begin{eqnarray} \label{lowb}
\max\{|x|,|y|\} \geq \frac{1}{2} \cdot \left(\frac{1}{p(p-1)} \cdot \left(\frac{q^{\frac{q-2}{q-1}}}{2 }\right)^{p-2}\right)^{q}. \end{eqnarray}
\end{theorem}
Here $h(p,q)$ is a divisor of the relative class number $h_{pq}^{-}$
which is explicitly computable by means of generalized Bernoulli
numbers and will be defined below. Assuming the stronger condition
$q \not \hspace{0.25em} \mid h_{pq}^{-}$, the relative class number of the \nth{pq}
cyclotomic field, we have: \begin{theorem} \label{main2} Let $p, q > 3$ be primes such that \rf{FC} has a solution and suppose that $-1 \in < p \mod q>$, $\max\{p,\frac{p(p-20)}{16}\} > q$ and $q \not \hspace{0.25em} \mid h_{pq}^-$. Then either \begin{eqnarray} \label{Wifcases} a^{q-1} \equiv 1 \mod q^{2} \quad \hbox{ for some } \quad a \in \{ 2, p, 2^{p-1} \cdot p^{p} \}, \end{eqnarray} or \begin{itemize}
\item[A.] $p \not \hspace{0.25em} \mid z$ and $q^{2} | x y$ if $q \not \equiv 1 \mod p$ and
$q^{3} | x y$, if $q \equiv 1 \mod p$. \item[B.] If $q \not \equiv 1 \mod p$, then \begin{eqnarray} \label{bound}
\min(|x|, |y|) > c_{1}(q) \left(\frac{q^{p-1}}{p} \right)^{q-2}, \quad \hbox{if} \quad q \not \equiv 1 \mod p, \end{eqnarray} and \begin{eqnarray} \label{bound1}
\min(|x|, |y|) > c_{1}(q) \left(\frac{q^{2(p-1)}}{p} \right)^{q-2}, \end{eqnarray} otherwise. Here $c_{1}(q)$ is am effectively computable, strictly increasing function with $c_{1}(5) > 1/2$. \end{itemize} \end{theorem}
An immediate consequence of this Theorem is the following generalization of Catalan's conjecture: \begin{corollary} \label{catg1} Let $p, q$ be odd primes such that \begin{enumerate} \item $-1 \not \in <p \mod q> $ and $q \not \hspace{0.25em} \mid h_{pq}^-$, \item $\max\{p,\frac{p(p-20)}{16}\} > q$, \item $a^{q-1} \not \equiv 1 \mod q^{2}$ for $a \in \{ 2, p, 2^{p-1}
\cdot p^{p} \}$. \end{enumerate}
Then the equation \[ X^{p} + C^{p} = Z^{q} \]
has no integer solution for fixed $C$ with $|C| < \frac{1}{2} \cdot
\left(\frac{q^{p-1}}{p} \right)^{q-2}$. If $q \equiv 1 \mod p$, there are no solutions with $|C| < \frac{1}{2} \cdot \left(\frac{q^{2(p-1)}}{p} \right)^{q-2}$ \end{corollary} \begin{proof} The premises allow us to apply Theorem \ref{main2} and the claim follows from \rf{bound} and \rf{bound1}. \end{proof}
Finally, by restricting the results above to the rational Catalan equation \rf{ratcat}, we are able to give a criterion which only depends on the exponents, namely the following: \begin{theorem} \label{trc} Let $p, q > 3$ be distinct primes for which the following conditions are true: \begin{itemize} \item[1.] $-1 \in < p \mod q>$ and $-1 \in < q \mod p>$, \item[2.] $\left(pq, h_{pq}^-\right) = 1$, \item[3.] $2^{p-1} \not \equiv 1 \mod p^{2}$ and $2^{q-1} \not \equiv 1 \mod q^{2}$, \item[4.] $\left(2^{p-1} p^{p}\right)^{q-1} \not \equiv 1 \mod q^{2}$
and $\left(2^{q-1} q^{q}\right)^{p-1} \not \equiv 1 \mod p^{2}$, \item[5.] $p^{q-1} \not \equiv 1 \mod q^{2}$ and $q^{p-1}
\not \equiv 1 \mod p^{2}$, \item[6.] $\max\{p,\frac{p(p-20)}{16}\} > q$ and
$\max\{q,\frac{q(q-20)}{16}\} > p$. \end{itemize} Then the equation $X^{p}+Y^{q} = 1$ has no rational solutions. \end{theorem}
\section{Generalities} The following lemma of Euler is used in both equations of Fermat and Catalan: \begin{lemma} \label{euler} Let $x, y$ be coprime integers and $n > 1$ be odd. Then \begin{eqnarray} \label{cop}
\left(\frac{x^{n}+y^{n}}{x+y}, x+y\right) \ | \ n. \end{eqnarray} \end{lemma} \begin{proof} Write $x = (x+y) - y$ and develop \begin{eqnarray} \label{dev} \frac{x^{n}+y^{n}}{x+y} & = & \frac{(x+y)^{n} + \sum_{k=1}^{n-2}
\binom{n}{k} (x+y)^{n-k} \cdot (-y)^{k} + n (x+y) y^{n-1} - y^{n} +
y^{n}}{x+y} \nonumber \\ & = & K \cdot (x+y) + n \cdot y^{n-1}, \quad \hbox{with } \quad K \in \mathbb{Z}. \end{eqnarray} The common divisor in \rf{cop} is consequently \[ D = \left(\frac{x^{n}+y^{n}}{x+y}, x+y\right) = \left(K \cdot (x+y) + n
\cdot y^{n-1}, x+y\right) = \left(n \cdot y^{n-1}, x+y\right). \] But
since $x, y$ are coprime, and consequently also $(x+y, y) = 1$, it
follows plainly that $D = (n, x+y) | n$. \end{proof} Like in Fermat's equation, the above lemma leads to the following case distinction for \rf{FC}:
\textbf{Case I}: The case in which $p \not \hspace{0.25em} \mid z$. Then \[ x^{p} + y^{p} = (x+y) \cdot \frac{x^{p}+y^{p}}{x+y} = z^{q}, \] and since by Lemma \ref{euler}, the two factors have no common divisor, they must simultaneously be \nth{q} powers. Thus \begin{eqnarray} \label{case1} x+y = A^{q}, \quad \frac{x^{p}+y^{p}}{x+y} = B^{q} \quad \hbox{with} \quad A, B \in \mathbb{Z}, \end{eqnarray} for this case.
\textbf{Case II}: The case in which $p \mid z$. Then $x^{p} + y^{p} \equiv x+y \equiv z^{q} \equiv 0 \mod p$. We show that in this case \[ v_{p}\left( \frac{x^{p}+y^{p}}{x+y}\right) = 1. \] The development in \rf{dev} yields \[ \frac{x^{p}+y^{p}}{x+y} = p \cdot y^{p-1} + \sum_{k=2}^{p-1} \binom{p}{k} (x+y)^{k-1} \cdot (-y)^{p-k} + (x+y)^{p-1}. \] The binomial coefficients in the above sum are all divisible by $p$. Since $(y, x) = (y, x+y) = 1$, it follows also that $(y,p) = 1$ and thus
$v_{p}(p y^{p-1}) = 1$. But $p | x+y$, so all the remaining terms in the expansion of $\frac{x^{p}+y^{p}}{x+y}$ are divisible (at least) by $p^{2}$, which confirms our claim. In the second case thus, a \nth{q} power of $p$ is split between the factors $x+y$ and $\frac{x^{p}+y^{p}}{x+y}$ in such a way that the latter is divisible exactly by $p$. In the second case we have herewith: \begin{eqnarray} \label{case2} x+y = p^{nq-1} \cdot A^{q} = (A')^{q}/p^{e}, & \quad & \frac{x^{p}+y^{p}}{x+y} = p \cdot B^{q} \\
\hbox{with} \quad (A, B, p) = 1 & \quad & \hbox{and} \quad n =
v_{p}(z) \geq 1. \nonumber \end{eqnarray}
The relations \rf{case1} and \rf{case2} are the analogs of the Barlow - Abel relations for the Fermat - Catalan equations. Fermat's equation is homogeneous; thus if $x^{p}+y^{p}+z^{p} = 0$ is a solution in which $(x,y,z)$ are not coprime, one may divide by the \nth{p} power of the common divisor, thus obtaining a solution with coprime $(x,y,z)$. The equation \rf{FC} is {\em not} homogeneous and thus one may ask whether the requirement that $(x,y,z)$ be coprime is not restrictive\footnote{I owe David Masser the observation that the
condition $(x,y,z) = 1$ is not obvious for the equation \rf{FC}}. The following lemma addresses this question and shows that one can construct arbitrary many solutions of $x^{p}+y^{p} = z^{q}$, if common divisors are allowed. It appears that the condition $(x,y,z) = 1$ is thus plausible.
\begin{lemma}
Let $x, y$ be coprime integers and $p, q$ be distinct odd primes.
Then there is an integer $D \in \mathbb{Z}$ such that \[ (D \cdot x)^{p} + (D \cdot y)^{p} = z^{q}, \quad \hbox{with} \quad z \in \mathbb{Z}. \] Furthermore, every integer solution of $X^{p}+Y^{p}=Z^{q}$ with $(X, Y, Z) > 1$ arises in this way. \end{lemma} \begin{proof} Let $x, y$ be coprime integers and \[ x^{p} + y^{p} = C \cdot z^{q}, \] where $C \in \mathbb{Z}$ and $z$ is the largest \nth{q} power dividing the left hand side ($z = 1$ is possible); i.e. $C$ is \nth{q} power -
free. Let $\ell | C$ be a prime and $n = v_{\ell}(C)$. We show that there is an integer $D(\ell)$ such that $x' = D(\ell) \cdot x$ and $y' = D(\ell) \cdot y$ verify ${x'}^{p} + {y'}^{p} = C' \cdot {z'}^{q}$
and $z | z', C' | C$ while $(C', \ell) = 1$. Indeed, let $a \in \mathbb{N}$ be such that $n + a\cdot p \equiv 0 \mod q$; such an integer exists since $(p, q) = 1$. We define $D(\ell) = \ell^{a}$ and $b = (n+ap)/q$. Then \begin{eqnarray*} {x'}^{p} + {y'}^{p} = \ell^{ap} \cdot C \cdot z^{q} = \ell^{ap+n} \cdot C/\ell^{n} \cdot z^{q}
= \frac{C}{\ell^{a}} \cdot (z \cdot \ell^{b})^{q}. \end{eqnarray*}
Setting $C' = C/\ell^{a}$ and $z' = z \cdot \ell^{b}$, the claim follows. By repeating the procedure recursively for all the prime divisors of $C$ one obtains $D = \prod_{\ell | C} \ D(\ell)$ for which the claim of the Lemma holds.
Conversely, let $x^{p}+y^{p} = z^{q}$ hold for a triple with $(x,y,z)
= G$. For each prime $\ell | G$ let $a = v_{\ell}(x,y)$. Then $a p \leq q \cdot v_{\ell}(z)$. If $G' = (x, y)$ it follows that $w = z^{q}/{G'}^{p}$ is an integer. The integers $x' = x/G', y' = y/G'$ are coprime; if $C$ is the \nth{q} power - free part of $w$ and ${z'}^{q} = w/C$, then \begin{eqnarray} \label{above}
{x'}^{p} + {y'}^{p} = C \cdot {z'}^{q} . \end{eqnarray} But then the initial equality can be derived from \rf{above} by the procedure described above. Thus all non trivial solutions of $x^{p}+y^{p} = z^{q}$ have coprime $x, y, z$. \end{proof} We finally prove the relation between \rf{ratcat} and \rf{FC}: \begin{lemma} \label{rcfc} Let $p, q$ be odd primes. The equation \rf{ratcat} has non trivial rational solutions if and only if \begin{eqnarray} \label{homfc} x^{p}+y^{q} = z^{pq}, \quad (x, y, z) = 1 \quad \hbox{ and } \quad x, y, z \in \mathbb{Z} \end{eqnarray} has non trivial solutions. \end{lemma} \begin{proof}
Suppose first that \rf{homfc} has some non trivial solution $x, y,
z$. Then one easily verifies that \rf{ratcat} has the solution $X =
x/z^{q}, Y = y/z^{p}$.
Conversely, let $\ X = a/c, Y = b/d$ be a non trivial solution of \rf{ratcat} with $(a, c) = (b, d) = 1; \ a, b, c, d, \in \mathbb{Z}$. Clearing denominators we find \[ a^{p} d^{q} + b^{q} c^{p} = c^{p} d^{q}. \]
Since $(a, c) = (b, d) = 1 $, by comparing the two sides of the identity, we find $c^{p} | d^{q}$ and $d^{q} | c^{p}$; thus $c^{p} = d^{q}$. But
$p$ and $q$ are distinct primes, thus for each prime $\ell | c$, we have $pq | v_{\ell}(c)$ and we may write $c^{p} = d^{q} = u^{pq}$. The equation \rf{ratcat} becomes $a^{p}+b^{q}=u^{pq}$, as claimed. \end{proof} \section{Cyclotomy and Fermat - Catalan} We start by fixing some notations which shall be used throughout the rest of this paper. \subsection{Notation} We shall let $p, q$ be two odd primes and $\zeta, \xi \in \mathbb{C}$ be primitive \nth{p} and \nth{q} roots of unity and $\mathbb{K} = \mathbb{Q}(\zeta), \mathbb{K}' = \mathbb{Q}(\xi), \mathbb{L}= \mathbb{Q}(\zeta, \xi)$ the respective cyclotomic fields. Furthermore, the Galois groups will be \begin{eqnarray*}
G_{p} & = & \mbox{ Gal }(\mathbb{K}/\mathbb{Q}) = < \sigma > , \quad G_{q} = \mbox{ Gal }(\mathbb{K}')/\mathbb{Q} = < \tau
>\quad \hbox{ and } \\
G & = & \mbox{ Gal }(\mathbb{L}/\mathbb{Q}) = < \sigma \tau > = < \sigma > \times < \tau > . \end{eqnarray*} Unless stated otherwise in some particular context, $\sigma, \tau$ are thus generators of the Galois groups $G_{p}, G_{q}$, respectively. Furthermore, if $0< a < p; 0 < b < q$ we shall use the notation $\sigma_{a} \in G_{p}, \tau_{b} \in G_{q}$ for the elements of the Galois groups given by $\zeta \mapsto \zeta^{a}$ and $ \xi \mapsto \xi^{b}$, respectively. The map $G_{p} \rightarrow \ZMs{p}$ given by $\sigma_{a} \mapsto a$ will be denoted by $\widehat{\sigma_{a}} = a$, and likewise for the analog for $G_{q}$. Complex conjugation is denoted also by $\jmath \in G$, while $\jmath_{p}, \jmath_{q}$ are the complex conjugation maps of $G_{p}, G_{q}$ respectively, lifted to $G$. Thus $\jmath_{p}$ acts on $\zeta$ but fixes $\xi$ and $\jmath_{q}$ does the reverse.
Finally, the ramified primes are $\wp = (1-\zeta), \eu{q} = (1-\xi)$. We use the notations $\lambda = (\xi - \overline \xi)$ and $\lambda' = (\zeta - \overline \zeta)$ for generators of these ramified primes; this is due to the nice behavior under complex conjugation. In several contexts it will come natural to use the classical $\lambda = (1-\xi)$, etc.; this deviation from the general use will be mentioned in place.
We now start with some classical results, adapted to the present equation \rf{FC}. \subsection{First Consequences of Class Field Theory} We assume $q \not \hspace{0.25em} \mid h_{p}^{-}$ and deduce some consequences, starting from a presumed non - trivial solution $(x, y, z)$ of \rf{FC}. With $e$ defined like above, we shall let \begin{eqnarray} \label{alphadef}
\alpha & = & \frac{x+\zeta \cdot y}{(1-\zeta)^{e}} \quad \hbox{ and }
\\ \eu{A} & = & \left(\alpha, \frac{x^{p}+y^{p}}{p^{e}(x+y)}\right)
\subset \id{O}(\mathbb{K})^{\times} = \mathbb{Z}[\zeta]^{\times}. \nonumber \end{eqnarray}
\begin{lemma} \label{euaq} Let $x, y, z$ be coprime integers verifying \rf{FC}. Then \begin{eqnarray} \label{firstalpha} \left(\sigma(\alpha), \sigma'(\alpha)\right) &= & 1 \quad \forall \ \sigma, \sigma' \in G_{p}, \sigma \neq \sigma' \\ \eu{A}^{q} & = & (\alpha).\nonumber \end{eqnarray} \end{lemma} \begin{proof} We cumulate the two Cases of the Fermat - Catalan equation in \begin{eqnarray} \label{cum1}
x+y & = & A^{q}/p^{e} \quad \hbox{and} \\ \label{cum2}
\frac{x^{p}+y^{p}}{x+y} & = & p^{e} \cdot B^{q} \quad \hbox{with} \quad e
\in \{ 0, 1\} \end{eqnarray}
Here $e = 0$ corresponds to the First Case and $e = 1$ to the Second Case. If $e = 1$, it is understood that $p | A$, so the right hand side in \rf{cum1} is an integer. Then \rf{cum2} is equivalent to \begin{eqnarray} \label{neq} \mbox{\bf N}_{\mathbb{Q}(\zeta)/\mathbb{Q}}(\alpha) = B^{q} \quad \hbox{ and } \quad \eu{A} = (\alpha, B). \end{eqnarray} We write $P = \{1, 2, \ldots, p-1\}$ and note that, for $c, d \in P$ we have \begin{eqnarray} \label{copr} \Delta(c,d) = (\sigma_{c}(\alpha), \sigma_{d}(\alpha)) = (1) \end{eqnarray} whenever $c \neq d$. Indeed \begin{eqnarray*}
y(\zeta^{c}-\zeta^{d}) = (1-\zeta^{c})^{e} \sigma_{c}(\alpha) - (1-\zeta^{d})^{e} \sigma_{d}(\alpha) \in \Delta(c,d) \quad \hbox{and} \\
x(\zeta^{-c}-\zeta^{-d}) = \overline \zeta^{c} (1-\zeta^{c})^{e} \sigma_{c}(\alpha) - \overline \zeta^{d }(1-\zeta^{d})^{e} \sigma_{d}(\alpha) \in \Delta(c,d). \end{eqnarray*} Since $(x, y) = 1$ it follows that $\Delta(c,d) \supset \wp$, the ramified prime above $p$ in $\mathbb{K}$. But $\alpha = (x+\zeta y)/(1-\zeta)^{e} = \frac{x+y}{(1-\zeta)^{e}} - y \cdot (1-\zeta)^{1-e}$. If $e = 0$, the first term is coprime to $\wp$ and the second is not. If $e = 1$, the second term is coprime to $\wp$ and the first is not. Thus in both cases, $(\alpha, \wp) = 1$ and $\Delta(c,d) = (1)$, as claimed. The second relation in \rf{firstalpha} follows now easily from the definition of $\eu{A}$: \[ \eu{A}^{q}/(\alpha) = \left(\alpha^{q-1}, B \cdot \alpha^{q-2},
\ldots, B^{q-1}, \mbox{\bf N}(\alpha)/\alpha \right) , \] and one verifies that the integer ideal on the right hand side is equal to the ideal $\left(\alpha, \mbox{\bf N}(\alpha)/\alpha\right) = (1)$. \end{proof}
An immediate consequence is: \begin{corollary} \label{crhodef} If $p, q$ are odd primes with $q \not \hspace{0.25em} \mid h_p$, the class number of $\mathbb{Q}(\zeta_p)$, $x,y,z$ verify \rf{FC} and $\alpha = (x+\zeta y)/(1-\zeta)^e$ with $e$ as above, then \begin{eqnarray} \label{algq} \alpha & = & \varepsilon \cdot \rho^q \quad \hbox{for some} \quad \varepsilon \in \mathbb{Z}[\zeta + \overline \zeta]^{\times} , \ \rho \in \mathbb{Z}[\zeta]. \end{eqnarray} and if only $q \not \hspace{0.25em} \mid h_{p}^{-}$ holds, then \begin{eqnarray} \label{rhod} \frac{x+\zeta y}{x + \overline \zeta y} = \pm \left(\frac{\rho_{1}}{\overline
\rho_{1}}\right)^{q}, \quad \hbox{for some} \quad \rho_{1} \mathbb{Z}[\zeta]. \end{eqnarray} The two algebraic integers $\rho, \rho_{1}$ may, but need not be equal. \end{corollary} \begin{proof}
The statement \rf{algq} is a direct consequence of \rf{firstalpha},
since the second relation implies that $\eu{A}$ is principal if $q
\not \hspace{0.25em} \mid h_{p}$. Since $\eu{A}^{q} = (\alpha)$ by \rf{firstalpha}, if
$q \not \hspace{0.25em} \mid h_{p}^{-}$, then there is a real ideal $\eu{B} \subset
\mathbb{Z}[\zeta]$ together with an algebraic number $\nu \in \mathbb{K}$ such that
$\eu{A} = (\nu) \cdot \eu{B}$. By dividing through the complex
conjugate of this identity, one finds \[ \left(\frac{\eu{A}}{\overline{\eu{A}}}\right)^{q} = (\alpha/\overline \alpha) = (\nu/\overline \nu)^{q}, \] and there is a unit $\eta$ such that $\alpha/\overline \alpha = \eta (\nu/\overline \nu)^{q}$ and thus \[ \frac{x+\zeta y}{x+\overline \zeta y} = \left(\frac{1-\overline
\zeta}{1-\zeta}\right)^{e} \cdot \eta \cdot \left(\frac{\nu}{\overline \nu}\right)^{q} = \eta' \cdot \left(\frac{\nu}{\overline \nu}\right)^{q}. \] But then $\eta' \cdot \overline \eta' = 1$ and Dedekind's unit Theorem implies that $\eta'$ is a root of unity of $\mathbb{K}$. Since all roots of unity of this field have order dividing $2p$, and $(2p, q) = 1$, the statement \rf{rhod} follows. \end{proof}
We now prove a lemma concerning ideals related to the above $\eu{A}$, in a more general setting. \begin{lemma} \label{idealq} Let $\rg{k} \subset \mathbb{L}$ be some field such that $q \not \hspace{0.25em} \mid h(\rg{k}$, the class number of the field $\rg{k}$. Let $\phi_i \in \id{O}(\rg{k}), i=1, 2, \ldots, n$ be such that $(\phi_i, \phi_j) = (1)$ for $1 \leq i \neq j < n$. Suppose that for some $m \geq 0$ all $\phi_i, i = m+1, m+2, \ldots, $ are not units, while $\phi_j, j = 1, 2, \ldots, m$ are units. Furthermore, there is a $C \in \rg{k}, (C, pq) = 1$ such that \begin{eqnarray} \label{prd} \prod_{i=1}^n \phi_i = C^{q}. \end{eqnarray} Then there are $\eta_i \in \id{O}(\rg{k})^{\times}$ and $\mu_i \in \id{O}(\rg{k})$ such that \begin{eqnarray} \label{etamu} \phi_i = \eta_i \cdot \mu_i^q \quad \hbox{for} \quad i = m+1, m+2, \ldots, n. \end{eqnarray} \end{lemma} \begin{proof}
Let $\eu{A}_{i} = (\phi_{i}, C)$ be ideals in $\id{O}(\rg{k})$. For
$i > m $ these ideals are not trivial, while for $i \leq m$ they are
equal to $\id{O}(\rg{k})$. We assume thus $i > m$ and claim that \begin{eqnarray} \label{qpow} \eu{A}_{i}^{q} = (\phi_{i}). \end{eqnarray} Indeed, \begin{eqnarray*} \eu{A}_{i}^{q}/(\phi_{i}) = \left(\phi_{i}^{q-1}, C \cdot \phi_{i}^{q-2},
\ldots, C^{q-1}, C^{q}/\phi_{i} \right). \end{eqnarray*} It follows from \rf{prd} that the right hand side ideal is integer and $(\phi_{i}) \mid \eu{A}_{i}^{q}$. On the other hand, since $(\phi_{i}, \phi_{j}) = (1)$ for $i \neq j$, we have $\left(C^{q}/\phi_{i},
\phi_{i}\right) = (1)$ and thus $\eu{A}_{i}^{q} = (\phi_{i})$, as claimed. Furthermore, $q \not \hspace{0.25em} \mid h(\rg{k})$ implies that the ideals $\eu{A}_i, i > m$ must be principal. There are $\mu_{i} \in \id{O}(\rg{k})$ such that $\eu{A}_{i} = (\mu_{i})$. It follows then from \rf{qpow} that \begin{eqnarray} \label{alg} ({\mu}_{i}^{q}) & = & \eu{A}_{i}^{q} = (\phi_{i}) \quad \hbox{and} \\ \phi_{i} & = & \eta_{i} \cdot {\mu}_{i}^{q} \quad \hbox{for some} \quad \eta_{i} \in \left(\id{O}(\rg{k})\right)^{\times}.\nonumber \end{eqnarray} This completes the proof of the lemma. \end{proof} \section{Fermat Equations and Proof of Theorem \ref{genfer}} Suppose that $x, y, z$ is a non trivial solution of \rf{FC} and $q \not \hspace{0.25em} \mid h_{p} = h(\mathbb{K})$. Then \rf{algq} holds by Corollary \ref{crhodef}. This leads to a reduction of the initial Fermat - Catalan equation \rf{FC} to a Fermat - like equation (i.e. involving only \textit{one } prime exponent) in extension fields. It is likely that this reduction may bring some progress in the \textit{general program } ennounced by Darmon \cite{Da} for the solution of \rf{FC}. Indeed, in this programatic paper, Darmon suggests that in order to solve general cases of \rf{FC}, "\textit{one is naturally led to replace elliptic
curves by certain 'hypergeometric Abelian varieties', so named
because their periods are related to values of hypergeometric
functions}". Our result shows however that there is a solid region of the $(p,q)$ plane, in which the simple elliptic curves, albeit defined over totally real (cyclotomic) extensions of $\mathbb{Q}$, can still do the job \footnote{ I thank Jordan Ellenberg for pointing out to me
that it is important to have Fermat equations defined over
\textit{totally real} fields, thus opening a door to the use of
Hilbert modular forms. Consequently, the Fermat equations which we
deduce here will have this property and be defined over the simplest
totally real fields available in the context. It should be mentioned
here, that a large variety of Fermat-like equations can be deduced
from our results; there are reasons to believe that the ones we
display may be the best point of departure for further
(non-cyclotomic) investigations.}.
The result in this direction was ennounced in Theorem \ref{genfer}, of which we give a proof below. \begin{proof}
We shall treat Case I and Case II separately, using the definitions
in \rf{cum1}, \rf{cum2}. Suppose first that $e = 0$ (Case I). Then \[ \alpha \cdot \overline \alpha = (x+\zeta y)(x+\overline \zeta y) = (x+y)^{2} - \mu xy = A^{2q} - \mu xy = \delta \cdot \nu^{q}, \] where $\delta = \varepsilon \cdot \overline \varepsilon$, $\nu = \rho \cdot \overline \rho$ and $\mu = (1-\zeta)(1-\overline \zeta)$ is the ramified prime above $p$ in $\mathbb{K}^{+}$. Since $p > 3$ there is at least one non trivial automorphism $\sigma \in G_{p}$; we may apply this automorphism to the above equation and eliminate $x y$ from the resulting two identities: \[ \left(\sigma(\mu) - \mu \right) \cdot A^{q} = \sigma(\mu) \cdot \delta \cdot \nu^{q} - \mu \cdot \sigma\left(\delta \cdot
\nu^{q}\right).\] If $\sigma(\zeta+\overline \zeta) = \zeta^{c}+\overline \zeta^{c}$, we note that \[ \mu-\sigma(\mu) = (\zeta-\zeta^{c}) + \overline{\zeta-\zeta^{c}} = (1-\zeta^{c-1}) \cdot (\zeta - \overline \zeta^{c}) = \delta_{1} \mu,\] with $\delta_{1} \in \mathbb{Z}[\zeta+\overline \zeta]^{\times}$. After division by $\mu$ in the previous identity, we find there are three units $\delta_{1}, \delta_{2} = -\delta \cdot \frac{\sigma(\mu)}{\mu}, \delta_{3} = \sigma(\delta)$ such that \[ \delta_{1} A^{2q} + \delta_{2} \nu^{q} + \delta_{3} \sigma(\nu)^{q} = 0 .\] In this Case, \rf{fereq} holds with $a, b, c$ being all units.
Suppose now that $e = 1$, so $x+y=p^{q-1} \cdot A^{q}$ and $\alpha \cdot (1-\zeta) = x+\zeta y$. In this case, \[ \mu \cdot \alpha \cdot \overline \alpha = (x+\zeta y)(x+\overline \zeta y) = (x+y)^{2} - \mu xy = A^{2q} \cdot p^{2(q-1)} - \mu xy = \mu \cdot \delta \cdot \nu^{q}, \] where again $\delta = \varepsilon \cdot \overline \varepsilon$ and $\nu = \rho \cdot \overline \rho$. We eliminate, like previously, the term in $x y$, thus obtaining: \[ \left(\sigma(\mu) - \mu \right) \cdot A^{2q} p^{2(q-1)} = \left(\mu \cdot \sigma(\mu)\right) \cdot \left( \delta \cdot
\nu^{q} - \sigma\left(\delta \cdot \nu^{q} \right) \right), \] and with the same $\delta_{1}$ as above, upon division by $\mu \cdot \sigma(\mu)$, \[ -\delta_{1} A^{2q} \cdot \frac{p^{2(q-1)}}{\sigma(\mu)} = \delta \cdot \nu^{q} - \sigma\left(\delta \cdot \nu^{q} \right) .\] In this Case, \rf{fereq} holds with $b, c$ being units, while $a = \delta_{1} \cdot \frac{p^{2(q-1)}}{\sigma(\mu)}$, so $(a)$ is a power of the ramified prime above $p$. This completes the proof of the Theorem. \end{proof}
In the Second Case, one may wish a Fermat equation with \textit{all - units} coefficients. This can be achieved at the cost of imposing $p \geq 7$ and the fact that all three unknowns will be non - rational. With this one has the following \begin{proposition} In the premises of Theorem \ref{genfer} and assuming that $e = 1$ (the Second Case, thus), let $p \geq 7$ and $\mathbb{K} = \mathbb{Q}(\zeta_p)^+, \rg{A} = \id{O}(\mathbb{K})$. Then for any $\sigma \in \mbox{ Gal }(\mathbb{K}/\mathbb{Q})$, there are three units $\varepsilon_j \in \rg{A}$ and a $\nu \in \rg{A}$, such that the ecuation: \begin{eqnarray} \label{fcase2}
\varepsilon_1 \cdot X^q + \varepsilon_2 Y^q + \varepsilon_3 Z^q = 0 \end{eqnarray} has the solution $(X, Y, Z) = (\nu, \sigma(\nu), \sigma^2(\nu)) \in \rg{A}^3$. \end{proposition} \begin{proof} We start form the identity \[ \delta_{1} A^{2q} \cdot \frac{p^{2(q-1)}}{\sigma(\mu)} = \delta \cdot \nu^{q} - \sigma\left(\delta \cdot \nu^{q} \right) \] derived above for this Case, in the proof of the Theorem \ref{genfer}. We shall need precise information about the units, and thus trace them back in the proof. Let $\sigma$ be fixed and $\chi = \mu^{\sigma-1}$; then $\delta_1 = \chi - 1$ is also a unit. The unit $\delta$ is fixed by its $q$ - adic expansion, but we shall not require more detail here.
Now apply $\sigma$ to the previous identity and use the definition of $\chi$: \begin{eqnarray*} (\chi -1 )^{-1} \left(\delta \cdot \nu^{q} - \sigma(\delta \cdot \nu^{q}) \right) & = & C/\sigma(\mu) \\ \sigma \left((\chi -1 )^{-1} \left(\delta \cdot \nu^{q} \right)- \sigma\left(\delta \cdot \nu^{q} \right) \right) & = & C/\sigma(\mu) \times \sigma(\chi^{-1}), \end{eqnarray*} and notice the crucial identity among units: \begin{eqnarray} \label{unitlc}
\frac{1}{\chi - 1} - \sigma\left(\frac{\chi}{\chi-1}\right) = \frac{1}{\sigma(\chi^{-1}) - 1} =: \Delta \in \rg{A}^{\times} . \end{eqnarray} Thus, a linear combination of the previous equations yields: \[ \varepsilon_1 \nu^q + \varepsilon_2 \sigma(\nu)^q + \varepsilon_3 \sigma^2(\nu)^q = 0, \] with the units: \[ \varepsilon_1 = \frac{\delta}{\chi-1}, \quad \varepsilon_2 = \frac{\sigma(\chi \cdot \delta)}{\sigma(\chi) - 1} \quad \varepsilon_3 = \sigma\left(\frac{\sigma(\delta)}{\chi-1}\right) . \] This completes the proof. \end{proof} \begin{remark} One may use above two distinct Galois actions $\sigma, \tau$ rather then just one and its square. The corresponding linear combination of units in \rf{unitlc} remains a unit and one thus obtains a more general Fermat equation with conjugate solutions. \end{remark} \subsection{The Case $p = 3$} According to Beukers \cite{Be}, this case has been solved for $n = 4, 5, 17 \leq n \leq 10000$ by N. Bruin \cite{Br2} and A. Kraus \cite{Kr}, respectively (note that here, composite exponents are taken into consideration too). Since this leaves the general case open, it may be interesting to deduce the associated Fermat equations. They are given by the following: \begin{proposition} \label{fer3} Let $q > 3$ be a prime for which the equation $x^{3}+y^{3} = z^{q}$ has non trivial coprime solutions in the integers. If $\mathbb{K} = \mathbb{Q}(\sqrt{-3})$ is the third cyclotomic field, $\rg{E} \subset \mathbb{K}$ are the Eisentstein integers and $\rho \in \rg{E}$ is a third root of unity, then there is a $\beta \in \rg{E}$ such that one of the following alternatives hold: \begin{eqnarray} \label{caseI3}
\beta^{q} + \overline \beta^{q} & = & A^{q},\\ \label{caseII3}
3(\rho-\rho^{2}) \cdot \left( \beta^{q} - \overline \beta^{q}\right) & =
& A^{q}, \end{eqnarray} where $A \in \mathbb{Z}$. \end{proposition} \begin{proof}
The alternative above corresponds to the two cases of \rf{FC}. In
the first case, $x+y=A^{q}$ and $\eu{A} = (x+\rho y, z)$ is a
principal ideal. Since all the units of $\rg{E}$ are (sixth) roots
of unity, and thus \nth{q} powers, it follows that there is a $\beta
\in \rg{E}$ such that $\beta^{q} = \rho x + \overline \rho y$. Then
$\beta^{q}+\overline \beta^{q} = (\rho+\overline \rho) (x+y) =
-A^{q}$, which proves \rf{caseI3}. In the second case $x+y =
A^{q}/3$ and \rf{caseII3} follows by a similar computation, with
details left to the reader. \end{proof}
It is also useful to know that the case $q | z$ can be ruled out in both \rf{caseI3} and \rf{caseII3} by using a generalized form of Kummer descent. We shall give details for $p > 3$ in a later section, leaving this case as an open remark for the reader. \section{Consequences of Class Field Theory} In this section we shall deduce some consequences of class number conditions in the cyclotomic fields of our interest. Of most value for our investigation, these conditions give some control on local properties of units of the \nth{pq} field and its subfields.
\subsection{Primary Numbers and Reflection} We shall be interested in the sequel in the algebraic integers of the field $\mathbb{L} = \mathbb{Q}(\zeta, \xi)$ and its subfields. Let $\rg{R}$ be one of the rings of integers $\mathbb{Z}[\zeta], \mathbb{Z}[\xi], \mathbb{Z}[\zeta, \xi]$ and $\rg{K}$ its quotient-field. An element $\alpha \in \rg{R}$ is $q$ - \textbf{primary} if it is a $q$ - adic \nth{q} power. The $q$ - primary numbers build a subring $\rg{R}_{q} \subset \rg{R}$ and it is immediately verified that $\rg{R}^{q} \subset \rg{R}_{q}$. If $E(\rg{R}) = E = \rg{R}^{\times} \subset \rg{R}$ are the units of the respective field, then we write $E_{q} = E \cap \rg{R}_{q}$. The ring $\rg{R}_{q}$ induces the following equivalence relation: \begin{eqnarray} \label{eqvq} \alpha =_{q} \beta \quad \Leftrightarrow \quad \exists \ \mu, \nu \in \rg{R}_{q} \ : \ \mu \cdot \alpha = \nu \cdot \beta. \end{eqnarray} If $\eu{q} = (q)$ or $\eu{q} = (1-\xi)$, depending on whether $\xi \not \in \rg{R}$ or $\xi \in \rg{R}$, we let $S = \rg{R} \setminus (\eu{q})$ and $\rg{R}' = S^{-1} \rg{R}$ be the corresponding localization. One may extend the definition of $q$ - primary numbers to $\rg{R}'$, thus obtaining the ring $\rg{R}'_{q} \subset \rg{R}'$. The equivalence relation in \rf{eqvq} may then also be written as \[ \alpha =_{q} \beta \ \Leftrightarrow \ \alpha = \gamma \cdot \beta \quad \hbox{ for some } \gamma \in \rg{R}'_{q}. \]
A number $\alpha \in \rg{R}'$ is called $q$ - \textbf{singular} if there is a \textit{non - principal} ideal $\eu{B} \subset \mathbb{K}$ such that $(\alpha) = \eu{B}^{q}$ as ideals. Let $\rg{R}"_{q}$ be the ring of the $q$ - primary numbers which are also singular. The degenerate case $\eu{B} = \rg{R}$ suggests allowing $E_{q} \subset \rg{R}''_{q}$. By class field theory, the singular primary numbers $\alpha \in \rg{R}''_{q}$ have the property that the extension $\mathbb{Q}(\zeta, \xi)[\alpha^{1/q}]$ is unramified Abelian. There is thus, by Hilbert's Theorem 94, an ideal of order $q$ of $\mathbb{Q}(\zeta, \xi)$ which capitulates in this extension (see e.g. \cite{Wa}, Exercise 9.3).
We now define the number $h(p, q), h_{pq}^-$ ennounced in the introduction. The definition involves an explicit use of Leopoldt's reflection theorem (see e.g. \cite{Lo}, \cite{Mi2}). The number $h(p, q)$ will be defined so that the condition $\left(q, h(p, q)\right) = 1$ becomes the tightest \textit{easily computable} condition which implies $q \not \hspace{0.25em} \mid h_{p}$. Let $X = \{ \chi_{0} : G_{p} \rightarrow \overline \mathbb{F}_{q} \} $ be the set of Dirichlet characters of order $q$ and $\omega_{0} : G_{q} \rightarrow \mathbb{F}_{q}$ the Teichm\"uller character. Then $\psi_{0} = \chi_{0}^{-1} \cdot \omega_{0}$ is a well defined Dirichlet character of $G_{pq}$ in $\overline \mathbb{F}_{q}$ which corresponds by reflection to $\chi$. If $\rg{k} = \mathbb{F}_{q}[\Im(X)]$ in the obvious sense (with $\Im(\chi)$ being the image of the character $\chi$ in $\mathbb{Q}(\zeta_{q-1})$), then one may chose a subfield $\rg{K} \subset \mathbb{Q}(\zeta_{q-1})$ of the \nth{q-1} cyclotomic extension and an integer ideal $\eu{Q}$ in this field, in such a way that $\rg{k} = \id{O}(\rg{K})/\eu{Q}$. This allows to lift $\psi_{0}, \chi_{0}$ and $\omega_{0}$ to characters $\chi, \psi, \omega$ with images in $\rg{k}$ \cite{Mi2}. In particular, the generalized Bernoulli number $B_{1, \psi}$ is given by \cite{Wa}: \[ B_{1, \psi} = \frac{1}{(p-1)(q-1)} \sum_{(a,pq) = 1; \
0 < a < p q } \ a \psi^{-1} \gamma_{a} , \] where $\gamma_{a} \in G$ with $\gamma_{a}(\zeta \xi) = (\zeta \xi)^{a}$. With this, we define \begin{eqnarray} \label{hpqdef} B_{\omega} & = & \prod_{\chi_{0} \in X} \ B_{1, \chi^{-1} \omega} \quad \hbox{ and } \nonumber \\ h(p, q) = h_{p}^{-} \cdot B_{\omega}. \end{eqnarray} Note that the Bernoulli numbers can be computed explicitly and in general $B_{1,\psi} \not \in \mathbb{Z}$, but $B_{\omega} \in \mathbb{Z}$, since it is the norm of an algebraic integer in $\mathbb{Q}(\zeta_{q-1})$. It is also true \cite{Mi2}, that $q \mid h_{p}^{+}$ implies $q \not \hspace{0.25em} \mid B_{\omega}$.
We shall see that certain ideals of $\mathbb{L}$ occurring in the subsequent proofs have order dividing $q$; they are thus principal is $q \not \hspace{0.25em} \mid h_{pq}^{-}$ (by reflection then $q \not \hspace{0.25em} \mid h_{pq}$!); if the ideals belong to $\mathbb{K}$, then they are already principal if $q \not \hspace{0.25em} \mid h(p, q)$. The following Proposition reflects these and further useful consequences of the above class number conditions.
\begin{proposition} \label{lemmaA} Let $E, E_{q} \subset \mathbb{L}$ be the units, respectively the $q$ - primary units of the \nth{pq} cyclotomic extension and $E', E'_{q} \subset \mathbb{K}$ be the respective sets in the \nth{p} cyclotomic extension. If $q \not \hspace{0.25em} \mid h_{pq}^-$ then $E_{q}=E^{q}$ and in particular, if $\varepsilon =_{q} 1$ is a unit, then it is a \nth{q} power. Likewise, if $q \not \hspace{0.25em} \mid h(p, q)$, then $E'_{q} = {E'}^{q}$ and all $q$ - primary units in $E'$ are \nth{q} powers. Furthermore, $q \not \hspace{0.25em} \mid h_{pq}$ in the first case and $q \not \hspace{0.25em} \mid h_{p}$ in the second. \label{qprim} \end{proposition} \begin{proof}
We start with the implication $q \not \hspace{0.25em} \mid h_{pq}^{-} \Rightarrow q \not \hspace{0.25em} \mid
h_{pq}$; this follows directly by reflection in $\mathbb{L}$. Let $q \not \hspace{0.25em} \mid
h_{pq}^-$ and $\varepsilon \in E_{q} \setminus E^{q}$ be a $q$ -
primary unit, which is not a \nth{q} power. Then $\rg{k} =
\mathbb{L}(\varepsilon^{1/q})$ is an Abelian unramified extension (see e.g
\cite{Wa}, lemma 9.1, 9.2) and $[\rg{k}:\mathbb{L}] = q$; by Hilbert's
Theorem 94, there is an ideal of order $q$ from $\mathbb{L}$ which
capitulates in $\rg{k}$, in contradiction with $q \not \hspace{0.25em} \mid h_{pq}^{-}$.
We now consider the case $q \not \hspace{0.25em} \mid h(p, q)$. The crucial remark here
is that even Dirichlet characters of $\mathbb{K}$ correspond by reflection
to the characters indexing Bernoulli numbers which divide
$B_{\omega}$ above. The claims for this case follow then in analogy
to the ones for $q \not \hspace{0.25em} \mid h_{pq}^{-}$. \end{proof}
\subsection{Units} We start the analysis of local properties of units in $\id{O}(\mathbb{L})$ under certain eventual restriction on the class number, by several simple basic Lemmata. \begin{lemma} \label{lru} Let $\delta = < -\zeta \cdot \xi >$ be a root of unity of $\mathbb{L}$. If $\delta \equiv 1 \mod q$, then $\delta = 1$.
Furthermore, if $\varepsilon \in \mathbb{Z}[\zeta, \xi]^{\times}$ is a unit such that $\varepsilon = a + b(\zeta) q + O(q\lambda)$ and $a \in \mathbb{Z}$ and $b(\zeta) \in \mathbb{Z}[\zeta]$, then $b(\zeta) = \overline b(\zeta)$. \end{lemma} \begin{proof} Let $\delta^2 = \zeta^a \cdot \xi^b$, with $0 \leq a < p, \ 0 \leq b < q$ - squaring cancels the sign. If $a \cdot b \neq 0$, then $\delta^2$ is a primitive \nth{pq} root of unity and thus \[ P(G) = \prod_{\psi \in G} \left(1-\psi(\delta) \right) = \Phi_{pq}(1) = 1.\] But since $\delta^2 \equiv 1 \mod q$ we should have $P(G) \equiv 0 \mod q^{\varphi(pq)}$, so $a \cdot b \neq 0$ is impossible. In the cases $a = 0, b \neq 0$ and $a \neq 0, b= 0$, the root $\delta^2$ is primitive of order $q$, resp. $p$ and the value of $P(G)$ is $q^{p-1}$ and $p^{q-1}$, respectively. In both cases $P(G) \not \equiv 0 \mod q^{\varphi(pq)}$, so we must have plainly $\delta^2 = 1$ and since $\delta \equiv 1 \mod q$, also $\delta = 1$.
If $\varepsilon$ is like in the claim of the lemma, then $\delta = \varepsilon/\overline \varepsilon \equiv 1 \mod q$ is a root of unity, and thus $\varepsilon = \overline \varepsilon$. The claim on $b(\zeta)$ follows. \end{proof}
\begin{lemma} \label{unitp} Let the $p, q$ be odd primes with $q \not \hspace{0.25em} \mid h_{pq}^{-}. p \not \hspace{0.25em} \mid (q-1)$ and suppose that $\varepsilon \in \id{O}(\mathbb{L})^{\times}$ is a unit such that $\varepsilon =_{q} a$, with $a \in \mathbb{Z}$. Then $\varepsilon$ is a \nth{q} power. \end{lemma} \begin{proof}
Let $\sigma \in G = \mbox{ Gal }(\mathbb{Q}(\zeta, \xi)/\mathbb{Q})$. Then $\delta_{\sigma}
= \varepsilon^{1-\sigma} =_{q} 1$ and by Proposition \ref{lemmaA} it
is a \nth{q} power. Since this holds for all $\sigma$, \[ \varepsilon^{(p-1)(q-1)} = \prod_{\sigma \in G} \delta_{\sigma} \in
\mathbb{L}^{q} . \] But $\left(q, (p-1)(q-1)\right) = 1$ by hypothesis and
consequently $\varepsilon$ in a \nth{q} power, as claimed. \end{proof}
Finally, we have: \begin{lemma} \label{lemmaC} \label{lamunit} Let $p, q$ be primes and $\varepsilon \in \mathbb{Z}[\zeta_{p}]^{*}$ be a unit such that $\varepsilon =_{q} c(1-\zeta)$, with $c \in \mathbb{Q}$. If $p \not \equiv 1 \mod q$ and $(p-1) m \equiv 1 \mod q$, then $p =_{q} 1$ and \begin{eqnarray} \label{uc}
\varepsilon =_{q} \left(\frac{1-\zeta}{p^{m}}\right) =_{q}\left(
\frac{(1-\zeta)^{p-1}}{p}\right)^{m} = \gamma \in \mathbb{Z}[\zeta]^{\times} . \end{eqnarray} \end{lemma} \begin{proof}
Let $\sigma \in G = \mbox{ Gal }(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ be a generator and
$\Omega = \sum_{i=0}^{p-3} (p-2-i) \sigma^{i} \in \mathbb{Z}[G]$, so that
$(\sigma-1) \Omega + (p -1)= \mbox{\bf N}_{\mathbb{Q}(\zeta)/\mathbb{Q}}$. By hypothesis,
$\varepsilon^{\sigma-1} =_{q} \eta = (1-\zeta)^{\sigma-1}$, so \[ 1 = \mbox{\bf N}(\varepsilon) = \varepsilon^{p-1+\Omega(\sigma-1)} =_{q} \varepsilon^{p-1} \cdot \eta^{\Omega}, \] and thus $\varepsilon^{p-1} =_{q} \eta^{-\Omega}$ and $\varepsilon =_{q} \eta^{- m \Omega}$. Note also that $\eta^{\Omega} = (1-\zeta)^{(\sigma-1)\Omega} = (1-\zeta)^{\mathbf{N}-p+1} = p/(1-\zeta)^{p-1}$, a simple expression for this unit. \end{proof}
The deeper results on local properties of units in $\mathbb{L}$ (and subfields), given class number restraints, are summarized in the following Proposition. The proof of the proposition is quite lengthy and involves, along with the previous class field results, some interesting properties of cyclotomic units in fields of composite order. The result is interesting in itself, but it shall be used only for improving an estimate in Theorem \ref{main2} for the special case when $q \equiv 1 \mod p$. Given the lack of generality of its application, the reader who is more interested in an overview of the main ideas and proofs, can thus skip to the next section. \begin{proposition} \label{un1q} Let $p, q$ be odd primes with $q \not \hspace{0.25em} \mid h_{pq}^-$, $p \not \equiv 1 \mod q$ and $\mathbb{L} = \mathbb{Q}[\zeta, \xi]$ be the \nth{pq} cyclotomic extension of $\mathbb{Q}$. If $\varepsilon \in \mathbb{L}$ is a unit for which there is a $a \in \mathbb{Z}[\zeta]$ such that $\varepsilon \equiv a \mod q$, then there is a unit $\delta \in \mathbb{Z}[\zeta]$ such that $\varepsilon =_q \delta$. \end{proposition} We first prove some Lemmata: \begin{lemma} \label{linindep} Let $p, q$ be odd primes with $p \not \equiv 1 \mod q$ and $q \not \hspace{0.25em} \mid h_p^-$; let $\zeta \in \mathbb{C}$ be a primitive \nth{p} root of unity. Then $\gamma_c = 1/(1-\zeta^c), c=1, 2, \ldots, p-1$ form a basis for the Galois ring $\mathbb{Z}[\zeta]/\left(q \cdot \mathbb{Z}[\zeta]\right)$. \end{lemma} \begin{proof}
We must show that $\gamma_c$ are linear independent. For this, we
shall show that the discriminant $\Delta$ of the $\mathbb{Z}$ - module $M =
[\gamma_1, \gamma_2, \ldots, \gamma_{p-1}]$ is coprime with $q$
under the given conditions. Let $\sigma$ be a generator of
$G_{p}=\mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q})$ and the matrix $\mathbf{A} =
\left(\sigma^{i+j} \left(\frac{1}
{1-\zeta}\right)\right)_{i,j=1}^{p-1}$; then $\Delta =
\det^2(\mathbf{A})$. The matrix $\mathbf{A}$ is a circulant matrix;
if $\omega \in \mathbb{C}$ is a primitive \nth{p-1} root of unity, then the
vectors $\vec{f}_k = (\omega^{jk})_{j=0}^{p-2}$, for $k = 0, 1,
\ldots, p-2$ are eigenvectors of $\mathbf{A}$. In the base spanned
by these vectors, $\mathbf{A}$ is diagonal. The base transform
matrix having $\vec{f}_k$ as columns is a Vandermonde matrix with
discriminant $D = \prod (\omega^i - \omega^j)$, which is a power of
$p-1$ and thus a unit modulo $q$, since $p \not \equiv 1 \mod q$.
The base transform is thus regular modulo $q$ and $\Delta$ is
invertible modulo $q$ iff its conjugate matrix in the eigenvector
base is so.
We now compute the determinant of the conjugate diagonal matrix
$\rg{D}$ of $\rg{A}$. If $\chi : \ZM{p}^{\times} \rightarrow <
\omega >$ is a character with $\chi(\widehat{\sigma}) = \omega$ -
where $\sigma(\zeta) = \zeta^{\widehat{\sigma}}$ - then one verifies
that the representation of $\mathbf{A}$ in the eigenvector base is \[ \mathbf{A} \sim \mbox{ \bf{Diag} }\left(\tau'(\chi^j)\right)_{j=0}^{p-2}, \] where, for $\psi \in < \chi > $, we defined the Lagrange resolvents \[ \tau'(\psi) = \sum_{x \in \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}} \frac{\psi(x)}{1-\zeta^x} .\] Let $\tau(\psi) = \sum_{x=1}^{p-1} \psi(x) \zeta^x$ be a regular Gauss sum; a known formula (see e.g. \cite{La}) implies $\psi^{-1}(i) \tau(\psi) = \sum_{x=1}^{p-1} \psi(x) \zeta^{ix}$. By an easy calculation, $-p/(1-\zeta^x) = \sum_{i=1}^{p-1} i \cdot \zeta^{ix}$. Finally, assembling these formulae we find: \begin{eqnarray*}
-p \cdot \tau'(\psi) & = & \sum_{x \in \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}} \psi(x) \cdot \sum_{i=1}^{p-1} i \cdot \zeta^{xi} \\ & = & \sum_{i=1}^{p-1} i \cdot \left(\sum_{x \in \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}} \psi(x) \cdot \zeta^{xi}\right) = \tau(\psi) \cdot \sum_{i=1}^{p-1} i \cdot \psi^{-1}(i). \end{eqnarray*} But $B_{1,\psi^{-1}} = \frac{1}{p} \cdot \sum_{i = 1}^{p-1} i \cdot \psi^{-1}(i)$ is a generalized Bernoulli number and
$\mbox{\bf N}(B_{1,\psi^{-1}}) | h_p^-$. But since $q \not \hspace{0.25em} \mid h_{p}^{-}$, this is a unit modulo $q$. Furthermore, $\tau(\psi) \cdot \overline \tau(\psi) = p$, so $\tau(\psi)$ is a unit too. Finally $\tau'(\psi) = - \tau(\psi) \cdot B_{1,\psi^{-1}}$ is a unit modulo $q$ for all $\psi \in < \chi >$.
But \[\det(\mathbf{A})= \det \left(\mbox{ \bf{Diag} } \left(\tau'(\chi^j)\right)_{j=0}^{p-2}\right) = \prod_{j=0}^{p-2} \ \tau'(\chi^j) . \] Since all the factors have been shown to be units modulo $q$, it follows that $(\Delta, q) = \left(\det^2(\mathbf{A}),q\right) = 1$, which completes, the proof. \end{proof}
We study next the cyclotomic units of $\mathbb{L}$. Let $\delta = (1 - \zeta \xi)$ and $C_{1} = < - \ \zeta \xi > \cdot \left(\mathbb{Z}[G]^{+} \delta \right) \subset \mathbb{Z}[\zeta, \xi]^{\times}$ be the $Z[G]$ module generated by the unit $\delta$ together with the roots of unity. If $q \not \equiv 1 \mod p$ we let $C_{2} = \{1\}$; otherwise, let $C_{2} = \mathbb{Z}[G_{p}] \cdot (1-\zeta)^{(1+\jmath)(\sigma-1)}$ be the
$\mathbb{Z}[G_{p}]^{+}$ module of units of $\mathbb{K}^{+}$ generated by the cyclotomic unit $\eta = |(1-\zeta)^{\sigma-1}|^{2}$. Thus $C_{2}$ is in this case \textit{essentially} equal to the cyclotomic units of $\mathbb{K}^{+}$ \cite{Wa}, Chapter VIII; it has in fact an index $2$ in this group, which is of no importance in our context, since we are focusing on $q$ - parts of unit groups.
Note that for $q \equiv 1 \mod p$, the norm $\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{K}^{+}}(\delta) = \frac{1-\zeta^{q}}{1-\zeta} = 1$ and we always have \begin{eqnarray} \label{disjoint} C_{1} \cap C_{2} = \{1\}. \end{eqnarray}
For $q \not \equiv 1 \mod p$, the statement is trivial. Suppose now that $p | (q-1)$ and let $\varepsilon = \gamma_{1} \cdot \gamma_{2} \in C_{1} \cap C_{2}$, with $\gamma_{i} \in C_{i}$. Since $\varepsilon \in C_{1}$, $\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{K}^{+}}(\varepsilon) = 1 = \gamma_{2}^{q-1}$ and since $\gamma_{2}$ is real, it follows that $\gamma_{2} = \pm 1$. But $\varepsilon \in C_{2}$ implies, by taking norms again, that $(\varepsilon/\gamma_{2})^{q-1} = \gamma_{1}^{q-1} = 1$ and eventually $\varepsilon = \gamma_{1} \cdot \gamma_{2} = 1$, as claimed. \begin{lemma} \label{q1p} Let $p, q$ be odd primes with $q \not \hspace{0.25em} \mid h_{pq}^-$ and $p \not \equiv 1 \mod q$. If $C' = C_{1}$ for $q \not \equiv 1 \mod p$ and $C' = C_{1} \cdot C_{2}$ otherwise, then $C'$ has finite index in $E$, the group of units of $\mathbb{L}$ and $q \not \hspace{0.25em} \mid \kappa = [E : C']$. In particular, \begin{eqnarray} \label{EC} E = C' \cdot E^{q}. \end{eqnarray} \end{lemma} \begin{proof}
In the case $q \not \equiv 1 \mod p$, there are no multiplicative
dependencies in $C_{1}$ and the claims are a direct consequence of
\cite{Wa} Corollary 8.8 (note that both $C'$ and $E$ contain the
same torsion, the roots of unity of $\mathbb{L}$). Indeed, since $p \not
\equiv 1 \mod q$ and $q \not \equiv 1 \mod p$, the Euler factor in
this corollary is not vanishing and also coprime to $q$. Thus
$E/E^{q}$ and $C'/{C'}^{q}$ have the same rank and annihilators and
the subsequent claims follow from this observation, since $C_{2}$ is
trivial in this case.
We now consider the case $q \equiv 1 \mod p$, for which we apply the
Theorem 8.3 in \cite{Wa}. Note that the Ramachandra units are, up to
roots of unity and an index $4$, exactly $C' = C_{1} \cdot C_{2}$ in
this case. By the Theorem of Ramachandra, it follows that $q \not \hspace{0.25em} \mid
\kappa = [ E : C']$, which also implies \rf{EC}. \end{proof}
Next we investigate the structure of the cyclotomic units as group ring modules. For this we note that the map $\iota : \mathbb{Z}[G] \rightarrow \id{Z} = \mathbb{Z}[X, Y]/\left(X^{p-1} - 1, Y^{q-1}-1\right)$ given by $\sigma \mapsto X$ and $\tau \mapsto Y$ - where $\sigma, \tau$ are generators of $G_{p}, G_{q}$, as usual - is an isomorphism of rings. We shall consider next various restrictions of this map to subrings and quotient rings of $\mathbb{Z}[G]$, without changing the notation.
The image of $\mathbb{Z}[G]^{+}$ under this map is $\id{Z}^{+} = \id{Z}/(X^{(p-1)/2} - Y^{(q-1)/2})$, since the partial conjugations $\jmath_{p} = \jmath_{q}$ in the real subfield.
We are interested in the $q$ - parts $W' = C'/{C'}^{q}$ and the components $W_{i} = C_{i}/C_{i}^{q}$, for $i = 1, 2$. These are obviously $\mathbb{F}_{q}[G]^{+}$ - modules and as a consequence of \rf{disjoint} we also have \[ W' = W_{1} \oplus W_{2}. \] Since the $\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{Q}}$ annihilates the units, they are also $\rg{R} = \mathbb{F}_{q}[G]^{+}/ \left(\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{Q}}\right)$ - modules. We have \begin{eqnarray*} \rg{R} & \cong & \id{R} = \id{Z}^{+}/\left(q, \frac{X^{p-1} \cdot
Y^{q-1} - 1} {X \cdot Y - 1}\right) \\ & = & \mathbb{F}_{q}[X, Y]/\left(X^{p-1} - 1, Y^{q-1}-1,
X^{(p-1)/2} - Y^{(q-1)/2}, \frac{X^{p-1} \cdot Y^{q-1}-1}{X
\cdot Y - 1} \right), \end{eqnarray*} under the isomorphism $\iota$. The above isomorphism illustrates that $\rg{R}$ is a semi-simple module, which is not cyclic.
Suppose now that $q \not \equiv 1 \mod p$ so $C' = C_{1}$, a cyclic $\rg{R}$ - module. By comparing ranks in \rf{EC}, it follows in fact that $C_{1} = \rg{R} \cdot \delta$ in this case. If $q \equiv 1 \mod p$ then $\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{K}^{+}}(\delta) = 1$ yields some multiplicative dependencies in $C_{1}$. If $\rg{R}_{1} = \rg{R}/\left(\mbox{\bf N}_{\mathbb{L}^{+}/\mathbb{K}^{+}}\right) \cong \iota\left(\rg{R}\right)/(X^{(q-1)/2}-1)$, then one verifies that $W_{1} = \rg{R}_{1} \cdot \delta$ in this case; in order to keep a uniform notation, we shall also write $\rg{R}_{1} = \rg{R}$, if $q \not \equiv 1 \mod p$, so that $W_{1} = \rg{R}_{1} \cdot \delta$ in both cases.
As to $W_{2}$, by \cite{Wa}, Theorem 8.11, one simply has $W_{2} = \mathbb{F}_{q}[G_{p}]^{+} \cdot \eta$. Note that the ranks of $\rg{R}_{1}$ and $\mathbb{F}_{q}[G_{p}]^{+}$ add up to $(p-1)(q-1)/2 = \rk{W'}$. We now apply the gained structure for analyzing some particular cyclotomic units. \begin{lemma} \label{ltau}
The notations being like above, let $\delta_{1} \in C_{1}$ be a unit
which verifies $\delta_{1}^{\tau-\hat{\tau}} \in C_{1}^{q}$. Then
$\delta_{1}^{\sigma \tau -1} = \delta^{\Theta}$ for some $\Theta \in
\rg{R}_{1}$ such that \begin{eqnarray} \label{annih1} \Theta = \varepsilon_{1} \cdot \theta, \quad \hbox{with} \quad \theta \in \mathbb{F}_{q}[G_{p}]^{+}, \end{eqnarray} and $\varepsilon_{1} = \frac{1}{q-1} \sum_{b=1}^{q-1} b \cdot \tau_{b}^{-1} \in G_{q}$ is the first orthogonal idempotent of $\mathbb{F}_{q}[G_{q}]$. Furthermore, \begin{eqnarray} \label{taum1} \delta_{1} \in C_{1} \quad \hbox{and} \quad \delta_{1}^{\tau-1} \in C_{1}^{q} \quad \Rightarrow \quad \delta_{1} \cdot \mbox{\bf N}(\delta_{1}) \in C_{1}. \end{eqnarray} \end{lemma} \begin{proof}
We let $\widetilde{\delta}_{1} = \delta_{1} \mod C_{1}^{q}$ be the
image in $W_{1}$. Then the hypothesis on $\delta_{1}$ translates to
$\widetilde{\delta}_{1}^{\tau-\hat{\tau}} = 1$. If $\delta_{1} =
\delta^{\Theta_{0}}$ for some $\Theta_{0} \in \mathbb{Z}[G]^{+}$, then
$\Theta(\tau-\hat{\tau})$ lays thus in the kernel of the map
$\mathbb{Z}[G]^{+} \rightarrow \rg{R}_{1}$.
We shall have, like usual, to distinguish whether $q \equiv 1 \mod
p$ or not. In the latter, simple case, we know that $W_{1} = \rg{R}
\cdot \delta$ and the previous remark on $\Theta$ implies that \[ \iota(\Theta_{0}) \cdot (\tau-\hat{\tau}) \equiv 0 \mod \left(q,
\frac{X^{p-1} \cdot Y^{q-1}-1}{X \cdot Y - 1} \right).\] In the second case, we have \[ \iota(\Theta_{0}) \cdot (\tau-\hat{\tau}) \equiv 0 \mod \left(q,
\frac{Y^{q-1}-1}{Y - 1} \right).\] We let $\Theta = \Theta_{0} \cdot (\sigma \tau -1) $. The second generators of the ideals in the kernels of the last two congruences are images of norms and they are annihilated in $\id{Z}^{+}$ by $\iota(\sigma \tau-1)$. It follows that $\Theta \cdot (\tau-\hat{\tau}) \equiv 0 \mod q$. Let $\theta_i \in \mathbb{Z}[G_p], i = 0, 1 , 2, \ldots, q-2$ be such that \begin{eqnarray*}
\Theta & = & \sum_{n=1}^{q-1} \tau^n \cdot \theta_{n-1} \quad \hbox{and} \\ \Theta \cdot (\tau-\widehat{\tau}) & = & \sum_{n=1}^{q-1} \left(\theta_{n-2} - \widehat{\tau} \theta_{n-1} \right) \tau^{n} \equiv 0 \mod q, \end{eqnarray*} where the indices in the last sum are taken modulo $q-1$. Since ${\tau^n}$ are independent over $\mathbb{Z}[G_p]$, the sum vanishes modulo $q$ if all of the coefficients do. Thus, inductively, \[ \theta_n \equiv \left(\widehat{\tau}\right)^{-n} \cdot \theta_0, \quad n = 1, 2, \ldots, q-2. \] But then \[\Theta \equiv -\theta_0 \cdot \varepsilon_1 \mod q\mathbb{Z}[G^+], \] with $\varepsilon_1 \equiv -\sum_{n=1}^{q-1} (\tau/\widehat{\tau})^n = -\sum_{a=1}^{q-1} a \tau_a^{-1} \mod q$ being the first orthogonal idempotent of $\ZM{q}[G_q]$.
We now prove \rf{taum1}. For this we note the following decomposition in $\mathbb{Z}[G_{q}]$: $\mbox{\bf N} = \mbox{\bf N}_{\mathbb{K}'/\mathbb{Q}} = \mbox{\bf N}_{\mathbb{L}/\mathbb{K}} = (\tau-1) \cdot \Omega + (q-1)$, for some $\Omega \in \mathbb{Z}[G_{q}]$; the verification is a simple computation and is left to the reader. But then, given $\delta_{1}$ in \rf{taum1}, we have: \[ \mbox{\bf N}(\delta_{1}) = \delta_{1}^{(\tau-1) \Omega + q}/\delta_{1} \quad \hbox{and} \quad \delta_{1} \cdot \mbox{\bf N}(\delta_{1}) \in C_{1}^{q}. \] This completes the proof. \end{proof}
The main result towards the proof of the Proposition is the following: \begin{lemma} \label{unitq1} Let $p, q$ be odd primes with $q \not \hspace{0.25em} \mid h_{pq}^-$, $p \not \equiv 1 \mod q$, $\mathbb{L}= \mathbb{Q}(\zeta, \xi)$ be the \nth{pq} cyclotomic extension and $G = \mbox{ Gal }(\mathbb{L}/\mathbb{Q}) = G_p \times G_q$ with $G_p = \mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q}), G_q = \mbox{ Gal }(\mathbb{Q}(\xi)/\mathbb{Q})$. If $\varepsilon \equiv 1 \mod q \lambda$ is a unit of $\mathbb{L}$, then $\varepsilon$ is a \nth{q} power. In particular, if $\varepsilon = 1 + a q \lambda + O(q \lambda^{2})$, with $a \in \mathbb{Z}[\zeta]$, then there is a $\beta \in \mathbb{Z}[\zeta]$ such that \begin{eqnarray} \label{addh90} a \equiv \sigma_{q}(\beta) - \beta \mod q. \end{eqnarray} \end{lemma} \begin{proof}
Let $E \subset \mathbb{L}$ be the real units, $C' \subset E$ the cyclotomic
units defined above and $\varepsilon \equiv 1 + a q \lambda \mod q
\lambda^2$, for $a \in \mathbb{Z}[\zeta]$. If $\kappa = [E : C'] \in \mathbb{N}$,
then $\varepsilon^{\kappa} \in C'$ is a unit with the same type of
$\lambda$ - expansion as $\varepsilon$, since $(\kappa, q) = 1$. We
may thus assume, for simplicity, that $\varepsilon \in C'$ to start
with and thus \[ \varepsilon = \delta_{1} \cdot \delta_{2}, \quad \hbox{with} \quad \delta_{i} \in C_{i}. \]
Note that $\varepsilon^{\tau-1} = \delta' = \delta_{1}^{\tau-1} \equiv 1 \mod q \lambda$. Since $\tau(\lambda) \equiv \widehat{\tau} \cdot \lambda \mod \lambda^2$, we have \[\psi = {\delta'}^{\tau-\hat{\tau}} = \varepsilon^{(\tau-1)(\tau-
\widehat{\tau})} \equiv \frac{1 + a' \cdot q \cdot \tau(\lambda)} {1
+ a' \cdot q \cdot \widehat{\tau} \cdot \lambda} \cdot \lambda \equiv 1 \mod q \lambda^2, \] where $a' = \widehat{\tau}-1 \in \mathbb{Z}^{\times}$; thus $\psi=_q 1$ and so $\psi \in {C'}^q \cap C_{1} = {C_{1}}^{q}$ by Proposition \ref{lemmaA} and \rf{EC}.
We are thus in the context of Lemma \ref{ltau}, which implies that \[ \delta' = \delta^{\ - \varepsilon_{1} \cdot \theta} \equiv 1 \mod q \lambda.\] We now estimate the unit $\delta^{\varepsilon_{1}}$ up to $\lambda^2$ and compare the result with the above. \begin{eqnarray*} \delta & = & 1-\zeta \xi = (1-\zeta) + \zeta(1-\xi) = (1-\zeta) \cdot \left(1 - \frac{1-\xi}{1-\overline \zeta}\right), \quad \hbox{so} \\ \delta^{-\varepsilon_1} & \equiv & (1-\zeta)^{-\varepsilon_1} \cdot \left(1 -\sum_{a=1}^{q-1} a \cdot \tau_a^{-1}\left(\frac{1-\xi}{1-\overline \zeta}\right) + O(\lambda^2)\right) \\ & \equiv & (1-\zeta)^{-q(q-1)/2} \cdot \left(1+\frac{1-\xi}{1-\overline \zeta} + O(\lambda^2)\right) \ \ \mod q \mathbb{Z}[\zeta, \xi]. \end{eqnarray*}
If $\theta = \sum_{c=1}^{p-1} n_c \sigma_c$ and $A = (1-\zeta)^{-q(q-1)/2} \in \mathbb{Z}[\zeta]^{q}$, then \begin{eqnarray*} \delta^{-\varepsilon_1} & = & A \left(1 +\frac{1-\xi}{1-\overline \zeta} + O(\lambda^2)\right) \quad \hbox{and} \\ \delta' = \delta^{\Theta} & = & \delta^{-\varepsilon_1 \theta} = A^{\theta} \left(1+\frac{1-\xi}{1-\overline \zeta} + O(\lambda^2)\right)^{\theta} \\ & = & A^{\theta} \left(1 + (1-\xi) \cdot \sum_{c=1}^{p-1} \frac{n_c}{1-\zeta^{-c}} + O(\lambda)^2\right). \end{eqnarray*} Lemma \ref{linindep} implies that the sum in the last equation only vanishes modulo $q$ if all the coefficients $n_c$ vanish, so $\theta \equiv 0 \mod q$ and also $\Theta \equiv 0 \mod q$, to start with. But then $\delta_{1}^{\tau-1} = \delta' = \delta^{\Theta}$ is a \nth{q} power. If $q \equiv 1 \mod p$, then \rf{taum1} implies that $\delta_{1}$ is a \nth{q} power, and since we have already shown that $\delta_{2}$ is a \nth{q} power, we have $\varepsilon = \delta_{1} \delta_{2} \in E^{q}$. Oddly, the case $q \not \equiv 1 \mod p$ requires now more attention - this is not an intrinsic problem, but rather a consequence of the build up of the auxiliary Lemmata, where the load was taken away from the second case. For the case $q \not \equiv 1 \mod p$ we have thus, again using \rf{taum1}, that $\varepsilon = \psi \cdot \gamma^{q}$, with $\psi = \mbox{\bf N}(\delta_{1})^{-1} \in \mathbb{Z}[\zeta]$ and $\gamma \in C_{1} = C'$. If $\gamma = a + b \lambda$, with $a \in \mathbb{Z}[\zeta]$ and $b \in \mathbb{Z}[\zeta, \xi]$, then the definition of $\varepsilon$ implies that \[ \psi \equiv a^{-q} \mod q \lambda ,\] and since $x \equiv 0 \mod \lambda$ implies $x \equiv 0 \mod q$ for $x \in \mathbb{Z}[\zeta]$, it follows that $\psi =_{q} 1$ and by Proposition \ref{lemmaA} it follows that $\psi$ is a \nth{q} power.
We still have to prove \rf{addh90}. Let $\gamma^{q} = \varepsilon = 1 + a q \lambda + O(q\lambda^{2})$. Then $\gamma = 1 + b \lambda + O(\lambda^{2})$, with $b \in \mathbb{Z}[\zeta]$ and raising to the \nth{q} power we find that \[ a \equiv (b^{q} \lambda^{q-1}/q + b) \mod \lambda .\] But $\frac{\lambda^{q-1}}{q} = \prod_{i=1}^{q-1} \frac{1-\xi}{1-\xi^{i}} \equiv \prod_{i=1}^{q-1} (1/i) \equiv -1 \mod \lambda$, where the last congruence is derived from Wilson's theorem. Thus $a \equiv b^{q} - b \mod \lambda$ and since $a, b \in \mathbb{Z}[\zeta]$, it follows that $a \equiv b^{q} - b \mod q$. If $\eu{Q}$ is a prime of $\mathbb{Z}[\zeta]$ over $q$, then it is fixed by $\sigma_{q}$ and $b^{q} \equiv \sigma_{q}(b) \mod \eu{Q}$, so the previous equivalence implies $a \equiv \sigma_{q}(b) - b \mod \eu{Q}$. This holds for all primes
$\eu{Q} | (q)$ uniformly, and it follows that $a \equiv \sigma_{q}(b) - b \mod q$. This completes the proof. \end{proof}
We can now prove Proposition \ref{un1q}: \begin{proof}
By the hypothesis of the Proposition, one can write $\varepsilon = a
+ b \cdot q + q \lambda \nu$, with $a, b \in \mathbb{Z}[\zeta]$ and $\nu \in
\mathbb{L}$; then \[ \varepsilon = C \cdot (1 + q \lambda \nu'), \quad \hbox{with } \quad C = a + b \cdot q \in \mathbb{Z}[\zeta], \ \nu' = \nu/C. \] We let $\delta^{-1} = \mbox{\bf N}_{\mathbb{L}/\mathbb{K}}( \varepsilon) = C^{q-1} \cdot \mbox{\bf N}_{\mathbb{L}/\mathbb{K}} (1+ q\lambda \nu') =_q C^{-1}$ and thus $\delta =_q C \equiv \varepsilon \mod q \lambda$. Obviously, $\delta \in \mathbb{Z}[\zeta]^{\times}$ and $\varepsilon/\delta = 1+q \lambda \nu''$ is a unit verifying the hypothesis of Lemma \ref{unitq1}. The first claim follows by applying the Lemma to $\varepsilon/\delta$ (note that the claim is trivial if $\varepsilon \in \mathbb{Z}[\zeta]$).
\end{proof}
\section{An Improved Case Distinctions} In this section we derive some easy consequences from the conditions deduced in the previous one. Finally, the methods developed in this section will be sharpened in the next one, thus leading to a proof of Theorem 1. In this Theorem, the two Cases discussed above, and which depend on congruences modulo $p$, analogous to the Abel-Barlow Cases in the classical Fermat equation, split into three additional cases each, and these additional cases rely upon congruences modulo $q$. \begin{lemma} \label{tdev}
Let $p, q$ be odd primes and $x, y$ coprime integers with $x \cdot y
\not \equiv 0 \mod q$ and such that there is a $\beta \in \mathbb{Q}(\zeta)$
with \begin{eqnarray} \label{premq}
\frac{x+\zeta^{q} \cdot y}{x+\overline \zeta^{q} \cdot y} = \pm \left(\frac{\beta}{\overline \beta}\right)^{q} . \end{eqnarray} Then \begin{eqnarray} \label{ponder} -(\zeta^{q}-\overline \zeta^{q}) \varphi(t) \equiv
\sum_{k=1}^{q-1} \ \frac{t^{k}-t^{2-k}}{k} \cdot (\zeta^{k}-\overline \zeta^{k}) \mod q. \end{eqnarray} \end{lemma} \begin{proof}
A development of \rf{premq} up to the second power of $q$ yields: \[ \frac{x+\zeta^{q} \cdot y}{x+\overline \zeta^{q} \cdot y} \equiv \left(\frac{x+\zeta \cdot y}{x+\overline \zeta \cdot y}\right)^{q} \mod q \mathbb{Z}[\zeta] . \] Combining with \rf{premq} we find \[ \pm \frac{\beta}{\overline \beta} = \frac{x+\zeta \cdot
y}{x+\overline \zeta \cdot y} + q \cdot \mu, \] with $\mu \in \mathbb{Q}(\zeta)$ being a $q$ - adic integer. Raising to the power $q$, it follows that in fact \begin{eqnarray} \label{q2}
\frac{x+\zeta^{q} \cdot y}{x+\overline \zeta^{q} \cdot y} \equiv \pm \left(\frac{x+\zeta \cdot y}{x+\overline \zeta \cdot y}\right)^{q} \mod q^{2} \mathbb{Z}[\zeta] . \end{eqnarray}
We write $\varphi(a) = \frac{a^{q}-a}{q} \mod q$, for $(a,q) = 1$ and let $t \equiv -y/x \mod q^{2}$, so $-(y/x)^{q} \equiv t + q \varphi(t) \mod q^{2}$. Now \begin{eqnarray*} (x+\zeta \cdot y)^{q} & \equiv & x^{q} \cdot (1 - t\cdot \zeta)^{q}
\equiv (x+q \varphi(x)) \cdot (1-t\zeta)^{q} \\ & \equiv & (x+q \varphi(x)) \cdot \left(1-t\zeta^{q} + q
f(\zeta)\right) \mod q^{2} \quad \hbox{where} \\ f(\zeta) & = & -\zeta^{q} \cdot \varphi(t) + \sum_{k=1}^{q-1} \binom{q}{k} (-t\zeta)^{k} \equiv - \left(\zeta^{q} \cdot \varphi(t) + \sum_{k=1}^{q-1} \frac{ t^{k} \zeta^{k}}{k} \right) \mod q. \end{eqnarray*}
Writing $x+\zeta^{q} y = x(1-t\zeta^{q}) = x \cdot \alpha$ and eliminating denominators in \rf{q2} we find that \begin{eqnarray*} \alpha \cdot (x+\varphi(x)) \left(\overline \alpha + q \cdot f(\overline
\zeta)\right) & \equiv & \overline \alpha \cdot (x+\varphi(x))
\cdot \left(\alpha + q \cdot
f(\zeta) \right) \mod q^{2} \quad \hbox{and} \\ \alpha \cdot f(\overline \zeta) & \equiv & \overline \alpha \cdot
f(\zeta) \mod q . \end{eqnarray*} We let $S = \sum_{k=1}^{q-1} \frac{ t^{k} \zeta^{k}}{k}$ and regroup the terms, finding: \begin{eqnarray*} (1-t\overline \zeta^{q}) \cdot (\varphi(t) \cdot \zeta^{q} + S) & \equiv & (1-t\zeta^{q}) \cdot (\varphi(t) \cdot \zeta^{q} + \overline S) \mod q, \quad \hbox{so} \\ -(\zeta^{q}-\overline \zeta^{q})\varphi(t) & \equiv & (1-t\overline \zeta^{q}) S - (1-t \zeta^{q}) \overline S \mod q, \end{eqnarray*} and \[ -(\zeta^{q}-\overline \zeta^{q}) \varphi(t) \equiv \sum \frac{t^{k}}{k}(\zeta^{k}-\overline \zeta^{k}) - \sum \frac{t^{k+1}}{k }(\zeta^{k-q}-\overline \zeta^{k-q}) \mod q . \] We regroup the powers of $\zeta$ using $q-k \equiv -k \mod q$, thus $\zeta^{k-q}/k \equiv -\overline \zeta^{q-k}/(q-k)$, which can be applied in the above for $k = 1, 2, \ldots, q-1$: \begin{eqnarray*} -(\zeta^{q}-\overline \zeta^{q}) \varphi(t) \equiv
\sum_{k=1}^{q-1} \ \frac{t^{k}-t^{2-k}}{k} \cdot (\zeta^{k}-\overline \zeta^{k}) \mod q, \end{eqnarray*} the statement of \rf{ponder}. \end{proof}
Lemma \ref{tdev} yields essentially a system of equations modulo $q$ in the unknown $t$. It turns out that under some additional conditions on $p$ and $q$, there are only three possible values for $t$ (one of which is $t = 0$). The \textit{light } version of this condition was presented in \cite{Mi1}; it reflects the main ideas which will subsequently lead, by a more in depth study of the system \rf{ponder}, to a sharper inequality between $p$ and $q$. The light result is the following: \begin{proposition} \label{qdivs} Assume that $p > q$ are odd primes and there is a $\beta \in \mathbb{Q}(\zeta)$ such that \rf{premq} holds. Then \begin{eqnarray} \label{qdiv} x+f \cdot y \equiv 0 \mod q^{2} \quad \hbox{ for some } \quad f \in \{ -1, 0, 1 \} . \end{eqnarray} \end{proposition} \begin{proof}
Assume first that $x \equiv 0 \mod q$ and $x = q u$ with $(u, q)=1$.
Since $(x,y) = 1$ and $p \neq$, it follows that $(x+\zeta^{a} y, q)
= 1$, so the right hand side of \rf{premq} is a $q$ - adic integer.
The equation is Galois - invariant, so we can replace $\zeta$ by
$\zeta^{q}$. Thus \rf{premq} becomes \[ \frac{y+ q \overline \zeta^{q} u}{y+q \zeta^{q} u} = \gamma^{q}, \] with $\gamma = \pm \zeta^{2} \cdot \beta/\overline \beta$. Obviously the above implies $\gamma \equiv 1 \mod q$, so $\gamma^{q} \equiv 1 \mod q^{2}$ and $y+q u \zeta^{2} \equiv y+ q u \overline \zeta^{2} \mod q^{2}$, so $u \cdot (\zeta^{2}-\overline \zeta^{2}) \equiv 0 \mod q$. This is only possible if $u \equiv 0 \mod q$ and thus $x \equiv 0 \mod q^{2}$. Since we can interchange $x$ and $y$, this proves that if $x$ or $y$ is divisible by $q$, then it is divisible by $q^{2}$, which takes care of $f = 0$ in this case.
We may now assume that $x \cdot y \not \equiv 0 \mod q$ and use the previous lemma, which implies that \rf{ponder} holds under the given premises. Since the set $\{\zeta, \zeta^{2}, \ldots, \zeta^{p-1}\}$ builds a base of the algebra $\mathbb{Z}[\zeta]/(q \cdot \mathbb{Z}[\zeta])$, the coefficients of the single powers in the above identity must all vanish and $p > q+1$ implies that the coefficient of $\zeta$ is $a_{1} = t(1-t^{-4})$ and thus \[ t^{4} \equiv 1 \mod q \] must hold. Furthermore, if $q+2 < p$, then the coefficient of $\zeta^{2}$ is \[ 2 \cdot a_{2} = (t^{2}-t^{-4}) \equiv 0 \quad \hbox{hence} \quad t^{6} - 1 \equiv 0 \mod q. \] The last two congruences in $t$ have the only common solution $t^{2} = 1 \mod q$. One easily verifies that if this holds, then the right hand side in \rf{ponder} vanishes and thus $\varphi(t) \equiv 0 \mod q$. This leads to the possible solution $x \pm y \equiv 0 \mod q^{2}$; inserting the value back shows that this is indeed a solution of \rf{premq}. If $p = q+2$, then we still have $a_{1} = t^{-3}(t^{4}-1)$ so $t^{4} \equiv 1 \mod q$. If $t^{2} - 1 \equiv 0 \mod q$, we find the previous solution. So let us assume that $t^{2} \equiv -1 \mod q$ and consider the second coefficient: but $\varphi(t) \overline \zeta^{q} = \varphi(t) \zeta^{2}$ has in this case a contribution to $a_{2}$. We estimate this coefficient by using $t^{2} \equiv -1 \mod q$: \begin{eqnarray*}
2 \cdot a_{2} & \equiv & t^{2}-t^{-4} - 2 \varphi(t) \equiv -t^{-4}\left(t^{6} - t^{2}+t^{2}-1+2 t^{4} \varphi(t)\right) \\ & \equiv & t^{2} - 1 + 2\varphi(t) \equiv 2(\varphi(t) -1)\mod q , \end{eqnarray*} a congruence which is satisfied by $\varphi(t) \equiv 1 \mod q$. We have to consider also \begin{eqnarray*} 3 \cdot a_{3} & = & (t^{3} - t^{-5}) - (t^{q-1} - t^{-q-1}) \equiv 0 \mod q \quad \Leftrightarrow \\ 0 & \equiv & t^{-5}(t^{8}-1) - (1 - t^{-2}) \mod q. \end{eqnarray*} But if $t^{2} \equiv -1 \mod q$, then the first term vanishes while the second is $-2 \not \equiv 0 \mod q$, so $t^{2} \equiv -1 \mod q$ is not possible. This takes care also of the case $p = q+2$, thus completing the proof of the proposition. \end{proof}
It follows from Corollary \ref{crhodef} that \begin{corollary} \label{c0}
If $p > q > 3$ are odd primes for which \rf{FC} has non trivial
solutions and such that $q \not \hspace{0.25em} \mid h_{p}^{-}$, then \rf{qdiv} holds. \end{corollary} \begin{proof} The premises of Corollary \ref{crhodef} are given and thus \rf{rhod} holds. By setting $\beta = \rho_{1}$ in this equation, we find that the hypotheses of Proposition \ref{qdivs} also hold, and by its proof it follows that \rf{qdiv} must be true. \end{proof} \subsection{Sharpening} Let $\rg{k}$ be a field and $\id{T}$ be the space of sequences on $\rg{k}(t)$. We define the following operators on $\id{T}$: \begin{eqnarray} \label{thetadef} b_{n} & = & \theta_{+}(a_{n}) = a_{n} - t \cdot a_{n-1} \nonumber \\ c_{n} & = & \theta_{-}(a_{n}) = t \cdot a_{n} - a_{n-1} \nonumber \\ d_{n} & = & \Theta(a_{n}) = \theta_{+} (\theta_{-}(a_{n})). \nonumber \end{eqnarray} Furthermore we let $\Delta$ be, classically, the forward difference operator $\Delta \ a_{n} \ = \ a_{n} - a_{n-1}$ and $n^{\underline{k}} = n \cdot (n-1) \ldots (n-k+1)$ be the \nth{k} falling power of $n$, so $\Delta n^{\underline{k}} = k \cdot (n-1)^{\underline{k-1}}$. With, the main properties of the operators in \rf{thetadef} are given by \begin{lemma} \label{thetaprop} The operators $\theta_{+}, \theta_{-}$ are linear and they commute, thus $\Theta = \theta_{+} \circ \theta_{-} = \theta_{-} \circ \theta_{+}$. Furthermore, \begin{eqnarray} \label{op1} \begin{array}{c c c c c c c} \theta_{+}(t^{n}) & = & 0 & \hbox{ and } & \theta_{+}(t^{-n}) & = & (1-t^{2}) t^{-n}, \\ \theta_{-}(t^{-n}) & = & 0 & \hbox{ and } & \theta_{-}(t^{n}) & = & -(1-t^{2}) t^{n-1}, \end{array} \\ \label{op2} \begin{array}{c c l} \theta_{+}^{l}(n^{\underline{k}} \cdot t^{n}) & = & \frac{k!}{l!} \cdot (n-l)^{\underline{k-l}} \cdot t^{n}, \\ \theta_{-}^{l}(n^{\underline{k}} \cdot t^{-n}) & = & \frac{k!}{l!} \cdot (n-l)^{\underline{k-l}} \cdot t^{-(n-l)}, \end{array} \end{eqnarray} where we set $a^{\underline{k-l}} = 0$ if $k < l$. In particular, we have: \begin{eqnarray} \label{op3} \begin{array}{c c l} \theta_{+}^{k}(n^{\underline{k}} \cdot t^{n}) & = & k! \cdot t^{n}, \\ \theta_{-}^{k}(n^{\underline{k}} \cdot t^{-n}) & = & k! \cdot t^{-(n-k)}, \\ \Theta^{k}(n^{\underline{k}} \cdot t^{n}) & = & k! \cdot (t^{2}-1)^{k} \cdot t^{n-k}, \\ \Theta^{k}(n^{\underline{k}} \cdot t^{-n}) & = & k! \cdot (-1)^{k} \cdot (t^{2}-1)^{k} \cdot t^{-(n-k)}. \end{array} \end{eqnarray} \end{lemma} \begin{proof} Commutativity follows by a straight forward computation from \[ \theta_{+} \circ \theta_{-} (a_{n} ) = \theta_{-} \circ \theta_{+} (a_{n} ) = t \cdot (a_{n}+a_{n-2}) - (t^{2}+1) a_{n-1} .\] The rules \rf{op1} are also easily verified and they yield \rf{op2} by induction on $k$. Finally, the first two actions in \rf{op3} are obtained by setting $l = k$ in \rf{op2}, while the action of $\Theta$ is obtained due to commutativity, by setting $\Theta^{k} = \theta_{-}^{k} \circ \theta^{k}_{+}$ or $\Theta^{k} = \theta_{+}^{k} \circ \theta^{k}_{-}$, depending whether the operand is $t^{n}$ or $t^{-n}$. Note that $k+1$ consecutive values of $a_{n}$ are necessary for applying $\theta_{\pm}^{k}$, while $\Theta^{k}$ requires $2k+1$ consecutive values. \end{proof} The task we pursue is to improve our estimates on pairs $p, q$ for which the system \rf{ponder} has no other solutions except \rf{qdiv}; in particular, we are concerned with $p < q$ - since Proposition \ref{qdivs} deals already with $p > q$. We shall use the fact on which the proof of Proposition \ref{qdivs} relays: $(\zeta^{k})_{k=1}^{p-1}$ form a base of the algebra $\mathbb{Z}[\zeta]/(q \mathbb{Z}[\zeta])$ and this allows one consider \rf{ponder} as a linear system modulo $q$. Concretely, the coefficients of $\zeta^{k} - \overline \zeta^{k}$ in that equation must vanish, for $k = 1, 2, \ldots, \frac{p-1}{2}$. Let $0 < \nu < \frac{p-1}{2}$ be the value for which $\nu \equiv q \mod p$ or $\nu \equiv -q \mod p$; then, with $\delta_{ij}$ the Kronecker $\delta$, the above remark yields the equations: \begin{eqnarray} \label{lsys} -\delta_{\nu,k} \cdot \varphi(t) & \equiv & \sum_{j \geq 0; jp + k < q} \frac{t^{k+pj} - t^{2-(k+pj)}}{pj+k} \\ & - & \sum_{j \geq 0; jp + (p-k) < q} \frac{t^{p-k+pj} - t^{2-(p-k+pj)}}{p-k+jp} \mod q. \nonumber \end{eqnarray} The index value $\nu$ is singular for the equations above; first, it is the only index for which the equations are not homogeneous. Second the number of terms in the sums of the right hand side changes between $0 <k < \nu$ and $p/2 > k > \nu$. In these two intervals \rf{lsys} yields homogeneous equations which manifest in the vanishing of polynomials of fixed degree in $k$. This suggests the use of the difference operators defined above. Let $5 \leq p < q$ be primes. We shall take the approach of choosing the one of the intervals $0 < k < \nu$ or $\nu < k < p/2$, which has more elements: in these intervals \rf{lsys} translates into polynomial equations of the type $f_{q}(k; t) = 0$. Having a contiguous interval on which this equation holds, one can use the iteration of $\Theta$ in order to reduce the degree in $k$ of the polynomial $f_{q}$. We have thus to distinguish the cases $\nu < p/4$ and $\nu > p/4$ \footnote{\ One may also take the approach
of considering the whole interval $0 < k < p/2$; in this case the
polynomials $f_{q}(k; t)$ change the degree and shape when $k$
passes the "singular" value $k = \nu$. The computations become more
intricate, for a gain of a factor at most $2$. We choose to analyze
here the simpler approach.}. \begin{proposition} \label{p+} Let $5 \leq p < q$ be primes such that \rf{ponder} holds and $\nu$ be defined above. Suppose that $\nu > p /4$; if additionally, $q < \frac{p^{2}}{16}$, then \rf{qdiv} holds. \end{proposition} \begin{proof} Let $n = \left[q/p\right]$. The equation \rf{lsys} yields on the interval $0 < k < \nu$: \begin{eqnarray*} \sum_{0 \leq j \leq n} \frac{t^{k+pj} - t^{2-(k+pj)}}{pj+k} \equiv \sum_{0 \leq j < n} \frac{t^{p-k+pj} - t^{2-(p-k+pj)}}{p-k+jp} \mod q. \end{eqnarray*} After eliminating denominators, this yields a polynomial equation: \begin{eqnarray*} (-1)^{n} k^{2n} & \cdot & \sum_{0 \leq j \leq n} \left(t^{k+pj} -
t^{2-(k+pj)} \right) + O(k^{2n-1}) \equiv \\ (-1)^{n-1} k^{2n} & \cdot & \sum_{0
\leq j < n} \left( t^{p-k+pj} - t^{2-(p-k+pj)} \right) + O(k^{2n-1}). \end{eqnarray*} In order to eliminate the lower order terms in $k$, we may take $\Theta^{2n}$ on both sides of the congruence. This requires at least $2(2n)+1$ contiguous points, so $1 \leq k -2n < k+2n < p/4$, which means $2(2n)+1 < p/4$. If this is provided, the equation reduces, after simplifying by $(-1)^{n} \cdot (2n)! \cdot (1-t^{2})^{2n}$, to : \begin{eqnarray} \label{redgen1} & & \sum_{0 \leq j \leq n} \left(t^{k+pj-2n} - t^{2-(k+pj-2n)}\right) \\ & + & \sum_{0 \leq j < n} \left( t^{p-k+2n+pj} - t^{2-(p+2n-k+pj)} \right)
\equiv 0 \mod q. \nonumber
\end{eqnarray} If $t \not \in \{-1, 0, 1\}$ then we can apply $\theta_{+}$ and $\theta_{-}$ independently to the above congruence. This yields: \begin{eqnarray*} 0 & \equiv & \sum_{0 \leq j \leq n} t^{2-(k+pj-2n)} - \sum_{0 \leq j < n} t^{p-k+2n+pj} \quad \hbox{and } \\ 0 & \equiv & \sum_{0 \leq j \leq n} t^{k+pj-2n}- \sum_{0 \leq j < n}t^{2-(p+2n-k+pj)}, \end{eqnarray*} and, upon multiplication by the lowest power of $t$, \begin{eqnarray} \label{toadd} 0 & \equiv & \sum_{0 \leq j \leq n} t^{pj} - \sum_{0 \leq j < n} t^{p(n+1)-2+pj} \mod q \quad \hbox{and } \\ 0 & \equiv & \sum_{0 \leq j \leq n} t^{pn+pj-2}- \sum_{0 \leq j < n} t^{pj} \mod q . \nonumber \end{eqnarray} Adding up the two congruences, we obtain $t^{pn} \equiv -t^{pn-2} \mod q$ with the solutions $t \equiv 0$ and $t^{2} \equiv -1 \mod q$. We show that the latter solution is impossible by reinserting it in \rf{redgen1}; this yields, after simple computations, $t^{k}+t^{-k} \equiv 0 \mod q$. Since we assumed $t \not \equiv 0$, it follows that $(-1)^{k}+1 \equiv 0 \mod q$. It suffices to take $k$ even in order to reach a contradiction. Let us finally examine all the conditions on $\nu$ (and thus on $p$ and $q$), which allowed us to reach this contradiction. Adding the points necessary for the final application of $\theta_{\pm}$ together with the condition that $k$ be even, we find: \[ 2n+1 \leq k \leq p/4-(2n+1) ,\] condition which is satisfied by the even value $k = 2(n+1)$, provided that $4n+3 < p/4$. On the other hand, we find from the definition of $\nu$ and the fact that $\nu > p/4$, that $p(4n+3) > 4q$, and thus \[ p^{2}/4 > p(4n+3) > 4q, \] as claimed. \end{proof} \begin{proposition} \label{p-} Let $5 \leq p < q$ be primes such that \rf{ponder} holds and $\nu$ be defined above. Suppose that $\nu < p /4$; if additionally, $q < \frac{p(p-20)}{16}$, then \rf{qdiv} holds. \end{proposition} \begin{proof} The proof of this proposition follows the same line as the previous, but raises few particular obstructions. We shall let \[ n = \begin{cases} \lfloor q/p \rfloor & \hbox{if} \quad (q \mod p) < p/4, \\ \lfloor q/p \rfloor + 1 & \hbox{if} \quad (q \mod p) > 3 p/4. \end{cases} \]
The equation \rf{lsys} yields now on the interval $ \nu < k < p/4$: \begin{eqnarray*} \sum_{0 \leq j \leq n} \frac{t^{k+pj} - t^{2-(k+pj)}}{pj+k} \equiv \sum_{0 \leq j \leq n} \frac{t^{p-k+pj} - t^{2-(p-k+pj)}}{p-k+jp} \mod q. \end{eqnarray*} Note that there are equally many terms in the sums of both sides of the above congruences, unlike the case of the previous proposition. This perpetuates down to the analog of \rf{toadd}, in which the two congruences become identical; they both yield the condition \begin{eqnarray} \label{condcum}
t^{p(n+1)} \equiv 1 \mod q \quad \hbox{or} \quad t^{p(n+1)} \equiv
t^{2} \mod q, \end{eqnarray} whose deduction is left to the reader. Note that this condition is equivalent to applying any of $\theta_{+} \Theta^{2n+1}$ or $\theta_{-} \Theta^{2n+1}$ to the original system \rf{lsys}.
In order to draw a contradiction we shall have to consider lower order terms in $k$. Let \[ \sigma_{j} = t^{k+pj} - t^{2-(k+pj)} \quad \hbox{ and } \quad \tau_{j} = t^{p-k+pj} -t^{2-(p-k+pj)}; \] with some additional work, the first congruence yields, after elimination of denominators: \begin{eqnarray*} \sum_{0 \leq j \leq n} \sigma_{j} \cdot
\left(k^{\underline{2n+1}}-\left[(n+j+1)p -(2n+1)n\right] \cdot
k^{\underline{2n}}\right) & + & \\ \sum_{0 \leq j \leq n} \tau_{j} \cdot
\left(k^{\underline{2n+1}}-\left[(n-j)p -(2n+1)n\right] \cdot
k^{\underline{2n}}\right) & + & O(k^{2n-1}) \equiv 0 \mod q. \end{eqnarray*} We apply $\Theta^{2n}$ to the above and let \[ \sigma'_j = t^{k-2n+pj} - t^{2-(k-2n+pj)} \quad \hbox{ and } \quad \tau'_{j} = t^{p-k+2n+pj} -t^{2-2n-(p-k+pj)}. \] With this we obtain \begin{eqnarray} \label{theta2n} \sum_{j=0}^{n} & & \sigma'_{j} \left((2n+1) k - \left[(n+j+1)p
-(2n+1)n\right] \right) + \\ \sum_{j=0}^{n} & & \tau'_{j} \left((2n+1) k - \left[(n-j)p -(2n+1)n\right] \right) \equiv 0 \mod q. \nonumber \end{eqnarray} We now apply $\theta_{+}^{2}$ to the above relation; this cancels the terms in $t^{k}$ and modifies the terms in $t^{-k}$. Note that by commutativity, $\theta^{-} \theta_{+}^{2} \Theta^{2n} = \theta_{+} \Theta^{2n+1}$, which yields \rf{condcum}. But applying $\theta^{-}$ after $\theta_{+}^{2}$ to \rf{theta2n} yields to a cancellation of all but the terms in $\theta_{+}^{2} (k t^{-k})$; conversely, it is precisely these terms which are canceled if the condition \rf{condcum} holds. Since $\theta_{+}^{2} t^{-k} \not \equiv 0 \mod q$ if $t \mod q \not \in \{-1,0,1\}$, this eventually leads to the congruence: \begin{eqnarray*} \begin{array}{l c c c c} p \cdot t^{2n} \cdot \left(\sum_{j=0}^{n} \left((j+1)t^{2-pj} + j
t^{p+pj}\right)
\theta_{+}^{2}(t^{-k}) \right) & \equiv & 0 & \mod q, & \hbox{so} \\ \sum_{j=0}^{n} \left((j+1)t^{2-pj} + j t^{p+pj}\right) & \equiv & 0 &
\mod q, & \hbox{and} \\ t^{2-pn} (n+1) \cdot \sum_{j=0}^{n} t^{pj} + t^{p} \cdot
(1-t^{2-p(n+1)}) \cdot \sum_{j=0}^{n} j t^{pj} & \equiv & 0 & \mod
q. & \end{array} \end{eqnarray*} We can now reintroduce the alternative \rf{condcum} in the last congruence above. If $t^{p(n+1)} \equiv 1 \mod q$, then the first sum $\sum_{j=0}^{n} t^{pj}$ vanishes; furthermore, since $(n,q) = 1$, one easily verifies that $\sum_{j=0}^{n} t^{pj} \equiv \sum_{j=0}^{n} j t^{pj} \equiv 0 \mod q$ cannot simultaneously hold, and thus it follows that $t^{2-p(n+1)} \equiv 1$. Since we also assumed $t^{(n+1)p} \equiv 1 \mod q$, it follows that $t^{2} \equiv 1 \mod q$, as required. Suppose now that in \rf{condcum} it is the condition $t^{p(n+1)} \equiv t^{2} \mod q$ which holds; by inserting this in the last congruence above, we find (since $t(n+1) \not \equiv 0 \mod q$) that $\sum_{j=0}^{n} t^{pj} \equiv 0 \mod q$ and we are in the previous case. Both ways, it follows that $t \mod q \in \{-1, 0, 1\}$.
We finally have to derive the inequality between $p$ and $q$, for which the proof above holds. The condition is that the interval $(p/4, p/2)$ contains sufficient contiguous points for applying both $\theta_{\pm} \Theta^{2n+1}$ and $\theta_{+}^{2} \Theta^{2n}$; i.e. $4n+5 < p/4$. Note that by definition of $n$, we always have $np > q$ and thus the previous inequality amounts to $4q < 4np < p(p/4-5)$ and thus \[ \frac{p(p-20)}{16} > q. \] This completes the proof of the proposition. \end{proof} \subsection{Proof of Theorem \ref{main}} The statement of Theorem \ref{main} follows directly from Corollary \ref{c0} together with the sharpening Propositions \ref{p+} and \ref{p-}.
\subsection{The Resulting Case Analysis} We suppose that the Fermat - Catalan equation \rf{FC} has a solution for odd primes $p, q$ with $p \not \equiv 1 \mod q$, $q \not \hspace{0.25em} \mid h_{p}^{-}$ and $\max \{p, \frac{p(p-20)}{16}\} > q$. Then Theorem \ref{main} holds and we are reduced to investigate the case $e = 0$ (Case I) or $e = 1$ (Case II) each with three subcases: $f = -1$ (case a), $f = 0$ (case b) and $f = 1$ (case c); together, this yields the Table 1, with six cases.
\begin{table} \caption{Cases of Fermat - Catalan}
\begin{tabular} {|c|c|c|c|} \hline & {\bf a ( f = -1)}& {\bf b (f = 0)} & {\bf c (f = 1)}
\\[2pt] \hline {\bf I (e = 0)} & I a & I b & I c \\ {\bf II (e = 1)} & II a & II b & II c \\[2pt] \hline \end{tabular} \end{table} Furthermore, if either $q \not \hspace{0.25em} \mid h(p, q)$ or $q \not \hspace{0.25em} \mid h_{pq}^-$, Corollary \ref{crhodef} holds and in particular the identity \rf{rhod}. We aim next to eliminate the unit $\varepsilon$ in this identity, using the fact that, by Proposition \ref{lemmaA}, the $q$ - primary units of $\mathbb{Q}(\zeta)$ are global \nth{q} powers. This shall be done by a case by case study. In view of Lemma \ref{lemmaC}, we let $0 < m < q$ with $m(p-1) \equiv 1 \mod q$ and $\gamma = \left(
\frac{(1-\zeta)^{p-1}}{p}\right)^{m}$ be the unit in \rf{uc}.
Suppose first that $e = 0$ and thus $\alpha = x+y \zeta$. In case a, $x \equiv y \mod q^{2}$ and \[ \varepsilon \cdot \rho^{q} = \alpha \equiv x (1+\zeta) \mod q^{2} .\] But then $\delta = \frac{\varepsilon}{1+\zeta} =_{q} x$ and by Lemma \ref{unitp} it follows that $\delta \in \mathbb{Z}[\zeta]^{q}$; consequently $\alpha = (1+\zeta) \rho^{q}$ in this case. If $f = 0$ (case b)), then $x \equiv 0 \mod q^{2}$ and $\varepsilon =_{q} y$ and Lemma \ref{unitp} shows that $\varepsilon$ is a \nth{q} power, so $\alpha = \rho^{q}$ in this case. Finally, if $f = 1$, then $\varepsilon =_{q} -y(1-\zeta)$. Since $\delta = \varepsilon : \gamma =_{q} y$, the Lemma \ref{lemmaC} implies that $\delta$ is a \nth{q} power. It follows that $\alpha = \gamma \cdot \rho^{q}$ in this case.
We assume next that $e = 1$ and consider the three possible values of $f$. In case a, $\alpha = \frac{x+\zeta y}{1-\zeta} \equiv \frac{x(1+\zeta)}{1-\zeta}$ and combining the Lemmata \ref{unitp} and \ref{lemmaC} we find that $\alpha = (1+\zeta) / \gamma \rho^{q}$. Likewise, $\alpha = \rho^{q}/\gamma$ in case b) and $\alpha = \rho^{q}$ in case c.
\begin{table} \caption{Values of the unit $\varepsilon$ in the six Cases}
\begin{tabular} {|c|c|c|c|} \hline & {\bf a }& {\bf b } & {\bf c }
\\[2pt] \hline {\bf I } & $(1+\zeta)$ & $1$ & $\gamma$ \\ {\bf II } & $\frac{1+\zeta}{\gamma} $ & $1/\gamma$ & $1$ \\[2pt] \hline \end{tabular} \end{table} We combine all these results in Table 2 and the following \begin{proposition} \label{propqpow} Let $p, q$ be odd primes with $p \not \equiv 1 \mod q$, $q \not \hspace{0.25em} \mid h(p, q)$ and suppose that $\max \{p, \frac{p(p-20)}{16}\} > q$ and the Fermat - Catalan equation \rf{FC} has a non trivial solution. Furthermore, let $0 < m < q$ be an integer with $m(p-1) \equiv 1 \mod p$ and $\gamma = \left( \frac{(1-\zeta)^{p-1}}{p}\right)^{m} \in \mathbb{Z}[\zeta]^{\times}$. Then for $e \in \{0, 1\}$ and $f \in \{-1, 0, 1\}$ like in Table 1, the following identity holds (with $\delta_{a,b}$ being the Kronecker $\delta$ symbol): \begin{eqnarray} \label{sixqpow} \alpha = \frac{x+\zeta y}{(1-\zeta)^{e}} = (1+\zeta)^{\delta_{f,-1}} \cdot \gamma^{\delta_{f,1}-e} \cdot \rho^{q}, \quad \hbox{with} \quad \rho \in \mathbb{Q}(\zeta). \label{qpowcomb} \end{eqnarray} \end{proposition}
We proceed with a case by case analysis of possible solutions in the above six cases. The results come in different levels of complexity and require different class number conditions - essentially the two possibilities $q \not \hspace{0.25em} \mid h(p,q)$ or $q | h_{pq}^-$, mentioned above.
The simplest fact is that in three out of six cases, one deduces some Wieferich - type local conditions, involving only the exponents $p$
and $q$. This is the topic of the next section. In the following section, keeping the same class number condition, we show that one can give lower bounds on $\max\{|x|, |y|\}$: this is Theorem \ref{main1}. We sharpen subsequently the class number condition to $q \not \hspace{0.25em} \mid h_{pq}^{-}$ and prove, by a generalization of Kummer descent - as used by Kummer in his Theorem on the Second Case of Fermat's Last Theorem, \cite{Wa} - that two additional cases ($f = -1$) are impossible. This leaves on last case - which we called the \textit{Ast\'erisque} - Case
- untreated by conditions involving only the exponents $p, q$. By using the sharper class number condition, we are able to improve the lower bound in this case to one on the \textit{minimum} $\min\{|x|,
|y|\}$; this is Theorem \ref{main2}, which is a first generalization of Catalan's conjecture. Finally, by applying this Theorem together with an additional consequence of the Kummer descent, we prove the Theorem \ref{trc} on the rational case of Catalan's equation, which is the most exhaustive result of this paper, since it shows the lack of solutions of Catalan's equation in the rationals, provided some conditions hold, which are related only to the exponents.
\section{The Wieferich Cases} We assume in this section that \rf{FC} has non-trivial solutions for odd prime exponents $p, q$ for which the premises of Theorem \ref{main} hold. Based on this theorem, we can thus assume that the solutions are in one of the cases given in the above tables. The three simplest cases lead to some Wieferich - type (see \cite{Ri}) condition. \begin{proposition} \label{p1} Notations being as above, if $e = 0$ and $f = -1$, then \[ 2^{q-1} \equiv 1 \mod q^2. \] Furthermore, $X =_{q} Y =_{q} 1$. \end{proposition} \begin{proof}
Since $e = 0$, we are in Case I and $X+Y = A^q$; also, $f = -1$
means $X-Y \equiv 0 \mod q^2$, so $X =_q Y$ and $X+Y =_q 2X =_q A^q
=_q 1$. But from \rf{FC}, $X^p + Y^p = Z^q =_q 2X^p =_q 1$. Dividing
the last two relations, we find $X^{p-1} =_q 1$ and since $q \not \hspace{0.25em} \mid
p-1$ by hypothesis, it follows also that $X =_q 1$. Combined with
$2X =_q = 1$ this yields the statement of the proposition. Since $X
=_{q} Y$ by definition of this case and $2X =_{q} 2 =_{q} 1$, the
second statement follows too. \end{proof} \begin{proposition} \label{p2} Notations being as above, if $e = 1$ and $f = 0$, then \[ p^{q-1} \equiv 1 \mod q^2 \] and $Y =_{q} 1$. \end{proposition} \begin{proof} Since $e = 1$, we are in Case II and $\frac{X^p+Y^p}{X+Y} = p \cdot B^q =_q p$; also, $f = 0$ means $X \equiv 0 \mod q^2$, so \[ \frac{X^p+Y^p}{X+Y} =_q Y^{p-1} =_q p. \] But $X^p +Y^p = Z^q =_q Y^p =_q 1$ and since $(p,q) = 1$, we must have $Y =_q 1$, which is the second statement of the Proposition. Combined with the previous equivalence, this yields $Y =_q p =_q 1$, which leads to the first claim. \end{proof} The third Wieferich case has a more complex statement. This is: \begin{proposition} \label{p3} Notations being as above, if $e = 1$ and $f = -1$, then \[ \left(2^{p-1} \cdot p^{p}\right)^{q-1} \equiv 1 \mod q^2 \] and $X =_{q} Y =_{q} p^{m}$, where $m (p-1) \equiv 1 \mod q$. \end{proposition} \begin{proof}
Since $e = 1$, we are still in Case II, so $\frac{X^p+Y^p}{X+Y} = p
\cdot B^q =_q p$; also, $f = -1$ implies $X =_{q} Y$ and
$\frac{X^p+Y^p}{X+Y}=_{q} X^{p-1} =_{q} p$, the second claim of the
Proposition. Furthermore, $X^{p}+Y^{p} =_{q} 2X^{p} =_{q} z^{q}
=_{q} 1$. Raising this to the power $p-1$ and the preceding
equivalence to the power $p$, we find after division that: \[ 2^{p-1} X^{p(p-1)} =_{q} 2^{p-1} p^{p} =_{q} 1. \] This is the first statement of the Proposition and completes the proof. \end{proof}
\section{Lower Bounds and Proof of Theorem \ref{main1}} We assume in this section that \rf{FC} has non trivial solutions for odd primes $p, q$ with $p \not \equiv 1 \mod q$, $q < \max\{\frac{p(p-20)}{16}\}$ and such that $q \not \hspace{0.25em} \mid h(p, q)$. The purpose of this section is to prove Theorem \ref{main1}.
The following $q$ - adic expansion will serve for gaining estimates in all six cases under investigation. \begin{lemma} \label{vanish} Let $\rho \in \id{O}(\mathbb{L})^{\times}$ be an algebraic integer with $q$ - adic expansion \[ \rho = a \cdot \sum_{m=0}^{\infty} \binom{1/q}{m} (\mu b)^{m}, \quad \hbox{with} \quad \mu \in \{\zeta, 1/(1\pm \zeta)\}, \quad a \in
\mathbb{Z}_{q}, \ b \in \mathbb{Q}^{\times}, \ v_{q}(b) = k \geq 2. \] Furthermore, suppose there is a real number such that $|\sigma(\rho)| \geq M > 0$ for all $\sigma \in G_{p}$. Then \begin{eqnarray} \label{lbgen} M \geq \frac{1}{p-1} \cdot \frac{q^{(p-2)\left(k-\frac{q}{q-1}\right)}}{2^{p-2} } . \end{eqnarray} \end{lemma} \begin{proof} Let \[ \nu = (\zeta^{2} - \zeta) \cdot \begin{cases} 1 & \hbox{if} \quad \mu = \zeta \\ \mu^{-(p-3)} & \hbox{otherwise}. \end{cases} \] It is an easy verification, that $\mbox{\bf Tr}_{\mathbb{K}/\mathbb{Q}} \left(\nu \cdot
\mu^{i}\right) = 0$ for $j = 0, 1, \ldots, p-3$. The case $\mu = \zeta$ is trivial, since $\mbox{\bf Tr}\left(\zeta^{i+2}-\zeta^{i+1}\right) = (-1)-(-1) = 0$ for $0 \leq i \leq p-2$. If $\mu = 1/(1 \pm \zeta)$, then \begin{eqnarray*}
\mbox{\bf Tr}\left(\nu \cdot \mu^{j}\right) & = & \mbox{\bf Tr}\left((\zeta^{2}-\zeta) \cdot
(1\pm\zeta)^{p-3-j}\right) \\ & = & \sum_{i=0}^{p-3-j} (\pm 1)^{i}
\binom{p-3-j}{i} \cdot \mbox{\bf Tr}\left((\zeta^{2}-\zeta) \zeta^{i}\right)
= 0, \end{eqnarray*} as claimed. Let now $\delta = \nu \cdot \rho \in \id{O}(\mathbb{K})^{\times}$. The $q$ - adic expansion of $\rho$ together with the above remark on the trace of $\nu \cdot \mu^{j}$ shows that the first $p-2$ terms in the $q$ - adic expansion of $\Delta = \mbox{\bf Tr}_{\mathbb{K}/\mathbb{Q}}(\delta) \in \mathbb{Z}$ vanish. Thus \begin{eqnarray} \label{Delta} \Delta = \binom{1/q}{p-2} b^{p-2} \cdot \left(\mbox{\bf Tr}
\left(\mu^{p-2}\right)+O(q)\right). \end{eqnarray} Note that \begin{eqnarray} \label{binomvq} v_{q}\left(\binom{1/q}{n}\right) = v_{q}\left(\frac{1}{q^{n}} \cdot
\frac{(1-q) \cdot \ldots \cdot (1-(n-1)q)}{n!}\right) = -n-v_{q}(n!), \end{eqnarray} and since $v_{q}(b) \geq k$, it follows that \begin{eqnarray*}
v_{q}\left(\binom{1/q}{p-2} b^{p-2} \right) & \geq & k(p-2) - (p-2) - v_{q} \left((p-2)!\right) \\ & > & (p-2)(k-1-1/(q-1)) = (p-2)\left(k-\frac{q}{q-1}\right) . \end{eqnarray*} We now show that $\Delta \neq 0$. Assume first that $\mu = \zeta$. Then from \rf{Delta} we find that $\Delta = \binom{1/q}{p-2} b^{p-2} \cdot (p + o(q))$. Since $b \neq 0$ the first $q$ - adic term is non vanishing, and so $\Delta \neq 0$. The proof is similar for $\mu = 1 \pm \zeta$. Finally, $\Delta$ is a rational integer and by assembling all the information, we find:
\[ q^{(p-2)\left(k-\frac{q}{q-1}\right)} \leq |\Delta| \leq
\sum_{\sigma \in G_{p}} | \sigma(\nu \cdot \rho) | \leq 2^{p-2} \cdot (p-1) \cdot M. \] The claim follows from these inequalities. \end{proof}
The proof of Theorem \ref{main1} follows now from the Lemma and the fact that $x+f y \equiv 0 \mod q^{2}$. \begin{proof} We shall show in a case by case analysis, that a $q$ - adic expansion such as required by Lemma \ref{vanish} exists. If $f = 1$, then \rf{sixqpow} and \rf{uc} yield: \[ \frac{x+\zeta y}{(1-\zeta)^{e}} =_{q} \frac{1+\zeta}{\gamma^{e} } = p^{(m-q)e} \frac{1+\zeta}{(1-\zeta)^{e} } \cdot \rho^{q} , \] and thus $\rho \in \mathbb{Z}[\zeta]$ with \[ \rho^{q} = p^{(q-m)e} \cdot y \left(1 + \frac{x-y}{y(1+\zeta)}\right) . \] If $e = 0$, the expansion follows by Proposition \ref{p1}, since the leading term is $y =_{q} 1$; if $e = 1$ the leading term is $p^{(q-m)} y =_{q} 1$, by Proposition \ref{p3}. We still have to deduce the bound $M$ from the expression of $\rho$. But \begin{eqnarray*}
\left| \rho \right| & = & \left| p^{(q-m)e} \cdot \frac{x+\zeta
y}{1+\zeta} \right|^{1/q} \leq p^{(1-m/q) e} \cdot
\left(\frac{|x|+|y|}{1/p}\right)^{1/q} \\
& \leq & \left(2p^{1+(q-m) e} \cdot \max\{|x|,|y|\}\right)^{1/q} . \end{eqnarray*} the last estimate is obviously Galois invariant, so we can replace $M$ in the Lemma \ref{vanish} by this value. It follows that \[ M = \left(2p^{1+(q-m) e}
\cdot \max\{|x|,|y|\}\right)^{1/q} \geq \frac{1}{p-1} \cdot \frac{q^{(p-2)\left(\frac{q-2}{q-1}\right)}}{2^{p-2} } , \] and \begin{eqnarray*}
\max\{|x|,|y|\} \geq \frac{1}{2} \cdot \left(\frac{1}{p(p-1)} \cdot \left(\frac{q^{\frac{q-2}{q-1}}}{2 }\right)^{p-2}\right)^{q} \end{eqnarray*} for both cases, as claimed in the first inequality of \rf{lowb}.
Let now $f = 0$ and $y \equiv 0 \mod q$, to fix the ideas. Then $\frac{x+\zeta y}{(1-\zeta)^{e}} = \gamma^{-e} \rho$ and $\rho^{q} = p^{(q-m)e} (x+\zeta y)$. If $e = 1$, the leading term is $p^{(q-m)e} x =_{q} 1$ by Proposition \ref{p2}, otherwise, $A^{q} = x+y =_{q} x =_{q} 1$, so the expansion of $\rho$ follows in both cases. Furthermore,
\[ \left| \rho \right| = \left| p^{(q-m)e} \cdot (x+\zeta
y) \right|^{1/q} \leq \left(p^{(q-m) e} \cdot 2 \cdot \max\{|x|,
|y|\}\right)^{1/q} = M .\] Like before, by applying the Lemma \ref{vanish}, we find that \rf{lowb} holds.
Finally, if $f = 1$, some usual computations yield \[ \rho^{q} = \frac{x+\zeta y}{1-\zeta} \cdot p^{me} = -y \cdot p^{me} \cdot \left(1 - \frac{x+y}{1-\zeta}\right). \] It follows immediately from the fact that the parenthesis on the right hand side of the last identity is a $q$ - adic \nth{q} power (since $x+y \equiv 0 \mod q^{2}$) that so must then be the cofactor $yp^{me}$; this shows the existence of the $q$ - adic expansion of $\rho$ required by Lemma \ref{vanish}. The details for the estimation of $M$ are analogous to the case $f = -1$ and are left to the reader. \end{proof}
\section{Kummer Descent} We shall prove in this section the following main Theorem, which generalizes Kummer's descent method to the present context. \begin{theorem} \label{tdesc} Let $p, q > 3$ be primes such that $-1 \in <p \mod q>$, $\zeta, \xi \in \mathbb{C}$ be respectively \nth{p} and \nth{q} primitive roots of unity, $\mathbb{L} = \mathbb{Q}(\zeta, \xi)$ and $\mathbb{L}^{++}$ the fixed field of the partial complex conjugations $\jmath_p, \jmath_q$. Suppose that the equation \begin{equation} \label{desc}
X^{q} + Y^{q} = \varepsilon \cdot \lambda^{N} \cdot {\lambda'}^{M} \cdot Z^{q} \end{equation} admits solutions with $X, Y, Z \in \id{O}\left( \mathbb{L}^{++}\right), X \cdot Y \cdot Z \neq 0$ and $(X \cdot Y \cdot Z, p \cdot q) = (1)$ and $X, Y$ are not units. Here $\lambda = (\xi-\overline \xi), \lambda' = (\zeta - \overline \zeta)$ and $\varepsilon \in \id{O}\left(
\mathbb{L}^{++}\right)^{\times}$; $M, N$ are integers with $N > 2q, N$ even and $M = 0$ or $M \geq 2$. Then $Z$ is not a unit and $q | h_{pq}^-$. \end{theorem}
The next Lemma will explain the condition $-1 \in <p \mod q>$: \begin{lemma} \label{realp} Let $p, q$ be odd primes and $\mathbb{K}' = \mathbb{Q}(\xi)$ be the \nth{q} cyclotomic extension. Then $p$ splits in $\mathbb{K}'$ in real prime ideals iff $-1 \in <p \mod q>$. \end{lemma} \begin{proof}
This is a direct consequence of Kummer's Theorem on the splitting of
primes in extensions with a power base for the ring of algebraic
integers \cite{La}. Let $\Phi_{q}(X) = \prod_{i=1}^{q-1}
(X-\xi^{i})$ be the \nth{q} cyclotomic polynomial, $n = {\rm ord}_{q}(p)
= |< p \mod q>| $ and $F(X) = \prod_{j=0}^{n-1} (X-\xi^{p^j}) \in
\mathbb{Z}[\xi][X]$. If $\rg{k} = \mathbb{F}_{p^{n}}$ is the finite field with
$p^{n}$ elements, then $\rg{k}$ is the smallest field of
characteristic $p$ which contains a non trivial \nth{q} root of
unity. Let $\rho \in \rg{k}$ be such a root of unity. Then there is
a natural map $\iota : \id{O}(\mathbb{K}') \rightarrow \rg{k}$ given by $\xi
\mapsto \rho$. Let then $\tilde{f}(X) = \iota(F(X)) \in \mathbb{F}_{p}(X)$
and $f \in \mathbb{Z}[X]$ be some polynomial with $\tilde{f} = f \mod p$.
Then $\tilde{f} \in \mathbb{F}_{p}[X]$ is an irreducible factor of $\Phi(X)
\mod p $; if $\eu{p} = (f(\xi), p)$, then $\eu{p}$ is a prime above
$(p)$ and each prime above $(p)$ arises in this way, by a choice of
$\rho \in \rg{k}$. In particular, $\xi \mod \eu{p} = \rho$ and
$\iota$ is in fact the reduction $\mod \eu{p}$ map. Furthermore,
$\tilde{f} = \iota\left(F(X)\right) = f(X) \mod p$, where in general
only the second polynomial has rational integer coefficients.
After this exposition of Kummer's Theorem, we can proceed with the
proof of our Lemma. First note that since $\mathbb{K}'/\mathbb{Q}$ is a CM Galois
extension, all the primes above $(p)$ are simultaneously real or not
real. Let us first suppose that $\eu{p} = (f(\xi), p)$ is a real
ideal. Since $\tilde{f}(X) = F(X) \mod \eu{p}$ and $\overline
{\eu{p}} = \eu{p}$, it follows that $F(X) = \overline {F(X)}$ and
under the action of $\iota$, \[ \tilde{f}(X) = \prod_{i=0}^{n-1} \left(X-\rho^{p^{i}}\right) = \prod_{i=0}^{n-1} \left(X-\rho^{-p^{i}}\right)\] But $\rho \in \rg{k}$, which is a field in which $\tilde{f}(X)$ has unique decomposition. Thus $\rho^{-1} \in \{ \rho^{p^{i}} : i = 0, 1, 2 \ldots, n-1\} $ and by the definition of $n$ it follows that $-1 \in < p\mod q>$, as claimed. Conversely, if $-1 \in <p \mod q>$, then $F(X) = \overline {F(X)}$ and it follows that $\eu{p}$ is invariant under complex conjugation. \end{proof} We proceed with the proof of the Theorem, assume that $q \not \hspace{0.25em} \mid h_{pq}^{-}$ holds under the given hypotheses and will derive a contradiction. The two statements of the theorem are apparently contradictory: if we show that $q \not \hspace{0.25em} \mid h_{pq}^-$ is impossible, then it is irrelevant whether $Z$ is a unit or not. For technical reasons, however, it will be useful to show that under the given premises, if $X, Y$ are not units, then neither is $Z$. Note that by Proposition \ref{lemmaA}, it follows that $E_{q} = E^{q}$ and $\id{C} = \id{C}^{q}$, with $\id{C}$ the ideal class group of $\mathbb{L}$. The quite lengthy proof is a straightforward adaption of the descent method used by Kummer in the proof of his fundamental Theorem on the Second Case of Fermat's Last Theorem (see \cite{Wa}, Chapter 9). The additional problems are linked to the fact that we work in a larger field.
We start with a simple fact: \begin{lemma} \label{snorm} Let $X, Y \in \mathbb{L}^{++}$ verify the premises of the theorem; in particular, suppose that $X^q+Y^q \equiv 0 \mod \lambda^N \cdot {\lambda'}^M$ and $v_{\lambda}(X^{q}+Y^{q}) = N, v_{\lambda'}(X^{q}+Y^{q}) = M$. Then $v_{\lambda}(X+Y) = N-(q-1)$ and $v_{\lambda'}(X+Y) = M$. \end{lemma} \begin{proof}
Any integer $\gamma \in \id{O}(\mathbb{L})$ has the $\lambda$ - development \[ \gamma = \sum_{i=0}^G g_i \cdot \lambda^i, \quad \hbox{for some } \quad G \in \mathbb{N}, \] and \[ g_i = \sum_{j=1}^{p-1} g_{i,j} \zeta^j\in \mathbb{Z}[\zeta], \quad 0 \leq g_{i,j} < q .\] Let $X = x_0 + x_1 \cdot \lambda + O(\lambda^2)$. Since $\jmath_q(\lambda) = -\lambda$ and $X \in \mathbb{L}^{++}$, so $\jmath_q(X) = X$, we must have $x_1 = 0$, so $X = x_0 + O(\lambda^2)$. Likewise, $Y = y_0 + O(\lambda^2)$. Thus \rf{desc} implies that $x_0^q+y_0^q \equiv 0 \mod \lambda^2$ and since $(X \cdot Y, q) = 1$, we have $(x_0/y_0)^q \equiv -1 \mod \lambda^2$. Since $\mathbb{Z}[\zeta]/(q \mathbb{Z}[\zeta])$ contains no \nth{q} roots of unity except $1$, it follows that $x_0/y_0 \equiv -1 \mod \lambda^2$ and $x_0 + y_0 \equiv X+Y \equiv 0 \mod \lambda^2$. The algebraic integers \begin{eqnarray} \label{phis} \phi'_{i} & = & \xi^{i} X + \overline \xi^{i}
Y \in \mathbb{L}, \quad \hbox{for} \quad i=1, 2, \ldots, q-1 \end{eqnarray} have the common divisor \[ (\phi'_{i}, \phi'_{j}) = \left((\xi^{i}-\xi^{j}) \cdot Y,
(\overline \xi^{i}-\overline \xi^{j}) \cdot X\right) = (\lambda) . \] But $(X+Y) \cdot \prod_{i=1}^{q-1} \phi'_{i} = X^{q}+Y^{q} \equiv 0 \mod \lambda^{N}$. Thus $v_{\lambda}(X+Y) = N-(q-1)$, as claimed.
Due to $(\phi'_{i}, \phi'_{j}) = (\lambda)$, it follows also that if
$\eu{P} | (\lambda')$ is a prime ideal of $\mathbb{L}^{++}$ with $\eu{P} |
(X+Y)$, then $\eu{P}^{M} | (X+Y)$. The primes above $(p)$ in $\eu{\mathbb{L}}$ are $(\eu{p}, (1-\zeta))$ for some prime $\eu{p} \in \mathbb{K}'$ and the hypothesis together with Lemma \ref{realp} imply that they are real primes. Suppose that $M > 0$ (there is nothing to prove for $M =
0$!); then for $\eu{P} | (p)$ we have $X^{q}+Y^{q} \equiv 0 \mod \eu{P}$ and $(-X/Y)^{q} \equiv 1 \mod \eu{P}$. Since $\eu{P}$ is a prime ideal, $\id{O}(\mathbb{L}^{++})/\eu{P}$ is a field and there is an integer $0 \leq a < q$ such that $-X/Y \equiv \xi^{a} \mod \eu{P}$. Taking complex conjugates - under consideration of the fact that $X/Y$ is invariant under conjugation - we also have $-X/Y \equiv \xi^{-a} \mod \overline {\eu{P}}$. But since $\eu{P} = \overline {\eu{P}}$, it follows that $\xi^{2a} \equiv 1 \mod \eu{P}$ and $a = 0$. This holds for all primes above $(p)$ and together with the previous remark implies that $v_{\lambda'}(X+Y) = M$, as claimed. \end{proof}
We wish to normalize the algebraic integers defined in \rf{phis}, eliminating all primes above $q$ and $p$. Using the result of the Lemma \ref{snorm}, this can be done as follows: \begin{eqnarray} \label{phis1} \phi_{i} & = & \frac{\xi^{i} X + \overline \xi^{i}
Y}{\xi^{i}-\overline \xi^{i}} , \quad \hbox{for} \quad i=1, 2,
\ldots, q-1, \\ \phi_{0} & = & \frac{q(X+Y)}{\varepsilon \cdot
\lambda^{N} \cdot {\lambda'}^M}. \end{eqnarray} It follows from the Lemma \ref{snorm} that $(\phi_{i}, p \cdot q) = (1)$ for $i=0, 1, \ldots, q-1$ and \begin{eqnarray*} \prod_{i=0}^{q-1} \phi_{i} & = & \frac{1}{q} \cdot \phi_{0} \cdot \prod_{i=1}^{q-1} \phi'_{i} = \frac{q}{q \cdot \varepsilon \cdot \lambda^{N}\cdot {\lambda'}^M} \cdot (X+Y) \cdot \prod_{i=1}^{q-1} \phi'_{i} = \frac{X^{q}+Y^{q}}{\varepsilon \cdot \lambda^{N}\cdot {\lambda'}^M}. \end{eqnarray*} Finally, this yields \begin{eqnarray} \label{prod2} \prod_{i=0}^{q-1} \ \phi_{i} = Z^{q}. \end{eqnarray} If $\mathbb{L}_p \subset \mathbb{L}$ is the subfield fixed by $\jmath_p$, the definition of $\phi_i$ implies $\phi_i \in \mathbb{L}_p$ for $i > 0$ and \rf{prod2} shows that this holds also for $\phi_0$: \begin{eqnarray} \label{inkp} \phi_i \in \mathbb{L}_p \quad i = 0, 1, \ldots, q-1. \end{eqnarray} From $(\phi'_{i}, \phi'_{j}) = (\lambda)$ and the definition of $\phi_{i}$ we deduce that \begin{eqnarray} \label{coprim} (\phi_{i}, \phi_{j}) = 1 \quad \hbox{for} \quad i, j \in \{0, 1, \ldots, q-1\} , \quad \hbox{and} \quad i \neq j. \end{eqnarray}
We want to show that $\phi_i$ are not units. This implies that $Z$ is not a unit, as a consequence of \rf{prod2}. For $\phi_0$, this fact will follow indirectly, with more work. We prove it first only for $i > 0$ and investigate the $q$ - expansion of $\phi_i$: \begin{eqnarray} \label{modq}
\phi_{i} & = & \frac{\xi^{i} X + \xi^{-i} Y}{\xi^{i}-\xi^{-i}} = \frac{\xi^{i}(X+Y)-(\xi^{i}-\xi^{-i})\cdot Y}{\xi^{i}-\xi^{-i}} \\ & = & -Y + \frac{\xi^{i}(X+Y)}{\xi^{i}-\xi^{-i}} =_{q} -Y \nonumber. \end{eqnarray} Note that $\phi_{q-i} = -\overline \phi_i$. If $\phi_i$ is a unit, then $\delta = \phi_i/\phi_{q-i} = \phi_i/\overline \phi_i$ is a root of unity and by \rf{modq}, since $Y \in \mathbb{R}$, it follows that $\delta =_q 1$. By Lemma \ref{lru} it follows that $\delta = \pm \zeta^a$ for some $a \in \ZM{p}$. Then $\zeta^{-a/2} \phi_i = \pm \zeta^{a/2} \phi_{q-i}$ and a short computation shows that \[ X = - Y \cdot \frac{\overline \zeta^{a/2} \cdot \overline \xi \mp \zeta^{a/2} \cdot \xi}{\overline \zeta^{a/2} \cdot \xi \mp \zeta^{a/2} \cdot \overline \xi} = \gamma \cdot Y. \] But $\gamma$ is a unit and $X = \gamma Y$ is a contradiction to $(X, Y) = (1)$, since $X$ and $Y$ are not units. The contradiction confirms our claim that $\phi_i$ are not units, for $i > 0$; thus $Z$ is not a unit.
We can now apply Lemma \ref{idealq} with $n = q, C = Z, \mathbb{L}' = \mathbb{L}_p$, thus obtaining: \begin{lemma} \label{l2} Let the premises of Theorem \ref{tdesc} hold, the normalized elements $\phi_{i}, i > 0$ be defined by \rf{phis1} and the ideals $\eu{A}_{i} = (\phi_{i}, Z)$. If $\ \mathbb{L}_p \subset \mathbb{Q}(\zeta, \xi)$ is the subfield fixed by $\jmath_p$, then $\eu{A}_{i}$ are principal and there are $\mu'_{i} \in \id{O}(\mathbb{L}_p)$ and $\eta_{i} \in \left(\id{O}(\mathbb{L}_p)\right)^{\times}$, such that \rf{alg} holds. \end{lemma} Note that if $\phi_0$ is not a unit, the result of Lemma \ref{l2} also holds for $\phi_0$. We proceed our proof, allowing for both possibilities. It will turn out that the same computations which allow descent also imply in the long run that $\phi_0$ is not a unit.
From \rf{prod2}, $\phi_{0} = \frac{Z^{q}}{\prod_{i=1}^{q-1} \phi_{i}} =_{q}(-Y)^{1-q} =_{q} -Y$. If $\phi_0$ is not a unit, we saw that we can write $\phi_0 = \eta_0 \cdot \mu_0^q$, with $\eta_i, \mu_i$ like in \rf{alg}; otherwise we may set $\phi_0 = \eta_0$. In both cases, the unit $\eta_0$ is defined and $\eta_0 =_q \phi_0 =_q -Y$.
Thus, for all $i=0, 1, \ldots, q-1$, we have $\eta_i =_q -Y$; the Lemma \ref{unitp} implies that $\eta_{i}$ must be \nth{q} powers, so \begin{eqnarray} \label{alg1} \phi_{i} & = & \mu_{i}^{q}, \quad \hbox{for} \quad i = 0, \ldots, q-1. \end{eqnarray} If $\phi_0$ is a unit, then the previous remarks imply that $\phi_{0} = \mu_{0}^{q}$, with $\mu_{0}$ a unit of the same field. Otherwise, by the same reasoning as in the case $i > 0$, $\mu_0 \in \id{O}\left(\mathbb{L}_{p}\right)$.
We are prepared for the main computations which will allow to perform the descent. We evaluate $\phi_{i} \times \phi_{-i}$ for $i > 0$, using the identity in \rf{modq}: \begin{eqnarray*} \psi_{i} = \phi_{i} \times \phi_{-i} & = & \left(-Y +
\frac{\xi^{i}(X+Y)}{\xi^{i}-\xi^{-i}} \right) \cdot \left(-Y +
\frac{\xi^{-i}(X+Y)}{\xi^{-i}-\xi^{i}} \right) \\ & = &\left(\frac{X+Y}{1-\overline{\xi}^{2i}}-Y\right) \cdot \left(\frac{X+Y}{1-\xi^{2i}}-Y\right) \\
& = & Y^{2} + \left(\frac{X+Y}{|1-\xi^{2i}|}\right)^{2} -Y\cdot(X+Y)
\cdot
\left(\frac{1}{1-\overline{\xi}^{2i}}+\frac{1}{1-\xi^{2i}}\right) \\
& = & \left(\frac{X+Y}{|1-\xi^{2i}|}\right)^{2} - X\cdot Y. \end{eqnarray*} The last equation above shows that $\psi_i \in \id{O}(\mathbb{L}^{++})$. We let $\psi_0 = \phi_0^2$, so we also have $\psi_0 \in \id{O}(\mathbb{L}^{++})$: \begin{eqnarray} \label{real} \psi_i \in \id{O}(\mathbb{L}^{++}) \quad \hbox{for} \quad i = 0, 1, \ldots, (q-1)/2. \end{eqnarray}
By subtracting the values of $\psi$ for two indices $i \not \equiv \pm j \mod q$ we find $\psi_{i}-\psi_{j} = \delta_{i,j} \cdot (X+Y)^{2}$, with $\delta_{i, j} = 1/|(1-\xi^{2i}| - 1/|(1-\xi^{2j}|$. For the choice of such indices we need here that $q \geq 5$. We claim that $\lambda^{2} \cdot \delta_{i,j} = \eta_{i,j} \in \id{O}\left(\mathbb{L}^{++}\right)^{\times}$. Indeed, \begin{eqnarray} \label{deltaij} \lambda^{2} \cdot \delta_{i,j} & = &
\frac{\lambda^{2}}{|(1-\xi^{2i})(1-\xi^{2j})|^{2}} \cdot
\left(|1-\xi^{2j}|^{2} - |1-\xi^{2i}|^{2}\right) \nonumber \\
& = & \frac{\lambda^{2}}{|(1-\xi^{2i})(1-\xi^{2j})|^{2}} \cdot
\left((2-\xi^{2i}-\xi^{-2i})-(2-\xi^{2j}-\xi^{-2j})\right) \\ & = & \frac{\lambda^{2} \cdot (\xi^{2j}-\xi^{2i}) \cdot (1-\overline
\xi^{2(i+j)})}{|(1-\xi^{2i})(1-\xi^{2j})|^{2}} \nonumber \end{eqnarray} In our definition $\lambda = \xi - \overline \xi$ is an imaginary number, so $\lambda^{2}$ is real and so is $\lambda^{2} \cdot \delta_{i,j}$. The last equality above shows that $v_{\eu{q}}\left(\lambda^{2} \cdot \delta_{i,j}\right) = 0$ and since it is real and invariant under $\jmath_{q}$ it follows that $\eta_{i,j} \in \id{O}(\mathbb{L}^{++})^{\times}$, as claimed.
We now substitute the definition \rf{phis} of $\phi_{i}$ and \rf{alg1} in the recent results, finding: \begin{eqnarray*} \psi_{i} - \psi_{j} & = & \eta^{2} \cdot \left((\mu_{i} \cdot
\mu_{q-i})^{q} - (\mu_{j} \cdot \mu_{q-j})^{q}\right) \\ & = &
\eta_{i,j} \cdot \lambda^{-2} \cdot (X+Y)^{2} = \eta_{i,j} \cdot
\lambda^{-2} \cdot \left(\frac{\varepsilon \cdot \lambda^{N} \cdot
{\lambda'}^M \cdot \phi_{0}}{q}\right)^{2} \\ & = & \eta_{i,j}
\cdot \lambda^{-2} \cdot \left(\frac{\varepsilon \cdot \lambda^{N}
\cdot {\lambda'}^M \cdot \eta \cdot \mu_{0}^{q}}{q}\right)^{2} \\ & = &
\left(\eta_{i,j} \cdot \eta^{2} \cdot
\left(\frac{ \lambda^{q-1}}{q}\right)^{2}\right) \times
{\lambda'}^{2M} \cdot\lambda^{2(N-q)} \cdot \mu_{0}^{2q}. \end{eqnarray*} After division by $\eta^{2}$ this yields: \begin{eqnarray} \label{down} (\mu_{i} \cdot \mu_{q-i})^{q} - (\mu_{j} \cdot \mu_{q-j})^{q} = \eta' \cdot {\lambda'}^{M'} \cdot \lambda^{N'} \cdot \mu_{0}^{2q}, \end{eqnarray} where $\eta' = \left(\delta_{i,j} \cdot
\left(\frac{\lambda^{q-1}}{q}\right)^{2}\right) \in \id{O}\left(\mathbb{L}^{++}\right)^{\times}$ and $N' = 2(N-q) = N + (N-2q) > N$ is even, $M' = 2M$. Also, by \rf{real}, the numbers occurring at the \nth{q} power in \rf{down} are elements of $\mathbb{L}^{++}$.
We have shown that for $i > 0$, $\phi_i$ are not units, and thus the \nth{q} powers on the left hand side of \rf{down} are neither units. We still have to show that $\mu_0$ is not a unit. For this we write $X' = \mu_{i} \cdot \mu_{q-i}, Y' = -\mu_{j} \cdot \mu_{q-j} \not \in \mathbb{Z}[\zeta, \xi]^{\times}$ and $Z' = \mu_0^2$ and anticipate the next descent step. We start from \rf{down}, which can be rephrased to \begin{eqnarray*} {X'}^{q} + {Y'}^{q} = \eta' \cdot {\lambda'}^{M'} \cdot \lambda^{N'} \cdot {Z'}^{q}. \end{eqnarray*} Since $X', Y'$ are not units, we have proved above that it follows that $Z'$ is not a unit either, so $\mu_0, \phi_0$ are indeed not units, as claimed.
In order to control the descent, we let $I$ be the group of integer ideals of $\mathbb{Z}[\zeta,\xi]$ and consider $\omega : I \rightarrow \mathbb{N}$, the function counting the number of distinct prime ideals which divide an ideal $\eu{A} \in I$. Thus $\omega$ generalizes the analogous function defined on the integers. We show that $0 < \omega(\mu_{0}^{2}) = \omega(\mu_{0}) < \omega(Z)$. Indeed, since the $\phi_{i}$ are coprime and not units, relation \rf{prod2} together with \rf{alg1} imply that \[ \omega(Z) = \sum_{i=0}^{q-1} \omega(\phi_{i}) = \sum_{i=0}^{q-1} \omega(\mu_{i}) ,\] thus $\omega(\mu_{0}) < \omega(Z)$. The inequality $\omega(\mu_{0}) > 0$ rephrases the fact the $\mu_0$ is not a unit, which we have proved. We thus have the main argument of descent: \begin{proposition} \label{pdesc} Let $p, q; X, Y, Z; \varepsilon, \lambda, \lambda', N, M$ be like in the statement of Theorem \ref{tdesc} and suppose that $q \not \hspace{0.25em} \mid h_{pq}^-$. Then there are $X^{(1)}, Y^{(1)}, Z^{(1)} \in \id{O}(\mathbb{L}^{++})$, a unit $\varepsilon^{(1)} \in \id{O}\left(\mathbb{L}^{++}\right)^{\times}$, an even integer $N^{(1)} > N$ and $M^{(1)} \geq M$, such that \begin{eqnarray} \label{desca}
\left(X^{(1)}\right)^{q} + \left(Y^{(1)}\right)^{q} =
\varepsilon^{(1)} \cdot \lambda^{N^{(1)}} \cdot {\lambda'}^{M^{(1)}}
\cdot \left(Z^{(1)}\right)^{q} \end{eqnarray} and $Z^{(1)} \mid Z, X^{(1)}, Y^{(1)}, Z^{(1)} \not \in \mathbb{Z}[\zeta, \xi]^{\times}$. Finally, $\omega\left(Z^{(1)}\right) < \omega(Z)$, where $\omega$ is the distinct prime factor counting function. \end{proposition} \begin{proof}
With the notations above, we let $X^{(1)} = \mu_i \cdot \mu_{q-i}$,
$Y^{(1)} = -\mu_j \cdot \mu_{q-j}$ and $Z^{(1)} = \mu_0^2$; also
$\varepsilon^{(1)} = \eta'$ and $N^{(1)} = N'$. We have proved that
$X^{(1)}, Y^{(1)}, Z^{(1)} \in \id{O}(\mathbb{L}^{++})$, they are coprime
an non vanishing and $(X^{(1)} \cdot Y^{(1)} \cdot Z^{(1)}, p \cdot
q) = (\phi_0, \cdot q) = (1)$. Also, $\eta' \in
\id{O}\left(\mathbb{L}^{++}\right)^{\times}$ and $N' = 2(N - q) > N > 2q$
is even; $M' = 2M \geq M$, trivially. It was shown that $X^{(1)},
Y^{(1)}, Z^{(1)} \not \in \mathbb{Z}[\zeta, \xi]^{\times}$. Thus all the
conditions of Theorem \ref{tdesc} are verified. The equation
\rf{desca} is then a reformulation of \rf{down}. \end{proof} The proof of Theorem \ref{tdesc} follows now easily: \begin{proof} The Proposition \ref{pdesc} can be applied recursively to \rf{desca}, thus generating an infinite sequence \[ (Z) = (Z^{(0)}) \subset (Z^{(1)}) \subset (Z^{(2)}) \subset \ldots \subset (Z^{(k)}) \subset \ldots , \] such that $\omega(Z^{(k)}) > \omega(Z^{(k+1)})$ for all $k \geq 0$. But $Z=Z^{(0)}$ has only a finite number of prime factors and the function $\omega$ is positive integer valued, so it cannot decrease indefinitely. This is a contradiction which shows that the hypothesis $q \not \hspace{0.25em} \mid h_{pq}^-$ of Proposition \ref{pdesc} is untenable, thus proving Theorem \ref{tdesc}. \end{proof}
\section{Case Analysis} We consider two primes $p, q > 3$ such that $q \not \hspace{0.25em} \mid h_{pq}^-$ and $-1 \in <p \mod q>$ and suppose that \rf{FC} holds for these values of $p, q$. The Barlow - Abel relations imply then that \[ \frac{X^p+Y^p}{X+Y} = p^e \cdot A^q, \] for some $e \in \{0,1\}$ and $A \in \mathbb{Z}$. By theorem \ref{main}, \begin{eqnarray*} x+f \cdot y & \equiv & 0 \mod q^{2} \quad \hbox{ with } \quad f \in \{-1,0,1\}. \end{eqnarray*} Together this yields six cases, three of which have been dealt with above, by means of Wieferich relations. We shall investigate below the remaining cases.
\subsection{The Descent Cases} \begin{theorem} \label{desc1} Notations being as above and assuming the premises of Theorem \ref{main2}, the equation \rf{FC} has no solution with $e=1, f = -1$. \end{theorem} \begin{proof}
Assume that \rf{FC} has a solution with $e=1, f=-1$. Then $(X+Y)/p$
is a \nth{q} power, divisible by $p \cdot q$. Let $v_q(X+Y) = nq$
and $v_p(X+Y) = mq-1$, so \[ X+Y = p^{mq-1} \cdot q^{nq} \cdot C^q \quad C \in \mathbb{Z}, \ (C, p q) = 1. \] By \rf{sixqpow} we have in the present case: \begin{eqnarray} \label{rholaq}
\frac{x+\zeta y}{1-\zeta} = -y + \frac{x+y}{1-\zeta} = \rho^{q} . \end{eqnarray} Note that \begin{eqnarray*}
\alpha - \overline \alpha &= & \frac{x+y}{1-\zeta} - \frac{x+y}{1-\overline \zeta} = \frac{(x+y)(1+\zeta)}{1-\zeta} = - \frac{(x+y)(\zeta^a+\overline \zeta^a)}{\zeta^a-\overline \zeta^a} \quad \hbox{so} \\ \rho^{q} - \overline \rho^{q} & = & \prod_{j=0}^{q-1} (\xi^{j} \cdot \rho -\xi^{-j} \cdot \overline \rho) = - C^{q} \cdot (p^m \cdot q^n)^q \cdot \frac{\zeta^a+\overline \zeta^a}{p(\zeta^a-\overline \zeta^a)}, \end{eqnarray*} with $a =(p+1)/2$.
We define the following system of normed divisors of $C^q$: \begin{eqnarray} \label{phisA} \phi_i & = & \frac{\xi^i \rho - \overline \xi^i \overline \rho}{\xi^i - \overline \xi^i} \\ \label{phi0A} \phi_0 & = & -\frac{\rho-\overline \rho}{p^{mq-1} \cdot q^{nq-1}} \cdot \frac{\zeta^a-\overline \zeta^a}{\zeta^a+\overline \zeta^a}. \end{eqnarray} Let \begin{eqnarray*} \varepsilon_1 & = & \prod_{c=1}^{p-1} \frac{\zeta^c - \overline \zeta^c}{\lambda'} \in \mathbb{Z}[\zeta+\overline \zeta]^{\times} \subset \id{O}\left(\mathbb{L}^{++}\right)^{\times} , \\ \varepsilon_2 & = & \prod_{c=1}^{q-1} \frac{\xi^c - \overline \xi^c}{\lambda} \in \mathbb{Z}[\xi+\overline \xi]^{\times} \subset \id{O}\left(\mathbb{L}^{++}\right)^{\times}. \end{eqnarray*} Then $p = \varepsilon_1 \cdot {\lambda'}^{p-1}$ and $q = \varepsilon_2 \cdot {\lambda}^{q-1}$ and \rf{phi0A} can be rewritten as \[\rho-\overline \rho = \varepsilon \cdot \lambda^{(q-1)(nq-1)} \cdot {\lambda'}^{(p-1)(mq-1)-1} \cdot \phi_0, \] with \[\varepsilon = \varepsilon_1^{mq-1} \cdot \varepsilon_2^{nq-1} \cdot \frac{(\zeta^a+\overline \zeta^a) \lambda'}{\zeta^a-\overline \zeta^a} \in \id{O}\left(\mathbb{L}^{++}\right)^{\times} .\] Finally, with $N = (q-1)(nq-1)$ and $M = (p-1)(mq-1)-1$, we have \begin{eqnarray} \label{phi0B} \rho-\overline \rho = \varepsilon \cdot \lambda^N \cdot {\lambda'}^M \cdot \phi_0 \quad \hbox{with} \quad \varepsilon \in \id{O}\left(\mathbb{L}^{++}\right)^{\times}. \end{eqnarray} The rest of the proof goes through a series of steps which were proved in detail in the previous section, so we list the arguments, leaving it to the reader to check the details.
We have by construction $\prod_{i=0}^{q-1} \phi_i = C^q$ and since $(C, pq) = 1$, a fortiori $(\phi_i, pq) = (1)$. Since $(\rho, \overline \rho) = (1)$, one verifies that $(\phi_i, \phi_j) = (1)$ for $0 \leq i \neq j < q$. We can apply Lemma \ref{idealq} to $\phi_i$, with $m = q, \mathbb{L}' = \mathbb{L}_p$, and find \[ \phi_i = \eta_i \cdot \mu_i^q \quad \eta_i \in \id{O}(\mathbb{L}_p)^{\times},
\ \mu_i \in \id{O}(\mathbb{L}_p). \] We have from \rf{rholaq} that $\rho^q =_q -y$; then there is an integer $t$ with $t^q \equiv -y \mod q^{nq}$ and the $q$-adic development based on \rf{rholaq} yields $\rho \equiv \overline \rho \equiv t \mod q^{nq-1}$. But for $i > 0$ we have $\eta_i =_q \phi_i =_q t$ and since $\phi_0 = C^q/\prod_{i > 0} \phi_i$, we also have $\eta_0 =_q \phi_0 =_q t$. Consequently, we may assume that \[ \phi_i = \eta_0 \mu_i^q, \] and $\eta_0 \in \mathbb{Z}[\zeta+\overline \zeta]^{\times} \subset \id{O}(\mathbb{L}^{++})^{\times}$.
Finally we define $\psi_i = \phi_i \cdot \phi_{q-i} \in \id{O}(\mathbb{L}^{++})$ and verify that for $i \not \equiv \pm j \mod q$ we have \begin{eqnarray*}
\psi_i + \psi_j & = & \eta_0^2 \cdot
\left(\left(\mu_i\cdot\mu_{q-i}\right)^q + \left(\mu_j \cdot
\mu_{q-j}\right)^q\right) = \eta_{i,j} \cdot \lambda^{-2} \cdot (\rho
- \overline \rho)^2 \\ & = & \eta_{i,j} \cdot \left(
\frac{\varepsilon \cdot \lambda^N \cdot {\lambda'}^M}{ \lambda }
\right)^2 \times \left(\eta_0^2 \cdot \mu_0^{2q} \right), \end{eqnarray*} where $\eta_{i,j} = \lambda^2 \cdot \delta_{i,j}$ is the unit in \rf{deltaij}. After dividing by $\eta_0^2$, we set $X = \mu_i \cdot \mu_{q-i}, Y = \mu_j \cdot \mu_{q-j}, Z = \mu_0^2, N' = 2(N-1), M' = 2M, \varepsilon' = \varepsilon^2 \cdot \eta_{i,j}$ and find: \[X^q + Y^q = \varepsilon' \cdot \lambda^{N'} \cdot {\lambda'}^{M'} \cdot Z^q. \] The hypotheses $(X,Y,Z) = (XYZ,pq) = (1)$, $X, Y, Z, \varepsilon' \in \mathbb{L}^{++}$, $N > 2 q$ is even and $M \geq 0$ being all fulfilled, as has been showed above, we can apply the Kummer descent Theorem \ref{tdesc}. This raises a contradiction to $q \not \hspace{0.25em} \mid h_{pq}^-$, which proves the statement of this Proposition. \end{proof}
Next we treat the case $p \not \hspace{0.25em} \mid z, q | x+y$: \begin{theorem} \label{desc2} Notations being as above and assuming the premises of Theorem \ref{main2}, the equation \rf{FC} has no solution with $e=0, f = 1$. \end{theorem} \begin{proof} This is a case with $(X+Y,p) = (1)$ and $M = 0$ in the descent theorem. By Corollary \ref{crhodef}, we have $\alpha = \varepsilon \cdot \rho^q$. The $q$ - adic development of $\rho$ is more delicate in this case and we shall work it out in detail - the rest of the proof being exempt of surprises. We are in the First Case and \begin{eqnarray} \label{xpy} x+y \equiv 0\mod q^q, \end{eqnarray} so \[\frac{x^p+y^p}{x+y} =_q p y^{p-1} = B^q =_q 1. \] If $m(p-1) = 1 + nq$, \rf{sixqpow} yields in this case, for a $\rho$ twisted by a root of unity: \begin{eqnarray*}
\rho^q = - \zeta^{-m/2} \cdot (1-\zeta)^{-nq} \cdot y p^m \cdot \left(1 - \frac{x+y}{(1-\zeta) y p^m}\right). \end{eqnarray*} Since the cofactor of $y p^{m}$ is a $q$ - adic \nth{q} power, it follows from the above equation that $y p^m =_q 1$, so there is a $t \in \mathbb{Z}$ with $t^q \equiv -y p^m \mod q^{2q}$. The ring $\ZM{q}[\zeta]$ contains no non trivial \nth{q} roots of unity (since $q$ in not ramified in $\mathbb{Z}_{q}[\zeta]$), so the resulting $q$ - adic extension of $\rho$ starts as follows: \[ \rho = \frac{t}{\zeta^{m/2q} (1-\zeta)^m} \cdot \left(1 - \frac{x+y}{(1-\zeta) q y p^m} + O(q^{2(q-1)}) \right). \] From the definition it follows that $n$ is odd and one verifies that $\rho$ verifies the necessary condition $(\rho/\overline\rho)^q \equiv -\zeta \mod q^{q}$.
We now investigate an adequate factoring of $x+y = B^q$. We have \[ (\zeta^{-1/2} + \zeta^{1/2}) (x+y) = \zeta^{-1/2} \alpha + \zeta^{1/2} \overline \alpha = \varepsilon \cdot \left(\zeta^{-1/2} \rho^q + \zeta^{1/2} \overline \rho^q\right). \] Defining $\rho_1 = \overline \zeta^{1/2q} \cdot \rho$, we have $\rho_1/\overline \rho_1 \equiv -1 \mod q^{q-1}$ and \[ \varepsilon \cdot (\rho_1^q + \overline \rho_1^q) = (\zeta^{1/2}+ \overline \zeta^{1/2}) \cdot B^q: \]
this looks like a good starting point. Let $q^{nq} || (x+y)$, $C = B/q^n$ with $(C, pq) = 1$ and define \begin{eqnarray*} \phi_i & = & \frac{\xi^i \rho_1 + \overline \xi^i \overline \rho_1 }{\xi^i-\overline \xi^i} \quad \hbox{ for } \quad i = 1, 2, \ldots, q-1 \quad \hbox{ and } \\ \phi_0 & = & \frac{\varepsilon (\rho_1 + \overline \rho_1)}{q^{nq-1} \cdot (\zeta^{1/2}+\overline \zeta^{1/2})}. \end{eqnarray*} Note that $\phi_0$ is an algebraic integer, since $\rho_1/\overline \rho_1 \equiv -1 \mod q^{nq-1}$. Then \begin{eqnarray*} \prod_{i=0}^{q-1} \phi_i = \left(\frac{\rho_1^q + \overline \rho_1^q}{q}\right) \times \left(\frac{\varepsilon}{q^{nq-1} \cdot (\zeta^{1/2}+\overline \zeta^{1/2})}\right) = (B/q^n)^q = C^q \end{eqnarray*} According to the usual frame, one verifies that $(\phi_i, \phi_j) = (1)$ for $0 \leq i \neq j < q$ and by Lemma \ref{idealq} it follows that \[ \phi_i = \eta_i \cdot \mu_i^q, \quad \hbox{ with } \quad \eta_i \in \id{O}(\mathbb{L}_p)^{\times}, \ \mu_i \in \id{O}(\mathbb{L}_p). \] Since $\phi_i \equiv -\overline \rho_1 \mod \left(\frac{\rho_1 + \overline \rho_1}{\lambda}\right)$ and $\phi_0 = C^q/\prod_{i > 0} \phi_i$, it follows that $\phi_i =_q \eta_i =_q -\rho_1$ and, using Lemma \ref{lru}, one deduces, after eventual modification of $\mu_i$, that \[ \phi_i = \eta_0 \cdot \mu_i^q \quad i = 0, 1, \ldots, q-1. \] Next we choose $i \not \equiv \pm j \mod q$, let \[ \psi_i = \phi_i \cdot \phi_{q-i} \quad \hbox{ and } \quad \psi_j = \phi_j \cdot \phi_{q-j} , \] and verify \begin{eqnarray} \label{todiv} \psi_i - \psi_j & = & \eta_{i,j} \cdot \lambda^{-2} \cdot (\rho_1+\overline \rho_1)^2 \\ & = & \eta_{i,j} \cdot \lambda^{-2} \cdot \phi_0^2 \cdot \left(\frac{ q^{nq-1} \cdot (\zeta^{1/2}+\overline \zeta^{1/2})}{\varepsilon}\right)^2. \nonumber \end{eqnarray} We let $X = \mu_i \cdot \mu_{q-i}, Y = -\mu_j \cdot \mu_{q-j}$ and $Z = \mu_0^2$. While $\mu_i$ are imaginary numbers, $X, Y$ are real, so $X, Y \in \mathbb{L}_p \cap \mathbb{R} = \mathbb{L}^{++}$; trivially, $Z \in \mathbb{Z}[\zeta+\overline \zeta] \subset \mathbb{L}^{++}$. Now write $q = \varepsilon_1 \cdot \lambda^{q-1}$ for the obvious real unit $\varepsilon_1 \in \id{O}(\mathbb{L}^{++})^{\times}$, set $N = 2 \left((q-1)(nq-1)-1\right)$ and \[ \delta = \eta_{i,j} \cdot \left(\frac{\zeta^{1/2}+\overline \zeta^{1/2}}{\varepsilon}\right)^2 \cdot \varepsilon_1^N \in \id{O}(\mathbb{L}^{++})^{\times} . \] Inserting these new notations in \rf{todiv} leads, after division by $\eta_0^2$, to: \[ X^q + Y^q = \delta \cdot \lambda^N \cdot Z^q. \] Once again we can apply Theorem \ref{tdesc}, obtaining a contradiction with $q \not \hspace{0.25em} \mid h_{pq}^-$. This completes the proof of this case. \end{proof} \subsection{The Ast\'{e}risque Case} We denote the case $e=f=0$ by \textit{ast\'erisque} case. This is the only case in which our results still depend on $x$ and $y$ -- the obstruction to a more general result.
We suppose that $x \equiv 0 \mod q^{2}$ and since $e=0$, then $A^{q} = y+x =_{q} y$, so that $y$ is a $q$-adic \nth{q} power. By \rf{sixqpow} there is in this case a $\rho \in \mathbb{Z}[\zeta]$ such that $\rho^{q} = y + x \cdot \overline \zeta^{2q}$ and \[ \frac{\zeta^{q} \cdot \rho^{q} + \overline{\zeta^{q} \cdot
\rho^{q}}}{\zeta^{q} + \overline \zeta^{q}} = x+y = A^{q} . \] By the usual argument of Lemma \ref{idealq}, we have then \[ \phi_{i} = \frac{\zeta \xi^{i} \cdot \rho + \overline{\zeta
\xi^{i} \cdot \rho}}{\zeta \xi^{i} + \overline {\zeta \xi^{i}}} = \delta_{i} \cdot \mu_{i}^{q}, \] for some units $\delta_{i} \in \mathbb{Z}[\zeta, \xi]$ and $i = 0, 1, \ldots, q-1$. Next, we investigate these units $q$ - adically. Let $t^{q} \equiv y \mod q^{N}$, for some integer $t \equiv A \mod q^{v_{q}(x)-1}$, or, likewise, $t$ be a $q$ - adic approximation of the \nth{q} root of $y$. Then \[ \delta_{i} =_{q} t \cdot \left(1 + \frac{\zeta^{1-2q} \cdot \xi^{i}
+ \overline{\zeta^{1-2q} \cdot \xi^{i}}}{\zeta \xi^{i} + \overline
{\zeta \xi^{i}}} \cdot \frac{x}{q y}\right).\] In particular \[ \delta_{0} =_{q} t \cdot \left(1 + \frac{\zeta^{2q-1}+\overline
\zeta^{2q-1}}{\zeta+\overline\zeta } \cdot \frac{x}{q \cdot y} \right) \in \mathbb{Z}[\zeta]. \]
Let $\lambda = (1-\xi)$ (note the deviation from the usual definition of $\lambda!$), so $\xi^{i} \equiv 1 -i \lambda \mod \lambda^{2}$. A further investigation of the units $\psi_{i} = \delta_{i}/\delta_{0}$ shows that $\psi_{i} =_{q} 1 + \frac{x}{q y} \cdot c(\zeta) \cdot \lambda + O(q \lambda^{2})$, with \begin{eqnarray*}
\lambda c(\zeta) \equiv \lambda \frac{\zeta^{1-2q} \cdot \xi^{i}
+ \overline{\zeta^{1-2q} \cdot \xi^{i}}}{\zeta \xi^{i} + \overline
{\zeta \xi^{i}}} - \frac{\zeta^{2q-1}+\overline
\zeta^{2q-1}}{\zeta+\overline\zeta }
\equiv 2 i \cdot \lambda \cdot \frac{\zeta^{2q}-\overline
\zeta^{2q}}{(\zeta+\overline \zeta)^{2}} \mod \lambda^{2}, \end{eqnarray*} and thus $c(\zeta) = 2i \cdot \frac{\zeta^{2q}-\overline
\zeta^{2q}}{(\zeta+\overline \zeta)^{2}} \neq 0$. Thus Lemma \ref{unitq1} implies that $\psi_{i}$ must be a \nth{q} power and $c(\zeta) \equiv \sigma_{q}(\beta) - \beta \mod q$, for some $\beta \in \mathbb{Z}[\zeta]$. In particular, if $q \equiv 1 \mod p$, then $\sigma_{q}(\beta) \equiv \beta \mod \eu{Q}$ for all degree one primes
$\eu{Q} | q$ of $\mathbb{Z}[\zeta]$. But then $c(\zeta) \equiv 0 \mod q$, which is in contradiction with $\mbox{\bf N}_{\mathbb{K}/\mathbb{Q}}(c(\zeta)) = (2i)^{p-1} \cdot p \not \equiv 0 \mod q$. In this case we should have $x/q \equiv 0 \mod q^{2}$. A fortiori, $\delta_{0} \equiv t \mod q^{2}$ and so by Proposition \ref{un1q} it must be a \nth{q} power. We have $\delta_{0} \equiv t \equiv A \mod q^{2}$ and, since it is a \nth{q} power, also $\delta_{0} = \gamma^{q}$ for some unit $\gamma$. But then $y \equiv \gamma^{q^{2}} \mod q^{3}$, and so $y$ is a $q$ - adic \nth{q^{2}} power with $y^{p-1} \equiv \mbox{\bf N}(\gamma^{q^{2}}) \equiv 1 \mod q^{3}$.
One notes that if $-1 \in <q \mod p>$ and thus $q$ splits in real primes in $\mathbb{K}$, then $c(\zeta)$ always verifies the condition $c = \sigma_{q}(\beta) - \beta$ - since in fact $c(\zeta)+\overline c(\zeta) = 0$ and the congruence holds modulo the real primes above $q$. What can be said more generally, if $-1 \not \in < q \mod p> $? One answer is that one can prove the statement of Lemma \ref{ast1} in this case, provided that additionally $p \equiv 1 \mod 4$. We have thus our first partial result: \begin{lemma} \label{ast1} Suppose that \rf{FC} has a solution with $e = f = 0$ and $q \equiv 1 \mod p$ of $-1 \not \in <q \mod p>$ and $p \equiv 1 \mod 4$. Then
$q^{3} | x$ and $y^{p-1} \equiv 1 \mod q^{3}$. \end{lemma} \begin{proof}
The case $q \equiv 1 \mod p$ was already explained above. If $-1
\not \in < q \mod p>$, then let $g$ generate $\ZMs{p}$ and $H = \{
g^{i} : 0 \leq i < (p-1)/2 \}$ be a set of represnetatives of
$\ZMs{p}/\{-1,1\}$, and $H' = \ZMs{p} \setminus H$. Thus $x \in H
\Leftrightarrow (p-x) \in H'$ and by hypothesis, $<q \mod p> \subset
H$. Let $\eu{q}$ be some prime above $p$, so $\eu{q} \neq \overline
{\eu{q}}$. The condition $c(\zeta) = \sigma_{q}(b)-b$ implies then
$\sum_{x \in H} \sigma_{x}(c(\zeta)) \equiv \sum_{x \in < q \mod p>}
\sigma_{x}(c(\zeta) \equiv 0 \mod q$. Let \[ a = \frac{\zeta^{2q}}{(\zeta+\overline \zeta)^{2}} = \frac{\zeta^{2(q+1)}}{(1+\zeta^{2})^{2}} = \sigma_{2} \left(\frac{\zeta^{q+1}}{(1+\zeta)^{2}}\right) ,\] so that $c(\zeta) = a - \overline a$ and the previous condition amounts to \begin{eqnarray} \label{con1} \sum_{x \in H} \sigma_{x}(a) \equiv \sum_{x \in H'} \sigma_{x}(a) \mod q. \end{eqnarray} We shall show that this condition cannot be fulfilled if $p \equiv 1 \mod 4$. \end{proof} \begin{remark}
The above result also implies that for all $q$ we have $\phi_{i} =
\delta_{0} \cdot \mu_{i}^{q}$ and $\mu_{0}^{q} =
\mu_{i}^{q}+\mu_{q-i}^{q}$; this fact is noteworthy but leads
unfortunately to no further descent.
It may also be observed that if $-1 \in <q \mod p>$ and thus $q$
splits in real primes in $\mathbb{K}$, then $c(\zeta)$ always verifies the
condition $c = \sigma_{q}(\beta) - \beta$. We have already shown
that this is not the case if $q \equiv 1 \mod p$. What can be said
more generally, if $-1 \not \in < q \mod p> $? One answer is that
one can prove the statement of Lemma \ref{ast1} in this case,
provided that additionally $p \equiv 1 \mod 4$. \end{remark}
We proceed with some global estimates. We shall assume that $|x| >
|y|$, which is allowed since $x, y$ are interchangeable for the global estimates; furthermore, we assume that $x > 0$. Note that this choice does not allow any more to choose which of $x$ and $y$ is divisible by $q^{3}$; this is of no relevance for the global estimates we are about to prove. We have \begin{lemma} \label{astyest}
Suppose that \rf{FC} has a solution with $e = f = 0$ and $x > |y| > 0$. Then \begin{eqnarray} \label{yest}
|y| > c(q) \cdot x^{1-2/q}, \end{eqnarray} for some absolutely computable, strictly increasing function $c(q)$ with $c(5) > 1$. \end{lemma} \begin{proof}
Suppose first that $y > 0$ and let $\psi = \rho \cdot \overline \rho
= (x+\zeta y)(x+\overline \zeta y) \in \mathbb{R} \cap \mathbb{Z}[\zeta]$. Then \[ \psi^{q} = \left((x+y)^{2} - \mu x y\right) = A^{2q} \cdot \left(1 - \frac{ \mu \cdot x y}{(x+y)^{2}}\right) \quad \hbox{
with } \quad \mu = (1-\zeta)(1-\overline \zeta) .\] Note that
$\left| \frac{ \mu \cdot x y}{(x+y)^{2}}\right| \leq |\mu|/4 < 1$ in this case, so there is a converging global binomial expansion of $f(x, y) = \left(1 - \frac{ \mu \cdot x
y}{(x+y)^{2}}\right)^{1/q}$. The expressions $\psi$ and $A^{2}
f(x, y)$ have the same \nth{q} power, so they differ by a \nth{q}
root of unity. But since they are both real, they must coincide:
$\psi = A^{2} \cdot f(x, y)$. Furthermore, the series summation commutes with the action of $\mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q})$, for the same reason \footnote{ \ For more detail on this kind of argument,
see for instance \cite{Mi}}. Thus, for all $\sigma \in G_{p}$, \[ \sigma(\psi) = A^{2} \cdot \left(1 + \sum_{n=1}^{\infty}
\binom{1/q}{n} \cdot \left(\frac{-\sigma(\mu) \cdot x
y}{(x+y)^{2}}\right)^{n} \right) . \] An easy computation (see e.g. \cite{Mi}) shows that the binomial coefficients are bounded by
$\left|\binom{1/q}{n} \right| < \frac{1}{qn}$, while
$|\sigma(\mu)| < 4$ for all $\sigma \in \mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q})$. If $R_{2}$ is the second order remainder of the above series, one finds from these estimates that
\[ \left|\sigma R_{2} \right| < \left(\frac{4 A \cdot x
y}{(x+y)^{2}}\right)^{2} \cdot \frac{2}{q}
\ln\left(\frac{x+y}{x-y}\right), \] uniformly for $\sigma \in \mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q})$.
For a fixed $\sigma_{0} \in \mbox{ Gal }(\mathbb{Q}(\zeta)/\mathbb{Q})$, we now give a uniform estimate of the difference $\delta = |\psi - \sigma_{0}(\psi)| \in
\mathbb{Z}[\zeta]$. Since $\delta | \left(\psi^{q} - \sigma_{0}(\psi)^{q} \right) = \left(\mu-\sigma_{0}(\mu)\right) \cdot x y \neq 0$, it follows that $\delta$ is a non vanishing algebraic integer. Its absolute value is: \begin{eqnarray*}
|\delta| & = & \left| A^{2} \cdot \frac{(\mu-\sigma_{0}(\mu))
x y}{q (x+y)^{2}} + (R_{2} - \sigma_{0}(R_{2}))\right| \\ & < & A^{2} \cdot \frac{4xy}{q(x+y)^{2}} + 2 \left(\frac{4 A \cdot x
y}{(x+y)^{2}}\right)^{2} \cdot \frac{2}{q}
\ln\left(\frac{x+y}{x-y}\right) \\ & = &\frac{4 A^{2} x y}{q(x+y)^{2}} \cdot \left(1 + \frac{16
xy}{(x+y)^{2}} \cdot \ln\left(\frac{x+y}{x-y}\right)\right). \end{eqnarray*}
Note that the above estimate holds for all $\sigma \delta$ uniformly; one then verifies that for $|y| < c(q) \cdot |x|^{1-2/q}$ and, say, $c^{-1}(q) = 4/q \cdot \left(1 + 16 q^{-2} \cdot \ln\left(1+2/q^{q}
\right) \right)$ (use the lower bound on $|x|$, above!), then $0 <
|\sigma \delta| < 1$ and thus $\mbox{\bf N} |\delta| < 1$, in contradiction with the fact that $\delta$ is a non vanishing algebraic integer. This completes the proof for $y > 0$.
If $y < 0$, one lets $y' = -y$ in the previous proof and sets $\mu = (1+\zeta)(1+\overline \zeta)$. Concretely, we have $\psi^{q} = (x+y')^{2} \cdot \left(1 - \mu
\frac{xy}{x+y'}\right)$. As a result, the factor $(x+y')^{2}$ is not the \nth{q} power of an integer any more, one replaces $A^{2}$ by the real positive value of $(x-y)^{2/q}$. This has no impact on the estimates of the algebraic integer $\delta$, which are perfectly analog, and lead to the same result. \end{proof} \subsection{The Proof of Theorem \ref{main2}} \begin{proof}
Suppose that \rf{FC} has a non trivial solution for odd primes $p,
q$ verifying the conditions of the Theorem. Then by Theorem
\ref{main} it follows that $p^{e} | z$ and $x+f y \equiv 0 \mod
q^{2}$, for some $f \in \{-1,0,1\}$. The cases $f = 1$ are
impossible, as proved in Theorems \ref{desc1} and \ref{desc2}. Three
of the remaining cases are dealt by some Wieferich condition, as
proved in Propositions \ref{p1}, \ref{p2} and \ref{p3}, while for
the Ast\'erisque case we have shown in Lemma \ref{ast1} that $q^{3}|
x$ if $q \equiv 1 \mod p$. We still have to prove the lower bound
\rf{bound}.
Since the lower bound on $\max\{|x|,|y|\}$ is slightly better in the
particular case Ast\'erisque then the general bounds in Theorem
\ref{main1}, we deduce this bound separately.
We assume now that $q^{2}|x$ and $k = v_{q}(x) \geq 2$ - thus
dropping the assumption $|x| > |y|$. By letting $t^{q} = y$ as
elements of $\mathbb{Z}_{q}$, the $q$ - adic expansion of $\rho = (y + \zeta
x)^{1/q}$ is then \[ \rho = t \cdot \left(1 + \sum_{n=1}^{\infty} \binom{1/q}{n} \cdot
(\zeta x/y)^{n} \right) . \] If $A^{q} = x+y$, we now consider the algebraic integer $\delta = A + \mbox{\bf Tr}_{\mathbb{Q}(\zeta)/\mathbb{Q}} ( \zeta \rho) \in \mathbb{Z}$. Since $A^{q} = y(1+x/y)$, one observes that the $q$-adic expansion of $A$ results from the one for $\rho$ by replacing $\zeta$ with $1$. An easy computation yields (note the factor $\zeta$ of $\rho$ in the definition of $\delta$) the $q$ - adic expansion: \[ \delta = p t \cdot \binom{1/q}{p-1} \cdot (x/y)^{p-1} + O\left(\binom{1/q}{p} \cdot (x/y)^{p}\right) .\] Obviously, $\delta \neq 0$. In order to see this, note that $V(j) = -v_{q}\left(\binom{1/q}{j}\right) = j+v_{q}(j!)$ by \rf{binomvq}. Furthermore, let $W(j) = v_{q}\left(\binom{1/q}{j} \cdot (x/y)^{j} \right) = k j - V(j)$ and thus \[ W(p) - W(p-1) = k + (V(p-1) - V(p)) = k-1+v_{q}((p-1)!)-v_{q}(p!) = k-1 > 0 . \] It follows that $\delta \equiv \binom{1/q}{p-1} \cdot
(x/y)^{p-1} \mod q^{W(p)}$ and thus $\delta \neq 0$. It follows in particular that $\delta \equiv 0 \mod q^{(k-1)(p-1)-v_{q}((p-1)!)}$. Let now $B = |x|+|y|$; then $A^{q} \leq B$ and $|\sigma(\zeta \rho)|^{q} < B$, so
$|\delta| < p B^{1/q}$. The last inequalities combine to: \begin{eqnarray} \label{bd1}
\max(|x|, |y|) > (|x|+|y|)/2 > \frac{1}{2} \cdot \left(\frac{q^{(k-1)(p-1)}}{p} \right)^{q} . \end{eqnarray}
If $|x| > |y|$, then one can use the bound in \rf{bd1} which is stronger then \rf{bound}. Otherwise, \rf{bd1} implies $|y| >
\frac{1}{2} \cdot \left(\frac{q^{(k-1)(p-1)}}{p} \right)^{q}$ and by interchanging $x$ and $y$ in \rf{yest} (the maximum is now $|y|$), we obtain the claim \rf{bound}, where $c_{1}(q) = c(q)/2$, with $c(q)$ from \rf{yest}. If $q \not \equiv 1 \mod p$, all we know is $k \geq 2$, which yields \rf{bound}; otherwise, by Lemma \ref{ast1}, we have $k \geq 3$ and \rf{bound1}.
This result improves upon \rf{lowb}. It is however due to \rf{yest}
that one obtains a lower bound on $\min\{|x|, |y|\}$, which allows to assert that $x^{p}+C^{p} = z^{q}$ has no solutions for $|C|$ below this lower bound, as we have explicitly shown in the Corollary \ref{catg1}. \end{proof}
\section{The Equation of Catalan in the Rationals} We have proved in Lemma \ref{rcfc} that the rational Catalan equation \rf{ratcat} is equivalent to \rf{homfc}: \[ X^{p}+Y^{q} = Z^{pq} .\] Note that this equation is \textit{symmetric} in $p, q$ in the sense that it splits in the two equations: \begin{eqnarray} \label{eqp} X^{p} + (-Z^{q})^{p} & = & (-Y)^{q}, \\ \label{epq} Y^{q} + (-Z^{p})^{q} & = & (-X)^{p}, \end{eqnarray} which are both of type \rf{FC}. Thus Theorem \ref{main2} applies to both equations. The task we still have to achieve for proving Theorem \ref{trc} consists in eliminating the Ast\'erisque Case, by using the symmetry in the above equations. This is a consequence of the following: \begin{proposition} \label{AstCat}
Suppose that $p, q$ are two odd primes verifying the premises of Theorem \ref{main2} and for which \rf{eqp} holds. Then $q | X$. \end{proposition} \begin{proof}
Under the given premises, Theorem \ref{main2} implies that $q |
\left(X \cdot Z^{q}\right)$. For clarity, we use the substitution $x
= X, y = (-Z)^{q}$ and $z = -Y$ in order to bring \rf{eqp} in the
form of the reference Fermat - Catalan equation \rf{FC}. We will
show that the assumption $q | Z$ -- and thus $q | y$ -- leads to a
contradiction. For this we use again the Descent Theorem and the
fact $y = (-Z)^{q}$ is a \nth{q} power.
We assume thus that $q | y = (-Z)^{q}$ and $p \not \hspace{0.25em} \mid xyz$. From \rf{sixqpow} we have in this case $\rho^{q} = x + \zeta y$ and thus \[ (\zeta - \overline \zeta) y = -(\zeta - \overline \zeta) Z^{q} = \rho^{q}-\overline \rho^{q}. \] Let $\phi'_{i} = \xi^{i} \rho - \overline \xi^{i} \overline \rho$ and $\phi'_{0} = \rho-\overline \rho$. Then $\prod_{i=0}^{q-1} \phi'_{i} = (\zeta - \overline \zeta) y$ and since $(y, p) = 1$ and $\phi'_{i} = \tau_{i}(\phi'_{1})$, while
$p \not \equiv 1 \mod q$ and thus $\wp = (1-\zeta)$ does not split completely in $\mathbb{L}/\mathbb{K}$. It follows that $\wp | \phi'_{0}$ and $(\phi'_{i}, \wp) = (1)$. Let $y = q^{nq} \cdot C^{q}$, with $(C, pq) = 1$. By introducing the normalization \[ \phi_{0} = \frac{\rho-\overline \rho}{q^{nq-1} \cdot (\zeta
-\overline \zeta)} \quad \hbox{ and } \quad \phi_{i} = \frac{\xi
\rho - \overline \xi \overline \rho}{\xi - \overline \xi}, \] the arguments use in the proof of the Descent Theorem yield here: \begin{eqnarray*} \prod_{i=0}^{q-1} \phi_{i} & = & C^{q}, \quad \hbox{and} \\ (\phi_{i}, p \cdot q) & = & (\phi_{i}, \phi_{j}) = (1), \quad i \neq j \geq 0 . \end{eqnarray*} We can apply now Lemma 5 and find that $\phi_{i} = \delta_{i} \cdot
\mu_{i}^{q}$, for $i = 0, 1, \ldots, q-1$. If $t \in \mathbb{Z}_{q}$ is such that $t^{q} = x$ (existence is provided by Proposition \ref{p1}), then the $q$ - adic expansion of $\delta_{i}$, given that $q^{2} | y$, yields: $\delta_{i} =_{q} t$. This must then be a \nth{q} power, by Lemma \ref{unitp} and it follows plainly that $\phi_{i} = \mu_{i}^{q}$. The proof proceeds like in the one for the first descent case and shall be sketched here. We define $\psi_{i} = \phi_{i} \cdot \phi_{q-i} = (\rho+\overline \rho)^{2}/(\xi^{i}-\overline \xi^{i})^{2} - \rho \cdot \overline \rho $ and find that \[ \psi_{i} - \psi_{j} = (\mu_{i} \cdot \mu_{q-i})^{q} - (\mu_{j} \cdot \mu_{q-j})^{q} = \delta_{i,j} (\zeta - \overline \zeta)^{2} \cdot q^{2(nq-1)} \cdot \mu_{0}^{2q} ,\] with $\delta_{i,j}$ defined in the proof of Theorem \ref{desc1}, so that $(\xi-\overline \xi)^{2} \delta_{i, j} \in \mathbb{Z}^{\times}[\zeta, \xi]$. The descent argument is in place and the claim of our Theorem follows from the assumption by means of Theorem \ref{desc}. \end{proof} \subsection{Proof of Theorem \ref{trc}} We can now complete the proof of Theorem \ref{trc}. \begin{proof} We know by Lemma \ref{rcfc} that \rf{eqp} and \rf{epq} hold simultaneously. The additional conditions ensure that the premises of Theorem \ref{main2} hold for both equations, considered as equations of the type \rf{FC} (e.g. by substitutions like in the proof of the previous Proposition). We analyze the consequences of the six conditions in Theorem \ref{trc}; for this we refer the reader to the case analysis made for the proof of Theorem \ref{main2}.
The conditions 1., 2. and 6. are sufficient for eliminating the descent cases $f = 1$ in both equations \rf{eqp} and \rf{epq}. The conditions 3. and 4. then show that the cases with $f = -1$ cannot occur for either \rf{eqp} or \rf{epq}. The only cases left are thus the ones with $f = 0$. Finally, condition 5. implies that the case $e
= 1, f = 0$ does also not occur and the only case left is the Ast\'erisque case $e = f= 0$, for both \rf{epq} and \rf{eqp}. However, by Proposition \ref{AstCat}, this implies that $q | X$. But this is exactly the case $e = 1$ in \rf{epq}, which is granted not to have solutions by the same condition 5. The contradiction completes the proof of the Theorem. \end{proof}
\vspace*{0.3cm}
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September 2012, 5(3): 615-638. doi: 10.3934/krm.2012.5.615
Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$
Zhong Tan 1, , Yong Wang 1, and Xu Zhang 1,
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China, China, China
Received January 2012 Revised February 2012 Published August 2012
We are concerned with the long-time behavior of global strong solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$, where the electric field is governed by the self-consistent Poisson equation. When the regular initial perturbations belong to $H^{4}(\mathbb{R}^{3})\cap \dot{B}_{1,\infty}^{-s}(\mathbb{R}^{3})$ with $s\in [0,1]$, we show that the density and momentum of the system converge to their equilibrium state at the optimal $L^2$-rates $(1+t)^{-\frac{3}{4}-\frac{s}{2}}$ and $(1+t)^{-\frac{1}{4}-\frac{s}{2}}$ respectively, and the decay rate is still $(1+t)^{-\frac{3}{4}}$ for temperature which is proved to be not optimal.
Keywords: optimal decay rates., Compressible Navier-Stokes-Poisson system.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615
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\begin{document}
\setlength{\parindent}{0pt} \title{Random projections for curves in high dimensions} \begin{abstract}
Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by ``discrete'' notions of distance, e.g. the dynamic time warping or the discrete Fr\'echet distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fr\'echet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence.
We study the ability of random projections \`a la Johnson and Lindenstrauss to preserve the continuous Fr\'echet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to $O(\varepsilon^{-2} \log N)$, where $N$ is the total number of input points while preserving the continuous Fr\'echet distance between any two determined polygonal curves within a factor of $1\pm \varepsilon$. We conclude with applications on clustering. \end{abstract} \pagenumbering{gobble}
\pagenumbering{arabic} \setcounter{page}{1} \section{Introduction}
Time series analysis lies in the core of various modern applications. Typically, a time series consists of various (physical) measurements over time. Formally, it is a finite sequence of points in $\mathbb{R}^d$. Depending on the use case, the ambient space may be extremely high-dimensional, for example $d \in 2^{\Omega(\log n)}$, or even $d \in 2^{\Omega(n)}$, where $n$ is the number of given sequences. For example, large facilities nowadays supervise their production lines using a plethora of sensors. Another concrete example are climatology applications, where data consist of measurements from multiple sensors, each one corresponding to a different dimension.
Many analysis techniques are based on (dis-)similarity between time series, c.f.~\cite{clustering_time_series}. This is often measured by distance functions such as the Euclidean distance, which however requires the time series to be of same length and does not include any form of alignment between the sequences. This is of course less expressive than distances which are indeed defined over an optimal alignment, e.g.~the dynamic time warping, or the discrete Fr\'echet distance, which are based on Euclidean distances between the points but enable compensation of differences in phase. A common downside of these distances is that they take into account solely the points of a time series. Hence, they are sensitive to differences in sampling rates or data gaps. Here, the continuous Fr\'echet distance offers a popular alternative which aims to alleviate this issue by assuming that time series are discretizations of continuous functions of time. It is an extension of the discrete Fr\'echet distance that takes into account all points on the polygonal curves obtained by linearly interpolating between any two consecutive points in a sequence (where the interpolation is carried out only implicitly).
Two main parameters typically govern computational tasks associated with the Fr\'echet distance: the lengths of the time series and the number of dimensions of the ambient space. In this paper, we study the problem of compressing the input with respect to the latter parameter using a dimension reducing linear transform that preserves Euclidean distances within a factor of $(1\pm \varepsilon)$. These transforms, which are usually named Johnson-Lindenstrauss (JL) transforms or embeddings, are a popular tool in dimensionality reduction. The preservation of pairwise distances within a factor of $(1\pm\varepsilon)$ is sometimes called JL guarantee. Recent work has provided various probability distributions over JL transforms~\cite{Dasgupta2003,jl_sphericity,Achlioptas03,Linial1995,KaneN14}, which are efficient to sample from and which yield the JL guarantee with at least constant positive probability while the target dimension is only $O(\varepsilon^{-2}\log n)$, where $n$ is the size of the input point set. Towards applying this result on time series, one can easily guarantee that all Euclidean distances between points of the time series are preserved. While this has direct implications on ``discrete'' notions of distances between time series, the case of the continuous Fr\'echet distance is far more intriguing.
\subsection{Related Work}
In their seminal paper \cite{johnson_lindenstrauss}, Johnson and Lindenstrauss proved the following statement, which is commonly known as the Johnson-Lindenstrauss lemma and coined the term JL embedding. \begin{theorem}[\cite{johnson_lindenstrauss}] \label{JLlemma}
For any $n \in {\mathbb N}$ and $\varepsilon \in (0,1)$ there exists a probability distribution over linear maps $f \colon {\mathbb R}^d \to {\mathbb R}^{d^{\prime}}$, where $d^{\prime} \in O(\varepsilon^{-2}\log n)$, such that for any $n$-point set $X \subset \mathbb{R}^d$ the following holds with high probability over the choice of $f$: \[\forall p,q \in X:~(1-\varepsilon) \| p-q\| \leq \|f(p)-f(q)\| \leq (1+\varepsilon) \|p-q\|.\] \end{theorem} In their proof, Johnson and Lindenstrauss \cite{johnson_lindenstrauss} show that this can be achieved by orthogonally projecting the points onto a random linear subspace of dimension $O(\varepsilon^{-2} \log n)$ -- and indeed there are point sets that require $\Omega(\varepsilon^{-2} \log n)$ dimensions \cite{DBLP:journals/dm/Alon03,DBLP:conf/icalp/LarsenN16,DBLP:conf/focs/LarsenN17}. Several proofs of their statement followed, these however don't require a proper projection but only a multiplying the points with a certain random matrix, cf. ~\cite{DBLP:conf/stoc/IndykM98,Dasgupta2003,Achlioptas03,KaneN14,AilonC09}.
The impact of a JL embedding on higher-dimensional objects other than points has already been studied. Magen \cite{DBLP:conf/random/Magen02,DBLP:journals/dcg/Magen07} shows that applying a (scaled) JL embedding not only to a given set $P \subset \mathbb{R}^d$ of points, but to $P \cup W$, where $W \subset \mathbb{R}^d$ is a well-chosen set of points determined by $P$, approximately preserves the height and angles of all triangles determined by any three points in $P$. Magen even extends this result and shows that by a clever choice of $W$, the volume (Lebesgue measure) of the convex hull of any $k-1$ points from $P$ is approximately preserved when the target dimension is in $\Theta(\varepsilon^{-2} k \log \lvert P \rvert)$. Furthermore, in this case the distance of any point from $P$ to the affine hull of any $k-1$ other points from $P$ is also approximately preserved. Furthermore, JL embeddings can even be utilized to preserve all pairwise Euclidean and geodesic distances on a smooth manifold \cite{BW09}.
Fr\'echet distance preserving embeddings are a relatively unexplored topic. Recently Driemel and Krivosija \cite{DriemelK18} studied the first Fr\'echet distance preserving embedding for $c$-packed curves, which are curves whose intersections with any ball of radius $r$ are of length at most $cr$. This class of curves was introduced by Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} and has so far been considered a viable assumption for realistic curves, see e.g. \cite{DBLP:conf/compgeom/AgarwalFPY16,DBLP:conf/focs/Bringmann14,DBLP:journals/siamcomp/DriemelH13}. Driemel and Krivosija consider projections on random lines, where curves are orthogonally projected on a vector which is sampled uniformly at random from the unit sphere. They observed that in any case (even if the curves are not $c$-packed), the discrete Fr\'echet distance between the curves decreases. Furthermore, they show that with high probability the discrete Fr\'echet distance between two curves $\sigma$ and $\tau$, of complexity (number of vertices of the curve) at most $m$, decreases by a factor in $O(m)$. Finally, they proved that there exist $c$-packed curves such that the discrete Fr\'echet distance decreases by a factor in $\Omega(m)$. The latter also holds for the continuous Fr\'echet distance and for the dynamic time warping distance.
More recently, Meintrup et al. \cite{DBLP:conf/nips/MeintrupMR19} studied JL embeddings in the context of preserving the Fr\'echet distance to facilitate $k$-median clustering of curves in a high-dimensional ambient space. They show that when the dimension is reduced to $\Theta(\varepsilon^{-2} \log N)$, where $N$ is the total number of vertices of the given curves, the Fr\'echet distances are preserved up to a combined multiplicative error of $(1\pm\varepsilon)$ and additive error of $\pm\varepsilon L$, where $L$ is the largest arclength of any input curve. For their proof, they only use the JL guarantee, i.e., the $(1\pm\varepsilon)$-preservation of Euclidean distances, and properties of the polygonal curves and the Fr\'echet distance, while linearity is not taken into account. In this setting, it seems that the additive error is possible -- Meintrup et al. give a simple example where some vertex-to-vertex distances expand and others contract, which induces an additive error to the Fr\'echet distance. Meintrup et al. complement their results with experimental evaluation showing that in real world data and using a JL transform (which is a \textit{linear} map), the Fr\'echet distance is preserved within the multiplicative error only in almost any case. The other cases can not be distinguished between a failed attempt to obtain a JL embedding (recall the probabilistic nature) and a successful attempt to obtain a JL embedding with the additive error occuring.
\subsection{Our Contributions}
We study the ability of random projections à la Johnson and Lindenstrauss to preserve the continuous Fr\'echet distances among a given set of $n$ polygonal curves, each of complexity (number of vertices of the curve) at most $m$. We show that there exists a set $X$ of vectors (in $\mathbb{R}^d$), of size polynomial in $n$ and $m$ and depending only on the given curves, such that any JL transform for the curves vertices and $X$ also preserves the continuous Fr\'echet distance between any two of the given polygonal curves within a factor of $(1\pm \varepsilon)$, without additional additive error. This effectively extends the JL guarantee to pairwise Fr\'echet distances. By plugging in any known JL transform from one of \cite{Dasgupta2003,jl_sphericity,Achlioptas03,Linial1995,KaneN14} we obtain our main dimension reduction result which states that one can reduce the number of dimensions to $O(\varepsilon^{-2} \log(nm))$. We achieve our result using a completely different approach than Meintrup et al. \cite{DBLP:conf/nips/MeintrupMR19}. Our approach relies on Fr\'echet distance predicates originating from \cite{DBLP:conf/soda/AfshaniD18}. These allow a reduction from deciding the continuous distance to a finite set of events occurring. Using only the predicates, it is relatively easy to prove that the Fr\'echet distance between two curves does not expand by more than a factor of $(1+\varepsilon)$ under a (\textit{linear}) JL transform. To prove that the Fr\'echet distance does not contract by more than a factor of $(1-\varepsilon)$ is however much more challenging. We achieve this by proving that \textit{all} distances between one fixed point and any point on a fixed line do not contract by more than a factor of $(1-\varepsilon)$ when a JL transform is applied to a well-chosen set of four vectors determined by the point and the line, which is then applied to any vertex of any curve and any line determined by an edge of any curve. We note that this result is comparable to a result by Magen \cite{DBLP:conf/random/Magen02,DBLP:journals/dcg/Magen07}, but our statement is stronger since it takes into account \textit{all} distances between the fixed point and the line and not only the affine distance, i.e., the distance between the point and its orthogonal projection onto the line.
Our motivation is that distance preserving dimensionality reductions imply improved algorithms for various tasks. Best-known algorithms for many proximity problems under the continuous Fr\'echet distance have exponential dependency on the dimension, in at least one of their performance parameters. Such algorithms either directly employ the continuous Fr\'echet distance, e.g.~the approximation algorithms for $k$-clustering problems~\cite{DBLP:conf/soda/BuchinDR21}, or approximate it with the discrete Fr\'echet distance by resampling the time series to a higher granularity. For example, to the best of our knowledge, the best solution for the approximate near neighbor (ANN) problem in general dimensions derives from building the data structure of Filtser et al. \cite{FFK20}, which originally solves the problem for the discrete Fr\'echet distance, on a modified input. The idea is that a new dense set of vertices can be added to each input polygonal curve so that the discrete Fr\'echet distance of the resulting curves approximates the continuous Fr\'echet distance of the original curves. Under the somewhat restrictive assumption that the arclength of each curve is short, a small number of new vertices suffices. Even in this case though, the space and preprocessing time of the data structure depends exponentially on the number of dimensions. Obviously, polynomial-time algorithms (e.g.~\cite{DBLP:conf/soda/BuchinDGHKLS19}) can also benefit from reducing the number of dimensions, especially when it comes to real applications.
Our embedding naturally inherits desired properties of the JL transforms like the fact that they are oblivious to the input. This makes it directly applicable to data structure problems like the above-mentioned ANN problem. Moreover, we show that our embedding is also applicable to estimating clustering costs. First, we show that one can approximate the optimal $k$-center cost within a constant factor, with an algorithm that has no dependency on the original dimensionality apart from an initial step of randomly projecting the input curves. Second, we show that one can use any algorithm for computing the $k$-median cost in the dimensionality-reduced space to get a constant factor approximation of the $k$-median cost in the original space.
\subsection{Organization}
The paper is organized as follows. In \cref{sec:preliminaries} we introduce the necessary notation, definitions and the concept of Fr\'echet distance predicates. In \cref{section:embedding} we prove our main result in two steps. First, as a warm-up, we prove that an application of any JL transformation for the given curves vertices and a polynomial-sized set $X$ determined by these does not increase Fr\'echet distances by more than a factor of $(1+\varepsilon)$. In \cref{ss:lowerbound} we prove the challenging part that this also does not decrease Fr\'echet distances by less than a factor of $(1-\varepsilon)$. Interestingly, here a different polynomial-sized set $X^\prime$ is used. In \cref{ss:mainresults} we combine both to our main result. Finally, in \cref{section:clustering} we apply our main result to clustering of curves; we modify an existing approximation algorithm for the $(k,\ell)$-center problem (see~\cite{DBLP:conf/soda/BuchinDGHKLS19}) which has negligibly decreased approximation quality compared to the original and we prove that applying \textit{any} algorithm for the $(k,\ell)$-median problem (such as the one from \cite{DBLP:conf/soda/BuchinDR21}) on the embedded curves leads to a constant factor approximation in terms of clustering cost. \cref{section:conclusions} concludes the paper.
\section{Preliminaries} \label{sec:preliminaries}
For $n \in \mathbb{N}$ we define $[n] = \{1, \dots, n\}$. By $\lVert \cdot \rVert$ we denote the Euclidean norm, by $\langle \cdot, \cdot \rangle$ we denote the Euclidean dot product and by $\mathbb{S}^{d-1} = \{ p \in \mathbb{R}^d \mid \lVert p \rVert = 1 \}$ we denote the unit sphere in $\mathbb{R}^{d}$. We define line segments, the building blocks of polygonal curves.
\begin{definition}
A line segment between two points $p_1, p_2 \in \mathbb{R}^d$, denoted by $\overline{p_1p_2}$, is the set of points $\{ (1-\lambda)p_1 + \lambda p_2 \mid \lambda \in [0,1] \}$. For $\lambda \in \mathbb{R}$ we denote by $\lp{\overline{p_1p_2}}{\lambda}$ the point $(1-\lambda)p_1 + \lambda p_2$, lying on the line supporting the segment $\overline{p_1p_2}$. \end{definition}
We formally define polygonal curves.
\begin{definition}
\label{def:polygonal_curve}
A (parameterized) curve is a continuous mapping $\tau \colon [0,1] \rightarrow \mathbb{R}^d$. A curve $\tau$ is polygonal, if and only if, there exist $v_1, \dots, v_m \in \mathbb{R}^d$, no three consecutive on a line, called $\tau$'s vertices and $t_1, \dots, t_m \in [0,1]$ with $t_1 < \dots < t_m$, $t_1 = 0$ and $t_m = 1$, called $\tau$'s instants, such that $\tau$ connects every two consecutive vertices $v_i = \tau(t_i), v_{i+1} = \tau(t_{i+1})$ by a line segment. \end{definition}
We call the line segments $\overline{v_1v_2}, \dots, \overline{v_{m-1}v_m}$ the edges of $\tau$ and $m$ the complexity of $\tau$, denoted by $\lvert \tau \rvert$. Sometimes we will argue about a sub-curve $\tau[i,j]$ of a given curve $\tau$, which is the polygonal curve determined by the vertices $v_{i}, \dots, v_j$. We define two notions of continuous Fr\'echet distances. We note that the weak Fr\'echet distances is however used only rarely.
\begin{definition}
Let $\sigma, \tau$ be curves. The weak Fr\'echet distance between $\sigma$ and $\tau$ is \[ \dwf(\sigma,\tau) = \inf_{\substack{f \colon [0,1] \rightarrow [0,1]\\ g \colon [0,1] \rightarrow [0,1]}} \max_{t \in [0,1]} \lVert \sigma(f(t)) - \tau(g(t)) \rVert, \]
where $f$ and $g$ are continuous functions with $f(0) = g(0) = 0$ and $f(1) = g(1) = 1$.
The Fr\'echet distance between $\sigma$ and $\tau$ is \[ \df(\sigma, \tau) = \inf_{\substack{f \colon [0,1] \rightarrow [0,1]\\ g \colon [0,1] \rightarrow [0,1]}} \max_{t \in [0,1]} \lVert \sigma(f(t)) - \tau(g(t)) \rVert, \] where $f$ and $g$ are continuous bijections with $f(0) = g(0) = 0$ and $f(1) = g(1) = 1$. \end{definition}
We define the type of embedding we are interested in. Since we want to keep our results general, we do not specify the target number of dimensions. As a consequence, we drop the JL-terminology and call these $(1\pm\varepsilon)$-embeddings.
\begin{definition}
\label{def:embedding}
Given a set $P \subset \mathbb{R}^d$ of points and $\varepsilon \in (0,1)$, a function $f\colon \mathbb{R}^d \rightarrow \mathbb{R}^{d^\prime}$ is a $(1 \pm \varepsilon)$-embedding for $P$, if it holds that \[ \forall p,q \in P: (1-\varepsilon) \lVert p - q \rVert \leq \lVert f(p) - f(q) \rVert \leq (1+\varepsilon) \lVert p - q \rVert. \] \end{definition}
We note that if $f$ is linear and $0 \in P$, then $\forall p \in P$: $(1-\varepsilon) \lVert p \rVert \leq \lVert f(p) \rVert \leq (1+\varepsilon) \lVert p \rVert$.
We now define valid sequences with respect to two polygonal curves. Such a sequence can be seen as a discrete skeleton in deciding the continuous Fr\'echet distance and is derived from the free space diagram concept used in Alt and Godau's algorithm \cite{alt_godau}.
\begin{definition}
Let $\sigma, \tau$ be polygonal curves with vertices $v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}$, respectively $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$. A valid sequence with respect to $\sigma$ and $\tau$ is a sequence $\mathcal{F} = (i_1, j_1), \dots, (i_k, j_k)$ with
\begin{itemize}
\item $i_1 = j_1 = 1$, $i_k = \lvert \sigma \rvert - 1$, $j_k = \lvert \tau \rvert - 1$,
\item $(i_l, j_l) \in [\lvert\sigma\rvert-1] \times [\lvert\tau\rvert-1]$,
\item $(i_l-i_{l-1}, j_l-j_{l-1}) \in \{ (0,1), (1,0), (0,-1), (-1,0) \}$ for all $1 < l < k$ and
\item any pair $(i_l,j_l) \in [\lvert\sigma\rvert-1] \times [\lvert\tau\rvert-1]$ appears at most once in $\mathcal{F}$.
\end{itemize}
A valid sequence is said to be monotone if $(i_l-i_{l-1}, j_l-j_{l-1}) \in \{ (0,1), (1,0) \}$ for all $1 < l < k$. \end{definition}
Further decomposing the free space diagram concept, any valid sequence for two curves $\sigma,\tau$ and any radius $r \geq 0$ induces a set of predicates which truth values in conjunction determine whether $\df(\sigma,\tau) \leq r$, respectively $\dwf(\sigma,\tau) \leq r$.
\begin{definition}[\cite{DBLP:conf/soda/AfshaniD18,DBLP:journals/dcg/DriemelNPP21}]
\label{def:predicates}
Let $\sigma, \tau$ be polygonal curves with vertices $v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}$, respectively $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$ and $r \in \mathbb{R}_{\geq 0}$. We define the Fr\'echet distance predicates for $\sigma$ and $\tau$ with respect to $r$.
\begin{itemize}
\item $(P_1)^{\sigma,\tau,r}$: This predicate is true, iff $\lVert \sigma_1 - \tau_1 \rVert \leq r$.
\item $(P_2)^{\sigma,\tau,r}$: This predicate is true, iff $\lVert \sigma_{\lvert \sigma \rvert} - \tau_{\lvert \tau \rvert} \rVert \leq r$.
\item $(P_3)^{\sigma,\tau,r}_{(i,j)}$: This predicate is true, iff there exists a point $p \in \overline{v^\sigma_{i}v^\sigma_{i+1}}$ with $\lVert p - v^\tau_{j} \rVert \leq r$.
\item $(P_4)^{\sigma,\tau,r}_{(i,j)}$: This predicate is true, iff there exists a point $p \in \overline{v^\tau_{j}v^\tau_{j+1}}$ with $\lVert p - v^\sigma_{i} \rVert \leq r$.
\item $(P_5)^{\sigma,\tau,r}_{(i,j,k)}$: This predicate is true, iff there exist $p_1 = \lp{\overline{v^\sigma_jv^\sigma_{j+1}}}{t_1}$ and $p_2 = \lp{\overline{v^\sigma_jv^\sigma_{j+1}}}{t_2}$ with $\lVert v^\tau_{i} - p_1 \rVert \leq r$, $\lVert v^\tau_k - p_2 \rVert \leq r$ and $t_1 \leq t_2$.
\item $(P_6)^{\sigma,\tau,r}_{(i,j,k)}$: This predicate is true, iff there exist $p_1 = \lp{\overline{v^\tau_iv^\tau_{i+1}}}{t_1}$ and $p_2 = \lp{\overline{v^\tau_iv^\tau_{i+1}}}{t_2}$ with $\lVert v^\sigma_{j} - p_1 \rVert \leq r$, $\lVert v^\sigma_k - p_2 \rVert \leq r$ and $t_1 \leq t_2$.
\end{itemize} \end{definition}
The following two theorems state the aforementioned facts. These will be one of our main tools in obtaining our main results. We note that these are rephrased here to fit our needs.
\begin{theorem}[\cite{DBLP:journals/dcg/DriemelNPP21}]
\label{theo:predicates_weak_frechet}
Let $\sigma, \tau$ be polygonal curves and $r \in \mathbb{R}_{\geq 0}$. There exists a valid sequence $\mathcal{F}$ with respect to $\sigma$ and $\tau$, such that $(P_1)^{\sigma,\tau,r} \wedge (P_2)^{\sigma,\tau,r} \wedge \Psi_w^{\sigma,\tau,r}(\mathcal{F})$ is true, where \[ \Psi_w^{\sigma,\tau,r}(\mathcal{F}) = \bigwedge_{\substack{(i,j) \in [\lvert \sigma \rvert] \times [\lvert \tau \rvert]\\(i,j-1),(i,j) \in \mathcal{F}}} (P_3)^{\sigma,\tau,r}_{(i,j)} \bigwedge_{\substack{(i,j) \in [\lvert \tau \rvert] \times [\lvert \sigma \rvert]\\(i-1,j),(i,j) \in \mathcal{F}}} (P_4)^{\sigma,\tau,r}_{(i,j)}, \] if, and only if, $\dwf(\sigma, \tau) \leq r$. \end{theorem}
\begin{theorem}[\cite{DBLP:conf/soda/AfshaniD18,DBLP:journals/dcg/DriemelNPP21}]
\label{theo:predicates_frechet}
Let $\sigma, \tau$ be polygonal curves and $r \in \mathbb{R}_{\geq 0}$. There exists a monotone valid sequence $\mathcal{F}$ with respect to $\sigma$ and $\tau$, such that $(P_1)^{\sigma,\tau,r} \wedge (P_2)^{\sigma,\tau,r} \wedge \Psi^{\sigma,\tau,r}(\mathcal{F})$ is true, where \[ \Psi^{\sigma,\tau,r}(\mathcal{F}) = \hspace{-1.5em} \bigwedge_{\substack{(i,j)\in [\lvert \sigma \rvert] \times [\lvert \tau \rvert]\\(i,j-1),(i,j) \in \mathcal{F}}} \hspace{-1.5em} (P_3)^{\sigma,\tau,r}_{(i,j)} \bigwedge_{\substack{(i,j)\in [\lvert \tau \rvert] \times [\lvert \sigma \rvert]\\(i-1,j),(i,j) \in \mathcal{F}}} \hspace{-1.5em} (P_4)^{\sigma,\tau,r}_{(i,j)} \bigwedge_{\substack{(i,j,k)\in [\lvert \tau \rvert] \times [\lvert \sigma \rvert] \times [\lvert \tau \rvert]\\(i,j-1),(i,k) \in \mathcal{F}\\j < k}} \hspace{-2em} (P_5)^{\sigma,\tau,r}_{(i,j,k)} \bigwedge_{\substack{(i,j,k) \in [\lvert \tau \rvert] \times [\lvert \sigma \rvert] \times [\lvert \sigma \rvert] \\(i-1,j),(k,j) \in \mathcal{F}\\i < k}} \hspace{-2em} (P_6)^{\sigma,\tau,r}_{(i,j,k)},\] if, and only if, $\df(\sigma, \tau) \leq r$. \end{theorem}
\section{Linear Embeddings Preserve Fr\'echet Distances} \label{section:embedding}
In this section we prove our main results on embeddings of polygonal curves that approximately preserve the Fr\'echet distance. In the following \cref{lem:upper_bound}, we show that linear $(1\pm \varepsilon)$-embeddings for a polynomial number of points determined by the input polygonal curves imply embeddings for the curves that are not expansive by a factor greater than $(1+\varepsilon)$. Similarly, in \cref{ss:lowerbound}, we show that linear $(1\pm \varepsilon)$-embeddings for a polynomial number of points determined by the curves, imply embeddings for the curves that are not contractive by a factor smaller than $(1-\varepsilon)$. Combining these two bounds yields our main results in \cref{ss:mainresults}. Our main dimensionality reduction result states that one can embed a set of $n$ polygonal curves of complexity at most $m$ into a Euclidean space of dimensions $d^\prime \in O(\varepsilon^{-2}\log (nm))$, so that all Fr\'echet distances are preserved within a factor of $(1\pm \varepsilon)$. The embedding is implemented by mapping the vertices of each polygonal curve with a JL transform. The image of each input curve is a curve in ${\mathbb R}^{d'}$ having as vertices the images of the original vertices.
\begin{lemma}
\label{lem:upper_bound}
Let $\sigma, \tau$ be polygonal curves with vertices $v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}$, respectively $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$, and let $f$ be a linear $(1 \pm \varepsilon)$-embedding for $P = \{v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}, v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert} \} \cup P^\prime$, where $P^\prime$ is a set of points determined by $\sigma$ and $\tau$ with $\lvert P^\prime \rvert \in O(\lvert \sigma \rvert^2 \cdot \lvert \tau \rvert + \lvert \tau \rvert^2 \cdot \lvert \sigma \rvert)$.
Let $\sigma^\prime$ and $\tau^\prime$ be polygonal curves with vertices $f(v^\sigma_1), \dots, f(v^\sigma_{\lvert \sigma \rvert})$, respectively $f(v^\tau_1), \dots, f(v^\tau_{\lvert \tau \rvert})$. It holds that
\begin{itemize}
\item $\dwf(\sigma^\prime, \tau^\prime) \leq (1+\varepsilon) \dwf(\sigma, \tau)$ and
\item $\df(\sigma^\prime, \tau^\prime) \leq (1+\varepsilon) \df(\sigma, \tau)$.
\end{itemize} \end{lemma}
The proof follows by an application of the $(1 \pm \varepsilon)$-embedding to all points determined by the (weak) Fr\'echet distance predicates. The proof can be found in \cref{sec:appendix}.
\subsection{Lower bound} \label{ss:lowerbound} In this section, we show that we can use linear $(1\pm \varepsilon)$-embeddings for a polynomial number of points determined by the input polygonal curves to define embeddings for the curves that are not contractive with respect to their Fr\'echet distance by a factor smaller than $(1-\varepsilon)$.
We first introduce a few necessary technical lemmas and then we proceed with the main result. We make use of the following lemma, which indicates that inner products are (weakly) concentrated in $(1\pm \varepsilon)$-embeddings. Slightly different versions of this lemma have been used before (see e.g.~\cite{DBLP:conf/focs/ArriagaV99,DBLP:journals/jcss/PapadimitriouRTV00,Sarlos06}). Since our statement is a bit more generic, because it holds for any linear $(1\pm \varepsilon)$-embedding, and we make use of the involved scaling factors, we include a proof in the appendix for completeness.
\begin{lemma}
\label{lem:inner_product}
Let $f$ be a linear $(1 \pm \varepsilon)$-embedding for a finite set $P \subset \mathbb{R}^d$ with $0 \in P$. For all $p,q \in P$ it holds that \[\langle p, q \rangle - 16 \varepsilon(\lVert p \rVert \cdot \lVert q \rVert) \leq \langle f(p), f(q) \rangle \leq \langle p, q \rangle + 14 \varepsilon(\lVert p \rVert \cdot \lVert q \rVert).\] \end{lemma}
Next, we prove that $(1\pm \varepsilon)$-embeddings for a specific point set do not contract distances between any point on a fixed ray starting from the origin and a fixed point lying in a certain halfspace by a factor smaller than $(1-3\varepsilon)$.
\begin{lemma} \label{lemma:segmentnoncontraction0} Let $x \in {\mathbb R}^d$ and $u\in {\mathbb S}^{d-1}$ such that $\langle x,u\rangle \leq 0$. Let $f$ be a linear $(1 \pm \varepsilon/16)$-embedding for $\{0,x,u\}$. For any $\lambda \geq 0$, we have \[
\left\|f(x)-\lambda \cdot f(u)\right\| \geq (1-3\varepsilon)\|x-\lambda u \|. \] \end{lemma} \begin{proof} By \cref{lem:inner_product,def:embedding}:
\begin{enumerate*}[label=\roman*)]
\item $\langle f(x),f(u) \rangle \in \langle x, u\rangle \pm \varepsilon \|x\|$,
\item $\|f(x)\| \in (1\pm\varepsilon) \|x\|$,
\item $\|f(u)\| \in (1\pm \varepsilon)$.
\end{enumerate*} For any $\lambda \geq 0$ we have: \begin{align}
\left\|f(x)-\lambda\cdot f( u) \right\|^2 & = \|f(x)\|^2 + \lambda^2\cdot \|f(u)\|^2 - 2\lambda \cdot \langle f(x),f(u)\rangle \nonumber \\
&\geq (1-\varepsilon)^2 \|x\|^2 +(1-\varepsilon)^2 \lambda^2 - 2\lambda \langle x,u\rangle - 2\lambda \varepsilon \|x\| \label{eq:sarlosapplication} \\
&\geq (1-\varepsilon)^2 \|x\|^2 +(1-\varepsilon)^2\lambda^2 - (1-\varepsilon)^2\cdot {2\lambda} \cdot \langle x,u\rangle - 2\lambda \varepsilon \cdot \|x\| \label{eq:negativeinner}\\
&\geq (1-\varepsilon)^2 \|x-\lambda u\|^2 - 2\varepsilon\lambda \|x\| \nonumber\\
&\geq (1-\varepsilon)^2 \|x-\lambda u\|^2 - 2\varepsilon \|x-\lambda u\|^2 \label{eq:lambdabound}\\
&\geq (1-3\varepsilon)^2 \|x-\lambda u\|^2, \nonumber \end{align} where the last inequality holds, since $\varepsilon/16 \in (0,1/4]$. In \cref{eq:sarlosapplication} we use events i), ii), iii),
in \cref{eq:negativeinner} we use the fact that $\langle x, u \rangle \leq 0$, and in \cref{eq:lambdabound} we use the fact that $\langle x, u \rangle \leq 0$ and $\lambda \geq 0$ implies that $\|x-\lambda u \| \geq \lambda$ and $\|x-\lambda u \| \geq \|x\|$. \end{proof}
We now prove our main technical lemma. This says that given a fixed line and a fixed point $p$, there is a set $P$ of points such that any linear $(1\pm\varepsilon)$-embedding for $P$ does not contract distances between $p$ and any point on the line by a factor smaller than $(1-3\varepsilon)$. A somewhat similar statement appears in \cite{DBLP:conf/random/Magen02} which however focuses on the distortion of point-line distances, i.e., how the distance between a point and its orthogonal projection onto the line changes after the embedding.
\begin{lemma}
\label{lem:affine_line_distance_contraction}
Let $x,y,z \in \mathbb{R}^d$ and $\ell = \{ \lp{\overline{yz}}{\lambda} \mid \lambda \in \mathbb{R} \}$ be the line supporting $\overline{yz}$. Let $f$ be a linear $(1\pm\varepsilon/16)$-embedding for $\{0,u,-u,x-(t+\langle x,u \rangle \cdot u)\}$, where $u \in \mathbb{S}^{d-1}$ and $t \in \mathbb{R}^d$, such that $\langle u, t \rangle = 0$ and $\{ t + \lambda u \mid \lambda \in \mathbb{R} \} = \ell$. For all $\lambda \in \mathbb{R}$ it holds that \[ \lVert f(x) - f(t + \lambda u) \rVert \geq (1-3\varepsilon) \lVert x - (t+\lambda u) \rVert.\] \end{lemma} \begin{proof} We first note that such an element $t$ exists, namely the orthogonal projection of $0$ onto $\ell$. Let $p = t + \langle x - t, u \rangle \cdot u = t + \langle x, u \rangle \cdot u$ be the projection of $x$ onto $\ell$ and let $x' = x-p$. Notice that \[\langle x',u \rangle =\langle x,u \rangle-\langle t+\langle x,u\rangle \cdot u,u\rangle = \langle x,u\rangle-\langle t ,u \rangle - \langle x , u \rangle = 0.\] We apply \cref{lemma:segmentnoncontraction0} on the vectors $x',u$. This implies that for any $\lambda\geq 0$, \begin{align*}
& \|f(x')-\lambda f(u) \| && \geq (1-3\varepsilon)\|x' -\lambda u\| \\
\iff & \|f(x-p)-\lambda f(u) \| && \geq (1-3\varepsilon)\|x-p -\lambda u\| \\
\iff & \|f(x-(t+\langle x, u \rangle \cdot u))-\lambda f(u) \| && \geq (1-3\varepsilon)\|x-(t+\langle x, u \rangle \cdot u) -\lambda u\| \\
\iff & \|f(x)-f(t)-\langle x, u \rangle \cdot f(u)-\lambda f(u) \|&& \geq (1-3\varepsilon)\|x-t-\langle x, u \rangle \cdot u -\lambda u\| \ \\
\iff & \|f(x)-f(t+(\langle x, u \rangle +\lambda) \cdot u) \|&& \geq (1-3\varepsilon)\|x-(t+(\langle x, u \rangle +\lambda) \cdot u) \|. \end{align*} Now by reparametrizing $\lambda^\prime \gets \langle x,u \rangle +\lambda$, we conclude that for any $\lambda^\prime \geq \langle x, u \rangle $, \begin{align} \label{eq:largelambdas}
\|f(x)-f(t+\lambda^\prime \cdot u) \|& \geq (1-3\varepsilon)\|x-(t+\lambda^\prime \cdot u) \| . \end{align}
Finally, we apply \cref{lemma:segmentnoncontraction0} on the vectors $x',-u$. Notice that $\langle x' ,-u\rangle = -\langle x',u \rangle =0$. This implies that for any $\lambda\geq 0$, \begin{align*}
& \|f(x')-\lambda f(-u) \| && \geq (1-3\varepsilon)\|x' -\lambda (-u)\| \\
\iff & \|f(x-p)-\lambda f(-u) \| && \geq (1-3\varepsilon)\|x-p -\lambda (-u)\| \\
\iff & \|f(x-(t+\langle x, u \rangle \cdot u))-\lambda f(-u) \| && \geq (1-3\varepsilon)\|x-(t+\langle x, u \rangle \cdot u) -\lambda (-u)\| \\
\iff & \|f(x)-f(t)-\langle x, u \rangle \cdot f(u)-\lambda f(-u) \| && \geq (1-3\varepsilon)\|x-t-\langle x, u \rangle \cdot u -\lambda(- u)\| \\
\iff & \|f(x)-f(t+(\langle x, u \rangle -\lambda) \cdot u) \| && \geq (1-3\varepsilon)\|x-(t+(\langle x, u \rangle -\lambda) \cdot u) \|. \end{align*} Now by reparametrizing $\lambda^\prime \gets \langle x,u \rangle -\lambda$, we conclude that for any $\lambda^\prime \leq \langle x, u \rangle $, \begin{align} \label{eq:smalllambdas}
\|f(x)-f(t+\lambda^\prime \cdot u) \|& \geq (1-3\varepsilon)\|x-(t+\lambda^\prime \cdot u) \| . \end{align} \cref{eq:largelambdas} and \cref{eq:smalllambdas} conclude the lemma. \end{proof}
Using the lemma above we can finally prove the main result of this section.
\begin{lemma}
\label{lem:lower_bound}
Let $\sigma, \tau$ be polygonal curves with vertices $v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}$, respectively $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$, and let $f$ be a linear $(1 \pm \varepsilon/48)$-embedding for $P = \{v^\sigma_1, \dots, v^\sigma_{\lvert \sigma \rvert}, v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert} \} \cup P^\prime$, where $P^\prime$ is a set of points determined by $\sigma$ and $\tau$ with $\lvert P^\prime \rvert \in O(\lvert \sigma \rvert \cdot \lvert \tau \rvert)$.
Let $\sigma^\prime$ and $\tau^\prime$ be polygonal curves with vertices $f(v^\sigma_1), \dots, f(v^\sigma_{\lvert \sigma \rvert})$, respectively $f(v^\tau_1), \dots, f(v^\tau_{\lvert \tau \rvert})$. It holds that
\begin{itemize}
\item $\dwf(\sigma^\prime, \tau^\prime) \geq (1-\varepsilon) \dwf(\sigma, \tau)$ and
\item $\df(\sigma^\prime, \tau^\prime) \geq (1-\varepsilon) \df(\sigma, \tau)$.
\end{itemize} \end{lemma} \begin{proof}
For the first claim, let $r = \dwf(\sigma, \tau)$, for the second claim let $r = \df(\sigma, \tau)$. In both cases, let $r^\prime = (1-\varepsilon)r$.
\\
In the following, we prove that for any (monotone) valid sequence $\mathcal{F}$ and any $\delta > 0$ we have that $(P_1)^{\sigma^\prime, \tau^\prime, r^\prime-\delta} \wedge (P_2)^{\sigma^\prime, \tau^\prime, r^\prime-\delta} \wedge \Psi_w^{\sigma^\prime, \tau^\prime, r^\prime-\delta}(\mathcal{F})$, respectively $(P_1)^{\sigma^\prime, \tau^\prime, r^\prime-\delta} \wedge (P_2)^{\sigma^\prime, \tau^\prime, r^\prime-\delta} \wedge \Psi^{\sigma^\prime, \tau^\prime, r^\prime-\delta}(\mathcal{F})$, is false and therefore $\dwf(\sigma^\prime, \tau^\prime) > r^\prime - \delta$, respectively $\df(\sigma^\prime, \tau^\prime) > r^\prime - \delta$ by \cref{theo:predicates_weak_frechet}, respectively \cref{theo:predicates_frechet}.
Now, let $\mathcal{F}$ be an arbitrary (monotone) valid sequence. By definition of $r$ and \cref{theo:predicates_weak_frechet}, respectively \cref{theo:predicates_frechet}, we know that for any $\delta > 0$ it holds that $(P_1)^{\sigma,\tau,r-\delta} \wedge (P_2)^{\sigma,\tau,r-\delta} \wedge \Psi_w^{\sigma,\tau,r-\delta}(\mathcal{F})$, respectively $(P_1)^{\sigma,\tau,r-\delta} \wedge (P_2)^{\sigma,\tau,r-\delta} \wedge \Psi^{\sigma,\tau,r-\delta}(\mathcal{F})$, is false. If $(P_1)^{\sigma,\tau,r-\delta}$ or $(P_2)^{\sigma,\tau,r-\delta}$ is false then clearly $(P_1)^{\sigma^\prime,\tau^\prime,r^\prime-\delta}$ or $(P_2)^{\sigma^\prime,\tau^\prime,r^\prime-\delta}$ is also false by \cref{def:predicates,def:embedding}. In the following, we assume that $\Psi_w^{\sigma,\tau,r-\delta}(\mathcal{F})$, respectively $\Psi^{\sigma,\tau,r-\delta}(\mathcal{F})$ is false.
Since the arguments for predicates of type $P_3$ and $P_4$ are analogous, we focus on the former type. Assume that $\Psi_w^{\sigma,\tau,r-\delta}(\mathcal{F})$ is false because a predicate $(P_3)^{\sigma,\tau,r-\delta}_{(i,j)}$ is false. This means that there does not exist a point $p \in \overline{v^\sigma_i v^\sigma_{i+1}}$ with $\lVert p - v^\tau_j \rVert \leq r - \delta$. At this point, recall that since $f$ is linear, any points $\lp{\overline{pq}}{t_1}, \dots, \lp{\overline{pq}}{t_n}$, where $p,q \in \mathbb{R}^d$, are still collinear when $f$ is applied and the relative order on the directed lines supporting $\overline{pq}$ is preserved, which is immediate since $f(\lp{\overline{pq}}{t_i}) = \lp{\overline{f(p)f(q)}}{t_i}$. By \cref{lem:affine_line_distance_contraction} for any $t \in \mathbb{R}$ and the determined point $p = \lp{\overline{v^\sigma_i v^\sigma_{i+1}}}{t}$ on the line supporting $\overline{v^\sigma_i v^\sigma_{i+1}}$ it holds that $\lVert f(v^\tau_j) - f(p) \rVert \geq (1-\varepsilon) \lVert p - v^\tau_j \rVert$. Thus, for any $f(p) \in \overline{f(v^\sigma_i) f(v^\sigma_{i+1})}$ we have $p \in \overline{v^\sigma_i v^\sigma_{i+1}}$ and $\lVert f(v^\tau_j) - f(p) \rVert \geq (1-\varepsilon) \lVert p - v^\tau_j \rVert$, which in conclusion is larger than $r^\prime - \delta$, hence $(P_3)^{\sigma^\prime,\tau^\prime,r^\prime-\delta}_{(i,j)}$ is false and therefore $\Psi_w^{\sigma^\prime,\tau^\prime,r^\prime-\delta}(\mathcal{F})$ is false. The first claim follows by \cref{theo:predicates_weak_frechet}.
Now, since again the arguments for predicates of type $P_5$ and $P_6$ are also analogous, we focus on the former. Assume that $\Psi^{\sigma, \tau, r - \delta}(\mathcal{F})$ is false, because a predicate $(P_5)^{\sigma,\tau,r-\delta}_{(i,j,k)}$ is false. This means that for any two $t_1, t_2 \in \mathbb{R}$ with $t_1 \leq t_2$, the points $p_1 = \lp{\overline{v^\sigma_j v^\sigma_{j+1}}}{t_1}$ and $p_2 = \lp{\overline{v^\sigma_j v^\sigma_{j+1}}}{t_2}$ do not satisfy $\lVert v^\tau_i - p_1 \rVert \leq r - \delta$ or $\lVert v^\tau_k - p_2 \rVert \leq r - \delta$. Since by \cref{lem:affine_line_distance_contraction} we have $\lVert f(v^\tau_i) - f(p_1) \rVert \geq (1-\varepsilon) \lVert v^\tau_i - p_1 \rVert$ and $\lVert f(v^\tau_k) - f(p_2) \rVert \geq (1-\varepsilon) \lVert v^\tau_k - p_2 \rVert$, one of these distances must be larger than $r^\prime - \delta$ and it follows that $(P_5)^{\sigma^\prime, \tau^\prime, r^\prime - \delta}_{(i,j,k)}$ is false. Therefore, $\Psi^{\sigma^\prime, \tau^\prime, r^\prime-\delta}(\mathcal{F})$ is false and the second claim follows by \cref{theo:predicates_frechet}.
Finally, for the above statements to hold, the set $P^\prime$ contains $0$, both directions $u,-u \in \mathbb{S}^{d-1}$ determined by an edge of $\sigma$ or $\tau$ and all points $x - (t+\langle x, u \rangle \cdot u)$, where $x$ is a vertex of a curve $\sigma$ or $\tau$, and $t,u$ determine a line supporting an edge of $\tau$ or $\sigma$. \end{proof}
\subsection{Main result}
\label{ss:mainresults} We now prove our main result which combines the upper and lower bounds on the distortion and \cref{lem:upper_bound}.
\begin{theorem}
\label{coro:JLembedding}
Let $T = \{ \tau_1, \dots, \tau_n \}$ be a set of polygonal curves in $\mathbb{R}^d$, each of complexity at most $m$. There exists a probability distribution over linear maps $f \colon {\mathbb R}^d \to {\mathbb R}^{d^{\prime}}$, where $d^{\prime} \in O(\varepsilon^{-2}\log (nm))$,
such that with high probability over the choice of $f$, the following is true for all $\sigma, \tau \in T$:
\begin{itemize}
\item $\lvert \dwf(\sigma, \tau) - \dwf(F(\sigma), F(\tau)) \rvert \leq \varepsilon \cdot \dwf(\sigma, \tau)$ and
\item $\lvert \df(\sigma, \tau) - \df(F(\sigma), F(\tau)) \rvert \leq \varepsilon \cdot \df(\sigma, \tau)$,
\end{itemize}
where for any $\tau \in T$ with vertices $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$ we let $F(\tau)$ be the curve with vertices $f(v^\tau_1), \dots, f(v^\tau_{\lvert \tau \rvert})$. \end{theorem} \begin{proof} We apply \cref{JLlemma} on the set $P$ of $O(n^2 m^3)$ points determined by an application of \cref{lem:upper_bound,lem:lower_bound} on all pairs of curves in $T$. \end{proof}
\section{Application to Clustering} \label{section:clustering}
In this section, we study the effect of randomized $(1\pm\varepsilon)$-embeddings on the cost of $k$-clustering of polygonal curves. In particular, we show that a constant factor approximation of the cost of the optimal $k$-center solution can be computed with an algorithm, which, except for the time needed to embed the input curves, runs in time independent of the input dimensionality. Moreover, we show that the optimal cost of the $k$-median problem is preserved within a constant factor in the target space. This means that running any algorithm for the $k$-median problem in the target space, yields an algorithm for estimating the cost in the original space.
This effectively reduces the computational effort required for approximating the clustering cost, and it directly assists analytical tasks like estimating the optimal number of clusters -- where cost estimations for multiple values of $k$ are typically performed.
\subsection{Clustering Under the Fr\'echet Distance}
In 2016, Driemel et al. \cite{DBLP:conf/soda/DriemelKS16} introduced clustering under the Fr\'echet distance, for the purpose of clustering (one-dimensional) time series. The objectives, named $(k,\ell)$-center and $(k,\ell)$-median, are derived from the well-known $k$-center and $k$-median objectives in Euclidean $k$-clustering. Both are $\mathrm{NP}$-hard \cite{DBLP:conf/soda/DriemelKS16,DBLP:conf/soda/BuchinDGHKLS19,DBLP:conf/swat/BuchinDS20}, even if $k=1$ and $d=1$, and the $(k,\ell)$-center problem is even $\mathrm{NP}$-hard to approximate within a factor of $(2.25 - \varepsilon)$ in general dimensions~\cite{DBLP:conf/soda/BuchinDGHKLS19}. One particularity of these clustering approaches is that the obtained center curves should be of low complexity. In detail, while the given curves have complexity at most $m$ each, the centers should be of complexity at most $\ell$ each, where $\ell \ll m$ is a constant. The idea behind is that due to the linear interpolation, a \textit{compact} summary of the cluster members through an aggregate center curve is enabled. A nice side effect is that overfitting, which may occur without the complexity restriction, is suppressed. For further details see~\cite{DBLP:conf/soda/DriemelKS16}.
We now present a modification of the constant factor approximation algorithm for $(k,\ell)$-center clustering from \cite{DBLP:conf/soda/BuchinDGHKLS19}. We note that due to its appealing complexity, this algorithm is used vastly in practice (c.f. \cite{DBLP:conf/gis/BuchinDLN19}) and therefore constitutes a prime candidate to be combined with dimensionality reduction.
\subsection{\texorpdfstring{$(k,\ell)$}{(k,l)}-Center Clustering} \label{ss:center}
We formally define the $(k,\ell)$-center clustering objective.
\begin{definition}
The $(k,\ell)$-center clustering problem is to compute a set $C$ of $k$ polygonal curves in $\mathbb{R}^d$, of complexity at most $\ell$ each, which minimizes the cost $\max_{\tau \in T} \min_{c\in C} \df(\tau, c)$, where $T = \{ \tau_1, \dots, \tau_n \}$ is a given set of polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each, and $k \in \mathbb{N}, \ell \in \mathbb{N}_{\geq 2}$ are constant parameters of the problem. \end{definition}
The following algorithm largely makes use of simplifications of input curves. We formally define this concept.
\begin{definition}
An $\alpha$-approximate minimum-error $\ell$-simplification of a curve $\tau$ in $\mathbb{R}^d$ is a curve $\sigma = \simpl(\tau)$ in $\mathbb{R}^d$ with at most $\ell$ vertices, where $\ell \in \mathbb{N}_{\geq 2}$ and $\alpha \geq 1$ are given parameters, such that $\df(\tau, \sigma) \leq \alpha \cdot \df(\tau, \sigma^\prime)$ for all other curves $\sigma^\prime$ with $\ell$ vertices. \end{definition}
A simplification $\sigma = \simpl(\tau)$ is vertex-restricted if the sequence of its vertices is a subsequence of the sequence of $\tau$s vertices. Crucial in our modification of the algorithm by Buchin et al. \cite{DBLP:conf/soda/BuchinDGHKLS19} is that we want to compute simplifications in the dimensionality-reduced ambient space to spare running time. In the following, we give a thorough analysis of the effect of dimensionality reduction before simplification. The proof can be found in \cref{sec:appendix}.
\begin{theorem}
\label{theo:simpl_embedding}
Let $F$ be the embedding of \cref{coro:JLembedding} with parameter $\varepsilon \in (0,1/2]$, for a given set $T$ of $n$ polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each, all segments $\overline{v^\tau_i v^\tau_j}$, all subcurves $\tau[i,j]$ as well as all vertex-restricted $\ell$-simplifications of all $\tau \in T$ (where $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$ are the vertices of $\tau$ and $i,j \in [\lvert \tau \rvert]$ with $i < j$). For each $\tau \in T$, a $4$-approximate minimum-error $\ell$-simplification $\simpl(F(\tau))$ of $F(\tau)$ can be computed in time $O(d^\prime \cdot \lvert \tau \rvert^3 \log \lvert \tau \rvert)$ and for all $\sigma \in T$ it holds that \[(1-\varepsilon) \df(\sigma, \simpl(\tau)) \leq \df(F(\sigma), \simpl(F(\tau)) \leq (1+\varepsilon) \df(\sigma, \simpl(\tau)),\]where $\simpl(\tau)$ denotes a $(4+16\varepsilon)$-approximate minimum-error $\ell$-simplification of $\tau$. \end{theorem}
We now present our modification of the algorithm. Let $F$ denote the embedding from \cref{coro:JLembedding} for $T \cup T^\prime \cup C^\ast$, where $T^\prime$ is the set of all segments $\overline{v^\tau_i v^\tau_j}$, all subcurves $\tau[i,j]$ as well as all vertex-restricted $\ell$-simplifications of all $\tau \in T$ (where $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$ are the vertices of $\tau$ and $i,j \in [\lvert \tau \rvert]$ with $i < j$), and $C^\ast$ is an optimal set of $k$ centers for $T$.
The algorithm first sets $C = \{ \simpl(F(\tau)) \}$ for an arbitrary $\tau \in T$. Then, until $\lvert C \rvert = k$ it computes a curve $\tau \in T$ that maximizes $\min_{c \in C} \df(F(\tau), c)$ and sets $C = C \cup \{ \simpl(F(\tau)) \}$. Finally, it returns $C$.
We now prove the approximation guarantee and analyse the running time of this algorithm, thereby we adapt parts of the analysis in \cite{DBLP:conf/soda/BuchinDGHKLS19}. The proof can be found in \cref{sec:appendix}.
\begin{theorem}
\label{theo:klcenter}
Given a set $T$ of $n$ polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each, and a parameter $\varepsilon \in (0,1/2]$, the above algorithm returns a solution $C$ to the $(k,\ell)$-center clustering problem, consisting of $k$ curves in $\mathbb{R}^{O(\varepsilon^{-2}\ell\log(knm))}$ of complexity at most $\ell$ each, such that \[ (1-3\varepsilon) r^\ast \leq \max_{\tau \in T} \min_{c \in C} \df(F(\tau), c) \leq (6+38\varepsilon) r^\ast,\]
where $r^\ast$ denotes the cost of an optimal solution. The algorithm has running time \[O(\varepsilon^{-2} k \ell \log(nm+k) m^3 \log m + \varepsilon^{-2} \ell \log(nm+k) k^2 nm \log m).\] \end{theorem}
\subsection{\texorpdfstring{$(k,\ell)$}{(k,l)}-Median Clustering}
In this section, we show that the cost of the optimal $(k,\ell)$-median solution is preserved within a constant factor, when projecting the input curves as described in \cref{section:embedding}. We first define the $(k,\ell)$-median clustering problem.
\begin{definition}
The $(k,\ell)$-median clustering problem is to compute a set $C$ of $k$ polygonal curves in $\mathbb{R}^d$ of complexity at most $\ell$ each, which minimizes the cost $\sum_{\tau \in T} \min_{c\in C} \df(\tau, c)$, where $T = \{ \tau_1, \dots, \tau_n \}$ is a given set of polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each, and $k \in \mathbb{N},\ell \in \mathbb{N}_{\geq 2}$ are constant parameters of the problem. \end{definition}
In \cref{sssunrestrictedmedians}, we focus on the case $\ell \geq m$, and we bound the distortion of the optimal cost by a factor of $2+O(\varepsilon)$. In \cref{sssrestrictedmedians}, we discuss case $\ell < m$, and we bound the distortion of the optimal cost by a factor of $6+O(\varepsilon)$.
\subsubsection{Unrestricted medians} \label{sssunrestrictedmedians} In this section, we present our results on the $(k,\ell)$-median clustering problem, when $\ell\geq m$.
Computing medians of complexity $\ell=m$ is a widely accepted scenario following, for example, from the wide acceptance of local search methods for clustering, which explore candidate solutions from the set of input curves. The proof follows a similar reasoning as in \cref{sssrestrictedmedians} and is diverted to \cref{appendixrestrictedmedians}. Comparing to \cref{sssrestrictedmedians}, we obtain an improved bound on the approximation factor. This is mainly because simplifications are no longer needed in order to obtain a meaningful bound. \begin{restatable}{theorem}{unrestrictedcorollary} Let $T = \{ \tau_1, \dots, \tau_n \}$ be a set of polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each and let $\ell \geq m $. There exists a probability distribution over linear maps $f \colon {\mathbb R}^d \to {\mathbb R}^{d^{\prime}}$, where $d^{\prime} \in O(\varepsilon^{-2}\log (n\ell))$, such that with high probability over the choice of $f$, the following is true. For any polygonal curve $\tau$ with vertices $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$, we define $F(\tau)$ to be the curve with vertices $f(v^\tau_1), \dots, f(v^\tau_{\lvert \tau \rvert})$. Then, \[ \frac{1-\varepsilon}{2}\cdot r^{\ast}\leq r_f^{\ast} \leq (1+\varepsilon)\cdot r^{\ast}, \] where $r^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $T$, and $r_f^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $F(T)$. \end{restatable}
\subsubsection{Restricted medians} \label{sssrestrictedmedians}
To bound the cost of the optimal $(k,\ell)$-median in the projected space, we use the notion of simplifications which was introduced in Section~\ref{ss:center}. By an averaging argument, for each cluster, there exists an input curve $\sigma_i$ which is within distance $ \frac{1}{|T_i|}\cdot \sum_{\tau\in T_i}\df(F(\tau),c_i^f)$ from the optimal median $c_i^f$, where $T_i$ is the input curves associated with the $i$\textsuperscript{th} cluster in the projected space. To lower bound the optimal cost in the projected space, we repeatedly apply the triangle inequality on distances involving a vertex-restricted $\ell$-simplification of $\sigma_i$ and a vertex-restricted
$\ell$-simplification of $F(\sigma_i)$. The upper bound simply follows by the non-contraction guarantee of JL transforms, on distances between input curves and the optimal medians in the original space. The complete proof can be found in \cref{appendixrestrictedmedians}. \begin{restatable}{theorem}{restrictedcorollary} Let $T = \{ \tau_1, \dots, \tau_n \}$ be a set of polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each and let $\ell < m $. There exists a probability distribution over linear maps $f \colon {\mathbb R}^d \to {\mathbb R}^{d^{\prime}}$, where $d^{\prime} \in O(\varepsilon^{-2} \ell \log (nm))$, such that with high probability over the choice of $f$, the following is true. For any polygonal curve $\tau$ with vertices $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$, we define $F(\tau)$ to be the curve with vertices $f(v^\tau_1), \dots, f(v^\tau_{\lvert \tau \rvert})$. Then, \[ \frac{1-\varepsilon}{6\cdot(1+\varepsilon)}\cdot r^{\ast}\leq r_f^{\ast} \leq (1+\varepsilon)\cdot r^{\ast}, \] where $r^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $T$, and $r_f^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $F(T)$. \end{restatable}
\section{Conclusion} \label{section:conclusions}
Our results are in line with the results by Magen \cite{DBLP:conf/random/Magen02,DBLP:journals/dcg/Magen07} in the sense that by increasing the constant hidden in the $O$-notation specifying the number of dimensions of the dimensionality-reduced space, JL transforms become more powerful and do not only preserve pairwise Euclidean distances but also affine distances, angles and volumes, and as we have proven, Fr\'echet distances.
Concerning JL transforms we have improved the work by Meintrup et al. \cite{DBLP:conf/nips/MeintrupMR19} by proving that no additive error is involved in the resulting Fr\'echet distances. To facilitate this result, we had to incorporate the linearity of these transforms, which is not done in \cite{DBLP:conf/nips/MeintrupMR19}. Interestingly, this shows that when one uses a terminal embedding instead (see e.g. \cite{DBLP:conf/stoc/NarayananN19}) -- for example to handle a dynamic setting involving queries -- this may induce an additive error to the Fr\'echet distance, as the results by Meintrup et al. \cite{DBLP:conf/nips/MeintrupMR19} can still be applied but ours can not since terminal embeddings are non-linear. Consequently, in contrast to Euclidean distances where a terminal embedding constitutes a proper extension of a JL embedding, this may not be the case when it comes to Fr\'echet distances.
One open question of practical importance is whether one can improve our result for polygonal curves that satisfy some realistic structural assumption, e.g., $c$-packness~\cite{DBLP:journals/dcg/DriemelHW12}. Moreover, it is possible that our implications on clustering can be improved. One question there is whether one can reduce (or eliminate) the dependence on $n$ from the target dimension, in the same spirit as with the analogous results for the Euclidean distance~\cite{MMR19}.
\appendix
\section{Missing Proofs} \label{sec:appendix}
\begin{proof}[Proof of \cref{lem:upper_bound}]
For the first claim, let $r = \dwf(\sigma, \tau)$, for the second claim let $r = \df(\sigma, \tau)$. In both cases, let $r^\prime = (1+\varepsilon)r$. Since $\sigma$ and $\tau$ have (weak) Fr\'echet distance $r$, there exists a (monotone) valid sequence $\mathcal{F}$, such that $(P_1)^{\sigma,\tau,r} \wedge (P_2)^{\sigma,\tau,r} \wedge \Psi_w^{\sigma,\tau,r}(\mathcal{F})$, respectively $(P_1)^{\sigma,\tau,r} \wedge (P_2)^{\sigma,\tau,r} \wedge \Psi^{\sigma,\tau,r}(\mathcal{F})$, is true, by \cref{theo:predicates_weak_frechet}, respectively \cref{theo:predicates_frechet}.
\\
Clearly, since $f$ is a $(1\pm \varepsilon)$-embedding, $(P_1)^{\sigma^\prime,\tau^\prime,r^\prime}$ and $(P_2)^{\sigma^\prime,\tau^\prime,r^\prime}$ are both true.
\\
We now denote \[\mathcal{I}_3 = \{ (i,j) \mid (i,j-1),(i,j) \in \mathcal{F} \},\ \mathcal{I}_4 = \{ (i,j) \mid (i-1,j),(i,j) \in \mathcal{F} \} \] and \[ \mathcal{I}_5 = \{ (i,j,k) \mid (i,j-1),(i,k) \in \mathcal{F}, j < k \},\ \mathcal{I}_6 = \{ (i,j,k) \mid (i-1,j),(k,j) \in \mathcal{F}, i < k \}.\] Furthermore, let $\mathcal{P}_3$ be the set of points $p \in \overline{v^\sigma_i v^\sigma_{i+1}}$ with $\lVert p - v^\tau_j \rVert \leq r$ guaranteed by $(P_3)^{\sigma,\tau,r}_{(i,j)}$, for all $(i,j) \in \mathcal{I}_3$, $\mathcal{P}_4$ be the set of points $p \in \overline{v^\tau_jv^\tau_{j+1}}$ with $\lVert p - v^\sigma_i \rVert \leq r$ guaranteed by $(P_4)^{\sigma,\tau,r}_{(i,j)}$, for all $(i,j) \in \mathcal{I}_4$, $\mathcal{P}_5$ be the set of points $p_1 = \lp{\overline{v^\sigma_jv^\sigma_{j+1}}}{t_1}, p_2 = \lp{\overline{v^\sigma_jv^\sigma_{j+1}}}{t_2}$ with $\lVert v^\tau_i - p_1 \rVert \leq r$, $\lVert v^\tau_k - p_2 \rVert \leq r$ and $t_1 \leq t_2$ guaranteed by $(P_5)^{\sigma,\tau,r}_{(i,j,k)}$, for all $(i,j,k) \in \mathcal{I}_5$, and $(P_6)^{\sigma,\tau,r}_{(i,j,k)}$ be the set of points $p_1 = \lp{\overline{v^\tau_i v^\tau_{i+1}}}{t_1}, p_2 = \lp{\overline{v^\tau_i v^\tau_{i+1}}}{t_2}$ with $\lVert v^\sigma_j - p_1 \rVert \leq r$, $\lVert v^\sigma_k - p_2 \rVert \leq r$ and $t_1 \leq t_2$ guaranteed by $(P_6)^{\sigma,\tau,r}_{(i,j,k)}$ for all $(i,j,k) \in \mathcal{I}_6$.
\\
We let $P^\prime = \mathcal{P}_3 \cup \mathcal{P}_4 \cup \mathcal{P}_5 \cup \mathcal{P}_6$. Clearly, for any $(i,j) \in \mathcal{I}_3$, respectively $(i,j) \in \mathcal{I}_4$, $(P_3)^{\sigma^\prime,\tau^\prime, r^\prime}_{(i,j)}$, respectively $(P_4)^{\sigma^\prime,\tau^\prime, r^\prime}_{(i,j)}$ is true, since there exist points $p,q \in f(P^\prime)$ with $p \in \overline{f(v^\sigma_i)f(v^\sigma_{i+1})}$ and $\lVert p - f(v^\tau_j) \rVert \leq r^\prime$, respectively $q \in \overline{f(v^\tau_j)f(v^\tau_{j+1})}$ and $\lVert p - f(v^\sigma_i) \rVert \leq r^\prime$. At this point, it can be observed that $(P_1)^{\sigma^\prime,\tau^\prime,r^\prime} \wedge (P_2)^{\sigma^\prime,\tau^\prime,r^\prime} \wedge \Psi_w^{\sigma^\prime,\tau^\prime, r^\prime}(\mathcal{F})$ is true and therefore the first claim follows by \cref{theo:predicates_weak_frechet}.
\\
For the following, observe that since $f$ is linear, any points $\lp{\overline{pq}}{t_1}, \dots, \lp{\overline{pq}}{t_n}$, where $p,q \in \mathbb{R}^d$, are still collinear when $f$ is applied and the relative order on the directed lines supporting $\overline{pq}$ is preserved -- this is immediate since $f(\lp{\overline{pq}}{t_i}) = \lp{\overline{f(p)f(q)}}{t_i}$. We conclude that for any $(i,j,k) \in \mathcal{P}_5$, respectively $(i,j,k) \in \mathcal{P}_6$, $(P_5)^{\sigma^\prime,\tau^\prime,r^\prime}_{(i,j,k)}$, respectively $(P_6)^{\sigma^\prime,\tau^\prime,r^\prime}_{(i,j,k)}$, is true, since there exists points $p_1,p_2,q_1,q_2 \in f(P^\prime)$ with $p_1 = \lp{\overline{f(v^\sigma_j)f(v^\sigma_{j+1})}}{t_1}$, $p_2 = \lp{\overline{f(v^\sigma_j)f(v^\sigma_{j+1})}}{t_2}$, $\lVert f(v^\tau_i) - p_1 \rVert \leq r^\prime$, $\lVert f(v^\tau_k) - p_2 \rVert \leq r^\prime$ and $t_1 \leq t_2$, respectively $q_1 = \lp{\overline{f(v^\tau_i)f(v^\tau_{i+1})}}{t_3}$, $q_2 = \lp{\overline{f(v^\tau_i)f(v^\tau_{i+1})}}{t_4}$, $\lVert f(v^\sigma_j) - q_1 \rVert \leq r^\prime$, $\lVert f(v^\sigma_k) - q_2 \rVert \leq r^\prime$ and $t_3 \leq t_4$. Thus, $(P_1)^{\sigma^\prime,\tau^\prime,r^\prime} \wedge (P_2)^{\sigma^\prime,\tau^\prime,r^\prime} \wedge \Psi^{\sigma^\prime,\tau^\prime, r^\prime}(\mathcal{F})$ is true and by \cref{theo:predicates_frechet} the second claim follows. The cardinality of $P^\prime$ is determined by the cardinalities of $\mathcal{I}_3, \dots \mathcal{I}_6$, which in turn can be bounded as stated in the theorem statement. \end{proof}
\begin{proof}[Proof of \cref{lem:inner_product}]
In the following, we assume that $\lVert p \rVert = \lVert q \rVert = 1$. To prove the upper bound, observe that
\begin{align*}
2 \langle f(p), f(q) \rangle & = \lVert f(p) - f(0) \rVert^2 + \lVert f(q) - f(0) \rVert^2 - \lVert f(p) - f(q) \rVert^2 \\
& \leq (1+\varepsilon)^2\lVert p \rVert^2 + (1+\varepsilon)^2\lVert q \rVert^2 - (1-\varepsilon)^2 \lVert p - q \rVert^2 \\
& \leq 2 \langle p, q \rangle + 6 \varepsilon + 2\varepsilon \lVert p - q \rVert^2 \\
& \leq 2 \langle p, q \rangle + 6 \varepsilon + 2\varepsilon(\lVert p \rVert + \lVert q \rVert)^2 = 2 \langle p, q \rangle + 14 \varepsilon,
\end{align*}
where the last inequality follows from the triangle inequality. To prove the lower bound, observe that
\begin{align*}
2 \langle f(p), f(q) \rangle & \geq (1-\varepsilon)^2 \lVert p \rVert^2 + (1-\varepsilon)^2 \lVert q \rVert^2 - (1+\varepsilon)^2\lVert p - q \rVert^2 \\
& \geq 2 \langle p, q \rangle - 2\varepsilon \lVert p \rVert^2 - 2\varepsilon \lVert q \rVert^2 - 3 \varepsilon \lVert p - q \rVert^2 \\
& \geq 2 \langle p, q \rangle - 4 \varepsilon - 3\varepsilon(\lVert p \rVert + \lVert q \rVert)^2 = 2 \langle p, q \rangle - 16 \varepsilon,
\end{align*}
where the last inequality again follows from the triangle inequality. Using the linearity of the dot product and $f$, we have that $\langle f(p), f(q) \rangle = \lVert p \rVert \cdot \lVert q \rVert \cdot \langle f(p/\lVert p \rVert), f(q/\lVert q \rVert) \rangle$, which yields the claim. \end{proof}
\begin{proof}[Proof of \cref{theo:simpl_embedding}]
We use the approach from \cite[Lemma 7.1]{DBLP:conf/soda/BuchinDGHKLS19}. Here, the algorithms by Imai and Iri \cite{imai_polygonal_1988} and Alt and Godau \cite{alt_godau} are combined to obtain a vertex-restricted simplification. In detail, for the given curve $\tau$ with vertices $v^\tau_1, \dots, v^\tau_{\lvert \tau \rvert}$ a directed graph $G(\tau)$ is constructed. The vertices of the graph are the vertices of $\tau$ and it has an edge $(v^\tau_i, v^\tau_j)$ for $i,j \in [\lvert \tau \rvert]$ and $i < j$, assigned with weight $\df(\tau[i,j], \overline{v^\tau_i v^\tau_j})$. The simplification is determined by the path from $v^\tau_1$ to $v^\tau_{\lvert \tau \rvert}$ of $\ell$ vertices and with cost, i.e., maximum edge weight, minimized. Observe that the cost of the path is $\df(\tau, \simpl(\tau))$. The approximation factor and the running time follow from \cite[Lemma 7.1]{DBLP:conf/soda/BuchinDGHKLS19} (and by incorporating the dimension, which is assumed to be constant in these works).
\\
Let $\tau \in T$ and consider $\simpl(F(\tau))$ returned by the above algorithm. There exists a vertex-restricted $\ell$-simplification $\tau^\prime$ of $\tau$, such that $\simpl(F(\tau)) = F(\tau^\prime)$.
Now, since $(1-\varepsilon) \df(\tau[i,j], \overline{v^\tau_i, v^\tau_j}) \leq \df(F(\tau[i,j]), F(\overline{v^\tau_i v^\tau_j}) \leq (1+\varepsilon) \df(\tau[i,j], \overline{v^\tau_i, v^\tau_j})$ by \cref{coro:JLembedding}, it may be that the minimum cost path in $G(F(\tau))$ from $f(v^\tau_1)$ to $f(v^\tau_{\lvert \tau \rvert})$ of $\ell$ vertices is by a factor of $(1-\varepsilon)$ cheaper than the corresponding path in $G(\tau)$, and that the path in $G(F(\tau))$ corresponding to the minimum cost path in $G(\tau)$ from $v^\tau_1$ to $v^\tau_{\lvert \tau \rvert}$ of $\ell$ vertices is by a factor of $(1+\varepsilon)$ more expensive. Thus, $\df(\tau, \tau^\prime) \leq (1+\varepsilon)/(1-\varepsilon) \df(\tau, \simpl(\tau)) \leq (1+4\varepsilon) \df(\tau, \simpl(\tau))$, for the given range of $\varepsilon$. We conclude that $\tau^\prime$ is a $(4+16\varepsilon)$-approximate minimum-error $\ell$-simplification of $\tau$.
\\
The remainder follows by the embedding guarantee of \cref{coro:JLembedding}. \end{proof}
\begin{proof}[Proof of \cref{theo:klcenter}]
For $i \in [k]$ we denote by $C_i$ the set of centers computed after the $i$\textsuperscript{th} iteration of the algorithm and let $r_i = \max_{\tau \in T} \min_{c \in C_i} \df(F(\tau), c)$. Clearly, $r_1 \geq \ldots \geq r_k$ and $r_k$ is the cost of the solution $C_k$. Furthermore, we let $C^\ast = \{c^\ast_1, \dots, c^\ast_k\}$ denote an optimal set of centers with cost $r^\ast$ and for $i \in [k]$ we let $T^\ast_i = \{\tau \in T \mid \forall j \in [k]: \df(\tau,c^\ast_i) \leq \df(\tau, c^\ast_j)\}$ denote the $i$\textsuperscript{th} optimal cluster, where we assume that ties are broken arbitrarily, such that $T^\ast_1, \dots, T^\ast_k$ form a partition of $T$.
\\
We prove the upper bound. Let $\sigma \in T$ be a curve that maximizes $\min_{c \in C_k} \df(F(\tau), c)$ among all $\tau \in T$, i.e., $\min_{c \in C_k} \df(F(\sigma), c) = r_k$ and let $C_{k+1} = C_{k} \cup \{\simpl(F(\sigma))\}$. By the pigeonhole principle there are two curves $c = \simpl(F(\tau)), c^\prime = \simpl(F(\tau^\prime)) \in C_{k+1}$, such that $\tau$ and $\tau^\prime$ that lie in the same optimal cluster $T^\ast_j$. W.l.o.g. assume that $c$ is added in an earlier iteration of the algorithm. We have
\begin{align*}
r_k \leq & \df(c, F(\tau^\prime)) \leq \df(\simpl(F(\tau)), F(\tau)) + \df(F(\tau), F(\tau^\prime)) \\
\leq & (1+\varepsilon)(\df(\tau,c^\ast_j) + \df(c^\ast_j, \tau^\prime)) + \df(\simpl(F(\tau)), F(\tau)) \\
\leq & 2(1+\varepsilon) r^\ast + \df(\simpl(F(\tau)), F(\tau)) \leq (6+38\varepsilon) r^\ast,
\end{align*}
where the last inequality follows from \cref{theo:simpl_embedding} and by the fact that $\simpl(\tau)$ is a $(4+16\varepsilon)$-approximate minimum-error $\ell$-simplification of $\tau$, hence
\begin{align*}
\df(\simpl(F(\tau)), F(\tau)) & \leq (1+\varepsilon) \cdot \df(\simpl(\tau), \tau) \leq (4+16\varepsilon) (1+\varepsilon) \df(c^\ast_j, \tau) \\
& \leq (4+16\varepsilon) (1+\varepsilon) r^\ast \leq (4+36\varepsilon)r^\ast.
\end{align*}
We prove the lower bound. For $i \in [k]$, let $c_i = \simpl(F(\tau_i))$ denote the single element of $C_i \setminus C_{i-1}$, where we let $C_0 = \emptyset$, and let $\sigma \in T$ be a curve that maximizes $\min_{i \in [k]} \df(F(\tau), c_i)$ among all $\tau \in T$. Thus, $\min_{c \in C_k} \df(F(\sigma), c) = r_k$.
\\
Now, let $C = \{\simpl(\tau_1), \dots, \simpl(\tau_k)\}$ and $\sigma^\prime \in T$ be a curve that maximizes $\min_{i \in [k]} \df(\tau, \simpl(\tau_i))$ among all $\tau \in T$. Also, let $r = \df(\sigma^\prime, \simpl(\tau_i))$. Clearly, $r \geq r^\ast$ must hold, since $C^\ast$ is an optimal solution, so $C$ must have equal or larger cost.
\\
Furthermore, by \cref{theo:simpl_embedding} and by definition of $\sigma$ and $\sigma^\prime$ it holds that \[(1+\varepsilon) \min_{i \in [k]} \df(\simpl(\tau_i), \sigma) \geq \min_{i \in [k]} \df(c_i, F(\sigma)) \geq \min_{i \in [k]} \df(c_i, F(\sigma^\prime)) \geq (1-\varepsilon) \min_{i \in [k]} \df(\simpl(\tau_i), \sigma^\prime).\]
We have
\begin{align*}
r_k = & \min_{i \in [k]} \df(c_i, F(\sigma)) = \min_{i \in [k]} \df(\simpl(F(\tau_i)), F(\sigma)) \geq (1-\varepsilon) \min_{i \in [k]} \df(\simpl(\tau_i), \sigma) \\
& \geq \frac{(1-\varepsilon)^2}{(1+\varepsilon)} r \geq (1-3\varepsilon)r^\ast,
\end{align*}
where the first inequality follows from \cref{theo:simpl_embedding}.
\\
We now discuss the running time. First, exactly $k$ simplifications are computed during the execution of the algorithm. This has running time $O(k \cdot d^\prime m^3 \log m)$ by \cref{theo:simpl_embedding}. Then, in $k$ rounds, the algorithm computes $i \cdot n$ Fr\'echet distances to the $i$ centers already computed using Alt and Godau's algorithm. This takes time $O(k^2 n d^\prime m \log m)$. \end{proof}
\section{Unrestricted medians} \label{appendixunrestrictedmedians} We begin by lower bounding the cost of the optimal solution on the set of projected curves.
\begin{lemma} \label{lemma:continuousmedianlowerbound} Let $T$ be a set of $n$ polygonal curves in $\mathbb{R}^d$ of complexity at most $m$ each. Let $\ell \geq m$ and let $F$ be the embedding of \cref{coro:JLembedding} for $T$, with parameter $\varepsilon \in (0,1)$. Then, \[ r^{\ast}\leq \frac{2}{1-\varepsilon}\cdot r_f^{\ast}, \] where $r^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $T$, and $r_f^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $F(T)$. \end{lemma} \begin{proof}
Let $\{c_1^{f},\ldots,c_k^{f}\}$ be an optimal $(k,\ell)$-median solution on $F(T)$, let $T_1^f,\ldots, T_k^f$ be the corresponding subsets (clusters) of $F(T)$ associated with them and let $T_i = F^{-1}(T_i^f)$.
By an averaging argument, for each $i\in[k]$ there exists a $\sigma_i\in T_i $ such that $\df(F(\sigma_i), c_i^f)\leq \frac{1}{|T_i|}\cdot \sum_{\tau\in T_i}\df(F(\tau),c_i^f)$. Then, \begin{align*} r^{\ast} &\leq \sum_{i=1}^k \sum_{\tau\in T_i} \d_F(\tau,\sigma_i) \leq \frac{1}{1-\varepsilon}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \df(F(\tau),F(\sigma_i)) \\&\leq \frac{1}{1-\varepsilon}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(F(\tau),{c}_i^f)+\df(c_i^f,F(\sigma_i))\right) \\&\leq \frac{1}{1-\varepsilon}\cdot \left( r_f^{\ast}+
\sum_{i=1}^k \sum_{\tau\in T_i} \frac{1}{|T_i|}\cdot \sum_{\tau\in T_i}\df(F(\tau),c_i^f)\right) \leq \frac{2}{1-\varepsilon} \cdot r_f^{\ast}. \end{align*} \end{proof}
Now, if we add the assumption that distances between the optimal medians in the original space and the input curves are approximately preserved, we obtain our main result. \begin{theorem} \label{theo:unremedians} Let $T$ be a set of $n$ polygonal curves in ${\mathbb R}^d$ of complexity at most $m$ each. Let $\ell\geq m$ and let $C^{\ast}$ be an optimal solution to the $(k,\ell)$-median problem with cost $r^{\ast}$. Let $F$ be the embedding of \cref{coro:JLembedding} for $T\cup C^{\ast}$, with parameter $\varepsilon \in (0,1)$. Then, \[ \frac{1-\varepsilon}{2}\cdot r^{\ast}\leq r_f^{\ast} \leq (1+\varepsilon)\cdot r^{\ast}, \] where $ r_f^{\ast}$ is the optimal cost of the $(k,\ell)$-median clustering problem with input $F(T)$. \end{theorem} \begin{proof} \cref{lemma:continuousmedianlowerbound} implies $\frac{1-\varepsilon}{2}\cdot r^{\ast}\leq r_f^{\ast} $. Moreover, since $F$ is as in \cref{coro:JLembedding}, it satisfies $\forall \tau, \sigma \in T \cup C^{\ast}: (1-\varepsilon) \df(\tau,\sigma) \leq \df(F(\tau),F(\sigma)) \leq (1+\varepsilon)\df(\tau,\sigma)$, we conclude that $F(C^{\ast})$ is a solution with cost at most $(1+\varepsilon) r^{\ast}$. Therefore $r_f^{\ast}\leq (1+\varepsilon)\cdot r^{\ast}$. \end{proof} Finally, we can apply a JL transform to obtain the following result on dimension reduction of curves. \unrestrictedcorollary* \begin{proof} The theorem follows by combining \cref{JLlemma}, \cref{coro:JLembedding} and \cref{theo:unremedians}. In particular, we apply \cref{JLlemma} for a set of $O((n+k)^2 \cdot \ell^3)$ points determined by $T \cup C^{\ast}$, where $C^\ast$ is an optimal $(k,\ell)$-median solution for $T$, to obtain the $(1\pm \varepsilon)$-embedding required by \cref{coro:JLembedding}. Then, \cref{coro:JLembedding} combined with \cref{theo:unremedians}, imply the statement. \end{proof}
\section{Restricted median} \label{appendixrestrictedmedians} We start with a lower bound on the optimal cost of the embedded curves. \begin{lemma} \label{lemma:continuousmedianlowerboundrestricted} Let $T$ be a set of $n$ polygonal curves in ${\mathbb R}^d$ of complexity at most $m$ each and let $k \in \mathbb{N},\ell \in {\mathbb N}_{\geq 2}$. Let $F$ be the embedding of \cref{coro:JLembedding} for $T \cup S(T)$, with parameter $\varepsilon \in (0,1)$, where $S(T)$ is the set of all vertex-restricted $\ell$-simplifications of all polygonal curves in $T$. Then, \[ r^{\ast}\leq \frac{6(1+\varepsilon)}{1-\varepsilon}\cdot r_f^{\ast}, \] where $r^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $T$, and $r_f^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $F(T)$. \end{lemma} \begin{proof} Let $\{c_1^{f},\ldots,c_k^{f}\}$ be an optimal $(k,\ell)$-median solution for $F(T)$, let $T_1^f,\ldots, T_k^f$ be the corresponding subsets (clusters) of $F(T)$ associated with them and let $T_i = F^{-1}(T_i^f)$.
By an averaging argument, for each $i\in[k]$ there exists a curve $\sigma_i\in T_i $ such that $\df(F(\sigma_i), c_i^f)\leq \frac{1}{|T_i|}\cdot \sum_{\tau\in T_i}\df(F(\tau),c_i^f)$. Let $\tilde{\sigma_i}$ be an optimal vertex-restricted minimum-error $\ell$-simplification of $\sigma_i$ and let $\tilde{\sigma}_i^f$ be an optimal vertex-restricted minimum-error $\ell$-simplification of $F(\sigma_i)$. Then, \begin{align} r^{\ast} &\leq \sum_{i=1}^k \sum_{\tau\in T_i} \df(\tau,\tilde{\sigma}_i) \nonumber\\& \leq
\sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(\tau,{\sigma}_i)+\df({\sigma}_i,\tilde{\sigma}_i) \right) \label{eq:trineq1}
\\& \leq \frac{1}{1-\varepsilon}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(F(\tau),F(\sigma_i)) +\df(F(\sigma_i),F(\tilde{\sigma}_i)) \right) \nonumber \\&\leq \frac{1}{1-\varepsilon}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(F(\tau),{c}_i^f)+\df(c_i^f,F(\sigma_i))+\df(F(\sigma_i),F(\tilde{\sigma}_i))\right) \label{eq:trineq2} \\&\leq \frac{1}{1-\varepsilon}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(F(\tau),{c}_i^f)+\df(c_i^f,F(\sigma_i))+\frac{1+\varepsilon}{1-\varepsilon}\cdot \df(F(\sigma_i),\tilde{\sigma}_i^f)\right) \label{eq:simplpres}\\ &\leq \frac{1+\varepsilon}{(1-\varepsilon)^2}\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \left(\df(F(\tau),{c}_i^f)+5\cdot \df(c_i^f,F(\sigma_i))\right) \label{eq:weaksimpl}\\ & \leq \frac{1+\varepsilon}{(1-\varepsilon)^2}\cdot \left( r_f^{\ast}+
5\cdot \sum_{i=1}^k \sum_{\tau\in T_i} \frac{1}{|T_i|}\cdot \sum_{\tau\in T_i}\df(F(\tau),c_i^f)\right)\nonumber\\ &\leq \frac{6\cdot(1+\varepsilon)}{(1-\varepsilon)^2} \cdot r_f^{\ast}\nonumber, \end{align} where in (\ref{eq:trineq1}), and (\ref{eq:trineq2}), we apply the triangle inequality, (\ref{eq:simplpres}) follows by the fact that $F$ approximately preserves distances to all vertex-restricted $\ell$-simplifications, and (\ref{eq:weaksimpl}) follows from \cite[Lemma 7.1]{DBLP:conf/soda/BuchinDGHKLS19} which states that there exists a vertex-restricted $\ell$-simplification which is within distance at most $4$ times that of any non-restricted $\ell$-simplification. \end{proof}
By assuming that distances between the input curves, the optimal medians and all vertex-restricted $\ell$-simplifications are approximately preserved, we obtain the following theorem. \begin{theorem}
Let $T$ be a set of $n$ polygonal curves in ${\mathbb R}^d$ of complexity at most $m$ each and let $k \in {\mathbb N}, \ell \in {\mathbb N}_{\geq 2}$. Let $F$ be the embedding of \cref{coro:JLembedding} for $T \cup S(T)\cup C^{\ast}$ with parameter $\varepsilon \in (0,1)$, where $S(T)$ is the set of all vertex-restricted $\ell$-simplifications of all polygonal curves in $T$ and $C^{\ast}$ is an optimal $(k,\ell)$-median solution for $T$. Then, \[ \frac{1-\varepsilon}{6\cdot(1+\varepsilon)}\cdot r^{\ast}\leq r_f^{\ast} \leq (1+\varepsilon)\cdot r^{\ast}, \] where $r^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $T$, and $r_f^{\ast}$ is the cost of an optimal solution to the $(k,\ell)$-median problem on $F(T)$. \end{theorem} \begin{proof} \cref{lemma:continuousmedianlowerboundrestricted} implies $\frac{1-\varepsilon}{6\cdot(1+\varepsilon)}\cdot r^{\ast}\leq r_f^{\ast} $. Moreover, since $F$ is as in \cref{coro:JLembedding}, it satisfies $\forall \tau, \sigma \in T \cup C^{\ast}: (1-\varepsilon) \df(\tau,\sigma) \leq \df(F(\tau),F(\sigma)) \leq (1+\varepsilon)\df(\tau,\sigma)$, we conclude that $F(C^{\ast})$ is a solution with cost at most $(1+\varepsilon) r^{\ast}$. Therefore $r_f^{\ast}\leq (1+\varepsilon)\cdot r^{\ast}$. \end{proof} Finally, we can effectively reduce the dimension by applying a JL transform. \restrictedcorollary* \begin{proof} The result follows by combining \cref{JLlemma}, \cref{coro:JLembedding} and \cref{theo:unremedians}. In particular, we apply \cref{JLlemma} for a set of $O(((n+nm^{\ell}+k)^2 \cdot m^3)$ points determined by $T \cup S(T)\cup C^{\ast}$, where $S(T)$ is the set of all vertex-restricted $\ell$-simplifications of all polygonal curves in $T$, and $C^{\ast}$ is an optimal $(k,\ell)$-median solution for $T$, to obtain the $(1\pm \varepsilon)$-embedding required by \cref{coro:JLembedding}. Then, \cref{coro:JLembedding} combined with \cref{theo:unremedians}, imply the statement. \end{proof}
\end{document} | arXiv |
Asian Pacific Journal of Cancer Prevention
Asian Pacific Organization for Cancer Prevention (아시아태평양암예방학회)
Health Sciences > Development of Pharmaceutical
The Asian Pacific Journal of Cancer Prevention is a monthly electronic journal publishing papers in all areas of cancer control. Its is indexed on PubMed (Impact factor for 2014 : 2.514) and the scope is wide-ranging: including descriptive, analytical and molecular epidemiology; experimental and clinical histopathology/biology of preneoplasias and early neoplasias; assessment of risk and beneficial factors; experimental and clinical trials of primary preventive measures/agents; screening approaches and secondary prevention; clinical epidemiology; and all aspects of cancer prevention education. All of the papers published are freely available as pdf files downloadable from www.apjcpcontrol.org, directly or through PubMed, or obtainable from the first authors. The APJCP is financially supported by the UICC Asian Regional Office and the National Cancer Center of Korea, where the Editorial Office is housed.
KSCI
Volume 17 Issue sup3
Phage Particles as Vaccine Delivery Vehicles: Concepts, Applications and Prospects
Jafari, Narjes;Abediankenari, Saeid 8019
https://doi.org/10.7314/APJCP.2015.16.18.8019 PDF KSCI
The development of new strategies for vaccine delivery for generating protective and long-lasting immune responses has become an expanding field of research. In the last years, it has been recognized that bacteriophages have several potential applications in the biotechnology and medical fields because of their intrinsic advantages, such as ease of manipulation and large-scale production. Over the past two decades, bacteriophages have gained special attention as vehicles for protein/peptide or DNA vaccine delivery. In fact, whole phage particles are used as vaccine delivery vehicles to achieve the aim of enhanced immunization. In this strategy, the carried vaccine is protected from environmental damage by phage particles. In this review, phage-based vaccine categories and their development are presented in detail, with discussion of the potential of phage-based vaccines for protection against microbial diseases and cancer treatment. Also reviewed are some recent advances in the field of phagebased vaccines.
β-Adrenergic Receptors : New Target in Breast Cancer
Wang, Ting;Li, Yu;Lu, Hai-Ling;Meng, Qing-Wei;Cai, Li;Chen, Xue-Song 8031
Background: Preclinical studies have demonstrated that ${\beta}$-adrenergic receptor antagonists could improve the prognosis of breast cancer. However, the conclusions of clinical and pharmacoepidemiological studies have been inconsistent. This review was conducted to re-assess the relationship between beta-adrenoceptor blockers and breast cancer prognosis. Materials and Methods: The literature was searched from PubMed, EMBASE and Web of Nature (Thompson Reuters) databases through using key terms, such as breast cancer and beta-adrenoceptor blockers. Results: Ten publications met the inclusion criteria. Six suggested that receiving beta-adrenoceptor blockers reduced the risk of breast cancer-specific mortality, and three of them had statistical significance (hazard ratio (HR)=0.42; 95% CI=0.18-0.97; p=0.042). Two studies reported that risk of recurrence and distant metastasis (DM) were both significantly reduced. One study demonstrated that the risk of relapse-free survival (RFS) was raised significantly with beta-blockers (BBS) (HR= 0.30; 95% CI=0.10-0.87; p=0.027). One reported longer disease-free interval (Log Rank (LR)=6.658; p=0.011) in BBS users, but there was no significant association between overall survival (OS) and BBS (HR= 0.35; 95% CI=0.12-1.0; p=0.05) in five studies. Conclusions: Through careful consideration, it is suggested that beta-adrenoceptor blockers use may be associated with improved prognosis in breast cancer patients. Nevertheless, larger size studies are needed to further explore the relationship between beta-blocker drug use and breast cancer prognosis.
Identification of HPV Integration and Genomic Patterns Delineating the Clinical Landscape of Cervical Cancer
Akeel, Raid-Al 8041
Cervical cancer is one of the most common cancers in women worldwide. During their life time the vast majority of women become infected with human papillomavirus (HPV), but interestingly only a small portion develop cervical cancer and in the remainder infection regresses to a normal healthy state. Beyond HPV status, associated molecular characterization of disease has to be established. However, initial work suggests the existence of several different molecular classes, based on the biological features of differentially expressed genes in each subtype. This suggests that additional risk factors play an important role in the outcome of infection. Host genomic factors play an important role in the outcome of such complex or multifactor diseases such as cervical cancer and are also known to regulate the rate of disease progression. The aim of this review was to compile advances in the field of host genomics of HPV positive and negative cervical cancer and their association with clinical response.
Potential Roles of Protease Inhibitors in Cancer Progression
Yang, Peng;Li, Zhuo-Yu;Li, Han-Qing 8047
Proteases are important molecules that are involved in many key physiological processes. Protease signaling pathways are strictly controlled, and disorders in protease activity can result in pathological changes such as cardiovascular and inflammatory diseases, cancer and neurological disorders. Many proteases have been associated with increasing tumor metastasis in various human cancers, suggesting important functional roles in the metastatic process because of their ability to degrade the extracellular matrix barrier. Proteases are also capable of cleaving non-extracellular matrix molecules. Inhibitors of proteases to some extent can reduce invasion and metastasis of cancer cells, and slow down cancer progression. In this review, we focus on the role of a few proteases and their inhibitors in tumors as a basis for cancer prognostication and therapy.
Potential Benefit of Metformin as Treatment for Colon Cancer: the Evidence so Far
Abdelsatir, Azza Ali;Husain, Nazik Elmalaika;Hassan, Abdallah Tarig;Elmadhoun, Wadie M;Almobarak, Ahmed O;Ahmed, Mohamed H 8053
Metformin is known as a hypoglycaemic agent that regulates glucose homeostasis by inhibiting liver glucose production and increasing muscle glucose uptake. Colorectal cancer (CRC) is one of the most common cancers worldwide, with about a million new cases diagnosed each year. The risk factors for CRC include advanced age, smoking, black race, obesity, low fibre diet, insulin resistance, and the metabolic syndrome. We have searched Medline for the metabolic syndrome and its relation to CRC, and metformin as a potential treatment of colorectal cancer. Administration of metformin alone or in combination with chemotherapy has been shown to suppress CRC. The mechanism that explains how insulin resistance is associated with CRC is complex and not fully understood. In this review we have summarised studies which showed an association with the metabolic syndrome as well as studies which tackled metformin as a potential treatment of CRC. In addition, we have also provided a summary of how metformin at the cellular level can induce changes that suppress the activity of cancer cells.
DNA Methylation Biomarkers for Nasopharyngeal Carcinoma: Diagnostic and Prognostic Tools
Jiang, Wei;Cai, Rui;Chen, Qiu-Qiu 8059
Nasopharyngeal carcinoma (NPC) is a common tumor in southern China and south-eastern Asia. Effective strategies for the prevention or screening of NPC are limited. Exploring effective biomarkers for the early diagnosis and prognosis of NPC continues to be a rigorous challenge. Evidence is accumulating that DNA methylation alterations are involved in the initiation and progression of NPC. Over the past few decades, aberrant DNA methylation in single or multiple tumor suppressor genes (TSGs) in various biologic samples have been described in NPC, which potentially represents useful biomarkers. Recently, large-scale DNA methylation analysis by genome-wide methylation platform provides a new way to identify candidate DNA methylated markers of NPC. This review summarizes the published research on the diagnostic and prognostic potential biomarkers of DNA methylation for NPC and discusses the current knowledge on DNA methylation as a biomarker for the early detection and monitoring of progression of NPC.
Long Non-coding RNAs and Drug Resistance
Pan, Jing-Jing;Xie, Xiao-Juan;Li, Xu;Chen, Wei 8067
Background: Long non-coding RNAs (lncRNAs) are emerging as key players in gene expression that govern cell developmental processes, and thus contributing to diseases, especially cancers. Many studies have suggested that aberrant expression of lncRNAs is responsible for drug resistance, a substantial obstacle for cancer therapy. Drug resistance not only results from individual variations in patients, but also from genetic and epigenetic differences in tumors. It is reported that drug resistance is tightly modulated by lncRNAs which change the stability and translation of mRNAs encoding factors involved in cell survival, proliferation, and drug metabolism. In this review, we summarize recent advances in research on lncRNAs associated with drug resistance and underlying molecular or cellular mechanisms, which may contribute helpful approaches for the development of new therapeutic strategies to overcome treatment failure.
Sarcopenia in Cancer Patients
Chindapasirt, Jarin 8075
Sarcopenia, characterized by a decline of skeletal muscle plus low muscle strength and/or physical performance, has emerged to be an important prognostic factor for advanced cancer patients. It is associated with poor performance status, toxicity from chemotherapy, and shorter time of tumor control. There is limited data about sarcopenia in cancer patients and associated factors. Moreover, the knowledge about the changes of muscle mass during chemotherapy and its impact to response and toxicity to chemotherapy is still lacking. This review aimed to provide understanding about sarcopenia and to emphasize its importance to cancer treatment.
Benefits of Metformin Use for Cholangiocarcinoma
Kaewpitoon, Soraya J;Loyd, Ryan A;Rujirakul, Ratana;Panpimanmas, Sukij;Matrakool, Likit;Tongtawee, Taweesak;Kootanavanichpong, Nusorn;Kompor, Ponthip;Chavengkun, Wasugree;Kujapun, Jirawoot;Norkaew, Jun;Ponphimai, Sukanya;Padchasuwan, Natnapa;Pholsripradit, Poowadol;Eksanti, Thawatchai;Phatisena, Tanida;Kaewpitoon, Natthawut 8079
Metformin is an oral anti-hyperglycemic agent, which is the most commonly prescribed medication in the treatment of type-2 diabetes mellitus. It is purportedly associated with a reduced risk for various cancers, mainly exerting anti-proliferation effects on various human cancer cell types, such as pancreas, prostate, breast, stomach and liver. This mini-review highlights the risk and benefit of metformin used for cholangiocarcinoma (CCA) prevention and therapy. The results indicated metformin might be a quite promising strategy CCA prevention and treatment, one mechanism being inhibition of CCA tumor growth by cell cycle arrest in both in vitro and in vivo. The AMPK/mTORC1 pathway in intrahepatic CCA cells is targeted by metformin. Furthermore, metformin inhibited CCA tumor growth via the regulation of Drosha-mediated expression of multiple carcinogenic miRNAs. The use of metformin seems to be safe in patients with cirrhosis, and provides a survival benefit. Once hepatic malignancies are already established, metformin does not offer any therapeutic potential. Clinical trials and epidemiological studies of the benefit of metformin use for CCA should be conducted. To date, whether metformin as a prospective chemotherapeutic for CCA is still questionable and waits further atttention.
HPV Infection and Cervical Abnormalities in HIV Positive Women in Different Regions of Brazil, a Middle-Income Country
Freitas, Beatriz C;Suehiro, Tamy T;Consolaro, Marcia EL;Silva, Vania RS 8085
Human papillomavirus is a virus that is distributed worldwide, and persistent infection with high-risk genotypes (HR-HPV) is considered the most important factor for the development of squamous cell cervical carcinoma (SCC). However, by itself, it is not sufficient, and other factors may contribute to the onset and progression of lesions. For example, infection with other sexually transmitted diseases such as human immunodeficiency virus (HIV) may be a factor. Previous studies have shown the relationship between HPV infection and SCC development among HIV-infected women in many regions of the world, with great emphasis on low- and middle-income countries (LMICs). Brazil is considered a LMIC and has great disparities across different regions. The purpose of this review was to highlight the current knowledge about HPV infection and cervical abnormalities in HIV+ women in Brazil because this country is an ideal setting to evaluate HIV impact on SCC development and serves as model of LMICs and low-resource settings.
Evaluation of the MTHFR C677T Polymorphism as a Risk Factor for Colorectal Cancer in Asian Populations
Rai, Vandana 8093
Background: Genetic and environmental factors play important roles in pathogenesis of digestive tract cancers like those in the esophagus, stomach and colorectum. Folate deficiency and methylenetetrahydrofolate reductase (MTHFR) as an important enzyme of folate and methionine metabolism are considered crucial for DNA synthesis and methylation. MTHFR variants may cause genomic hypomethylation, which may lead to the development of cancer, and MTHFR gene polymorphisms (especially C677T and A1298C) are known to influence predispositions for cancer development. Several case control association studies of MTHFR C677T polymorphisms and colorectal cancer (CRC) have been reported in different populations with contrasting results, possibly reflecting inadequate statistical power. Aim: The present meta-analysis was conducted to investigate the association between the C677T polymorphism and the risk of colorectal cancer. Materials and Methods: A literature search of the PubMed, Google Scholar, Springer link and Elsevier databases was carried out for potential relevant articles. Pooled odds ratio (OR) with corresponding 95 % confidence interval (95 % CI) was calculated to assess the association of MTHFR C677T with the susceptibility to CRC. Cochran's Q statistic and the inconsistency index (I2) were used to check study heterogeneity. Egger's test and funnel plots were applied to assess publication bias. All statistical analyses were conducted by with MetaAnalyst and MIX version 1.7. Results: Thirty four case-control studies involving a total of 9,143 cases and 11,357 controls were retrieved according to the inclusion criteria. Overall, no significant association was found between the MTHFR C677T polymorphism and colorectal cancer in Asian populations (for T vs. C: OR=1.03; 95% CI= 0.92-1.5; p= 0.64; for TT vs CC: OR=0.88; 95%CI= 0.74-1.04; p= 0.04; for CT vs. CC: OR = 1.02; 95%CI= 0.93-1.12; p=0.59; for TT+ CT vs. CC: OR=1.07; 95%CI= 0.94-1.22; p=0.87). Conclusions: Evidence from the current meta-analysis indicated that the C677T polymorphism is not associated with CRC risk in Asian populations. Further investigations are needed to offer better insight into any role of this polymorphism in colorectal carcinogenesis.
Psychometric Validation of the Bahasa Malaysia Version of the EORTC QLQ-CR29
Magaji, Bello Arkilla;Moy, Foong Ming;Roslani, April Camilla;Law, Chee Wei;Raduan, Farhana;Sagap, Ismail 8101
Background: This study examined the psychometric properties of the Bahasa Malaysia (BM) version of the European Organization for Research and Treatment of Cancer (EORTC) Colorectal Cancer-specific Quality Of Life Questionnaire (QLQ-CR29). Materials and Methods: We studied 93 patients recruited from University Malaya and Universiti Kebangsaan Medical Centers, Kuala Lumpur, Malaysia using a self-administered method. Tools included QLQ-C30, QLQ-CR29 and Karnofsky Performance Scales (KPS). Statistical analyses included Cronbach's alpha, test-retest correlations, multi-traits scaling and known-groups comparisons. A p vaue ${\leq}0.05$ was considered significant. Results: The internal consistency coefficients for body image, urinary frequency, blood and mucus and stool frequency scales were acceptable (Cronbach's alpha ${\alpha}{\geq}0.65$). However, the coefficients were low for the blood and mucus and stool frequency scales in patients with a stoma bag (${\alpha}=0.46$). Test-retest correlation coefficients were moderate to high (range: r = 0.51 to 1.00) for most of the scales except anxiety, urinary frequency, buttock pain, hair loss, stoma care related problems, and dyspareunia (r ${\leq}0.49$). Convergent and discriminant validities were achieved in all scales. Patients with a stoma reported significantly higher symptoms of blood and mucus in the stool, flatulence, faecal incontinence, sore skin, and embarrassment due to the frequent need to change the stoma bag (p < 0.05) compared to patients without stoma. None of the scales distinguished between patients based on the KPS scores. There were no overlaps between scales in the QLQ-C30 and QLQ-CR29 (r < 0.40). Conclusions: the BM version of the QLQ-CR29 indicated acceptable psychometric properties in most of the scales similar to original validation study. This questionnaire could be used to complement the QLQ-C30 in assessing HRQOL among BM speaking population with colorectal cancer.
Psychometric Validation of the Malaysian Chinese Version of the EORTC QLQ-C30 in Colorectal Cancer Patients
Magaji, Bello Arkilla;Moy, Foong Ming;Roslani, April Camilla;Law, Chee Wei;Sagap, Ismail 8107
Background and Aims: Colorectal cancer is the second most frequent cancer in Malaysia. We aimed to assess the validity and reliability of the Malaysian Chinese version of European Organization for Research and Treatment of Cancer (EORTC) Quality of Life Questionnaire core (QLQ-C30) in patients with colorectal cancer. Materials and Methods: Translated versions of the QLQ-C30 were obtained from the EORTC. A cross sectional study design was used to obtain data from patients receiving treatment at two teaching hospitals in Kuala Lumpur, Malaysia. The Malaysian Chinese version of QLQ-C30 was self-administered in 96 patients while the Karnofsky Performance Scales (KPS) was generated by attending surgeons. Statistical analysis included reliability, convergent, discriminate validity, and known-groups comparisons. Statistical significance was based on p value ${\leq}0.05$. Results: The internal consistencies of the Malaysian Chinese version were acceptable [Cronbach's alpha (${\alpha}{\geq}0.70$)] in the global health status/overall quality of life (GHS/QOL), functioning scales except cognitive scale (${\alpha}{\leq}0.32$) in all levels of analysis, and social/family functioning scale (${\alpha}=0.63$) in patients without a stoma. All questionnaire items fulfilled the criteria for convergent and discriminant validity except question number 5, with correlation with role (r = 0.62) and social/family (r = 0.41) functioning higher than with physical functioning scales (r = 0.34). The test-retest coefficients in the GHS/QOL, functioning scales and in most of the symptoms scales were moderate to high (r = 0.58 to 1.00). Patients with a stoma reported statistically significant lower physical functioning (p=0.015), social/family functioning (p=0.013), and higher constipation (p=0.010) and financial difficulty (p=0.037) compared to patients without stoma. There was no significant difference between patients with high and low KPS scores. Conclusions: Malaysian Chinese version of the QLQ-C30 is a valid and reliable measure of HRQOL in patients with colorectal cancer.
Glehnia littoralis Root Extract Induces G0/G1 Phase Cell Cycle Arrest in the MCF-7 Human Breast Cancer Cell Line
de la Cruz, Joseph Flores;Vergara, Emil Joseph Sanvictores;Cho, Yura;Hong, Hee Ok;Oyungerel, Baatartsogt;Hwang, Seong Gu 8113
Glehnia littoralis (GL) is widely used as an oriental medicine for cough, fever, stroke and other disease conditions. However, the anti-cancer properties of GL on MCF-7 human breast cancer cells have not been investigated. In order to elucidate anti-cancer properties and underlying cell death mechanisms, MCF-7cells ($5{\times}10^4/well$) were treated with Glehnia littoralis root extract at 0-400 ug/ml. A hot water extract of GL root inhibited the proliferation of MCF-7 cells in a dose-dependent manner. Analysis of the cell cycle after treatment of MCF-7 cells with increasing concentrations of GL root extract for 24 hours showed significant cell cycle arrest in the G1 phase. RT-PCR and Western blot analysis both revealed that GL root extract significantly increased the expression of p21 and p27 with an accompanying decrease in both CDK4 and cyclin D1. Our reuslts indicated that GL root extract arrested the proliferation of MCF-7 cells in G1 phase through inhibition of CDK4 and cyclin D1 via increased induction of p21 and p27. In summary, the current study showed that GL could serve as a potential source of chemotherapeutic or chemopreventative agents against human breast cancer.
Anti-tumor and Chemoprotective Effect of Bauhinia tomentosa by Regulating Growth Factors and Inflammatory Mediators
Kannan, Narayanan;Sakthivel, Kunnathur Murugesan;Guruvayoorappan, Chandrasekaran 8119
Cancer is a leading cause of death worldwide. Due to the toxic side effects of the commonly used chemotherapeutic drug cyclophosphamide (CTX), the use of herbal medicines with fewer side effects but having potential use as inducing anti-cancer outcomes in situ has become increasingly popular. The present study sought to investigate the effects of a methanolic extract of Bauhinia tomentosa against Dalton's ascites lymphoma (DAL) induced ascites as well as solid tumors in BALB/c mice. Specifically, B. tomentosa extract was administered intraperitonealy (IP) at 10 mg/kg. BW body weight starting just after tumor cell implantation and thereafter for 10 consecutive days. In the ascites tumor model hosts, administration of extract resulted in a 52% increase in the life span. In solid tumor models, co-administration of extract and CTX significantly reduced tumor volume (relative to in untreated hosts) by 73% compared to just by 52% when the extract alone was provided. Co-administration of the extract also mitigated CTX-induced toxicity, including decreases in WBC count, and in bone marrow cellularity and ${\alpha}$-esterase activity. Extract treatment also attenuated any increases in serum levels of $TNF{\alpha}$, iNOS, IL-$1{\beta}$, IL-6, GM-CSF, and VEGF seen in tumor-bearing hosts. This study confirmed that, the potent antitumor activity of B.tomentosa extract may be associated with immune modulatory effects by regulating anti-oxidants and cytokine levels.
Back Massage to Decrease State Anxiety, Cortisol Level, Blood Prsessure, Heart Rate and Increase Sleep Quality in Family Caregivers of Patients with Cancer: A Randomised Controlled Trial
Pinar, Rukiye;Afsar, Fisun 8127
Background: The objective of this study was to evaluate the effect of back massage on the anxiety state, cortisol level, systolic/diastolic blood pressure, pulse rate, and sleep quality in family caregivers of patients with cancer. Materials and Methods: Forty-four family caregivers were randomly assigned to either the experimental or control group (22 interventions, 22 controls) after they were matched on age and gender. The intervention consisted of back massage for 15 minutes per day for a week. Main research outcomes were measured at baseline (day I) and follow-up (day 7). Unpaired t-test, paired t test and chi-square test were used to analyse data. Results: The majority of the caregivers were women, married, secondary school educated and housewife. State anxiety (p<0.001), cortisol level (p<0.05), systolic/diastolic blood pressure (p<0.001, p<0.01 respectively), and pulse rate (p<0.01) were significantly decreased, and sleep quality (p<0.001) increased after back massage intervention. Conclusions: The study results show that family caregivers for patients with cancer can benefit from back massage to improve state anxiety, cortisol level, blood pressure and heart rate, and sleep quality. Oncology nurses can take advantage of back massage, which is non-pharmacologic and easily implemented method, as an independent nursing action to support caregivers for patients with cancer.
Cell Cycle Modulation of MCF-7 and MDA-MB-231 by a Sub-Fraction of Strobilanthes crispus and its Combination with Tamoxifen
Yaacob, Nik Soriani;Kamal, Nik Nursyazni Nik Mohamed;Wong, Kah Keng;Norazmi, Mohd Nor 8135
Background: Cell cycle regulatory proteins are suitable targets for cancer therapeutic development since genetic alterations in many cancers also affect the functions of these molecules. Strobilanthes crispus (S. crispus) is traditionally known for its potential benefits in treating various ailments. We recently reported that an active sub-fraction of S. crispus leaves (SCS) caused caspase-dependent apoptosis of human breast cancer MCF-7 and MDA-MB-231 cells. Materials and Methods: Considering the ability of SCS to also promote the activity of the antiestrogen, tamoxifen, we further examined the effect of SCS in modulating cell cycle progression and related proteins in MCF-7 and MDA-MB-231 cells alone and in combination with tamoxifen. Expression of cell cycle-related transcripts was analysed based on a previous microarray dataset. Results: SCS significantly caused G1 arrest of both types of cells, similar to tamoxifen and this was associated with modulation of cyclin D1, p21 and p53. In combination with tamoxifen, the anticancer effects involved downregulation of $ER{\alpha}$ protein in MCF-7 cells but appeared independent of an ER-mediated mechanism in MDA-MB-231 cells. Microarray data analysis confirmed the clinical relevance of the proteins studied. Conclusions: The current data suggest that SCS growth inhibitory effects are similar to that of the antiestrogen, tamoxifen, further supporting the previously demonstrated cytotoxic and apoptotic actions of both agents.
Extended Low Anterior Resection with a Circular Stapler in Patients with Rectal Cancer: a Single Center Experience
Talaeezadeh, Abdolhasan;Bahadoram, Mohammad;Abtahian, Amin;Rezaee, Alireza 8141
Background: to evaluate the outcome of stapled colo-anal anastomoses after extended low anterior resection for distal rectal carcinoma. Materials and Methods: A retrospective study of fifty patients who underwent coloanal anastomoses after extended low anterior resection was conducted at Imam Hospital from September 2007 up to July 2012. Results: The distance of the tumor from anal verge was 3 to 8 cm. Anastomotic leakage developed in 6% of patients and defecation problems in 16%. One-year local recurrence was 6% while three-year local recurrence was 4%. One-year systemic recurrence was seen in 22% while three-year systemic recurrence was seen in 20%. Conclusions: Colo-anal anastomoses after extended low anterior resection for distal rectal carcinoma can be conducted safely.
Clinical Significance of Atypical Squamous Cells of Undetermined Significance among Patients Undergoing Cervical Conization
Nishimura, Mai;Miyatake, Takashi;Nakashima, Ayaka;Miyoshi, Ai;Mimura, Mayuko;Nagamatsu, Masaaki;Ogita, Kazuhide;Yokoi, Takeshi 8145
Background: Atypical squamous cells of undetermined significance (ASCUS) feature a wide variety of cervical cells, including benign and malignant examples. The management of ASCUS is complicated. Guidelines for office gynecology in Japan recommend performing a high-risk human papillomavirus (HPV) test as a rule. The guidelines also recommend repeat cervical cytology after 6 and 12 months, or immediate colposcopy. The purpose of this study was to determine the clinical significance of ASCUS. Materials and Methods: Between January 2012 and December 2014, a total of 162 patients underwent cervical conization for cervical intraepithelial neoplasia grade 3 (CIN3), carcinoma in situ, squamous cell carcinoma, microinvasive squamous cell carcinoma, and adenocarcinoma in situ at our hospital. The results of cervical cytology prior to conization, the pathology after conization, and high-risk HPV testing were obtained from clinical records and analyzed retrospectively. Results: Based on cervical cytology, 31 (19.1%) of 162 patients were primarily diagnosed with ASCUS. Among these, 25 (80.6%) were positive for high-risk HPV, and the test results of the remaining 6 patients (19.4%) were uncertain. In the final pathological diagnosis after conization, 27 (87.1%) and 4 patients (12.9%) were diagnosed with CIN3 and carcinoma in situ, respectively. Conclusions: Although ASCUS is known as a low-risk abnormal cervical cytology, approximately 20% of patients who underwent cervical conization had ASCUS. The relationship between the cervical cytology of ASCUS and the final pathological results for CIN3 or invasive carcinoma should be investigated statistically. In cases of ASCUS, we recommend HPV tests or colposcopic examination rather than cytological follow-up, because of the risk of missing CIN3 or more advanced disease.
Malignant Neoplasm Prevalence in the Aktobe Region of Kazakhstan
Bekmukhambetov, Yerbol;Mamyrbayev, Arstan;Jarkenov, Timur;Makenova, Aliya;Imangazina, Zina 8149
An oncopathological state assessment was conducted among adults, children and teenagers in Aktobe region for 2004-2013. Overall the burden of mortality was in the range of 94.8-100.2 per 100,000 population, without any obvious trend over time. Ranking by pathology, the highest incidences among women were registered for breast cancer (5.8-8.4), cervix uteri (2.9-4.6), ovary (2.4-3.6) and corpus uteri, stomach, esophagus, without any marked change over time except for a slight rise in cervical cancer rates. In males, the first place in rank was trachea, bronchus and lung, followed by stomach and esophagus, which are followed by bladder, lymphoid and hematopoietic tissues pathology. Agian no clear trends were apparent over time. In children, main localizations in cancer incidence blood (acute lymphocytic leukemia, lymphosarcoma, acute myeloid leukemia, Hodgkin's disease), brain and central nervous system, bones and articular cartilages, kidneys, and eye and it's appendages, in both sexes. Similarly, in young adults, the major percentage was in blood and lymphatic tissues (acute myeloid leukemia, acute lymphocytic leukemia, Hodgkin's disease) a significant percentage accruing to lymphosarcoma, lymphoma, other myeloid leukemia and hematological malignancies as well as tumors of brain and central nervous system, bones and articular cartilages. This initial survey provides the basis for more detailed investigation of cancer epidemiology in Aktobe, Kazakhstan.
Prognostic Significance of Two Dimensional AgNOR Evaluation in Local Advanced Rectal Cancer Treated with Chemoradiotherapy
Gundog, Mete;Yildiz, Oguz G;Imamoglu, Nalan;Aslan, Dicle;Aytekin, Aynur;Soyuer, Isin;Soyuer, Serdar 8155
The prognostic significance of AgNOR proteins in stage II-III rectal cancers treated with chemoradiotherapy was evaluated. Silver staining was applied to the $3{\mu}m$ sections of parafin blocked tissues from 30 rectal cancer patients who received 5-FU based chemoradiotherapy from May 2003 to June 2006. The microscopic displays of the cells were transferred into the computer via a video camera. AgNOR area (nucleolus organizer region area) and nucleus area values were determined as a nucleolus organizer regions area/total nucleus area (NORa/TNa). The mean NORa/TNa value was found to be $9.02{\pm}3.68$. The overall survival and disease free survival in the high NORa/TNa (>9.02) patients were 52.2 months and 39.4 months respectively, as compared to 100.7 months and 98.4 months in the low NORa/TNa (<9.02) cases. (p<0.001 and p<0.001 respectively). In addition, the prognosis in the high NORa/TNa patients was worse than low NORa/TNa patients (p<0.05). In terms of overall survival and disease-free survival, a statistically significant negative correlation was found with the value of NORa/TNa in the correlations tests. Cox regression analyses demostrated that overall survival and disease-free survival were associated with lymph node status (negative or positive) and the NORa/TNa value. We suggest that two-dimensional AgNOR evaluation may be a safe and usable parameter for prognosis and an indicator of cell proliferation instead of AgNOR dots.
Epidemiological Study on Breast Cancer Associated Risk Factors and Screening Practices among Women in the Holy City of Varanasi, Uttar Pradesh, India
Paul, Shatabdi;Solanki, Prem Prakash;Shahi, Uday Pratap;Srikrishna, Saripella 8163
Background: Breast cancer is the second most cause of death (1.38 million, 10.9% of all cancer) worldwide after lung cancer. In present study, we assess the knowledge, level of awareness of risk factors and screening practices especially breast self examination (BSE) among women, considering the non-feasibility of diagnostic tools such as mammography for breast screening techniques of breast cancer in the holy city Varanasi, Uttar Pradesh, India. Materials and Methods: A cross-sectional population based survey was conducted. The investigation tool adopted was self administrated questionnaire format. Data were analysed using SPSS 20 version and Chi square test to determine significant association between various education groups with awareness and knowledge, analysis of variance was applied in order to establish significance. Results: The attitude of participants in this study, among 560 women 500 (89%) responded (age group 18-65 years), 53.8% were married. The knowledge about BSE was very low (16%) and out of them 15.6% were practised BSE only once in life time. study shown that prominent age at which women achieve their parity was 20 yrs, among 500 participants 224 women have achieved their parity from age 18 to 30 yrs. Very well known awareness about risk factors of breast cancer were alcohol (64.6%), smoking (64%) and least known awareness risk factors were early menarche (17.2%) and use of red meat (23%). The recovery factors of breast cancer cases were doctors support (95%) and family support (94.5%) as most familiar responses of the holy city Varanasi. Conclusions: The study revealed that the awareness about risk factors and practised of BSE among women in Varanasi is extremely low in comparison with other cities and countries as well (Delhi, Mumbai, Himachal Pradesh, Turkey and Nigeria). However, doctors and health workers may promote the early diagnosis of breast cancer.
Evaluation of Nutritional Status of Cancer Patients during Treatment by Patient-Generated Subjective Global Assessment: a Hospital-Based Study
Sharma, Dibyendu;Kannan, Ravi;Tapkire, Ritesh;Nath, Soumitra 8173
Cancer patients frequently experience malnutrition. Cancer and cancer therapy effects nutritional status through alterations in the metabolic system and reduction in food intake. In the present study, fifty seven cancer patients were selected as subjects from the oncology ward of Cachar Cancer Hospital and Research Centre, Silchar, India. Evaluation of nutritional status of cancer patients during treatment was carried out by scored Patient-Generated Subjective Global Assessment (PG-SGA). The findings of PG-SGA showed that 15.8% (9) were well nourished, 31.6% (18) were moderately or suspected of being malnourished and 52.6% (30) were severely malnourished. The prevalence of malnutrition was highest in lip/oral (33.33%) cancer patients. The study showed that the prevalence of malnutrition (84.2%) was high in cancer patients during treatment.
HeLa Cells Containing a Truncated Form of DNA Polymerase Beta are More Sensitized to Alkylating Agents than to Agents Inducing Oxidative Stress
Khanra, Kalyani;Chakraborty, Anindita;Bhattacharyya, Nandan 8177
The present study was aimed at determining the effects of alkylating and oxidative stress inducing agents on a newly identified variant of DNA polymerase beta ($pol{\beta}{\Delta}_{208-304}$) specific for ovarian cancer. $Pol{\beta}{\Delta}_{208-304}$ has a deletion of exons 11-13 which lie in the catalytic part of enzyme. We compared the effect of these chemicals on HeLa cells and HeLa cells stably transfected with this variant cloned into in pcDNAI/neo vector by MTT, colony forming and apoptosis assays. $Pol{\beta}{\Delta}_{208-304}$ cells exhibited greater sensitivity to an alkylating agent and less sensitivity towards $H_2O_2$ and UV when compared with HeLa cells alone. It has been shown that cell death in $Pol{\beta}{\Delta}_{208-304}$ transfected HeLa cells is mediated by the caspase 9 cascade. Exon 11 has nucleotidyl selection activity, while exons 12 and 13 have dNTP selection activity. Hence deletion of this part may affect polymerizing activity although single strand binding and double strand binding activity may remain same. The lack of this part may adversely affect catalytic activity of DNA polymerase beta so that the variant may act as a dominant negative mutant. This would represent clinical significance if translated into a clinical setting because resistance to radiation or chemotherapy during the relapse of the disease could be potentially overcome by this approach.
Detection of High-Risk Human Papillomaviruses in the Prevention of Cervical Cancer in India
Baskaran, Krishnan;Kumar, P Kranthi;Karunanithi, Santha;Sethupathy, Subramanian;Thamaraiselvi, B;Swaruparani, S 8187
Human papillomaviruses (HPVs) are small, non-enveloped, double-stranded DNA viruses that infect epithelial tissues. Specific genotypes of human papillomavirus are the single most common etiological agents of cervical intraepithelial lesions and cervical cancer. Cervical cancer usually arises at squamous metaplastic epithelium of transformation zone (TZ) of the cervix featuring infection with one or more oncogenic or high-risk HPV (HR-HPV) types. A hospital-based study in a rural set up was carried out to understand the association of HR-HPV with squamous intraepithelial lesions (SILs) and cervical cancer. In the present study, HR-HPV was detected in 65.7% of low-grade squamous intraepithelial lesions (LSILs), 84.6% of high-grade squamous intraepithelial lesions (HSILs) and 94% of cervical cancer as compared to 10.7% of controls. The association of HPV infection with SIL and cervical cancer was analyzed with Chi square test (p<0.001). The significant association found confirmed that detection of HR-HPV is a suitable candidate for early identification of cervical precancerous lesions and in the prevention of cervical cancer in India.
Identification and Pharmacological Analysis of High Efficacy Small Molecule Inhibitors of EGF-EGFR Interactions in Clinical Treatment of Non-Small Cell Lung Carcinoma: a Computational Approach
Gudala, Suresh;Khan, Uzma;Kanungo, Niteesh;Bandaru, Srinivas;Hussain, Tajamul;Parihar, MS;Nayarisseri, Anuraj;Mundluru, Hema Prasad 8191
Inhibition of EGFR-EGF interactions forms an important therapeutic rationale in treatment of non-small cell lung carcinoma. Established inhibitors have been successful in reducing proliferative processes observed in NSCLC, however patients suffer serious side effects. Considering the narrow therapeutic window of present EGFR inhibitors, the present study centred on identifying high efficacy EGFR inhibitors through structure based virtual screening strategies. Established inhibitors - Afatinib, Dacomitinib, Erlotinib, Lapatinib, Rociletinib formed parent compounds to retrieve similar compounds by linear fingerprint based tanimoto search with a threshold of 90%. The compounds (parents and respective similars) were docked at the EGF binding cleft of EGFR. Patch dock supervised protein-protein interactions were established between EGF and ligand (query and similar) bound and free states of EGFR. Compounds ADS103317, AKOS024836912, AGN-PC-0MXVWT, GNF-Pf-3539, SCHEMBL15205939 were retrieved respectively similar to Afatinib, Dacomitinib, Erlotinib, Lapatinib, Rociletinib. Compound-AGN-PC-0MXVWT akin to Erlotinib showed highest affinity against EGFR amongst all the compounds (parent and similar) assessed in the study. Further, AGN-PC-0MXVWT brought about significant blocking of EGFR-EGF interactions in addition showed appreciable ADMET properties and pharmacophoric features. In the study, we report AGN-PC-0MXVWT to be an efficient and high efficacy inhibitor of EGFR-EGF interactions identified through computational approaches.
FHIT Gene Expression in Acute Lymphoblastic Leukemia and its Clinical Significance
Malak, Camelia A Abdel;Elghanam, Doaa M;Elbossaty, Walaa Fikry 8197
Background: To investigate the expression of the fragile histidine triad (FHIT) gene in acute lymphoblastic leukemia and its clinical significance. Materials and Methods: The level of expressed FHIT mRNA in peripheral blood from 50 patients with acute lymphoblastic leukemia (ALL) and in 50 peripheral blood samples from healthy volunteers was measured via RT-PCR. Correlation analyses between FHIT gene expression and clinical characteristics (gender, age, white blood count, immunophenotype of acute lymphoblastic leukemia and percentage of blast cells) of the patients were performed. Results: The FHIT gene was expressed at $2.49{\pm}7.37$ of ALL patients against $14.4{\pm}17.9$ in the healthy volunteers. The difference in the expression levels between ALL patients and healthy volunteers was statistically significant. The rate of gene expression did not significantly vary with immunophenotype subtypes. Gene expression was also found to be correlated with increase of total leukocyte and decrease in platelets, but not with age, gender, immunophenotyping or percentage of blast cells. Conclusions: FHIT gene expression is low in acute lymphoblastic leukemia and could be a useful marker to monitor minimal residual disease. This gene is also a candidate target for the immunotherapy of acute lymphoblastic leukemia.
Is Immunohistochemical Sex Hormone Binding Globulin Expression Important in the Differential Diagnosis of Adenocarcinomas?
Bulut, Gulay;Kosem, Mustafa;Bulut, Mehmet Deniz;Erten, Remzi;Bayram, Irfan 8203
Adenocarcinomas (AC) are the most frequently encountered carcinomas. It may be quite challenging to detect the primary origin when those carcinomas metastasize and the first finding is a metastatic tumor. This study evaluated the role of sex hormone binding globulin (SHBG) positivity in tumor cells in the subclassification and detection of the original organ of adenocarcinomas. Between 1994 and 2008, 64 sections of normal tissue belonging to ten organs, and 116 cases diagnosed as adenoid cystic carcinoma and mucoepidermoid carcinoma of the salivary gland, lung adenocarcinoma, invasive ductal carcinoma of the breast, adenocarcinoma of stomach, colon, gallbladder, pancreas and prostate, endometrial adenocarcinoma and serous adenocarcinoma and mucinous adenocarcinoma of the ovary, were sent to the laboratory at the Department of Pathology at the Yuzuncu Yil University School of Medicine, where they were stained immunohistochemically, using antibodies against SHBG. The SHBG immunoreactivity in both the tumor cells and normal cells, together with the type, diffuseness and intensity of the staining were then evaluated. In the differential diagnosis of the adenocarcinomas of the organs, including the glandular structures, impressively valuable results are encountered in the tumor cells, whether the SHBG immunopositivity is evaluated alone or together with other IHC markers. Further extensive research with a larger number of cases, including instances of cholangiocarcinoma and cervix uteri AC [which we could not include in the study for technical reasons] should be performed, in order to appropriately evaluate the role of SHBG in the differential diagnosis of AC.
Levels of Conscience and Related Factors among Iranian Oncology Nurses
Gorbanzadeh, Behrang;Rahmani, Azad;Mogadassian, Sima;Behshid, Mojhgan;Azadi, Arman;Taghavy, Saied 8211
Background: Having a conscience is one of the main pre-requisite of providing nursing care. The knowledge regarding levels of conscience among nurses in eastern countries is limited. So, the purpose of this study was to examine the level of conscience and its related factors among Iranian oncology nurses. Materials and Methods: This descriptive-correlational study was conducted in 3 hospitals in Tabriz, Iran. Overall, 68 nurses were selected using a non-probability sampling method. The perceptions of conscience questionnaire was used to identify the levels of conscience among nurses. The data were analyzed using SPSS version 13.0. Results: The mean nurses' level of conscience scores was 72.7. In the authority and asset sub-scales nurses acquired higher scores. The mean of nurses' scores in burden and depending on culture sub-scales were the least. Also, there were no statistical relationship between some demographic characteristics of participants and their total score on the perceptions of conscience questionnaire. Conclusions: According to study findings Iranian nurses had high levels of conscience. However, understanding all the factors that affect nurses' perception of conscience requires further studies.
Oral non Squamous Cell Malignant Tumors in an Iranian Population: a 43 year Evaluation
Mohtasham, Nooshin;Saghravanian, Nasrollah;Goli, Maryam;Kadeh, Hamideh 8215
Background: The prevalence of non-squamous cell malignant tumors of the oral cavity has not been evaluated in Iran extensively. The aim of this study was to evaluate epidemiological aspects of the oral malignancies with non-squamous cell origin during a 43-year period in the Faculty of Dentistry, Mashhad University of Medical Sciences, Iran. Materials and Methods: In this retrospective study, the records of all patients referred to dental school of Mashhad university of medical sciences in northeast of Iran, during the period 1971-2013 were evaluated. All confirmed samples of oral non squamous cell malignant tumors were included in this study. Demographic information including age, gender and location of the lesions were extracted from patient's records. Data were analyzed using SPSS statistical soft ware, Chi-square and Fisher's exact tests. Results: Among 11,126 patients, 188 (1.68%) non squamous cell malignant tumors were found, with mean age of 39.9 years ranging from 2 to 92 years. The most common tumors were mucoepidermoid carcinoma (33 cases) and lymphoma (32 cases). Non squamous cell malignant tumors occurred almost equally in men (94 cases) and women (93 cases). Most (134 cases) of them were located peripherally with high frequency in salivary glands (89 cases) and 52 cases were centrally with high frequency in the mandible (38 cases). Conclusions: More findings in this survey were similar to those reported from other studies with differences in some cases; it may be due to variation in the sample size, geographic and racial differences in tumors.
Misclassification Adjustment of Family History of Breast Cancer in a Case-Control Study: a Bayesian Approach
Moradzadeh, Rahmatollah;Mansournia, Mohammad Ali;Baghfalaki, Taban;Ghiasvand, Reza;Noori-Daloii, Mohammad Reza;Holakouie-Naieni, Kourosh 8221
Background: Misreporting self-reported family history may lead to biased estimations. We used Bayesian methods to adjust for exposure misclassification. Materials and Methods: A hospital-based case-control study was used to identify breast cancer risk factors among Iranian women. Three models were jointly considered; an outcome, an exposure and a measurement model. All models were fitted using Bayesian methods, run to achieve convergence. Results: Bayesian analysis in the model without misclassification showed that the odds ratios for the relationship between breast cancer and a family history in different prior distributions were 2.98 (95% CRI: 2.41, 3.71), 2.57 (95% CRI: 1.95, 3.41) and 2.53 (95% CRI: 1.93, 3.31). In the misclassified model, adjusted odds ratios for misclassification in the different situations were 2.64 (95% CRI: 2.02, 3.47), 2.64 (95% CRI: 2.02, 3.46), 1.60 (95% CRI: 1.07, 2.38), 1.61 (95% CRI: 1.07, 2.40), 1.57 (95% CRI: 1.05, 2.35), 1.58 (95% CRI: 1.06, 2.34) and 1.57 (95% CRI: 1.06, 2.33). Conclusions: It was concluded that self-reported family history may be misclassified in different scenarios. Due to the lack of validation studies in Iran, more attention to this matter in future research is suggested, especially while obtaining results in accordance with sensitivity and specificity values.
Polymorphisms in Heat Shock Proteins A1B and A1L (HOM) as Risk Factors for Oesophageal Carcinoma in Northeast India
Saikia, Snigdha;Barooah, Prajjalendra;Bhattacharyya, Mallika;Deka, Manab;Goswami, Bhabadev;Sarma, Manash P;Medhi, Subhash 8227
Background: To investigate polymorphisms in heat shock proteins A1B and A1L (HOM) and associated risk of oesophageal carcinoma in Northeast India. Materials and Methods: The study includes oesophageal cancer (ECA) patients attending general outpatient department (OPD) and endoscopic unit of Gauhati Medical College. Patients were diagnosed based on endoscopic and histopathological findings. Genomic DNA was typed for HSPA1B1267 and HSPA1L2437 SNPs using the polymerase chain reaction with restriction fragment length polymorphisms. Results: A total of 78 cases and 100 age-sex matched healthy controls were included in the study with a male: female ratio of 5:3 and a mean age of $61.4{\pm}8.5years$. Clinico-pathological evaluation showed 84% had squamous cell carcinoma and 16% were adenocarcinoma. Dysphagia grades 4 (43.5%) and 5 (37.1%) were observed by endoscopic and hispathological evaluation. The frequency of genomic variation of A1B from wild type A/A to heterozygous A/G and mutant G/G showed a positive association [chi sq=19.9, p=<0.05] and the allelic frequency also showed a significant correlation [chi sq=10.3, with cases vs. controls, OR=0.32, $p{\leq}0.05$]. The genomic variation of A1L from wild T/T to heterozygous T/C and mutant C/C were found positively associated [chi sq=7.02, p<0.05] with development of ECA. While analyzing the allelic frequency, there was no significant association [chi sq=3.19, OR=0.49, p=0.07]. Among all the risk factors, betel quid [OR=9.79, Chi square=35.0, p<0.05], tobacco [OR=2.95, chi square=10.6, p<0.05], smoking [OR=3.23, chi square=10.1, p<0.05] demonstrated significant differences between consumers vs. non consumers regarding EC development. Alcohol did not show any significant association [OR=1.34, chi square=0.69, p=0.4] independently. Conclusions: It can be concluded that the present study provides marked evidence that polymorphisms of HSP70 A1B and HSP70 A1L genes are associated with the development of ECA in a population in Northeast India, A1B having a stronger influence. Betel quid consumption was found to be a highly significant risk factor, followed by smoking and tobacco chewing. Although alcohol was not a potent risk factor independently, alcohol consumption along with tobacco, smoking and betel nut was found to contribute to development of ECA.
Clinico-Pathological Profile and Haematological Abnormalities Associated with Lung Cancer in Bangalore, India
Baburao, Archana;Narayanswamy, Huliraj 8235
Background: Lung cancer is one of the most common types of cancer causing high morbidity and mortality worldwide. An increasing incidence of lung cancer has been observed in India. Objectives:To evaluate the clinicpathological profile and haematological abnormalities associated with lung cancer in Bangalore, India. Materials and Methods: This prospective study was carried out over a period of 2 years. A total of 96 newly diagnosed and histopathologically confirmed cases of lung cancer were included in the study. Results: Our lung cancer cases had a male to female ratio of 3:1. Distribution of age varied from 40 to 90 years, with a major contribution in the age group between 61 and 80 years (55.2%). Smoking was the commonest risk factor found in 69.7% of patients. The most frequent symptom was cough (86.4%) followed by loss of weight and appetite (65.6%) and dyspnea (64.5%). The most common radiological presentation was a mass lesion (55%). The most common histopathological type was squamous cell carcinoma (47.9%), followed by adenocarcinoma (28.1%) and small cell carcinoma (12.5%). Distant metastasis at presentation was seen in 53.1% patients. Among the haematological abnormalities, anaemia was seen in 61.4% of patients, leucocytosis in 36.4%, thrombocytosis in 14.5% and eosinophilia in 19.7% of patients. Haematological abnormalities were more commonly seen in non small cell lung cancer. Conclusions: Squamous cell carcinoma was found to be the most common histopathological type and smoking still remains the major risk factor for lung cancer. Haematological abnormalities are frequently observed in lung cancer patients, anaemia being the commonest of all.
Human Papillomavirus E6 Knockdown Restores Adenovirus Mediated-estrogen Response Element Linked p53 Gene Transfer in HeLa Cells
Kajitani, Koji;Ken-Ichi, Honda;Terada, Hiroyuki;Yasui, Tomoyo;Sumi, Toshiyuki;Koyama, Masayasu;Ishiko, Osamu 8239
The p53 gene is inactivated by the human papillomavirus (HPV) E6 protein in the majority of cervical cancers. Treatment of HeLa S3 cells with siRNA for HPV E6 permitted adenovirus-mediated transduction of a p53 gene linked to an upstream estrogen response element (ERE). Our previous study in non-siRNA treated HHUA cells, which are derived from an endometrial cancer and express estrogen receptor ${\beta}$, showed enhancing effects of an upstream ERE on adenovirus-mediated p53 gene transduction. In HeLa S3 cells treated with siRNA for HPV E6, adenovirus-mediated transduction was enhanced by an upstream ERE linked to a p53 gene carrying a proline variant at codon 72, but not for a p53 gene with arginine variant at codon 72. Expression levels of p53 mRNA and Coxsackie/adenovirus receptor (CAR) mRNA after adenovirus-mediated transfer of an ERE-linked p53 gene (proline variant at codon 72) were higher compared with those after non-ERE-linked p53 gene transfer in siRNA-treated HeLa S3 cells. Western blot analysis showed lower ${\beta}$-tubulin levels and comparatively higher p53/${\beta}$-tubulin or CAR/${\beta}$-tubulin ratios in siRNA-treated HeLa S3 cells after adenovirus-mediated ERE-linked p53 gene (proline variant at codon 72) transfer compared with those in non-siRNA-treated cells. Apoptosis, as measured by annexin V binding, was higher after adenovirus-mediated ERE-linked p53 gene (proline variant at codon 72) transfer compared with that after non-ERE-linked p53 gene transfer in siRNA-treated cells.
Promoter Methylation Status of Two Novel Human Genes, UBE2Q1 and UBE2Q2, in Colorectal Cancer: a New Finding in Iranian Patients
Mokarram, Pooneh;Shakiba-Jam, Fatemeh;Kavousipour, Soudabeh;Sarabi, Mostafa Moradi;Seghatoleslam, Atefeh 8247
Background: The ubiquitin-proteasome system (UPS) degrades a variety of proteins which attach to specific signals. The ubiquitination pathway facilitates degradation of damaged proteins and regulates growth and stress responses. This pathway is altered in various cancers, including acute lymphoblastic leukemia, head and neck squamous cell carcinoma and breast cancer. Recently it has been reported that expression of newly characterized human genes, UBE2Q1 and UBE2Q2, putative members of ubiquitin-conjugating enzyme family (E2), has been also changed in colorectal cancer. Epigenetics is one of the fastest-growing areas of science and nowadays has become a central issue in biological studies of diseases. According to the lack of information about the role of epigenetic changes on gene expression profiling of UBE2Q1 and UBE2Q2, and the presence of CpG islands in the promoter of these two human genes, we decided to evaluate the promoter methylation status of these genes as a first step. Materials and Methods: The promoter methylation status of UBE2Q1 and UBE2Q2 was studied by methylation-specific PCR (MSP) in tumor samples of 60 colorectal cancer patients compared to adjacent normal tissues and 20 non-malignant controls. The frequency of the methylation for each gene was analyzed by chi-square method. Results: MSP results revealed that UBE2Q2 gene promoter were more unmethylated, while a higher level of methylated allele was observed for UBE2Q1 in tumor tissues compared to the adjacent normal tissues and the non malignant controls. Conclusions: UBE2Q1 and UBE2Q2 genes show different methylation profiles in CRC cases.
Plasma Soluble CD30 as a Possible Marker of Adult T-cell Leukemia in HTLV-1 Carriers: a Nested Case-Control Study
Takemoto, Shigeki;Iwanaga, Masako;Sagara, Yasuko;Watanabe, Toshiki 8253
Elevated levels of soluble CD30 (sCD30) are linked with various T-cell neoplasms. However, the relationship between sCD30 levels and the development of adult T-cell leukemia (ATL) in human T-cell leukemia virus type 1 (HTLV-1) carriers remains to be clarified. We here investigated whether plasma sCD30 is associated with risk of ATL in a nested case-control study within a cohort of HTLV-1 carriers. We compared sCD30 levels between 11 cases (i.e., HTLV-1 carriers who later progressed to ATL) and 22 age-, sex- and institution-matched control HTLV-1 carriers (i.e., those with no progression). The sCD30 concentration at baseline was significantly higher in cases than in controls (median 65.8, range 27.2-134.5 U/mL vs. median 22.2, range 8.4-63.1 U/mL, P=0.001). In the univariate logistic regression analysis, a higher sCD30 (${\geq}30.2U/mL$) was significantly associated with ATL development (odds ratio 7.88 and the 95% confidence intervals 1.35-45.8, P = 0.02). Among cases, sCD30 concentration tended to increase at the time of diagnosis of aggressive-type ATL, but the concentration was stable in those developing the smoldering-type. This suggests that sCD30 may serve as a predictive marker for the onset of aggressive-type ATL in HTLV-1 carriers.
Upregulation of Mir-34a in AGS Gastric Cancer Cells by a PLGA-PEG-PLGA Chrysin Nano Formulation
Mohammadian, Farideh;Abhari, Alireza;Dariushnejad, Hassan;Zarghami, Faraz;Nikanfar, Alireza;Pilehvar-Soltanahmadi, Yones;Zarghami, Nosratollah 8259
Background: Nano-therapy has the potential to revolutionize cancer therapy. Chrysin, a natural flavonoid, was recently recognized as having important biological roles in chemical defenses and nitrogen fixation, with anti-inflammatory and anti-oxidant effects but the poor water solubility of flavonoids limitstheir bioavailability and biomedical applications. Objective: Chrysin loaded PLGA-PEG-PLGA was assessed for improvement of solubility, drug tolerance and adverse effects and accumulation in a gastric cancer cell line (AGS). Materials and Methods: Chrysin loaded PLGA-PEG copolymers were prepared using the double emulsion method (W/O/W). The morphology and size distributions of the prepared PLGA-PEG nanospheres were investigated by 1H NMR, FT-IR and SEM. The in vitro cytotoxicity of pure and nano-chrysin was tested by MTT assay and miR-34a was measured by real-time PCR. Results: 1H NMR, FT-IR and SEM confirmed the PLGA-PEG structure and chrysin loaded on nanoparticles. The MTT results for different concentrations of chrysin at different times for the treatment of AGS cell line showed IC50 values of 68.2, 56.2 and $42.3{\mu}M$ and 58.2, 44.2, $36.8{\mu}M$ after 24, 48, and 72 hours of treatment, respectively for chrysin itslef and chrysin-loaded nanoparticles. The results of real time PCR showed that expression of miR-34a was upregulated to a greater extent via nano chrysin rather than free chrysin. Conclusions: Our study demonstrates chrysin loaded PLGA-PEG promises a natural and efficient system for anticancer drug delivery to fight gastric cancer.
Cost-Utility of "Doxorubicin and Cyclophosphamide" versus "Gemcitabine and Paclitaxel" for Treatment of Patients with Breast Cancer in Iran
Hatam, Nahid;Askarian, Mehrdad;Javan-Noghabi, Javad;Ahmadloo, Niloofar;Mohammadianpanah, Mohammad 8265
Purpose: A cost-utility analysis was performed to assess the cost-utility of neoadjuvant chemotherapy regimens containing doxorubicin and cyclophosphamide (AC) versus paclitaxel and gemcitabine (PG) for locally advanced breast cancer patients in Iran. Materials and Methods: This cross-sectional study in Namazi hospital in Shiraz, in the south of Iran covered 64 breast cancer patients. According to the random numbers, the patients were divided into two groups, 32 receiving AC and 32 PG. Costs were identified and measured from a community perspective. These items included medical and non-medical direct and indirect costs. In this study, a data collection form was used. To assess the utility of the two regimens, the European Organization for Research and Treatment of Cancer Quality of Life Questionnaire-Core30 (EORTC QLQ-C30) was applied. Using a decision tree, we calculated the expected costs and quality adjusted life years (QALYs) for both methods; also, the incremental cost-effectiveness ratio was assessed. Results: The results of the decision tree showed that in the AC arm, the expected cost was 39,170 US$ and the expected QALY was 3.39 and in the PG arm, the expected cost was 43,336 dollars and the expected QALY was 2.64. Sensitivity analysis showed the cost effectiveness of the AC and ICER=-5535 US$. Conclusions: Overall, the results showed that AC to be superior to PG in treatment of patients with breast cancer, being less costly and more effective.
Altered Cell to Cell Communication, Autophagy and Mitochondrial Dysfunction in a Model of Hepatocellular Carcinoma: Potential Protective Effects of Curcumin and Stem Cell Therapy
Tork, Ola M;Khaleel, Eman F;Abdelmaqsoud, Omnia M 8271
Background: Hepato-carcinogenesis is multifaceted in its molecular aspects. Among the interplaying agents are altered gap junctions, the proteasome/autophagy system, and mitochondria. The present experimental study was designed to outline the roles of these players and to investigate the tumor suppressive effects of curcumin with or without mesenchymal stem cells (MSCs) in hepatocellular carcinoma (HCC). Materials and Methods: Adult female albino rats were divided into normal controls and animals with HCC induced by diethyl-nitrosamine (DENA) and $CCl_4$. Additional groups treated after HCC induction were: Cur/HCC which received curcumin; MSCs/HCC which received MSCs; and Cur+MSCs/HCC which received both curcumin and MSCs. For all groups there were histopathological examination and assessment of gene expression of connexin43 (Cx43), ubiquitin ligase-E3 (UCP-3), the autophagy marker LC3 and coenzyme-Q10 (Mito.Q10) mRNA by real time, reverse transcription-polymerase chain reaction, along with measurement of LC3II/LC3I ratio for estimation of autophagosome formation in the rat liver tissue. In addition, the serum levels of ALT, AST and alpha fetoprotein (AFP), together with the proinflammatory cytokines $TNF{\alpha}$ and IL-6, were determined in all groups. Results: Histopathological examination of liver tissue from animals which received DENA-$CCl_4$ only revealed the presence of anaplastic carcinoma cells and macro-regenerative nodules. Administration of curcumin, MSCs; each alone or combined into rats after induction of HCC improved the histopathological picture. This was accompanied by significant reduction in ${\alpha}$-fetoprotein together with proinflammatory cytokines and significant decrease of various liver enzymes, in addition to upregulation of Cx43, UCP-3, LC3 and Mito.Q10 mRNA. Conclusions: Improvement of Cx43 expression, nonapoptotic cell death and mitochondrial function can repress tumor growth in HCC. Administration of curcumin and/or MSCs have tumor suppressive effects as they can target these mechanisms. However, further research is still needed to verify their effectiveness.
High Prevalence of Helicobacter pylori Resistance to Clarithromycin: a Hospital-Based Cross-Sectional Study in Nakhon Ratchasima Province, Northeast of Thailand
Tongtawee, Taweesak;Dechsukhum, Chavaboon;Matrakool, Likit;Panpimanmas, Sukij;Loyd, Ryan A;Kaewpitoon, Soraya J;Kaewpitoon, Natthawut 8281
Background: Helicobacter pylori is a cause of chronic gastritis, peptic ulcer disease, and gastric malignancy, infection being a serious health problem in Thailand. Recently, clarithromycin resistant H. pylori strains represent the main cause of treatment failure. Therefore this study aimed to determine the prevalence and pattern of H. pylori resistance to clarithromycin in Suranaree University of Technology Hospital, Suranree University of Technology, Nakhon Ratchasima, Northeastern Thailand, Nakhon Ratchasima province, northeast of Thailand. Materials and Methods: This hospital-based cross-sectional study was carried out between June 2014 and February 2015 with 300 infected patients interviewed and from whom gastric mucosa specimens were collected and proven positive by histology. The gastric mucosa specimens were tested for H. pylori and clarithromycin resistance by 23S ribosomal RNA point mutations analysis using real-time polymerase chain reactions. Correlation of eradication rates with patterns of mutation were analyzed by chi-square test. Results: Of 300 infected patients, the majority were aged between 47-61 years (31.6%), female (52.3%), with monthly income between 10,000-15,000 Baht (57%), and had a history of alcohol drinking (59.3%). Patient symptoms were abdominal pain (48.6%), followed by iron deficiency anemia (35.3%). Papaya salad consumption (40.3%) was a possible risk factor for H. pylori infection. The prevalence of H. pylori strains resistant to clarithromycin was 76.2%. Among clarithromycin-resistant strains tested, all were due to the A2144G point mutation in the 23S rRNA gene. Among mutations group, wild type genotype, mutant strain mixed wild type and mutant genotype were 23.8%, 35.7% and 40.5% respectively. With the clarithromycin-based triple therapy regimen, the efficacy decreased by 70% for H. pylori eradication (P<0.01). Conclusions: Recent results indicate a high rate of H. pylori resistance to clarithromycin. Mixed of wild type and mutant genotype is the most common mutant genotype in Nakhon Ratchasima province, therefore the use of clarithromycin-based triple therapy an not advisable as an empiric first-line regimen for H. pylori eradication in northeast region of Thailand.
Thymidylate Synthase Polymorphisms and Risk of Lung Cancer among the Jordanian Population: a Case Control Study
Qasem, Wiam Al;Yousef, Al-Motassem;Yousef, Mohammad;Manasreh, Ihab 8287
Background: Thymidylate synthase (TS) catalyzes the methylation of deoxyuridylate to deoxythymidylate and is involved in DNA methylation, synthesis and repair. Two common polymorphisms have been reported, tandem repeats in the promoter-enhancer region (TSER), and 6bp ins/del in the 5'UTR, that are implicated in a number of human diseases, including cancer. The association between the two polymorphisms in risk for lung cancer (LC) was here investigated in the Jordanian population. Materials and Methods: An age, gender, and smoking-matched case-control study involving 84 lung cancer cases and 71 controls was conducted. The polymerase chain reaction/restriction fragment length polymorphism (PCR-RFLP) technique was used to detect the polymorphism of interest. Results: Individuals bearing the ins/ins genotype were 2.5 times more likely to have lung cancer [(95%CI: 0.98-6.37), p=0.051]. Individuals who were less than or equal to 57 years and carrying ins/ins genotype were 4.6 times more susceptible to lung cancer [OR<57 vs >57years: 4.6 (95%CI: 0.93-22.5), p=0.059)]. Genotypes and alleles of TSER were distributed similarly between cases and controls. Weak linkage disequilibrium existed between the two loci of interest (Lewontin's coefficient [D']) (LC: D' =0.03, r2: 0. 001, p=0.8; Controls: D' =0.29, r2: 0.08, p=0.02). Carriers of the "3 tandem repeats_insertion" haplotype (3R_ins) were 2 times more likely to have lung cancer [2 (95%CI: 1.13-3.48), p=0.061]. Conclusions: Genetic polymorphism of TS at 3 'UTR and its haplotype analysis may modulate the risk of lung cancer in Jordanians. The 6bp ins/del polymorphism of TS at 3 'UTR is more informative than TSER polymorphism in predicting increased risk.
Retrospective Evaluation of Risk Factors and Immunohistochemical Findings for Pre-Neoplastic and Neoplastic lesions of Upper Urinary Tract in Patients with Chronic Nephrolithiasis
Desai, Fanny Sharadkumar;Nongthombam, Jitendra;Singh, Lisam Shanjukumar 8293
Background: Urinary stones are known predisposing factors for upper urinary tract carcinoma (UUTC) which are commonly detected at advanced stage with poor outcome because of rarity and lack of specific criteria for early detection. Aims and objectives: The main aim was to evaluate the impact of age, gender andstone characteristics on risk of developing UUTC in patients with chronic nephrolithiasis. We also discuss the role of aberrant angiogenesis (AA) and immunohistochemical expression of p53, p16INK4a, CK20 and Ki-67 in diagnosis of pelvicalyceal neoplastic (NL) and pre-neoplastic lesions (PNL) in these patients. Materials and Methods: Retrospective analysis of pelvicalyceal urothelial lesions from 88 nephrectomy specimens were carried out in a tertiary care centre from June 2012 to December 2014. Immunohistochemistry (IHC) was performed on 37 selected cases. Computed image analysis was performed to analyse aberrant angiogenesis. Results: All UUTC (5.7%) and metaplastic lesions were found to be associated with stones. Some 60% were pure squamous cell carcinoma and 40% were transitional cell carcinoma. Odd ratios for developing NL and PNL lesions in presence of renal stone, impacted stones, multiple and large stag horn stones were 9.39 (95% CI 1.15-76.39, p value 0.05), 6.28 (95% CI 1.59-24.85, p value 0.000) and 7.4 (95% CI, 2.29-23.94, p value 0.001) respectively. When patient age was ${\geq}55$, the odds ratio for developing NL was 3.43 (95% CI 1.19-9.88, p value 0.019). IHC analysis showed that mean Ki-67 indices were $3.15{\pm}3.63%$ for non-neoplastic lesions, $10.0{\pm}9.45%$ for PNL and $28.0{\pm}18.4%$ for NL. Sensitivity and specificity of CK20, p53, p16INK4a, AA were 76% and 95.9%; 100% and 27.5%; 100% and 26.5%; 92.3 % and 78.8% respectively. Conclusions: Age ${\geq}55years$, large stag horn stones, multiple stones and impacted stones are found to be associated with increased risk of NL and PNL in UUT. For flat lesions, a panel of markers, Ki 67 index >10 and presence of aberrant angiogenesis were more useful than individual markers.
Plasma Circulating Cell-free Nuclear and Mitochondrial DNA as Potential Biomarkers in the Peripheral Blood of Breast Cancer Patients
Mahmoud, Enas H;Fawzy, Amal;Ahmad, Omar K;Ali, Amr M 8299
Background: In Egypt, breast cancer is estimated to be the most common cancer among females. It is also a leading cause of cancer-related mortality. Use of circulating cell-free DNA (ccf-DNA) as non-invasive biomarkers is a promising tool for diagnosis and follow-up of breast cancer (BC) patients. Objective: To assess the role of circulating cell free DNA (nuclear and mitochondrial) in diagnosing BC. Materials and Methods: Multiplex real time PCR was used to detect the level of ccf nuclear and mitochondrial DNA in the peripheral blood of 50 breast cancer patients together with 30 patients with benign lesions and 20 healthy controls. Laboratory investigations, histopathological staging and receptor studies were carried out for the cancer group. Receiver operating characteristic curves were used to evaluate the performance of ccf-nDNA and mtDNA. Results: The levels of both nDNA and mtDNA in the cancer group were significantly higher in comparison to the benign and the healthy control group. There was a statistically significant association between nDNA and mtDNA levels and well established prognostic parameters; namely, histological grade, tumour stage, lymph node status andhormonal receptor status. Conclusions: Our data suggests that nuclear and mitochondrial ccf-DNA may be used as non-invasive biomarkers in BC.
Pharmacophore Development for Anti-Lung Cancer Drugs
Haseeb, Muhammad;Hussain, Shahid 8307
Lung cancer is one particular type of cancer that is deadly and relatively common than any other. Treatment is with chemotherapy, radiation therapy and surgery depending on the type and stage of the disease. Focusing on drugs used for chemotherapy and their associated side effects, there is a need to design and develop new anti-lung cancer drugs with minimal side effects and improved efficacy. The pharmacophore model appears to be a very helpful tool serving in the designing and development of new lead compounds. In this paper, pharmacophore analysis of 10 novel anti-lung cancer compounds was validated for the first time. Using LigandScout the pharmacophore features were predicted and 3D pharmacophores were extracted via VMD software. A training set data was collected from literature and the proposed model was applied to the training set whereby validating and verifying similar activity as that of the most active compounds was achieved. Therefore pharmacophore develoipment could be recommended for further studies.
In Vitro Anti-Neuroblastoma Activity of Thymoquinone Against Neuro-2a Cells via Cell-cycle Arrest
Paramasivam, Arumugam;Raghunandhakumar, Subramanian;Priyadharsini, Jayaseelan Vijayashree;Jayaraman, Gopalswamy 8313
We have recently shown that thymoquinone (TQ) has a potent cytotoxic effect and induces apoptosis via caspase-3 activation with down-regulation of XIAP in mouse neuroblastoma (Neuro-2a) cells. Interestingly, our results showed that TQ was significantly more cytotoxic towards Neuro-2a cells when compared with primary normal neuronal cells. In this study, the effects of TQ on cell-cycle regulation and the mechanisms that contribute to this effect were investigated using Neuro-2a cells. Cell-cycle analysis performed by flow cytometry revealed cell-cycle arrest at G2/M phase and a significant increase in the accumulation of TQ-treated cells at sub-G1 phase, indicating induction of apoptosis by the compound. Moreover, TQ increased the expression of p53, p21 mRNA and protein levels, whereas it decreased the protein expression of PCNA, cyclin B1 and Cdc2 in a dose-dependent manner. Our finding suggests that TQ could suppress cell growth and cell survival via arresting the cell-cycle in the G2/M phase and inducing apoptosis of neuroblastoma cells.
Epidemiology of Hydatidiform Moles in a Tertiary Hospital in Thailand over Two Decades: Impact of the National Health Policy
Wairachpanich, Varangkana;Limpongsanurak, Sompop;Lertkhachonsuk, Ruangsak 8321
Background: The incidence of hydatidiform mole (HM) differs among regions but has declined significantly over time. In Thailand, the initiation of universal health coverage in 2002 has resulted in a change of medical services countrywide. However, impacts of these policies on gestational trophoblastic disease (GTD) cases in Thailand have not been reported. This study aimed to find the incidence of hydatidiform mole (HM) in King Chulalongkorn Memorial Hospital (KCMH) from 1994-2013, comparing before and after the implementation of the universal coverage health policy. Materials and Methods: All cases of GTD in KCMH from 1994-2013 were reviewed from medical records. The incidence of HM, patient characteristics, treatment and remission rates were compared over two study decades between 1994-2003 and 2004-2013. Results: Hydatidiform mole cases decreased from 204 cases in the first decade to 111 cases in the seond decade. Overall incidence of HM was 1.70 per 1,000 deliveries. The incidence of HM in the first and second decades were 1.70 and 1.71 per 1,000 deliveries, respectively (p=0.65, 95%CI 1.54-1.88). Referred cases of nonmolar gestational trophoblastic neoplasia (GTN) increased from 12 (4.4%) to 23 (14.4%, p<0.01). Vaginal bleeding was the most common presenting symptom which decreased from 89.4% to 79.6% (p=0.02). Asymptomatic HM patients increased from 4.8% to 10.2% (p=0.07). Rate of postmolar GTN was 26%. Conclusions: The number of HM cases in this study decreased over 2 decades but incidence was unchanged. Referral rates of malignant cases were more common after universal health coverage policy initiation. Classic clinical presentation was decreased significantly in the last decade.
Breast Cancer in Lampang, a Province in Northern Thailand: Analysis of 1993-2012 Incidence Data and Future Trends
Lalitwongsa, Somkiat;Pongnikorn, Donsuk;Daoprasert, Karnchana;Sriplung, Hutcha;Bilheem, Surichai 8327
Background: The recent epidemiologic transition in Thailand, with decreasing incidence of infectious diseases along with increasing rates of chronic conditions, including cancer, is a serious problem for the country. Breast cancer has the highest incidence rates among females throughout Thailand. Lampang is a province in the upper part of Northern Thailand. A study was needed to identify the current burden, and the future trends of breast cancer in upper Northern Thai women. Materials and Methods: Here we used cancer incidence data from the Lampang Cancer Registry to characterize and analyze the local incidence of breast cancer. Joinpoint analysis, age period cohort model and Nordpred package were used to investigate the incidences of breast cancer in the province from 1993 to 2012 and to project future trends from 2013 to 2030. Results: Age-standardized incidence rates (world) of breast cancer in the upper parts of Northern Thailand increased from 16.7 to 26.3 cases per 100,000 female population which is equivalent to an annual percentage change of 2.0-2.8%, according to the method used. Linear drift effects played a role in shaping the increase of incidence. The three projection method suggested that incidence rates would continue to increase in the future with incidence for women aged 50 and above, increasing at a higher rate than for women below the age of 50. Conclusions: The current early detection measures increase detection rates of early disease. Preparation of a budget for treatment facilities and human resources, both in surgical and medical oncology, is essential.
Comparative Investigation of Single Voxel Magnetic Resonance Spectroscopy and Dynamic Contrast Enhancement MR Imaging in Differentiation of Benign and Malignant Breast Lesions in a Sample of Iranian Women
Faeghi, Fariborz;Baniasadipour, Banafsheh;Jalalshokouhi, Jalal 8335
Purpose: To make a comparison of single voxel magnetic resonance spectroscopy (SV-MRS) and dynamic contrast enhancement (DCE) MRI for differentiation of benign and malignant breast lesions in a sample of Iranian women. Materials and Methods: A total of 30 women with abnormal breast lesions detected in mammography, ultrasound, or clinical breast exam were examined with DCE and SV-MRS. tCho (total choline) resonance in MRS spectra was qualitatively evaluated and detection of a visible tCho peak at 3.2 ppm was defined as a positive finding for malignancy. Different types of DCE curves were persistent (type 1), plateau (type 2), and washout (type 3). At first, lesions were classified according to choline findings and types of DCE curve, finally being compared to pathological results as the standard reference. Results: this study included 19 patients with malignant lesions and 11 patients with benign ones. While 63.6 % of benign lesions (7 of 11) showed type 1 DCE curves and 36.4% (4 of 11) showed type 2, 57.9% (11of 19) of malignant lesions were type 3 and 42.1% (8 of 19) type 2. Choline peaks were detected in 18 of 19 malignant lesions and in 3 of 11 benign counterparts. 1 malignant and 8 benign cases did not show any visible resonance at 3.2 ppm so SV-MRS featured 94.7% sensitivity, 72.7 % specificity and 86.7% accuracy.Conclusions: The present findings indicate that a combined approach using MRS and DCE MRI can improve the specificity of MRI for differentiation of benign and malignant breast lesions.
Outcome and Cost Effectiveness of Ultrasonographically Guided Surgical Clip Placement for Tumor Localization in Patients undergoing Neo-adjuvant Chemotherapy for Breast Cancer
Masroor, Imrana;Zeeshan, Sana;Afzal, Shaista;Sufian, Saira Naz;Ali, Madeeha;Khan, Shaista;Ahmad, Khabir 8339
Background: To determine the outcome and cost saving by placing ultrasound guided surgical clips for tumor localization in patients undergoing neo-adjuvant chemotherapy for breast cancer. Materials and Methods: This retrospective cross sectional analytical study was conducted at the Department of Diagnostic Radiology, Aga Khan University Hospital, Karachi, Pakistan from January to December 2014. A sample of 25 women fulfilling our selection criteria was taken. All patients came to our department for ultrasound guided core biopsy of suspicious breast lesions and clip placement in the index lesion prior to neo-adjuvant chemotherapy. All the selected patients had biopsy proven breast cancer. Results: The mean age was $45{\pm}11.6years$. There were no complications seen after clip placement in terms of clip migration or hemorrhage. The cost of commercially available markers was approximately PKR 9,000 (US$ 90) and that of the surgical clip was PKR 900 (US$ 9). The cost of surgical clips in 25 patients was PKR 22,500 (US$ 225), when compared to the commercially available markers which may have incurred a cost of PKR 225,000 (US$ 2,250). The total cost saving for 25 patients was PKR 202,500 (US$ 2, 025), making it PKR 8100 (US$ 81) per patient. Conclusions: The results of our study show that ultrasound guided surgical clip placement in index lesions prior to neo-adjuvant therapy is a safe and cost effective method to identify tumor bed and response to treatment for further management.
Colorectal Cancer Awareness and Screening Preference: A Survey during the Malaysian World Digestive Day Campaign
Suan, Mohd Azri Mohd;Mohammed, Noor Syahireen;Hassan, Muhammad Radzi Abu 8345
Background: Although the incidence of colorectal cancer in Malaysia is increasing, awareness of this cancer, including its symptoms, risk factors and screening methods, remains low among Malaysian populations. This survey was conducted with the aim of (i) ascertaining the awareness level regarding colorectal cancer symptoms, risk factors and its screening among the general populations and (ii) assessing the public preference and willingness to pay for colorectal cancer screening. Materials and Methods: The questionnaire was distributed in eight major cities in West Malaysia during the World Health Digestive Day (WDHD) campaign. Two thousand four hundred and eight respondents participated in this survey. Results: Generally, awareness of colorectal cancer was found to be relatively good. Symptoms such as change in bowel habit, blood in the stool, weight loss and abdominal pain were well recognized by 86.6%, 86.9%, 83.4% and 85.6% of the respondents, respectively. However, common risk factors such as positive family history, obesity and old age were acknowledged only by less than 70% of the respondents. Almost 80% of the respondents are willing to take the screening test even without any apparent symptoms. Colonoscopy is the preferred screening method, but only 37.5% were willing to pay from their own pocket to get early colonoscopy. Conclusions: Continous cancer education should be promoted with more involvement from healthcare providers in order to make future colorectal cancer screening programs successful.
Automatic Electronic Cleansing in Computed Tomography Colonography Images using Domain Knowledge
Manjunath, KN;Siddalingaswamy, PC;Prabhu, GK 8351
Electronic cleansing is an image post processing technique in which the tagged colonic content is subtracted from colon using CTC images. There are post processing artefacts, like: 1) soft tissue degradation; 2) incomplete cleansing; 3) misclassification of polyp due to pseudo enhanced voxels; and 4) pseudo soft tissue structures. The objective of the study was to subtract the tagged colonic content without losing the soft tissue structures. This paper proposes a novel adaptive method to solve the first three problems using a multi-step algorithm. It uses a new edge model-based method which involves colon segmentation, priori information of Hounsfield units (HU) of different colonic contents at specific tube voltages, subtracting the tagging materials, restoring the soft tissue structures based on selective HU, removing boundary between air-contrast, and applying a filter to clean minute particles due to improperly tagged endoluminal fluids which appear as noise. The main finding of the study was submerged soft tissue structures were absolutely preserved and the pseudo enhanced intensities were corrected without any artifact. The method was implemented with multithreading for parallel processing in a high performance computer. The technique was applied on a fecal tagged dataset (30 patients) where the tagging agent was not completely removed from colon. The results were then qualitatively validated by radiologists for any image processing artifacts.
Breast Cancer in Lopburi, a Province in Central Thailand: Analysis of 2001-2010 Incidence and Future Trends
Sangkittipaiboon, Somphob;Leklob, Atit;Sriplung, Hutcha;Bilheem, Surichai 8359
Background: Thailand has come to an epidemiologic transition with decreasing infectious diseases and increasing burden of chronic conditions, including cancer. Breast cancer has the highest incidence rates among females throughout Thailand. This study aimed to identify the current burden and the future trends of breast cancer of Lopburi, a province in the Central Thailand. Materials and Methods: We used cancer incidence data from the Lopburi Cancer Registry to characterize and analyze the incidence of breast cancer in Central Thailand. With joinpoint and age-period-cohort analyses, the incidence of breast cancer in the province from 2001 to 2010 and project future trends from 2011 to 2030 was investigated. Results: Age-adjusted incidence rates of breast cancer in Lopburi increased from 23.4 to 34.3 cases per 100,000 female population during the period, equivalent to an annual percentage change of 4.3% per year. Both period and cohort effects played a role in shaping the increase in incidence. Joinpoint projection suggested that incidence rates would continue to increase in the future with incidence for women ages 50 years and above increasing at a higher rate than for women below the age of 50. Conclusions: The current situation where early detection measures are being promoted could increase detection rates of the disease. Preparation of sufficient budget for treatment facilities and human resources, both in surgical and medical oncology, is essential for future medical care.
Association between Shammah Use and Oral Leukoplakia-like Lesions among Adult Males in Dawan Valley, Yemen
Al-Tayar, Badr Abdullah;Tin-Oo, Mon Mon;Sinor, Modh Zulkarnian;Alakhali, Mohammed Sultan 8365
Background: Shammah is a traditional form of snuff dipping tobacco (a smokeless tobacco form) that is commonly used in Yemen. Oral mucosal changes due to the use of shammah can usually be observed in the mucosal surfaces that the product touches. The aim of this study was to determine the association between shammah use and oral leukoplakia-like lesions. Other associated factors were also determined. Materials and Methods: A cross sectional study was conducted on 346 randomly selected adult males. Multi-stage random sampling was used to select the study location. After completing the structured questionnaire interviews, all the participants underwent clinical exanimation for screening of oral leukoplakia-like lesions Clinical features of oral leukoplakia-like lesion were characterized based on the grades of $Ax{\acute{e}}ll$ et al (1976). Univariable logistic regression and multivariable logistic regression were used to assess the potential associated factors. Results: Out of 346 male participants aged 18 years and older, 68 (19.7%) reported being current shammah users. The multivariable analysis revealed that age, non-formal or primary level of education, former shammah user, current shammah user, and frequency of shammah use per day were statistically associated with the presence of oral leukoplakia-like lesions [Adjusted odds ratio (AOR) = 1.03; 95% confidence interval (CI) : 1.01, 1.06; P=0.006], (AOR=8.65; 95% CI: 2.81, 26.57; P=0.001), (AOR=3.65; 95% CI: 1.40, 9.50; P=0.008), (AOR=12.99; 95% CI: 6.34, 26.59; P=0.001), and (AOR=1.17; 95% CI: 1.02, 1.36; P=0.026), respectively. Conclusions: The results revealed oral leukoplakia-like lesions to be significantly associated with shammah use. Therefore, it is important to develop comprehensive shammah prevention programs in Yemen.
Factors Associated with Adherence to Colorectal Cancer Screening among Moderate Risk Individuals in Iran
Taheri-Kharameh, Zahra;Noorizadeh, Farsad;Sangy, Samira;Zamanian, Hadi;Shouri-Bidgoli, Ali Reza;Oveisi, Helaleh 8371
Background: Colorectal cancer is one of the most common neoplasms in Iran. Secondary prevention (colorectal cancer screening) is important and a most valuable method of early diagnosis of this cancer. The objectives of this study were to determine the factors associated with colorectal cancer screening adherence among Iranians 50 years and older using the Health Belief Model. Materials and Methods: This cross-sectional study was conducted from June 2012 to May 2013. A convenience sample of 200 individuals aged 50 and older was recruited from the population at outpatient clinics in teaching hospitals. Data gathering tools were the Champions health belief model scale (CHBMS) with coverage of socio-demographic background and CRC screening information. Multiple logistic regression was performed to identify factors associated with colorectal cancer screening adherence. Results: The mean age of participants was $62.5{\pm}10.8$ and 75.5% were women. A high percentage of the participants had not heard or read about colorectal cancer (86.5%) and CRC screening (93.5%). Perceived susceptibility to colorectal cancer had the lowest percentage of all of the subscales. Participants who perceived more susceptibility (OR =2.99; CI 95%: 1.23-5.45) and reported higher knowledge (OR =1.29; CI 95%: 1.86-3.40) and those who reported fewer barriers (OR =.37; CI 95%:.21-.89), were more likely to have carried out colorectal cancer screening. Conclusions: Our findings indicated that CRC knowledge, perceived susceptibility and barriers were significant predictors of colorectal cancer screening adherence. Strategies to increase knowledge and overcome barriers in risk individuals appear necessary. Education programs should be promoted to overcome knowledge deficiency and negative perceptions in elderly Iranians.
Association of PNPLA3 Polymorphism with Hepatocellular Carcinoma Development and Prognosis in Viral and Non-Viral Chronic Liver Diseases
Khlaiphuengsin, Apichaya;Kiatbumrung, Rattanaporn;Payungporn, Sunchai;Pinjaroen, Nutcha;Tangkijvanich, Pisit 8377
Background: The aim of this study was to evaluate any association between a single nucleotide polymorphism (SNP) in the patatin-like phospholipase domain containing 3 (PNPLA3) (rs738409, C>G) and the development and prognosis in patients with hepatocellular carcinoma (HCC). Materials and Methods: Two hundred heathy controls and 388 HCC cases were included: 211 with HBV, 98 patients with HCV, 29 with alcoholic steatohepatitis (ASH) and 52 with non-alcoholic steatohepatitis (NASH). The SNP was determined by real-time PCR based on TaqMan assays. Results: The prevalence of rs738409 genotypes CC, CG and GG in controls was 91 (45.5%), 88 (44.0%), and 21 (10.5%), respectively, while the corresponding genotypes in all patients with HCC was 158 (40.7%), 178 (45.9%), and 52 (13.4%). The GG genotype had significantly higher distribution in patients with ASH/NASH-related HCC compared with controls (OR=2.34, 95% CI=1.16-4.71, P=0.018), and viral-related HCC cases (OR=2.15, 95% CI=1.13-4.08, P=0.020). However, the frequency of the GG genotype was similar between controls and patients with viral-related HCC. At initial diagnosis, HBV-related HCC were larger and at more advanced BCLC stage than the other HCC groups. There were no significant differences between the GG and non-GG groups regarding clinical characteristics, tumor stage and overall survival. Conclusions: These data suggest an influence of the PNPLA3 polymorphism on the occurrence of HCC in patients with ASH/NASH but not among those with chronic viral hepatitis. However, the polymorphism was not associated with the prognosis of HCC.
Treatment of Oral Leukoplakia with Diode Laser: a Pilot Study on Indian Subjects
Kharadi, Usama A Rashid;Onkar, Sanjeev;Birangane, Rajendra;Chaudhari, Swapnali;Kulkarni, Abhay;Chaudhari, Rohan 8383
Background: To evaluate the safety, convenience and effectiveness of 940nm diode laser for treatment of homogenous leukoplakia. Materials and Methods: Ten patients having homogenous leukoplakia which were diagnosed clinically were selected from an Indian dental educational institution for the study. Toludine blue staining was applied locally over the lesion. The area where there was increased uptake of stain was excised using a 940 nm EZLASE TM diode laser (BIOLASE-USA). Results: Although various treatment modalities have been tried and the search continues for novel treatment modalities for complete removal of homogenous leukoplakia, from results of our preliminary pilot study it is clear that the use of 940 nm diode laser as a treatment modality for homogenous leukoplakia is a good substitute. Healing was perfect without any complication within a duration of 1 month. Pain intensity was also mild and absolutely zero on the VAS scale after 1 month follow up. Conclusions: 940 nm diode lasers are safe and can be effectively used as a treatment modality of homogenous leukoplakia, without any complication and without compromising health and oral function of patients. Considering recurrence factor, long term follow up for patients is a must.
Comparison between Use of PSA Kinetics and Bone Marrow Micrometastasis to Define Local or Systemic Relapse in Men with Biochemical Failure after Radical Prostatectomy for Prostate Cancer
Murray, Nigel P;Reyes, Eduardo;Fuentealba, Cynthia;Orellana, Nelson;Jacob, Omar 8387
Background: Treatment of biochemical failure after radical prostatectomy for prostate cancer is largely empirically based. The use of PSA kinetics has been used as a guide to determine local or systemic treatment of biochemical failure. We here compared PSA kinetics with detection of bone marrow micrometastasis as methods to determine local or systemic relapse. Materials and Methods: A transversal study was conducted of men with biochemical failure, defined as a serum PSA >0.2ng/ml after radical prostatectomy. Consecutive patients having undergone radical prostatectomy and with biochemical failure were enrolled and clinical and pathological details were recorded. Bone marrow biopsies were obtained from the iliac crest and touch prints made, micrometastasis (mM) being detected using anti-PSA. The clinical parameters of total serum PSA, PSA velocity, PSA doubling time and time to biochemical failure, age, Gleason score and pathological stage were registered. Results: A total of 147 men, mean age $71.6{\pm}8.2years$, with a median time to biochemical failure of 5.5 years (IQR 1.0-6.3 years) participated in the study. Bone marrow samples were positive for micrometastasis in 98/147 (67%) of patients at the time of biochemical failure. The results of bone marrow micrometastasis detected by immunocytochemistry were not concordant with local relapse as defined by PSA velocity, time to biochemical failure or Gleason score. In men with a PSA doubling time of < six months or a total serum PSA of >2,5ng/ml at the time of biochemical failure the detection of bone marrow micrometastasis was significantly higher. Conclusions: The detection of bone marrow micrometastasis could be useful in defining systemic relapse, this minimally invasive procedure warranting further studies with a larger group of patients.
Nutritional Status among Rural Community Elderly in the Risk Area of Liver Fluke, Surin Province, Thailand
Kaewpitoon, Soraya J;Namwichaisirikul, Niwatchai;Loyd, Ryan A;Churproong, Seekaow;Ueng-Arporn, Naporn;Matrakool, Likit;Tongtawee, Taweesak;Rujirakul, Ratana;Nimkhuntod, Porntip;Wakhuwathapong, Parichart;Kaewpitoon, Natthawut 8391
Thailand is becoming an aging society, this presenting as a serious problem situation especially regarding health. Chronic diseases found frequently in the elderly may be related to dietary intake and life style. Surin province has been reported as a risk area for liver fluke with a high incidence of cholangiocarcinma especially in the elderly. Therefore, this study aimed to determine the nutritional status and associated factors among elderly in Surin province, northeast of Thailand. A community-based cross-sectional study was conducted among 405 people aged 60 years and above, between September 2012 and July 2014. The participants were selected through a randomized systematic sampling method and completed a pre-designed questionnaire with general information, food recorded, weight, height, waist circumference, and behavior regarding to food consume related to liver fluke infection. The data were analyzed using descriptive statistics and Spearman's rank correlation coefficients. The majority of participants was female (63.5%), age between 60-70 years old (75.6%), with elementary school education (96.6%), living with their (78.9%), and having underlying diseases (38.3%). Carbohydrate (95.3%) was need to improve the consumption. The participants demonstrated under-nutrition (24.4%), over-nutrition (16.4%), and obesity (15.4%). Elderly had a waist circumference as the higher than normal level (34.0%). Gender, female, age 71-80 years old, elementary school and underlying diseases were significantly associated with poor nutritional status. The majority of them had a high knowledge (43.0%), moderate attitude (44.4%), and moderate practice (46.2%) regarding food consumption related to liver fluke infection. In conclusion, these findings data indicated that elderly age group often have an under- or over-nutritional status. Carbohydrate consumption needs to be improved. Some elderly show behavior regarding food consumption that is related to liver fluke infection hat needs to be improved, so that health education pertaining good nutrition is required.
Association of Histopathological Markers with Clinico-Pathological Factors in Mexican Women with Breast Cancer
Bandala, Cindy;De la Garza-Montano, Paloma;Cortes-Algara, Alfredo;Cruz-Lopez, Jaime;Dominguez-Rubio, Rene;Gonzalez-Lopez, Nelly Judith;Cardenas-Rodriguez, Noemi;Alfaro-Rodriguez, A;Salcedo, M;Floriano-Sanchez, E;Lara-Padilla, Eleazar 8397
Background: Breast cancer (BCa) is the most common malignancy in Mexican women. A set of histopathological markers has been established to guide BCa diagnosis, prognosis and treatment. Nevertheless, in only a few Mexican health services, such as that of the Secretariat of National Defense (SEDENA for its acronym in Spanish), are these markers commonly employed for assessing BCa. The aim of this study was to explore the association of Ki67, TP53, HER2/neu, estrogenic receptors (ERs) and progesterone receptors (PRs) with BCa risk factors. Materials and Methods: Clinical histories provided background patient information. Immunohistochemical (IHC) analysis was conducted on 48 tissue samples from women diagnosed with BCa and treated with radical mastectomy. The Chi square test or Fisher exact test together with the Pearson and Spearman correlation were applied. Results: On average, patients were $58{\pm}10.4$ years old. It was most common to find invasive ductal carcinoma (95.8%), histological grade 3 (45.8%), with a poor Nottingham Prognostic Index (NPI; 80.4%). ERs and PRs were associated with smoking and alcohol consumption, metastasis at diagnosis and Ki67 expression (p<0.05). PR+ was also related to urea and ER+ (p<0.05). Ki67 was associated with TP53 and elevated triglycerides (p<0.05), and HER2/neu with ER+, the number of pregnancies and tumor size (p<0.05). TP53 was also associated with a poor NPI (p<0.05) and CD34 with smoking (p<0.05). The triple negative status (ER-/PR-/HER2/neu-) was related to smoking, alcohol consumption, exposure to biomass, number of pregnancies, metastasis and a poor NPI (p<0.05). Moreover, the luminal B subty was associated with histological type (p=0.007), tumor size (p=0.03) and high cholesterol (p=0.02). Conclusions: Ki67, TP53, HER2/neu, ER and PR proved to be related to several clinical and pathological factors. Hence, it is crucial to determine this IHC profile in women at risk for BCa. Certain associations require further study to understand physiological/biochemical/molecular processes.
Single Nucleotide Polymorphisms in STAT3 and STAT4 and Risk of Hepatocellular Carcinoma in Thai Patients with Chronic Hepatitis B
Chanthra, Nawin;Payungporn, Sunchai;Chuaypen, Natthaya;Piratanantatavorn, Kesmanee;Pinjaroen, Nutcha;Poovorawan, Yong;Tangkijvanich, Pisit 8405
Hepatitis B virus (HBV) infection is the leading cause of hepatocellular carcinoma (HCC) development. Recent studies demonstrated that single nucleotide polymorphisms (SNPs) rs2293152 in signal transducer and activator of transcription 3 (STAT3) and rs7574865 in signal transducer and activator of transcription 4 (STAT4) are associated with chronic hepatitis B (CHB)-related HCC in the Chinese population. We hypothesized that these polymorphisms might be related to HCC susceptibility in Thai population as well. Study subjects were divided into 3 groups consisting of CHB-related HCC (n=192), CHB without HCC (n=200) and healthy controls (n=190). The studied SNPs were genotyped using polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP). The results showed that the distribution of different genotypes for both polymorphisms were in Hardy-Weinberg equilibrium (P>0.05). Our data demonstrated positive association of rs7574865 with HCC risk when compared to healthy controls under an additive model (GG versus TT: odds ratio (OR)=2.07, 95% confidence interval (CI)=1.06-4.03, P=0.033). This correlation remained significant under allelic and recessive models (OR=1.46, 95% CI=1.09-1.96, P=0.012 and OR=1.71, 95% CI=1.13-2.59, P=0.011, respectively). However, no significant association between rs2293152 and HCC development was observed. These data suggest that SNP rs7574865 in STAT4 might contribute to progression to HCC in the Thai population.
Inhibition of NF-ĸB, Bcl-2 and COX-2 Gene Expression by an Extract of Eruca sativa Seeds during Rat Mammary Gland Carcinogenesis
Abdel-Rahman, Salah;Shaban, Nadia;Haggag, Amany;Awad, Doaa;Bassiouny, Ahmad;Talaat, Iman 8411
The effect of Eruca sativa seed extract (SE) on nuclear factor kappa B (NF-${\kappa}B$), cyclooxygenase-2 (COX-2) and B-cell lymphoma-2 (Bcl-2) gene expression levels was investigated in rat mammary gland carcinogenesis induced by 7,12 dimethylbenz(${\alpha}$)anthracene (DMBA). DMBA increased NF-${\kappa}B$, COX-2 and Bcl-2 gene expression levels and lipid peroxidation (LP), while, decreased glutathione-S-transferase (GST) and superoxide dismutase (SOD) activities and total antioxidant concentration (TAC) compared to the control group. After DMBA administration, SE treatment reduced NF-${\kappa}B$, COX-2 and Bcl-2 gene expression levels and LP. Hence, SE treatment reduced inflammation and cell proliferation, while increasing apoptosis, GST and SOD activities and TAC. Analysis revealed that SE has high concentrations of total flavonoids, triterpenoids, alkaloids and polyphenolic compounds such as gallic, chlorogenic, caffeic, 3,4-dicaffeoyl quinic, 3,5-dicaffeoyl quinic, tannic, cinnamic acids, catechin and phloridzin. These findings indicate that SE may be considered a promising natural product from cruciferous vegetables against breast cancer, especially given its high antioxidant properties.
Aqueous Extract of Anticancer Drug CRUEL Herbomineral Formulation Capsules Exerts Anti-proliferative Effects in Renal Cell Carcinoma Cell Lines
Verma, Shiv Prakash;Sisoudiya, Saumya;Das, Parimal 8419
Purpose: Anti-cancer activity evaluation of aqueous extract of CRUEL (herbomineral formulation) capsules on renal cell carcinoma cell lines, and exploration of mechanisms of cell death. Materials and Methods: To detect the cytotoxic dose concentration in renal cell carcinoma (RCC) cells, MTT assays were performed and morphological changes after treatment were observed by inverted microscopy. Drug effects against RCC cell lines were assessed with reference to cell cycle distribution (flow cytometry), anti-metastatic potential (wound healing assay) and autophagy(RT-PCR). Results: CRUEL showed anti-proliferative effects against RCC tumor cell lines with an IC50 value of ${\approx}4mg/mL$ in vitro., while inducing cell cycle arrest at S-phase of cell cycle and inhibiting wound healing. LC3 was found to be up-regulated after drug treatment in RT-PCR resulting in an autophagy mode of cell death. Conclusions: This study provides the experimental validation for antitumor activity of CRUEL.
Safety and Prognostic Impact of Prophylactic Level VII Lymph Node Dissection for Papillary Thyroid Carcinoma
Fayek, Ihab Samy;Kamel, Ahmed Ahmed;Sidhom, Nevine FH 8425
Purpose: To study the safety of prophylactic level VII nodal dissection regarding hypoparathyroidism (temporary and permanent) and vocal cord dysfunction (temporary and permanent) and its impact on disease free survival. Materials and Methods: This prospective study concerned 63 patients with papillary thyroid carcinoma with N0 neck node involvement (clinically and radiologically) in the period from December 2009 to May 2013. All patients underwent total thyroidectomy and prophylactic central neck dissection including levels VI and VII lymph nodes in group A (31 patients) and level VI only in group B (32 patients). The thyroid gland, level VI and level VII lymph nodes were each examined histopathologically separately for tumor size, multicentricity, bilaterality, extrathyroidal extension, number of dissected LNs and metastatic LNs. Follow-up of both groups, regarding hypoparathyroidism, vocal cord dysfunction and DFS, ranged from 6-61 months. Results: The mean age was 34.8 and 34.3, female predominance in both groups with F: M 24:7 and 27:5 in groups A and B, respectively. Mean tumor size was 12.6 and 14.7mm. No statistical differences were found between both groups regarding age, sex, bilaterality, multicentricity or extrathyroidal extension. The mean no. of dissected level VI LNs was 5.06 and 4.72 and mean no. of metastatic level VI was 1 and 0.84 in groups A and B, respectively. The mean no. of dissected level VII LNs was 2.16 and mean no. of metastatic LNs was 0.48. Postoperatively temporary hypoparathyroidism was detected in 10 and 7 patients and permanent hypoparathyroidism in 2 and 3 patients; temporary vocal cord dysfunction was detected in 4 patients and one patient, and permanent vocal cord dysfunction in one and 2 patients in groups A and B, respectively. No significant statistical differences were noted between the 2 groups regarding hypoparathyroidism (P=0.535) or vocal cord dysfunction (P=0.956). The number of dissected LNs at level VI only significantly affected the occurrence of hypoparathyroidism (<0.001) and vocal cord dysfunction (<0.001).The DFS was significantly affected by bilaterality, multicentricity and extrathyroidal extension. Conclusions: Level VII nodal dissection is a safe procedure complementary to level VI nodal dissection with prophylactic central neck dissection for papillary thyroid carcinoma.
Combined Treatment with 2-Deoxy-D-Glucose and Doxorubicin Enhances the in Vitro Efficiency of Breast Cancer Radiotherapy
Islamian, Jalil Pirayesh;Aghaee, Fahimeh;Farajollahi, Alireza;Baradaran, Behzad;Fazel, Mona 8431
Doxorubicin (DOX) was introduced as an effective chemotherapeutic for a wide range of cancers but with some severe side effects especially on myocardia. 2-Deoxy-D-glucose (2DG) enhances the damage caused by chemotherapeutics and ionizing radiation (IR) selectively in cancer cells. We have studied the effects of $1{\mu}M$ DOX and $500{\mu}M$ 2DG on radiation induced cell death, apoptosis and also on the expression levels of p53 and PTEN genes in T47D and SKBR3 breast cancer cells irradiated with 100, 150 and 200 cGy x-rays. DOX and 2DG treatments resulted in altered radiation-induced expression levels of p53 and PTEN genes in T47D as well as SKBR3 cells. In addition, the combination along with IR decreased the viability of both cell lines. The radiobiological parameter (D0) of T47D cells treated with 2DG/DOX and IR was 140 cGy compared to 160 cGy obtained with IR alone. The same parameters for SKBR3 cell lines were calculated as 120 and 140 cGy, respectively. The sensitivity enhancement ratios (SERs) for the combined chemo-radiotherapy on T47D and SKBR3 cell lines were 1.14 and 1.16, respectively. According to the obtained results, the combination treatment may use as an effective targeted treatment of breast cancer either by reducing the single modality treatment side effects.
Incidence and Mortality of Breast Cancer and their Relationship with the Human Development Index (HDI) in the World in 2012
Ghoncheh, Mahshid;Mirzaei, Maryam;Salehiniya, Hamid 8439
Background: Breast cancer is the most common malignancy in women worldwide and its incidence is generally increasing. In 2012, it was the second most common cancer in the world. It is necessary to obtain information on incidence and mortality for health planning. This study aimed to investigate the relationship between the human development index (HDI), and the incidence and mortality rates of breast cancer in the world in 2012. Materials and Methods: This ecologic study concerns incidence rate and standardized mortality rates of the cancer from GLOBOCAN in 2012, and HDI and its components extracted from the global bank site. Data were analyzed using correlation tests and regression with SPSS software (version 15). Results: Among the six regions of WHO, the highest breast cancer incidence rate (67.6) was observed in the PAHO, and the lowest incidence rate was 27.8 for SEARO. There was a direct, strong, and meaningful correlation between the standardized incidence rate and HDI (r=0.725, $p{\leq}0.001$). Pearson correlation test showed that there was a significant correlation between age-specific incidence rate (ASIR) and components of the HDI (life expectancy at birth, mean years of schooling, and GNP). On the other, a non-significant relationship was observed between ASIR and HDI overall (r=0.091, p=0.241). In total, a significant relationship was not found between age-specific mortality rate (ASMR) and components of HDI. Conclusions: Significant positive correlations exist between ASIR and components of the HDI. Socioeconomic status is directly related to the stage of the cancer and patient's survival. With increasing the incidence rate of the cancer, mortality rate from the cancer does not necessariloy increase. This may be due to more early detection and treatment in developed that developing countries. It is necessary to increase awareness of risk factors and early detection in the latter.
Estudy the Effect of Breast Cancer on Tlr2 Expression in Nb4 Cell
Amirfakhri, Siamak;Salimi, Arsalan;Fernandez, Nelson 8445
Background: Breast cancer is the most common neoplasm in women and the most frequent cause of death in those between 35 and 55 years of age. All multicellular organisms have an innate immune system, whereas the adaptive or 'acquired' immune system is restricted to vertebrates. This study focused on the effect of conditioned medium isolated from cultured breast cancer cells on NB4 neutrophil-like cells. Materials and Methods: In the current study neutrophil-like NB4 cells were incubated with MCF-7 cell-conditioned medium. After 6 h incubation the intracellular receptor TLR2, was analyzed. Results: The results revealed that MCF-7 cell-conditioned medium elicited expression of TLR2 in NB4 cells. Conclusions: This treatment would result in the production of particular stimulants (i.e. soluble cytokines), eliciting the expression of immune system receptors. Furthermore, the flow cytometry results demonstrated that MCF-7 cell-conditioned medium elicited an effect on TLR2 intracellular receptors.
Epidemiological Characteristics of Gallbladder Cancer in Jeju Island: A Single-Center, Clinically Based, Age-Sex-Matched, Case-Control Study
Cha, Byung Hyo 8451
Background: Gallbladder cancer (GBC) is a rare but highly invasive malignancy characterized by poor survival. In a national cancer survey, the age-standardized incidence rate of GBC was highest in Jeju Island among the 15 provinces in South Korea. The aim of this descriptive epidemiological study was to suggest the modifiable risk factors for this rare malignant disease in Jeju Island by performing an age-sex-matched case-control study. Materials and Methods: The case group included patients diagnosed with GBC at the Department of Internal Medicine of Cheju Halla General Hospital, Jeju, South Korea, within the 5-year study period. The control group consisted of age-sex-matched subjects selected from among the participants of the health promotion center at the same institute and in the same period. We compared 78 case-control pairs in terms of clinical variables such as histories of hypertension, diabetes, vascular occlusive disorders, alcohol and smoking consumption, obesity, and combined polypoid lesions of the gallbladder (PLG) or gallstone diseases (GSDs). Results: Among the relevant risk factors, alcohol consumption, parity ${\geq}2$, PLG, and GSDs were significant risk factors in the univariate analysis. PLG (p < 0.01; OR, 51.1; 95% confidence interval [CI], 2.98-875.3) and GSD (p < 0.01; OR, 54.9; 95% CI, 3.00-1001.8) were associated risk factors of GBC in the multivariate analysis with the conditional logistic regression model. However, we failed to find any correlation between obesity and GBC. We also found a negative correlation between alcohol consumption history and GBC in the multivariate analysis (p < 0.01; OR, 0.06; 95% CI, 0.01-0.31). Conclusions: These results suggest that combined PLG and GSDs are strongly associated with the GBC in Jeju Island and mild to moderate alcohol consumption may negatively correlate with GBC risk.
Improving Participation in Colorectal Cancer Screening: a Randomised Controlled Trial of Sequential Offers of Faecal then Blood Based Non-Invasive Tests
Symonds, Erin L;Pedersen, Susanne;Cole, Stephen R;Massolino, Joseph;Byrne, Daniel;Guy, John;Backhouse, Patricia;Fraser, Robert J;LaPointe, Lawrence;Young, Graeme P 8455
Background: Poor participation rates are often observed in colorectal cancer (CRC) screening programs utilising faecal occult blood tests. This may be from dislike of faecal sampling, or having benign bleeding conditions that can interfere with test results. These barriers may be circumvented by offering a blood-based DNA test for screening. The aim was to determine if program participation could be increased by offering a blood test following faecal immunochemical test (FIT) non-participation. Materials and Methods: People were invited into a CRC screening study through their General Practice and randomised into control or intervention (n=600/group). Both groups were mailed a FIT (matching conventional screening programs). Participation was defined as FIT completion within 12wk. Intervention group non-participants were offered a screening blood test (methylated BCAT1/IKZF1). Overall participation was compared between the groups. Results: After 12wk, FIT participation was 82% and 81% in the control and intervention groups. In the intervention 96 FIT nonparticipants were offered the blood test - 22 completed this test and 19 completed the FIT instead. Total screening in the intervention group was greater than the control (88% vs 82%, p<0.01). Of 12 invitees who indicated that FIT was inappropriate for them (mainly due to bleeding conditions), 10 completed the blood test (83%). Conclusions: Offering a blood test to FIT non-participants increased overall screening participation compared to a conventional FIT program. Blood test participation was particularly high in invitees who considered FIT to be inappropriate for them. A blood test may be a useful adjunct test within a FIT program.
DPPA2 Protein Expression is Associated with Gastric Cancer Metastasis
Shabestarian, Hoda;Ghodsi, Mohammad;Mallak, Afsaneh Javdani;Jafarian, Amir Hossein;Montazer, Mehdi;Forghanifard, Mohammad Mahdi 8461
Gastric cancer (GC) as the fourth most common cause of malignancies shows high rate of morbidity appropriating the second leading cause of cancer-related death worldwide. Developmental pluripotency associated-2 (DPPA2), cancer-testis antigen (CT100), is commonly expressed only in the human germ line and pluripotent embryonic cells but it is also present in a significant subset of malignant tumors. To investigate whether or not DPPA2 expression is recalled in GC, our aim in this study was to elucidate DPPA2 protein expression in gastric cancer. Fifty five GC tumor and their related margin normal tissues were recruited to evaluate DPPA2 protein expression and its probable associations with different clinicopathological features of the patients. DPPA2 was overexpressed in GC cases compared with normal tissues (P < .005). While DPPA2 expression was detected in all GC samples, its high expression was found in 23 of 55 tumor tissues (41.8%). Interestingly, 50 of 55 normal samples (90.9%) were negative for DPPA2 protein expression and remained 5 samples showed very low expression of DPPA2. DPPA2 protein expression in GC was significantly correlated with lymph node metastasis (p = 0.012). The clinical relevance of DPPA2 in GC illustrated that high level expression of this protein was associated with lymph node metastasis supporting this hypothesis that alteration in DPPA2 was associated with aggressiveness of gastric cancer and may be an early event in progression of the disease. DPPA2 may be introduced as a new marker for invasive and metastatic GCs.
Genetic Variation in the ABCB1 Gene May Lead to mRNA Level Chabge: Application to Gastric Cancer Cases
Mansoori, Maryam;Golalipour, Masoud;Alizadeh, Shahriar;Jahangirerad, Ataollah;Khandozi, Seyed Reza;Fakharai, Habibollah;Shahbazi, Majid 8467
Background: One of the major mechanisms for drug resistance is associated with altered anticancer drug transport, mediated by the human-adenosine triphosphate binding cassette (ABC) transporter superfamily proteins. The overexpression of adenosine triphosphate binding cassette, sub-family B, member 1 (ABCB1) by multidrug-resistant cancer cells is a serious impediment to chemotherapy. In our study we have studied the possibility that structural single-nucleotide polymorphisms (SNP) are the mechanism of ABCB1 overexpression. Materials and Methods: A total of 101 gastric cancer multidrug resistant cases and 100 controls were genotyped with sequence-specific primed PCR (SSP-PCR). Gene expression was evaluated for 70 multidrug resistant cases and 54 controls by real time PCR. The correlation between the two groups was based on secondary structures of RNA predicted by bioinformatics tool. Results: The results of genotyping showed that among 3 studied SNPs, rs28381943 and rs2032586 had significant differences between patient and control groups but there were no differences in the two groups for C3435T. The results of real time PCR showed over-expression of ABCB1 when we compared our data with each of the genotypes in average mode. Prediction of secondary structures in the existence of 2 related SNPs (rs28381943 and rs2032586) showed that the amount of ${\Delta}G$ for original mRNA is higher than the amount of ${\Delta}G$ for the two mentioned SNPs. Conclusions: We have observed that 2 of our studied SNPs (rs283821943 and rs2032586) may elevate the expression of ABCB1 gene, through increase in mRNA stability, while this was not the case for C3435T.
Knowledge, Attitude and Practices Regarding HPV Vaccination Among Medical and Para Medical in Students, India a Cross Sectional Study
Swarnapriya, K;Kavitha, D;Reddy, Gopireddy Murali Mohan 8473
Background: High risk human papilloma virus (HPV) types 16 and 18 have been proven as central causes of cervical cancer and safety and immunogenicity of HPV vaccines are sufficiently established. Knowledge and practices of HPV vaccination among medical and paramedical students is vital as these may strongly determine intention to recommend vaccination to others in the future. The present study was therefore undertaken to assess the knowledge, attitude and practices regarding cervical cancer screening and HPV vaccination among medical and paramedical students and to analyze factors influencing them. Materials and Methods: The present cross sectional study, conducted in a tertiary care teaching hospital in south India, included undergraduate students aged 18 years and above, belonging to medical, dental and nursing streams, after informed written consent. Results: Out of 957 participants, only 430 (44.9%) displayed good knowledge and only 65 (6.8%) had received HPV vaccination. Among the unvaccinated, 433 (48.54%), were not willing to take the vaccine. Concerns regarding the efficacy (30.5%), safety (26.1%) and cost of the vaccine (21.7%) were responsible for this. Age, gender, family history of malignancy and mother's education had no influence on knowledge. Compared to medical students, nursing students had better knowledge (OR-1.49, 95% CI 0.96 to 2.3, p = 0.072) and students of dentistry had poor knowledge (OR-0.50 95% CI 0.36 to 0.70, p<0.001). Conclusions: The knowledge and uptake of HPV vaccination among medical and paramedical students in India is poor. Targeted health education interventions may have huge positive impact not only on the acceptance of vaccination among them, but also on their intention to recommend the vaccine in future.
Knowledge and Perceptions about Colorectal Cancer in Jordan
Taha, Hana;Jaghbeer, Madi Al;Shteiwi, Musa;AlKhaldi, Sireen;Berggren, Vanja 8479
Background: Colorectal cancer (CRC) is the third most common cancer globally. In Jordan, it is the number one cancer among men and the second most common cancer among women, accounting for 15% and 9.4% respectively of all male and female diagnosed cancers. This study aimed to evaluate the knowledge and perceptions about colorectal cancer risk factors, signs and symptoms in Jordan and to provide useful data about the best modes of disseminating preventive messages about the disease. Materials and Methods: A stratified clustered random sampling technique was used to recruit 300 males and 300 females aged 30 to 65 years without a previous history of CRC from four governorates in Jordan. A semi-structured questionnaire and face to face interviews were employed. Descriptive and multivariate analysis was applied to assess knowledge and perceptions about CRC. Results: Both males and females perceived their CRC risk to be low. They had low knowledge scores about CRC with no significant gender association (P=0.47). From a maximum knowledge score of 18 points, the median scores of males and females were 4 points (SD=2.346, range 0-13) and 4 points (SD=2.329, range 0-11) respectively. Better knowledge scores were associated with governorate, higher educational level, older age, higher income, having a chronic disease, having a family history of CRC, previously knowing someone who had CRC and their doctor's knowledge about their family history of CRC. Conclusions: There is a low level of knowledge about CRC and underestimation of risk among the study participants. This underlines the need for public health interventions to create awareness about the illness. It also calls for further research to assess the knowledge and perceptions about CRC early detection examinations in Jordan.
Improved Detection of Helicobacter pylori Infection and Premalignant Gastric Mucosa Using "Site Specific Biopsy": a Randomized Control Clinical Trial
Tongtawee, Taweesak;Dechsukhum, Chavaboon;Leeanansaksiri, Wilairat;Kaewpitoon, Soraya;Kaewpitoon, Natthawut;Loyd, Ryan A;Matrakool, Likit;Panpimanmas, Sukij 8487
Background: Helicobacter pylori infection and premalignant gastric mucosa can be reliably identified using conventional narrow band imaging (C-NBI) gastroscopy. The aim of our study was to compare standard biopsy with site specific biopsy for diagnosis of H. pylori infection and premalignant gastric mucosa in daily clinical practice. Materials and Methods: Of a total of 500 patients who underwent gastroscopy for investigation of dyspeptic symptoms, 250 patients underwent site specific biopsy using C-NBI (Group 1) and 250 standard biopsy (Group 2). Sensitivity, specificity, and positive and negative predictive values were assessed. The efficacy of detecting H. pylori associated gastritis and premalignant gastric mucosa according to the updated Sydney classification was also compared. Results: In group 1 the sensitivity, specificity, positive and negative predictive values for predicting H. pylori positivity were 95.4%, 97.3%, 98.8% and 90.0% respectively, compared to 92.9%, 88.6%, 83.2% and 76.1% in group 2. Site specific biopsy was more effective than standard biopsy in terms of both H. pylori infection status and premalignant gastric mucosa detection (P<0.01). Conclusions: Site specific biopsy using C-NBI can improve detection of H. pylori infection and premalignant gastric mucosa in daily clinical practice.
Comparison of Unsatisfactory Rates and Detection of Abnormal Cervical Cytology Between Conventional Papanicolaou Smear and Liquid-Based Cytology (Sure Path®)
Kituncharoen, Saroot;Tantbirojn, Patou;Niruthisard, Somchai 8491
Purpose: To compare unsatisfactory rates and detection of abnormal cervical cytology between conventional cytology or Papanicolaou smear (CC) and liquid-based cytology (LBC). Materials and Methods: A total of 23,030 cases of cervical cytology performed at King Chulalongkorn Memorial Hospital during 2012-2013 were reviewed. The percentage unsatisfactory and detection rates of abnormal cytology were compared between CC and LBC methods. Results: There was no difference in unsatisfactory rates between CC and LBC methods (0.1% vs. 0.1%, p = 0.84). The detection rate for squamous cell abnormalities was significantly higher with the LBC method (7.7% vs. 11.5%, p < 0.001), but those for overall abnormal glandular epithelium were similar (0.4% vs. 0.6%, p = 0.13). Low grade squamous lesion (ASC-US and LSIL) were more frequently detected by the LBC method (6.1% vs. 9.5%, p < 0.001). However, there was no difference in high gradd squamous lesions (1.1% vs. 1.1%, p = 0.95). When comparing between types of glandular abnormality, there was no significant difference the groups. Conclusions: There was no difference in unsatisfactory rates between the conventional smear and LBC. However, LBC could detect low grade squamous cell abnormalities more than CC, while there were similar rates of detection of high grade squamous cell lesions and glandular cell abnormalities.
Comparative Assessment of a Self-sampling Device and Gynecologist Sampling for Cytology and HPV DNA Detection in a Rural and Low Resource Setting: Malaysian Experience
Latiff, Latiffah A;Ibrahim, Zaidah;Pei, Chong Pei;Rahman, Sabariah Abdul;Akhtari-Zavare, Mehrnoosh 8495
Purpose: This study was conducted to assess the agreement and differences between cervical self-sampling with a Kato device (KSSD) and gynecologist sampling for Pap cytology and human papillomavirus DNA (HPV DNA) detection. Materials and Methods: Women underwent self-sampling followed by gynecologist sampling during screening at two primary health clinics. Pap cytology of cervical specimens was evaluated for specimen adequacy, presence of endocervical cells or transformation zone cells and cytological interpretation for cells abnormalities. Cervical specimens were also extracted and tested for HPV DNA detection. Positive HPV smears underwent gene sequencing and HPV genotyping by referring to the online NCBI gene bank. Results were compared between samplings by Kappa agreement and McNemar test. Results: For Pap specimen adequacy, KSSD showed 100% agreement with gynecologist sampling but had only 32.3% agreement for presence of endocervical cells. Both sampling showed 100% agreement with only 1 case detected HSIL favouring CIN2 for cytology result. HPV DNA detection showed 86.2%agreement (K=0.64, 95% CI 0.524-0.756, p=0.001) between samplings. KSSD and gynaecologist sampling identified high risk HPV in 17.3% and 23.9% respectively (p=0.014). Conclusion: The self-sampling using Kato device can serve as a tool in Pap cytology and HPV DNA detection in low resource settings in Malaysia. Self-sampling devices such as KSSD can be used as an alternative technique to gynaecologist sampling for cervical cancer screening among rural populations in Malaysia.
Early Activation of Apoptosis and Caspase-independent Cell Death Plays an Important Role in Mediating the Cytotoxic and Genotoxic Effects of WP 631 in Ovarian Cancer Cells
Gajek, Arkadiusz;Denel-Bobrowska, Marta;Rogalska, Aneta;Bukowska, Barbara;Maszewski, Janusz;Marczak, Agnieszka 8503
The purpose of this study was to provide a detailed explanation of the mechanism of bisanthracycline, WP 631 in comparison to doxorubicin (DOX), a first generation anthracycline, currently the most widely used pharmaceutical in clinical oncology. Experiments were performed in SKOV-3 ovarian cancer cells which are otherwise resistant to standard drugs such as cis-platinum and adriamycin. As attention was focused on the ability of WP 631 to induce apoptosis, this was examined using a double staining method with Annexin V and propidium iodide probes, with measurement of the level of intracellular calcium ions and cytosolic cytochrome c. The western blotting technique was performed to confirm PARP cleavage. We also investigated the involvement of caspase activation and DNA degradation (comet assay and immunocytochemical detection of phosphorylated H2AX histones) in the development of apoptotic events. WP 631 demonstrated significantly higher effectiveness as a pro-apoptotic drug than DOX. This was evident in the higher levels of markers of apoptosis, such as the externalization of phosphatidylserine and the elevated level of cytochrome c. An extension of incubation time led to an increase in intracellular calcium levels after treatment with DOX. Lower changes in the calcium content were associated with the influence of WP 631. DOX led to the activation of all tested caspases, 8, 9 and 3, whereas WP 631 only induced an increase in caspase 8 activity after 24h of treatment and consequently led to the cleavage of PARP. The lack of active caspase 3 had no outcome on the single and double-stranded DNA breaks. The obtained results show that WP 631 was considerably more genotoxic towards the investigated cell line than DOX. This effect was especially visible after longer times of incubation. The above detailed studies indicate that WP 631 generates early apoptosis and cell death independent of caspase-3, detected at relatively late time points. The observed differences in the mechanisms of the action of WP631 and DOX suggest that this bisanthracycline can be an effective alternative in ovarian cancer treatment.
Breast Cancer Survival at a Leading Cancer Centre in Malaysia
Abdullah, Matin Mellor;Mohamed, Ahmad Kamal;Foo, Yoke Ching;Lee, Catherine May Ling;Chua, Chin Teong;Wu, Chin Huei;Hoo, LP;Lim, Teck Onn;Yen, Sze Whey 8513
Background: GLOBOCAN12 recently reported high cancer mortality in Malaysia suggesting its cancer health services are under-performing. Cancer survival is a key index of the overall effectiveness of health services in the management of patients. This report focuses on Subang Jaya Medical Centre (SJMC) care performance as measured by patient survival outcome for up to 5 years. Materials and Methods: All women with breast cancer treated at SJMC between 2008 and 2012 were enrolled for this observational cohort study. Mortality outcome was ascertained through record linkage with national death register, linkage with hospital registration system and finally through direct contact by phone or home visits. Results: A total of 675 patients treated between 2008 and 2012 were included in the present survival analysis, 65% with early breast cancer, 20% with locally advanced breast cancer (LABC) and 4% with metastatic breast cancer (MBC). The overall relative survival (RS) at 5 years was 88%. RS for stage I was 100% and for stage II, III and IV disease was 95%, 69% and 36% respectively. Conclusions: SJMC is among the first hospitals in Malaysia to embark on routine measurement of the performance of its cancer care services and its results are comparable to any leading centers in developed countries.
Religion as an Alleviating Factor in Iranian Cancer Patients: a Qualitative Study
Rahnama, Mozhgan;Khoshknab, Masoud Fallahi;Maddah, Sadat Seyed Bagher;Ahmadi, Fazlollah;Arbabisarjou, Azizollah 8519
After diagnosis of cancer, many patients show more inclination towards religion and religious activities. This qualitative study using semi-structured interviews explored the perspectives and experiences of 17 Iranian cancer patients and their families regarding the role of religion in their adaptation to cancer in one of the hospitals in Tehran and a charity institute. The content analysis identified two themes: "religious beliefs" (illness as God's will, being cured by God's will, belief in God's supportiveness, having faith in God as a relieving factor, and hope in divine healing) and "relationship with God during the illness." In general, relationship with God and religious beliefs had a positive effect on the patients adapting to their condition, without negative consequences such as stopping their treatment process and just waiting to be cured by God. Thus a strengthening of such beliefs, as a coping factor, could be recommended through religious counseling.
Patterns of Cancer in Kurdistan - Results of Eight Years Cancer Registration in Sulaymaniyah Province-Kurdistan-Iraq
Khoshnaw, Najmaddin;Mohammed, Hazha A;Abdullah, Dana A 8525
Background: Cancer has become a major health problem associated with high mortality worldwide, especially in developing countries. The aim of our study was to evaluate the incidence rates of different types of cancer in Sulaymaniyah from January-2006 to January-2014. The data were compared with those reported for other middle east countries. Materials and Methods: This retrospective study depended on data collected from Hiwa hospital cancer registry unit, death records and histopathology reports in all Sulaymaniyah teaching hospitals, using international classification of diseases. Results: A total of 8,031 cases were registered during the eight year period, the annual incidence rate in all age groups rose from 38 to 61.7 cases/100,000 population/year, with averages over 50 in males and 50.7 in females. The male to female ratio in all age groups were 0.98, while in the pediatric age group it was 1.33. The hematological malignancies in all age groups accounted for 20% but in the pediatric group around half of all cancer cases. Pediatric cancers were occluding 7% of total cancers with rates of 10.3 in boys and 8.7 in girls. The commonest malignancies by primary site were leukemia, lymphoma, brain, kidney and bone. In males in all age groups they were lung, leukaemia, lymphoma, colorectal, prostate, bladder, brain, stomach, carcinoma of unknown primary (CUP) and skin, while in females they were breast, leukaemia, lymphoma, colorectal, ovary, lung, brain, CUP, and stomach. Most cancers were increased with increasing age except breast cancer where decrease was noted in older ages. High mortality rates were found with leukemia, lung, lymphoma, colorectal, breast and stomach cancers. Conclusions: We here found an increase in annual cancer incidence rates across the period of study, because of increase of cancer with age and higher rates of hematological malignancies. Our study is valuable for Kurdistan and Iraq because it provides more accurate data about the exact patterns of cancer and mortality in our region.
Role of Tumor Necrosis Factor-Producing Mesenchymal Stem Cells on Apoptosis of Chronic B-lymphocytic Tumor Cells Resistant to Fludarabine-based Chemotherapy
Valizadeh, Armita;Ahmadzadeh, Ahmad;Saki, Ghasem;Khodadadi, Ali;Teimoori, Ali 8533
Background: B-cell chronic lymphocytic leukemia B (B-CLL), the most common type of leukemia, may be caused by apoptosis deficiency in the body. Adipose tissue-derived mesenchymal stem cells (AD-MSCs) as providers of pro-apoptotic molecules such as tumor necrosis factor-related apoptosis-inducing ligand (TRAIL), can be considered as an effective anti-cancer therapy candidate. Therefore, in this study we assessed the role of tumor necrosis factor-producing mesenchymal stem cells oin apoptosis of B-CLL cells resistant to fludarabine-based chemotherapy. Materials and Methods: In this study, after isolation and culture of AD-MSCs, a lentiviral LeGO-iG2-TRAIL-GFP vector containing a gene producing the ligand pro-apoptotic with plasmid PsPAX2 and PMDG2 virus were transfected into cell-lines to generate T293HEK. Then, T293HEK cell supernatant containing the virus produced after 48 and 72 hours was collected, and these viruses were transduced to reprogram AD-MSCs. Apoptosis rates were separately studied in four groups: group 1, AD-MSCs-TRAIL; group 2, AD-MSCs-GFP; group 3, AD-MSCs; and group 4, CLL. Results: Observed apoptosis rates were: group 1, $42{\pm}1.04%$; group 2, $21{\pm}0.57%$; group 3, $19{\pm}2.6%$; and group 4, % $0.01{\pm}0.01$. The highest rate of apoptosis thus occurred ingroup 1 (transduced TRAIL encoding vector). In this group, the average medium-soluble TRAIL was 72.7pg/m and flow cytometry analysis showed a pro-apoptosis rate of $63{\pm}1.6%$, which was again higher than in other groups. Conclusions: In this study we have shown that tumor necrosis factor (TNF) secreted by AD-MSCs may play an effective role in inducing B-CLL cell apoptosis.
Low Coverage and Disparities of Breast and Cervical Cancer Screening in Thai Women: Analysis of National Representative Household Surveys
Mukem, Suwanna;Meng, Qingyue;Sriplung, Hutcha;Tangcharoensathien, Viroj 8541
Background: The coverage of breast and cervical cancer screening has only slightly increased in the past decade in Thailand, and these cancers remain leading causes of death among women. This study identified socioeconomic and contextual factors contributing to the variation in screening uptake and coverage. Materials and Methods: Secondary data from two nationally representative household surveys, the Health and Welfare Survey (HWS) 2007 and the Reproductive Health Survey (RHS) 2009 conducted by the National Statistical Office were used. The study samples comprised 26,951 women aged 30-59 in the 2009 RHS, and 14,619 women aged 35 years and older in the 2007 HWS were analyzed. Households of women were grouped into wealth quintiles, by asset index derived from Principal components analysis. Descriptive and logistic regression analyses were performed. Results: Screening rates for cervical and breast cancers increased between 2007 and 2009. Education and health insurance coverage including wealth were factors contributing to screening uptake. Lower or non-educated and poor women had lower uptake of screenings, as were young, unmarried, and non-Buddhist women. Coverage of the Civil Servant Medical Benefit Scheme increased the propensity of having both screenings, while the universal coverage scheme increased the probability of cervical screening among the poor. Lack of awareness and knowledge contributed to non-use of both screenings. Women were put off from screening, especially Muslim women on cervical screening, because of embarrassment, fear of pain and other reasons. Conclusions: Although cervical screening is covered by the benefit package of three main public health insurance schemes, free of charge to all eligible women, the low coverage of cervical screening should be addressed by increasing awareness and strengthening the supply side. As mammography was not cost effective and not covered by any scheme, awareness and practice of breast self examination and effective clinical breast examination are recommended. Removal of cultural barriers is essential.
Gene Expression Biodosimetry: Quantitative Assessment of Radiation Dose with Total Body Exposure of Rats
Saberi, Alihossein;Khodamoradi, Ehsan;Birgani, Mohammad Javad Tahmasebi;Makvandi, Manoochehr 8553
Background: Accurate dose assessment and correct identification of irradiated from non-irradiated people are goals of biological dosimetry in radiation accidents. Objectives: Changes in the FDXR and the RAD51 gene expression (GE) levels were here analyzed in response to total body exposure (TBE) to a 6 MV x-ray beam in rats. We determined the accuracy for absolute quantification of GE to predict the dose at 24 hours. Materials and Methods: For this in vivo experimental study, using simple randomized sampling, peripheral blood samples were collected from a total of 20 Wistar rats at 24 hours following exposure of total body to 6 MV X-ray beam energy with doses (0.2, 0.5, 2 and 4 Gy) for TBE in Linac Varian 2100C/D (Varian, USA) in Golestan Hospital, in Ahvaz, Iran. Also, 9 rats was irradiated with a 6MV X-ray beam at doses of 1, 2, 3 Gy in 6MV energy as a validation group. A sham group was also included. After RNA extraction and DNA synthesis, GE changes were measured by the QRT-PCR technique and an absolute quantification strategy by taqman methodology in peripheral blood from rats. ROC analysis was used to distinguish irradiated from non-irradiated samples (qualitative dose assessment) at a dose of 2 Gy. Results: The best fits for mean of responses were polynomial equations with a R2 of 0.98 and 0.90 (for FDXR and RAD51 dose response curves, respectively). Dose response of the FDXR gene produced a better mean dose estimation of irradiated "validation" samples compared to the RAD51 gene at doses of 1, 2 and 3 Gy. FDXR gene expression separated the irradiated rats from controls with a sensitivity, specificity and accuracy of 87.5%, 83.5% and 81.3%, respectively, 24 hours after dose of 2 Gy. These values were significantly (p<0.05) higher than the 75%, 75% and 75%, respectively, obtained using gene expression of RAD51 analysis at a dose of 2 Gy. Conclusions: Collectively, these data suggest that absolute quantification by gel purified quantitative RT-PCR can be used to measure the mRNA copies for GE biodosimetry studies at comparable accuracy to similar methods. In the case of TBE with 6MV energy, FDXR gene expression analysis is more precise than that with RAD51 for quantitative and qualitative dose assessment.
Radiofrequency Ablation in Treating Colorectal Cancer Patients with Liver Metastases
Xu, Chuan;Huang, Xin-En;Lv, Peng-Hua;Wang, Shu-Xiang;Sun, Ling;Wang, Fu-An 8559
Purpose: To evaluate efficacy of radiofrequency ablation (RFA) in treating colorectal cancer patients with liver metastases. Methods: During January 2010 to April 2012, 56 colorectal cancer patients with liver metastases underwent RFA. CT scans were obtained one month after RFA for all patients to evaluate tumor response. (CR+PR+SD)/n was used to count the disease control rates (DCR). Survival data of 1, 2 and 3 years were obtained from follow up. Results: Patients were followed for 10 to 40 months after RFA (mean time, $25{\pm}10months$). Median survival time was 27 months. The 1, 2, 3 year survival rate were 80.4%, 71.4%, 41%, 1 % respectively. 3-year survival time for patients with CR or PR after RFA was 68.8% and 4.3% respectively, the difference was statistically significant. The number of CR, PR, SD and PD in our study was 13, 23, 11 and 9 respectively. Conclusions: RFA could be an effective method for treating colorectal cancer patients with liver metastases, and prolong survival time, especially for metastatic lesions less than or equal to 3 cm. But this result should be confirmed by randomized controlled studies.
Effect of Root Extracts of Medicinal Herb Glycyrrhiza glabra on HSP90 Gene Expression and Apoptosis in the HT-29 Colon Cancer Cell Line
Nourazarian, Seyed Manuchehr;Nourazarian, Alireza;Majidinia, Maryam;Roshaniasl, Elmira 8563
Colorectal cancer is one of the most common lethal cancer types worldwide. In recent years, widespread and large-scale studies have been done on medicinal plants for anti-cancer effects, including Glycyrrhiza glabra. The aim of this study was to evaluate the effects of an ethanol extract Glycyrrhiza glabra on the expression of HSP90, growth and apoptosis in the HT-29 colon cancer cell line. HT-29 cells were treated with different concentrations of extract (50,100,150, and $200{\mu}g/ml$). For evaluation of cell proliferation and apoptosis, we used MTT assay and flow cytometry technique, respectively. RT-PCR was also carried out to evaluate the expression levels of HSP90 genes. Results showed that Glycyrrhiza glabra inhibited proliferation of the HT-29 cell line at a concentration of $200{\mu}g/ml$ and this was confirmed by the highest rate of cell death as measured by trypan blue and MTT assays. RT-PCR results showed down-regulation of HSP90 gene expression which implied an ability of Glycyrrhiza glabra to induce apoptosis in HT-29 cells and confirmed its anticancer property. Further studies are required to evaluate effects of the extract on other genes and also it is necessary to make an extensive in vivo biological evaluation and subsequently proceed with clinical evaluations.
Survival Analysis of Patients with Breast Cancer using Weibull Parametric Model
Baghestani, Ahmad Reza;Moghaddam, Sahar Saeedi;Majd, Hamid Alavi;Akbari, Mohammad Esmaeil;Nafissi, Nahid;Gohari, Kimiya 8567
Background: The Cox model is known as one of the most frequently-used methods for analyzing survival data. However, in some situations parametric methods may provide better estimates. In this study, a Weibull parametric model was employed to assess possible prognostic factors that may affect the survival of patients with breast cancer. Materials and Methods: We studied 438 patients with breast cancer who visited and were treated at the Cancer Research Center in Shahid Beheshti University of Medical Sciences during 1992 to 2012; the patients were followed up until October 2014. Patients or family members were contacted via telephone calls to confirm whether they were still alive. Clinical, pathological, and biological variables as potential prognostic factors were entered in univariate and multivariate analyses. The log-rank test and the Weibull parametric model with a forward approach, respectively, were used for univariate and multivariate analyses. All analyses were performed using STATA version 11. A P-value lower than 0.05 was defined as significant. Results: On univariate analysis, age at diagnosis, level of education, type of surgery, lymph node status, tumor size, stage, histologic grade, estrogen receptor, progesterone receptor, and lymphovascular invasion had a statistically significant effect on survival time. On multivariate analysis, lymph node status, stage, histologic grade, and lymphovascular invasion were statistically significant. The one-year overall survival rate was 98%. Conclusions: Based on these data and using Weibull parametric model with a forward approach, we found out that patients with lymphovascular invasion were at 2.13 times greater risk of death due to breast cancer.
Development and Evaluation of a Patient-Reported Outcome (PRO) Scale for Breast Cancer
Zhang, Jun;Yao, Yu-Feng;Zha, Xiao-Ming;Pan, Li-Qun;Bian, Wei-He;Tang, Jin Hai 8573
Background: This study was guided by principles of the theoretical system of evidence-based medicine. In particular, when searching for evidence of breast cancer, a measuring scale is an instrument for evaluating curative effects in accordance with the laws and characteristics of medicine and exploring the establishment of a system for medically assessing curative effects. At present, there exist few tools for evaluating curative effects. Patient-reported outcomes (PROs) refer to outcomes directly reported by patients (without input or explanations from doctors or other intermediaries) with respect to all aspects of their health. Data obtained from PROs provide evidence of treatment effects. Materials and Methods: In accordance with the tenets of theoretical medicine and ancient medical theory regarding breast cancer, principles for developing a PRO scale were established, and a theoretical model was developed and a literature review was performed, items from this pool were combined and split, and an initial scale was constructed. After a pilot survey and additional modifications, a pre-questionnaire scale was formed and used in a field investigation. After the application of statistical methods, the item pool was used to create a formal scale. The reliability, validity and feasibility of this formal scale were then assessed. Results: In a clinical investigation, 479 responses were recovered, with an acceptance rate of 95%. a combination of various methods was employed, and the items that were selected by all methods or more than half of the methods were employed in the questionnaire. In these cases, the screening methods were combined with certain features of the item, A total of four domains and 38 items were reserved. The reliability analysis indicated that the PRO scale was relatively reliable. Conclusions: Scientific assessment proved that the proposed scale exhibited good reliability and validity. This scale was readily accepted and could be used to assess the curative effects of medical therapy. However, given the limited scope of this investigation, the capacity for adapting this scale to incorporate other theories could not be determined.
Significance of Tissue Expression and Serum Levels of Angiopoietin-like Protein 4 in Breast Cancer Progression: Link to NF-κB /P65 Activity and Pro-Inflammatory Cytokines
Shafik, Noha M;Mohamed, Dareen A;Bedder, Asmaa E;El-Gendy, Ahmed M 8579
Background: The molecular mechanisms linking breast cancer progression and inflammation still remain obscure. The aim of the present study was to investigate the possible association of angiopoeitin like protein 4 (ANGPTL4) and its regulatory factor, hypoxia inducible factor-$1{\alpha}$ (HIF-$1{\alpha}$), with the inflammatory markers nuclear factor kappa B/p65 (NF-${\kappa}B$/P65) and interleukin-1 beta (IL-$1{\beta}$) in order to evaluate their role in inflammation associated breast cancer progression. Materials and Methods: Angiopoietin-like protein 4 (ANGPTL4) mRNA expressions were evaluated using quantitative real time PCR and its protein expression by immunohistochemistry. DNA binding activity of NF-${\kappa}B$/P65 was evaluated by transcription factor binding immunoassay. Serum levels of ANGPTL4, HIF-$1{\alpha}$ and IL-$1{\beta}$ were immunoassayed. Tumor clinico-pathological features were investigated. Results: ANGPTL4 mRNA expressions and serum levels were significantly higher in high grade breast carcinoma ($1.47{\pm}0.31$ and $184.98{\pm}18.18$, respectively) compared to low grade carcinoma ($1.21{\pm}0.32$ and $171.76{\pm}7.58$, respectively) and controls ($0.70{\pm}0.02$ and $65.34{\pm}6.41$, respectively), (p<0.05). Also, ANGPTL4 high/moderate protein expression was positively correlated with tumor clinico-pathological features. In addition, serum levels of HIF-$1{\alpha}$ and IL-$1{\beta}$ as well as NF-${\kappa}B$/P65 DNA binding activity were significantly higher in high grade breast carcinoma ($148.54{\pm}14.20$, $0.79{\pm}0.03$ and $247.13{\pm}44.35$ respectively) than their values in low grade carcinoma ( $139.14{\pm}5.83$, $0.34{\pm}0.02$ and $184.23{\pm}37.75$, respectively) and controls ($33.95{\pm}3.11$, $0.11{\pm}0.02$ and $7.83{\pm}0.92$, respectively), (p<0.001). Conclusion: ANGPTL4 high serum levels and tissue expressions in advanced grade breast cancer, in addition to its positive correlation with tumor clinico-pathological features and HIF-$1{\alpha}$ could highlight its role as one of the signaling factors involved in breast cancer progression. Moreover, novel correlations were found between ANGPTL4 and the inflammatory markers, IL-$1{\beta}$ and NF-${\kappa}B$/p65, in breast cancer, which may emphasize the utility of these markers as potential tools for understanding interactions for axes of carcinogenesis and inflammation contributed for cancer progression. It is thus hoped that the findings reported here would assist in the development of new breast cancer management strategies that would promote patients' quality of life and ultimately improve clinical outcomes. However, large-scale studies are needed to verify these results.
Age-Period-Cohort Analysis of Liver Cancer Mortality in Korea
Park, Jihwan;Jee, Yon Ho 8589
Background: Liver cancer is one of the most common causes of death in the world. In Korea, hepatitis B virus (HBV) is a major risk factor for liver cancer but infection rates have been declining since the implementation of the national vaccination program. In this study, we examined the secular trends in liver cancer mortality to distinguish the effects of age, time period, and birth cohort. Materials and Methods: Data for the annual number of liver cancer deaths in Korean adults (30 years and older) were obtained from the Korean Statistical Information Service for the period from 1984-2013. Joinpoint regression analysis was used to study the shapes of and to detect the changes in mortality trends. Also, an age-period-cohort model was designed to study the effect of each age, period, and birth cohort on liver cancer mortality. Results: For both men and women, the age-standardized mortality rate for liver cancer increased from 1984 to 1993 and decreased thereafter. The highest liver cancer mortality rate has shifted to an older age group in recent years. Within the same birth cohort group, the mortality rate of older age groups has been higher than in the younger age groups. Age-period-cohort analysis showed an association with a high mortality rate in the older age group and in recent years, whereas a decreasing mortality rate were observed in the younger birth cohort. Conclusions: This study confirmed a decreasing trend in liver cancer mortality among Korean men and women after 1993. The trends in mortality rate may be mainly attributed to cohort effects.
Electronic Risk Assessment System as an Appropriate Tool for the Prevention of Cancer: a Qualitative Study
Amoli, Amir hossein Javan;Maserat, Elham;Safdari, Reza;Zali, Mohammad Reza 8595
Background: Decision making modalities for screening for many cancer conditions and different stages have become increasingly complex. Computer-based risk assessment systems facilitate scheduling and decision making and support the delivery of cancer screening services. The aim of this article was to survey electronic risk assessment system as an appropriate tool for the prevention of cancer. Materials and Methods: A qualitative design was used involving 21 face-to-face interviews. Interviewing involved asking questions and getting answers from exclusive managers of cancer screening. Of the participants 6 were female and 15 were male, and ages ranged from 32 to 78 years. The study was based on a grounded theory approach and the tool was a semi-structured interview. Results: Researchers studied 5 dimensions, comprising electronic guideline standards of colorectal cancer screening, work flow of clinical and genetic activities, pathways of colorectal cancer screening and functionality of computer based guidelines and barriers. Electronic guideline standards of colorectal cancer screening were described in the s3 categories of content standard, telecommunications and technical standards and nomenclature and classification standards. According to the participations' views, workflow and genetic pathways of colorectal cancer screening were identified. Conclusions: The study demonstrated an effective role of computer-guided consultation for screening management. Electronic based systems facilitate real-time decision making during a clinical interaction. Electronic pathways have been applied for clinical and genetic decision support, workflow management, update recommendation and resource estimates. A suitable technical and clinical infrastructure is an integral part of clinical practice guidline of screening. As a conclusion, it is recommended to consider the necessity of architecture assessment and also integration standards.
Methylation Status and Expression of BRCA2 in Epithelial Ovarian Cancers in Indonesia
Pradjatmo, Heru 8599
Ovarian cancer is the main cause of mortality in gynecological malignancy and extensive studies have been conducted to study the underlying molecular mechanisms. The BRCA2 gene is known to be an important tumor suppressor in ovarian cancer, thereby BRCA2 alterations may lead to cancer progression. However, the BRCA2 gene is rarely mutated, and loss of function is suspected to be mediated by epigenetic regulation. In this study we investigated the methylation status and gene expression of BRCA2 in ovarian cancer patients. Ovarian cancer pateints (n=69) were recruited and monitored for 54 months in this prospective cohort study. Clinical specimens were used to study the in situ expression of aberrant BRCA2 proteins and the methylation status of BRCA2. These parameters were then compared with clinical parameters and overall survival rate. We found that BRCA2 methylation was found in the majority of cases (98.7%). However, the methylation status was not associated with protein level expression of BRCA2 (49.3%). Therefore in addition to DNA methylation, other epigenetic mechanisms may regulate BRCA2 expresison. Our findings may become evidence of BRCA2 inactivation mechanism through DNA methylation in the Indonesian population. More importantly, from multivariate analysis, BRCA2 expression was correlated with better overall survival (HR 0.32; p=0.05). High percentage of BRCA2 methylation and correlation of BRCA2 expression with overall survival in epithelial ovarian cancer cases may lead to development of treatment modalities specifically to target methylation of BRCA genes.
Assessment of Reliability when Using Diagnostic Binary Ratios of Polycyclic Aromatic Hydrocarbons in Ambient Air PM10
Pongpiachan, Siwatt 8605
The reliability of using diagnostic binary ratios of particulate carcinogenic polycyclic aromatic hydrocarbons (PAHs) as chemical tracers for source characterisation was assessed by collecting PM10 samples from various air quality observatory sites in Thailand. The major objectives of this research were to evaluate the effects of day and night on the alterations of six different PAH diagnostic binary ratios: An/(An + Phe), Fluo/(Fluo + Pyr), B[a]A/(B[a]A + Chry), B[a]P/(B[a]P + B[e]P), Ind/(Ind + B[g,h,i]P), and B[k]F/Ind, and to investigate the impacts of site-specific conditions on the alterations of PAH diagnostic binary ratios by applying the concept of the coefficient of divergence (COD). No significant differences between day and night were found for any of the diagnostic binary ratios of PAHs, which indicates that the photodecomposition process is of minor importance in terms of PAH reduction. Interestingly, comparatively high values of COD for An/(An + Phe) in PM10 collected from sites with heavy traffic and in residential zones underline the influence of heterogeneous reactions triggered by oxidising gaseous species from vehicular exhausts. Therefore, special attention must be paid when interpreting the data of these diagnostic binary ratios, particularly for cases of low-molecular-weight PAHs.
Clinical, Radiologic, and Endoscopic Manifestations of Small Bowel Malignancies: a First Report from Thailand
Tangkittikasem, Natthakan;Boonyaarunnate, Thiraphon;Aswakul, Pitulak;Kachintorn, Udom;Prachayakul, Varayu 8613
Background: The symptoms of small bowel malignancies are mild and frequently nonspecific, thus patients are often not diagnosed until the disease is at an advanced stage. Moreover, the lack of sufficient studies and available data on small bowel cancer makes diagnosis difficult, further delaying proper treatment for these patients. In fact, only a small number of published studies exist, and there are no studies specific to Thailand. Radiologic and endoscopic studies and findings may allow physicians to better understand the disease, leading to earlier diagnosis and improved patient outcomes. Objective: To retrospectively analyze the clinical, radiologic, and endoscopic characteristics of small bowel cancer patients in Thailand's Siriraj Hospital. Materials and Methods: This retrospective analysis included 185 adult patients (97 men, 88 women; mean age = $57.6{\pm}14.9$) with pathologically confirmed small bowel cancer diagnosed between January 2006 and December 2013. Clinical, radiologic, and endoscopic findings were collected and compared between each subtype of small bowel cancer. Results: Of the 185 patients analyzed, gastrointestinal stromal tumor (GIST) was the most common diagnosis (39.5%, n=73). Adenocarcinoma was the second most common (25.9%, n = 48), while lymphoma and all other types were identified in 24.3% (n = 45) and 10.3% (n = 19) of cases, respectively. The most common symptoms were weight loss (43.2%), abdominal pain (38.4%), and upper gastrointestinal bleeding (23.8%). Conclusions: Based on radiology and endoscopy, this study revealed upper gastrointestinal bleeding, an intra-abdominal mass, and a sub-epithelial mass as common symptoms of GIST. Obstruction and ulcerating/circumferential masses were findicative of adenocarcinoma, as revealed by radiology and endoscopy, respectively. Finally, no specific symptoms were related to lymphoma.
Assessing the Potential of Thermal Imaging in Recognition of Breast Cancer
Zadeh, Hossein Ghayoumi;Haddadnia, Javad;Ahmadinejad, Nasrin;Baghdadi, Mohammad Reza 8619
Background: Breast cancer is a common disorder in women, constituting one of the main causes of death all over the world. The purpose of this study was to determine the diagnostic value of the breast tissue diseases by the help of thermography. Materials and Methods: In this paper, we applied non-contact infrared camera, INFREC R500 for evaluating the capabilities of thermography. The study was conducted on 60 patients suspected of breast disease, who were referred to Imam Khomeini Imaging Center. Information obtained from the questionnaires and clinical examinations along with the obtained diagnostic results from ultrasound images, biopsies and thermography, were analyzed. The results indicated that the use of thermography as well as the asymmetry technique is useful in identifying hypoechoic as well as cystic masses. It should be noted that the patient should not suffer from breast discharge. Results: The accuracy of asymmetry technique identification is respectively 91/89% and 92/30%. Also the accuracy of the exact location of identification is on the 61/53% and 75%. The approach also proved effective in identifying heterogeneous lesions, fibroadenomas, and intraductal masses, but not ISO-echoes and calcified masses. Conclusions: According to the results of the investigation, thermography may be useful in the initial screening and supplementation of diagnostic procedures due to its safety (its non-radiation properties), low cost and the good recognition of breast tissue disease.
Could Tumor Size Be A Predictor for Papillary Thyroid Microcarcinoma: a Retrospective Cohort Study
Wang, Min;Wu, Wei-Dong;Chen, Gui-Ming;Chou, Sheng-Long;Dai, Xue-Ming;Xu, Jun-Ming;Peng, Zhi-Hai 8625
Background: Central lymph node metastasis(CLNM) is common in papillary thyroid microcarcinoma (PTMC). The aim of this study was to define the pathohistologic risk grading based on surgical outcomes. Materials and Methods: Statistical analysis was performed to figure out the optimal cut-off values of size in preoperative ultrasound images for defining the risk of CLNM in papillary thyroid microcarcinoma. Receiver operating characteristic curves (ROC) studies were carried out to determine the cutoff value(s) for the predictor(s). All the patients were divided into two groups according to the above size and the clinic-pathological and immunohistochemical parameters were compared to determine the significance of findings. Results: The optimal cut-off value of tumor size to predict the risk of CLNM in papillary thyroid microcarcinoma was 0.575 cm (area under the curve 0.721) according to the ROC curves. Significant differences were observed on the multifocality, extrathyroidal extension and central lymph node metastasis between two groups which were divided according to the tumor size by the cutoff values. Patients in two groups showed different positive rate and intensity of Ki67. Conclusions: The size of PTMC in ultrasound images are helpful to predict the aggressiveness of the tumors, it could be an easy predictor for PTMC prognosis and assist us to choose treatment.
Primary Idiopathic Myelofibrosis: Clinico-Epidemiological Profile and Risk Stratification in Pakistani Patients
Sultan, Sadia;Irfan, Syed Mohammed 8629
Background: Primary idiopathic myelofibrosis (PMF) is a clonal Philadelphia chromosome-negative myeloproliferative neoplasm characterized by extramedullary hematopoiesis and marrow fibrosis. It is an uncommon hematopoietic malignancy which primarily affects elderly individuals. The rational of this study was to determine its clinico-epidemiological profile along with risk stratification in Pakistani patients. Materials and Methods: In this retrospective cross sectional study, 20 patients with idiopathic myelofibrosis were enrolled from January 2011 to December 2014. Data were analyzed with SPSS version 22. Results: The mean age was $57.9{\pm}16.5years$ with 70% of patients aged above 50. The male to female ratio was 3:1. Overall only 10% of patients were asymptomatic and the remainder presented with constitutional symptoms. In symptomatic patients, major complaints were weakness (80%), weight loss (75%), abdominal discomfort (60%), night sweats (13%), pruritus (5%) and cardiovascular accidents (5%). Physical examination revealed splenomegaly as a predominant finding detected in 17 patients (85%) with the mean splenic span of $22.2{\pm}2.04cm$. The mean hemoglobin was $9.16{\pm}2.52g/dl$ with the mean MCV of $88.2{\pm}19.7fl$. The total leukocyte count of $17.6{\pm}19.2{\times}10^9/l$ and platelets count were $346.5{\pm}321.9{\times}10^9/l$. Serum lactate dehydrogenase, serum creatinine and uric acid were $731.0{\pm}154.1$, $0.82{\pm}0.22$ and $4.76{\pm}1.33$ respectively. According to risk stratification, 35% were in high risk, 40% in intermediate risk and 25% in low risk groups. Conclusions: The majority of PMF patients were male and presented with constitutional symptoms in our setting. Risk stratification revealed predominance of advanced disease in our series.
Expression and Clinical Significance of Sushi Domain-Containing Protein 3 (SUSD3) and Insulin-like Growth Factor-I Receptor (IGF-IR) in Breast Cancer
Zhao, Shuang;Chen, Shuang-Shuang;Gu, Yuan;Jiang, En-Ze;Yu, Zheng-Hong 8633
Background: To investigate the expression of insulin-like growth factor-I receptor (IGF-IR) and sushi domain containing protein 3 (SUSD3) in breast cancer tissue, and analyze their relationship with clinical parameters and the correlation between the two proteins. Materials and Methods: The expression of IGF-IR and SUSD3 in 100 cases of breast cancer tissues and adjacent normal breast tissues after surgery was detected by immunohistochemical technique MaxVisionTM, and the relationship with clinical pathological features was further analyzed. Results: The positive rate of IGF-IR protein was 86.0% in breast cancer, higher than 3.0% in adjacent normal breast tissue (P<0.05). The positive expression rate of SUSD3 protein was 78.0% in breast cancer, higher than 2.0% in adjacent normal breast tissue (P<0.05). The expression of IGF-IR and SUSD3 was related to estrogen receptor and pathological types (P<0.05),but not with age, stage, the expression of HER-2 and Ki-67 (P>0. 05). The expression of IGF-IR and SUSD3 in breast cancer tissue was positively related (r=0.553, P<0.01). Conclusions: The expression of IGF-IR and SUSD3 may be correlated to the occurrence and development of breast cancer. The combined detection of IGF-IR, SUSD3 and ER may play an important role in judging prognosis and guiding adjuvant therapy after surgery of breast cancer.
Intraperitoneal Perfusion Therapy of Endostar Combined with Platinum Chemotherapy for Malignant Serous Effusions: A Meta-analysis
Liang, Rong;Xie, Hai-Ying;Lin, Yan;Li, Qian;Yuan, Chun-Ling;Liu, Zhi-Hui;Li, Yong-Qiang 8637
Background: Malignant serous effusions (MSE) are one complication in patients with advanced cancer. Endostar is a new anti-tumor drug targeting vessels which exerts potent inhibition of neovascularization. This study aimed to systematically evaluate the efficacy and safety of intraperitoneal perfusion therapy of Endostar combined with platinum chemotherapy for malignant serous effusions (MSE). Materials and Methods: Randomized controlled trials (RCTs) on intraperitoneal perfusion therapy of Endostar combined with platinum chemotherapy for malignant serous effusions were searched in the electronic data of PubMed, EMBASE, Web of Science, CNKI, VIP, CBM and WanFang. The quality of RCTs was evaluated by two independent researchers and a meta-analysis was performed using RevMan 5.3 software. Results: The total of 25 RCTs included in the meta-analysis covered 1,253 patients, and all literature quality was evaluated as "B" grade. The meta-analysis showed that Endostar combined with platinum had an advantage over platinum alone in terms of response rate of effusions (76% vs 48%, RR=1.63, 95%CI: 1.50-1.78, P<0.00001) and improvement rate in quality of life (69% vs 44%, RR=1.57, 95%CI: 1.42-1.74, P<0.00001). As for safety, there was no significant difference between the two groups in the incidences of nausea and vomiting (35% vs 34%, RR=1.01, 95%CI: 0.87-1.18, P=0.88), leucopenia (38% vs 38%, RR=1, 95%CI: 0.87-1.15, P=0.99), and renal impairment (18% vs 20%, RR=0.86, 95%CI: 0.43-1.74, P=0.68). Conclusions: Endostar combined with platinum by intraperitoneal perfusion is effective for malignant serous effusions, and patient quality of life is significantly improved without the incidence of adverse reactions being obviously increased.
Turkish Adolescent Perceptions about the Effects of Water Pipe Smoking on their Health
Cakmak, Vahide;Cinar, Nursan 8645
Background: Consumption of tobacco in the form of a water pipe has recently increased, especially among young people. This study aimed to develop a scale which would be used in order to detect perceptions about the effects of water pipe smoking on health and to test its validity and reliability. Our scale named "a scale of perception about the effects of water pipe smoking on health" was developed in order to detect factors effecting the perception of adolescents about the effects of water pipe smoking on health. Materials and Methods: The sample consisted of 150 voluntary students in scale development and 750 voluntary students in the study group. Data were collected via a questionnaire prepared by researchers themselves and 5-pont Likert scale for "a scale of perception about the effects of water pipe smoking on health" which was prepared through the literature. Data evaluation was carried out on a computer with SPSS. Results: The findings of the study showed that "a scale of perception about the effects of water pipe smoking on health" was valid and reliable. Total score average of the adolescents participated in the study was $58.5{\pm}1.25$. The mean score of the ones who did not smoke water pipe ($60.1{\pm}11.7$) was higher than the mean score of the ones who smoked water pipe ($51.6{\pm}13.8$), the difference being statistically significant. Conclusions: It is established that "a scale of perception about the effects of water pipe smoking on health" was a reliable and valid measurement tool. It is also found out that individuals who smoked a water pipe had a lower level of perception of water pipe smoking effects on health than their counterparts who did not smoke a water pipe.
Heparanase mRNA and Protein Expression Correlates with Clinicopathologic Features of Gastric Cancer Patients: a Meta-analysis
Li, Hai-Long;Gu, Jing;Wu, Jian-Jun;Ma, Chun-Lin;Yang, Ya-Li;Wang, Hu-Ping;Wang, Jing;Wang, Yong;Chen, Che;Wu, Hong-Yan 8653
Background: Heparanase is believed to be involved in gastric carcinogenesis. However, the clinicopathologic features of gastric cancer with high heparanase expression remain unclear. Aim : The purpose of this study was to comprehensively and quantitatively summarize available evidence for the use of heparanase mRNA and protein expression to evaluate the clinicopathological associations in gastric cancer in Asian patients by meta-analysis. Materials and Methods: Relevant articles listed in MEDLINE, CNKI and the Cochrane Library databases up to MARCH 2015 were searched by use of several keywords in electronic databases. A meta-analysis was performed to clarify the impact of heparanase mRNA and protein on clinicopathological parameters in gastric cancer. Combined ORs with 95%CIs were calculated by Revman 5.0, and publication bias testing was performed by stata12.0. Results: A total of 27 studies which included 3,891 gastric cancer patients were combined in the final analysis. When stratifying the studies by the pathological variables of heparanase mRNA expression, the depth of invasion (633 patients) (OR=4.96; 95% CI=2.38-1.37; P<0.0001), lymph node metastasis (639 patients) (OR=6.22; 95%CI=2.70-14.34, P<0.0001), and lymph node metastasis (383 patients) (OR=6.85; 95% CI=2.04-23.04; P=0.002) were all significant. When stratifying the studies by the pathological variables of heparanase protein expression, this was the case for depth of invasion (1250 patients) (OR=2.76; 95% CI=1.52-5.03; P=0.0009), lymph node metastasis (1178 patients) (OR=4.79 ; 95% CI=3.37-6.80, P<0.00001), tumor size (727 patients) (OR=2.06 ; 95% CI=1.31-3.23; P=0.002) (OR=2.61; 95% CI=2.09-3.27; P=0.000), and TNM stage (1233 patients) (OR=6.85; 95% CI=2.04-23.04; P=0.002). Egger's tests suggested publication bias for depth of invasion, lymph node metastasis, lymph node metastasis and tumor size of heparanase mRNA and protein expression. Conclusions: This meta-analysis suggests that higher heparanase expression in gastric cancer is associated with clinicopathologic features of depth of invasion, lymph node metastasis and TNM stage at mRNA and protein levels, and of tumor size only at the protein level. Egger's tests suggested publication bias for these clinicopathologic features of heparanase mRNA and protein expression, and which may be caused by shortage of relevant studies. As a result, although abundant reports showed heparanase may be associated with clinicopathologic features in gastric cancer, this meta-analysis indicates that more strict studies were needed to evaluate its clinicopathologic significance.
Malignant Neoplasm Burden in Nepal - Data from the Seven Major Cancer Service Hospitals for 2012
Pun, Chin Bahadur;Pradhananga, Kishore K;Siwakoti, Bhola;Subedi, Krishna;Moore, Malcolm A 8659
In Nepal, while no population based cancer registry program exists to assess the incidence, prevalence, morbidity and mortality of cancer, at the national level a number of hospital based cancer registries are cooperating to provide relevant data. Seven major cancer diagnosis and treatment hospitals are involved, including the BP Koirala Memorial Cancer hospital, supported by WHO-Nepal since 2003. The present retrospective analysis of cancer patients of all age groups was conducted to assess the frequencies of different types of cancer presenting from January 1st to December 31st 2012. A total of 7,212 cancer cases were registered, the mean age of the patients being 51.9 years. The most prevalent age group in males was 60-64 yrs (13.6%), while in females it was 50-54 yrs (12.8%). The commonest forms of cancer in males were bronchus and lung (17.6%) followed by stomach (7.3%), larynx (5.2%) and non Hodgkins lymphoma (4.5%). In females, cervix uteri (19.1%) and breast (16.3%), were the top ranking cancer sites followed by bronchus and lung (10.2%), ovary (6.1%) and stomach (3.8%). The present data provide an update of the cancer burden in Nepal and highlight the relatively young age of breast and cervical cancer patients.
Efficacy of Prophylactic Entecavir for Hepatitis B Virus-Related Hepatocellular Carcinoma Receiving Transcatheter Arterial Chemoembolization
Li, Xing;Zhong, Xiang;Chen, Zhan-Hong;Wang, Tian-Tian;Ma, Xiao-Kun;Xing, Yan-Fang;Wu, Dong-Hao;Dong, Min;Chen, Jie;Ruan, Dan-Yun;Lin, Ze-Xiao;Wen, Jing-Yun;Wei, Li;Wu, Xiang-Yuan;Lin, Qu 8665
Background and Aims: Hepatitis B virus (HBV) reactivation was reported to be induced by transcatheter arterial chemoembolization (TACE) in HBV-related hepatocellular carcinonma (HCC) patients with a high incidence. The effective strategy to reduce hepatitis flares due to HBV reactivation in this specific group of patients was limited to lamivudine. This retrospective study was aimed to investigate the efficacy of prophylactic entecavir in HCC patients receiving TACE. Methods: A consecutive series of 191 HBV-related HCC patients receiving TACE were analyzed including 44 patients received prophylactic entecavir. Virologic events, defined as an increase in serum HBV DNA level to more than 1 log10 copies/ml higher than nadir the level, and hepatitis flares due to HBV reactivation were the main endpoints. Results: Patients with or without prophylactic were similar in host factors and the majorities of characteristics regarding to tumor factors, HBV status, liver function and LMR. Notably, cycles of TACE were parallel between the groups. Ten (22.7%) patients receiving prophylactic entecavir reached virologic response. The patients receiving prophylactic entecavir presented significantly reduced virologic events (6.8% vs 54.4%, p=0.000) and hepatitis flares due to HBV reactivation (0.0% vs 11.6%, p=0.039) compared with patients without prophylaxis. Kaplan-Meier analysis illustrated that the patients in the entecavir group presented significantly improved virologic events free survival (p=0.000) and hepatitis flare free survival (p=0.017). Female and Eastern Cooperative Oncology Group (ECOG) performance status 2 was the only significant predictors for virological events in patients without prophylactic antiviral. Rescue antiviral therapy did not reduce the incidence of hepatitis flares due to HBV reactivation. Conclusion: Prophylactic entecavir presented promising efficacy in HBV-related cancer patients receiving TACE. Lower performance status and female gender might be the predictors for HBV reactivation in these patients.
Deactivation of Telomerase Enzyme and Telomere Destabilization by Natural Products: a Potential Target for Cancer Green Therapy
Sasidharan, Sreenivasan;Jothy, Subramanion L;Kavitha, Nowroji;Chen, Yeng;Kanwar, Jagat R 8671
Clues to Identifying Risk Factors for Nasopharyngeal Carcinoma
Wang, Chuqiong;He, Jiman 8673 | CommonCrawl |
Find $\left \lceil \frac{\left \lfloor 52/3 \right. \rfloor}{5/23} \right \rceil$.
Since $17 = \frac{51}{3} < \frac {52}3 < \frac {54}3 = 18$, the floor of $52/3$ is $17$. The given quantity is thus equal to $$\left \lceil \frac{17}{5/23} \right \rceil = \left \lceil \frac{391}{5} \right \rceil = \left \lceil 78.2 \right. \rceil = \boxed{79}.$$ | Math Dataset |
Ultrabroad Microwave Absorption Ability and Infrared Stealth Property of Nano-Micro CuS@rGO Lightweight Aerogels
Special issue on EM Wave Functional Materials
Yue Wu1,
Yue Zhao1,
Ming Zhou1,
Shujuan Tan1,
Reza Peymanfar2,
Bagher Aslibeiki3 &
Guangbin Ji1
Nano-Micro Letters volume 14, Article number: 171 (2022) Cite this article
The CuS@rGO composite aerogel can achieve the broad effective absorption bandwidth (EAB) of 8.44 GHz with the filler content of 6 wt%.
The RLmin of CuS@rGO composite aerogel is -55.1 dB and EAB is 7.2 GHz with the filler content of 2 wt% by ascorbic acid thermal reduction. The radar cross-section reduction value of CuS@rGO composite aerogel can reach 53.3 dB m2.
The CuS@rGO composite aerogels possess lightweight, compression and recovery, radar-infrared compatible stealth properties.
Developing ultrabroad radar-infrared compatible stealth materials has turned into a research hotspot, which is still a problem to be solved. Herein, the copper sulfide wrapped by reduced graphene oxide to obtain three-dimensional (3D) porous network composite aerogels (CuS@rGO) were synthesized via thermal reduction ways (hydrothermal, ascorbic acid reduction) and freeze-drying strategy. It was discovered that the phase components (rGO and CuS phases) and micro/nano structure (microporous and nanosheet) were well-modified by modulating the additive amounts of CuS and changing the reduction ways, which resulted in the variation of the pore structure, defects, complex permittivity, microwave absorption, radar cross section (RCS) reduction value and infrared (IR) emissivity. Notably, the obtained CuS@rGO aerogels with a single dielectric loss type can achieve an ultrabroad bandwidth of 8.44 GHz at 2.8 mm with the low filler content of 6 wt% by a hydrothermal method. Besides, the composite aerogel via the ascorbic acid reduction realizes the minimum reflection loss (RLmin) of − 60.3 dB with the lower filler content of 2 wt%. The RCS reduction value can reach 53.3 dB m2, which effectively reduces the probability of the target being detected by the radar detector. Furthermore, the laminated porous architecture and multicomponent endowed composite aerogels with thermal insulation and IR stealth versatility. Thus, this work offers a facile method to design and develop porous rGO-based composite aerogel absorbers with radar-IR compatible stealth.
With the fast development of detection technology, stealth materials have attracted extensive attention [1,2,3]. However, single-waveband stealth materials are hard to satisfy the requirement of harsh environments, and multispectral compatible stealth is becoming the future direction of stealth materials [4,5,6]. Particularly, with the occurrence of advanced precision-guided weapons and infrared (IR) detectors, designing and exploring the radar-IR compatible stealth materials is of great significance with low IR emissivity and excellent microwave absorbing (MA) ability. Usually, microwave absorbers need low reflectivity and high absorptivity [7,8,9], while IR stealth materials require high reflectivity and low IR absorptivity [10]. Furthermore, outstanding thermal insulation ability is also required for IR stealth materials according to the Stefan-Boltzmann theory [11]. Thus, it seems to be challenging to integrate IR and radar stealth owing to the thoroughly opposite principles.
To achieve radar-IR compatible stealth, it is of significance to overcome the issue of conflict between IR and radar camouflage material requirements. CuS, a kind of semiconductor transition metal sulfide, has caused broad concern in the IR stealth field owing to the absorbance behavior of local surface plasmon resonance in the near-IR region [12]. At the same time, CuS has also been applied as microwave absorbers due to its exceptional electrical property and unique geometrical micromorphology. For instance, Cui et al. prepared a sandwich-like CuS/Ti3C2Tx MXene composites and got the RLmin value of − 45.3 dB and the effective absorption bandwidth (EAB) of 5.2 GHz with the filler content of 35 wt% [13]. Quaternary composite of CuS/RGO/PANI/Fe3O4 was fabricated and the influence of special microstructure on MA capacity was further studied by Wang's group [14]. The RLmin of the products was − 60.2 dB and absorption bandwidth below − 10 dB was up to 7.4 GHz. Liu and his team designed CuS nanoflakes aligned on magnetically decorated graphene via a solvothermal method [15], and found that the different morphologies of nanocomposites showed excellent MA capacity, that was the EAB of 4.5 GHz and RLmin value of -54.5 dB. Guan et al. synthesized a series of CuS/ZnS nanocomposites with a 3D hierarchical structure by a hydrothermal method [16]. The obtained nanocomposite possessed the RLmin value of − 22.6 dB at 9.7 GHz with the thickness of 3 mm and the EAB of 2.2 GHz (9.2–11.4 GHz). Therefore, CuS-based composites show the application prospects in the field of microwave absorption.
Integrating CuS into thermal-insulating materials is provided a new perspective to design the IR-radar compatible stealth materials. Carbon materials such as carbon nanotubes and graphene have been applied as building blocks to create lightweight and multifunctional microwave absorbers due to their lightweight, conspicuous chemical and mechanical properties, high stability, etc. [17, 18]. Numerous researchers have combined graphene with metallic compounds (ZnO, CeO2, MoS2, etc.) and magnetic nanoparticles (Ni, Fe, Co, or its alloys) or magnetic compounds (typical ferrites) to fabricate composite powder absorbers that can achieve the integration of dielectric/magnetic loss, and optimize the impedance mismatch owing to the poor impedance matching form single graphene [19, 20]. Although they have achieved excellent MA ability, these composites are hard to meet the other functions for unique applicated environments. Besides, common powder materials also have high filler contents and density. In recent years, aerogels with high porosity (> 95%) and extremely low density (< 0.1 g cm−3) have been attractive to researchers [21]. Among them, graphene-based aerogels consisting of interconnected 3D networks of graphene sheets are gained wide attention for their low cost and density, facile synthesis, unique porous structure, and large specific surface area. Moreover, the porous graphene-based aerogels possess the superior thermal-insulating effect for the existence of high porous, air phase, and 3D network structure. The studies on graphene/Ni aerogel [22], CoFe2O4/N-doped reduced graphene oxide aerogel [23], polyaniline/graphene aerogel [24], and SiC whiskers/reduced graphene oxide aerogel [25] have further confirmed that the composition regulation of graphene-based composite aerogels is conducive to achieving effective absorption bandwidth (EAB) and reducing the filler contents.
Currently, foams and aerogels with porous network structure, high porosity, high specific surface area, such as melamine hybrid foam [26], chitosan-derived carbon aerogels [27], porous carbon@CuS [11], antimony tin oxide/rGO aerogels [28], cobalt ferrite/carbon nanotubes/waterborne polyurethane hybrid aerogels [29], Fe/Fe2O3@porous carbon composites [30], cellulose-chitosan framework/polyaniline hybrid aerogel [31], rGO/MWCNT-melamine composite [32], organic-inorganic hybrid aerogel [33], and rGO/Fe3O4 [34], are commonly applied as radar-IR stealth materials. Although the reported carbon-based radar-IR compatible stealth materials can achieve MA performance and thermal/IR stealth, it is difficult to gain a wide EAB (> 8 GHz) and low IR emissivity (< 6.5) with a low filler content (< 5 wt%).
In this work, two kinds of 3D porous CuS@rGO composite aerogels were synthesized by hydrothermal and ascorbic acid thermal reduction methods and subsequent freeze-drying technique. Thanks to the bicomponent synergistic effect and their unique porous architecture, the obtained composite aerogels achieved MA performance and IR stealth ability. By modulating the additive amounts of CuS powders and thermal reduction ways, the porous CuS@rGO aerogels manifested adjustable MA capacity and IR emissivity. Notably, an excellent MA performance of CuS@rGO (30 mg) aerogel with the widest EAB of 8.44 GHz and RLmin of − 40.2 dB at an extremely low filler content of merely 6 wt% could be achieved. Besides, the low IR emissivity of 0.6442 was also obtained by adjusting the additive amounts of CuS. Furthermore, the MA and IR stealth mechanisms of CuS@rGO composite aerogels were investigated in detail. This work exploits a novel path in the design and development of radar-IR compatible stealth materials that can work in the today's complex environment.
2 Experimental Section
Copper chloride dihydrate (CuCl2·2H2O), ethylene glycol (EG), thiourea (CH4N2S), ascorbic acid and anhydrous ethanol (C2H5OH) were all bought from the Nanjing Chemical Reagent Co., Ltd. Graphite oxide was provided by Suzhou TANFENG Graphene Tech Co., Ltd. (Suzhou, China). All of the chemical reagents were analytically pure and employed without further purification.
2.2 Preparation of CuS Microspheres
The CuS microspheres were prepared via an ordinary solvothermal strategy. CuCl2·2H2O (6 mmol) was dissolved in 30 mL of EG, which was named solution A that was quickly turned from blue to dark green. CH4N2S (24 mmol) was dispersed in another 30 mL of EG that was marked as solution B at the same time. Then, solution B was poured into solution A, and continuously stirred for 0.5 h until the solution became transparent. Next, the final solution was transformed into a Teflon-lined autoclave (100 mL) and maintained at 170 °C for 5 h. The products were collected by centrifugation with distilled water and anhydrous ethanol several times. Finally, the products were dried at 60 °C in a vacuum oven.
2.3 Preparation via the Hydrothermal Method
The 3D porous CuS@rGO composite aerogels were synthesized via a hydrothermal method. First, a certain amount of CuS powders (0, 15, 30, 60, and 120 mg) and 120 mg of multilayer graphite oxide were dispersed into distilled water (30 mL) under ultrasonication for 1 h and subsequently stirred for 0.5 h. Then, the dispersions were placed into a Teflon-lined autoclave (50 mL) and lasted at 120 °C for 12 h. Finally, the obtained CuS@rGO composite hydrogels were dialyzed in anhydrous ethanol/distilled water solution with a volume ratio of 1:9 for 48 h and then freeze-drying at − 50 °C for 48 h to obtain CuS@rGO composite aerogels. The composite aerogels were marked as rC-1, rC-2, rC-3, rC-4, and rC-5.
2.4 Preparation via the Ascorbic Acid Reduction Method
The 3D porous CuS@rGO composite aerogels were synthesized via the ascorbic acid reduction method. First, a certain amount of CuS powders (0, 10, 20, 30, and 40 mg), 80 mg of multilayer graphite oxide and 1.2 g ascorbic acid were dispersed into distilled water (20 mL) under the ultrasonication treatment for 1 h and stirred for 0.5 h. Then, the dispersions were poured into a custom silicone mold (25 mL) at 95 °C for 12 h. Finally, the obtained CuS@rGO composite hydrogels were dialyzed in anhydrous ethanol/distilled water solution with a volume ratio of 1:9 for 48 h and then freeze-drying at − 50 °C for 48 h to obtain CuS@rGO composite aerogels. The composite aerogels were labeled as RC-1, RC-2, RC-3, RC-4, and RC-5.
2.5 Characterization
The composition and crystal structure of CuS@rGO aerogels were investigated by X-ray diffraction (XRD, Bruker D8 ADVANCE, equipped with Cu-Kα radiation). X-ray photoelectron spectroscopy (XPS) was carried out on a Kratos AXIS Ultra spectrometer with the Al Kα X-rays as the excitation source. The micromorphology was characterized by a Hitachi S4800 field emission scanning electron microscope (SEM) and a Talos F200X transmission electron microscopy (TEM) equipped with energy dispersive spectrum (EDS).
2.6 Microwave Absorption Measurements
The EM parameters of complex permeability (\(\mu_{r} = \mu^{\prime}{-} \, j\mu^{\prime\prime}\)) and complex permittivity (\(\varepsilon_{r} = \varepsilon^{\prime}{-} \, j\varepsilon^{\prime\prime}\)) were measured by the vector network analyzer (VNA, Agilent PNA N5244A) adopting the coaxial line method. The rC aerogels (6 wt%) were mixed with 94 wt% paraffin, and RC aerogels (1 and 2 wt%) respectively mixed with 99 and 98 wt% paraffin, and then pressed into a toroidal ring of the inner diameter of 3.04 mm and out diameter of 7.00 mm.
2.7 Computer Simulation Technology
Computer simulation technology (CST) studio Suite 2018 was applied to simulate the RCS values of as-prepared CuS@rGO composite aerogels under open boundary conditions. The simulation model consisted of the perfect electric conductor (PEC) layer with a thickness of 1.0 mm at the bottom and an absorbing layer with a thickness of 2.0 mm on the top. The dimension of length was equal to the width of 200 mm. Then, the created model was placed on the xOy plane, and the linear polarized plane EMW was added with the incidence direction on Z-axis positive to negative, and the electric polarization was along the X-axis. In addition, the far-field monitor frequency was set as 15.7 GHz. The RCS values could be computed as follows [35]:
$$ \sigma = 10{\text{log}}\left( {\frac{4\pi S}{{\lambda^{2} }}\left| {\frac{{E_{{\text{S}}} }}{{E_{{\text{i}}} }}} \right|} \right)^{2} $$
where λ and S are the wavelength of incident wave and area of the simulation model, Ei and Es are the intensity of electric field of the incident and scatted EMWs, respectively.
2.8 IR Stealth Measurement
The IR-2 dual-band IR emissivity meter was used to test the IR emissivity in the waveband of 3 ~ 5 and 8 ~ 14 μm. Thermal IR imaging digital images were recorded by TVS-2000 MK with a heating platform, and the temperature was set as 120 °C.
3 Results and Discussion
3.1 Preparation and Reduction Mechanism
The synthetic processes of CuS@rGO composite aerogels are depicted in Fig. 1. The first step is to fabricate CuS flower-like microspheres via a solvothermal method in Fig. 1a. Then, the 3D porous CuS@rGO composite aerogels were fabricated through complexing CuS in graphene/deionized water dispersion and combining with freeze-drying technique. Hydrothermal (Fig. 1b) and ascorbic acid reduction (Fig. 1c) methods were employed for the preparation of CuS@rGO composite hydrogels, and the freeze-drying technique was applied to obtain the corresponding aerogels with 3D porous architecture. The reduction processes of hydrothermal method can be illustrated in Fig. S1a–b [36]. The carboxyl functional groups can be reduced through a hydrothermal method. As depicted in Fig. S1a, the decarboxylation reaction is accompanied by the production of carbon dioxide. The deoxidation processes of epoxide groups to form a carbon-carbon double bond can be divided into two steps (Fig. S1b). The first step is that the ring of epoxide groups is opened in the existence of formic acid by the acid-catalyzed reaction to produce alcohol in the decarboxylation reaction. The nucleophilic reagent or strong bases can attack the ternary ring of epoxide groups and then relieve the strain energy. Under the circumstances, the hydride ions of formic acid work as nucleophiles at the hydrothermal reaction temperature. First, the epoxide groups are protonated, which activates them to attack the nucleophile. Then, the carbocation is formed that is attacked by hydride ions from formic acid, and the ring is opened to generate alcohol. The second step refers to the dehydration reaction of alcohol to carbon-carbon double bonds with the help of an acidic medium. The -OH (weak leaving groups) needs the protonation reaction to transform it to H2O which is easy to leave. A carbocation is formed by water loss, and the water then absorbs the protons to generate carbon-carbon double bonds in rGO. The reduction mechanisms for rGO under the action of ascorbic acid are depicted in Fig. S1c [37]. The carboxyl, epoxy, carbonyl and hydroxyl groups are existed on the surfaces or at the edge of the graphene oxide (GO) sheet. The ascorbic acid can liberate two protons to obtain dehydroascorbic acid, while the protons usually possess a strong affinity with the oxygen-containing groups that can react to form water molecules during the reduction of GO to rGO. At the same time, a number of the neighboring carbon atoms will be taken away as the oxygen-containing functional groups are removed, which can cause vacancy defects in the rGO. Due to the difference in reduction strategies, it can be inferred that the structure of CuS@rGO composite aerogels is also different. Thus, we have further measured the physical parameters of rC composite aerogels. It can be found that the as-prepared aerogels have a few differences in size, including the length, radius and even the mass weight (Table S1). The density of rC composite aerogels is approximate 0.01 g cm−3, and is increased with the additive amounts of CuS. The results are that the pure rGO aerogel possesses the lowest density of 0.0110 g cm−3, while the rC-5 has the largest density of 0.0160 g cm−3.
Schematic diagram of preparation processes of a flower-like CuS microspheres, and b, c CuS@rGO composite aerogels through a hydrothermal method (b), and via the ascorbic acid thermal reduction (c)
To confirm the characteristic of lightweight, it is observed that the CuS@rGO composite aerogel can stand on the petals without damaging them at all, demonstrating excellent lightweight feature (Fig. 2a). Besides, the aerogel is observed to express good thermal insulation when placed over the flame of the alcohol lamp. When the aerogel is further compressed with tweezers, it can be well compressed. While the tweezers are released, it can return to its original shape in Fig. 2b, indicating its good compression and recovery characteristic.
CuS@rGO aerogel characteristics of a light weight, b compression and recovery. c XRD patterns of CuS, rGO, rC-4, RGO and RC-4. d XPS full spectrum of rC-4. e–f TEM images, and g–k EDS mapping images
The crystalline structure of the prepared CuS@rGO composite aerogels is characterized through XRD analysis. In Fig. 2c, the diffraction peaks at 54.8°, 46.4°, 32.3°, 29.4°, and 27.8° are ascribed to the (108), (110), (103), (102), and (101) crystal planes of CuS (JCPDS No.06–0464) [13]. The rC-1 and RC-1 samples show a broad peak corresponding to the (002) plane of rGO. Besides, the peak intensity becomes weaker with the addition of CuS, and the peak intensity of rGO is too strong, resulting in the relatively weak intensity of CuS. The chemical valence state and surface composition of rC-3 aerogel were measured through XPS. The full spectrum depicted in Fig. 2d confirms the occurrence of S, O, C, and Cu elements that is consistence with the composition of aerogel. From Fig. S2a, the C 1s spectrum shows three peaks at 288.9, 285.5, and 284.6 eV, which are assigned to the O–C = O, C–OH, and C–C/C = C bonds, severally [38]. Figure S2b is the Cu 2p high-resolution spectrum with two typical peaks at 932.0 and 952.3 eV, corresponding to the Cu 2p3/2 and Cu 2p1/2 orbitals of S–Cu bonds [13]. From Fig. S2c, the S 2p spectrum can be divided into three peaks, i.e., S–C (168.3 eV), S 2p1/2 (163.6 eV), and S 2p3/2 (162.0 eV) [38]. For the O 1s spectrum illustrated in Fig. S2d, the obvious peaks at 532.8 and 531.9 eV are indexed to the –OH and lattice oxygen, respectively [38]. The above XPS results further verify the high purity of CuS@rGO composite aerogel.
The morphology and microstructure of CuS and CuS@rGO are observed by SEM. Figure S2e shows a hierarchical flower-like structure of CuS with an around diameter of 5 μm. From Fig. S2f–j, the rC composite aerogels present a typical 3D porous structure composed of overlapping neighboring rCO sheets. Furthermore, the surface of the rGO sheet occurs some holes marked as white boxes. The CuS was wrapped by the rGO sheet when the additive amounts of CuS powders reached 15 mg. In addition, the surface of rGO becomes rougher compared with rC-1 (pure rGO aerogel), which may be the formation of interfaces between CuS and rGO that is conducive to attenuating the incident EMWs. From Fig. S2p, it is more evident that the CuS microspheres are wrapped by rGO sheet from RC-5 (marked by a red dotted box). Interestingly, the rC-3 possesses a larger porous structure than that of other aerogels. The geometrical structure of CuS and rGO of rC-4 was further investigated by the TEM. As depicted in Fig. 2e–f, the rGO and CuS can be easily distinguished from TEM images. The flower-like CuS structure was assembled by 2D nanoflakes, and there are many voids between the interwoven CuS nanosheets. Besides, the rGO exhibits sparse lamellar structure duo to the almost transparent nature of rGO in the CuS@rGO composite aerogel. From Fig. 2g–k, the EDS mapping images of rC-4 demonstrate that the Cu and S elements are chiefly distributed on the CuS microsphere. In addition, C and O elements are distributed throughout the region, indicating the structure of CuS wrapped by rGO sheets. All of these results can well distinguish and see rGO from CuS.
3.2 Microwave Absorption Performance
EM parameters of CuS@rGO composite aerogels synthesized by two different reduction strategies are investigated to deduce the effects of the defects and porous structure on MA performance. The EM parameters and reflection loss of CuS@rGO composite aerogels by hydrothermal reduction and ascorbic acid reduction two methods are calculated as follows [39, 40]:
$$ {\text{RL}} = 20{\text{lg}}\left| {\frac{{Z_{{{\text{in}}}} - Z_{0} }}{{Z_{{{\text{in}}}} + Z_{0} }}} \right| $$
$$ Z_{{{\text{in}}}} = Z_{0} \sqrt {\frac{{\mu_{{\text{r}}} }}{{\varepsilon_{{\text{r}}} }}} {\text{tanh}}\left( {j\frac{{2\pi fd\sqrt {\mu_{{\text{r}}} \varepsilon_{{\text{r}}} } }}{c}} \right) $$
Herein the physical parameters of Zin, Z0, c, f, d, μr and εr represent the input impedance, free space impedance, speed of light, frequency, matching thickness, relative complex permeability and relative complex permittivity, respectively. As depicted in Fig. S3b1–b5, the RLmin values of rC aerogels show a trend of increasing first and then declining, that is the RLmin values of − 12.3 (2.0 mm), − 16.1 (2.0 mm), − 40.2 (2.3 mm), − 50.4 (2.0 mm), and − 38.4 (3.0 mm) dB, respectively. It is worth noting that the complexing with CuS microspheres is beneficial to improving MA capacity. As depicted in Fig. 3a1–a2, the rC-3 can achieve the RLmin of − 40.2 dB and a narrow EAB of 4.7 GHz at 2.0 mm. Furthermore, the broadest EAB is up to 8.44 GHz at 2.8 mm. When the additive content of CuS is 60 mg, the rC-4 obtains the EAB of 7.16 GHz at 2.3 mm and the RLmin of − 50.4 dB at 2.0 mm in Fig. 3b1–b2. Interestingly, the RLmin values show a shift to low frequency as the thicknesses increase.
RL curves: a1 rC-3, b1 rC-4, c1 RC-4 (2 wt%) and d1 RC-4 (1 wt%). 2D RL contour maps of a2 rC-3, b2 rC-4, c2 RC-4 (2 wt%) and d2 RC-4 (1 wt%). e RLmin and f EAB at different thickness of rC-4 and RC-4. g Selected RL-f curves at various frequency wavebands. h Comparison of MA performance considering the EAB and filler contents with reported rGO-based composite aerogels
The EM parameters include the \(\varepsilon^{\prime}\), \(\varepsilon^{\prime\prime}\), \(\mu^{\prime}\) and \(\mu^{\prime\prime}\). The \(\mu^{\prime}\) and \(\varepsilon^{\prime}\) denote the storage ability of magnetic and electric energy, while \(\mu^{\prime\prime}\) and \(\varepsilon^{\prime\prime}\) denote the dissipation capacity of magnetic and electric energy, respectively [41]. Owning to the rGO and CuS@rGO without magnetic components (\(\mu^{\prime\prime} = 0\) and \(\mu^{\prime} = 1\)), we merely pay attention on the εr and dielectric loss tangent (tanδe). From Fig. S4, the dielectric constants (\(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\)) descend as the frequency goes up, indicating an obvious frequency dispersion effect that is conducive to attenuating incident EMWs. In addition, with the increase in additive amounts of CuS, the \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) generally present a decreasing trend. The tanδe of rC aerogels with the order of rC-1 > rC-2 > rC-3 > rC-4 > rC-5 is depicted in Fig. S4c.
Besides, the effects of additive contents of CuS on EM parameters and MA performance of RC composite aerogels via the ascorbic acid reduction strategy with the lower filler content of 2 wt% are also investigated in Fig. S5–S6, Tables S2 and S3. The RLmin values are − 32.0 (4.0 mm), − 16.8 (2.5 mm), − 21.5 (2.5 mm), − 60.3 (3.5 mm), and − 16.2 (2.5 mm) dB, respectively. In general, the RC-4 possesses the optimal MA behavior considering the low thickness, strong absorption, and broad bandwidth, i.e., the RLmin of -55.1 dB and the EAB of 7.2 GHz can be achieved under 2.45 mm. Furthermore, a lower RLmin value is − 60.3 dB at 3.5 mm as shown in Fig. 3c1–c2.
From Fig. S6a, the RC-1 has the largest \(\varepsilon^{\prime}\) values than that of other RC aerogels, and the range of ε′ values for other aerogels is small. The \(\varepsilon^{\prime\prime}\) curves of RC aerogels show a familiar downward trend with multiple polarization peaks in 6–18 GHz (Fig. S6b), manifesting the existence of conduction loss and polarization loss. Figure S6c displays the frequency-dependent curves of tanδe, which implies that the RC-4 has relatively stronger dielectric loss capacity and the RC aerogels occur polarization peak in high frequency of 11–15 GHz.
Furthermore, the RC composite aerogels with the lower filler content of 1 wt% are studied in Fig. S7. It is seen that the RC composite aerogels show an enhanced MA capacity than pure rGO aerogel (RC-1). Figure S7f more intuitively observed that the absolute values of RLmin (|RLmin|) enhance first and then decline, and RC-4 has the biggest |RLmin| of 63.5 dB. It is interesting that by changing filler content, the final result of RC-4 has the optimal reflection loss.
The Cole-Cole curves of CuS@rGO aerogel were investigated to further elucidate the polarization relaxation processes. Based on the Debye theory, the \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) are described as follows:
$$ \varepsilon^{\prime} = \varepsilon_{\infty } + \frac{{\varepsilon_{s} - \varepsilon_{\infty } }}{{1 + \left( {2\pi f} \right)^{2} \tau^{2} }} $$
$$ \varepsilon^{\prime\prime} = \frac{{2\pi f\tau \left( {\varepsilon_{s} - \varepsilon_{\infty } } \right)}}{{1 + \left( {2\pi f} \right)^{2} \tau^{2} }} $$
Based on the above equations, the correlation between \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) could be calculated [42, 43]:
$$ \left( {\varepsilon^{\prime} - \frac{{\varepsilon_{s} - \varepsilon_{\infty } }}{2}} \right) + \left( {\varepsilon ^{\prime\prime}} \right)^{2} = \left( {\frac{{\varepsilon_{s} + \varepsilon_{\infty } }}{2}} \right)^{2} $$
Herein ε∞, εs, and τ are relative complex permittivity at infinite frequency limit, static permittivity, and relaxation time, respectively. Therefore, the curve of \(\varepsilon^{\prime\prime}\) vs \(\varepsilon^{\prime}\) should be a semicircle, called the Cole-Cole semicircle. Generally, each semicircle is on behalf of one Debye relaxation process. From Fig. S4d–h, the curves of all rC aerogels are made up of distorted semicircles and straight tails. The distorted semicircle may be ascribed to polarization relaxation like dipole polarization and interfacial polarization, while the straight line in tail is relevant to conduction loss. It can be discovered that all rC aerogels have at least two semicircles. From Fig. S6d–h, all RC aerogels also have at least two semicircles, indicating the polarization relaxation loss. Compared with rC aerogels, the conduction loss of RC aerogels is much lower from the tail straight. The polarization loss of CuS@rGO aerogels primarily comes from the following aspects. On the one hand, complexing CuS with rGO can be considered as a "capacitor-like" structure that leads to the inhomogeneous distribution and accumulation of free electrons at the heterogeneous interface, enhancing the interfacial polarization to attenuate incident EMWs. On the other hand, CuS, a p-type semiconductor, has ample Cu vacancies, which can result in the unbalance of charges located at the defect sites and then induces dipole polarization. In addition, the –COOH, –OH, etc. on the surface or edge of rGO can also cause dipole polarization.
To compare the effect of reduction way on MA performance, the RL and EAB of RC-4 (1 wt%), RC-4 (2 wt%), rC-4 and rC-3 are drawn in Fig. 3e–g. Figure 3e depicts the RLmin values of rC-4, RC-4 (2 wt%) at 1.0–4.0 mm. The RC-4 (2 wt%) possesses overall lower RLmin values than rC-4. In addition to RLmin, EAB also should be taken into consideration. From Fig. 3f, RC-4 (1 wt%) has the smallest EAB at 2.4–3.0 mm, and rC-3 reaches the highest EAB at 2.6–3.0 mm. As presented in Fig. 3g, the RL curves of the selected thickness for rC-4 and RC-4 (2 wt%) can occur in different frequency wavebands (C band, X band, and Ku band). The performance comparison about EAB and filler content of this work to other reported rGO-based aerogels has been given in Fig. 3h [23, 44,45,46,47,48,49,50]. Most of reported works had higher filler contents or smaller EAB. However, this work can realize the wider EAB and the lower filler content simultaneously.
According to the structure of rC composite aerogels (rC-3 and rC-4) and RC-4, the EM parameters and dielectric loss have been further explored in detail. As depicted in Fig. 4a, d, g, the rC-4 has the largest average dielectric constant (\(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\)), implying the stronger dielectric loss behavior. Due to the difference in additive amounts of CuS and reduction methods, the CuS@rGO composite aerogels display the various structures in Fig. 4b, e, h. Compared with rC-3, rC-4 has a higher content of CuS, which is beneficial to forming the more interfacial polarization. As for rC-4 and RC-4, rGO in rC-4 is reduced at 120 °C, while the RC-4 at 95 °C. Therefore, it is deduced that more defects could be formed in rC-4 than RC-4. Besides, the pore diameter of rC-4 is much larger than RC-4 according to the SEM results, which is more help to attenuate the EMWs. From Cole-Cole curves in Fig. 4c, f, i, the upward tails of rC composite aerogels become longer, suggesting the enhanced conduction loss. So, the structure difference of CuS@rGO composite aerogels with two various reduction methods is presented in Fig. 4j–k. The hydrothermal strategy with the higher temperature can generate more defects and form larger pores than that of the ascorbic acid reduction method.
\(\varepsilon^{\prime}\), \(\varepsilon^{\prime\prime}\), tanδe ~ f curves: a rC-3, d rC-4, and g RC-4. Structure diagram: b rC-3, e rC-4, and h RC-4. Cole–Cole curves: c rC-3, f rC-4, and i RC-4. Structure difference of rC and RC composite aerogels in j pore size and k number of defects
Usually, attenuation constant (α) and impedance matching have a decisive impact on MA capability. The α denotes the dissipation capacity of EMWs, which is described as follows [51,52,53].
$$ \alpha = (\sqrt 2 \pi f/c \times \sqrt {\left( {\varepsilon^{\prime\prime}\mu^{\prime\prime} - \varepsilon^{\prime}\mu^{\prime}} \right) + \sqrt {\left( {\varepsilon^{\prime\prime}\mu^{\prime\prime} - \varepsilon^{\prime}\mu^{\prime}} \right)^{2} + \left( {\varepsilon^{\prime\prime}\mu^{\prime} + \varepsilon ^{\prime}\mu ^{\prime\prime}} \right)^{2} } } $$
The larger \(\varepsilon^{\prime\prime}\) values can lead to the improved α values from Eq. (7) for the \(\mu^{\prime} = 1\) and \(\mu^{\prime\prime} = 0\). The α curves of rC aerogels are shown in Fig. S4i, which keep an escalating tendency at 2–18 GHz. The α values with the order of rC-5 < rC-3 < rC-4 < rC-2 < rC-1 reveal that the introduction of low dielectric component CuS would reduce the α values. From Fig. S6j, RC aerogels demonstrate the same variation as the frequency increases, while the order of α values is RC-2 < RC-5 < RC-3 < RC-4 < RC-1. Since the RC-4 possesses relatively attenuation capacity among composite aerogels, leading to superior MA behavior.
In addition to attenuation loss, another factor, impedance matching (Z) also can affect MA performance. Impedance matching is on behalf of the EMWs entering into the absorbents, which can be accessed as follows [54].
$$ Z = Z_{{{\text{in}}}} /Z_{0} = \sqrt {\frac{{\mu_{{\text{r}}} }}{{\varepsilon_{{\text{r}}} }}} {\text{tanh}}\left( {j\frac{2\pi fd}{c}\sqrt {\varepsilon_{{\text{r}}} \mu_{{\text{r}}} } } \right) $$
Generally, the optimal impedance matching needs that the Z is equal to or close to 1, that is, the input impedance equal to free space impedance (Zin = Z0). As illustrated in Fig. S4j–n, it can be discovered that the |Zin/Z0| of rC-1 and rC-2 are much lower than 1, indicating poor impedance matching, and other rC samples are much closer to 1, which is accordance with the reflection loss results that they possess better MA performance than the other two samples. Figure S4o further draws the impedance matching curves of rC aerogels at the thickness of 2.0 mm, which shows the rC-4 is closest to 1 compared with other samples. For RC aerogels, the RC-1 and RC-4 are pretty close to 1 in Fig. S6k–o, manifesting their good absorbing performance (Figs. S5b1–d1 and S5b4–d4). The superior performance may be owing to the more defects and functional groups (Fig. S6p).
According to the above results, the RLmin absorption peaks shift to the low frequency with increasing thicknesses, which can use the explanation of λ/4 cancellation theory [55, 56].
$$ t_{m} = \frac{nc}{{4f_{m} \sqrt {\varepsilon_{r} \mu_{r} } }}\left( {n = 1,3,5, \ldots } \right) $$
From Fig. 5c-d, compared with rC-3, RC-4 shows the perfect matching point as the RLmin is achieved at 8.56 GHz at 3.5 mm that the impedance match is just at 1. Therefore, the RC-4 can satisfy the λ/4 wavelength model and perfect impedance matching at the same time, which is conducive to the formation of RLmin. Besides, the RL, tm and |Zin/Z0| curves of rC-4 and RC-4 composite aerogels are given in Fig. S8. It is clear that all tmexp (experimental tm) values fall perfectly on the λ/4 curve, which suggests that the λ/4 cancellation model plays a leading role in the relationship between tm and fm.
a RL ~ f curve of rC-3 with the broadest EAB at 2.8 mm. b RL ~ f curve of RC-4 with the RLmin at 3.5 mm. RL, tm and |Zin/Z0| curves: c rC-3 and d RC-4. Possible MA mechanism of CuS@rGO composite aerogels: e dipole polarization, f interfacial polarization and g conduction loss
Based on the discussion of composition, structure and performance, the EMW absorbing mechanism of CuS@rGO is demonstrated in Fig. 5e–g. Firstly, the complex of low dielectric CuS can optimize the impedance matching of pure rGO aerogel. rGO with microporous structure can availably reduce the permittivity for the incorporation of the high-volume fraction of air (\(\varepsilon_{r} = 1\)), which is helpful to improve impedance matching. The effective permittivity (εeff) can be described based on the Maxwell-Garnett model [57, 58].
$$ \varepsilon_{{{\text{eff}}}}^{{{\text{MG}}}} = \left[ {\frac{{\left( {\varepsilon_{2} + 2\varepsilon_{1} } \right) + 2p\left( {\varepsilon_{2} - \varepsilon_{1} } \right)}}{{\left( {\varepsilon_{2} + 2\varepsilon_{1} } \right) - p\left( {\varepsilon_{2} - \varepsilon_{1} } \right)}}} \right]\varepsilon_{1} $$
Herein \(\varepsilon_{2} ,\varepsilon_{1}\) and p are the permittivity of the air phase and solid phase, and the volume fraction of air phase in the porous structure. Typically, the incident EMWs are uninterested in the hole lower than the wavelength, so the micropore and nanopore can act as the effective medium to reduce the \(\varepsilon_{eff}\) value for the existence of air. Secondly, the surface or edge of rGO has defects and functional groups, which can induce the formation of dipole polarization [59]. Thirdly, the combination of CuS micro-flower with rGO aerogel can promote the generation of multiple heterogeneous interfaces like CuS/rGO, rGO/paraffin, and CuS/paraffin, causing the stronger interfacial polarization than pure CuS or rGO aerogel [60]. Finally, the interconnected conductive network constructed by rGO sheet can form microcurrents by means of electron migration and hopping, endowing CuS@rGO composite aerogel with excellent conduction loss [61, 62]. As a result, it can be concluded that the CuS@rGO composite aerogels can achieve excellent MA performance due to the unique merits of lightweight, low filler content, compression and recovery, wide absorption bandwidth and strong absorption, which integrates the "thin, light, wide and strong" properties of absorbers.
3.3 Microwave Dissipation Capacity Evaluated by RCS through CST Simulation
Microwave dissipation capacity of rC composite aerogels in the far-field condition is assessed by the RCS values of rC aerogels covered with the PEC model that are calculated by CST simulation. Figure 6a–f depicts the 3D radar wave scattering signals of PEC and rC aerogels. It is distinct that the rC-4 covered with PEC displays the weakest scattering intensity than other rC aerogels and PEC model, suggesting that the rC-4 possesses the lowest RCS. The detailed RCS value in the − 60° < θ < 60° angle range are presented in Fig. 6g. The PEC has the biggest RCS values, manifesting that rC aerogels can reduce the radar scattering intensities of the pure PEC plate. Besides, RCS value of PEC larger than 0 at 0° is owing to the interference between the reflected EMW and the incident EMW that is perpendicular to the absorber (Fig. 6h). RCS reduction values are further calculated in Fig. 6i. All five samples realize the reduced RCS values compared with the simulated PEC modes, and rC-4 exhibits the highest RCS reduction values at each primary angle. It is up to the maximum value of 53.3 dB m2, which is in accord with the minimum reflection loss of rC-4. These results confirmed that with the synergistic effect of dipole polarization, interfacial polarization, conduction loss, and unique porous structure, the EM energy can be effectively dissipated, and the radar scattering intensities are reduced at the same time.
3D radar wave scattering signals of a PEC, b rC-1, c rC-2, d rC-3, e rC-4 and f rC-5. g RCS simulated curves of PEC and RC composite aerogels. h Schematic diagram of CST simulation. i RCS reduction values of RC composite aerogels at the scanning angles of 0°, 20°, 40° and 60°
3.4 IR Stealth Performance
To satisfy the demand for radar-IR compatible stealth, the as-prepared CuS@rGO composite aerogels with excellent thermal insulation performance due to the unique porous structure are also necessary in addition to the superior MA ability. The IR radiation will be emitted from the target when the temperature is above absolute zero, which can be detected by the IR detector. Besides, once the target has a high contrast with the background IR radiation, it will be exposed. Reducing the IR radiation energy is the main strategy to achieve IR stealth, originating from the Stefan-Boltzmann equation [63].
$$ E\left( T \right) = \int\limits_{0}^{\infty } {\varepsilon \left( {\lambda ,T} \right)c_{1} \lambda ^{{ - 5}} \left[ {\exp \left( {\frac{{c_{2} }}{{\lambda T}}} \right) - 1} \right]^{{ - 1}} d\lambda = \varepsilon \left( T \right)\sigma T^{4} } $$
Herein E, ε, T and σ mean IR radiation energy, IR emissivity, surface temperature and Stefan-Boltzmann constant, c1 and c2 represent the first and second radiation constant, respectively. Superior thermal stealth can protect targets from detection in the military field. Thus, the IR stealth performance of CuS@rGO composite aerogels was studied by a thermal IR camera. Besides, the IR emissivity is also characterized at 3–5 and 8–14 m via IR-2 Emissometer. The thermal IR images of rC-4 at 10-min intervals are depicted in Fig. 7a. The rC-4 aerogel is placed in the center of a circular heating platform (Fig. 7d), and the heating temperature is set to 120 °C. The surface temperature of rC-4 is 26.6 °C at the beginning. From Fig. 7b, it is interesting that the surface temperature will go up at a tiny temperature difference (surface temperature and maximum temperature, ΔT < 0.8 °C), and then it can maintain almost its original temperature after 30 min heating, indicating its stable thermal stealth capability. The other CuS@rGO aerogels are tested with the same condition and their results are depicted in Figs. S9–S12 and Table S4. It can be more intuitively seen from Figs. S13 and 7c that the ΔT is decreasing, and rC-5, in particular, has almost no temperature difference, suggesting that the surface temperature of rC composite aerogels is much closer to the beginning temperature after 30 min heating with the increase in CuS content. These results further confirm that complexing low-emissivity CuS with 3D porous rGO aerogel is conducive to thermal stealth ability. The abundant air with lower thermal conductivity can take the place of solid phase with higher thermal conductivity. Besides, 3D aerogels endow with a low density and porous structure, and a large number of pores inside hinder the heat transfer. The existence of CuS microspheres also obstruct the heat transfer between rGO sheets. Therefore, the CuS@rGO composite aerogels have excellent thermal insulation performance. Furthermore, low IR emissivity is another way to realize IR stealth. The IR radiation energy can be reduced by modulating the emissivity with unchanged surface temperature. There are currently two atmospheric window regions of 3 ~ 5 and 8 ~ 14 m adopted by IR detectors. As presented in Fig. 7e and Table S2, the IR emissivity of rC composite aerogels shows a downward trend on the IR waveband of both 3 ~ 5 and 8 ~ 14 m, which is consistence with the results of thermal IR images. Besides, the emissivity at 3 ~ 5 m is much lower than 8 ~ 14 m. The possible IR stealth mechanism is summarized in Fig. 7f. The forms of thermal transfer consist of thermal radiation, thermal conduction and thermal convection, which all occur in CuS@rGO aerogels. Owing to the low density of porous aerogels, the gas-phase components can reduce the thermal conduction for their low thermal conductivity. Moreover, the 3D network structure is conducive to prolonging the thermal transfer path and reducing the thermal conduction in the solid phase, leading to a perfect insulation performance. Figure 7g shows the ideal double-layer radar-IR stealth coating. The EMWs can pass through the IR stealth layer, and enter the MA layer, then be dissipated. Impedance matching is one of the most significant factors in minimizing the radar reflectivity of IR stealth coating.
a Thermal IR images of rC-4 at different heating times. b Surface temperature curve of rC-4. c Difference between heating temperature and surface temperature of rC aerogels. d Schematic diagram of IR thermal imaging test. e IR emissivity of rC composite aerogels at 3 ~ 5 and 8 ~ 14 m. f Thermal transfer processes of porous CuS@rGO composite aerogels. g Schematic diagram of radar-IR compatible stealth
In this work, we developed an effective composite-structure-performance strategy to enhance MA performance and reduce IR emissivity. Two types of CuS@rGO composite aerogels were successfully fabricated via hydrothermal reduction and ascorbic acid thermal reduction. The reduction mechanisms involved the decarboxylation process, dehydroxylation process, and deoxidation process of epoxy groups, which could lead to the defects. In addition, adjacent graphene sheets wrapped by numerous tiny CuS are stacked with each other to form a 3D porous structure during the thermal reduction process. The porous structure and defects could be modulated by the thermal reduction and additive amounts of CuS. Because of the balanced attenuation capability and impedance matching, the as-prepared CuS@rGO aerogels depicted impressive microwave absorbing performance. The CuS@rGO aerogels achieved the broadest EAB of 8.44 GHz (2.8 mm) with the additive amount of 30 mg. The samples realized the RLmin of − 50.4 dB (2.0 mm) with the additive amount of 60 mg through the hydrothermal reduction method under the filler content of 6 wt%. Besides, the CuS@rGO aerogel (RC-4) could achieve the EAB of 7.2 GHz and RLmin of − 55.1 dB at 2.45 mm with the filler content of 2 wt%, in addition, the RLmin of − 48.1 dB and EAB of 5.96 GHz could be obtained at 2.2 mm with the lowest filler content of 1 wt%. The CST simulated results also demonstrated that the CuS@rGO composite aerogels could effectively reduce the radar scattering intensity. Furthermore, thermal IR images and IR emissivity could confirm that the GuS@rGO composite aerogels had the ability to reduce the surface temperature and IR emissivity. Thus, these results will lead to the development of radar-IR compatible stealth materials composed of carbon-based aerogels, which can make them a considerable application prospect in a harsh military environment.
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We are thankful for financial support from the National Nature Science Foundation of China (No. 51971111).
Open access funding provided by Shanghai Jiao Tong University.
College of Materials Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People's Republic of China
Yue Wu, Yue Zhao, Ming Zhou, Shujuan Tan & Guangbin Ji
Department of Chemical Engineering, Energy Institute of Higher Education, Saveh, Iran
Reza Peymanfar
Faculty of Physics, University of Tabriz, Tabriz, 51666-16471, Iran
Bagher Aslibeiki
Yue Wu
Yue Zhao
Shujuan Tan
Guangbin Ji
Correspondence to Shujuan Tan or Guangbin Ji.
Below is the link to the electronic supplementary material.
Supplementary file1 (PDF 2343 KB)
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Microwave absorption
Ultrabroad bandwidth
Composite aerogel
Radar cross section
Radar-infrared compatible stealth | CommonCrawl |
Genome-wide joint analysis of single-nucleotide variant sets and gene expression for hypertension and related phenotypes
Xiaoran Tong1,
Changshuai Wei2 &
Qing Lu1
With the advance of next-generation sequencing technologies, the study of rare variants in targeted genome regions or even the whole genome becomes feasible. Nevertheless, the massive amount of sequencing data brings great computational and statistical challenges for association analyses. Aside from sequencing variants, other high-throughput omic data (eg, gene expression data) also become available, and can be incorporated into association analysis for better modeling and power improvement. This motivates the need of developing computationally efficient and powerful approaches to model the joint associations of multilevel omic data with complex human diseases.
A similarity-based weighted U approach is used to model the joint effect of sequencing variants and gene expression. Using a Mexican American sample provided by Genetic Analysis Workshop 19 (GAW19), we performed a whole-genome joint association analysis of sequencing variants and gene expression with systolic (SBP) and diastolic blood pressure (DBP) and hypertension (HTN) phenotypes.
The whole-genome joint association analysis was completed in 80 min on a high-performance personal computer with an i7 4700 CPU and 8 GB memory. Although no gene reached statistical significance after adjusting for multiple testing, some top-ranked genes attained a high significance level and may have biological plausibility to hypertension-related phenotypes.
The weighted U approach is computationally efficient for high-dimensional data analysis, and is capable of integrating multiple levels of omic data into association analysis. Through a real data application, we demonstrate the potential benefit of using the new approach for joint association analysis of sequencing variants and gene expression.
Next-generation sequencing technology provides denser genetic profiles than previous microarray-based genotyping technology [1]. It could effectively capture rare variants with low minor allele frequency (MAF). Driven by the advance of sequencing technology and limited heritability explained by the genome-wide association studies (GWAS) findings [2, 3], current research focus has shifted toward studying rare variants associated with common complex diseases. Although these studies hold great promise for finding new genetic variants predisposing to human disease, they also face great challenges, for example, low power for detecting rare variants because of their low frequency. The dramatic increase in numbers of single nucleotide variants (SNVs) also raises computational and statistical challenges (eg, multiple testing issue). One practical strategy is to group multiple SNVs according to known functional information (eg, variants in a gene or a pathway) or location (eg, variants in a fix-sized bin [4]), and jointly analyze these SNVs [5, 6]. By grouping and testing multiple SNVs, we are able to aggregate association signals and reduce the number of tests.
Besides SNVs, other omic data, such as gene expression, could also be collected. These intermediate omic data can be integrated into sequencing studies for improved power and better biological interpretation. While the conventional analysis only links SNVs or gene expression to disease phenotypes, the emergence of multilevel data brings the possibility of jointly analyzing SNVs and other omic data. By fully utilizing the information, the joint analysis has great potential to improve power [7]. Nevertheless, how to efficiently analyze the high-dimensional sequencing data and other omic data remains a challenge.
In this empirical study, we used a similarity based weighted U approach to jointly model SNVs and gene expression data of 142 unrelated Mexican American samples provided by Genetic Analysis Workshop 19 (GAW19). By using the weighted U approach, we performed a genome-wide joint association analysis, evaluating the association of 17,558 genes with three phenotypes (ie,, systolic blood pressure [SBP], diastolic blood pressure [DBP], and hypertension [HTN]).
For the integrative analysis, we extended previously developed nonparametric approaches [8] to handle both SNVs and gene expression. To aggregate the rare variants in a gene, a weighted sum approach is used [8]. Let p k denote the MAF of the k th SNV (k = 1,2,…,K), the weight for the k th SNV can be defined as \( {w}_k=1/\sqrt{p_k\left(1-{p}_k\right)} \). Let K be the total number of SNVs in a gene region, the weighted sum score for the j th sample can be obtained by,
$$ {a}_j=\frac{{\displaystyle {\sum}_{k=1}^K{w}_k{v}_{jk}}}{2{\displaystyle {\sum}_{k=1}^K{w}_k}}, $$
where v jk is the genotype value of the k th SNV for the j th sample, coded by the minor allele count (ie, 0, 1, and 2). We then define a weighted U statistic to assess the joint effect of SNVs and gene expression on the disease phenotype,
$$ U={\sum}_{i\ne j}f\left({a}_i,{a}_j\right)\mathit{\mathsf{g}}\left({t}_i,{t}_j\right)h\left({y}_i,{y}_j\right), $$
where f(a i ,a j ), g(t i ,t j ), and h(y i ,y j ) measure the similarities of SNVs, gene expression, and phenotypes, respectively. Phenotypic similarity h(y i ,y j ) serves as the U kernel,
$$ h\left({y}_i,{y}_j\right)=\frac{\left({y}_i-E(Y)\right)\left({y}_j-E(Y)\right)}{Var(Y)}, $$
where y i and y j are ranks of the i th and j th samples' phenotypes. The genetic and gene expression similarities are weight functions, defined based on the Gaussian distance,
$$ f\left({a}_i,{a}_j\right)={e}^{-\frac{{\left({a}_i-{a}_j\right)}^2}{2N}}\kern0.5em \mathit{\mathsf{g}}\left({t}_i,{t}_j\right)={e}^{-\frac{{\left({t}_i-{t}_j\right)}^2}{2N}}, $$
where a i (a j ) and t i (t j ) denote the weighted sum score and the gene expression value of the individual i(j), respectively.
Under the null hypothesis of no association, phenotypic similarity is unrelated to genetic or gene expression similarities. Because phenotypic similarity is symmetric, that is, E (h(y i ,y j )) = 0, the expectation of U statistic is 0. Under the alternative, phenotypic similarity increases with the increase of genetic or gene expression similarities. Therefore, the positive phenotypic similarities are heavier weighted and the negative phenotypic similarities are lighter weighted, leading to a positive value of U. Because the U kernel satisfies the finite second moment condition, E(h 2(y i ,y j )) < ∞, and is degenerate (ie, Var(E(h(y i ,y j ))) = 0), the limiting distribution of U can be approximated as a linear combination of chi-squared random variables with one degree of freedom [8], and its p value can be obtained by using the Davis method [9].
The weighted U approach is also flexible for testing other hypothesis. In addition to evaluating the joint effect of genetic markers and gene expression (G + T), it could be used for testing genetic effect (G) alone or gene expression (T) effect alone. For example, we can modify the approach by setting the gene expression similarity as constant (eg, g(t i ,t j ) ≡ 1) to test genetic effect.
Genome screening
We applied three tests (ie, G + T, G, and T) to 142 unrelated Mexican American samples from the San Antonio Family Heart Study (SAFHS) and the San Antonio Family Diabetes/Gallbladder Study (SAFDGS). All analyses were based on SNVs on the odd-numbered autosomes and gene expression data provided by GAW19. In this study, we assembled multiple SNVs based on the functional unit (ie, gene) to facilitate the joint modeling of gene and gene expression. We obtained primary and alternative assembles from Genome Reference Consortium release version 38 (GRCh38) and identified 32,436 gene regions in correspondence to 17,264 RNA probes. The number of gene regions exceeds the probes because multiple assembles of one gene can share one nucleotide sequence, as well as the RNA probes designed to capture such sequence. SNVs that are not within or near a gene (±5 kb at both ends) were removed. Gene regions with no SNVs or RNA probes were also discarded. SNVs with no variation (ie, MAF = 0) were dropped, as were gene regions containing only such SNVs. A total of 6,956,910 SNVs, corresponding to 17,558 gene regions, remained for the joint analysis. The first, second, and third quartiles of the SNV counts in these regions are 115, 205, and 411, respectively. We used SBP, DBP, and HTN measurements at the first examination year as phenotypes, and age, gender, medication use, and smoking status as covariates. To account for population stratification, we performed principal components (PCs) analysis by using the EIGENSTRAT software [10]. The first 20 PCs were used in the analysis to adjust for potential confounding bias because of population stratification.
The whole-genome joint analysis of 3 phenotypes was completed in 80 min using a single core of i7 4700 CPU with 8 GB memory. Table 1 summarizes the top genes from the analysis, which were selected based on the smallest p value of three tests. In general, we observed that the G + T test either attained the smallest p value or a p value close to the smallest one. After adjusting for multiple testing, none of the genes were significantly associated with the phenotypes. However, if we used a significant threshold of 0.05, 4 of 15 genes were missed by considering SNVs alone (ie, G) and six genes were missed by considering gene expression alone (ie, T), while all 15 genes could be captured by the joint association analysis (ie, G + T). This suggests that there are potential advantages to combining genetic and gene expression information in the association analysis. The quantile–quantile (QQ) plot was also drawn, which showed no evidence of systematically inflation of the G + T test (Fig. 1).
Table 1 Summary of top 5 genes associated with SBP, DBP, and HTN
QQ plot for the joint association analysis (G + T) of SBP, DBP, and HTN
Further investigation of the top genes also found biological plausibility of several genes related to blood pressure. For instance, the product of PED4A hydrolyzes the second messenger cyclic adenosine monophosphate (cAMP), which plays a crucial role in controlling blood pressure [11]. PHOX2A is also important for the development of autonomic nervous system, which controls the involuntary functions, such as heart rate and blood pressure [12].
The study has certain limitations. Out of 8,348,674 SNVs, 1,391,764 (17 %) were unused because they are not in or near any gene. We could group these SNVs by physical location and also incorporate them into the analysis [4]. We found limited association evidence of single-nucleotide polymorphisms (SNPs) identified from previous GWAS, possibly because of differences in study samples (ie, whites vs. Mexican Americans). Another possibility is that majority of SNVs in our study are rare (MAF <0.01), whereas previous GWAS mainly focus on common variants (MAF >0.05).
The analysis of a large number of genes raised the issue of multiple testing. In our analysis, the false discovery rate approach was used to account for the issue of multiple testing. After adjusting for multiple testing, none of the genes could reach statistical significance. By using the biology knowledge and statistical tools, we might be able to further reduce the number of tests and increase our chance to detect an association. For instance, all assembles of one gene have high correlation, and we can either exclusively use the primary assemble or adjust p values for multiple correlated tests to better solve the multiple testing issue.
The emerging sequencing data and other omic data provide invaluable source for genetic study of human diseases, yet integrating and modeling these high-dimensional data remain a great challenge. By integrating both sequencing variants and gene expression into the association analysis, the weighted U approach provides a powerful and computationally efficient way for screening disease-associated genes. By applying the approach to the GAW19 data, we showed that the joint analysis of sequencing variants and gene expression could have some advantages over association analysis only using sequence variants or gene expression.
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Wen G, Wessel J, Zhou W, Ehret GB, Rao F, Stridsberg M, Mahata SK, Gent PM, Das M, Cooper RS, et al. An ancestral variant of Secretogranin II confers regulation by PHOX2 transcription factors and association with hypertension. Hum Mol Genet. 2007;16(14):1752–64.
The Genetic Analysis Workshop is supported by NIH grant R01 GM031575. The authors wish to thank the editor and two anonymous referees for their helpful comments that improved the manuscript.
XT conducted the data analysis and drafted the manuscript. CW helped write the R code for the joint analysis. QL conceived of the study and helped finalize the manuscript. All authors read and approved the final manuscript.
The authors declare they have no competing interests.
Department of Epidemiology and Biostatistics, Michigan State University, East Lansing, MI, 48824, USA
Xiaoran Tong
& Qing Lu
Health Science Centre, University of North Texas, Fort Worth, TX, 76107, USA
Changshuai Wei
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Correspondence to Qing Lu.
Minor Allele Frequency
Rare Variant
Omic Data
Gene Expression Similarity | CommonCrawl |
Works by Keita Yokoyama
( view other items matching `Keita Yokoyama`, view all matches )
Reverse mathematics and Peano categoricity.Stephen G. Simpson & Keita Yokoyama - 2013 - Annals of Pure and Applied Logic 164 (3):284-293.details
We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i∈A and f:A→A. A subset X⊆A is said to be inductive if i∈X and ∀a ∈X). The system A,i,f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to be (...) an inductive system such that f is one-to-one and i∉the range of f. The standard example of a Peano system is N,0,S where N={0,1,2,…,n,…}=the set of natural numbers and S:N→N is given by S=n+1 for all n∈N. Consider the statement that all Peano systems are isomorphic to N,0,S. We prove that this statement is logically equivalent to WKL0 over RCA0⁎ source. From this and similar equivalences we draw some foundational/philosophical consequences. (shrink)
Areas of Mathematics in Philosophy of Mathematics
On the strength of Ramsey's theorem without Σ1 -induction.Keita Yokoyama - 2013 - Mathematical Logic Quarterly 59 (1-2):108-111.details
In this paper, we show that equation image is a equation image-conservative extension of BΣ1 + exp, thus it does not imply IΣ1.
Proof Theory in Logic and Philosophy of Logic
On principles between ∑1- and ∑2-induction, and monotone enumerations.Alexander P. Kreuzer & Keita Yokoyama - 2016 - Journal of Mathematical Logic 16 (1):1650004.details
We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a (...) far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text]. (shrink)
Propagation of partial randomness.Kojiro Higuchi, W. M. Phillip Hudelson, Stephen G. Simpson & Keita Yokoyama - 2014 - Annals of Pure and Applied Logic 165 (2):742-758.details
Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to a (...) PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)
The reverse mathematics of theorems of Jordan and lebesgue.André Nies, Marcus A. Triplett & Keita Yokoyama - 2021 - Journal of Symbolic Logic 86 (4):1657-1675.details
The Jordan decomposition theorem states that every function $f \colon \, [0,1] \to \mathbb {R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function of bounded variation is differentiable almost everywhere in the sense of Lebesgue measure. We analyze the strength of these theorems in the setting of reverse mathematics. Over $\mathsf {RCA}_{0}$, a stronger version of Jordan's result where all functions are continuous is equivalent to (...) $\mathsf {ACA}_0$, while the version stated is equivalent to ${\textsf {WKL}}_{0}$. The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to ${\textsf {WWKL}}_{0}$. To state this equivalence in a meaningful way, we develop a theory of Martin–Löf randomness over $\mathsf {RCA}_0$. (shrink)
A Nonstandard Counterpart of WWKL.Stephen G. Simpson & Keita Yokoyama - 2011 - Notre Dame Journal of Formal Logic 52 (3):229-243.details
In this paper, we introduce a system of nonstandard second-order arithmetic $\mathsf{ns}$-$\mathsf{WWKL_0}$ which consists of $\mathsf{ns}$-$\mathsf{BASIC}$ plus Loeb measure property. Then we show that $\mathsf{ns}$-$\mathsf{WWKL_0}$ is a conservative extension of $\mathsf{WWKL_0}$ and we do Reverse Mathematics for this system.
Categorical characterizations of the natural numbers require primitive recursion.Leszek Aleksander Kołodziejczyk & Keita Yokoyama - 2015 - Annals of Pure and Applied Logic 166 (2):219-231.details
The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic.Nobuyuki Sakamoto & Keita Yokoyama - 2007 - Archive for Mathematical Logic 46 (5-6):465-480.details
In this paper, we show within ${\mathsf{RCA}_0}$ that both the Jordan curve theorem and the Schönflies theorem are equivalent to weak König's lemma. Within ${\mathsf {WKL}_0}$ , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of ${\mathsf {WKL}_0}$ has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).
Formalizing non-standard arguments in second-order arithmetic.Keita Yokoyama - 2010 - Journal of Symbolic Logic 75 (4):1199-1210.details
In this paper, we introduce the systems ns-ACA₀ and ns-WKL₀ of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACA₀ and WKL₀, respectively. Then, we give direct transformations from non-standard proofs in ns-ACA₀ or ns-WKL₀ into proofs in ACA₀ or WKL₀.
Nonstandard second-order arithmetic and Riemann's mapping theorem.Yoshihiro Horihata & Keita Yokoyama - 2014 - Annals of Pure and Applied Logic 165 (2):520-551.details
In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemann's mapping theorem, and show several different versions of Riemann's mapping theorem.
Non-standard analysis in ACA0 and Riemann mapping theorem.Keita Yokoyama - 2007 - Mathematical Logic Quarterly 53 (2):132-146.details
This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).
Reverse mathematical bounds for the Termination Theorem.Silvia Steila & Keita Yokoyama - 2016 - Annals of Pure and Applied Logic 167 (12):1213-1241.details
Complex analysis in subsystems of second order arithmetic.Keita Yokoyama - 2007 - Archive for Mathematical Logic 46 (1):15-35.details
This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy's integral theorem. We show that a weak version of Cauchy's integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, (...) we show that a full version of Cauchy's integral theorem cannot be proved in RCAo but is equivalent to weak König's lemma over RCAo. (shrink)
The Dirac delta function in two settings of Reverse Mathematics.Sam Sanders & Keita Yokoyama - 2012 - Archive for Mathematical Logic 51 (1-2):99-121.details
The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. We show that (...) the Dirac Delta Theorem is equivalent to weak König's Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the 'Big' systems of Reverse Mathematics. In the context of ERNA's Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA's Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA's Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength. (shrink)
Intuitionism and Constructivism in Philosophy of Mathematics
The strength of ramsey's theorem for pairs and arbitrarily many colors.Theodore A. Slaman & Keita Yokoyama - 2018 - Journal of Symbolic Logic 83 (4):1610-1617.details
How Strong is Ramsey's Theorem If Infinity Can Be Weak?Leszek Aleksander Kołodziejczyk, Katarzyna W. Kowalik & Keita Yokoyama - forthcoming - Journal of Symbolic Logic:1-20.details
We study the first-order consequences of Ramsey's Theorem for k-colourings of n-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of $\mathrm {RCA}^*_0$ that does not satisfy $\Sigma ^0_1$ induction, $\mathrm {RT}^n_k$ is equivalent to its relativization to any proper $\Sigma ^0_1$ -definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give (...) a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of $\mathrm {PA}$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _{\ell +3}$ fragment for $\ell \ge 1$ lies between $\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$ and $\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of $\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$ form a subtheory of $\mathrm {I} \Sigma _2$ whose $\Pi _3$ fragment coincides with $\mathrm {B} \Sigma _1 + \exp $ and whose $\Pi _4$ fragment is strictly weaker than $\mathrm {B} \Sigma _2$ but not contained in $\mathrm {I} \Sigma _1$. Additionally, we consider a principle $\Delta ^0_2$ - $\mathrm {RT}^2_2$ which is defined like $\mathrm {RT}^2_2$ but with both the $2$ -colourings and the solutions allowed to be $\Delta ^0_2$ -sets rather than just sets. We show that the behaviour of $\Delta ^0_2$ - $\mathrm {RT}^2_2$ over $\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$ is in many ways analogous to that of $\mathrm {RT}^2_2$ over $\mathrm {RCA}^*_0$, and that $\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$ - $\mathrm {RT}^2_2$ is $\Pi _4$ - but not $\Pi _5$ -conservative over $\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of $\Pi _5$ -conservativity is not provable in $\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)
Logics, Misc in Logic and Philosophy of Logic | CommonCrawl |
\begin{document}
\begin{frontmatter}
\title{Analyticity of Bounded Solutions of Analytic State-Dependent Delay Differential Equations }
\author[]{Qingwen Hu}\ead{\tt [email protected]} \address[]{Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, 75080}
\small{\today}
\maketitle
\begin{abstract} We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract ordinary differential equation in a subspace of the sequence space $l^{\infty}(\mathbb{R}^{N+1})$ and prove the existence of complex extension of the bounded solutions. An example is given to illustrate the general results. \end{abstract}
\begin{keyword} State-dependent delay\sep analyticity\sep bounded solutions
\end{keyword}
\end{frontmatter}
\thispagestyle{plain}
\pagestyle{plain}
\section{Introduction}\label{SOPS-4-1}
The analyticity of bounded solutions of delay differential equations with constant delay such as the well-known Wright's equation was established in work of Nussbaum \cite{Nussbaum-analyticity}. It is natural to conjecture that this analyticity result holds true for many differential equations with state-dependent delay such as \begin{align}\label{eqn-2-oldd}\left\{
\begin{aligned}
\dot{x}(t)&= f(x(t),\,x(t- \tau)),\\
\tau & =r(x(t)),
\end{aligned} \right. \end{align} with analytic $f$ and $r$. In this paper, we solve this conjecture.
We should remark that the work of Mallet-Paret and Nussbaum \cite{Nussbaum-2} also presented some examples where bounded solutions are no-longer analytic, while Krisztin \cite{Tibor} showed that globally defined bounded solutions of threshold type delay equations are analytic. Then an important theoretical problem is what would be the most general form of state-dependent delay differential equations for which the conjecture remains true for differential equations with state-dependent delay. We also notice that establishing the analyticity of bounded solutions such as periodic solutions is essential for describing the global dynamics of some state-dependent delay differential equations. For example, in \cite{HWZ} the nonexistence of a nonconstant $p$-periodic real-valued solution which is constant in a small interval in $\mathbb{R}$ was assumed in order to obtain the global continuation of periodic solutions of the following system \begin{align}\label{eqn-2-old}\left\{
\begin{aligned}
\dot{x}(t)=& f(x(t),\,x(t- \tau(t))),\\
\dot{\tau}(t)=& g(x(t), \tau (t)),
\end{aligned} \right. \end{align} with analytic $f$ and $g$. Specifically, it was needed to exclude the case where there is a nonconstant $p-$periodic solution for which \begin{align}
\tau(t)=\tau_0,\,t\in I+kp, k\in\mathbb{Z} \end{align} where $\tau_0>0$ is a constant and $I$ is an interval in $\mathbb{R}$ with length less than $p$. On the one hand, if there is such a periodic solution and if this solution is analytic on $\mathbb{R}$, then the delay $\tau$ must be a constant on the whole real line $\mathbb{R}$. On the other hand, under certain technical conditions, it can be ruled out the existence of such a periodic solution with constant delay by considering a cyclic system of ordinary differential equations (see \cite{HWZ} for more details) and hence these technical conditions can ensure the nonexistence of a nonconstant $p$-periodic solution for which $\tau $ remains to be a constant in a small interval in $\mathbb{R}$.
In this paper, we first note that bounded solutions of system (\ref{eqn-2-oldd}) and system (\ref{eqn-2-old}) and many others including those with ``threshold delay" must satisfy the following differential equations with state-dependent delays \begin{align}\label{eqn-2}\left\{
\begin{aligned}
\dot{x}(t)=& f(x(t),\,x(t- \tau(t))),\\
\dot{\tau}(t)=& g(x(t),\,x(\eta(t)),\,\cdots,\,x(\eta^{M-1}(t)),\,\,\tau(t)),
\end{aligned} \right. \end{align} where
$\eta^0(t)=t$, $\eta(t)=t-\tau(t)$, $\eta^j(t)=\eta(\eta^{j-1}(t))$ for $j=1,\,2\,\,\cdots,\,M$ with $M\in\mathbb{N}$, and we assume \begin{description} \item[(A1)]The maps $f$: $U\times U\ni (\theta_1,\theta_2) \rightarrow f(\theta_1,\theta_2)\in\mathbb{C}^N$ and $g$: $U^M\times V \ni (\gamma_1,\,\gamma_2)\rightarrow g(\gamma_1,\,\gamma_2)\in\mathbb{C}$ are analytic with respect to $(\theta_1,\theta_2)$ and $(\gamma_1,\,\gamma_2)$, respectively, where $U\subset \mathbb{C}^N,\, V\subset\mathbb{C}$ are bounded open sets, $U^M=\underbrace{U\times U\times\cdots\times U}_{M}$.
\item[(A2)] There exist $l\in (0,\,1)$ and $c>1$ such that
$ |1-g(\gamma_1,\,\gamma_2)-\frac{c+l}{2}|<\frac{c-l}{2}$ for all $(\gamma_1,\,\gamma_2)\in \overline{U}^M \times \overline{V}$, where $\overline{U}^M \times \overline{V}$ is the closure of $U^M\times V$.
\end{description}
(A1) is a natural assumption on the analyticity of $f$ and $g$ on their domains. (A2) is assuming that $g$ satisfies $l<|1-g|<c$ which ensures that the mapping $\mathbb{R}\ni t\rightarrow t-\tau(t)\in\mathbb{R}$ is increasing with a bounded rate.
Let $(x,\,\tau)\in C(\mathbb{R};\mathbb{R}^{N+1})$ be a bounded solution of system~(\ref{eqn-2}) and define the sequence $\left((y_1,\,z_1),\,(y_2,\,z_2,),\,\cdots\right)$ by \begin{align*}(y_j(t),\, z_j(t))=\left(\frac{1}{c^{j}}x(\eta^{j-1}(t)),\,\frac{1}{c^{j}}\tau(\eta^{j-1}(t))\right) \mbox{ for $j\geq 1,\,j\in\mathbb{N},\,t\in\mathbb{R}$.} \end{align*} The reason that we carry a term $\frac{1}{c^{j}}$ will be clear by the end of this section. For $j=1$ we have for every $t\in\mathbb{R}$, \begin{align}
\frac{d}{dt}y_1(t)& = \dot{x}(t) =\frac{1}{c}{f(cy_1(t),\, c^2y_{2}(t) )},\label{analyticity-ODE-1-J} \\ \frac{d}{dt}z_1(t)& =\dot{\tau}(t) =\frac{1}{c}g(cy_1(t),\,c^2y_2(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\, cz_{1}(t) ).\label{analyticity-ODE-2-J} \end{align} For $j\geq 2,\,j\in\mathbb{N}$, we have for every $t\in\mathbb{R}$, \begin{align}
\frac{d}{dt}y_j(t)& =\frac{1}{c^j}\dot{x}(\eta^{j-1}(t))\prod_{i=0}^{j-2}\dot{\eta}(\eta^i(t))\notag\\
& =\dot{x}(\eta^{j-1}(t))\frac{1}{c^j} \prod_{i=0}^{j-2}(1-g(x(\eta^i(t)), \,x(\eta^{i+1}(t)),\,\cdots,\,x(\eta^{i+M-1}(t)),\,\tau(\eta^i(t)) )\notag\\
& =\frac{f((c^{j}y_j(t),\, {c^{j+1}}y_{j+1}(t) )}{1-g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t)) }\notag\\
& \quad \times\frac{1}{c^j} \prod_{i=0}^{j-1}(1-g(c^{i+1}y_{i+1}(t), \,c^{i+2}y_{i+2}(t),\,\cdots,\,c^{i+M}y_{i+M}(t),\,c^{i+1}z_{i+1}(t))),\label{analyticity-ODE-1} \intertext{and} \frac{d}{dt}z_j(t)& =\frac{1}{c^j}\dot{\tau}(\eta^{j-1}(t))\prod_{i=0}^{j-2}\dot{\eta}(\eta^i(t))\notag\\
& =\dot{\tau}(\eta^{j-1}(t))\frac{1}{c^j} \prod_{i=0}^{j-2}(1-g(x(\eta^i(t)), \,x(\eta^{i+1}(t)),\,\cdots,\,x(\eta^{i+M-1}(t)),\,\tau(\eta^i(t)) )\notag\\
& =\frac{g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t))}{1-g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t))}\notag\\
& \quad \times\frac{1}{c^j} \prod_{i=0}^{j-1}(1-g(c^{i+1}y_{i+1}(t), \,c^{i+2}y_{i+2}(t),\,\cdots,\,c^{i+M}y_{i+M}(t),\,c^{i+1}z_{i+1}(t))).\label{analyticity-ODE-2} \end{align} Then the sequence $((y_1,\,z_1),\,(y_2,\,z_2),\,\cdots)$, $t\in\mathbb{R}$ satisfies a system of ordinary differential equations of (\ref{analyticity-ODE-1-J}), (\ref{analyticity-ODE-2-J}), (\ref{analyticity-ODE-1}) and (\ref{analyticity-ODE-2}). Namely, for every $t\in\mathbb{R}$ and for $j\geq 1$, we have \begin{align} \label{New-Eqn-1} \left\{ \begin{aligned} \frac{d}{dt}y_j(t) & =\frac{f((c^{j}y_j(t),\, {c^{j+1}}y_{j+1}(t) )}{1-g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t)) } \\
& \quad \times\frac{1}{c^j} \prod_{i=0}^{j-1}(1-g(c^{i+1}y_{i+1}(t), \,c^{i+2}y_{i+2}(t),\,\cdots,\,c^{i+M}y_{i+M}(t),\,c^{i+1}z_{i+1}(t))),\\ \frac{d}{dt}z_j(t) & =\frac{g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t))}{1-g(c^{j}y_j(t), \,c^{j+1}y_{j+1}(t),\,\cdots,\,c^{j+M-1}y_{j+M-1}(t),\,c^{j}z_j(t))} \\
& \quad \times\frac{1}{c^j} \prod_{i=0}^{j-1}(1-g(c^{i+1}y_{i+1}(t), \,c^{i+2}y_{i+2}(t),\,\cdots,\,c^{i+M}y_{i+M}(t),\,c^{i+1}z_{i+1}(t))).
\end{aligned}\right. \end{align} It remains to decide an appropriate space where $(y_1(t),\,z_1(t),\,(y_2,\,z_2),\,\cdots)$, $t\in\mathbb{R}$ lives in. With the arguments of $f$ and $g$ in the right hand side of (\ref{New-Eqn-1}), it turns out that we can set $w(t)=((y_1(t),\,z_1(t)),\,(y_2(t),\,z_2(t)),\,\cdots)$ to be in the sequence space $l_c^{\infty} (\mathbb{R}^{N+1})$ defined by \begin{align}\label{l-c-def}
l_c^{\infty} (\mathbb{R}^{N+1})=\{v=(v_1,\,v_2,\,\cdots,v_j,\cdots)\in l^{\infty}(\mathbb{R}^{N+1}): \sup_{j\in\mathbb{N}} c^j|v_j|<+\infty\}, \end{align} where we can find a subset such that the terms of $f$ and $g$ in system~(\ref{New-Eqn-1}) are well-defined. Besides, the product terms
in system~(\ref{New-Eqn-1}) need to be treated so that the right hand side of system~(\ref{New-Eqn-1}) always remains bounded as $j\rightarrow\infty$. We address this issue at Lemma~\ref{Lemma-l-infty}.
With the above preparations we can represent system~(\ref{New-Eqn-1}) by the following abstract ordinary differential equation: \begin{align}\label{Abs-ODE-S-infty-new}
\frac{d}{dt}w(t)= H(Tw(t)), \end{align} where
the mapping $T: l_c^{\infty} (\mathbb{R}^{N+1})\rightarrow l^{\infty}(\mathbb{R}^{N+1})$ is defined by \[
T(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)=(cv_1,\,c^2v_2,\,\cdots,c^{j}v_j,\cdots), \] and $H: l^{\infty} (\mathbb{R}^{N+1})\rightarrow l^{\infty}(\mathbb{R}^{N+1})$ is defined by the right hand side of system~(\ref{New-Eqn-1}).
To obtain the analyticity of bounded solutions $(x(t),\,\tau(t))$, $t\in\mathbb{R}$ of system~(\ref{eqn-2}), we follow the idea of \cite{Nussbaum-analyticity} to show that the solution $w(t)$ to system~(\ref{Abs-ODE-S-infty-new}) has a complex extension and hence $(x(t),\,\tau(t))$, $t\in\mathbb{C}$ satisfies system~(\ref{eqn-2}) on the complex domain. We remark that there are significant new challenges not present in \cite{Nussbaum-analyticity} but in this paper. First, the operator $T$ is not a self mapping on $l_c^{\infty} (\mathbb{C}^{N+1})$ and the range of $H$ is in $l^{\infty} (\mathbb{R}^{N+1})$. This means that the right hand side of system~(\ref{Abs-ODE-S-infty-new}) does not define a vector field on $l_c^{\infty} (\mathbb{C}^{N+1})$ while we are looking for solutions in $l_c^{\infty} (\mathbb{C}^{N+1})$; Secondly, when we transform system~(\ref{Abs-ODE-S-infty-new}) into an integral form and consider the associated fixed point problem on $l^{\infty} (\mathbb{C}^{N+1})$ using the uniform contraction principle in Banach spaces, we can not obtain a contractive mapping on $l^{\infty} (\mathbb{C}^{N+1})$ unless we introduce a small perturbation. The problem is then reduced to show that the solution of the initial value problem associated with system~(\ref{Abs-ODE-S-infty-new}) is the limit of that of the perturbed system.
We organize the remaining part of the paper as follows: in section~\ref{preliminary}, we will develop results on analyticity of $H$ in the right hand side of system~(\ref{Abs-ODE-S-infty-new}) and some basic functional analysis necessary for proving the existence of complex extension of solutions to system~(\ref{Abs-ODE-S-infty-new}), using the the uniform contraction principle in Banach spaces; We present the main results in section~\ref{Main-results} and will illustrate this general result with an example in the last section.
\section{Notations and Preliminary Results}\label{preliminary}
Let $E$ be a complex Banach space, $D$ an open subset of the complex plane $\mathbb{C}$. A continuous mapping $u: D\ni t\rightarrow u(t)\in E $ is called analytic if for every $t\in D$, $
\lim_{t\rightarrow t_0} \frac{u(t)-u(t_0)}{t-t_0}=u'(t_0) $ exists. If $W$ is an open subset of $E$, $\tilde{E}$ is a complex Banach space, a continuous mapping $G: W \ni u\rightarrow G(u)\in \tilde{E}$ is called analytic if for all $u_0\in W$, and for all $h\in E$, the mapping $t\rightarrow G(u_0+th)$ is analytic in the neighbourhood of $0\in \mathbb{C}$.
Let $\mathbb{K}$ stand for the space of real numbers ($\mathbb{R}$) or complex numbers ($\mathbb{C}$). In the following, we develop some basic properties of the map $T$ and the spaces $l_c^{\infty} (\mathbb{K}^{N+1})$ and $l^{\infty} (\mathbb{K}^{N+1})$. We denote by $(v_j)_{j=1}^\infty$ the element $(v_1,\,v_2,\,\cdots,v_j,\cdots) $ in the sequence spaces.
\begin{lemma}\label{Banach-S-space}
Let $c>1$ be a constant and $l_c^{\infty} (\mathbb{K}^{N+1})$ be defined by
\[l_c^{\infty} (\mathbb{K}^{N+1})=\{v=(v_j)_{j=1}^\infty\in l^{\infty}(\mathbb{K}^{N+1}): \sup_{j\in\mathbb{N}} c^j|v_j|<+\infty\}. \] Then $l_c^{\infty} (\mathbb{K}^{N+1})$ is a Banach space under the norm \mbox{$\|\cdot\|_{l_c^{\infty} (\mathbb{K}^{N+1})}$} defined by \[
\|v\|_{l_c^{\infty} (\mathbb{K}^{N+1})}=\sup_{j\in\mathbb{N}} c^j|v_j|. \] \end{lemma}
\begin{lemma}\label{l-m-space}
Let $m\in\mathbb{N}, m\geq 2$ be a constant and $l_m^{\infty} (\mathbb{K}^{N+1})$ be defined by
\[l_m^{\infty} (\mathbb{K}^{N+1})=\{v=(v_j)_{j=1}^\infty\in l^{\infty}(\mathbb{K}^{N+1}): \sup_{j\in\mathbb{N}} j^m|v_j|<+\infty\}. \] Then $l_m^{\infty} (\mathbb{K}^{N+1})$ is a Banach space under the norm \mbox{$\|\cdot\|_{l_m^{\infty} (\mathbb{K}^{N+1})}$} defined by \[
\|v\|_{l_m^{\infty} (\mathbb{K}^{N+1})}=\sup_{j\in\mathbb{N}} j^m|v_j|. \] Moreover, the embedding $I_m: l_m^{\infty} (\mathbb{K}^{N+1})\rightarrow l^{\infty} (\mathbb{K}^{N+1})$ is compact.
\end{lemma} \begin{proof}
It is clear that $l_m^{\infty} (\mathbb{K}^{N+1})$ is a subspace of $l^{\infty}(\mathbb{K}^{N+1})$ and $\|v\|_{l_m^{\infty} (\mathbb{K}^{N+1})}=\sup_{j\in\mathbb{N}} j^m|v_j|$ defines a norm on $l_m^{\infty} (\mathbb{K}^{N+1})$. Let $\{v^n\}_{n=1}^{\infty}$ be a Cauchy sequence in $l_m^{\infty} (\mathbb{K}^{N+1})$. For every $n\in\mathbb{N}$, let $b^n=(v_1^{n},\,2^mv_2^{n},\,\cdots,j^mv_j^{n}, \cdots)$. Then $\{b^n\}_{n=1}^{\infty}$ is a Cauchy sequence in $l^{\infty}(\mathbb{K}^{N+1})$. Since $l^{\infty}(\mathbb{K}^{N+1})$ is a Banach space, there exists $b^*\in l^{\infty}(\mathbb{K}^{N+1})$ so that \[
\lim_{n\rightarrow+\infty}|b^n-b^*|_{l^{\infty}(\mathbb{K}^{N+1})}=0. \] Then we have $v^*=(\frac{b_1^{*}}{1},\,\frac{b_2^{*}}{2^m},\,\cdots,\frac{b_j^{*}}{j^m}\cdots)\in l_m^{\infty} (\mathbb{K}^{N+1})$ and \[
\lim_{n\rightarrow+\infty}|v^n-v^*|_{l_m^{\infty} (\mathbb{K}^{N+1})}=\lim_{n\rightarrow+\infty}|b^n-b^*|_{l^{\infty}(\mathbb{K}^{N+1})}=0.\] Next we show that the embedding $I_m: l_m^{\infty} (\mathbb{K}^{N+1})\rightarrow l^{\infty} (\mathbb{K}^{N+1})$ is compact.
For every $k\in\mathbb{N}$ we define the ``cut-off'' operator $H_k: l_m^{\infty} (\mathbb{K}^{N+1})\rightarrow l^{\infty} (\mathbb{K}^{N+1})$
by \[
H_k(v_1,\,v_2,\,\cdots,\,v_k,\,\cdots)=(v_1,\,v_2,\,\cdots, v_k,\,0,\cdots). \] Then $H_k$ is compact since the dimension of the range is finite. Moreover we have \[
\|(I_m-H_k)(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)\|_{_{l^{\infty} (\mathbb{K}^{N+1})}}=\sup_{j\geq k+1}|v_j|, \]
which implies that $\|I_m-H_k\|\rightarrow 0$ as $k\rightarrow +\infty$ and hence $I_m$ is compact.
\hspace*{1em}\qed\end{proof}
\begin{lemma}\label{Banach-spaces}Let $c>1$ be a constant. The closed unit ball of
$l_c^{\infty} (\mathbb{K}^{N+1})$ is closed under the norm $\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})}$. \end{lemma}
\begin{proof} Let $B_c(1)=\{v\in l_c^{\infty}(\mathbb{K}^{N+1}): \|v\|_{l_c^{\infty}(\mathbb{K}^{N+1})}\leq 1\}.$ Let $\{v^n\}_{n=1}^{+\infty}\subset B_c(1)$ be a Cauchy sequence in the norm $\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})}$. Since $l_c^{\infty} (\mathbb{K}^{N+1})$ is a subspace of the Banach space $(l^{\infty} (\mathbb{K}^{N+1}),\,|\cdot|_{l^{\infty}(\mathbb{K}^{N+1})})$. There exists $v^0\in l^{\infty} (\mathbb{K}^{N+1})$ such that \begin{align}\label{Bs-0}
\lim_{n\rightarrow +\infty}\|v^n-v^0\|_{l^{\infty}(\mathbb{K}^{N+1})}=0. \end{align}Now we show that $v^0\in B_c(1)$. By way of contradiction, assume that $v^0\not\in B_c(1)$. Then we distinguish the following two cases:\\ \textit{Case 1}. $v^0\not\in l_c^{\infty}(\mathbb{K}^{N+1})$.
Then for every $K>0$, there exists $j_0\in\mathbb{N}$ such that $c^{j_0}|(v^0)_{j_0}|>K$. That is, \begin{align}\label{Bs-1}
|(v^0)_{j_0}|>\frac{K}{c^{j_0}}. \end{align} On the other hand, it follows from (\ref{Bs-0}) that for every $\epsilon>0$, there exists $N_0\in\mathbb{N}$ such that for every $n>N_0$, we have $
\sup_{j\in\mathbb{N}}|(v^0)_{j}-(v^n)_{j}|<\epsilon $ which leads to $
|(v^0)_{j}|- |(v^n)_{j}|<\epsilon,\,\mbox{for every $j\in\mathbb{N}$}, n>N_0. $ It follows that \begin{align}\label{Bs-tri-inequality-0}
|(v^n)_{j}|> |(v^0)_{j}|-\epsilon,\,\mbox{for every $j\in\mathbb{N}$}, n>N_0. \end{align} Choosing $j=j_0$ and $\epsilon= \frac{K}{2c^{j_0}}$ in (\ref{Bs-tri-inequality-0}), then by (\ref{Bs-0}) and (\ref{Bs-1}) we obtain that $
|(v^n)_{j_0} | \geq |(v^0)_{j_0}|-\frac{K}{2c^{j_0}} >\frac{K}{2c^{j_0}}, $ which leads to
$ |c^{j_0}(v^n)_{j_0}|>K/2$ for every $n>N_0$. That is, $\lim_{n\rightarrow+\infty}\|v^n\|_{l_c^{\infty}(\mathbb{K}^{N+1})}=+\infty.$
This is a contradiction since $\{v^n\}_{n=1}^{+\infty}\subset B_c(1)$. \\
\textit{Case 2}. $v^0\in l_c^{\infty}(\mathbb{K}^{N+1})$ but $\|v^0\|_{l_c^{\infty}(\mathbb{K}^{N+1})}>1$. Let $s=\|v^0\|_{l_c^{\infty}(\mathbb{K}^{N+1})}$. Then $s>1$ and there exists $j_1\in\mathbb{N}$ such that $c^{j_1}|(v^0)_{j_1}|>1$. That is, \begin{align}\label{Bs-2}
\frac{s}{c^{j_1}}=|(v^0)_{j_1}|>\frac{1}{c^{j_1}}. \end{align} On the other hand, it follows from (\ref{Bs-0}) that for every $\epsilon>0$, there exists $N_1\in\mathbb{N}$ such that for every $n>N_1$, we have $
\sup_{j\in\mathbb{N}}|(v^n)_{j}-(v^0)_{j}|<\epsilon $ which leads to $
|(v^0)_{j}|- |(v^n)_{j}|<\epsilon,\,\mbox{for every $j\in\mathbb{N}$}, n>N_1. $ It follows that \begin{align}\label{Bs-tri-inequality}
|(v^n)_{j}|> |(v^0)_{j}|-\epsilon,\,\mbox{for every $j\in\mathbb{N}$}, n>N_0. \end{align} Note that $\{v^n\}_{n=1}^{+\infty}\subset B_c(1))$. Then by (\ref{Bs-tri-inequality}) we have \begin{align}\label{Bs-tri-inequality-1}
\frac{1}{c^j}\geq |(v^n)_{j}|> |(v^0)_{j}|-\epsilon,\,\mbox{for every $j\in\mathbb{N}$}, n>N_0. \end{align}
Choosing $j=j_1$, $\epsilon=\frac{s-1}{2c^{j_1}}$ in (\ref{Bs-tri-inequality-1}) we obtain from (\ref{Bs-2}) that
\begin{align}\label{Bs-tri-inequality-2}
\frac{1}{c^{j_1}}\geq |(v^n)_{j_1}|> |(v^0)_{j_1}|-\epsilon=\frac{s}{c^{j_1}}-\frac{s-1}{2c^{j_1}},\,\mbox{for every }\, n>N_0. \end{align} Then we have $s<1$. This is a contradiction.
\hspace*{1em}\qed \end{proof}
We remark that the unit sphere of $l_c^{\infty} (\mathbb{K}^{N+1})$ is not closed under the norm $\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})}$.
In light of Lemma~\ref{Banach-spaces} we will equip bounded sets of $l_c^{\infty} (\mathbb{K}^{N+1})$ with the norm $\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})}$. The following three lemmas discuss the properties of a linear operator on $l_c^{\infty} (\mathbb{K}^{N+1})$ equipped with the norm $\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})}$.
\begin{lemma}\label{OMT}Let $c>1$ be a constant.
The mapping $T: (l_c^{\infty} (\mathbb{K}^{N+1},\,\|\cdot\|_{l^{\infty} (\mathbb{K}^{N+1})})\rightarrow (l^{\infty}(\mathbb{K}^{N+1}),\,\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})})$ defined by \[
T(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)=(cv_1,\,c^2v_2,\,\cdots,c^{j}v_j,\cdots), \]
has a compact inverse $T^{-1}$ with norm $\|T^{-1}\|=\frac{1}{c}$. Moreover, $T$ is a closed operator. \end{lemma}
\begin{proof} We first show that $T^{-1}$ exists and is continuous. By definition of $T$ and that $c>1$, we know that $T$ is 1-1 and onto. Therefore $T^{-1}: (l^{\infty} (\mathbb{K}^{N+1},\,\|\cdot\|_{l^{\infty} (\mathbb{K}^{N+1})})\rightarrow (l_c^{\infty}(\mathbb{K}^{N+1}),\,\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})})$ exists and is given by \[
T^{-1}(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)=(c^{-1}v_1,\,c^{-2}v_2,\,\cdots,c^{-j}v_j,\cdots). \] Moreover, we have \begin{align*}
\|T^{-1}\|
= & \sup_{v\in l^{\infty} (\mathbb{K}^{N+1})}\frac{\|T^{-1}v\|_{l^{\infty}(\mathbb{K}^{N+1})}}{\|v\|_{l^{\infty}(\mathbb{K}^{N+1})}}
= \sup_{\|v\|_{l^{\infty}(\mathbb{K}^{N+1})}=1}\|T^{-1}v\|_{l^{\infty}(\mathbb{K}^{N+1})}
= \frac{1}{c}. \end{align*}
Next we show that $T^{-1}$ is compact. For every $m\in\mathbb{N}$ we define an operator $H_m: (l^{\infty} (\mathbb{K}^{N+1},\,\|\cdot\|_{l^{\infty} (\mathbb{K}^{N+1})})\rightarrow (l_c^{\infty}(\mathbb{K}^{N+1}),\,\|\cdot\|_{l^{\infty}(\mathbb{K}^{N+1})})$
by \[
H_m(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)=(c^{-1}v_1,\,c^{-2}v_2,\,\cdots, c^{-m}v_m,\,0,\cdots). \] Then $H_m$ is compact since the dimension of the range is finite. Moreover we have \[
\|(T^{-1}-H_m)(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)\|_{_{l^{\infty} (\mathbb{K}^{N+1})}}=\sup_{j\geq m+1}c^{-j}\|(v_1,\,v_2,\,\cdots,\,v_j,\,\cdots)\|_{_{l^{\infty} (\mathbb{K}^{N+1})}}, \]
which implies that $\|T^{-1}-H_m\|\rightarrow 0$ as $m\rightarrow +\infty$ and hence $T^{-1}$ is compact.
Next we show that $T$ is a closed operator. Let $\{v^n\}_{n=1}^\infty\subset {l_c^{\infty} (\mathbb{K}^{N+1})}$ be a convergent sequence such that $\lim_{n\rightarrow+\infty}\|v^n-v\|_{l^{\infty} (\mathbb{K}^{N+1})}=0$ for some $v\in l^{\infty} (\mathbb{K}^{N+1})$, and such that $\lim_{n\rightarrow+\infty}\|Tv^n-u\|_{l^{\infty} (\mathbb{K}^{N+1})}=0$ for some $u\in l^{\infty} (\mathbb{K}^{N+1})$. Then we have \begin{align*}
\|T^{-1}u-v\|_{l^{\infty} (\mathbb{K}^{N+1})} & =\|T^{-1}u-v^n+v^n-v\|_{l^{\infty} (\mathbb{K}^{N+1})} \\
& =\|T^{-1}u-v^n\|_{l^{\infty} (\mathbb{K}^{N+1})} +\|v^n-v\|_{l^{\infty} (\mathbb{K}^{N+1})} \\
& \leq \|T^{-1}\|\cdot\|u-Tv^n\|_{l^{\infty} (\mathbb{K}^{N+1})} +\|v^n-v\|_{l^{\infty} (\mathbb{K}^{N+1})} \\ &\rightarrow 0 \mbox{ as } n\rightarrow+\infty. \end{align*} Therefore we have $T^{-1}u-v=0$. That is, $Tv=u$. $T$ is closed.
\hspace*{0.05em}\qed \end{proof}
Denote by $\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))$ the space of bounded linear operators from $l^{\infty}(\mathbb{K}^{N+1})$ to $l^{\infty}(\mathbb{K}^{N+1}))$. We have the following two lemmas which will be used when we deal with the integral forms of the relevant abstract ordinary differential equations.
\begin{lemma}\label{Compact-operator} Let the mapping $T: l_c^{\infty} (\mathbb{K}^{N+1})\rightarrow l^{\infty}(\mathbb{K}^{N+1})$ be as in Lemma~\ref{OMT} and $\lambda\geq 0$. Then the mappings $I - T^{-1}$ and $\lambda I+ T^{-1}: l^{\infty}(\mathbb{K}^{N+1})\rightarrow l^{\infty}(\mathbb{K}^{N+1})$ are bounded linear operators with \begin{align*}
\| I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}&=1,\\
\| \lambda I + T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}&=\lambda+\frac{1}{c}. \end{align*}
Moreover, if $\lambda\in (0,\,1-1/c)$ then \[\| (1-\lambda) I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=1-\lambda. \] \end{lemma} \begin{proof}
Let $S(1)=\{v\in l^{\infty}(\mathbb{K}^{N+1}):\sup_{j\in\mathbb{N}}|v_j|=1\}\subset l^{\infty}(\mathbb{K}^{N+1}).$ Note that \begin{align*}
\| I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}(1-c^{-j})|v_j|\\
\leq & \sup_{v\in S(1)} \left(\sup_{j\in\mathbb{N}}|v_j|-\inf_{j\in\mathbb{N}} c^{-j} |v_j|\right)\\ =& 1. \end{align*}Taking $v_0=\{\frac{j}{j+1}\vec{e}\}_{j=1}^{\infty}\in S(1)$ where $\vec{e}$ is a unit vector on the boundary of the unit ball of $\mathbb{K}^{N+1}$, we have \begin{align*}
\| I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}(1-c^{-j})|v_j|\\ \geq & \sup_{v=v_0} \left( \sup_{j\in\mathbb{N}}\frac{j}{j+1}(1-c^{-j}) \right)\\ =& 1.
\end{align*} It follows that $\| I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=1$. Moreover, we have \begin{align*}
\| \lambda I + T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}(\lambda+c^{-j})|v_j|\\
\leq & \sup_{v\in S(1)} \left(\lambda \sup_{j\in\mathbb{N}}|v_j|+\sup_{j\in\mathbb{N}} c^{-j} |v_j|\right)\\ =& \lambda+\frac{1}{c}. \end{align*}Taking $v'_0=\{c^{-(j-1)}\vec{e}\}_{j=1}^{\infty}\in S(1)$, we have \begin{align*}
\|\lambda I + T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}(\lambda+c^{-j})|v_j|\\ \geq & \sup_{v=v'_0} \left( \sup_{j\in\mathbb{N}}c^{-(j-1)}(\lambda+c^{-j}) \right)\\ =& \lambda+\frac{1}{c}.
\end{align*} It follows that $\|\lambda I + T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}= \lambda+\frac{1}{c}$.
Finally, we show that $\| (1-\lambda) I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=1-\lambda.$ Note that we have $1-\lambda-c^{-j}>0$ for all $j\in\mathbb{N}$ since $\lambda\in (0,\,1-1/c)$. Then on the one hand we have \begin{align*}
\| (1-\lambda) I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=& \sup_{v\in B(1)} \sup_{j\in\mathbb{N}}(1-\lambda-c^{-j})|v_j|\\ \leq & \sup_{j\in\mathbb{N}}(1-\lambda-c^{-j})\\ =& 1-\lambda. \end{align*}On the other hand, \begin{align*}
\| (1-\lambda) I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=& \sup_{v\in B(1)} \sup_{j\in\mathbb{N}}(1-\lambda-c^{-j})|v_j|\\
\geq & \sup_{v=v'_0}\sup_{j\in\mathbb{N}}(1-\lambda-c^{-j})|v_j|\\ =& \sup_{j\in\mathbb{N}}(1-\lambda-c^{-j})\frac{j}{j+1}\\ =& 1-\lambda.
\end{align*}It follows that $\|(1-\lambda) I - T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=1-\lambda.$
\qed \end{proof}
\begin{lemma}\label{Extension-operator} Let the mapping $T: l_c^{\infty} (\mathbb{K}^{N+1})\rightarrow l^{\infty}(\mathbb{K}^{N+1})$ be as in Lemma~\ref{OMT}. Then for every $\lambda\geq 0$, the mapping
$(\lambda T + I)^{-1}: l^{\infty}(\mathbb{K}^{N+1})\rightarrow l_c^{\infty}(\mathbb{K}^{N+1})\subset l^{\infty}(\mathbb{K}^{N+1})$ is continuous with norm \begin{align*}
\| (\lambda T + I)^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}&=\frac{1}{c\lambda+1}. \end{align*} \end{lemma}
\begin{proof} We compute $\|(\lambda T + I)^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}$. Let $S(1)=\{v\in l^{\infty}(\mathbb{K}^{N+1}):\sup_{j\in\mathbb{N}}|v_j|=1\}\subset l^{\infty}(\mathbb{K}^{N+1}).$ Note that \begin{align*}
\|(\lambda T + I)^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}\frac{|v_j|}{\lambda {c^j}+1}\\
=& \sup_{v\in S(1)} \sup_{j\in\mathbb{N}}\frac{|v_j|}{\lambda c^j +1}\\ \leq & \sup_{j\in\mathbb{N}} \frac{1}{\lambda c^j +1}\\ =& \frac{1}{\lambda c +1}. \end{align*} Taking $v_0=\{c^{-(j-1)}\vec{e}\}_{j=1}^{\infty}\in B_c(1)$, we have
\begin{align*}\|(\lambda T + I)^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}=&\sup_{v\in S(1)} \sup_{j\in\mathbb{N}}\frac{|v_j|}{\lambda {c^j}+1}\\
\geq & \sup_{v=v_0} \sup_{j\in\mathbb{N}}\frac{1}{\lambda {c^j}+1}|v_j|\\ \geq & \sup_{j\in\mathbb{N}}\frac{c^{-(j-1)}}{\lambda {c^j}+1} \\ =& \frac{1}{\lambda c +1}.
\end{align*} It follows that $\|(\lambda T + I)^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1});\, l^{\infty}(\mathbb{K}^{N+1}))}= \frac{1}{\lambda c +1}$.
\hspace*{1em}\qed \end{proof}
The following three lemmas address the well-posedness of system~(\ref{Abs-ODE-S-infty-new}) and the analyticity of the map $H$.
\begin{lemma}\label{Lemma-l-infty}Assume $\textrm{(A1)-(A2)}$. For every sequence $\{(u_i,\,v_i)\}_{i=0}^{+\infty}\subset U\times V$, let $\mu_i=(u_i,\,u_{i+1},\,\cdots,\,u_{i+M-1},\,v_i)\in U^M\times V$. Then we have \begin{align*}
\lim_{j\rightarrow+\infty} \frac{1}{c^j}\prod_{i=0}^{j-1}|1-g(\mu_i)|=0, \end{align*} Moreover, for every $m\in\mathbb{N}$, we have \[
\lim_{j\rightarrow+\infty} \frac{j^m}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|=0. \] \end{lemma}
\begin{proof} By (A2), we have $|1-g(\gamma_1,\,\gamma_2)|<c$ and $|1-g(\gamma_1,\,\gamma_2)|$ with $(\gamma_1,\,\gamma_2)\in \overline{U}^M\times \overline{V}$ has a supremum less than $c$. Let $s>0$ be such that $c=e^s$. Then there exists $N_0\geq 1$, $N_0\in\mathbb{N}$ so that $|1-g(\mu_i)|\leq e^{s(1-\frac{1}{N_0})}$ for all $i\in\mathbb{N}$. Then for every $n\in\mathbb{N}$ we have
\[|1-g(\mu_i)|\leq e^{s(1-\frac{1}{N_0})}\leq e^{s(1-\frac{n}{i})}\textrm { for all } i\geq nN_0. \]
It follows that $\ln \left( \frac{|1-g(\mu_i)|}{c}\right)\leq -\frac{ns}{i} $ for all $ i\geq nN_0$. Then for $j> nN_0$ we have \begin{align}
\sum_{i=0}^{j-1}\ln \left(\frac{|1-g(\mu_i )|}{c}\right)& =\sum_{i=0}^{nN_0-1}\ln \left(\frac{|1-g(\mu_i)|}{c}\right)+\sum_{i=nN_0}^{j-1}\ln \left(\frac{|1-g(\mu_i)|}{c}\right)\notag\\
& \leq \sum_{i=0}^{nN_0-1}\ln \left(\frac{|1-g(\mu_i)|}{c}\right)+s\sum_{i=nN_0}^{j-1} \left(-\frac{n}{i}\right).\label{analyticity-infty-new} \end{align}
Let $c_0=\sum_{i=0}^{nN_0-1}\ln \left(\frac{|1-g(\mu_i)|}{c}\right)$. Then by (\ref{analyticity-infty-new}) and (A2), we have \begin{align}
0<\frac{1}{c^j}\prod_{i=0}^{j-1}|1-g(\mu_i)|& = \exp{\sum\limits_{i=0}^{j-1}\ln \left(\frac{|1-g(\mu_i)|}{c}\right)}\notag\\ & \leq e^{c_0}\exp{\left(s\sum\limits_{i=nN_0}^{j-1} -\frac{n}{i}\right)}.\label{analyticity-infty-inequality} \end{align} Taking limits as $j\rightarrow +\infty$ in (\ref{analyticity-infty-inequality}) we have \[
\lim_{j\rightarrow+\infty} \frac{1}{c^j}\prod_{i=0}^{j-1}|1-g(\mu_i)|=0. \] Choosing $n=m$ in the inequality (\ref{analyticity-infty-inequality}), we have \begin{align}
0< \frac{j^m}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|& \leq j^m e^{c_0}\exp{\left(s\sum\limits_{i=mN_0}^{j-1} -\frac{m}{i} \right)}\notag\\ & = \exp\left(c_0+\sum\limits_{i=1}^{mN_0-1}\left(\frac{m}{i}\right)+\frac{m}{j}\right) \frac{j^m}{\exp{(mH_j)}}\notag\\ & = \exp\left(c_0+\sum\limits_{i=1}^{mN_0-1}\left(\frac{m}{i}\right)+\frac{m}{j}\right)\exp(m\ln j-mH_j),\label{analyticity-infty-inequality-2} \end{align} where $H_j=1+\frac{1}{2}+\cdots+\frac{1}{j}$ and $\sum_{i=1}^{mN_0-1}\left(\frac{m}{i}\right)$ is regarded 0 if $mN_0=1$. We note that $\lim_{j\rightarrow+\infty}\ln j-H_j=-\gamma$ where $\gamma>0$ is the Euler-M\'{a}scheroni constant. Taking supremum limits as $j\rightarrow +\infty$ in (\ref{analyticity-infty-inequality-2}) we have \[
0<\limsup_{j\rightarrow+\infty}\frac{j^m}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|\leq \exp\left(c_0+\sum\limits_{i=1}^{mN_0-1}\left(\frac{m}{i}\right)-m\gamma\right)<+\infty. \]Then we have \begin{align*}
& \lim_{j\rightarrow+\infty}\frac{j^{m-1}}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|\\
\leq& \limsup_{j\rightarrow+\infty}\frac{j^m}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|\lim_{j\rightarrow+\infty}\frac{1}{j}\\
= &\,0. \end{align*}
Since $m\in\mathbb{N}$ is arbitrary, it follows that $ \lim_{j\rightarrow+\infty} \frac{j^m}{c^j} \prod_{i=0}^{j-1}|1-g(\mu_i)|=0$ for all $m\in\mathbb{N}$.
\qed \end{proof}
Let $ l^{\infty}(U\times V)$ be the subset of $l^{\infty}(\mathbb{K}^{N+1})$ defined by
\[
l^{\infty}(U\times V)=\prod_{j=0}^\infty (U\times V).
\] Note that $ l^{\infty}(U\times V)$ is not an open set of $l^{\infty}(\mathbb{K}^{N+1})$ if $l^{\infty}(\mathbb{K}^{N+1})$ is equipped with the product topology. However, we are concerned with the following set: \begin{align}\label{set-A} A=\{w=(w_0,\,w_1,\,\cdots)\in l^{\infty}(U\times V): \mbox{$\{w_j\}_{j=0}^\infty\subset Q_0$ for some compact $Q_0\subset U\times V$} \}. \end{align} For every $w=(w_0,\,w_1,\,\cdots)\in A$, we can find an open set $P$ and a compact set $Q$ such that $\{w_j\}_{j=0}^\infty\subset P\subset Q\subset U\times V$. Then $w\in l^\infty(P)\subset A\subset l^{\infty}(U\times V)$. Namely, $A$ is open under the box topology.
We also define the projections $\chi_i: l^{\infty}(U\times V)\rightarrow U^M\times V$ with $i\in\{0,\,1,\,2,\,\cdots\}$ by
\begin{align}\label{chi}
\chi_i (w)=(u_i,\,u_{i+1},\,\cdots,\,u_{i+M-1},\,v_i)
\end{align}for every $w=((u_i,\,v_i))_{j=1}^\infty\in l^{\infty}(U\times V).$ \begin{lemma}\label{Lemma-G}Let $A$ be defined at (\ref{set-A}). Assume $($\textrm{A1 -- A2}$\,)$.
The mapping $G$ defined by \[
G: A\ni w=(w_0,\,w_1,\,w_2,\,\cdots,\,w_i,\,\cdots)\rightarrow G(w)=\left(\frac{1}{c^j}\prod_{i=0}^{j-1}(1-g(\chi_i (w)))\right)_{j=1}^{+\infty}, \]where $w_i=(u_i,\,v_i)\in U\times V$, is continuous and analytic from $A\subset l^{\infty}(U\times V)$ to $ l^{\infty}(\mathbb{C}^{N+1})$. \end{lemma} \begin{proof}
By Lemma~\ref{Lemma-l-infty}, we know that $G$ is a mapping from $l^{\infty}(\mathbb{C}^{N+1})$ to $ l^{\infty}(\mathbb{C}^{N+1})$. Note that for every $i,\,j\in\mathbb{N}$ with $0\leq i\leq j-1$, we have \begin{align}\label{product-derivative}
\frac{\partial }{\partial \mu_i}\prod_{i=0}^{j-1}(1-g(\mu_i ))&=
\frac{-\frac{\partial }{\partial \mu_i}g(\mu_i)}{(1-g(\mu_i))}\prod_{i=0}^{j-1}(1-g(\mu_i )).
\end{align} Let $(\mu_0,\,\mu_1,\,\cdots,\,\mu_{j-1})$ denote a column vector in $\bigoplus\limits_{i=0}^{j-1}\mathbb{C}^{MN+1}$. Then we have \begin{align*}
&\frac{\partial }{\partial (\mu_0,\,\mu_1,\,\cdots,\,\mu_{j-1})}\prod_{i=0}^{j-1}(1-g(\mu_i ))\\ =&\left(\prod_{i=0}^{j-1}(1-g(\mu_i))\right) \left(\frac{-\frac{\partial }{\partial \mu_0} g(\mu_0)}{(1-g(\mu_0))},\,\frac{-\frac{\partial }{\partial \mu_1} g(\mu_1)}{(1-g(\mu_1))},\,\cdots,\,\frac{-\frac{\partial }{\partial \mu_{j-1}} g(\mu_{j-1})}{(1-g(\mu_{j-1}))}\right), \end{align*} which is also regarded as a column vector in $\bigoplus\limits_{i=0}^{j-1}\mathbb{C}^{MN+1}$.
For every $\epsilon>0$, choose $\delta=\epsilon$, for every $w_1=({w_1}_i),\,w_2=({w_2}_i)\in A$ with $|w_1-w_2|_{l^{\infty}(\mathbb{C}^{N+1})}<\delta$, by (\ref{product-derivative}) and the Integral Mean Value Theorem, we have \begin{align*}
|G(w_1)-G(w_2)|_{l^{\infty}(\mathbb{C}^{N+1})} &=\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\prod_{i=0}^{j-1}(1-g(\chi_i(w_1) ))-\prod_{i=0}^{j-1}(1-g(\chi_i(w_2)))\right|\\
&\leq \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\bar{\chi}_i ))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\bar{\chi}_i )}{(1-g(\bar{\chi}_i ))}\left(\chi_i(w_1)-\chi_i(w_2)\right)\right|, \end{align*} where $\bar{\chi}_i=\chi_i(w_1)+\theta (\chi_i(w_1)-\chi_i(w_2))$ for some $\theta\in [0,\,1]$. By (A2) we have
$l<|1-g(\bar{\chi}_i )|<c$. By (A1), there exists $M_0>0$ so that $ |\frac{\partial }{\partial \chi_i} g(\bar{\chi}_i )|<M_0$. By Lemma~\ref{Lemma-l-infty}, there exists $M_1>0$ so that $\sup_{j\in\mathbb{N}}\frac{j}{c^j} \prod_{i=0}^{j-1}|1-g(\bar{\chi}_i )| <M_1$. It follows that \begin{align}\label{G-continuity}
|G(w_1)-G(w_2)|_{l^{\infty}(\mathbb{C}^{N+1})} &\leq \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left(\prod_{i=0}^{j-1}|1-g(\bar{\chi}_i )|\right)
\sum_{i=0}^{j-1}\frac{M_0}{l}\left| (\chi_i(w_1)-\chi_i(w_2))\right|\notag\\
&\leq \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left(\prod_{i=0}^{j-1}|1-g(\bar{\chi}_i )|\right)
\frac{jM_0}{l}|w_1-w_2|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\
&= \sup_{j\in\mathbb{N}}\frac{j}{c^j}\left(\prod_{i=0}^{j-1}|1-g(\bar{\chi}_i )|\right)
\frac{M_0}{l}|w_1-w_2|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\ &= \frac{M_0M_1}{l} \epsilon, \end{align}which implies that $G$ is continuous. Next, we show that for every $w=(w_i)\in A\subset l^{\infty}(U\times V)$, and for all $h=(h_i)\in l^{\infty}(\mathbb{C}^{N+1})$, the mapping $\mathscr{G}:t\rightarrow G(w+th)$ is analytic in the neighborhood of $0\in \mathbb{C}$. Denote by $\bar{G}h$ the sequence \[ \left(\frac{1}{c^j} \left(\prod_{i=0}^{j-1}(1-g(\chi_i(w)))\right) \sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))}{(1-g(\chi_i(w) ))}\chi_i(h)\right)_{j=1}^{\infty}.\] Then by the same argument leading to (\ref{G-continuity}), we know that $\bar{G}h\in l^{\infty}(\mathbb{C}^{N+1})$ and \begin{align}\label{analyticity-G}
&\left|\frac{G(w+th)-G(w)}{t}-\bar{G}h\right|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\
= & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\frac{1}{t}\left(\prod_{i=0}^{j-1}(1-g(\chi_i(w+th) ))-\prod_{i=0}^{j-1}(1-g(\chi_i(w)))\right)\right.\notag\\
& \left.-\left(\prod_{i=0}^{j-1}(1-g(\chi_i(w)))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w) )\chi_i(h)}{ 1-g(\chi_i(w) ) }\right|\notag\\
= & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right) \sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{1-g(\tilde{\chi}_i )}\right.\notag\\ & \left.-\left(\prod_{i=0}^{j-1}(1-g(\chi_i(w) ))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(h)}{1-g(\chi_i(w) )}\right|\notag\\
\leq & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))-\prod_{i=0}^{j-1}(1-g({\chi}_i(w)))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{1-g(\tilde{\chi}_i)}\right|\notag\\
& +\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\chi_i(w)))\right) \left(\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial\chi_i} g(\tilde{\chi}_i)\chi_i(h)}{1-g(\tilde{\chi}_i)}-
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial u_i} g(\chi_i(w))\chi_i(h)}{1-g(\chi_i(w))}\right)\right|, \end{align} where $\tilde{\chi}_i=\chi_i(w+t\theta\, h)$ for some $\theta\in [0,\,1]$. By applying the same argument leading to (\ref{G-continuity}) on the first term of the last inequality of (\ref {analyticity-G}) and by Lemma~\ref{Lemma-l-infty} we have \begin{align}\label{Analyticity-G-1}
& \lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))-\prod_{i=0}^{j-1}(1-g(\chi_i(w) ))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{1-g(\chi_i(w) )}\right|\notag\\
\leq & \lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))-\prod_{i=0}^{j-1}(1-g(\chi_i(w)))\right)
\right|\frac{jM_0}{l}|h|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\
\leq & \lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\tilde{\chi}}_i ))\right)
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\tilde{\chi}}_i )}{(1-g(\tilde{\tilde{\chi}}_i ))}\theta \, t \chi_i(h)\right|\frac{jM_0}{l}|h|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\
= & \lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left| \prod_{i=0}^{j-1}(1-g(\tilde{\tilde{\chi}}_i ))
\right|\frac{j^2M_0^2}{l^2}|h|_{l^{\infty}(\mathbb{C}^{N+1})}^2\cdot |t|\notag\\
= & \lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{j^2}{c^j}\left| \prod_{i=0}^{j-1}(1-g(\tilde{\tilde{\chi}}_i ))
\right|\frac{M_0^2}{l^2}|h|_{l^{\infty}(\mathbb{C}^{N+1})}^2\cdot |t|\notag\\ = & \,0. \end{align}
where $\tilde{\tilde{\chi}}_i=\chi_i(w+ t\theta\theta'h)$ for some $\theta'\in [0,\,1].$ By (A1), there exists $M_2>0$ so that $ |\frac{\partial^2 }{\partial \chi_i^2} g({\chi}_i(w) )|<M_2$ for every $w\in A$ and $i\in\mathbb{N}$. Then it follows from the Integral Mean Value Theorem that the second term of the last inequality of (\ref {analyticity-G}) satisfies that \begin{align*}
& \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i))\right) \left(\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{(1-g(\tilde{\chi}_i ))}-
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(h)}{(1-g(\chi_i(w) ))}\right)\right| \notag\\
= & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right)\left[ \left(\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{(1-g(\tilde{\chi}_i ))}- \sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(h)}{(1-g(\tilde{\chi}_i ))} \right)\right.\right. \notag\\ &\left.\left. +\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(w)}{(1-g(\tilde{\chi}_i ))}-
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(w)}{(1-g(\chi_i(w) ))}\right] \right|\notag\\
\leq & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right)\left[
\frac{jM_2t}{l}|h|_{l^{\infty}(\mathbb{C}^{N+1})}^2\right.\right. \notag\\ &\left.\left. +\left(-\frac{\partial }{\partial \chi_i} g(\chi_i(w) )\chi_i(h)\right)\sum_{i=0}^{j-1}\left(\frac{1}{(1-g(\tilde{\chi}_i ))}-
\frac{1}{(1-g(\chi_i(w)))}\right)\right] \right|\notag\\
= & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right)\left[
\frac{jM_2t}{l}|h|_{l^{\infty}(\mathbb{C}^{N+1})}^2\right.\right. \notag\\
&\left.\left. +M_0|h|_{l^{\infty}(\mathbb{C}^{N+1})}\sum_{i=0}^{j-1}\left(\frac{g(\tilde{\chi}_i )-g(\chi_i(w))}{(1-g(\tilde{\chi}_i ))(1-g(\chi_i(w)))}
\right)\right] \right|\notag\\
\leq & \sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right)\left[
\frac{jM_2 |h|_{l^{\infty}(\mathbb{C}^{N+1})}^2t}{l} +\frac{jM_0^2|h|_{l^{\infty}(\mathbb{C}^{N+1})}^2t}{l^2} \right] \right|. \end{align*} Then by Lemma~\ref{Lemma-l-infty} we have \begin{align}\label{Analyticity-G-2}
\lim_{t\rightarrow 0}\sup_{j\in\mathbb{N}}\frac{1}{c^j}\left|\left(\prod_{i=0}^{j-1}(1-g(\tilde{\chi}_i ))\right) \left(\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\tilde{\chi}_i )\chi_i(h)}{(1-g(\tilde{\chi}_i ))}-
\sum_{i=0}^{j-1}\frac{-\frac{\partial }{\partial \chi_i} g(\chi_i(w))\chi_i(h)}{(1-g(\chi_i(w) ))}\right)\right|=0. \end{align} By (\ref{analyticity-G}), (\ref{Analyticity-G-1}) and (\ref{Analyticity-G-2}) we have \[
\lim_{t\rightarrow 0}\left|\frac{G(w+th)-G(w)}{t}-\bar{G}h\right|_{l^{\infty}(\mathbb{C}^{N+1})}=0. \]
\hspace*{1em}\qed \end{proof}
\begin{lemma}\label{complete-continuity}Assume $($\textrm{A1 -- A2}$\,)$. Let the set $A$ and the map $G$ be as in Lemma~\ref{Lemma-G}.
Define $H: A\subset l^{\infty} (U\times V)\rightarrow l^{\infty} (\mathbb{K}^{N+1})$ by
\[
H(\theta)=(F_j(\theta)G_j(\theta))_{j=1}^{\infty}\in l^{\infty} (\mathbb{K}^{N+1}),
\]
where \begin{align*} \theta = & (\theta_1,\,\theta_2,\,\cdots,\,\theta_j,\,\cdots)=((u_1,\,v_1),\,(u_2,\,v_2),\,\cdots,\,(u_j,\,v_j),\,\cdots)\in l^{\infty} (\mathbb{K}^{N+1}),\\ F_j(\theta)= & \,\left(\frac{f(u_j,\, u_{j+1} )}{1-g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}, \frac{g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}{1-g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}\right), \end{align*} for $j\geq 1,\,j\in\mathbb{N}$. Then $H: \bar{A}\rightarrow l^{\infty} (\mathbb{K}^{N+1})$ is completely continuous and is analytic. \end{lemma} \begin{proof} According to Lemma~\ref{l-m-space}, we only need to show that for every bounded set $B\subset l^{\infty} (\mathbb{K}^{N+1})$, $H(B)$ is bounded in $ l_m^{\infty} (\mathbb{K}^{N+1})$. By (A1)-(A2), we know that $F(B)$ is bounded in $ l^{\infty} (\mathbb{K}^{N+1})$. Then by Lemma~\ref{Lemma-l-infty}, we have \begin{align*}
\lim_{j\rightarrow+\infty}{j^m} \left|F_j(\theta)G_j(\theta)\right|=0. \end{align*}Therefore we have $H(\theta)\in l_m^{\infty} (\mathbb{K}^{N+1})$. By Lemma~\ref{l-m-space}, $H$ is completely continuous. Then by (A1)--(A2), $F: l^{\infty} (\mathbb{C}^{N+1})\ni \theta\rightarrow F(\theta)\in l^{\infty} (\mathbb{C}^{N+1})$ is analytic. Then by the similar procedure for the proof of Lemma~\ref{Lemma-G}, we can show that for every $w=(w_i)\in A\subset l^{\infty}(U\times V)$, and for all $h=(h_i)\in l^{\infty}(\mathbb{C}^{N+1})$, the mapping $\mathscr{G}:t\rightarrow H(w+th)$ is analytic in the neighborhood of $0\in \mathbb{C}$.
\hspace*{1em}\qed \end{proof}
\section{Main Results}\label{Main-results}
Let $\Omega$ be a bounded closed ball in $\mathbb{K}$. We denote by $C(\Omega; l^{\infty}(\mathbb{K}^{N+1}))$ the space of continuous functions $u: \Omega\ni t\rightarrow u(t)\in l^{\infty}(\mathbb{K}^{N+1})$ and denote by $C^1(\Omega; l^{\infty}(\mathbb{K}^{N+1}))$ the space of continuously differentiable functions $u: \Omega\ni t\rightarrow u(t)\in l^{\infty}(\mathbb{K}^{N+1})$. Then it is clear that $C(\Omega; l^{\infty}(\mathbb{K}^{N+1}))$
and $C^1(\Omega; l^{\infty}(\mathbb{K}^{N+1}))$ are Banach spaces equipped, respectively, with the norms
$\|u\| =\max_{t\in \Omega }|u(t)|_{l^{\infty}(\mathbb{K}^{N+1}))}$ and
\[\|u\| =\max\{\max_{t\in \Omega }|u(t)|_{l^{\infty}(\mathbb{K}^{N+1})},\,\max_{t\in \Omega }|u'(t)|_{ {l}^{\infty}(\mathbb{K}^{N+1})}\}. \]
\begin{theorem}\label{Analyticity-th}Assume $($\textrm{A1 -- A2}$\,)$. Let $(x,\,\tau)\in\mathbb{R}^{N+1}$ be a bounded solution of system (\ref{eqn-2}). Suppose that there exists a compact set $Q\subset U\times V$ such that $(x(t),\,\tau(t))\in Q$ for all $t\in\mathbb{R}$. Then $(x,\,\tau)$ is analytic on $\mathbb{R}$.
\end{theorem}
\begin{proof} We define $\left((y_j,\,z_j)\right)_{j=1}^\infty\in C(\mathbb{R}; l_c^{\infty}(\mathbb{R}^{N+1}))$ by \begin{align*}(y_j(t),\, z_j(t))=\left(\frac{1}{c^{j}}x(\eta^{j-1}(t)),\,\frac{1}{c^{j}}\tau(\eta^{j-1}(t))\right) \mbox{ for $j\geq 1,\,j\in\mathbb{N},\,t\in\mathbb{R}$.} \end{align*} Then by the derivation in Section~\ref{SOPS-4-1}, for every $t\in\mathbb{R}$, $((y_j(t),\,z_j(t)))_{j=1}^{\infty}\in l_c^{\infty}(\mathbb{R}^{N+1})$ satisfies system~(\ref{New-Eqn-1}).
Let \begin{align}\label{map-F}
F(\theta)=(F_1(\theta),\,F_2(\theta),\,\cdots,\,F_j(\theta),\,\cdots), \end{align} where \begin{align*} \theta = & (\theta_1,\,\theta_2,\,\cdots,\,\theta_j,\,\cdots)=((u_1,\,v_1),\,(u_2,\,v_2),\,\cdots,\,(u_j,\,v_j),\,\cdots)\in l^{\infty} (\mathbb{R}^{N+1}),\\ F_j(\theta)= & \,\left(\frac{f(u_j,\, u_{j+1} )}{1-g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}, \frac{g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}{1-g(u_j,\,u_{j+1},\,\cdots,u_{j+M-1}, \,v_j )}\right), \end{align*} for $j\geq 1,\,j\in\mathbb{N}$.
Let $T$ be as in Lemma~\ref{OMT}, $G$ in Lemma~\ref{Lemma-G}. Then $w=\left((y_1,\,z_1),\, (y_2,\,z_2),\,\cdots)\right)\in C(\mathbb{R};l_c^{\infty} (\mathbb{R}^{N+1}))$ is a solution of the following ordinary differential equation \begin{align}\label{Abs-ODE-S-infty}
\frac{d}{dt}w(t)= H(Tw(t)), \end{align} where $H(T(w))=(F_1(T(w))G_1(T(w)),\,F_2(T(w))G_2(T(w)),\,\cdots)\in l^{\infty} (\mathbb{R}^{N+1})$ and $G_j$ is the $j$-th coordinate of $G$. Moreover, we notice that for every $j\in\mathbb{N}$, \[ \{(c^jy_j(t),\,c^jz_j(t)): t\in\mathbb{R}\}=\{(x(t),\,y(t)):t\in\mathbb{R}\}\subset Q. \] Then we have $\{(c^jy_j(t),\,c^jz_j(t))_{j=1}^\infty: t\in\mathbb{R}\}\subset Q$ and \[ Tw(t)=(c^jy_j(t),\,c^jz_j(t))_{j=1}^\infty\in A \]for every $t\in\mathbb{R}$, where $A$ is defined by (\ref{set-A}).
Let $w_{t_0}=\left((y_j(t_0),\,z_j(t_0))\right)_{j=1}^\infty\in l_c^{\infty} (\mathbb{R}^{N+1})$, $t_0\in\mathbb{R}$. Then $w(t)$ is a solution of the following initial value problem \begin{align}\label{Abs-ODE-IVP}\left\{ \begin{aligned}
\frac{d}{dt}w(t)& = H(Tw(t)), \\ w(t_0)&= w_{t_0}. \end{aligned} \right. \end{align} To prove the existence of complex extension of $w(t)\in l_c^{\infty} (\mathbb{R}^{N+1})$,
we put $\nu(t)=T w(t)\in l^{\infty} (\mathbb{C}^{N+1})$ and consider equation (\ref{Abs-ODE-IVP}) in $l^{\infty} (\mathbb{C}^{N+1})$. Then equation (\ref{Abs-ODE-IVP}) is transformed into the following integral equation \begin{align}\label{Abs-Integral-Eq}
T^{-1}{\nu}(t)= w_{t_0}+\int_{t_0}^tH(\nu(s)) ds, \end{align}where the integral is taken along the linear path $\xi\rightarrow t_0+\xi(t-t_0)$, $0\leq \xi\leq 1$.
Denote by $\Omega_h=\{t\in\mathbb{C}: |t-t_0|\leq h\}$ for some $h>0$. To prove the existence and uniqueness of the solution using the Uniform Contraction Principle, we consider fixed point problem associated with the following mapping \begin{align}\label{L0-lambda-operator}
L_0(\nu)(t)=(I-T^{-1})\nu(t) + w_{t_0}+\int_{t_0}^tH(\nu(s)) ds, \end{align} on $C(\Omega_h; A)$. However, by Lemma~\ref{Compact-operator}, $L_0$ is not contractive on $C(\Omega_h; l^\infty(\mathbb{C}^{N+1}))$ in general, but its perturbation
$L: C(\Omega_h; l^\infty(\mathbb{C}^{N+1}))\times [0,\,1] \rightarrow C(\Omega_h; l^\infty(\mathbb{C}^{N+1}))$ defined by \begin{align}\label{L-lambda-operator}
L(\nu,\,\lambda)(t)=((1-\lambda)I-T^{-1})\nu(t) + (T^{-1}+\lambda I)\nu_{t_0}+\int_{t_0}^tH(\nu(s)) ds, \end{align} is contractive on $C(\Omega_h; l^\infty(\mathbb{C}^{N+1}))$ for $\lambda\in (0,\,1-1/c)$ and some $h>0$, where $\nu_{t_0}=Tw_{t_0}$.
If there exists a $\nu\in C(\Omega_h; A)$ such that $L(\nu)=\nu$, then $\nu$ is a solution of the following initial value problem: \begin{align}\label{Abs-ODE-IVP-lambda}\left\{ \begin{aligned}
(\lambda I+T^{-1})\frac{d}{dt}\nu(t)& = H(\nu(t)), \\ \nu(t_0)&=T w_{t_0}. \end{aligned} \right. \end{align} Writing (\ref{Abs-ODE-IVP-lambda}) in integral form, we have
\begin{align}\label{Abs-ODE-IVP-integral-lambda} \begin{aligned} T^{-1}\nu (t)& = w_{t_0}+\int_{t_0}^t(\lambda T+ I)^{-1} H\left(\nu (s)\right)ds. \end{aligned} \end{align}
By Lemma~\ref{complete-continuity}, $H$ is analytic and $L$ is an analytic mapping from $A\subset l^{\infty} (\mathbb{C}^{N+1})$ to $ l^{\infty} (\mathbb{C}^{N+1})$.
We organize the remaining part of the proof as follows: We first show with claims 1 and 2 the existence and uniqueness of solutions $\nu_\lambda$ of system~(\ref{Abs-ODE-IVP-integral-lambda}) and with claim 3 $\nu_\lambda$ satisfies that $\lim_{\lambda\rightarrow 0^+}T^{-1}\nu_\lambda =w_0\in C(\Omega_{h_0}; T^{-1}(\bar{A}))$ for some $h_0>0$. Secondly, we show with claim 4 that the right hand side of system~(\ref{Abs-ODE-IVP-integral-lambda}) is coordinate-wise convergent to that of system~(\ref{Abs-Integral-Eq}) with
$\lim_{n\rightarrow+\infty}H(\nu_{\lambda_n}(t))=H(\nu_0(t))$, where $\nu_0: \Omega_{h_0}\rightarrow\bar{A}$ is a map such that $H(\nu_0)$ is continuous in $t\in \Omega_{h_0}$ and the sequence $\{\lambda_n\}_{n=1}^{\infty}$ is in $(0,\,1-\frac{1}{c})$ with $\lim_{n\rightarrow+\infty}\lambda_n=0$. Lastly, we show with claim 5 that $H(Tw_0)=H(\nu_0)$ which implies that $w_0$ satisfies system~(\ref{Abs-Integral-Eq}) and hence it is the solution of the initial value problem (\ref{Abs-ODE-S-infty}).
Now we show the following
\textbf{Claim 1}: For every $\lambda\in (0,\,1)$, there exists $h>0$ such that there exists one and only one point $\nu_\lambda\in C(\Omega_h; \bar{A})$ such that $L(\nu_\lambda,\,\lambda)=\nu_\lambda$ and $\nu_\lambda$ is analytic and is differentiable with respect to $\lambda$.
{\textit{Proof of Claim 1:}} We only need to show that $L$ is a contractive mapping in some closed neighborhood of $w_{t_0}$ in $C(\Omega_h; l^{\infty}(\mathbb{C}^{N+1}))$ where $\Omega_h=\{t\in\mathbb{C}: |t-t_0|\leq h\}$ for some $h>0$ to be determined. Denote by $\|\cdot\|_{C}$ the supremum norm on the Banach space ${C(\Omega_h; l^{\infty}(\mathbb{C}^{N+1}))}$. For every $w_1,\,w_2\in C(\Omega_h; l^{\infty}(\mathbb{C}^{N+1}))$ we have \begin{align}\label{eqn-condensing}
&\|L(w_1,\,\lambda)-L(w_2,\,\lambda)\|_{C}\notag\\
=& \max_{t\in \Omega_h} \left\|((1-\lambda)I-T^{-1})w_1(t)+\int_{t_0}^tH(w_1(s)) ds\right)\notag\\
& \left.-((1-\lambda)I-T^{-1})w_2(t)-\int_{t_0}^tH(w_2(s)) ds \right\|_{l^{\infty}(\mathbb{C}^{N+1})}\notag\\
\leq & \max_{t\in \Omega_h} \|(1-\lambda) I -T^{-1}\|_{\mathscr{L}(l^{\infty}(\mathbb{K}^{N+1};\, l^{\infty}(\mathbb{K}^{N+1})} |w_1(t)-w_2(t)|_{l^{\infty}(\mathbb{C}^{N+1})}\notag \\
& +\max_{t\in \Omega_h}\int_{t_0}^t \left\|H(w_1(s))-H(w_2(s))\right \|_{l^{\infty}(\mathbb{C}^{N+1})} ds. \end{align} Since $H$ is analytic on $A$, there exist constants $\delta>0$ and $l_0>0$ so that
$|H(\nu_1 )-H(\nu_2) |_{l^{\infty}(\mathbb{C}^{N+1})}\leq l_0 |\nu_1-\nu_2|_{l^{\infty}(\mathbb{C}^{N+1})}$ for every $\nu_1,\,\nu_2\in A$ with $|\nu_1-\nu({t_0})|_{l^{\infty}(\mathbb{C}^{N+1})}\leq \delta, |\nu_2-\nu(t_0)|_{l^{\infty}(\mathbb{C}^{N+1})}\leq \delta$.
Let $X=\{\nu\in C(\Omega_h; \bar{A}): \max_{t\in\Omega_h}|\nu(t)-\nu({t_0})|_{l^{\infty}(\mathbb{C}^{N+1})}\leq \delta\}$. Then $X$ is a closed subset of the Banach space $C(\Omega_h; l^{\infty}(\mathbb{C}^{N+1})$. By (\ref{eqn-condensing}) and by Lemma~\ref{Compact-operator}, we have \begin{align*}
\|L(w_1,\,\lambda)-L(w_2,\,\lambda)\|_{C}\leq & (1-\lambda) \|w_1-w_2\|_{C}+ l_0 h \|w_1-w_2\|_{C}\\ = & (1-\lambda+l_0 h) \|w_1-w_2\|_{C}, \end{align*} for every $w_1,\,w_2\in X$. Therefore, if $h\in (0,\,\frac{\lambda}{l_0})$, then $1-\lambda+l_0 h\in (0,\,1)$. Moreover, we choose $h>0$
small enough so that \begin{align*}
& \max_{t\in\Omega_h}\|L(\nu(t),\,\lambda)-\nu({t_0})\|_{l^{\infty}(\mathbb{C}^{N+1})}\\
=&\max_{t\in\Omega_h}\left\|((1-\lambda)I-T^{-1})(\nu(t)-\nu({t_0}))+\int_{t_0}^tH(\nu(s)) ds\right\|_{l^{\infty}(\mathbb{C}^{N+1})}\\ \leq & \,\delta. \end{align*} Then by the Uniform Contraction Principle in Banach spaces, we know that $L(\cdot,\,\lambda): X\rightarrow X$ is a contractive mapping with a unique fixed point $\nu_\lambda\in C(\Omega_h; \bar{A})$ and $\nu_{\lambda}$ is analytic. Noticing that $L$ is linear in $\lambda$, $\nu_{\lambda}$ is differentiable with respect to $\lambda$. This completes the proof of Claim 1.
\textbf{Claim 2}: There exists $h_0>0$ and so that $\Omega_{h_0}$ is the common existence region of the fixed points $\nu_\lambda$ of $L(\nu,\,\lambda)$ for all $\lambda\in (0,\,1-1/c).$
{\textit{Proof of Claim 2:}} Let $w_\lambda= T^{-1}\nu_\lambda,\,\nu_\lambda\in X$ where $X$ is as in Claim 1. Note that $\nu_{\lambda}\in C(\Omega_{h_\lambda}; \bar{A})$ where $h_\lambda>0$ is a constant depending on $\lambda$.
Let $\widetilde{M}>0$ be the supremum of $\|H(\nu)\|_{l^{\infty}(\mathbb{C}^{N+1})}$ on $\bar{A}$. Let $0< \beta\leq +\infty$ be such that $\{t\in\mathbb{C}: |t-t_0|<\beta\}$ is the maximal existence region of $\nu_\lambda(t)$ on $\bar{A}$. If $\beta=+\infty$, then $\nu_\lambda$ can be extended to the whole complex plane $\mathbb{C}$ with $\nu_\lambda(t)\in l^{\infty}(\bar{U}\times \bar{V}))$ for all $t\in\mathbb{C}$. Otherwise, by Theorem 10.5.5 of \cite{Dieudonne}, there exists $t_1 \in\{t\in\mathbb{C}: |t-t_0|<\beta\}$ so that $\nu_\lambda$ achieves value in the boundary of $A$. Let $B$ denote the boundary of $A$. Let $r$ be defined by \begin{align*}
r= \inf_{\nu\in B}\|T^{-1}(\nu-\nu_{t_0})\|_{l^{\infty}(\mathbb{C}^{N+1})}. \end{align*}
Now we show that $r>0$. Suppose not. Note that by Lemma~\ref{OMT}, $T^{-1}$ is compact and $B$ is closed and bounded in $l^{\infty}(\mathbb{C}^{N+1})$. Therefore $r$ is the minimum norm of a compact set. There exists $\nu^*\in B$ such that $r= \|T^{-1}(\nu^*-\nu_{t_0})\|_{l^{\infty}(\mathbb{C}^{N+1})}=0.$
Then
we have $\nu_{t_0}=\nu(t_0)=\nu^*\in B$. This is a contradiction since $\nu(t_0)$ is in the interior of $A$. It follows that $r>0$.
By Lemma~\ref{Compact-operator}, we know that $\lambda I +T^{-1}\in \mathscr{L}(l^{\infty}(\mathbb{C}^{N+1}); l^{\infty}(\mathbb{C}^{N+1}))$ has norm equal to $ \lambda+\frac{1}{c}$. Then we have \begin{align*}
r = & \inf_{\nu\in B}\|T^{-1}(\nu-\nu(t_0))\|_{l^{\infty}(\mathbb{C}^{N+1})}\\
\leq &\inf_{\nu\in B}\|(\lambda I +T^{-1})(\nu-\nu(t_0))\|_{l^{\infty}(\mathbb{C}^{N+1})}.\\
\leq & \left\|(\lambda I +T^{-1})(\nu_\lambda(t_1)-\nu(t_0))\right\|_{l^{\infty}(\mathbb{C}^{N+1})}\\
= & \left\| \int_{t_0}^{t_1} (\lambda I +T^{-1}) \nu_\lambda' (s)ds\right\|_{l^{\infty}(\mathbb{C}^{N+1})}\\
\leq & \sup_{t\in\Omega_h}\int_{t_0}^t \left\| H(\nu_\lambda (s)) \right\|_{l^{\infty}(\mathbb{C}^{N+1})}ds\\
\leq &\widetilde{M}\beta. \end{align*} It follows that $\beta\geq \frac{r}{\widetilde{M}}$. Let $h_0=\frac{r}{\widetilde{M}}$. Then $\Omega_{h_0}$ is the common existence region of $\nu_\lambda$ for all $\lambda\in (0,\,1-1/c).$ This completes the proof of Claim 2.
\textbf{Claim 3}: Let $\nu_\lambda$, and $h_0$ be as in Claim 2. There exists an analytic function $w_0\in C(\Omega_{h_0};T^{-1}(\bar{A}))$ so that $\lim_{\lambda\rightarrow 0^+} \|T^{-1}\nu_{\lambda}-w_0\|_{C(\Omega_{h_0}; l^{\infty}(\mathbb{C}^{N+1}))}=0$.
{\textit{Proof of Claim 3:}} By Claim 2, we have $w_\lambda=T^{-1}\nu_\lambda\in C(\Omega_{h_0}; T^{-1}(\bar{A}))$. Moreover, the uniformly bounded set $\left\{w_{\lambda}: \lambda\in (0,\,1-1/c)\right\}$ is compact in $C(\Omega_{h_0}; T^{-1}(\bar{A}))$, by the Arzel\'{a}--Ascoli theorem, since for every $\varepsilon>0$ there exists $\tilde{\delta}=\frac{\varepsilon}{\widetilde{M}} >0$ so that $|t-t'|<\tilde{\delta}$ implies that \begin{align*}
\left\|w_{\lambda}(t)-w_{\lambda}(t')\right\|_{l^{\infty}(\mathbb{C}^{N+1})}
\leq & \left\|\int_{t'}^t (\lambda T+I)^{-1} H(Tw_{\lambda} (s))ds \right\|_{l^{\infty}(\mathbb{C}^{N+1})}\\ \leq & \widetilde{M} \tilde{\delta}\\ =&\varepsilon, \end{align*} where $\widetilde{M}>0$ was defined in the proof of Claim 2, and Lemma~\ref{Extension-operator} was applied to obtain the second inequality. Therefore, there exists $w_0\in C(\Omega_{h_0}; \bar{A}) $ so that
\begin{align}\label{w-uniform}\lim_{\lambda\rightarrow 0}\|w_{\lambda}-w_0\|_{C(\Omega_{h_0}; \bar{A})}=0.
\end{align} Since $\left\{w_{\lambda}\right\}_{\lambda\in (0,\,1-1/c)}$ is a set of analytic functions in norm $\|\cdot\|_{ C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1}))}$ and analytic in norm $\|\cdot\|_{C(\Omega_{h_0}; l^{\infty}(\mathbb{C}^{N+1}))}$, $w_0$ is also analytic in norm $\|\cdot\|_{C(\Omega_{h_0}; l^{\infty}(\mathbb{C}^{N+1}))}$.
Now we show that $w_0\in C(\Omega_{h_0}; T^{-1}(A))$. First we show that $w_0\in C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1})$. Suppose that $w_0\not\in C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1})$. Then for every $K>0$ there exists $j_0\in\mathbb{N}$ such that $\sup_{t\in \Omega_{h_0}}c^{j_0}|(w_0)_{j_0}(t)|>K$. That is, \begin{align}\label{MJ0}
\sup_{t\in \Omega_{h_0}}|(w_0)_{j_0}(t)|>\frac{K}{c^{j_0}}. \end{align} On the other hand, it follows from
$\lim_{\lambda\rightarrow 0^+}\|w_{\lambda}-w_0\|_C=0$, that for every $\epsilon>0$, there exists $\delta>0$ such that for every $\lambda\in (0,\,\delta)$, we have \begin{align*}
\sup_{t\in \Omega_{h_0}}\sup_{j\in\mathbb{N}}|(w_0)_{j}(t)-(w_\lambda)_{j}(t)|<\epsilon, \end{align*} which leads to \begin{align*}
\sup_{t\in \Omega_{h_0}} |(w_0)_{j}(t)|-\sup_{t\in \Omega_{h_0}}|(w_\lambda)_{j}(t)|<\epsilon,\,\mbox{for every $j\in\mathbb{N}$}. \end{align*} It follows that \begin{align}\label{tri-inequality}
\sup_{t\in \Omega_{h_0}}|(w_\lambda)_{j}(t)|>\sup_{t\in \Omega_{h_0}} |(w_0)_{j}(t)|-\epsilon,\,\mbox{for every $j\in\mathbb{N}$}. \end{align} Choosing $j=j_0$ and $\epsilon=\frac{K}{2c^{j_0}}$ in (\ref{tri-inequality}), then by (\ref{MJ0}) we obtain that \begin{align*}
\sup_{t\in \Omega_{h_0}}|(w_\lambda)_{j_0}(t)|& \geq \sup_{t\in \Omega_{h_0}} |(w_0)_{j_0}(t)|-\frac{K}{2c^{j_0}}\\
& >\frac{K}{2c^{j_0}}, \end{align*}which leads to
$ \sup_{t\in \Omega_{h_0}}|c^{j_0}(w_\lambda)_{j_0}(t)|>K/2$ for every $\lambda\in (0,\,\delta)$. That is, $w_\lambda\not\in C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1}))$ as $\lambda\rightarrow 0$ and hence $\nu_\lambda=Tw_\lambda\not\in C(\Omega_{h_0}; l^{\infty}(\mathbb{C}^{N+1}))$. This is a contradiction and hence $w_0\in C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1}))$.
Next we show that $w_0\in C(\Omega_{h_0}; T^{-1}(\bar{A}))$. Suppose not. Since $w_0\in C(\Omega_{h_0}; l_c^{\infty}(\mathbb{C}^{N+1}))$, there exists $t^*\in\Omega_{h_0}$ so that $w_0(t^*)\in l_c^{\infty}(\mathbb{C}^{N+1})\setminus T^{-1}(\bar{A})$. By (\ref{w-uniform}) we have \begin{align}\label{w-t-star}
\lim_{\lambda\rightarrow 0}\|w_\lambda(t^*)-w_0(t^*)\|_{l^{\infty}(\mathbb{C}^{N+1})}=0. \end{align} Since $\left\{w_{\lambda}: \lambda\in (0,\,1-1/c)\right\}$ is uniformly bounded in $C(\Omega_{h_0}; T^{-1}(\bar{A}))$, there exists a closed ball
$B'$ in $T^{-1}(\bar{A})$ which contains the closure of $\{w_\lambda(t^*)\}_{\lambda\in (0,\,1-1/c)}$. Then by Lemma~\ref{Banach-spaces} and by (\ref{w-t-star}), we have $w_0(t^*)\in B'\subset T^{-1}(\bar{A})$ which is a contradiction. This completes the proof of Claim 3.
{\bf Claim 4:} Let $h_0$ be as in Claim 2. There exists a map $\nu_0: \Omega_{h_0}\rightarrow \bar{A}$ such that $H(\nu_0)$ is continuous and is such that for every $t\in \Omega_{h_0}$,
there exists a sequence $\{\lambda_n\}_{n=1}^{\infty}\subset (0,\,1-\frac{1}{c})$ with $\lim_{n\rightarrow+\infty}\lambda_n=0$ such that
$\lim_{n\rightarrow+\infty}H(\nu_{\lambda_n}(t))=H(\nu_0(t)).$
{\it Proof of Claim 4}: Note that by Claim 1, $\nu_\lambda \in C(\Omega_{h_0};T^{-1}(\bar{A}))$ is uniformly bounded with respect to $\lambda\in (0,\,1-\frac{1}{c})$. Since by Lemma~\ref{complete-continuity} $H$ is completely continuous, for every $t\in \Omega_{h_0}$, the set \[ \left\{H(\nu_\lambda(t)): \lambda\in \left(0,\,1-\frac{1}{c}\right)\right\}, \]is pre-compact in $l^\infty(\mathbb{C}^{N+1})$.
So there exists a sequence $\{\lambda_n\}_{n=1}^{\infty}\subset (0,\,1-\frac{1}{c})$ with $\lim_{n\rightarrow+\infty}\lambda_n=0$ and $\nu_0(t)\in T^{-1}(\bar{A})$, where $T^{-1}(\bar{A})$ is compact, such that
\begin{align}\label{new-limit-01}
\lim_{n\rightarrow+\infty}H(\nu_{\lambda_n}(t))=\lim_{n\rightarrow+\infty} H(Tw_{\lambda_n}(t))=H(\nu_0(t)). \end{align}
Next we show that $H(\nu_0): \Omega_{h_0}\ni t\rightarrow H(\nu_0(t)) \in l^\infty(\mathbb{C}^{N+1})$ is continuous in $t\in \Omega_{h_0}$. Let $t\in \Omega_{h_0}$. By (\ref{new-limit-01}), for every $\epsilon>0$, there exists $N_1\in\mathbb{N}$ such that for every $n>N_1$,
\begin{align}\label{nu-1}
\|H(\nu_{\lambda_n}(t))-H(\nu_0(t))\|_{ l^\infty(\mathbb{C}^{N+1})}<\frac{\epsilon}{3},
\end{align}
Since $H(\nu_{\lambda_n})$ is continuous, there exists $\delta>0$ such that for every $t'\in \Omega_{h_0} $ with $|t-t'|<\delta$ we have
\begin{align}\label{nu-2}
\|H(\nu_{\lambda_n}(t))-H(\nu_{\lambda_n}(t'))\|_{ l^\infty(\mathbb{C}^{N+1})}<\frac{\epsilon}{3}.
\end{align} Taking subsequence of $\{\lambda_n\}$ if necessary, by (\ref{new-limit-01}) there exists $N'$ such that for every $n>N'$, we have
\begin{align}\label{nu-3}
\|H(\nu_{\lambda_n}(t'))-H(\nu_0(t'))\|_{ l^\infty(\mathbb{C}^{N+1})}<\frac{\epsilon}{3}.
\end{align}
By (\ref{nu-1}), (\ref{nu-2}) and (\ref{nu-3}) we have for $n>\max\{N_1,\,N'\}$,
\[
\|H(\nu_0(t))-H(\nu_0(t'))\|_{ l^\infty(\mathbb{C}^{N+1})}<\epsilon.
\]That is $H(\nu_0)$ is continuous. This completes the proof of Claim 4.
{\bf Claim 5:} Let $h_0$ be as in Claim 2, $w_0$ be as in Claim 3, $\nu_0$ be as in Claim 4. Then $H(Tw_0)=H(\nu_0)$ and $w_0$ is the solution of the initial value problem (\ref{Abs-ODE-IVP}).
{\it Proof of Claim 5}:
It follows from Claim 3 that $w_0$ is in $C(\Omega_{h_0}; T^{-1}(\bar{A}))$ and $w_0$ is the limit of $w_\lambda$ as $\lambda\rightarrow 0^+$ in the norm $\|\cdot\|_{C(\Omega_{h_0}; l^\infty(\mathbb{C}^{N+1}))}$. We first show that $Tw_{\lambda} $ converges to $Tw_0$ coordinate-wise. That is, for every $j\in\mathbb{N}$,
\begin{align}\label{claim-5-1}
\lim_{\lambda\rightarrow 0}\sup_{t\in \Omega_{h_0}}|(Tw_{\lambda})_j(t)-(Tw_0)_j(t)|= 0.
\end{align} If not, there exists $j_0\in\mathbb{N}$ and $\epsilon_0>0$ and a sequence $\{\lambda_{n}\}_{n=1}^{\infty}\subset (0,\,1-\frac{1}{c}) $ converging to 0 such that \[
\sup_{t\in \Omega_{h_0}}|(w_{\lambda_{n}})_{j_0}(t)-(w_0)_{j_0}(t)|\geq \frac{\epsilon_0}{c^{j_0}}, \mbox{for all } n\in\mathbb{N}, \]which leads to \[
\sup_{t\in \Omega_{h_0}}\sup_{j\in\mathbb{N}}|(w_{\lambda_{n}})_{j}(t)-(w_0)_{j}(t)|\geq\sup_{t\in \Omega_{h_0}}|(w_{\lambda_{n}})_{j_0}(t)-(w_0)_{j_0}(t)|\geq \frac{\epsilon_0}{c^{j_0}}, \]
for all $n\in\mathbb{N}$. This is a contradiction, since $w_0$ is the limit of $w_\lambda$ as $\lambda\rightarrow 0^+$ in the norm $\|\cdot\|_{C(\Omega_{h_0}; l^\infty(\mathbb{C}^{N+1}))}$.
Noticing that each coordinate of $H(Tw_\lambda)$ involves only finitely many coordinates of $Tw_\lambda$ and $H$ is analytic. $H(Tw_{\lambda})$ converges to $H(Tw_0)$ coordinate-wise as $Tw_{\lambda} $ converges to $Tw_0$ coordinate-wise with $\lambda\rightarrow 0^+$. By Claim 4, we have \begin{align}\label{eqn-last0} H(\nu_0)=H(Tw_0). \end{align} Noting that by Lemma~\ref{Extension-operator}, $(\lambda T+I)^{-1}\in \mathscr{L}(l^{\infty}(\mathbb{C}^{N+1}); l^{\infty}(\mathbb{C}^{N+1}))$ is bounded for every $\lambda\in[0,\,1)$. Notice that $\nu_\lambda$, $\lambda\in (0,\,1-\frac{1}{c})$, satisfies (\ref{Abs-ODE-IVP-integral-lambda}). On the one hand, by Claim 3 we have \begin{align}\label{eqn-last2}
\lim_{\lambda\rightarrow 0^+}\|T^{-1}\nu_{\lambda} (t)-w_0(t)\|_{l^\infty(\mathbb{C}^{N+1})}=0. \end{align}On the other hand, for every $j\in\mathbb{N}$ and $t\in\Omega_{h_0}$ we have \begin{align}\label{eqn-last}
& \left|\int_{t_0}^t\left[(\lambda T+ I)^{-1} H\right]_j\left(\nu_{\lambda}(s)\right)ds-\int_{t_0}^t H_j(\nu_0(s))ds\right|\notag\\
= & \left|\int_{t_0}^t\left[(\lambda T+ I)^{-1} H\right]_j\left(\nu_{\lambda}(s)\right)- H_j(\nu_0(s))ds\right|\notag\\
= &\left|\int_{t_0}^t \frac{1}{(\lambda c^j+ 1)} H_j\left(\nu_{\lambda}(s)\right)- H_j(\nu_0(s)) ds\right|\notag\\
= &\left|\int_{t_0}^t \left( \frac{1}{(\lambda c^j+ 1)} (H_j\left(\nu_{\lambda}(s)\right)- H_j(\nu_0(s)))-\frac{\lambda c^j}{\lambda c^j+1} H_j(\nu_0(s))\right)ds\right|\notag\\
\leq &\left|\int_{t_0}^t \frac{1}{(\lambda c^j+ 1)} (H_j\left(\nu_{\lambda}(s)\right)- H_j(\nu_0(s)))ds\right|+\left|\int_{t_0}^t\frac{\lambda c^j}{\lambda c^j+1} H_j(\nu_0(s)) ds\right|\notag\\
= & \frac{1}{(\lambda c^j+ 1)} \left|\int_{t_0}^t(H_j\left(\nu_{\lambda}(s)\right)- H_j(\nu_0(s)))ds\right|+\frac{\lambda c^j}{\lambda c^j+1}\left|\int_{t_0}^t H_j(\nu_0(s)) ds\right|\notag\\
= & \frac{1}{(\lambda c^j+ 1)} \left|\int_{t_0}^t(H_j\left(Tw_{\lambda}(s)\right)- H_j(Tw_0(s)))ds\right|+\frac{\lambda c^j}{\lambda c^j+1}\left|\int_{t_0}^t H_j(\nu_0(s)) ds\right| \end{align} where $\xi\in\Omega_{h_0}$ and $H_j$ denotes the $j$-th coordinate of $H$. Since $H(Tw_{\lambda})$ converges to $H(Tw_0)$ coordinate-wise as $Tw_{\lambda} $ converges to $Tw_0$ coordinate-wise with $\lambda\rightarrow 0^+$, uniformly with respect to $t\in\Omega_{h_0}$. Letting $\lambda \rightarrow 0^+$ in (\ref{eqn-last}), we have for every $j\in\mathbb{N}$ and $t\in\Omega_{h_0}$,
\begin{align}\label{eqn-last3} \left|\int_{t_0}^t\left[(\lambda_n T+ I)^{-1} H\right]_j\left(\nu_{\lambda}(s)\right)ds-\int_{t_0}^t H_j(\nu_0(s))ds\right|\rightarrow 0 \mbox{ as $\lambda\rightarrow0^+$}.
\end{align}
By (\ref{eqn-last2}) and (\ref{eqn-last3}), we have for every $t\in\Omega_{h_0}$,
\[
w_0(t)=w_{t_0}+\int_{t_0}^tH(\nu_0(s))ds,
\]
which combined with (\ref{eqn-last0}) gives
\[
w_0(t)=w_{t_0}+\int_{t_0}^tH(Tw_0(s))ds.
\] That is, $w_0$ is a solution of the initial value problem (\ref{Abs-ODE-IVP}). By analyticity of $w_0$, it is the unique solution of (\ref{Abs-ODE-IVP}) which is the complex extension of the real-valued solution $w=\left((y_1,\,z_1),\,(y_2,\,z_2),\,\cdots\right)\in C(\mathbb{R}; l_c^{\infty}(\mathbb{R}^{N+1}))$ at $t=t_0\in\mathbb{R}$. It follows that $(x,\,\tau)=(cy_1,\,cz_1)$ is analytic at $t_0$. Since $t_0\in\mathbb{R}$ is arbitrary, $(x,\,\tau)$ is analytic on $\mathbb{R}$. This completes the proof of Claim 5 and that of the theorem.\qed
\end{proof}
\section{Example}\label{section-3} In this section, we present an example from important applications. We now study the analyticity of periodic solutions for the following delay differential equations with adaptive delay: \begin{align}\label{ch3-eqn-4-23} \left\{ \begin{aligned} \dot{x}_1(t)&=-\mu x_1(t)+\sigma b( x_2(t-\tau(t))),\\ \dot{x}_2(t)&=-\mu x_2(t)+\sigma b( x_1(t-\tau(t))),\\ \dot{\tau}(t)&=1-h(x(t))\cdot(1+\tanh\tau(t)), \end{aligned} \right. \end{align} where $x(t)=(x_1(t),\,x_2(t))\in\mathbb{R}^2$, $\tau(t)\in\mathbb{R}$, $\tanh(\tau)= (e^{2\tau}-1)/(e^{2\tau}+1)$ and $\mu>0$ is a constant. We make the following assumptions: \begin{description} \item[$(\alpha_1)$] $b: \mathbb{R}\rightarrow\mathbb{R}$ and $h: \mathbb{R}^2\rightarrow\mathbb{R}$ are continuously differentiable functions with $b'(0)=-1$; \item[$(\alpha_2)$] There exist $h_0<h_1$ in $(1/2,\,1)$ such that $h_1> h(x)> h_0$ for all $x\in \mathbb{R}^2$; \item[$(\alpha_3)$] $b$ is decreasing on $\mathbb{R}$ and the map $\mathbb{R}\ni y\rightarrow yb(y)\in\mathbb{R}$ is injective; \item[$(\alpha_4)$]$yb(y)<0$ for $y\neq 0$, and there exists a continuous function $M:\mathbb{R}\ni \sigma\rightarrow M(\sigma)\in (0,\,+\infty)$ so that \[
\frac{b(y)}{y}>-\frac{\mu}{2|\sigma|}, \]
for $|y|\geq M(\sigma)$;
\item[$(\alpha_5)$] $h_0>(1+e^{-\pi})/2$ and there exists $\epsilon>0$ so that $b$ and $h$ have analytic complex extensions on \[
U_0\times V_0=\{(p,\,q)\in \mathbb{C}^2\times\mathbb{C}: \Re(p,\,q)\in \overline{\Omega}_1,\, |\Im(p,\,q)|\leq\epsilon\} \] where $\Omega_1= (-M(\sigma),\,M(\sigma))\times (-M(\sigma),\,M(\sigma))\times \left(0,\,-\frac{\ln (2h_0-1)}{2}\right).$ \end{description} \begin{lemma}[\cite{HWZ}]\label{ch3-lemma-4-6} Assume $(\alpha_1)$--$(\alpha_4)$ hold. Then the range of every periodic solution $(x_1,\,x_2,\,\tau)$ of $($\ref{ch3-eqn-4-23}$\,)$ with $\sigma\in\mathbb{R}$ is contained in \[ \Omega_1=(-M(\sigma),\,M(\sigma))\times(-M(\sigma),\,M(\sigma))\times \left(0, -\frac{\ln(2h_0-1)}{2}\right). \] \end{lemma} \begin{theorem}
Assume that $(\alpha_1)$--$(\alpha_5)$ hold. Then all the periodic solutions of (\ref{ch3-eqn-4-23}) are analytic on $\mathbb{R}$. \end{theorem} \begin{proof} By Lemma~\ref{ch3-lemma-4-6}, the range of every periodic solution $(x_1,\,x_2,\,\tau)$ of $($\ref{ch3-eqn-4-23}$\,)$ with $\sigma\in\mathbb{R}$ is contained in $\Omega_1$. Now we apply Theorem~\ref{Analyticity-th}. Let $l=1/2\in (0,\,1)$. For every $(x(t),\,\tau(t))=(x_1(t),\,x_2(t),\,\tau(t))\in \overline{\Omega}_1$, $t\in\mathbb{R}$, we have \[1\leq 1+\tanh \tau(t)\leq\frac{1}{h_0}<\frac{2}{1+e^{-\pi}}\] and hence by $(\alpha_2)$ and $(\alpha_5)$ we obtain \begin{align*}
& 1-(1-h(x(t))\cdot(1+\tanh\tau(t)))-\frac{e+l}{2}\\ = &\, h(x(t))\cdot(1+\tanh\tau(t)) -\frac{e+l}{2}\\ < &\, 1+\tanh \tau(t)-\frac{e+l}{2}\\ < &\frac{2}{1+e^{-\pi}}-\frac{e+l}{2}\\ < & \, \frac{e-l}{2}, \intertext{and}
& 1-(1-h(x(t))\cdot(1+\tanh\tau(t)))-\frac{e+l}{2}\\ = &\, h(x(t))\cdot(1+\tanh\tau(t)) -\frac{e+l}{2}\\ > &\, \frac{1+e^{-\pi}}{2}(1+\tanh \tau(t))-\frac{e+l}{2}\\ \geq &\frac{1+e^{-\pi}}{2}-\frac{e+l}{2}\\ >& \, -\frac{e-l}{2}.
\end{align*}Therefore, we have $| 1-(1-h(x(t))\cdot(1+\tanh\tau(t)))-\frac{e+l}{2}|<\frac{e-l}{2}$ for all $(x(t),\,\tau(t))\in\overline{\Omega}_1$. Note that $1>h_0>(1+e^{-\pi})/2$ and $(x(t),\,\tau(t))\in\overline{\Omega}_1$ imply that $0< \tau(t)<\frac{\pi}{2}$. And the complex extension of $1+\tanh q$ is analytic for $|q|<\frac{\pi}{2},\,q\in \mathbb{C}$. Then by $(\alpha_5)$ we can choose $\epsilon_0\in (0,\,\epsilon)$
small enough so that $| 1-(1-h(p)\cdot(1+\tanh q))-\frac{e+l}{2}|<\frac{e-l}{2}$ for all $(p,\,q)\in U\times V$ where \begin{align*}
U\times V= \{(p,\,q)\in \mathbb{C}^2\times\mathbb{C}: \Re(p,\,q)\in {\Omega}_1,\, |\Im(p,\,q)|<\epsilon_0\}\subset U_0\times V_0. \end{align*} Then by applying Theorem~\ref{Analyticity-th} on $U\times V$, analyticity of all the periodic solutions of (\ref{ch3-eqn-4-23}) follows.
\hspace*{1em}\qed \end{proof}
\end{document} | arXiv |
How can a substance be diluted to nothing?
If I start with a small amount of a substance, and I dilute that substance with billions of gallons of water, what is the scientific explanation for there being no (actually zero) remaining molecules of that original substance?
This question is related to the practice of homeopathic dilutions. I was reading some articles which stated that homeopathic dilutions don't work because at most homeopathic levels of dilution, there are very few or no molecules or the original substance.
What I'm confused about is how there can no longer be any molecules of the original substance (or even fewer molecules) of an original substance when diluted.
If there are X number of molecules in a solution, and Y liters of water are poured into the solution for dilution, don't the X number of molecules remain? How can a molecule just "disappear"? How can even one molecule just "disappear"? It makes more sense if these molecules are undergoing a chemical reaction, but if two kind of molecules (the substance, and water) are not undergoing any chemical reaction, then how can one molecule of either kind simply vanish?
everyday-chemistry concentration
wythagoras
You correctly point out that the number of molecules in a solution is finite and constant, however the volumetric concentration (that is, how many molecules per litre) changes upon dilution. If, for instance, you take one liter of a 1 mol/L solution of ethanol in water ($\approx{6.02\times{10}^{23}}$ ethanol molecules per liter) and add 9 liters of distilled water, you now have a 1 mole of ethanol in 10 liters, giving a concentration of 0.1 mol/L. If you were to draw out 1 liter of this and count the number of ethanol molecules in that liter, you would find approximately $\approx{6.02\times{10}^{22}}$ - a tenfold reduction. The rest of the ethanol is retained in the other 9 liters of solution.
Equivalently, if you were to measure out 100 mL of 1 mol/L ethanol solution and then make it up to 1 liter by adding distilled water, you would also get a 0.1 mol/L solution. This is the kind of thing taken to absurd extremes in homoeopathy, where tiny quantities of solution are repeatedly diluted, with some shaking. Obviously, they don't literally add billions of liters of water to their solutions, but rather pipette out a tiny amount of solution and discard the remainder, adding water to make up the desired volume. This process is repeated many times.
The finite number of molecules present in a solution leads to the concept of the dilution limit, which is where you have carried out sufficient dilutions that statistically speaking, your chances of finding even a single molecule of the dissolved ingredient become vanishingly small and that by extrapolation, you would have to drink approximately a zillion liters of the preparation to get a single molecule of it.
Where did all those other molecules go? They didn't simply disappear - they were poured down the drain during the dilution process, which curiously is the same place money goes when spent on homoeopathy.
Bonus: homoeopathic plutonium! (courtesy of Theodore Gray's periodic table)
Richard TerrettRichard Terrett
$\begingroup$ Any treatment has a placebo effect, which means it will cure about 1/3 of patients anyway. Homoeopathy is based on principles that are out of place here. $\endgroup$ – f p Mar 4 '13 at 12:43
$\begingroup$ Homeopathic plutonium? backs off $\endgroup$ – ManishEarth Mar 4 '13 at 14:30
$\begingroup$ Yeah I realized after I posted this question, that my mistake was in thinking a person simply adds more water to the molecules. This is false; homeopathic dilution works by taking a small amount of the original solution, in which case, I agree that there will be no molecules left after a certain number of dilutions. Thanks! $\endgroup$ – Jason Mar 5 '13 at 8:38
$\begingroup$ @fp - "placebo effect [...] will cure about 1/3 of patients anyway" - This is incorrect. If it were true, placebos would be the most important drug in medicine. $\endgroup$ – Richard Terrett Mar 7 '13 at 7:01
$\begingroup$ @Manishearth - As someone quipped, 'it doubles in strength every 88 years'. $\endgroup$ – Richard Terrett Mar 7 '13 at 7:04
They don't vanish. It's just that if there are X molecules and Z bottles with Z > X, part of the bottles will not have even one molecule.
f pf p
A substance can be diluted to "no substance" because:
No substance present means we are sure there are no molecules of it in some finite volume.
Being sure means we assume it to be true to some finite level of probability.
We start with a finite number of molecules. And we can infinitely dilute the the solution.
We have only a finite amount of water available, because even in an infinite universe, we can reach only a finite amount of mass/energy because it expands.
But we can simply clean and reuse our water.
It's not even slow to do that: It is exponential, and limited only by how much water we can move: Imagine repeatedly diluting a drop in a Olympic size swimming pool. Filling the pool can be done in parallel to selecting a drop and mixing it in, and takes longer.
And there is industrial level experience available, ask your Civil Engineering Hydraulics expert. For off the shelf components, see hydro power stations.
Volker SiegelVolker Siegel
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What is the sum of 26 and 52, rounded to the nearest ten?
We want to find the sum of $26$ and $52$, or $26+52$ and round that number to the nearest ten. We get $26+52=78$. Rounding $78$ to the nearest ten we get $\boxed{80}$. | Math Dataset |
Ludwig Staiger
Ludwig Staiger is a German mathematician and computer scientist at the Martin Luther University of Halle-Wittenberg.
He received his Ph.D. in mathematics from the University of Jena in 1976; Staiger wrote his doctoral thesis, Zur Topologie der regulären Mengen, under the direction of Gerd Wechsung and Rolf Lindner.[1]
Previously he held positions at the Academy of Sciences in Berlin (East), the Central Institute of Cybernetics and Information Processes, the Karl Weierstrass Institute for Mathematics and the Technical University Otto-von-Guericke Magdeburg. He was a visiting professor at RWTH Aachen University, the universities Dortmund, Siegen, and Cottbus in Germany and the Technical University Vienna, Austria. He is a member of the Managing Committee of the Georg Cantor Association and an external researcher of the Center for Discrete Mathematics and Theoretical Computer Science at the University of Auckland, New Zealand.[2]
He co-invented with Klaus Wagner the Staiger–Wagner automaton. Staiger is an expert in ω-languages, an area in which he wrote more than 19 papers [3] including the paper on this topic in the monograph.[4] He found surprising applications of ω-languages in the study of Liouville numbers.
Staiger is an active researcher in combinatorics on words, automata theory, effective dimension theory,[5] and algorithmic information theory.
Notes
1. Ludwig Staiger at the Mathematics Genealogy Project
2. CDMTCS External Researchers
3. Ludwig Staiger at DBLP Bibliography Server
4. Handbook of Formal Languages
5. Electronic Colloquium on Computational Complexity Reports of Ludwig Staiger
Bibliography
• L. Staiger. Quasiperiods of infinite words. In Alexandra Bellow, Cristian S. Calude, Tudor Zamfirescu, editors, Mathematics Almost Everywhere: In Memory of Solomon Marcus, pages 17–36, World Scientific, Singapore, 2018.
• C. S. Calude, L. Staiger. Liouville numbers, Borel normality and algorithmic randomness, Theory of Computing Systems, First online 27 April 2017, doi:10.1007/s00224-017-9767-8.
• Staiger, L. "Exact Constructive and Computable Dimensions", Theory of Computing Systems 61 (2017) 4, 1288-1314.
• C. S. Calude, L. Staiger, F. Stephan. Finite state incompressible infinite sequences, Information and Computation 247 (2016), 23-36.
• Staiger, L. "On Oscillation-Free Chaitin h-Random Sequences". In M. Dinneen, B. Khoussainov and A. Nies, editors, Computation, Physics and Beyond, pages 194-202. Springer-Verlag, 2012.
• Staiger, L. The Kolmogorov complexity of infinite words, Electronic Colloquium on Computational Complexity (EECC) 13, 70 (2006).
• Staiger, L. "ω-Languages". In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Volume 3, pages 339-387. Springer-Verlag, Berlin, 1997.
External links
• Ludwig Staiger Home Page
• CDMTCS at the University of Auckland
• Ludwig Staiger at DBLP Bibliography Server
• Ludwig Staiger publications indexed by Google Scholar
• Algorithmic Complexity and Applications: Special issue of Fundamenta Informaticae (83, 1-2, 2008), dedicated to Professor L. Staiger 60's birthday.
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| Wikipedia |
\begin{document}
\begin{frontmatter}
\title{Enumerating path diagrams in connection with $q$-tangent and $q$-secant numbers} \author{Anum Khalid} \author{Thomas Prellberg} \address{School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom} \ead{[email protected]} \begin{abstract} We enumerate height-restricted path diagrams associated with $q$-tangent and $q$-secant numbers by considering convergents of continued fractions, leading to expressions involving basic hypergeometric functions. Our work generalises some results by M. Josuat-Verg\'es for unrestricted path diagrams [European Journal of Combinatorics {\bf 31} (2010) 1892]. \end{abstract}
\begin{keyword} path diagrams, continued fractions, q-tangent numbers and q-secant numbers\\ 05A15\sep 05A30 \end{keyword} \end{frontmatter} \section{Introduction and Statement of Results} Much work has been done on the enumeration of non-crossing directed lattice paths in both the mathematics and the physics communities, see e.g.~\cite{walkers,dwalks}. The work here takes into account two paths given by a path diagram, i.e.~a Dyck path and a general directed path restrained to lie between the $x$-axis and this Dyck path. In particular, we shall consider Dyck paths restricted by height.
A Dyck path is a lattice path on $\mathbb{N}^2$ from $(0,0)$ to $(2n,0)$ consisting of $n$ steps in the northeast direction of the form $(1,1)$ and $n$ steps in the southeast direction of the form $(1,-1)$ such that the path never goes below the line $y=0$. We encode a Dyck path in terms of labelled steps where each step is indexed with the height of the point from where it starts. For example, the labelled path shown in Figure \ref{fig:dp_0} is encoded as $(a_0,b_1,a_0,a_1,a_2,b_3,b_2,a_1,b_2,b_1)$, where $a_i$ is a northeast step starting at height $i$ and $b_j$ is a southeast step starting at height $j$. So we can say that there is a set $X=\{a_0,a_1,a_2, \ldots\} \cup \{b_1,b_2,b_3,\ldots\}$, the elements of which, as an ordered finite sequence, are associated with a Dyck path. We consider path diagrams \cite{Flajolet2006992} which are represented by a Dyck path and the set of points under it subjected to some conditions expressed using the above encoding. \begin{definition}[\cite{Flajolet2006992}]{\textit{Path Diagrams.}} A system of path diagrams is defined by a possibility function $$pos:X\rightarrow \mathbb{N}_0.$$ Path diagrams are composed of the Dyck path $u=u_1 u_2 u_3 \ldots u_n$ where for $j=1,2,\ldots,n$ each $u_j \in X$, and the corresponding sequence of integers $s=s_1 s_2 s_3 \ldots s_n$ where for $i=1,2,\ldots,n$ each $0 \leq s_i \leq pos(u_j)$. We get $n$ points corresponding to a path of length $n$. \end{definition} We consider two types of path diagrams. In the first case we consider all possible lattice points bounded by the $x$-axis and a Dyck path by using the possibility function \begin{equation} pos(a_j)=j, \quad pos(b_k)=k, \quad\text{for}\quad j \geq0 \quad\text{and}\quad k \geq1. \label{tangentfunc} \end{equation} In the second case we restrict this set of points by excluding the points which are in contact with the Dyck path at a southeast step, leading to \begin{equation} pos(a_j)=j, \quad pos(b_k)=k-1, \quad\text{for}\quad j \geq0 \quad\text{and}\quad k \geq1. \label{secantfunc} \end{equation} These two possibility functions map labelled steps onto a set of integers. These integers can be visualised as column heights, and a path is then formed by joining the peaks of the columns.
\begin{figure}
\caption{A Dyck path of half length $N=5$ (solid blue line), together with columns of heights formed by the integers $(0,0,0,0,1,3,1,0,1,1)$ (dashed green lines). The dotted red line combined with the Dyck path represents the path diagram.}
\label{fig:dp_0}
\end{figure}
Figure \ref{fig:dp_0} shows an example of one such path diagram given the Dyck path example used above. Columns of heights are formed by the sequence of integers $(0,0,0,0,1,3,1,0,1,1)$, with the associated path shown as a dotted line. When restricting the height of the Dyck path, we can interpret this as a model of two non crossing paths in a finite slit.
Let $a^{(w)}_{N,m}$ be the number of path diagrams defined by the possibility function \eqref{tangentfunc} and $b^{(w)}_{N,m}$ be the number of path diagrams formed by the possibility function \eqref{secantfunc}, bounded by a Dyck path of length $2N$ in a slit of width $w$. Then we define the associated generating functions \begin{equation} G_w(t,q)=\sum_{N,m=0}^{\infty}a^{(w)}_{N,m} t^{2N} q^m \label{1} \end{equation} and \begin{equation} G'_w(t,q)=\sum_{N,m=0}^{\infty}b^{(w)}_{N,m} t^{2N} q^m, \label{2} \end{equation} with the variable $q$ conjugate to the sum of column heights $m$ and the variable $t$ conjugate to the length $2N$ of the Dyck path.
To state our results, we define \begin{equation} \phi(\lambda,x)=\sum_{k=0}^{\infty}\frac{(i\lambda;q)_k(-i\lambda;q)_k x^k}{(\lambda^2q;q)_k(q;q)_k}=\,_2\phi_1(i\lambda,-i\lambda;\lambda^2 q;q,x) \end{equation} and \begin{equation} \psi(\lambda,x)=\sum_{k=0}^{\infty}\frac{(i\lambda \sqrt{q};q)_k(-i\lambda\sqrt{q};q)_k x^k}{(\lambda^2q;q)_k(q;q)_k}=\,_2\phi_1(i\lambda\sqrt{q},-i\lambda\sqrt{q};\lambda^2 q;q,x), \label{psi} \end{equation} where $_2\phi_1(a,b;c;q,x)=\sum_{k=0}^{\infty}\frac{(a;q)_k(b;q)_k\,x^k}{(c;q)_k (q;q)_k}$ is a basic hypergeometric function. Here, $(a;q)_n=\prod\limits_{k=0}^{n-1}(1-aq^k)$ is the standard notation for the $q$-Pochhammer symbol.
For the path diagrams defined via (\ref{tangentfunc}), we obtain the following theorem. \begin{theorem} For $w \geq 0$, \begin{equation} G_w(t,q)=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\left[\bar{\lambda}^w\phi(\lambda,q^{3})\phi\left(\bar{\lambda},q^{w+3}\right)-\lambda^w\phi\left(\bar{\lambda},q^{3}\right)\phi(\lambda,q^{w+3})\right]}{{(1+\lambda^2)\left[\bar{\lambda}^w\phi(\lambda,q^{2})\phi\left(\bar{\lambda},q^{w+3}\right) -\lambda^{w+2}\phi\left(\bar{\lambda},q^{2}\right)\phi(\lambda,q^{w+3})\right]}}}\,, \label{full solution} \end{equation} where $\lambda$ is a root of $\lambda^2-\lambda(1-q)/t+1=0$ and $\bar\lambda=1/\lambda$. \label{theorem3} \end{theorem} Taking the limit $w\to\infty$ we obtain the generating function $G(t,q)$ for unrestricted path diagrams. \begin{corollary}\label{cort1} The generating function of $q$-tangent numbers is \begin{equation} G(t,q)=\dfrac{(1+\lambda)^2\left[1-(1+\lambda^2) \sum\limits_{k=0}^{\infty}\dfrac{(-i\lambda)^k}{(1-i\lambda q^{k})} \right]}{\lambda^2 (1-q)}\,, \label{halfplanefortangent} \end{equation} where $\lambda$ is the root of $\lambda^2-\lambda(1-q)/t+1=0$ with smallest modulus. \end{corollary} Extracting coefficients of this generating function, we can derive a result equivalent to one obtained previously by different methods \cite[Theorem 1.4]{JosuatVerges20101892}. \begin{corollary}\label{cor36} \begin{equation} [t^{2N}]G(t,q)=\dfrac{1}{(1-q)^{2N+1}}\sum\limits_{m=0}^{N}\dfrac{q^{m^2+2m}\left( \sum\limits_{l=-m}^{m+1}(-1)^l q^{-l^2+2l}\right) (2m+2)\binom{2N+1}{N+m+1}}{N+m+2} \label{tantcoeff} \end{equation} \end{corollary} For the path diagrams defined via (\ref{secantfunc}), we obtain the following theorem. \begin{theorem} For $w\geq0$, \begin{equation} G'_w(t,q)=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\left[\bar{\lambda}^w\psi(\lambda,q^{2})\psi\left(\bar{\lambda},q^{w+2}\right)-\lambda^w\psi\left(\bar{\lambda},q^{2}\right)\psi(\lambda,q^{w+2})\right]}{{(1+\lambda^2)\left[\bar{\lambda}^w\psi(\lambda,q)\psi\left(\bar{\lambda},q^{w+2}\right) -\lambda^{w+2}\psi\left(\bar{\lambda},q\right)\psi(\lambda,q^{w+2})\right]}}}\,, \label{full solution1} \end{equation} where $\lambda$ is the root of $\lambda^2-\lambda(1-q)/t+1=0$ and $\bar{\lambda}=1/\lambda$. \label{theoremnot3} \end{theorem} Taking the limit $w\to\infty$ we obtain the generating function $G'(t,q)$ for unrestricted path diagrams. \begin{corollary}\label{cors1} The generating function of $q$-secant numbers is \begin{equation} G'(t,q)=(1+ \lambda^2) \sum\limits_{k=0}^{\infty}\dfrac{(-i\lambda\sqrt{q})^k}{(1-i\lambda \sqrt{q} q^{k})}\;, \label{halfplaneofsecant} \end{equation} where $\lambda$ is the root of $\lambda^2-\lambda(1-q)/t+1=0$ with smallest modulus. \end{corollary} Extracting coefficients of this generating function, we can derive a result equivalent to one obtained previously by different methods \cite[Theorem 1.5]{JosuatVerges20101892}. \begin{corollary}\label{cors2} \begin{equation} [t^{2N}]G'(t,q)=\dfrac{1}{(1-q)^{2N}}\sum\limits_{m=0}^{N}\dfrac{q^{m^2+m}\left(\sum\limits_{l=-m}^{m}(-1)^l q^{-l^2}\right)(2m+1)\binom{2N}{N+m}}{N+m+1}. \label{sectcoeff} \end{equation} \end{corollary}
This paper is organized as follows. Section 2 contains further conventions and preliminaries. Section 3 contains the proofs of Theorem \ref{theorem3} and Corollaries \ref{cort1} and \ref{cor36}. Section 4 contains the proofs of Theorem \ref{theoremnot3} and Corollaries \ref{cors1} and \ref{cors2}. Section 5 contains some novel identities discovered.
\section{Conventions and Preliminaries} In \cite{Flajolet2006992}, the correspondence between generating functions and continued fractions has been discussed in detail. In particular, in \cite[Theorem 3A]{Flajolet2006992} and \cite[Theorem 3B]{Flajolet2006992} we find continued fraction expansions for the formal generating functions of path diagrams bounded by a Dyck path with possibility functions given by (\ref{tangentfunc}) and (\ref{secantfunc}), respectively. It turns out that the counting numbers in these generating functions are the Euler numbers $E_N$, with $G(t,1)$ and $G'(t,1)$ summing over the odd and even Euler numbers, i.e. \begin{equation} tG(t,1)=\sum_{N=0}^\infty E_{2N+1}t^{2N+1}=\cfrac{t}{1-\cfrac{1.2 t^2}{1-\cfrac{2.3 t^2}{\ddots}}}\,, \label{continued fraction} \end{equation} and \begin{equation} G'(t,1)=\sum_{N=0}^\infty E_{2N}t^{2N}=\cfrac{1}{1-\cfrac{1.1 t^2}{1-\cfrac{2.2 t^2}{\ddots}}} \label{continued fraction2} \end{equation} as formal non-convergent power series. The odd and even Euler numbers $E_{2N+1}$ and $E_{2N}$ are also known as tangent and secant numbers, respectively, as they occur in the Taylor expansion \begin{equation} \tan t+\sec t=\sum_{N=0}^\infty E_N\frac{t^N}{N!}\;. \end{equation} The formulas (\ref{continued fraction}) and (\ref{continued fraction2}) were generalised in \cite{JosuatVerges20101892,qtangent} by introducing a variable $q$ conjugate to the sum of the column heights. Briefly, this corresponds to replacing an integer $k$ in (\ref{continued fraction}) and (\ref{continued fraction2}) by the $q$-integer $[k]_q=1+q+\cdots q^k$ (more details are given in Proposition \ref{Prop21}), leading to \begin{equation} tG(t,q)=\sum_{N=0}^\infty E_{2N+1}(q)t^{2N+1}=\cfrac{\alpha(1-q)}{1-\cfrac{\alpha^2(1-q)(1-q^2)}{1-\cfrac{\alpha^2(1-q^2)(1-q^3)}{\ddots}}} \label{Igeneral continued fraction} \end{equation} and \begin{equation} G'(t,q)=\sum_{N=0}^\infty E_{2N}(q)t^{2N}=\cfrac{1}{1-\cfrac{\alpha^2(1-q)^2}{1-\cfrac{\alpha^2(1-q^2)^2}{\ddots}}}\,, \label{Igeneral continued fraction2} \end{equation} where we have introduced \begin{equation} \alpha=\dfrac{t}{1-q} \label{alpha1} \end{equation} and $E_N(q)$ are the $q$-Euler numbers. In particular $E_{2N+1}(q)$ and $E_{2N}(q)$ are known as $q$-tangent and $q$-secant numbers, respectively \cite{qtangent}. Below we shall use $\alpha$ and $t$ interchangably, as convenient.
The following proposition is the starting point of our analysis. It expresses the height-restricted path diagram generating functions $G_w(t,q)$ and $G'_w(t,q)$ as finite continued fractions. \begin{proposition} For $w \geq 0$, \begin{equation} G_w(t,q)=\cfrac{1}{1-\cfrac{\alpha^2(1-q)(1-q^2)}{1-\cfrac{\alpha^2(1-q^2)(1-q^3)}{{\ddots-}{\cfrac{\alpha^2(1-q^{w-1})(1-q^{w})}{1-\alpha^2(1-q^w)(1-q^{w+1})}}}}} \label{general continued fraction} \end{equation} and \begin{equation} G'_w(t,q)=\cfrac{1}{1-\cfrac{\alpha^2(1-q)^2}{1-\cfrac{\alpha^2(1-q^2)^2}{\ddots-\cfrac{\alpha^2(1-q^{w-1})^2}{1-\alpha^2(1-q^w)^2}}}}\, . \label{general continued fraction2} \end{equation} \label{Prop21} \end{proposition} \begin{proof}
From the combinatorial theory of continued fractions given in \cite{Flajolet2006992}, if $X=(a_0,a_1,a_2,..,b_0,b_1,..)$ then the Stieltjes type continued fraction is $$S_k(X,t)=\cfrac{1}{1-\cfrac{a_0 b_1 t^2}{1-\cfrac{a_1 b_2 t^2}{\ddots-\cfrac{a_{k-2}b_{k-1}t^2}{1-a_{k-1} b_k t^2}}}}$$ where $a_i$ corresponds to the weight of a northeast step starting at height $i$, $b_j$ corresponds to the weight of a southeast step starting at height $j$, and $t$ is conjugate to the length of the Dyck path. Hence, we only need to specify the weights $a_i$ and $b_j$.
Possible column heights below a northeast step starting at height $i$ range from $0$ to $i$, and hence $a_i=1+q+\ldots+q^i$. For $G_w$ possible column heights below a southeast step starting at height $j$ range from $0$ to $j$, and hence $b_j=1+q+\ldots+q^j$, whereas for $G'_w$ possible column heights below a southeast step starting at height $j$ range from $0$ to $j-1$, and hence $b_j=1+q+\ldots+q^{j-1}$. \end{proof}
It is obvious that we can write the right-hand sides of \eqref{general continued fraction} and \eqref{general continued fraction2} as rational functions. \begin{proposition} For $w\geq 0$, \begin{equation} G_w(t,q)=\dfrac{P_w(\alpha,q)}{Q_w(\alpha,q)}\quad\text{and}\quad G'_w(t,q)=\dfrac{P'_w(\alpha,q)}{Q'_w(\alpha,q)} \, , \label{rational function} \end{equation} where \begin{eqnarray}
P_w&=&
\begin{cases}
0 & w=-1 \\
1 & w=0\\
P_{w-1}-\alpha^2(1-q^w)(1-q^{w+1})P_{w-2}& w\geq 1
\end{cases}\;,
\label{recursion P} \\
Q_w&=&
\begin{cases}
1 & w=-1 \\
1 & w=0\\
Q_{w-1}-\alpha^2(1-q^w)(1-q^{w+1})Q_{w-2}& w\geq 1
\end{cases}\;,
\label{recursion Q} \\
P'_w&=&
\begin{cases}
0 & w=-1 \\
1 & w=0\\
P_{w-1}-\alpha^2(1-q^w)^2 P_{w-2}& w\geq 1
\end{cases}\qquad\text{and}
\label{recursion P1} \\
Q'_w&=&
\begin{cases}
1 & w=-1 \\
1 & w=0\\
Q_{w-1}-\alpha^2(1-q^w)^2 Q_{w-2}& w\geq 1
\end{cases}\;.
\label{recursion Q1} \end{eqnarray} \label{proposition2} \end{proposition} \begin{proof} The initial conditions follow from the fact that $G_{-1}(t,q)=G'_{-1}(t,q)=0/1$. This implies that $P_{-1}=P'_{-1}=0$ and $Q_{-1}=Q'_{-1}=1$. Also for $w=0$ we have $G_{0}(t,q)=G'_{0}(t,q)=1/1$. For $w \geq 1$ we compare with the $h$-th convergent of the $J$-fraction on page 152 of \cite{Flajolet2006992}. We have $z=t$ and $a_k=1$ for $k \geq 1 $ and $b_k=(1-q^w)(1-q^{w+1})$ and $c_k=0$ for $k \geq0$. This reduces to the recurrence equations given in \eqref{recursion P} and \eqref{recursion Q}. For the generating function $G'_w(t,q)$ we see that, instead, $b_k=(1-q^w)(1-q^{w})$, which results in the recurrence equations given in \eqref{recursion P1} and \eqref{recursion Q1}. \end{proof} \section{$q$-tangent numbers} We shall prove Theorem \ref{theorem3} by solving the recurrence relations \eqref{recursion P} and \eqref{recursion Q}. We can write $P_w$ and $Q_w$ as the linear combination of two basic hypergeometric functions and determine the coefficients from the initial conditions of the recurrences given in Proposition \ref{proposition2}.
\begin{proof}[Proof of Theorem \ref{theorem3}] For $w \geq 1$ the recurrence relations for $P_w(\alpha,q)$ and $Q_w(\alpha,q)$ are the same, so we represent them both by $R(w)$ and solve simultaneously. From the recursion given in \eqref{recursion P} and \eqref{recursion Q} we have for $w\geq1$, \begin{equation} R(w)=R(w-1)-\alpha^2(1-q^w)(1-q^{w+1})R(w-2). \label{recurrence} \end{equation} \indent Unlike a linear recurrence with constant coefficients, this cannot be solved by a standard method because we have $w$-dependent coefficients. Moreover, the occurrence of both $q^w$ and $q^{2w}$ poses a difficulty, so our next step will be to eliminate the term containing $q^{2w}$ by appropriate rewriting of the recurrences. It is evident from the coefficient of $R(w-2)$ that multiplying by a $q$-factorial will simplify \eqref{recurrence} appropriately. Rescaling the recursion \eqref{recurrence} by substituting \begin{equation} R(w)=\alpha^w(q;q)_{w+1}S(w) \label{ansatz1} \end{equation} leads to the recurrence \begin{equation} S(w)-\frac1\alpha S(w-1)+S(w-2)=q^{w+1}(S(w)+S(w-2)) \label{scaled recurrence} \end{equation} for $w \geq 1$. This eliminates $q^{2w}$ from the recurrence as intended, as the right hand side only contains a $q^w$ prefactor. The left hand side of equation \eqref{scaled recurrence} is a linear homogeneous recurrence relation with a characteristic polynomial \begin{equation} P(\lambda)=\lambda^2-\frac{\lambda}{\alpha}+1. \label{polynomial} \end{equation} The two roots $\lambda_1$ and $\lambda_2$ of the characteristic polynomial are reciprocal to each other, \begin{equation} \label{prod of roots} \lambda_1 \lambda_2=1, \end{equation} a fact that we will need to use below. If the right hand side of the recurrence relation \eqref{scaled recurrence} was zero then the solution could be written as a $q$-independent linear combination of the powers of the roots of the characteristic polynomial. To solve the recurrence \eqref{scaled recurrence} in general, we use the ansatz \begin{equation}
S(w)=\lambda^w\sum\limits_{k=0}^{\infty}c_k q^{kw},
\label{ansatz2} \end{equation} which has been shown to work when there are powers of $q^w$ in such a linear recurrence \cite{RSOS,RSOSslit}. The recurrence relation for $c_k$ can then be read off from \begin{equation} P(\lambda)c_0+\sum_{k=1}^{\infty}q^{kw-2k}\left(P(\lambda q^k)c_k-(\lambda^2 q^{2k}+ q^2)qc_{k-1}\right)=0. \label{Two term recurrence} \end{equation} This equation is satisfied if $P(\lambda)=0$ and all the coefficients in the sum vanish, i.e. $P(\lambda q^k)c_k-(\lambda^2 q^{2k}+q^2)qc_{k-1}=0$. The latter condition implies \begin{equation} c_k=\frac{(\lambda^2 q^{2k}+q^2)qc_{k-1}}{P(\lambda q^k)}\, . \label{ck} \end{equation} The condition $P(\lambda)=0$ enables us to express $\alpha$ in terms of $\lambda$ as $\alpha={\lambda}/(1+\lambda^2)$, and eliminating $\alpha$ in the characteristic polynomial \eqref{polynomial}, we find \begin{equation} P(\lambda q^k)=(1-q^k)(1-\lambda^2 q^k). \label{plambdaq} \end{equation} Now substituting in the value of $P(\lambda q^k)$ from \eqref{plambdaq} in \eqref{ck} and iterating it we have \begin{equation} c_k=\frac{(-\lambda^2;q^2)_k\, q^{3k}}{(q;q)_k(\lambda^2 q;q)_k} , \label{Hypergeometric Ck} \end{equation} where we choose to write all products in terms of the $q$-Pochhammer symbol. The full solution to the recurrence equation \eqref{scaled recurrence} is a linear combination of the ansatz (\ref{ansatz2}) over both roots of $P(\lambda)$. As $P(\lambda)=0$ implies $P(\bar{\lambda})=0$, we can write the general solution for $S(w)$ as \begin{equation} S(w)=A\lambda^w\sum_{k=0}^{\infty}c_k(\lambda,q) q^{kw}+B\bar{\lambda}^w\sum_{k=0}^{\infty}c_k\left(\bar{\lambda},q\right)q^{kw}. \end{equation} We can now write the general solution in terms of a basic hypergeometric series by defining \begin{equation} \phi(\lambda,x)=\sum_{k=0}^{\infty}\frac{(-\lambda^2;q^2)_k x^{k}}{(q;q)_k(\lambda^2 q;q)_k} =\,_2\phi_1(i\lambda,-i\lambda;\lambda^2 q;q,x), \label{phi} \end{equation} where $$_2\phi_1(a,b;c;q,x)=\sum_{k=0}^{\infty}\frac{(a;q)_k(b;q)_k\,x^k}{(c;q)_k (q;q)_k}.$$ Using this notation, the general solution $S(w)$ can simply be written as \begin{equation} S(w)=A\lambda^w\phi(\lambda,q^{w+3})+B\bar{\lambda}^w\phi\left(\bar{\lambda},q^{w+3}\right). \label{linear general solution} \end{equation} Using the initial conditions $$S(-1)=0\qquad S(0)=\frac{1}{1-q}$$ derived from \eqref{recursion P} and solving for $A$ and $B$, we get $$A=\frac{-\lambda^2\phi\left(\bar{\lambda},q^2\right)}{(1-q)\left(\phi(\lambda,q^2)\phi\left(\bar{\lambda},q^3\right)-\lambda^2\phi\left(\bar{\lambda},q^2\right)\phi(\lambda,q^3)\right)}$$ and $$B=\frac{\phi(\lambda,q^2)}{(1-q)\left(\phi(\lambda,q^2)\phi\left(\bar{\lambda},q^3\right)-\lambda^2\phi\left(\bar{\lambda},q^2\right)\phi(\lambda,q^3)\right)}\, .$$ Similarly, using the initial conditions $$ S(-1)=\alpha\qquad S(0)=\frac{1}{1-q} $$ derived from \eqref{recursion Q} we get $$A=\frac{(\alpha)(\lambda)(1-q)\phi\left(\bar{\lambda},q^3\right)-\lambda^2\phi\left(\bar{\lambda},q^2\right)}{(1-q)\left(\phi(\lambda,q^2)\phi\left(\bar{\lambda},q^3\right)-\lambda^2\phi\left(\bar{\lambda},q^2\right)\phi(\lambda,q^3)\right)}$$ and $$B=\frac{\phi(\lambda,q^2)-(\alpha)(\lambda)(1-q)\phi(\lambda,q^3)}{(1-q)\left(\phi(\lambda,q^2)\phi\left(\bar{\lambda},q^3\right) -\lambda^2\phi\left(\bar{\lambda},q^2\right)\phi(\lambda,q^3)\right)}\,.$$ Substituting the full solution for $P_w(\alpha,q)$ and $Q_w(\alpha,q)$ in \eqref{rational function}, we arrive at the expression given in \eqref{full solution}. This completes the proof. \end{proof} By taking the limit of infinite $w$ in the generating function $G_w$, we derive an expression for the generating function of $q$-tangent numbers. \begin{proof}[Proof of Corollary \ref{halfplanefortangent}]
We consider the right-hand side of \eqref{full solution}. We know that the basic hypergeometric functions converge when $|q| < 1$ using the ratio test. From \eqref{prod of roots} we see that one of the roots of the characteristic polynomial \eqref{polynomial} is less than one if $t$ is sufficiently small. We therefore choose the root $\lambda$ such that $|\lambda| < 1$. When $w \rightarrow \infty$,
$$\phi(\lambda,q^{w+3})=\,_2\phi_1(i\lambda,-i\lambda;\lambda^2q;q,q^{w+3})\rightarrow \,_2\phi_1(i\lambda,-i\lambda;\lambda^2q;q,0)=1.$$ Also $$ |\lambda^w| \rightarrow 0.$$ This implies \begin{equation} G(t,q)=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\phi(\lambda,q^{3})}{(1+\lambda^2)\phi(\lambda,q^{2})}}\, . \label{HP} \end{equation} Heine's transformation formula for $_2\phi_1$ series \cite{Hein} is given by \begin{equation} _2\phi_1(a,b;c;q,z)=\frac{(b;q)_\infty(az;q)_\infty}{(c;q)_\infty (z;q)_\infty}\,_2\phi_1(c/b,z;az;q,b). \label{Hein} \end{equation} Using this transformation we can write the basic hypergeometric functions in \eqref{HP} as follows \begin{equation} \phi(\lambda,q^2) =\frac{(-i\lambda;q)_\infty(i\lambda q^2;q)_\infty}{(\lambda^2q;q)_\infty (q^2;q)_\infty} \,_2\phi_1(i\lambda q,q^2;i\lambda q^2;q,-i\lambda) \label{fihein1} \end{equation} and \begin{equation} \phi(\lambda,q^3) =\frac{(-i\lambda;q)_\infty(i\lambda q^3;q)_\infty}{(\lambda^2q;q)_\infty (q^3;q)_\infty} \,_2\phi_1(i\lambda q,q^3;i\lambda q^3;q,-i\lambda). \label{fihein2} \end{equation} Further substituting the transformations of basic hypergeometric functions from \eqref{fihein1} and \eqref{fihein2} into \eqref{HP} yields \begin{equation} G(t,q)=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\dfrac{(-i\lambda;q)_\infty(i\lambda q^3;q)_\infty}{(\lambda^2q;q)_\infty (q^3;q)_\infty} \,_2\phi_1(i\lambda q,q^3;i\lambda q^3;q,-i\lambda)}{(1+\lambda^2)\dfrac{(-i\lambda;q)_\infty(i\lambda q^2;q)_\infty}{(\lambda^2q;q)_\infty (q^2;q)_\infty} \,_2\phi_1(i\lambda q,q^2;i\lambda q^2;q,-i\lambda)}}\, . \end{equation} Expressing these basic hypergeometric functions by their explicit sums, we find that many factors in the coefficients cancel: \begin{align} G(t,q)&=\cfrac{1}{1-\dfrac{\lambda^2(1-q)(1-q^2)\sum\limits_{k=0}^{\infty}\dfrac{(i\lambda q;q)_k(q^3;q)_k}{(i\lambda q^3;q)_k(q;q)_k}(-i\lambda)^k}{(1-i\lambda q^2)(1+\lambda^2)\sum\limits_{k=0}^{\infty}\dfrac{(i\lambda q;q)_k(q^2;q)_k}{(i\lambda q^2;q)_k(q;q)_k}(-i\lambda)^k }}\newline\\ &=\cfrac{1}{1-\dfrac{(\lambda^2)(1-q)\sum\limits_{k=0}^{\infty}\dfrac{(1-q^{k+1})(1-q^{k+2})}{(1-i\lambda q^{k+1})(1-i\lambda q^{k+2})}(-i\lambda)^k}{(1+\lambda^2)\sum\limits_{k=0}^{\infty}\dfrac{(1-q^{k+1})}{(1-i\lambda q^{k+1})}(-i\lambda)^k}}\, . \label{simple1} \end{align} We next aim to simplify the terms in the sums on the right hand side of \eqref{simple1}. For this we let \begin{equation} N=\frac{(1-q^{k+1})(1-q^{k+2})}{(1-i\lambda q^{k+1})(1-i\lambda q^{k+2})}(-i\lambda)^k \label{N} \end{equation} and \begin{equation} D=\frac{(1-q^{k+1})}{(1-i\lambda q^{k+1})}(-i\lambda)^k. \label{D} \end{equation} We substitute $x=-i\lambda$ and employ partial fraction expansion with respect to $q^k$. Shifting summation indices and combining fractions, we find \begin{multline} N=(-1)\dfrac{x^4\sum\limits_{k=0}^{\infty}\dfrac{x^k}{(1+x q^{k+1})}-2 x^2\sum\limits_{k=0}^{\infty}\dfrac{x^k}{(1+x q^{k+1})}}{x^4 (q-1)(x-1)}\\ -\dfrac{\sum\limits_{k=0}^{\infty}\dfrac{x^k}{(1+xq^{k+1})}+x^2 q+1}{x^4 (q-1)(x-1)} \label{p1} \end{multline} and \begin{equation} D=\dfrac{x^2\sum\limits_{k=0}^{\infty}\dfrac{x^k}{(1+x q^{k})}-\sum\limits_{k=0}^{\infty}\dfrac{x^k}{(1+x q^{k})}+1}{x^2 (x-1)}. \label{p2} \end{equation} Substituting \eqref{p1} and \eqref{p2} into \eqref{simple1} and simplifying, we get the final expression \eqref{halfplanefortangent}. \end{proof} Next, we extract the coefficient of $t^{2N}$ of $G(t,q)$ given in \eqref{halfplanefortangent}. To start, we need an identity which can be obtained from counting rectangles on the square lattice in two different ways, taking ideas from \cite{thomas}. \begin{lemma} \label{rectlemma} \begin{equation} \label{rectangles} \sum_{n=0}^\infty\frac{x^n}{1-yq^n}=\sum_{n=0}^\infty\frac{x^ny^nq^{n^2}(1-xyq^{2n})}{(1-xq^n)(1-yq^n)}\;. \end{equation} \end{lemma} \begin{proof} We consider the generating function of rectangles (including those of height or width zero) on the square lattice, counted with respect to height, width, and area, given by \begin{equation} R(x,y,q)=\sum_{n,m=0}^\infty x^ny^mq^{nm}\;. \end{equation} Summing over $m$ gives the left hand side of identity \eqref{rectangles}. If we instead sum over rectangles of fixed minimal width or height $N$, then this gives the right hand side of identity \eqref{rectangles}. \end{proof} \begin{proof}[Proof of Corollary \ref{cor36}] The sum in \eqref{halfplanefortangent} can be identified with $R(-i\lambda,i\lambda,q)$, so that using Lemma \ref{rectlemma} we get \begin{align} G(t,q)=&\dfrac{(1+\lambda)^2\left[1-(1+\lambda^2) R(-i\lambda,i\lambda,q) \right]}{\lambda^2 (1-q)}\nonumber\\ =&\dfrac{(1+\lambda)^2\left[1-(1+\lambda^2) \sum\limits_{n=0}^{\infty}\dfrac{q^{n^2} \lambda^{2n} (1-\lambda^2 q^{2n}) }{(1+ \lambda^2 q^{2n})} \right]}{\lambda^2 (1-q)}\;. \label{tangentt} \end{align} We remind that $G(t,q)$ is by definition an even function in $t$, and that the $t$-dependence on the right hand side is implicit in $\lambda=\lambda(t)$ by \eqref{polynomial} and \eqref{alpha1}. To extract the coefficient of $t^{2N}$, we evaluate the contour integral \begin{equation} [t^{2N}] G(t,q)=\frac{1}{2 \pi i} \oint \dfrac{G(t,q)}{t^{2N+1}}dt\;. \label{cit} \end{equation} Using the variable substitution \begin{equation} t=\frac{(1-q)\lambda}{1+\lambda^2}, \label{valuet} \end{equation} we get \begin{multline} [t^{2N}] G(t,q)=\\ \frac{1}{2 \pi i}\oint \left(\dfrac{(1+\lambda^2)^{2N} \left(1-(1+\lambda^2)\sum\limits_{n=0}^{\infty}\dfrac{q^{n^2} \lambda^{2n} (1-\lambda^2 q^{2n}) }{(1+\lambda^2 q^{2n})}\right)(1-\lambda^2)}{\lambda^{2N+2}(1-q)^{2N+1}}\right)\frac{d\lambda}{\lambda}. \end{multline} We thus have \begin{equation} [t^{2N}]G\left(t,q\right)=[\lambda^0]H_N(\lambda,q), \end{equation} and to extract the constant term in $\lambda$ on the right-hand side we now expand $H_N(\lambda,q)$ as a Laurent series in $\lambda$. For this, we write \begin{equation} [\lambda^0]H_N(\lambda,q)=[\lambda^0]\frac{T_1-T_2\sum\limits_{n=0}^{\infty}T_3}{((1-q)^{2N+1}}\;, \label{constantterm0} \end{equation} where $$ T_1=\left(\lambda+\dfrac{1}{\lambda}\right)^{2N} \left(\dfrac{1}{\lambda^2}-1\right), $$ $$T_2=\left(\lambda+\dfrac{1}{\lambda}\right)^{2N} \left(\dfrac{1}{\lambda^2}-1\right)(1+\lambda^2), $$ and $$T_3=\dfrac{q^{n^2} \lambda^{2n} (1-\lambda^2 q^{2n}) }{(1+\lambda^2 q^{2n})}.$$ We find the series expansions \begin{equation} T_1=\sum\limits_{k=0}^{2N+1}\dfrac{(2N)!(2N-2k+1)\lambda^{2k-2N-2}}{k!(2N-k+1)!}\, , \label{term0} \end{equation} \begin{equation} T_2=\sum\limits_{k=0}^{2N+2}\dfrac{(2N+1)!(2N-2k+2)\lambda^{2k-2N-2}}{k!(2N-k+2)!} \label{term1} \end{equation} and \begin{equation} T_3=q^{n^2} \lambda^{2n}\left(2\left(\sum\limits_{l=0}^{\infty}(-1)^l(\lambda^2 q^{2n})^l\right)-1\right). \label{term2} \end{equation} Next we substitute the expression \eqref{term0}, \eqref{term1} and \eqref{term2} in \eqref{constantterm0}, which after some simplification leads to \begin{multline} H_N(\lambda,q)=\dfrac{1}{(1-q)^{2N+1}}\left(\sum\limits_{k=0}^{2N+1}\dfrac{(2N)!(2N-2k+1)\lambda^{2k-2N-2}}{k!(2N-k+1)!}\right.\\ \left.-2\sum_{n=0}^{\infty}\sum_{l=0}^{\infty}\sum_{k=0}^{2N+2}(-1)^l q^{n^2+2nl} \,\dfrac{(2N+1)!(2N-2k+2)\lambda^{2k-2N-2+2n+2l}}{k!(2N-k+2)!}\right.\\ \left.+\sum_{n=0}^{\infty}\sum_{k=0}^{2N+2}q^{n^2}\dfrac{(2N+1)!(2N-2k+2)\lambda^{2k-2N-2+2n}}{k!(2N-k+2+2n)!}\right)\;. \label{2.65} \end{multline} We want to extract the constant term in $\lambda$, so we combine the powers of $\lambda$ and equate them to $0$. This fixes the summation index $k$, and we get \begin{multline} [t^{2N}]G\left(t,q\right) =\dfrac{1}{(1-q)^{2N+1}}\left(-\dfrac{(2N)!}{N!(N+1)!}\right.\\ \left. -2\sum_{n=0}^{N+1}\sum_{l=0}^{N-n+1}(-1)^l q^{n^2+2nl}\dfrac{(2N+1)!(2n+2l)}{(N-n-l+1)!(N+n+l+1)!}\right.\\ \left.+\sum_{n=0}^{N+1} q^{n^2}\dfrac{(2N+1)!(2n)}{(N-n+1)!(N+n+1)!}\right). \end{multline} Completing the square in the middle sum and changing summation indices leads to \begin{multline} [t^{2N}]G\left(t,q\right)=\dfrac{1}{(1-q)^{2N+1}}\left(-\dfrac{(2N)!}{N!(N+1)!}\right.\\ \left.-\sum_{m=0}^{N+1}q^{m^2}\dfrac{(2N+1)!(2m)}{(N-m+1)!(N+m+1)!}\left(\sum_{l=-(m+1)}^{m+1}(-1)^l q^{-l^2}\right)\right)\;, \end{multline} where in a final step we combined the last two sums. Shifting summation indices $m$ and $l$ gives \begin{multline} [t^{2N}]G\left(t,q\right)=\dfrac{1}{(1-q)^{2N+1}}\left(-\dfrac{(2N)!}{N!(N+1)!}\right.\\ \left.+\sum_{m=0}^{N}q^{m^2+2m}\dfrac{(2N+1)!(2m+2)}{(N-m)!(N+m+2)!}\left(\sum_{l=-m}^{m+2}(-1)^{l} q^{-l^2+2l}\right)\right). \end{multline} Performing the sum over $m$ with $l=m+2$ cancels the first term and we arrive at an expression equivalent to \eqref{tantcoeff}, \begin{multline} [t^{2N}]G\left(t,q\right)=\dfrac{1}{(1-q)^{2N+1}}\times\\ \sum_{m=0}^{N}q^{m^2+2m}\dfrac{(2N+1)!(2m+2)}{(N-m)!(N+m+2)!}\left(\sum_{l=-m}^{m+1}(-1)^{l} q^{-l^2+2l}\right). \end{multline} \end{proof} \section{$q$-secant numbers} We begin by giving the proof of Theorem \ref{theoremnot3}. We shall prove it by solving the recurrences \eqref{recursion P1} and \eqref{recursion Q1}. This is done along the same lines as in the proof of Theorem \ref{theorem3}. \begin{proof} It follows from the continued fraction expansion given in \eqref{general continued fraction2} that both the numerator $P'_w(\alpha,q)$ and denominator $Q'_w(\alpha,q)$ satisfy the recurrence relations given in \eqref{recursion P1} and \eqref{recursion Q1} respectively. As the recursions are the same for $w \geq 1$, we represent them both by $R(w)$ and solve simultaneously. It follows that \begin{equation} R(w)=R(w-1)-\alpha^2(1-q^w)^2 R(w-2). \end{equation} Expanding the coefficient of $R(w-2)$ gives three terms which cannot be solved explicitly using standard methods because we have $w$-dependent coefficients. Also the terms $q^w$ and $q^{2w}$ cause difficulty, so we will aim to eliminate the terms containing $q^{2w}$ by suitable rescaling. For this we use the ansatz \eqref{ansatz1}. This transformation of coefficients leads to \begin{equation} S(w)-\frac{1}{\alpha}S(w-1)+S(w-2)=q^{w+1}S(w)+q^w S(w-2) \label{scaled recurrence1} \end{equation} for $w \geq 1$. This eliminates the $q^{2w}$ from the recurrence as intended, with only $q^w$ factors on the right hand side. We see that this recurrence is very similar to \eqref{scaled recurrence}. The left hand side of \eqref{scaled recurrence1} is a linear homogeneous recurrence relation with the same characteristic polynomial \eqref{polynomial} as above, however the right hand side is slightly different, with a prefactor of $q^w$ in front of $S(w-2)$ instead of a prefactor $q^{w+1}$. We thus use the same ansatz \eqref{ansatz2} to solve the recurrence. Following a calculation identical to the one for $q$-tangent numbers, we find for $k>0$ \begin{equation} c_k=\frac{(\lambda^2 q^{2k}+q)qc_{k-1}}{P(\lambda q^k)}. \label{ck1} \end{equation} Now substituting the value of $P(\lambda q^k)$ in \eqref{ck1} and iterating it, we get \begin{equation} c_k=\frac{(-\lambda^2q;q^2)_k\, q^{2k}}{(q;q)_k(\lambda^2 q;q)_k}\;. \label{Hypergeometric ck1} \end{equation} The full solution to the recurrence equation \eqref{scaled recurrence1} is a linear combination of the ansatz over both the values of $\lambda$. Here $P(\lambda)=0$ and also $P(\bar{\lambda})=0$ (where $\bar{\lambda}=\frac{1}{\lambda}$). We can write the general solution for $S(w)$ as \begin{equation} S(w)=A\lambda^w\sum_{k=0}^{\infty}c_k(\lambda,q) q^{kw}+B\bar{\lambda}^w\sum_{k=0}^{\infty}c_k\left(\bar{\lambda},q\right)q^{kw}. \end{equation} We define \begin{align*} \psi(\lambda,x)=& \sum_{k=0}^{\infty}\frac{(-\lambda^2q;q^2)_k\, x^{k}}{(q;q)_k(\lambda^2 q;q)_k} =\sum_{k=0}^{\infty}\frac{(i\lambda \sqrt{q};q)_k(-i\lambda \sqrt{q};q)_k x^k}{(\lambda^2q;q)_k(q;q)_k}\\ =& \,_2 \phi_1(i\lambda \sqrt{q},-i\lambda \sqrt{q};\lambda^2 q;q,x) \end{align*} where $_2 \phi_1$ is a basic hypergeometric function. The general solution can be expressed as follows \begin{equation} S(w)=A\lambda^n\psi(\lambda,q^{w+2})+B\bar{\lambda}^w\psi\left(\bar{\lambda},q^{w+2}\right). \label{linear general solution1} \end{equation} Using the initial conditions, we can solve for $A$ and $B$. First we solve it for $P'_w$ with the initial conditions as $$S(-1)=0\qquad S(0)=\frac{1}{1-q}\;.$$ Substituting these initial conditions into equation \eqref{linear general solution1} and solving it for $A$ and $B$, we have $$A=\frac{-\lambda^2\psi\left(\bar{\lambda},q\right)}{(1-q)\left(\psi(\lambda,q)\psi\left(\bar{\lambda},q^2\right)-\lambda^2\psi\left(\bar{\lambda},q\right)\psi(\lambda,q^2)\right)}$$ and $$B=\frac{\psi(\lambda,q)}{(1-q)\left(\psi(\lambda,q)\psi\left(\bar{\lambda},q^2\right)-\lambda^2\psi\left(\bar{\lambda},q\right)\psi(\lambda,q^2)\right)}.$$ Similarly, we solve for $Q'_w$ using the initial conditions $$S(-1)=\alpha\qquad\text{and}\qquad S(0)=\frac{1}{1-q}.$$ We obtain $$A=\frac{(\alpha)(\lambda)(1-q)\psi\left(\bar{\lambda},q^2\right)-\lambda^2\psi\left(\bar{\lambda},q\right)}{(1-q)\left(\psi(\lambda,q)\psi\left(\bar{\lambda},q^2\right)-\lambda^2\psi\left(\bar{\lambda},q\right)\psi(\lambda,q^2)\right)}$$ and $$B=\frac{\psi(\lambda,q)-(\alpha)(\lambda)(1-q)\psi(\lambda,q^2)}{(1-q)\left(\psi(\lambda,q)\psi\left(\bar{\lambda},q^2\right)-\lambda^2\psi\left(\bar{\lambda},q\right)\psi(\lambda,q^2)\right)}.$$ Substituting the full solution for $P'_w(\alpha,q)$ and $Q'_w(\alpha,q)$ in \eqref{rational function} we have the expression in \eqref{full solution1}. \end{proof} By taking the limit of infinite $w$ in the generating function $G'_w$, we derive an expression for the generating function of $q$-secant numbers. \begin{proof}[Proof of Corollary \ref{cors1}] Consider the right hand side of \eqref{full solution1}. As above, we choose $\lambda$ to be the smaller root of the characteristic polynomial \eqref{polynomial}. For $w \rightarrow \infty$ we have \begin{multline} \psi(\lambda,q^{w+2})=\,_2\phi_1(i\lambda\sqrt{q},-i\lambda\sqrt{q};\lambda^2q;q,q^{w+2})\\ \rightarrow \,_2\phi_1(i\lambda\sqrt{q},-i\lambda\sqrt{q};\lambda^2q;q,0)=1\end{multline} and
$$ |\lambda^w| \rightarrow 0\;.$$ This implies \begin{equation} G'(t,q)=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\psi(\lambda,q^{2})}{(1+\lambda^2)\psi(\lambda,q)}}. \label{HP1} \end{equation} Using Heine's transformation formula given in \eqref{Hein}, we transform the basic hypergeometric functions given in \eqref{HP1} as follows \begin{align} \psi(\lambda,q) &=\frac{(-i\lambda\sqrt{q};q)_\infty(i\lambda q^{3/2};q)_\infty}{(\lambda^2q;q)_\infty(q;q)_\infty} \,_2\phi_1(i\lambda \sqrt{q},q;i\lambda q^{3/2};q,-i\lambda\sqrt{q}) \label{psiq} \end{align} and \begin{align} \psi(\lambda,q^2) &=\frac{(-i\lambda\sqrt{q};q)_\infty(i\lambda q^{5/2};q)_\infty}{(\lambda^2q;q)_\infty(q^2;q)_\infty} \,_2\phi_1(i\lambda \sqrt{q},q^2;i\lambda q^{5/2};q,-i\lambda\sqrt{q}). \label{psiq2} \end{align} Substituting the transformations \eqref{psiq} and \eqref{psiq2} into the limit \eqref{HP1} yields \begin{multline} G'(t,q)=\\\cfrac{1}{1-\dfrac{\lambda^2(1-q)\dfrac{(-i\lambda\sqrt{q};q)_\infty(i\lambda q^{5/2};q)_\infty}{(\lambda^2q;q)_\infty (q^2;q)_\infty} \,_2\phi_1(i\lambda\sqrt{q},q^2;i\lambda q^{5/2};q,-i\lambda\sqrt{q})}{(1+\lambda^2)\dfrac{(-i\lambda\sqrt{q};q)_\infty(i\lambda q^{3/2};q)_\infty}{(\lambda^2q;q)_\infty(q;q)_\infty} \,_2\phi_1(i\lambda \sqrt{q},q;i\lambda q^{3/2};q,-i\lambda\sqrt{q})}}\,. \end{multline} Expressing these basic hypergeometric functions by their explicit sums, we find \begin{align} G'(t,q)&=\cfrac{1}{1-\dfrac{\lambda^2(1-q)^2\sum\limits_{k=0}^{\infty}\dfrac{(i\lambda\sqrt{q};q)_k(q^2;q)_k}{(i\lambda q^{5/2};q)_k(q;q)_k}(-i\lambda\sqrt{q})^k}{(1+\lambda^2)(1-i\lambda q^{3/2})\sum\limits_{k=0}^{\infty}\dfrac{(i\lambda \sqrt{q};q)_k(q;q)_k}{(i\lambda q^{3/2};q)_k(q;q)_k}(-i\lambda\sqrt{q})^k }}\newline \\ &=\cfrac{1}{1-\dfrac{\lambda^2(1-q)\sum\limits_{k=0}^{\infty}\dfrac{(1-q^{k+1})}{(1-i\lambda q^{k+1/2})(1-i\lambda q^{k+3/2})}(-i\lambda\sqrt{q})^k}{(1+\lambda^2)\sum\limits_{k=0}^{\infty}\dfrac{(-i\lambda\sqrt{q})^k}{(1-i\lambda q^{k+1/2})} }}\;. \label{simple2} \end{align} To simplify further we consider the expression in \eqref{simple2}. We aim to simplify the terms in the sums on the right hand side of \eqref{simple2}. For this we let \begin{equation} N=\sum\limits_{k=0}^{\infty}\dfrac{(1-q^{k+1})}{(1-i\lambda q^{k+1/2})(1-i\lambda q^{k+3/2})}(-i\lambda\sqrt{q})^k \end{equation} and \begin{equation} D=\dfrac{(-i\lambda\sqrt{q})^k}{(1-i\lambda q^{k+1/2})} \;. \end{equation} We substitute $x=-i\lambda\sqrt{q}$ and apply partial fraction decomposition to get \begin{equation} N=\dfrac{(q-x^2)\sum\limits_{k=0}^{\infty}\dfrac{x^k}{1+xq^k}}{x^2(q-1)}-\frac{q}{x^2(q-1)}\; \label{N2} \end{equation} and \begin{equation} D=\sum\limits_{k=0}^{\infty}\frac{x^k}{1+xq^k}\;. \label{d2} \end{equation} Substituting \eqref{N2} and \eqref{d2} into \eqref{simple2} and simplifying, we get the final result as \eqref{halfplaneofsecant}. \end{proof} Next we extract the coefficient of $t^{2N}$ of $G'(t,q)$ given in \eqref{halfplaneofsecant}. \begin{proof}[Proof of corollary \ref{cors2}] To prove this corollary we will again use the Lemma \ref{rectlemma}. The sum in \eqref{halfplaneofsecant} can be identified with $R(-i\lambda \sqrt{q},i\lambda \sqrt{q},q)$, so we get \begin{align} G'(t,q)=&(1+\lambda)^2R(-i\lambda\sqrt{q},i\lambda\sqrt{q},q)\nonumber\\ =&(1+\lambda)^2 \sum\limits_{n=0}^{\infty}\dfrac{q^{n^2+n} \lambda^{2n} (1-\lambda^2 q^{2n+1}) }{(1+\lambda^2 q^{2n+1})}. \end{align} We remind that $t$-dependence on the right hand side is implicit in $\lambda=\lambda(t)$. To extract the coefficient of $t^{2N}$, we evaluate the contour integral \begin{equation} [t^{2N}]G'(t,q)=\frac{1}{2 \pi i}\oint \dfrac{G'(t,q)}{t^{2N+1}}dt. \end{equation} Using the variable substitution from \eqref{valuet}, we have \begin{multline} [t^{2N}]G'(t,q)=\\\frac{1}{2 \pi i}\oint \left(\dfrac{(1+\lambda)^{2N} \left(\sum\limits_{n=0}^{\infty}\dfrac{q^{n^2+n} \lambda^{2n} (1-\lambda^2 q^{2n+1}) }{(1+\lambda^2 q^{2n+1})}\right)(1-\lambda^2)}{\lambda^{2N}(1-q)^{2N})}\right)\frac{d\lambda}{\lambda}. \end{multline} We thus have \begin{equation} [t^{2N}]G'\left(t,q\right)=[\lambda^0]H'_N(\lambda,q) \end{equation} and to extract the constant term in $\lambda$ on the right hand side we now expand $H'_N(\lambda,q)$ as a Laurent series in $\lambda$. For this we write \begin{equation} [\lambda^0]H'_N(\lambda,q)=[\lambda^0]T_1 \sum \limits_{k=0}^{\infty}T_2\;, \label{constantterm} \end{equation} where \begin{equation*} T_1=\dfrac{\left(\lambda+\dfrac{1}{\lambda}\right)^{2N} (1-\lambda^2)}{(1-q)^{2N}} \end{equation*} and \begin{equation} T_2=\dfrac{q^{n^2+n} \lambda^{2n} (1-\lambda^2 q^{2n+1})}{(1+\lambda^2 q^{2n+1})}. \end{equation} We find the series expansions as \begin{equation} T_1=\dfrac{\sum\limits_{k=0}^{2N+1}\dfrac{(2N)!(2N-2k+1)\lambda^{2k-2N}}{k!(2N-k+1)!}}{(1-q)^{2N}} \label{bt1} \end{equation} and \begin{equation} T_2=q^{n^2+n} \lambda^{2n}\left(2\sum\limits_{l=0}^{\infty}(-1)^l(\lambda^2 q^{2n+1})^l-1\right). \label{bt2} \end{equation} Next we substitute the expression \eqref{bt1} and \eqref{bt2} in \eqref{constantterm}, which after some simplification leads to \begin{multline} H'_N(\lambda,q)=\dfrac{1}{(1-q)^{2N}}\\ \left(2\sum\limits_{k=0}^{2N+1}\sum\limits_{n=0}^{\infty}\sum\limits_{l=0}^{\infty} (-1)^lq^{n^2+n+2nl+l}\dfrac{(2N)!(2N-2k+1)\lambda^{2k-2N+2n+2l}}{k!(2N-k+1)!}\right.\\ \left.-\sum\limits_{k=0}^{2N+1}\sum\limits_{n=0}^{\infty}q^{n^2+n} \dfrac{(2N)!(2N-2k+1)\lambda^{2k-2N+2n}}{k!(2N-k+1)!}\right). \label{bt} \end{multline} We aim to get the constant term in $\lambda$, so we combine the powers of $\lambda$ and equate them to $0$. This fixes the summation index $k$, and we get \begin{multline} [t^{2N}]G'\left(t,q\right)=\dfrac{1}{(1-q)^{2N}}\times\\ \left(2\sum\limits_{n,l=0}^{\infty} (-1)^lq^{n^2+n+2nl+l}\dfrac{(2N)!(2n+2l+1)}{(N-n-l)!(N+n+l+1)!}\right.\\ \left.-\sum\limits_{n=0}^{\infty}q^{n^2+n} \dfrac{(2N)!(2n+1)}{(N-n)!(N+n+1)!}\right). \end{multline} Completing the square in the first sum and changing summation indices leads to \begin{multline} [t^{2N}]G'\left(t,q\right)=\dfrac{1}{(1-q)^{2N}}\left(\sum\limits_{m=0}^{N}q^{m^2+m}\dfrac{(2N)!(2m+1)}{(N-m)!(N+m+1)!}\right.\\\left.\left(2\sum\limits_{l=0}^{m} (-1)^l q^{-l^2}-1\right)\right). \end{multline} \end{proof} \section{Identities} The central results of this chapter have been given in Theorems \ref{theorem3} and \ref{theoremnot3}, which express finite continued fractions in terms of basic hypergeometric functions. For example, for $q$-tangent numbers we have \begin{multline*} \cfrac{1}{1-\cfrac{\alpha^2(1-q)(1-q^2)}{1-\cfrac{\alpha^2(1-q^2)(1-q^3)}{{\ddots-}{\cfrac{\alpha^2(1-q^{w-1})(1-q^{w})}{1-\alpha^2(1-q^w)(1-q^{w+1})}}}}}=\\ \cfrac{1}{1-\dfrac{\lambda^2(1-q)\left[\bar{\lambda}^w\phi(\lambda,q^{3})\phi\left(\bar{\lambda},q^{w+3}\right)-\lambda^w\phi\left(\bar{\lambda},q^{3}\right)\phi(\lambda,q^{w+3})\right]}{{(1+\lambda^2)\left[\bar{\lambda}^w\phi(\lambda,q^{2})\phi\left(\bar{\lambda},q^{w+3}\right) -\lambda^{w+2}\phi\left(\bar{\lambda},q^{2}\right)\phi(\lambda,q^{w+3})\right]}}} \end{multline*} and a similar result holds for $q$-secant numbers. The point we would like to make in this section is that these results can be interpreted as giving hierarchies of identities for basic hypergeometric functions. For $w$ small, the left hand side is a relatively simple rational function in $t$ and $q$, whereas the right hand side is a weighted ratio of products of basic hypergeometric functions at specific arguments. We make the resulting identities explicit for $w=1$ in the following corollary. \begin{corollary} \begin{multline} \tiny \frac{1-q^2}{1-\nu^2}=\dfrac{\left[ \splitdfrac{\,_2\phi_1(\nu,-\nu;-\nu^2 q;q,q^3)\,_2\phi_1\left(-\bar{\nu},\bar{\nu};-\nu^2\bar{ q};q,q^4\right)}{+\nu\,_2\phi_1\left(-\bar{\nu},\bar{\nu};-\nu^2\bar{ q};q,q^3\right)\,_2\phi_1(\nu,-\nu;-\nu^2 q;q,q^4)}\right]}{\left[\splitdfrac{\,_2\phi_1(\nu,-\nu;-\nu^2 q;q,q^2)\,_2\phi_1\left(-\bar{\nu},\bar{\nu};-\nu^2\bar{ q};q,q^4\right)}{-\nu^4\,_2\phi_1\left(-\bar{\nu},\bar{\nu};-\nu^2 \bar{ q};q,q^2\right)\,_2\phi_1(\nu,-\nu;-\nu^2 q;q,q^4)}\right]} \end{multline} where $\nu=i \lambda$, $\bar{\nu}=\frac{1}{\nu}$ and $\bar{q}=\frac{1}{q}$ , and \begin{multline} \tiny \frac{1-q}{1-\mu^2\bar{q}}=\dfrac{\left[\splitdfrac{\,_2\phi_1(\mu,-\mu;-\mu^2;q,q^2)\,_2\phi_1\left(-q\bar{\mu},q\bar{\mu};-q^2\bar{\mu}^2;q,q^3\right)}{+\left(\mu^2\bar{q}\right)\,_2\phi_1\left(-q\bar{\mu},q\bar{\mu};-q^2\bar{\mu}^2;q,q^2\right)\,_2\phi_1(\mu,-\mu;-\mu^2;q,q^3)}\right]}{\left[\splitdfrac{\,_2\phi_1(\mu,-\mu;-\mu^2;q,q)\,_2\phi_1\left(-q \bar{\mu},q\bar{\mu};-q^2\bar{\mu}^2;q,q^3\right)}{-\mu^4\bar q^2\,_2\phi_1\left(-q\bar{\mu},q\bar{\mu};-q^2\bar{\mu}^2;q,q\right)\,_2\phi_1(\mu,-\mu;-\mu^2;q,q^3)}\right]} \end{multline} where $\mu=i \lambda \sqrt{q}$, $\bar{\mu}=\frac{1}{\mu}$ and $\bar{q}=\frac{1}{q}$. \end{corollary} \begin{proof} Insert $w=1$ in Theorems \ref{theorem3} and \ref{theoremnot3} and simplify. \end{proof} To the best of our knowledge these identities are new. It would be interesting to find an alternative derivation and perhaps deeper understanding of their meaning.
\end{document} | arXiv |
\begin{document}
\title{ State generatings for Jones and Kauffman-Jones polynomials } \author{ Liangxia Wan \thanks{\it E-mail address: [email protected]$. }
\\
\small\it Department of Mathematics, Beijing Jiaotong University, Beijing $100044$, China}
\date{} \maketitle
\noindent{\small {\bf Abstract} A state generating is introduced to determine the Jones polynomial of a link. Formulae for two infinite families of knots are shown by applying this method, the second family of which are proved to be non-alternating. Moreover, the method is generalized to compute the Jones-Kauffman polynomial of a virtual link. As examples, formulae for one infinite family of virtual knots are given.
\vskip 2mm \noindent{\it keywords:} Link; virtual link; state generating; embedding presentation
\vskip 5mm \noindent {\bf $1.$ Introduction} \vskip 5mm
\noindent
Given a diagram $L$ of a link in $R^3$ (or $S^3$), denote a crossing by a letter, regard $e=(u^r,v^s)$ as an edge if no any other crossings along the line between $u^r$ and $v^s$, then an embedding presentation $L=(V,E)$ with a rotation ${\cal P}=\sum\limits_{u\in V}\sigma_u$ is obtained \cite{Wan}. Here, $V$ is the set of all crossings and $E$ is the set of edges. $\sigma_u$ is an anticlockwise rotation of edges incident with $u$. If $e$ is an overcrossing at $u$, then $r=+$ (omitted for brevity), otherwise $r=-$. Throughout this paper a link $L$ (or a virtual link) is always a corresponding embedding presentation, also a marked diagram (or a marked virtual diagram) unless otherwise specified. The link equivalent class $[L]$ is the corresponding link in $R^3$ (or $S^3$) and the virtual link equivalent class $[L]$ is the corresponding virtual link in $S\times I$.
The Jones polynomial is an invariant of $[L]$ which brought on major advances in knot theory \cite{Jo85}.
The Kauffman bracket polynomial of a link was introduced, which is a simple definition to calculate the corresponding Jones polynomial \cite{Ka87}. Based on the Kauffman bracket polynomial, several methods were proposed to compute Jones polynomials of links via Tutte polynomials \cite{Tu54,Ka89} and Bollob${\rm\acute{a}}$s- Riordan polynomials \cite{BR01,BR02} for some graphs.
A spanning tree expansion of Jones polynomial was first introduced by constructing a signed graph in \cite{Th87}. This method was extended in \cite{Ka89}. The Jones polynomial of any link equivalent class can also be calculated from the Bollob${\rm\acute{a}}$s-Riordan polynomial of the ribbon graph via a certain oriented ribbon graph \cite{DFKLS}. In addition, a matrix for calculating the Jones polynomial of a knot equivalent class was given \cite{Zu95}. However, since determining the Tutte polynomial of a graph is \#P-hard, it is still tough to calculate the Jones polynomial of a link equivalent class [L] \cite{JVW90}, especially a non-alternating link $L$ with a large crossings.
A virtual link in $S\times I$ and its Kauffman-Jones polynomial were introduced in \cite{Ka99}, which are the generalizations of a link in $R^3$ (or $S^3$) and its Jones polynomial. Similarly, given a virtual diagram $L$ of a virtual link in $S\times I$, denote a crossing by a letter, regard $e=(u^r,v^s)$ as an edge if no any other crossings along the line between $u^r$ and $v^s$, then an embedding presentation $L=(V,E)$ with a rotation ${\cal P}=\sum\limits_{u\in V}\sigma_u$ is obtained \cite{Wan}. Here, $V$ is the set of all crossings and $E$ is the set of edges. $\sigma_u$ is an anticlockwise rotation of edges incident with $u$. If $u$ is a classical crossing, then $u^+$ (omitted for brevity) and $u^-$ represent an overcrossing and an undercrossing at $u$ respectively, otherwise $u^+$ and $u^-$ represent two occurrences of $u$. Throughout this paper a virtual link $L$ is always a corresponding embedding presentation, also a marked virtual diagram unless otherwise specified. The virtual link equivalent class $[L]$ is the corresponding virtual link in $S\times I$.
Correspondingly, approaches for the Jones polynomial of a link equivalent class [L] were extended to compute the Kauffman-Jones polynomial of a virtual link equivalent class. Firstly, the Kauffman-Jones polynomial of a checkerboard colorable virtual link $L$ can be calculated via the Bollob${\rm\acute{a}}$s-Riordan poly- nomial of the corresponding ribbon graph \cite{CP07}. Secondly, a relative variant of the other generalization of the Tutte polynomial can be used to compute the Kauffman-Jones polynomials of some virtual links equivalent classes \cite{DH10}. Thirdly, the Kauffman-Jones polynomial of a virtual link equivalent class was computed via the the signed ribbon graph polynomial of its Seifert ribbon graph \cite{CV08}. In fact, the Jones polynomial of a link equivalent class and the Kauffman-Jones polynomial of a virtual link equivalent class can be computed from the signed ribbon graph polynomial of any of their signed ribbon graphs \cite{Ch09}.
This paper introduces a new method called a state generating to calculate Jones polynomials of links based on their bracket polynomials and generalizes this approach to calculate the Kauffman-Jones polynomial of a virtual link. \vskip 3mm
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Given a link $L$ with $n$ crossings for $n\ge 2$, let $\sigma_u=(e_1,e_2,e_3,e_4)$ be the rotation at $u\in V(L)$ where $e_1=(u,x_1^{r_1}),e_2=(u^-,x_2^{r_2}),e_3=(u,x_3^{r_3}),e_4=(u^-,x_4^{r_4})$, $r_i\in\{+,-\}$ for $1\le i\le 4$. If one replaces passes $x_1^{r_1}ux_3^{r_3},x_2^{r_2}u^-x_4^{r_4}$ with passes $x_1^{r_1}ux_4^{r_4},x_2^{r_2}ux_3^{r_3}$ respectively, then gets a state $A$ at $u$ denoted by $s_u=A$. Otherwise, if one replaces $x_1^{r_1}ux_3^{r_3},x_2^{r_2}u^-x_4^{r_4}$ with $x_1^{r_1}u^-x_2^{r_2},x_3^{r_3}u^-x_4^{r_4}$ respectively, then gets a state $A^{-1}$ at $u$ denoted by $s_u=A^{-1}$ (See Fig.$0$).
By assigning one and only one state of $A$ and $A^{-1}$ to each $u\in V(L)$, one obtains a state $s$ of $L$ and a corresponding graph called the {\it state graph} $G(s)$ of $s$ which consists of loops. Two states $s$ and $s'$ of $L$ are distinct if and only if there exists a crossing $u$ such that $s_u\ne s'_u$. Set $S_L$ to be the set of all of states of $L$. It is obvious that $ S_L$ contains $2^n$ elements. Let $c(s),b(s)$ and $l(s)$ denote the number of crossings, $A^{-1}$ and loops in a state $s$ of $L$ respectively. Then $a(s)=c(s)-b(s)$ is the number of state $A$ in $s$. Let $p_i(L)=\sum\limits_{s\in S_L,l(s)=i}A^{a(s)-b(s)}$. Then the Kauffman bracket polynomial is given below
$$<L>=\sum\limits_{i\ge 1}p_i(L)(-A^2-A^{-2})^{i-1}.$$
Thus, the Jones polynomial of $[L]$ is deduced as follow
$$V_L(t)=(-A)^{-3\omega(L)}<L>
$$
where $\omega(L)$ is the writhe of $L$ and $t=A^{-4}$. Let $\rho_h(V_L(t))$ and $\rho_l(V_L(t))$ denote the highest and lowest powers of $t$ occurring in $V_L(t)$ respectively. Then the value
$br(V_L(t))=\rho_h(V_L(t))$- $\rho_l(V_L(t))$ is called the {\it breath} of $V_L(t)$. Obviously,
it is enough to calculate $p_i(L)$ and $\omega(L)$ in order to obtain $V_L(t)$.
Similarly, set $L$ to be a virtual link and set $\Gamma L$ to be a set of its classical crossings with $|\Gamma L|=n$ for $n\ge 1$. Let $\sigma_u=(e_1,e_2,e_3,e_4)$ be the rotation at $u\in V(\Gamma L)$ where $e_1=(u,x_1^{r_1}),e_2=(u^-,x_2^{r_2}),e_3=(u,x_3^{r_3}),e_4=(u^-,x_4^{r_4})$, $r_i\in\{+,-\}$ for $1\le i\le 4$. If one replaces passes $x_1^{r_1}ux_3^{r_3},x_2^{r_2}u^-x_4^{r_4}$ with passes $x_1^{r_1}ux_4^{r_4},x_2^{r_2}ux_3^{r_3}$ respectively, then gets a state $A$ at $u$ denoted by $s_u=A$. Otherwise, if one replaces $x_1^{r_1}ux_3^{r_3},x_2^{r_2}u^-x_4^{r_4}$ with $x_1^{r_1}u^-x_2^{r_2},x_3^{r_3}u^-x_4^{r_4}$ respectively, then gets a state $A^{-1}$ at $u$ denoted by $s_u=A^{-1}$ (See Fig.$0$). By assigning one and only one state of states $A$ and $A^{-1}$ to each $u\in V(\Gamma L)$, one obtains a state $s$ of $L$ and then gets the corresponding {\it state graph} which consists of components. Two states $s$ and $s'$ of $L$ are distinct if and only if there exists a crossing $u\in V(\Gamma L)$ such that $s_u\ne s'_u$. Set $S_L$ to be the set of all of states of $L$. It is obvious that $ S_L$ contains $2^n$ elements. Let $c(s),b(s)$ and $l(s)$ denote the number of classical crossings, $A^{-1}$ and connected components in a state $s$ of $L$ respectively. Then $a(s)=c(s)-b(s)$ is the number of state $A$ in a state $s$. Let $p_i(L)=\sum\limits_{s\in S_L,l(s)=i}A^{a(s)-b(s)}$. So the Kauffman-Jones polynomial $f_L(A)$ for a virtual link is given below
$$f_L(A)=(-A)^{-3\omega(L)}\sum\limits_{i\ge 1}p_i(L)(-A^2-A^{-2})^{i-1}$$
where $\omega(L)$ is the writhe of $L$. \vskip 3mm
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\begin{footnotesize}
\put(-18.5,33){{$x_1$}}
\put(-18.5,27){{$x_2$}}
\put(-18.5,16){{$x_3$}}
\put(1.5,33){{$x_1$}}
\put(2,24.2){{$x_2^-$}}
\put(1.5,16){{$x_3$}}
\put(21.5,33){{$x_1$}}
\put(21.5,27){{$x_2$}}
\put(22,18.2){{$x_3^-$}}
\put(41.5,33){{$x_1$}} \put(42,24.2){{$x_2^-$}} \put(42,18.2){{$x_3^-$}}
\put(62,30.2){{$x_1^-$}}
\put(61.5,23){{$x_2$}}
\put(61.5,16){{$x_3$}}
\put(82,30.2){{$x_1^-$}}
\put(82,24.2){{$x_2^-$}}
\put(81.5,16){{$x_3$}}
\put(102,30.2){{$x_1^-$}}
\put(101.5,23){{$x_2$}}
\put(102,18.2){{$x_3^-$}}
\put(122,30.2){{$x_1^-$}}
\put(122,24.2){{$x_2^-$}}
\put(122,18.2){{$x_3^-$}}
\put(3,55){{$x_1$}}
\put(21.5,51){{$x_1$}}
\put(42.5,54.3){{$x_1^-$}}
\put(63.8,60.5){{$x_1$}} \put(63.8,54.5){{$x_2$}} \put(63.8,48.5){{$x_3$}}
\put(83.8,60.5){{$x_1$}} \put(83.8,54.5){{$x_2$}} \put(83.8,48.5){{$x_3$}} \put(0.5,41){{$O$}} \put(20.5,41){{$s_1$}} \put(40.5,41){{$s_2$}} \put(60.5,41){{$RT_0$}} \put(73.5,41){{oriented $RT_0$}}
\put(-21,11){{$s_1(0)$}} \put(-1,11){{$s_2(0)$}} \put(19,11){{$s_3(0)$}} \put(39,11){{$s_4(0)$}} \put(59,11){{$s_5(0)$}} \put(79,11){{$s_6(0)$}} \put(99,11){{$s_7(0)$}} \put(119,11){{$s_8(0)$}} \put(10,1){{Fig.1: Jones polynomial of the right handed trefoil $RT_0$}} \end{footnotesize} \end{picture} \end{center}
Now we introduce a state generating to calculate the Jones polynomial of a link and the Kauffman-Jones polynomial for a virtual link. In order to calculate the Jones polynomial of a link $L$ (or a Kauffman-Jones polynomial of a virtual link $L$), choose a link $L_1$ (or a virtual link $L_1$ ) with $|V(L_1)|<|V(L)|$ (or $|V(\Gamma L_1)|<|V(\Gamma L)|$) such that each state of $L$ is generated by some state of $L_1$. This method is called a {\it state generating}. If a state $s_1$ of $L_1$ generates a state of $s$ of $L$, then $s_1$ is called the {\it parent} of $s$ denoted by $par(s)$.
For example, in order to calculate the Jones polynomial of the right handed trefoil $RT_0$, we choose the unknot $O$ shown in Fig.1. $O$ contains two distinct states $s_j$ whose state graphs are $(x_1)(x_1)$ and $(x_1^-x_1^-)$, respectively, for $1\le j\le 2$. It is clear that $p_1(O)=A^{-1}$ and $p_2(O)=A$. The state $s_1$ generates four distinct states $s_j(0)$ of $RT_0$ for $1\le j\le 4$. The state $s_2$ generates four distinct states $s_j(0)$ of $RT_0$ for $5\le j\le 8$ (See Fig.1). We show their loops of state graphs of $s_j(0)$ with loop number in brackets in sequences below for $1\le j\le 8$
\vskip 2mm
\hskip 5mm $(x_1x_3x_2)(x_1x_2x_3)\{2\}$ \hskip 15mm $(x_1x_2^-x_1x_3x_2^-x_3)\{1\}$ \hskip 15mm
$(x_1x_2x_3^-x_2x_1x_3^-)\{1\}$
\hskip 5mm $(x_1x_2^-x_1x_3^-)(x_2^-x_3^-)\{2\}$ \hskip 12mm
$(x_1^-x_3x_2x_1^-x_2x_3)\{1\}$ \hskip 15mm $(x_1^-x_3x_2^-x_3)(x_1^-x_2^-)\{2\}$
\hskip 5mm $(x_1^-x_3^-)(x_1^-x_2x_3^-x_2)\{2\}$ \hskip 12mm $(x_1^-x_3^-)(x_1^-x_2^-)(x_2^-x_3^-)\{3\}$
\vskip 2mm
Obviously, $$\left\{ \begin{array}{ll}
p_1(RT_0)=A^{2}p_1(O)+2p_2(O)=A+2A=3A,\\
p_2(RT_0)=2p_1(O)+(A^2+A^{-2})p_2(O)=3A^{-1}+A^{3},\\
p_3(RT_0)=A^{-2}p_1(O)=A^{-3}.
\end{array}
\right.
$$
Then $$<RT_0>=3A+(3A^{-1}+A^{3})(-A^2-A^{-2})+A^{-3}(-A^2-A^{-2})^{2}=A^{-7}-A^{-3}-A^{5}.$$
Since $\omega(RT_0)=3$,
$$V_{RT_0}(t)=(-A)^{-9}(A^{-7}-A^{-3}-A^{5})=A^{-4}+A^{-12}-A^{-16}=t+t^{3}-t^{4}.$$ \vskip 3mm Consider $RT_0$. Add $2n$ crossings $y_i$ on $(x_1^-,x_2)$ in sequence and add $2n$ crossings $z_i$ on $(x_1^-,x_3)$ in sequence for $1\le i\le 2n$, delete edges $(x_1,x_2^-),(x_1^-,x_2),(x_1^-,x_3)$, and then add edges $(x_2,y_{2n}^-)$, $(x_2^-,z_{2n})$, $(x_3,z_{2n}^-)$, $(y_{2k},y_{2k-1}^-),(y_{2k}^-,y_{2k-1})$, $(z_{2k},z_{2k-1}^-),(z_{2k}^-,z_{2k-1})$, $(z_{2k},y_{2k+1}^-),(y_{2k},z_{2k-1}^-)$, $(z_{2k}^-,z_{2k+1})$ and $(y_{2k}^-,y_{2k+1})$ for $1\le k\le n$ where $y_{2n+1}^-=x_2^-$, $y_{2n+1}=x_2$ and $z_{2n+1}=x_3$. A type of knots $RT_n$ are obtained for $n\ge 1$, which belong to the first type of knots called {\it $2$-string alternating} knots. $RT_3$ is shown in Fig.2. The Jones polynomials of $RT_n$ are obtained for $n\ge 1$.
\vskip 3mm
\noindent{\bf Theorem $1.1$} {\it For $n\ge 1$ $$V_{RT_n}(t)=\frac{t^{3n}}{\alpha-\bar\alpha}((t+t^3-t^4)(\alpha^{n+1}-\bar\alpha^{n+1})- (1+t-t^2)(\alpha^{n}-\bar\alpha^{n}))$$ where $$ \left\{
\begin{array}{ll}
\alpha+\bar\alpha=t^{-2}-t^{-1}+2-t+t^2;\\
\alpha\cdot\bar\alpha=1.
\end{array} \right. $$} \vskip 3mm Given a knot $KV_0$ in Fig.3, delete edges $(x_3^-,x_6^-)$, add $2n$ crossings $y_i$ on $(x_2^-,x_3)$ in sequence, $2n$ crossings $z_i$ on $(x_6,x_4^-)$ in sequence for $1\le i\le 2n$, add edges $(x_2^-,y_{1})$, $(x_6^-,y_1^-)$, $(x_6,z_{1})$, $(y_{2k-1}^-,y_{2k})$, $(y_{2k-1},y_{2k}^-)$, $(y_{2k}^-,y_{2k+1})$, $(y_{2k},z_{2k-1})$, $(z_{2k-1}^-,z_{2k})$, $(z_{2k-1},z_{2k}^-)$, $(z_{2k},z_{2k+1}^-)$, $(z_{2k}^-,y_{2k+1})$ for $1\le k\le n-1$ where $y_{2n+1}=x_3$, $y_{2n+1}^-=x_3^-$ and $z_{2n+1}^-=x_4^-$. Then the second type of knots $KV_n$ are given for $n\ge 1$. $KV_1$ is the knot $10_{152}$ \cite{Ro76}. Each $KV_n$ is non-alternating and its Jones polynomial is shown for $n\ge 1$.
\vskip 3mm \noindent{\bf Theorem $1.2$}
{\it $KV_n$ are non-alternating knots for $n\ge 1$.}
\vskip 3mm \noindent{\bf Theorem $1.3$}
{\it For $n\ge 1$, $$V_{KV_n}(A)=A^{(12n+18)}\sum\limits_{i=1}^3g_i(n)$$ where \begin{eqnarray*} g_1(n) & = & (A^4+1+A^{-4})A^{-4n-6}+\sum\limits_{i=0}^{n-1}A^{-4i}((\alpha_1^{n-i}-\bar\alpha_1^{n-i})-(2A^4-A^{-4})
(\alpha_1^{n-1-i}-\bar\alpha_1^{n-1-i}))\\
& + & (A^{-2}-2A^{-6}+A^{-10})\sum\limits_{i=0}^{n-1}A^{-4i}(\alpha_1^{n-1-i}-\bar\alpha_1^{n-1-i})\\
& + & (A^{-6}-A^{-10})\sum\limits_{j=0}^{n-1}A^{-4j}(1+A^{8n-8j-4})+\displaystyle\frac{A^2-2A^{-2}
+A^{-6}}{1-A^8}\sum\limits_{j=0}^nA^{-4j}(1-A^{8n-8j}), \end{eqnarray*}
$ g_2(n) = \displaystyle\frac{A^2-A^{-2}+A^{-6}}{A^{4}+1}(1-A^{8n})+\displaystyle\frac{A^6+A^{-6}}{A^{4}+1}(A^{8n+4}-1), $
$g_3(n) = \displaystyle\frac{(A^2+A^{-2})(A^4-1+A^{-4})}{\alpha_2-\bar\alpha_2}((1-A^{12}+A^6-A^2)
(\alpha_2^{n+1}-\bar\alpha_2^{n+1})\\$
\hskip 9mm $+ (A^{12}-A^8+A^{4}-A^2)(\alpha_2^{n}-\bar\alpha_2^{n})), $ \vskip 1mm $ \left\{
\begin{array}{ll}
\alpha_1+\bar\alpha_1=A^{8}+2A^{4}+1-2A^{-4};\\
\alpha_1\cdot\bar\alpha_1=A^{12}+2A^{8}-2-A^{-4}+A^{-8},
\end{array} \right. $ $ \left\{
\begin{array}{ll}
\alpha_2+\bar\alpha_2=A^{8}+A^{4}-1-A^{-4};\\
\alpha_2\cdot\bar\alpha_2=A^{8}-2A^{4}-2A^{-4}-2A^{-8}.
\end{array} \right. $} \vskip 3mm Let $x_3$ be a virtual crossing in $RT_n$ for $n\ge 0$. Then a type of virtual knots $RT'_n$ are obtained. Their Kauffman-Jones polynomials are as follows for $n\ge 1$. \vskip 3mm \noindent{\bf Theorem $1.4$}
{\it For $n\ge 1$ $$f_{RT'_n}(A)=\frac{A^{-12n}}{\alpha-\bar\alpha}((2A^{-4}-A^{-10})(\alpha^{n+1}-\bar\alpha^{n+1})- (1-A^{-2}+A^{-6}+A^{-8}-A^{-10})(\alpha^{n}-\bar\alpha^{n}))$$ where $$ \left\{
\begin{array}{ll}
\alpha+\bar\alpha=A^8-A^4+2-A^{-4}+A^{-8};\\
\alpha\cdot\bar\alpha=1.
\end{array} \right. $$ }
\vskip 3mm
This paper is organized as follows. In Section $2$, we use the state generating method introduced in Section $1$ to study the properties of $RT_n$ and then prove Theorem $1.1$ for $n\ge 1$. In Section $3$, we prove Theorems $1.2$ and $1.3$. In Section $4$ we prove Theorem $1.4$ by applying the state generating method for an infinite family of virtual links $RT_n'$ for $n\ge 1$. Finally some open problems are given in Section $5$.
\vskip 5mm \noindent{\bf $2.$ Jones polynomials of $RT_n$ }
\vskip 5mm In this section, we divide the set $S(RT_n)$ of all of states for $RT_n$ into four set $S_j(n)$ for $j\in \{\hbox{\rm I},\hbox{\rm II},\hbox{\rm III},\hbox{\rm IV}\}$ for $n\ge 1$. We study the recursive formulae for $S_j(n)$ and then prove Theorem $1.1$. \vskip 12mm
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\begin{footnotesize}
\put(-7,45){{$x_{1}$}} \put(-7,41){{$z_{1}$}} \put(-7,37){{$z_{2}$}} \put(-7,32.5){{$z_{3}$}} \put(-7,26.5){{$z_{4}$}} \put(4.5,45){{$y_{1}$}} \put(4.5,41.5){{$y_{2}$}} \put(4.5,37){{$y_{3}$}} \put(4.5,30.5){{$y_{4}$}} \put(4.5,27){{$y_{2n-1}$}} \put(4.5,23.5){{$y_{2n}$}} \put(-12,23){{$z_{2n-1}$}} \put(-8,17){{$z_{2n}$}} \put(4,11){{$x_{3}$}} \put(4.5,19){{$x_{2}$}}
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\put(-5,4){{(a) $RT_{n}$ }} \put(40,4){{(b) oriented $RT_{n}$ }} \put(95,4){{(c) $RT_{3}$ }} \put(20,-5){{Fig.2: A type of $2$-string alternating knots $RT_{n}$ }} \end{footnotesize} \end{picture} \end{center} \vskip 3mm
$RT_0$ has eight distinct states $s_j(0)$ shown in Fig.$1$ for $1\le j\le 8$. Each state $s(0)$ of $RT_0$ generates sixteen distinct states of $RT_1$ according to distinct states of $y_i$ and $z_i$ for $1\le i\le 2$. Generally, each state $s(n-1)$ of $RT_{n-1}$ generates sixteen distinct states of $RT_n$ for $n\ge 1$ according to distinct states of $y_i$ and $z_i$ for $2n-1\le i\le 2n$. Let $S({RT}_n)$ denote the set of all of distinct states of $RT_n$ for $n\ge 0$ and set
$$ \begin{array}{ll}
S_{\hbox{\rm\scriptsize I}}(n)=\{s\in S(RT_n),s_{x_2}=s_{x_3}=A|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^2, s_{y_i}=A,\\
\hskip 18mm s_{z_{2k-1}}=s_{z_{2k}}=s_{z_i}=A^{-}\mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{z_i}=A^-, s_{y_i}=A \mbox{ for }1\le i\le 2n \}, \end{array} $$ $ \begin{array}{ll}
S_{\hbox{\rm\scriptsize II}}(n)=\{s\in S(RT_n),s_{x_2}=s_{x_3}=A|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2}, s_{y_i}=A, \\ \hskip 18mm s_{z_i}=A^{-} \mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{z_i}=A^-, s_{x_1}=s_{y_i}=A \mbox{ for }1\le i\le 2n \}, \end{array} $ $$ \begin{array}{ll}
S_{\hbox{\rm\scriptsize III}}(n)=\{s\in S(RT_n),s_{x_2}\cdot s_{x_3}\ne A^2|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^2, s_{y_i}=A,\\
\hskip 18mm s_{z_{2k-1}}=s_{z_{2k}}=s_{z_i}=A^{-}\mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{z_i}=A^-, s_{y_i}=A \mbox{ for }1\le i\le 2n\}, \end{array} $$ $ \begin{array}{ll}
S_{\hbox{\rm\scriptsize IV}}(n)=\{s\in S(RT_n),s_{x_2}\cdot s_{x_3}\ne A^2|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2}, s_{y_i}=A, \\ \hskip 18mm s_{z_i}=A^{-} \mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{y_i}=A, s_{z_i}=A^- \mbox{ for }1\le i\le 2n \}. \end{array} $
\vskip 2mm
Obviously, there exists one and only one $j\in \{\hbox{\rm I},\hbox{\rm II},\hbox{\rm III},\hbox{\rm IV}\}$ such that $s\in S_j(n)$ for each $s\in S({RT}_n)$. Given $s({n-1})\in S({RT}_{n-1})$, it generates sixteen distinct states $s_j({n})$ of $S({RT}_{n})$ as follows for $1\le j\le 16$: \vskip 2mm \hskip 15mm $s_1(n)$ with $s_{y_{2n-1}}=s_{y_{2n}}=s_{z_{2n-1}}=s_{z_{2n}}=A$
\hskip 15mm $s_2(n)$ with $s_{y_{2n-1}}=A^-$ and $s_{y_{2n}}=s_{z_{2n-1}}=s_{z_{2n}}=A$
\hskip 15mm $s_3(n)$ with $s_{y_{2n}}=A^-$ and $s_{y_{2n-1}}=s_{z_{2n-1}}=s_{z_{2n}}=A$
\hskip 15mm $s_4(n)$ with $s_{z_{2n-1}}=A^-$ and $s_{y_{2n-1}}=s_{y_{2n}}=s_{z_{2n}}=A$
\hskip 15mm $s_5(n)$ with $s_{z_{2n}}=A^-$ and $s_{y_{2n-1}}=s_{y_{2n}}=s_{z_{2n-1}}=A$
\hskip 15mm $s_6(n)$ with $s_{y_{2n-1}}=s_{y_{2n}}=A^-$ and $s_{z_{2n-1}}=s_{z_{2n}}=A$
\hskip 15mm $s_7(n)$ with $s_{y_{2n-1}}=s_{z_{2n-1}}=A^-$ and $s_{y_{2n}}=s_{z_{2n}}=A$
\hskip 15mm $s_8(n)$ with $s_{y_{2n-1}}=s_{z_{2n}}=A^-$ and $s_{y_{2n}}=s_{z_{2n-1}}=A$
\hskip 15mm $s_9(n)$ with $s_{y_{2n}}=s_{z_{2n-1}}=A^-$ and $s_{y_{2n-1}}=s_{z_{2n}}=A$
\hskip 15mm $s_{10}(n)$ with $s_{y_{2n}}=s_{z_{2n}}=A^-$ and $s_{y_{2n-1}}=s_{z_{2n-1}}=A$
\hskip 15mm $s_{11}(n)$ with $s_{z_{2n-1}}=s_{z_{2n}}=A^-$ and $s_{y_{2n-1}}=s_{y_{2n}}=A$
\hskip 15mm $s_{12}(n)$ with $s_{y_{2n-1}}=s_{y_{2n}}=s_{z_{2n-1}}=A^-$ and $s_{z_{2n}}=A$
\hskip 15mm $s_{13}(n)$ with $s_{y_{2n-1}}=s_{y_{2n}}=s_{z_{2n}}=A^-$ and $s_{z_{2n-1}}=A$
\hskip 15mm $s_{14}(n)$ with $s_{y_{2n-1}}=s_{z_{2n-1}}=s_{z_{2n}}=A^-$ and $s_{y_{2n}}=A$
\hskip 15mm $s_{15}(n)$ with $s_{y_{2n}}=s_{z_{2n-1}}=s_{z_{2n}}=A^-$ and $s_{y_{2n-1}}=A$
\hskip 15mm $s_{16}(n)$ with $s_{y_{2n-1}}=s_{z_{2n-1}}=s_{z_{2n}}=s_{y_{2n}}=A^-$
\vskip 3mm \noindent{\bf Lemma $2.1$} {\it Let $s({n-1})\in S_{\hbox{\rm\scriptsize I}}({n-1})$ and let its state graph have $i$ loops for $n,i\ge 1$. Suppose that $s_j(n)$ are states of $RT_n$ and that $par(s_j(n))=s({n-1})$ above for $1\le j\le 16$.
$(1)$ If $j=11$ and $14\le j\le 16$, then $s_j\in S_{\hbox{\rm\scriptsize I}}({n})$. Moreover, the state graph of $s_{11}(n)$ has $i$ loops, the state graph of $s_{j}(n)$ have $i+1$ loops for $14\le j\le 15$ and the state graph of $s_{16}(n)$ has $i+2$ loops.
$(2)$ Otherwise $s_j(n)\in S_{\hbox{\rm\scriptsize II}}({n})$. Moreover, the state graph of each $s_{j}(n)$ has $i+1$ loops for $4\le j\le 5$, the state graph of each $s_{j}(n)$ has $i+2$ loops for $j=1$ and $7\le j\le 10$, the state graph of each $s_{16}(n)$ has $i+3$ loops for $2\le j\le 3$ and $12\le j\le 13$ and the state graph of $s_{6}(n)$ has $i+4$ loops.}
\vskip 3mm
{\bf Proof.} Set $s({n-1})\in S_{\hbox{\rm\scriptsize I}}({n-1})$. Without loss of generality, suppose that there exists some $1\le k\le n$ such that $s_{z_{2k-1}}\cdot s_{z_{2k}}=s_{y_{2k-1}}=A^{-}, s_{y_{2k}}= A,\prod\limits_{j=2k+1}^{2n}s_{y_j}=A^{2n-2k},\prod\limits_{j=2k+1}^{2n}s_{z_j}=A^{-(2n-2k)}.$ Other cases are left to readers to verify.
Assume that the state graph of $s(n-1)$ has the loops $(x_1^{\epsilon}x_3x_2y_{2n-2}Az_{2n-2}^-x_2x_3z_{2n-2}^-B)C$ where $A$ and $B$ are linear sequences, $C$ is the product of $i-1$ loops for $i\ge 1$ and $\epsilon\in\{+,-\}$.
(1) Because $s_{z_{2k-1}}=s_{z_{2k}}=A^-$, $s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^2$, $s_{z_{i}}=A^-$ and $s_{y_{i}}=A$ for $2k+1\le i\le 2n$ in $s_{11}(n)$, $$s_{11}(n)\in S_{\hbox{\rm\scriptsize I}}({n}).$$ Because $s_{z_{2n-1}}=s_{z_{2n}}=A^-$ and $s_{y_{2n-1}}\cdot s_{y_{2n}}\ne A^2$ at $s_j(n)$ for $14\le j\le 16$, $$s_{j}(n)\in S_{\hbox{\rm\scriptsize I}}({n}).$$ Moreover, the state graph of $s_j(n)$ has loops with loop number in brackets in sequence below for $j=11$ and $14\le j\le 16$: \vskip 2mm $(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1}y_{2n}z_{2n-1}^-z_{2n}^-x_2x_3z_{2n}^-z_{2n-1}^- z_{2n-2}^-B)C\{i\}$
$(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}^-y_{2n}z_{2n-1}^-z_{2n}^-x_2x_3z_{2n}^-z_{2n-1}^- z_{2n-2}^-B)C\{i+1\}$
$(y_{2n}^-y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1})(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1}^-z_{2n}^-x_2x_3z_{2n}^-z_{2n-1}^- z_{2n-2}^-B)C\{i+1\}$
$(y_{2n}^-y_{2n-1}^-)(y_{2n-2}Az_{2n-2}^-y_{2n-1}^-)(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1}^-z_{2n}^-x_2x_3z_{2n}^-z_{2n-1}^- z_{2n-2}^-B)C\{i+2\}$ \vskip 2mm Thus the result is clear.
(2) Because $s_{z_{2n-1}}\cdot s_{z_{2n}}\ne A^{-2}$ for $1\le j\le 10$ and $12\le j\le 13$, $$s_j(n)\in S_{\hbox{\rm\scriptsize II}}({n}).$$ Moreover, the state graph of $s_j(n)$ has loops with loop number in brackets in sequence as follows for $1\le j\le 10$ and $12\le j\le 13$: \vskip 2mm $(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1}y_{2n}z_{2n-1} z_{2n-2}^-B)C\{i+2\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}^-y_{2n}z_{2n-1} z_{2n-2}^-B)C\{i+3\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(y_{2n}^-y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1})(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1} z_{2n-2}^-B)C\{i+3\}$
$(z_{2n}x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1}y_{2n}z_{2n-1}^-z_{2n}z_{2n-1}^- z_{2n-2}^-B)C\{i+1\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1}y_{2n}z_{2n-1} z_{2n-2}^-B)C\{i+1\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(y_{2n-1}^-y_{2n}^-)(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1} z_{2n-2}^-B)C\{i+4\}$
$(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}^-y_{2n}z_{2n-1}^-z_{2n}z_{2n-1}^- z_{2n-2}^-B)C\{i+2\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}^-y_{2n}z_{2n-1} z_{2n-2}^-B)C\{i+2\}$
$(z_{2n}x_3x_2)(y_{2n}^-y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1})(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1}^-z_{2n}z_{2n-1}^- z_{2n-2}^-B)C\{i+2\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(y_{2n}^-y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1})(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1} z_{2n-2}^-B)C\{i+2\}$
$(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}Az_{2n-2}^-y_{2n-1}y_{2n}z_{2n-1}^-z_{2n}^-x_2x_3z_{2n}^-z_{2n-1}^- z_{2n-2}^-B)C\{i\}$
$(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(y_{2n-1}^-y_{2n}^-)(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1}^-z_{2n}z_{2n-1}^- z_{2n-2}^-B)C\{i+3\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(y_{2n-1}^-y_{2n-2}Az_{2n-2}^-)(y_{2n-1}^-y_{2n}^-)(x_1^{\epsilon}x_3x_2y_{2n}^-z_{2n-1} z_{2n-2}^-B)C\{i+3\}$ \vskip 2mm Therefore the consequence holds. $\Box$
\vskip 3mm \noindent{\bf Lemma $2.2$} {\it Let $s({n-1})\in S_{\hbox{\rm\scriptsize II}}({n-1})$ and its state graph with loops $i$ for $n\ge 1$ and $i\ge 2$. Suppose that $s_j(n)$ are states of $RT_n$ above and that $par(s_j(n))=s({n-1})$ for $1\le j\le 16$.
$(1)$ If $14\le j\le 16$, then $s_j(n)\in S_{\hbox{\rm\scriptsize I}}({n})$. Moreover, the state graph of $s_{11}(n)$ has $i$ loops, the state graph of each $s_{j}(n)$ have $i-1$ loops for $14\le j\le 15$ and the state graph of $s_{16}(n)$ has $i$ loops.
$(2)$ Otherwise $s_j(n)\in S_{\hbox{\rm\scriptsize II}}({n})$. Moreover, the state graph of each $s_{j}(n)$ has $i$ loops for $7\le j\le 11$, the state graph of each $s_{j}(n)$ has $i+1$ loops for each $2\le j\le 5$ and $12\le j\le 13$ and the state graph of $s_{j}(n)$ has $i+2$ loops for $j=1,6$.}
\vskip 3mm
{\bf Proof.} Set $s({n-1})\in S_{\hbox{\rm\scriptsize II}}({n-1})$. Without loss of generality, suppose that there exists some $1\le k\le n-1$ such that $s_{z_{2k}}=s_{y_l}=A,s_{z_l}=A^{-}$ for $2k+1\le l\le 2n-2$. Other cases are left to readers to verify.
Assume that the state graph of $s(n-1)$ has the loops $(x_1^{\epsilon}x_3x_2y_{2n-2}A)(z_{2n-2}^-x_2x_3z_{2n-2}^-B)C$ where $A$ and $B$ are linear sequences, $C$ is the product of $i-2$ loops and $\epsilon\in\{+,-\}$.
(1) Because there exists $n$ such that $s_{z_{2n-1}}=s_{z_{2n}}=A^-$, $s_{y_{2n-1}}\cdot s_{y_{2n}}\ne A^2$ for $14\le j\le 16$, $$s_{j}(n)\in S_{\hbox{\rm\scriptsize I}}({n}).$$ Moreover, the state graph of $s_j(n)$ has loops with loop number in brackets in sequence below for $14\le j\le 16$: \vskip 2mm $
(z_{2n-2}^-z_{2n-1}^-z_{2n}^-x_3x_2z_{2n}^-z_{2n-1}^-y_{2n}y_{2n-1}^-
y_{2n}x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i-1\}$
$ (z_{2n-2}^-z_{2n-1}^-z_{2n}^-x_3x_2z_{2n}^-z_{2n-1}^-y_{2n}^-x_2x_3x_1^{\epsilon} A^{re}y_{2n-2}y_{2n-1}y_{2n}^-y_{2n-1}z_{2n-2}^-B)C\{i-1\}$
$(y_{2n-1}^-y_{2n}^-)
(z_{2n-2}^-z_{2n-1}^-z_{2n}^-x_3x_2z_{2n}^-z_{2n-1}^-y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}
y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i\}$
\vskip 2mm (2) Because $s_{z_{2n-1}}\cdot s_{z_{2n}}\ne A^-$ for $1\le j\le 13$ and $j\ne 11$, $$s_{j}(n)\in S_{\hbox{\rm\scriptsize II}}({n}).$$ Because there exists $k$ such that $s_{z_{2k}}=s_{y_{l}}=A$ and that $s_{z_{l}}=A^-$ for $2k+1\le l\le 2n$ in $s_{11}(n)$, $$s_{11}(n)\in S_{\hbox{\rm\scriptsize II}}({n}).$$ Moreover, $s_j(n)$ has loops with loop number in brackets in sequence below for $1\le j\le 13$: \vskip 2mm $(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}A)
(z_{2n-2}^-z_{2n-1}y_{2n}y_{2n-1}z_{2n-2}^-B)C\{i+2\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2)
(z_{2n-2}^-z_{2n-1}y_{2n}y_{2n-1}^-y_{2n}x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i+1\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2) (z_{2n-2}^-z_{2n-1}y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}y_{2n}^-y_{2n-1}z_{2n-2}^-B)C\{i+1\}$
$(z_{2n}x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}A)
(z_{2n-2}^-z_{2n-1}^-z_{2n}z_{2n-1}^-y_{2n}y_{2n-1}z_{2n-2}^-B)C\{i+1\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}A)
(z_{2n-2}^-z_{2n-1}y_{2n}y_{2n-1}z_{2n-2}^-B)C\{i+1\}$
$(z_{2n-1}z_{2n})(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n}^-)
(z_{2n-2}^-z_{2n-1}y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i+2\}$
$(z_{2n}x_3x_2)
(z_{2n-2}^-z_{2n-1}^-z_{2n}z_{2n-1}^-y_{2n}y_{2n-1}^-
y_{2n}x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)
(z_{2n-2}^-z_{2n-1}y_{2n}y_{2n-1}^-y_{2n}x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i\}$
$(z_{2n}x_3x_2) (z_{2n-2}^-z_{2n-1}^-z_{2n}z_{2n-1}^-y_{2n}^-x_2x_3x_1^{\epsilon} A^{re}y_{2n-2}y_{2n-1}y_{2n}^-y_{2n-1}z_{2n-2}^-B)C\{i\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2) (z_{2n-2}^-z_{2n-1}y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}y_{2n}^-y_{2n-1}z_{2n-2}^-B)C\{i\}$
$(x_1^{\epsilon}x_3x_2y_{2n}y_{2n-1}y_{2n-2}A)
(z_{2n-2}^-z_{2n-1}^-z_{2n}^-x_3x_2z_{2n}^-z_{2n-1}^-y_{2n}y_{2n-1}z_{2n-2}^-B)C\{i\}$
$(z_{2n}x_3x_2)(y_{2n-1}^-y_{2n}^-)
(z_{2n-2}^-z_{2n-1}^-z_{2n}z_{2n-1}^-y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}
y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i+1\}$
$(z_{2n}^-z_{2n-1}z_{2n}^-x_3x_2)(y_{2n-1}^-y_{2n}^-)
(z_{2n-2}^-z_{2n-1}y_{2n}^-x_2x_3x_1^{\epsilon}A^{re}y_{2n-2}y_{2n-1}^-z_{2n-2}^-B)C\{i+1\}$ \vskip 2mm Thus the result is clear. $\Box$
\vskip 3mm The following results hold by applying a similar way in the argument of the proof of Lemma $2.1$. \vskip 3mm \noindent{\bf Lemma $2.3$} {\it Let $s({n-1})\in S_{\hbox{\rm\scriptsize III}}({n-1})$ and let its state graph $i$ loops for $n\ge 1$ and $i\ge 2$. Suppose that $s_j(n)$ are states of $RT_n$ above and that $par(s_j(n))=s({n-1})$ for $1\le j\le 16$.
$(1)$ If $j=11$ and $14\le j\le 16$, then $s_j\in S_{\hbox{\rm\scriptsize III}}({n})$. Moreover, the state graph of $s_{11}(n)$ has $i$ loops, the state graph of each $s_{j}(n)$ have $i+1$ loops for $14\le j\le 15$ and the state graph of $s_{16}(n)$ has $i+2$ loops.
$(2)$ Otherwise $s_j(n)\in S_{\hbox{\rm\scriptsize IV}}({n})$. Moreover, the state graph of each $s_{j}(n)$ has $i-1$ loops for $4\le j\le 5$, the state graph of each $s_{j}(n)$ has $i$ loops for $j=1$ and $7\le j\le 10$, the state graph of each $s_{j}(n)$ has $i+1$ loops for $2\le j\le 3$ and $12\le j\le 13$ and the state graph of $s_{6}(n)$ has $i+2$ loops.}
\vskip 3mm \noindent{\bf Lemma $2.4$} {\it Let $s({n-1})\in S_{\hbox{\rm\scriptsize IV}}({n-1})$ with loops $i$ for $n,i\ge 1$. Suppose that $s_j(n)$ are states of $RT_n$ above and that $par(s_j(n))=s({n-1})$ for $1\le j\le 16$.
$(1)$ If $14\le j\le 16$, then $s_j(n)\in S_{\hbox{\rm\scriptsize III}}({n})$. Moreover, the state graph of each $s_{j}(n)$ has $i+1$ loops for $14\le j\le 15$ and the state graph of $s_{16}(n)$ has $i+2$ loops.
$(2)$ Otherwise $s_j(n)\in S_{\hbox{\rm\scriptsize IV}}({n})$. Moreover, the state graph of each $s_{j}(n)$ has $i$ loops for $7\le j\le 11$, the state graph of each $s_{j}(n)$ has $i+1$ loops for $2\le j\le 5$ and $12\le j\le 13$ and the state graph of $s_{j}(n)$ has $i+2$ loops for $j=1,6$.}
\vskip 3mm
Recursive relations are given below. \vskip 3mm
\noindent{\bf Lemma $2.5$} {\it Let $p_{1,\hbox{\rm\scriptsize I}}(RT_0)=A$, $p_{2,\hbox{\rm\scriptsize II}}(RT_0)=A^3$, $p_{2,\hbox{\rm\scriptsize III}}(RT_0)=2A^-$, $p_{3,\hbox{\rm\scriptsize III}}(RT_0)=A^{-3}$, $p_{1,\hbox{\rm\scriptsize IV}}(RT_0)=2A$ and $p_{2,\hbox{\rm\scriptsize IV}}(RT_0)=A^-$. Set $p_{i,\hbox{\rm\scriptsize I}}(RT_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize I}}(RT_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize II}}(RT_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize II}}(RT_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize III}}(RT_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize III}}(RT_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize IV}}(RT_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize IV}}(RT_n),l(s)=i}A^{a(s)-b(s)}$. Then for $n\ge 1$ $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}(RT_n)=p_{i,\hbox{\rm\scriptsize
I}}(RT_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize I}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})\\
\hskip 19mm +2A^{-2}p_{i+1,\hbox{\rm\scriptsize II}}(RT_{n-1})+A^{-4}p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1}); \hskip 41mm (1)\\ p_{i,\hbox{\rm\scriptsize II}}(RT_n)=2A^2p_{i-1,\hbox{\rm\scriptsize
I}}(RT_{n-1})+(A^4+4)p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})\\
\hskip 19mm +(2A^2+2A^{-2})p_{i-3,\hbox{\rm\scriptsize I}}(RT_{n-1})+p_{i-4,\hbox{\rm\scriptsize I}}(RT_{n-1})+5p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize II}}(RT_{n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}(RT_{n-1}); \hskip 18mm (2)\\ p_{i,\hbox{\rm\scriptsize III}}(RT_n)=p_{i,\hbox{\rm\scriptsize
III}}(RT_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize III}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize III}}(RT_{n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize IV}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize IV}}(RT_{n-1}); \hskip 35.5mm (3)\\ p_{i,\hbox{\rm\scriptsize IV}}(RT_n)=2A^2p_{i+1,\hbox{\rm\scriptsize
III}}(RT_{n-1})+(A^4+4)p_{i,\hbox{\rm\scriptsize III}}(RT_{n-1})\\
\hskip 19mm +(2A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize III}}(RT_{n-1})+p_{i-2,\hbox{\rm\scriptsize III}}(RT_{n-1})+5p_{i,\hbox{\rm\scriptsize IV}}(RT_{n-1})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize IV}}(RT_{n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize IV}}(RT_{n-1}).\hskip 16.5mm (4) \end{array} \right. $$} \vskip 3mm
{\bf Proof.} Based on Lemmas $2.1$ and $2.2$, for $n\geq 1$, $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}(RT_n)=p_{i,\hbox{\rm\scriptsize
I}}(RT_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize I}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})\\
\hskip 19mm +2A^{-2}p_{i+1,\hbox{\rm\scriptsize II}}(RT_{n-1})+A^{-4}p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1}); \\ p_{i,\hbox{\rm\scriptsize II}}(RT_n)=2A^2p_{i-1,\hbox{\rm\scriptsize
I}}(RT_{n-1})+(A^4+4)p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})+(2A^2+2A^{-2})p_{i-3,\hbox{\rm\scriptsize I}}(RT_{n-1})\\
\hskip 19mm +p_{i-4,\hbox{\rm\scriptsize I}}(RT_{n-1})+5p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1})+(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize II}}(RT_{n-1})\\
\hskip 19mm +(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}(RT_{n-1}). \end{array} \right. $$ Similarly, the following result is clear from Lemmas $2.3$ and $2.4$. $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize III}}(RT_n)=p_{i,\hbox{\rm\scriptsize
III}}(RT_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize III}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize III}}(RT_{n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize IV}}(RT_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize IV}}(RT_{n-1}); \\ p_{i,\hbox{\rm\scriptsize IV}}(RT_n)=2A^2p_{i+1,\hbox{\rm\scriptsize
III}}(RT_{n-1})+(A^4+4)p_{i,\hbox{\rm\scriptsize III}}(RT_{n-1})+(2A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize III}}(RT_{n-1})\\
\hskip 19mm +p_{i-2,\hbox{\rm\scriptsize III}}(RT_{n-1})+5p_{i,\hbox{\rm\scriptsize IV}}(RT_{n-1})+(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize IV}}(RT_{n-1})\\
\hskip 19mm +(A^4+1)p_{i-2,\hbox{\rm\scriptsize IV}}(RT_{n-1}). \end{array} \right. $$ \hskip 150mm $\Box$
\vskip 3mm
{\bf Proof of Theorem $1.1$.} Set $F_1(x,y)=\sum\limits_{i\ge 1,n\ge 0}p_{i,\hbox{\rm\scriptsize I}}(RT_{n})x^{i-1}y^n$, $F_2(x,y)=\sum\limits_{i\ge 1,n\ge 0}p_{i,\hbox{\rm\scriptsize II}}(RT_{n})x^{i-1}y^n$, $F_3(x,y)=\sum\limits_{i\ge 1,n\ge 0}p_{i,\hbox{\rm\scriptsize III}}(RT_{n})x^{i-1}y^n$, $F_4(x,y)=\sum\limits_{i\ge 1,n\ge 0}p_{i,\hbox{\rm\scriptsize IV}}(RT_{n})x^{i-1}y^n$. Let $f_1(x)=\sum\limits_{i\ge 1}p_{i,\hbox{\rm\scriptsize I}}(RT_{n})x^{i-1}$, let $f_2(x)=\sum\limits_{i\ge 1}p_{i,\hbox{\rm\scriptsize II}}(RT_{n})x^{i-1}$, let $f_3(x)=\sum\limits_{i\ge 1}p_{i,\hbox{\rm\scriptsize III}}(RT_{n})x^{i-1}$ and let $f_4(x)=\sum\limits_{i\ge 1}p_{i,\hbox{\rm\scriptsize IV}}(RT_{n})x^{i-1}$. It follows from equations (1-2) that $$ \left\{ \begin{array}{lll} \sum\limits_{i\ge 1,n\ge 1}p_{i,\hbox{\rm\scriptsize I}}(RT_n)x^{i}y^n=\sum\limits_{i\ge 1,n\ge 1}p_{i,\hbox{\rm\scriptsize
I}}(RT_{n-1})x^{i}y^n+2A^{-2}\sum\limits_{i\ge 1,n\ge 1}p_{i-1,\hbox{\rm\scriptsize I}}(RT_{n-1})x^{i}y^n\\
\hskip 26.5mm +A^{-4}\sum\limits_{i\ge 1,n\ge 1}p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})x^{i}y^n
+2A^{-2}\sum\limits_{i\ge 1,n\ge 1}p_{i+1,\hbox{\rm\scriptsize II}}(RT_{n-1})x^{i}y^n\\
\hskip 26.5mm +A^{-4}\sum\limits_{i\ge 1,n\ge 1}p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1})x^{i}y^n; \hskip 73mm (5)\\ \sum\limits_{i\ge 1,n\ge 1}p_{i,\hbox{\rm\scriptsize II}}(RT_n)x^{i-1}y^n=2A^2\sum\limits_{i\ge 1,n\ge 1}p_{i-1,\hbox{\rm\scriptsize
I}}(RT_{n-1})x^{i-1}y^n
+(A^4+4)\sum\limits_{i\ge 1,n\ge 1}p_{i-2,\hbox{\rm\scriptsize I}}(RT_{n-1})x^{i-1}y^n \\
\hskip 27mm +(2A^2+2A^{-2})\sum\limits_{i\ge 1,n\ge 1}p_{i-3,\hbox{\rm\scriptsize I}}(RT_{n-1})x^{i-1}y^n
+\sum\limits_{i\ge 1,n\ge 1}p_{i-4,\hbox{\rm\scriptsize I}}(RT_{n-1})x^{i-1}y^n\\
\hskip 27mm +5\sum\limits_{i\ge 1,n\ge 1}p_{i,\hbox{\rm\scriptsize II}}(RT_{n-1})x^{i-1}y^n
+(4A^2+2A^{-2})\sum\limits_{i\ge 1,n\ge 1}p_{i-1,\hbox{\rm\scriptsize II}}(RT_{n-1})x^{i-1}y^n\\
\hskip 27mm +(A^4+1)\sum\limits_{i\ge 1,n\ge 1}p_{i-2,\hbox{\rm\scriptsize II}}(RT_{n-1})x^{i-1}y^n. \\ \end{array} \right. $$ Since $p_{1,\hbox{\rm\scriptsize I}}(RT_0)=A$ and $p_{2,\hbox{\rm\scriptsize II}}(RT_0)=A^3$, the set (5) of equations is reduced to the following set of equations $$ \left\{ \begin{array}{lll} (xy+2A^{-2}x^2y+A^{-4}x^3y-x)F_1(x,y)+(2A^{-2}y+A^{-4}xy)F_2(x,y)=-Ax;\\ (2A^2xy+(A^4+4)x^2y+(2A^2+2A^{-2})x^3y+x^4y)F_1(x,y)+((5y+4A^2+2A^{-2})xy\\
\hskip 34.5mm +(A^4+1)x^2y-1)F_2(x,y)=-A^3x. \end{array} \right. $$ Let $$D=
\left| \begin{array}{ll} xy+2A^{-2}x^2y+A^{-4}x^3y-x & 2A^{-2}y+A^{-4}xy\\ 2A^2xy+(A^4+4)x^2y+(2A^2+2A^{-2})x^3y+x^4y & (5y+4A^2+2A^{-2})xy+(A^4+1)x^2y-1\\ \end{array}
\right| $$ \hskip 7.5mm $=-x+(6x-5x^3+x^5)y-xy^2.$
\noindent Then,
$$F_1(x,y)=\frac{1}{D}
\left| \begin{array}{ll} -Ax & 2A^{-2}y+A^{-4}xy\\ -A^{3}x & (5y+4A^2+2A^{-2})xy+(A^4+1)x^2y-1\\ \end{array}
\right|
$$ $\hskip 42mm =\displaystyle\frac{A(1-3y-A^{-2}xy-4A^2xy-(A^4+1)x^2y}{1-(6-5x^2+x^4)y+y^2}$
\noindent and
$$F_2(x,y)=\frac{1}{D}
\left| \begin{array}{ll} xy+2A^{-2}x^2y+A^{-4}x^3y-x & -Ax\\ 2A^2xy+(A^4+4)x^2y+(2A^2+2A^{-2})x^3y+x^4y & -A^{3}x \end{array}
\right|
$$ $\hskip 42mm =\displaystyle\frac{A^2x+A^2xy+(A^4+2)x^2y+(2A^2+A^{-2}x^3)y+x^4y}{1-(6-5x^2+x^4)y+y^2}.$
\noindent Suppose that $1-(6-5x^2+x^4)y+y^2=(1-\alpha y)(1-\bar\alpha y)$ where $$ \left\{
\begin{array}{ll}
\alpha+\bar\alpha=6-5x^2+x^4;\\
\alpha\cdot\bar\alpha=1.
\end{array} \right. $$ The following equalities can be obtained \begin{eqnarray*} F_1(x,y) & = & \displaystyle\frac{A(1+(-3-A^{-2}x-4A^2x-(A^4+1)x^2)y}{(1-\alpha y)(1-\bar\alpha y)} \\
& = & \frac{A}{\alpha-\bar\alpha}((\alpha-3-A^{-2}x-4A^2x-(A^4+1)x^2)\sum\limits_{n\ge 0}\alpha^ny^n\\
& + & (3+A^{-2}x+4A^2x+(A^4+1)x^2-\bar\alpha)\sum\limits_{n\ge 0}\bar\alpha^ny^n ) \end{eqnarray*} \noindent and \begin{eqnarray*} F_2(x,y) & = & \displaystyle\frac{A^3x(1+(1+(A^2+2A^{-2})x+(2+A^{-4})x^2+A^{-2}x^3)y)}{(1-\alpha y)(1-\bar\alpha y)} \\
& = & \displaystyle\frac{A^3x}{\alpha-\bar\alpha}((\alpha+1+(A^2+2A^{-2})x+(2+A^{-4})x^2+A^{-2}x^3)\sum\limits_{n\ge 0}\alpha^ny^n\\
& + & (-1-(A^2+2A^{-2})x-(2+A^{-4})x^2-A^{-2}x^3-\bar\alpha)\sum\limits_{n\ge 0}\bar\alpha^ny^n ). \end{eqnarray*} Thus, for $n\ge 1$ $$f_1(x)=\displaystyle\frac{A}{\alpha-\bar\alpha}((\alpha^{n+1}-\bar\alpha^{n+1})+(3+A^{-2}x+4A^2x+(A^4+1)x^2) (\alpha^{n}-\bar\alpha^{n})) \hskip 15mm (6)$$ and $$f_2(x)=\displaystyle\frac{A^3x}{\alpha-\bar\alpha}((\alpha^{n+1}-\bar\alpha^{n+1})+(1+(A^2+2A^{-2})x+(2+A^{-4})x^2+A^{-2}x^3) (\alpha^{n}-\bar\alpha^{n})). \hskip 5mm (7)$$ By a similar way, the following equalities can be concluded for $n\ge 1$ $$f_3(x)=\displaystyle\frac{2A^-x+A^{-3}x^2}{\alpha-\bar\alpha}((\alpha^{n+1}-\bar\alpha^{n+1})-(3+(4A^2+A^{-2})x+(A^4+1)x^2) (\alpha^{n}-\bar\alpha^{n})) \hskip 15mm (8)$$ and $$f_4(x)=\displaystyle\frac{2A+A^{-}x}{\alpha-\bar\alpha}((\alpha^{n+1}-\bar\alpha^{n+1}) +(1+(A^2+2A^{-2})x+(2+A^{-4})x^2+A^{-2}x^3) (\alpha^{n}-\bar\alpha^{n})).\hskip 5mm (9)$$ Since $RT_n$ contains $4n+3$ crossings and $\omega(v)=1$ for each $v\in V(RT_n)$, by setting $x=-A^2-A^{-2}$ and combining with the equalities (6-9), we conclude the following results for $n\ge 1$ \begin{eqnarray*} V_{RT_n}(t) & = & (-A)^{-(12n+9)}\sum\limits_{j=1}^4f_j(x)\\
& = & \displaystyle\frac{A^{-12n}}{\alpha-\bar\alpha}((A^{-4}+A^{-12}-A^{-16})(\alpha^{n+1}-\bar\alpha^{n+1}) -(1+A^{-4}-A^{-8}) (\alpha^{n}-\bar\alpha^{n}))\\ & = & \displaystyle\frac{t^{3n}}{\alpha-\bar\alpha}((t+t^{3}-t^{4})(\alpha^{n+1}-\bar\alpha^{n+1}) -(1+t-t^{2}) (\alpha^{n}-\bar\alpha^{n})) \end{eqnarray*} where $$ \left\{
\begin{array}{ll}
\alpha+\bar\alpha=t^{-2}-t^{-1}+2-t+t^2;\\
\alpha\cdot\bar\alpha=1.
\end{array} \right. $$ \hskip 150mm $\Box$
\vskip 5mm \noindent{\bf $3.$ Jones polynomials of $KV_n$ }
\vskip 5mm In this section, for each $KV_n$ with $n\ge 1$, we divide the set $S(KV_n)$ of all of its states into $S^{(j)}(n)$ for $1\le j\le 3$ and obtain some recursive relations. Based on these relations, $KV_n$ is proved to be non-alternating and Theorem 1.2 is concluded.
\vskip 12mm
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\begin{footnotesize}
\put(-14.5,23){{$x_{1}$}} \put(-5,33.5){{$x_{2}$}} \put(-14.5,16){{$x_{3}$}} \put(-1,37){{$x_{4}$}} \put(1.5,20.5){{$x_{5}$}} \put(-16,9.5){{$x_{6}$}} \put(-10,2){$KV_0$}
\put(52,36){{$x_{4}$}} \put(45,33.5){{$x_{2}$}} \put(31,23){{$x_{1}$}} \put(26.5,15){{$x_{3}$}} \put(33,14){{$y_{2n}$}} \put(58.5,29.5){{$x_{5}$}} \put(49,17){{$z_{2}$}} \put(48,29.5){{$y_{1}$}} \put(47,25){{$y_{2}$}} \put(53,26.5){{$x_{6}$}} \put(50.5,20.5){{$z_{1}$}} \put(41,23.5){{$y_{3}$}} \put(42,19){{$y_{4}$}} \put(51,14.5){{$z_{3}$}} \put(46,10){{$z_{4}$}} \put(33,9){{$z_{2n-1}$}} \put(26,8){{$z_{2n}$}} \put(36,16){{$y_{2n-1}$}} \put(40,2){$KV_n$}
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Let $S(KV_n)$ be the set of all of states of $KV_n$. Denote three sets below $$ \begin{array}{ll}
S^{(1)}(n)=\{s\in S(KV_n)|s_{x_i}= A^- \mbox{ for }1\le i\le 6\}, \end{array} $$ $$ \begin{array}{ll}
S^{(2)}(n)=\{s\in S(KV_n)|s_{x_i}= A^- \mbox{ for }1\le i\le 3, \mbox{ and }\prod\limits_{i=4}^{6}s_{x_i}=A^3, \prod\limits_{i=4}^{6}s_{x_i}=A \mbox{ or }\prod\limits_{i=4}^{6}s_{x_i}=A^-\}, \end{array} $$ $$ \begin{array}{ll} S^{(3)}(n)=S(KV_n)\setminus \bigcup\limits_{i=1}^2S^{(i)}(n). \end{array} $$ Set $p_{i}^{(j)}(n)=\sum\limits_{s\in S^{(j)}(n),l(s)=i}A^{a(s)-b(s)}$ for $1\le j\le 3$ and $i\ge 1$. Obviously,
$$p_{i}(KV_n)=\sum\limits_{i=1}^3p_{i}^{(j)}(n).$$
\vskip 3mm {\it Case $1.$} $s\in S^{(1)}(n)$.
Set
$ \begin{array}{ll}
S^{(1)}_{\hbox{\rm\scriptsize I}}(n)=\{s|s_{y_i}=s_{z_i}= A^- \mbox{ for }1\le i\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(1)}_{\hbox{\rm\scriptsize II}}(n)=\{s|\exists 1\le k_0\le k_1\le n \mbox{ such that }s_{z_{2k_0-1}}\cdot s_{z_{2k_0}}\ne A^{-2}, s_{z_{2k_1-1}}\cdot s_{z_{2k_1}}\ne A^{-2},\\ \hskip 18mm s_{y_{2k_0-1}}= s_{y_{2k_0}}=s_{y_l}=s_{z_l}= A^{-}, \mbox{ for }1\le l\le 2k_0-2\mbox{ and }2k_1+1\le l\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(1)}_{\hbox{\rm\scriptsize III}}(n)=\{s|\exists 1\le k_0\le k_1\le n \mbox{ such that }s_{y_{2k_0-1}}\cdot s_{y_{2k_0}}\ne A^{-2}, s_{y_{2k_1-1}}\cdot s_{y_{2k_1}}\ne A^{-2},\\ \hskip 18mm s_{z_{2k_1-1}}= s_{z_{2k_1}}=s_{y_l}=s_{z_l}= A^{-}, \mbox{ for }1\le l\le 2k_0-2\mbox{ and }2k_1+1\le l\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(1)}_{\hbox{\rm\scriptsize IV}}(n)=\{s|\exists 1\le k_0\le k_1-1\le n \mbox{ such that }s_{z_{2k_0-1}}\cdot s_{z_{2k_0}}\ne A^{-2}, s_{y_{2k_1-1}}\cdot s_{y_{2k_1}}\ne A^{-2},\\ \hskip 18mm s_{z_{2k_1-1}}= s_{z_{2k_1}}=s_{y_{2k_0-1}}= s_{y_{2k_0}}=s_{y_l}=s_{z_l}= A^{-}, \mbox{ for }1\le l\le 2k_0-2\mbox{ and }\\ \hskip 18mm 2k_1+1\le l\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(1)}_{\hbox{\rm\scriptsize V}}(n)=\{s|\exists 1\le k_0\le k_1\le n \mbox{ such that }s_{y_{2k_0-1}}\cdot s_{y_{2k_0}}\ne A^{-2}, s_{z_{2k_1-1}}\cdot s_{z_{2k_1}}\ne A^{-2},\\ \hskip 18mm s_{y_l}=s_{z_l}= A^{-}, \mbox{ for }1\le l\le 2k_0-2\mbox{ and }2k_1+1\le l\le 2n\}. \end{array} $ \vskip 3mm
\vskip 3mm By a similar way in the argument of the proof in Lemma 2.5, the following recursive relations are shown. \vskip 3mm \noindent{\bf Lemma $3.1$} {\it Let $p_{3,\hbox{\rm\scriptsize I}}^{(1)}(0)=A^{-6}$. Set $p_{i,\hbox{\rm\scriptsize I}}^{(1)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize I}}^{(1)}(n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize II}}^{(1)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize II}}^{(1)}(n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize III}}^{(1)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize III}}^{(1)}(n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize IV}}^{(1)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize IV}}^{(1)}(n),l(s)=i}A^{a(s)-b(s)}$ and $p_{i,\hbox{\rm\scriptsize V}}^{(1)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize V}}^{(1)}(n),l(s)=i}A^{a(s)-b(s)}$. Then $$p_{i}^{(1)}(n)=p_{i,\hbox{\rm\scriptsize I}}^{(1)}(n)+p_{i,\hbox{\rm\scriptsize II}}^{(1)}(n) +p_{i,\hbox{\rm\scriptsize III}}^{(1)}(n)+p_{i,\hbox{\rm\scriptsize IV}}^{(1)}(n)+ p_{i,\hbox{\rm\scriptsize V}}^{(1)}(n)$$ where $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}^{(1)}(n)=A^{-4}p_{i,\hbox{\rm\scriptsize
I}}^{(1)}({n-1}); \\ p_{i,\hbox{\rm\scriptsize II}}^{(1)}(n)=2A^{-2}p_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({n-1})+p_{i,\hbox{\rm\scriptsize I}}^{(1)}({n-1})\\
\hskip 19mm +(A^4+4)p_{i,\hbox{\rm\scriptsize II}}^{(1)}({n-1})+(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize II}}^{(1)}({n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}^{(1)}({n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize IV}}^{(1)}({n-1})+5p_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({n-1})\\
\hskip 19mm +4A^{2}p_{i-3,\hbox{\rm\scriptsize IV}}^{(1)}({n-1})+A^4p_{i-4,\hbox{\rm\scriptsize IV}}^{(1)}({n-1}); \\ p_{i,\hbox{\rm\scriptsize III}}^{(1)}(n)=2A^{-2}p_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({n-1})+p_{i,\hbox{\rm\scriptsize I}}^{(1)}({n-1})+A^{-4}p_{i,\hbox{\rm\scriptsize III}}^{(1)}({n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({n-1})+p_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize
V}}^{(1)}({n-1})+p_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({n-1}); \\ p_{i,\hbox{\rm\scriptsize IV}}^{(1)}(n)=2A^{-2}p_{i+1,\hbox{\rm\scriptsize
II}}^{(1)}({n-1})+p_{i,\hbox{\rm\scriptsize II}}^{(1)}({n-1})\\
\hskip 19mm +A^4p_{i,\hbox{\rm\scriptsize IV}}^{(1)}({n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize
IV}}^{(1)}({n-1})+p_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({n-1});\\ p_{i,\hbox{\rm\scriptsize V}}^{(1)}(n)=4p_{i+2,\hbox{\rm\scriptsize
I}}^{(1)}({n-1})+4A^2p_{i+1,\hbox{\rm\scriptsize I}}^{(1)}({n-1})+A^4p_{i,\hbox{\rm\scriptsize I}}^{(1)}({n-1})+2A^{-2}p_{i+1,\hbox{\rm\scriptsize III}}^{(1)}({n-1})\\
\hskip 19mm +5p_{i,\hbox{\rm\scriptsize III}}^{(1)}({n-1})+4A^2p_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({n-1})+A^4p_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({n-1})
+(4+A^{-4})p_{i,\hbox{\rm\scriptsize V}}^{(1)}({n-1})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize V}}^{(1)}({n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({n-1}).\\ \end{array} \right. $$} \vskip 3mm {\it Case $2$.} $s\in S^{(2)}(n).$
Let \vskip 3mm
$ \begin{array}{ll}
S^{(2)}_{\hbox{\rm\scriptsize I}}(n)=\{s|\exists 1\le k\le n \mbox{ such that }s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^{-2},
s_{z_{2k-1}}= s_{z_{2k}}=s_{y_l}=s_{z_l}= A^{-},\\
\hskip 19mm \mbox{ for }2k+1\le l\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(2)}_{\hbox{\rm\scriptsize II}}(n)=\{s|\mbox{ either }s_{y_i}=s_{z_i}= A^{-}\mbox{ or }\exists 1\le k\le n \mbox{ such that }s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2},\\
\hskip 20mm s_{y_l}=s_{z_l}= A^{-}, \mbox{ for } 1\le i\le 2n, 2k_1+1\le l\le 2n\}. \end{array} $
\vskip 3mm \noindent{\bf Lemma $3.2$} {\it Set $p_{2,\hbox{\rm\scriptsize II}}^{(2)}(0)=3A^{-4}$, $p_{3,\hbox{\rm\scriptsize II}}^{(2)}(0)=3A^{-2}$, $p_{4,\hbox{\rm\scriptsize II}}^{(2)}(0)=1$. Set $p_{i,\hbox{\rm\scriptsize I}}^{(2)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize I}}^{(2)}(n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize II}}^{(2)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize II}}^{(2)}(n),l(s)=i}A^{a(s)-b(s)}$. Then $$p_{i}^{(2)}(n)=p_{i,\hbox{\rm\scriptsize I}}^{(2)}(n)+p_{i,\hbox{\rm\scriptsize II}}^{(2)}(n)$$ where $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}^{(2)}(n)=A^{-4}p_{i,\hbox{\rm\scriptsize I}}^{(2)}({n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize
I}}^{(2)}({n-1})+p_{i-2,\hbox{\rm\scriptsize I}}^{(2)}({n-1})\\
\hskip 19mm +2A^{-2}p_{i+1,\hbox{\rm\scriptsize
II}}^{(2)}({n-1})+p_{i,\hbox{\rm\scriptsize II}}^{(2)}({n-1}); \\ p_{i,\hbox{\rm\scriptsize II}}^{(2)}(n)=2A^{-2}p_{i-1,\hbox{\rm\scriptsize
I}}^{(2)}({n-1})+5p_{i-2,\hbox{\rm\scriptsize I}}^{(2)}({n-1})+4A^{2}p_{i-3,\hbox{\rm\scriptsize I}}^{(2)}({n-1})\\
\hskip 19mm +A^{4}p_{i-4,\hbox{\rm\scriptsize I}}^{(2)}({n-1})+(4+A^{-4})p_{i,\hbox{\rm\scriptsize II}}^{(2)}({n-1})\\
\hskip 19mm +(4A^{2}+2A^{-2})p_{i-1,\hbox{\rm\scriptsize
II}}^{(2)}({n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}^{(2)}({n-1}).\\ \end{array} \right. $$} \vskip 3mm {\it Case $3$.} $s\in S^{(3)}(n)$. \vskip 3mm
Let
$ \begin{array}{ll}
S^{(3)}_{\hbox{\rm\scriptsize I}}(n)=\{s|\mbox{ either }s_{x_j}=A^-, \prod\limits_{i=1}^3s_{x_i}\ne A^{-3}\mbox{ for }4\le j\le 6\mbox{ or }\exists 1\le k\le n \mbox{ such that }\\ \hskip 20mm s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^{-2},
s_{z_{2k-1}}= s_{z_{2k}}=s_{y_l}=s_{z_l}= A^{-},
\mbox{ for }2k+1\le l\le 2n\}, \end{array} $
$ \begin{array}{ll}
S^{(3)}_{\hbox{\rm\scriptsize II}}(n)=\{s|\mbox{ either }s_{y_i}=s_{z_i}= A^{-}, \prod\limits_{i=1}^3s_{x_i}\ne A^{-3}, \prod\limits_{i=4}^6s_{x_i}\ne A^{-3}\mbox{ or }\exists 1\le k\le n \mbox{ such that }\\
\hskip 20mm s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2}, s_{y_l}=s_{z_l}= A^{-}, \mbox{ for } 1\le i\le 2n, 2k+1\le l\le 2n\}. \end{array} $ \vskip 3mm \noindent{\bf Lemma $3.3$} {\it Set $p_{2,\hbox{\rm\scriptsize I}}^{(3)}(0)=3A^{-4}$, $p_{3,\hbox{\rm\scriptsize I}}^{(3)}(0)=3A^{-2}$, $p_{4,\hbox{\rm\scriptsize I}}^{(3)}(0)=1$, $p_{1,\hbox{\rm\scriptsize II}}^{(3)}(0)=9A^{-2}$, $p_{2,\hbox{\rm\scriptsize II}}^{(3)}(0)=18$, $p_{3,\hbox{\rm\scriptsize II}}^{(3)}(0)=15A^2$, $p_{4,\hbox{\rm\scriptsize II}}^{(3)}(0)=6A^4$, $p_{5,\hbox{\rm\scriptsize II}}^{(3)}(0)=A^6$. Set $p_{i,\hbox{\rm\scriptsize I}}^{(3)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize I}}^{(3)}(n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize II}}^{(3)}(n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize II}}^{(3)}(n),l(s)=i}A^{a(s)-b(s)}$. Then $$p_{i}^{(3)}(n)=p_{i,\hbox{\rm\scriptsize I}}^{(3)}(n)+p_{i,\hbox{\rm\scriptsize II}}^{(3)}(n)$$ where $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}^{(3)}(n)=A^{-4}p_{i,\hbox{\rm\scriptsize I}}^{(3)}({n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize
I}}^{(3)}({n-1})+p_{i-2,\hbox{\rm\scriptsize I}}^{(3)}({n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize
II}}^{(3)}({n-1})+p_{i-2,\hbox{\rm\scriptsize II}}^{(3)}({n-1}); \\ p_{i,\hbox{\rm\scriptsize II}}^{(3)}(n)=2A^{-2}p_{i+1,\hbox{\rm\scriptsize
I}}^{(3)}({n-1})+5p_{i,\hbox{\rm\scriptsize I}}^{(3)}({n-1})+4A^{2}p_{i-1,\hbox{\rm\scriptsize I}}^{(3)}({n-1})\\
\hskip 19mm +A^{4}p_{i-2,\hbox{\rm\scriptsize I}}^{(3)}({n-1})+(4+A^{-4})p_{i,\hbox{\rm\scriptsize II}}^{(2)}({n-1})\\
\hskip 19mm +(4A^{2}+2A^{-2})p_{i-1,\hbox{\rm\scriptsize
II}}^{(3)}({n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}^{(3)}({n-1}).\\ \end{array} \right. $$} \hskip 150mm $\Box$
As an example, we calculate the Jones polynomial of $KV_1$ (also $10_{152}$). By Lemma $3.1$, the following results are obtained $$ \left\{ \begin{array}{lll} p_{3,\hbox{\rm\scriptsize I}}^{(1)}(1)=A^{-10}, p_{2,\hbox{\rm\scriptsize III}}^{(1)}(1)=p_{2,\hbox{\rm\scriptsize II}}^{(1)}(1)=2A^{-8},
\\ p_{3,\hbox{\rm\scriptsize III}}^{(1)}(1)=p_{3,\hbox{\rm\scriptsize II}}^{(1)}(1)=A^{-6}, p_{1,\hbox{\rm\scriptsize V}}^{(1)}(1)=4A^{-6}, \hskip 20mm (10)\\
p_{2,\hbox{\rm\scriptsize V}}^{(1)}(1)=4A^{-4}, p_{3,\hbox{\rm\scriptsize V}}^{(1)}(1)=A^{-2}.\\ \end{array} \right. $$ By Lemma $3.2$, the following conclusions are given $$ \left\{ \begin{array}{lll} p_{1,\hbox{\rm\scriptsize I}}^{(2)}(1)=6A^{-6}, p_{2,\hbox{\rm\scriptsize I}}^{(2)}(1)=9A^{-4}, p_{3,\hbox{\rm\scriptsize I}}^{(2)}(1)=5A^{-2},
p_{4,\hbox{\rm\scriptsize I}}^{(2)}(1)=1,
\\ p_{2,\hbox{\rm\scriptsize II}}^{(2)}(1)=12A^{-4}+3A^{-8}, p_{3,\hbox{\rm\scriptsize II}}^{(2)}(1)=24A^{-2}+9A^{-6}, p_{4,\hbox{\rm\scriptsize II}}^{(2)}(1)=19+10A^{-4}, \hskip 20mm (11)\\
p_{5,\hbox{\rm\scriptsize II}}^{(2)}(1)=7A^2+5A^{-2}, p_{6,\hbox{\rm\scriptsize II}}^{(2)}(1)=A^{4}+1.\\ \end{array} \right. $$ By Lemma $3.3$, we get $$ \left\{ \begin{array}{lll} p_{2,\hbox{\rm\scriptsize I}}^{(3)}(1)=18A^{-4}+3A^{-8}, p_{3,\hbox{\rm\scriptsize I}}^{(3)}(1)=9A^{-6}+45A^{-2}, p_{4,\hbox{\rm\scriptsize I}}^{(3)}(1)=48+10A^{-4},
\\ p_{5,\hbox{\rm\scriptsize I}}^{(3)}(1)=27A^{2}+5A^{-2}, p_{6,\hbox{\rm\scriptsize I}}^{(3)}(1)=8A^{4}+1, p_{7,\hbox{\rm\scriptsize I}}^{(3)}(1)=A^{6}, p_{1,\hbox{\rm\scriptsize II}}^{(3)}(1)=36A^{-2}+15A^{-6},\\ p_{2,\hbox{\rm\scriptsize II}}^{(3)}(1)=108+57A^{-4},
p_{3,\hbox{\rm\scriptsize II}}^{(3)}(1)=141A^{2}+89A^{-2},p_{4,\hbox{\rm\scriptsize II}}^{(3)}(1)=74+102A^{4}, \hskip 20mm (12)\\
p_{5,\hbox{\rm\scriptsize II}}^{(3)}(1)=43A^6+35A^{2},p_{6,\hbox{\rm\scriptsize II}}^{(3)}(1)=10A^{8}+9A^{4}, p_{7,\hbox{\rm\scriptsize II}}^{(3)}(1)=A^{10}+A^6.\\ \end{array} \right. $$ By combining with the equalities $(10-12)$, we have $$ \left\{ \begin{array}{lll} p_{1}(KV_1)=36A^{-2}+25A^{-6},\\
p_{2}(KV_1)(-A^2-A^{-2})=-108A^2-208A^{-2}-110A^{-6}-10A^{-10}, \\
p_{3}(KV_1)(-A^2-A^{-2})^2=141A^6+446A^{2}+469A^{-2}+205A^{-6}+22A^{-10}+A^{-14},\\
p_{4}(KV_1)(-A^2-A^{-2})^3=-102A^{10}-448A^6-752A^2-588A^{-2}-202A^{-6}-20A^{-10},\hskip 7mm (13)\\
p_{5}(KV_1)(-A^2-A^{-2})^4=43A^{14}+241A^{10}+337A^6+626A^2+379A^{-2}\\
\hskip 36mm +109A^{-6}+10A^{-10},\\
p_{6}(KV_1)(-A^2-A^{-2})^5=-10A^{18}-68A^{14}-192A^{10}-290A^6-250A^2\\
\hskip 36mm -120A^{-2}-28A^{-6}-2A^{-10},\\
p_{7}(KV_1)(-A^2-A^{-2})^6=A^{22}+8A^{18}+27A^{14}+50A^{10}+55A^6+36A^2+14A^{-2}+2A^{-6} \end{array} \right. $$
By applying the equalities (1) and (13), the Kauffman bracket polynomial of $KV_1$ is \begin{eqnarray*} <KV_1> & = & \sum\limits_{i=1}^7p_i(KV_1)(-A^2-A^{-2})^{i-1}\\
& = & A^{22}-2A^{18}+2A^{14}-3A^{10}+2A^6-2A^2+A^{-2}+A^{-6}+A^{-14}. \end{eqnarray*} Since $\omega(KV_1)=-10$, the Jones polynomial of $RV_1$ is deduced \begin{eqnarray*} V_{KV_1}(t) & = & A^{30}(A^{22}-2A^{18}+2A^{14}-3A^{10}+2A^6-2A^2+A^{-2}+A^{-6}+A^{-14})\\
& = & A^{52}-2A^{48}+2A^{44}-3A^{40}+2A^{36}-2A^{32}+A^{28}+A^{24}+A^{16}\\
& = & t^{-13}-2t^{-12}+2t^{-11}-3t^{-10}+2t^{-9}-2t^{-8}+t^{-7}+t^{-6}+t^{-4}. \end{eqnarray*} \vskip 3mm It is obvious that $RT_1$ is non-alternating. In order to prove that $RT_n$ are non-alternating for $n\ge 2$. We consider the highest power and the lowest power of $A$ in $f_{i}(n)=p_i(KV_n)(-A^2-A^{-2})^{i-1}$ for $i\ge 1$. \vskip 3mm \noindent{\bf Lemma $3.4$} {\it Set $f_{i}(n)=p_i(KV_n)(-A^2-A^{-2})^{i-1}$ for $n,i\ge 1$. Let $\rho_h(f_{n,i})$ and $\rho_l(f_{i}(n))$ denote the highest power and the lowest power of $A$ in $f_{i}(n)$ respectively. Then for $n,i\ge 1$ $$\rho_h(f_{i}(n))\le 8k+14$$ and $$\rho_l(f_{i}(n))\ge -4k-10.$$} \vskip 3mm {\bf Proof.} This conclusion will be verified by induction on $n$. By equalities of $(13)$, the result is obvious for $n=1$.
Assume that the result holds for the integer $k (k\ge 2)$. That is for $i\ge 1$ $$\rho_h(f_{i}(k))\le 8n+14 \mbox{ and }\rho_l(f_{i}(k))\ge -4n-10.$$ This implies for $i\ge 1$ $$\rho_h(f_{i,j}^{(r)}(k))\le 8n+14 \mbox{ and }\rho_l(f_{i,j}^{(r)}(k))\ge -4n-10 \hskip 10mm (14)$$ where $f_{i,j}^{(r)}(k)=p_i^{(r)}(KV_k)(-A^2-A^{-2})^{i-1}$ for $j\in \{\hbox{\rm I},\hbox{\rm II},\hbox{\rm III}, \hbox{\rm IV}\}$ and $1\le r\le 3$. By Lemma $3.6$, $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}^{(1)}(k+1)(-A^2-A^{-2})^{i-1}=A^{-4}p_{i,\hbox{\rm\scriptsize
I}}^{(1)}({k}) (-A^2-A^{-2})^{i-1}; \\ p_{i,\hbox{\rm\scriptsize II}}^{(1)}(k+1)(-A^2-A^{-2})^{i-1}=(2A^{-2}p_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({k})+p_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})\\
\hskip 19mm +(A^4+4)p_{i,\hbox{\rm\scriptsize II}}^{(1)}({k})+(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize II}}^{(1)}({k})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}^{(1)}({k})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize IV}}^{(1)}({k})+5p_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({k})\\
\hskip 19mm +4A^{2}p_{i-3,\hbox{\rm\scriptsize IV}}^{(1)}({k})+A^4p_{i-4,\hbox{\rm\scriptsize IV}}^{(1)}({k}))(-A^2-A^{-2})^{i-1}; \\ p_{i,\hbox{\rm\scriptsize III}}^{(1)}(k+1)(-A^2-A^{-2})^{i-1}=(2A^{-2}p_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({k})+p_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})+A^{-4}p_{i,\hbox{\rm\scriptsize III}}^{(1)}({k})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({k})+p_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({k})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize
V}}^{(1)}({k})+p_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({k}))(-A^2-A^{-2})^{i-1}; \\ p_{i,\hbox{\rm\scriptsize IV}}^{(1)}(k+1)(-A^2-A^{-2})^{i-1}=(2A^{-2}p_{i+1,\hbox{\rm\scriptsize
II}}^{(1)}({k})+p_{i,\hbox{\rm\scriptsize II}}^{(1)}({k})\\
\hskip 19mm +A^4p_{i,\hbox{\rm\scriptsize IV}}^{(1)}({k})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize
IV}}^{(1)}({k})+p_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({k}))(-A^2-A^{-2})^{i-1};\\ p_{i,\hbox{\rm\scriptsize V}}^{(1)}(k+1)(-A^2-A^{-2})^{i-1}=(4p_{i+2,\hbox{\rm\scriptsize
I}}^{(1)}({k})+4A^2p_{i+1,\hbox{\rm\scriptsize I}}^{(1)}({k})+A^4p_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})+2A^{-2}p_{i+1,\hbox{\rm\scriptsize III}}^{(1)}({k})\\
\hskip 19mm +5p_{i,\hbox{\rm\scriptsize III}}^{(1)}({k})+4A^2p_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({k})+A^4p_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({k})\\
\hskip 19mm +(4+A^{-4})p_{i,\hbox{\rm\scriptsize V}}^{(1)}({k})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize V}}^{(1)}({k})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({k}))(-A^2-A^{-2})^{i-1}.\\ \end{array} \right. $$ This implies the following equalities $$ \left\{ \begin{array}{lll} f_{i,\hbox{\rm\scriptsize I}}^{(1)}(k+1)=A^{-4}f_{i,\hbox{\rm\scriptsize
I}}^{(1)}({k}); \\ f_{i,\hbox{\rm\scriptsize II}}^{(1)}(k+1)=(2A^{-2}f_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({k})(-A^2-A^{-2})^{-1}+f_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})+(A^4+4)f_{i,\hbox{\rm\scriptsize II}}^{(1)}({k})\\
\hskip 19mm +(4A^2+2A^{-2})f_{i-1,\hbox{\rm\scriptsize II}}^{(1)}({k})(-A^2-A^{-2})+(A^4+1)f_{i-2,\hbox{\rm\scriptsize II}}^{(1)}({k})(-A^2-A^{-2})^{2}\\
\hskip 19mm +2A^{-2}f_{i-1,\hbox{\rm\scriptsize IV}}^{(1)}({k})(-A^2-A^{-2})+5f_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({k})(-A^2-A^{-2})^{2}\\
\hskip 19mm +4A^{2}f_{i-3,\hbox{\rm\scriptsize IV}}^{(1)}({k})(-A^2-A^{-2})^{3}+A^4f_{i-4,\hbox{\rm\scriptsize IV}}^{(1)}({k})(-A^2-A^{-2})^{4}; \\ f_{i,\hbox{\rm\scriptsize III}}^{(1)}(k+1)=2A^{-2}f_{i+1,\hbox{\rm\scriptsize
I}}^{(1)}({k})(-A^2-A^{-2})^{-1}+f_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})+A^{-4}f_{i,\hbox{\rm\scriptsize III}}^{(1)}({k})\hskip 30mm (15)\\
\hskip 19mm +2A^{-2}f_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({k})(-A^2-A^{-2})+f_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({k})(-A^2-A^{-2})^{2}\\
\hskip 19mm +2A^{-2}f_{i-1,\hbox{\rm\scriptsize
V}}^{(1)}({k})(-A^2-A^{-2})+f_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({k})(-A^2-A^{-2})^{2}; \\ f_{i,\hbox{\rm\scriptsize IV}}^{(1)}(k+1)=2A^{-2}f_{i+1,\hbox{\rm\scriptsize
II}}^{(1)}({k})(-A^2-A^{-2})^{-1}+f_{i,\hbox{\rm\scriptsize II}}^{(1)}({k})\\
\hskip 19mm +A^4f_{i,\hbox{\rm\scriptsize IV}}^{(1)}({k})+2A^{-2}f_{i-1,\hbox{\rm\scriptsize
IV}}^{(1)}({k})(-A^2-A^{-2})+f_{i-2,\hbox{\rm\scriptsize IV}}^{(1)}({k})(-A^2-A^{-2})^{2};\\ f_{i,\hbox{\rm\scriptsize V}}^{(1)}(k+1)=4f_{i+2,\hbox{\rm\scriptsize
I}}^{(1)}({k})(-A^2-A^{-2})^{-2}+4A^2f_{i+1,\hbox{\rm\scriptsize I}}^{(1)}({k})(-A^2-A^{-2})^{-1}+A^4f_{i,\hbox{\rm\scriptsize I}}^{(1)}({k})\\
\hskip 19mm +2A^{-2}f_{i+1,\hbox{\rm\scriptsize III}}^{(1)}({k})(-A^2-A^{-2})^{-1}+5f_{i,\hbox{\rm\scriptsize III}}^{(1)}({k})+4A^2f_{i-1,\hbox{\rm\scriptsize III}}^{(1)}({k})(-A^2-A^{-2})\\
\hskip 19mm +A^4f_{i-2,\hbox{\rm\scriptsize III}}^{(1)}({k})(-A^2-A^{-2})^{2}+(4+A^{-4})f_{i,\hbox{\rm\scriptsize V}}^{(1)}({k})\\
\hskip 19mm +(4A^2+2A^{-2})f_{i-1,\hbox{\rm\scriptsize V}}^{(1)}({k})(-A^2-A^{-2})+(A^4+1)f_{i-2,\hbox{\rm\scriptsize V}}^{(1)}({k})(-A^2-A^{-2})^{2}.\\ \end{array} \right. $$ By combining with (14-15), we get for $i\ge 1$ $$\rho_h(f_{i,j}^{(1)}(k+1))\le 8n+14 \mbox{ and }\rho_l(f_{i,j}^{(1)}(k+1))\ge -4n-10 \hskip 10mm (16)$$ where $j\in \{\hbox{\rm I},\hbox{\rm II},\hbox{\rm III}, \hbox{\rm IV}\}$. \noindent By applying Lemma $3.2-3$ with a similar way in the argument of the proof of (16), we obtain for $i\ge 1$ $$\rho_h(f_{i,j}^{(r)}(k+1))\le 8n+14 \mbox{ and }\rho_l(f_{i,j}^{(r)}(k+1))\ge -4n-10 \hskip 10mm (17)$$ where $j\in \{\hbox{\rm I},\hbox{\rm II},\hbox{\rm III}, \hbox{\rm IV}\}$ and $2\le r\le 3$.
Thus, it is obvious by combining (16-17) that for $i\ge 1$ $$\rho_h(f_{i}(k+1))\le 8(k+1)+14 \mbox{ and }\rho_l(f_{i}(k+1))\ge -4(k+1)-10.$$ Hence, the conclusion is implied. \hskip 15mm $\Box$ \vskip 3mm In 1987, Kauffman, Thistlethwaite and Murasugi independently proved the following result. \vskip 3mm \noindent{\bf Lemma $3.5$ ( \cite{Th87,Ka87,Mu87})} {\it If $L$ is connected, irreducible, alternating link, then the breadth of $V_L(t)$ is precisely $m$.}
\vskip 3mm \noindent{\bf Proof of Theorem $1.2$.} Since $$<KV_n>=\sum\limits_{i\ge 1}f_{i}(n),$$ it is obvious by Lemma $3.4$ that for $i\ge 1$ $$\rho_h(<KV_n>)\le 8n+14 \mbox{ and }\rho_l(<KV_n>)\ge -4n-10.$$ Thus, $$\rho_h(V_{KV_n}(t))\le \displaystyle\frac{-3\omega(KV_n)+8n+14}{4} \mbox{ and }\rho_l(V_{KV_n}(t))\ge \displaystyle\frac{-3\omega(KV_n)-4n-10}{4}.$$ Then $$br(V_{KV_n}(t))=\rho_h(V_{KV_n}(t))-\rho_l(V_{KV_n}(t))\le 3n+6. \hskip 10mm (18)$$ Because $KV_n$ is connected and irreducible with $4n+6$ crossings, the result is implied by Lemma 3.5 and the inequality (18). \hskip 10mm $\Box$ \vskip 3mm
\vskip 3mm \noindent{\bf Proof Theorem $1.3$.} Since $\omega(KV_n)=-4n-6$ and $p_{i}(n)=\sum\limits_{k=1}^3p_{i}^{k}(n)$, the result is implied by applying a similar way in the argument of the proof of Theorem $1.1$. $\Box$ \vskip 5mm \noindent{\bf $4.$ Kauffman-Jones polynomials for a type of virtual links }
\vskip 5mm
\vskip 12mm
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\begin{footnotesize}
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\put(-5,4){{(a) $RT'_{n}$ }} \put(40,4){{(b) oriented $RT'_{n}$ }} \put(95,4){{(c) $RT'_{3}$ }} \put(20,-5){{Fig.4: A type of virtual knots $RT'_{n}$ }} \end{footnotesize} \end{picture} \end{center} \vskip 3mm
In this section, we calculate Kauffman-Jones polynomials of the infinite family of virtual knots $RT'_n$ for $n\ge 1$. Set $S(RT'_n)$ to be the set of all of states of $RT'_n$ for $n\ge 0$. Denote the following set.
$$ \begin{array}{ll}
S_{\hbox{\rm\scriptsize I}}(RT'_n)=\{s\in S(RT'_n),s_{x_2}=A|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^2, s_{y_i}=A,\\
\hskip 18mm s_{z_{2k-1}}=s_{z_{2k}}=s_{z_i}=A^{-}\mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{z_i}=A^-, s_{y_i}=A \mbox{ for }1\le i\le 2n \}, \end{array} $$ $ \begin{array}{ll}
S_{\hbox{\rm\scriptsize II}}(RT'_n)=\{s\in S(RT'_n),s_{x_2}=A|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2}, s_{y_i}=A, \\ \hskip 18mm s_{z_i}=A^{-} \mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{z_i}=A^-, s_{x_1}=s_{y_i}=A \mbox{ for }1\le i\le 2n \}, \end{array} $ $$ \begin{array}{ll}
S_{\hbox{\rm\scriptsize III}}(RT'_n)=\{s\in S(RT'_n),s_{x_2}=A^-|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{y_{2k-1}}\cdot s_{y_{2k}}\ne A^2, s_{y_i}=A,\\
\hskip 18mm s_{z_{2k-1}}=s_{z_{2k}}=s_{z_i}=A^{-}\mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{z_i}=A^-, s_{y_i}=A \mbox{ for }1\le i\le 2n\} \end{array} $$ $ \begin{array}{ll}
S_{\hbox{\rm\scriptsize IV}}(RT'_n)=\{s\in S(RT'_n),s_{x_2}=A^-|\mbox{ either }\exists 1\le k\le n \mbox{ such that }s_{z_{2k-1}}\cdot s_{z_{2k}}\ne A^{-2}, s_{y_i}=A, \\ \hskip 18mm s_{z_i}=A^{-} \mbox{ for }2k+1\le i\le 2n\mbox{ or }s_{x_1}=s_{y_i}=A, s_{z_i}=A^- \mbox{ for }1\le i\le 2n \}. \end{array} $
\vskip 3mm
\noindent{\bf Lemma $4.1$} Let $p_{1,\hbox{\rm\scriptsize I}}(RT'_0)=1$, $p_{1,\hbox{\rm\scriptsize II}}(RT'_0)=A^2$, $p_{2,\hbox{\rm\scriptsize III}}(RT'_0)=A^{-2}$, $p_{1,\hbox{\rm\scriptsize IV}}(RT'_0)=1$. Set $p_{i,\hbox{\rm\scriptsize I}}(RT'_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize I}}(RT'_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize II}}(RT'_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize II}}(RT'_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize III}}(RT'_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize III}}(RT'_n),l(s)=i}A^{a(s)-b(s)}$, $p_{i,\hbox{\rm\scriptsize IV}}(RT'_n)=\sum\limits_{s\in S_{\hbox{\rm\scriptsize IV}}(RT'_n),l(s)=i}A^{a(s)-b(s)}$. Then for $n\ge 1$ $$ \left\{ \begin{array}{lll} p_{i,\hbox{\rm\scriptsize I}}(RT'_n)=p_{i,\hbox{\rm\scriptsize
I}}(RT'_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize I}}(RT'_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize I}}(RT'_{n-1})\\
\hskip 19mm +2A^{-2}p_{i,\hbox{\rm\scriptsize II}}(RT'_{n-1})+A^{-4}p_{i-1,\hbox{\rm\scriptsize II}}(RT'_{n-1}); \\ p_{i,\hbox{\rm\scriptsize II}}(RT_n)=2A^2p_{i,\hbox{\rm\scriptsize
I}}(RT'_{n-1})+(A^4+4)p_{i-1,\hbox{\rm\scriptsize I}}(RT'_{n-1})\\
\hskip 19mm +(2A^2+2A^{-2})p_{i-2,\hbox{\rm\scriptsize I}}(RT'_{n-1})+p_{i-3,\hbox{\rm\scriptsize I}}(RT'_{n-1})+5p_{i,\hbox{\rm\scriptsize II}}(RT'_{n-1})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize II}}(RT'_{n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize II}}(RT'_{n-1}); \\ p_{i,\hbox{\rm\scriptsize III}}(RT_n)=p_{i,\hbox{\rm\scriptsize
III}}(RT'_{n-1})+2A^{-2}p_{i-1,\hbox{\rm\scriptsize III}}(RT'_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize III}}(RT'_{n-1})\\
\hskip 19mm +2A^{-2}p_{i-1,\hbox{\rm\scriptsize IV}}(RT'_{n-1})+A^{-4}p_{i-2,\hbox{\rm\scriptsize IV}}(RT'_{n-1}); \\ p_{i,\hbox{\rm\scriptsize IV}}(RT'_n)=2A^2p_{i+1,\hbox{\rm\scriptsize
III}}(RT'_{n-1})+(A^4+4)p_{i,\hbox{\rm\scriptsize III}}(RT'_{n-1})\\
\hskip 19mm +(2A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize III}}(RT'_{n-1})+p_{i-2,\hbox{\rm\scriptsize III}}(RT'_{n-1})+5p_{i,\hbox{\rm\scriptsize IV}}(RT'_{n-1})\\
\hskip 19mm +(4A^2+2A^{-2})p_{i-1,\hbox{\rm\scriptsize IV}}(RT'_{n-1})+(A^4+1)p_{i-2,\hbox{\rm\scriptsize IV}}(RT'_{n-1}). \end{array} \right. $$
\vskip 3mm
\noindent{\bf Proof of Theorem $1.4$} By a similar way in the argument of the proof of Theorem $1.1$, the result holds.
\hskip 150mm $\Box$
\vskip 5mm
\hskip 55mm{\bf $5.$ Further study } \vskip 5mm
In this section, we introduce general $m$-string alternating links(or virtual links) and $m$-string tangle links(or virtual links) for $m\ge 2$. Several problems are proposed.
Set $n$ to be a positive integer in this section. Generally, given a link(or virtual link) $L_0$ with $m+1$ parallel edges, denote one of them by $e_0$ and denote others by $e_i=(u_i^{r_i},v_i^{\varepsilon_i})$ in sequence for $1\le i\le m$. Here $r_i\in\{+,-\}$, $r_0=r$. Assume that $e_0$ is the leftmost edge and leave other cases to readers to get in a similar way. Add $2n$ crossings $x_{i,l}$ on $e_i$ in sequence for $1\le i\le m$ and $1\le l\le 2n$ respectively. Let $(u_{i}^{r_i},x_{i,1}^{r})$, $(x_{i,1}^{r},x_{i,2}^{-r})$, $\cdots$, $(x_{i,2n-1}^{r},x_{i,2n}^{-r})$, $(x_{i,2n}^{-r},v_{i}^{\epsilon_{i}})$ be a subdivision of $e_i$ for odd $1\le i\le m$ and let $(u_{i}^{r_i},x_{i,1}^{-r})$, $(x_{i,1}^{-r},x_{i,2}^{r})$, $\cdots$, $(x_{i,2n-1}^{-r},x_{i,2n}^{r})$, $(x_{i,2n}^{r},v_{i}^{\epsilon_{i}})$ be a subdivision of $e_i$ for even $1\le i\le m$. Add edges $(u_{i}^{r_{i}},x_{i,1}^{-r})$ and $(v_{i}^{\varepsilon_{i}},x_{i,2n}^{r})$ for odd $1\le i\le m$, add edges $(u_{i}^{r_{i}},x_{i,1}^{r})$ and $(v_{i}^{\varepsilon_{i}},x_{i,2n}^{-r})$ for even $1\le i\le m$, add edges $(x_{i,l}^{-r},x_{i+1,l}^{r})$, $(x_{i,l}^{-r},x_{i-1,l}^{r})$ for odd $1\le i\le m$ and odd $1\le l\le 2n$, $(x_{i,l}^{r},x_{i+1,l}^{-r})$, $(x_{i,l}^{r},x_{i-1,l}^{-r})$ for odd $1\le i\le m$ and even $1\le l\le 2n$, and then add edges $(x_{m,l}^r,x_{m,l+1}^{-r})$ for even $m$ and odd $1\le l\le 2n$. A link (or a virtual link) $AL_n$ is constructed which is called an {\it $m$-string alternating} link(or virtual link). Here, $x_{m+1,l}^{r}=x_{m,l+1}^{r}$ and $x_{0,l}^{r}=x_{1,l-1}^{r}$ for odd $1\le l\le 2n$, $x_{m+1,l}^{-r}=x_{m,l-1}^{-r}$ and $x_{0,l}^{-r}=x_{1,l+1}^{-r}$ for even $1\le l\le 2n$, $x_{0,1}^r=x_{1,0}^r=u_0^r$, $x_{0,2n+1}^{-r}=v_0^{\varepsilon_0}$. An example is shown in Fig.5 (b) for $m=3$ and $n=2$.
\vskip 12mm
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\put(89,4){(c) $TL_4$} \put(29,-2){Fig.$5$:$L_0$, $AL_4$ and $TL_4$} \end{picture} \end{center}
\vskip 3mm Similarly, let $L_0$ be a link (or virtual link) above. Add $2n$ crossings $x_{i,l}$ on $e_i$ in sequence for $1\le i\le m$ and $1\le l\le 2n$ respectively. Let $(u_{i}^{r_i},x_{i,1}^{r})$, $(x_{i,1}^{r},x_{i,2}^{-r})$, $\cdots$, $(x_{i,2n-1}^{r},x_{i,2n}^{-r})$, $(x_{i,2n}^{-r},v_{i}^{\epsilon_{i}})$ be a subdivision of $e_i$. Add edges $(u_{i}^{r_{i}},x_{i,1}^{r})$ and $(v_{i}^{\varepsilon_{i}},x_{i,2n}^{-r})$ for $1\le i\le l$, add edges $(x_{i,l}^{-r},x_{i+1,l}^{-r})$, $(x_{i,l}^{-r},x_{i-1,l}^{-r})$ for odd $1\le i\le m$ and odd $1\le l\le 2n$, $(x_{i,l}^{r},x_{i+1,l}^{r})$, $(x_{i,l}^{r},x_{i-1,l}^{r})$ for odd $1\le i\le m$ and even $1\le l\le 2n$, and then add edges $(x_{m,l}^{-r},x_{m,l+1}^{r})$ for even $m$ and odd $1\le l\le 2n$. A link (or virtual link) $TL_n$ is constructed which is called an {\it $m$-string tangle} link (or virtual link). Here, $x_{m+1,l}^{-r}=x_{m,l+1}^{r}$ and $x_{0,l}^{-r}=x_{1,l-1}^{r}$ for odd $1\le l\le 2n$, $x_{m+1,l}^{r}=x_{m,l-1}^{-r}$ and $x_{0,l}^{r}=x_{1,l+1}^{-r}$ for even $1\le l\le 2n$, $x_{0,1}^{-r}=x_{1,0}^{-r}=u_0^r$, $x_{0,2n+1}^{r}=v_0^{\varepsilon_0}$. An example is also shown in Fig.5 (c) for $m=3$ and $n=2$.
\vskip 3mm \noindent{\bf Problem $5.1.$ }{\it Given a link $L_0$, let $AL_n$ is an $m$-string alternating link constructed from $L_0$ for $m\ge 3$. Determine $V_{AL_n}(t)$.}
\vskip 3mm \noindent{\bf Problem $5.2.$ }{\it Given a link $L_0$, let $TL_n$ is an $m$-string tangle link constructed from $L_0$ for $m\ge 3$. Determine $V_{TL_n}(t)$.}
\vskip 3mm \noindent{\bf Conjecture $5.3.$ }{\it Suppose that $L_0$ is connected and irreducible and that $TL_n$ is an $m$-string tangle link constructed from $L_0$ for $m\ge 2$. If $L_0$ is non-alternating, then $TL_n$ is also non-alternating. }
\vskip 3mm \noindent{\bf Conjecture $5.4.$ }{\it Suppose that a link $L_0$ is prime and that $L_n$ is an $m$-string alternating (or tangle) link constructed from $L_0$ for $m\ge 2$. Then $L_n$ is prime. }
\vskip 5mm
\end{document} | arXiv |
Tue, 11 Aug 2020 18:19:48 GMT
10.1: General Principles of Angular Momentum
[ "article:topic", "authorname:rfitzpatrick", "showtoc:no" ]
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Book: Introductory Quantum Mechanics (Fitzpatrick)
10: Addition of Angular Momentum
Contributed by Richard Fitzpatrick
Professor (Physics) at University of Texas at Austin
Contributors and Attributions
The three fundamental orbital angular momentum operators, \(L_x\), \(L_y\), and \(L_z\), obey the commutation relations ([e8.6])–([e8.8]), which can be written in the convenient vector form:
\[{\bf L}\times{\bf L} = {\rm i}\,\hbar\,{\bf L}.\] Likewise, the three funcdamental spin angular momentum operators, \(S_x\), \(S_y\), and \(S_z\), obey the commutation relations ([e10.1x])–([e10.2x]), which can also be written in vector form: that is,
\[{\bf S}\times{\bf S} = {\rm i}\,\hbar\,{\bf S}.\] Because the orbital angular momentum operators are associated with the electron's motion through space, whereas the spin angular momentum operators are associated with its internal motion, and these two types of motion are completely unrelated (i.e., they correspond to different degrees of freedom—see Section [sfuncon]), it is reasonable to suppose that the two sets of operators commute with one another: that is,
\[[L_i, S_j] = 0,\] where \(i,j=1,2,3\) corresponds to \(x,y,z\), respectively.
Let us now consider the electron's total angular momentum vector
\[{\bf J} = {\bf L} + {\bf S}.\] We have
\[\begin{aligned} {\bf J}\times{\bf J} &= ({\bf L}+{\bf S})\times({\bf L}+{\bf S})\nonumber \\[0.5ex] &= {\bf L}\times{\bf L} + {\bf S}\times{\bf S}+{\bf L}\times{\bf S}+{\bf S}\times{\bf L} ={\bf L}\times{\bf L} + {\bf S}\times{\bf S} \nonumber\\[0.5ex] &= {\rm i}\,\hbar\,{\bf L} + {\rm i}\,\hbar\,{\bf S}= {\rm i}\,\hbar\,{\bf J}.\end{aligned}\] In other words,
\[{\bf J}\times{\bf J} = {\rm i}\,\hbar\,{\bf J}.\]
It is thus evident that the three funcdamental total angular momentum operators, \(J_x\), \(J_y\), and \(J_z\), obey analogous commutation relations to the corresponding orbital and spin angular momentum operators. It therefore follows that the total angular momentum has similar properties to the orbital and spin angular momenta. For instance, it is only possible to simultaneously measure the magnitude squared of the total angular momentum vector,
\[J^{\,2} =J_x^{\,2}+J_y^{\,2}+J_z^{\,2},\]
together with a single Cartesian component. By convention, we shall always choose to measure \(J_z\). A simultaneous eigenstate of \(J_z\) and \(J^{\,2}\) satisfies
\[\begin{aligned} J_z\,\psi_{j,m_j}&= m_j\,\hbar\,\psi_{j,m_j},\\[0.5ex] J^{\,2}\,\psi_{j,m_j} &= j\,(j+1)\,\hbar^{\,2}\,\psi_{j,m_j},\end{aligned}\]
where the quantum number \(j\) can take positive integer, or half-integer, values, and the quantum number \(m_j\) is restricted to the following range of values:
\[-j, -j+1,\cdots, j-1, j.\]
\[J^{\,2} = ({\bf L}+{\bf S})\cdot({\bf L}+{\bf S}) = L^2+ S^{\,2} + 2\,{\bf L}\cdot{\bf S},\]
which can also be written as
\[\label{e11.12} J^{\,2} = L^2+S^{\,2} +2\,L_z\,S_z+ L_+\,S_-+L_-\,S_+.\]
We know that the operator \(L^2\) commutes with itself, with all of the Cartesian components of \({\bf L}\) (and, hence, with the raising and lowering operators \(L_\pm\)), and with all of the spin angular momentum operators. (See Section [s8.2].) It is therefore clear that
\[[J^{\,2},L^2] = 0.\] A similar argument allows us to also conclude that
\[[J^{\,2},S^{\,2}]=0.\] Now, the operator \(L_z\) commutes with itself, with \(L^2\), with all of the spin angular momentum operators, but not with the raising and lowering operators \(L_\pm\). (See Section [s8.2].) It follows that
\[[J^{\,2},L_z]\neq 0.\]
Likewise, we can also show that
\[[J^{\,2},S_z]\neq 0.\]
Finally, we have
\[J_z = L_z+S_z,\]
where \([J_z,L_z]=[J_z,S_z]=0\).
Recalling that only commuting operators correspond to physical quantities that can be simultaneously measured (see Section [smeas]), it follows, from the previous discussion, that there are two alternative sets of physical variables associated with angular momentum that we can measure simultaneously. The first set correspond to the operators \(L^2\), \(S^{\,2}\), \(L_z\), \(S_z\), and \(J_z\). The second set correspond to the operators \(L^2\), \(S^{\,2}\), \(J^{\,2}\), and \(J_z\). In other words, we can always measure the magnitude squared of the orbital and spin angular momentum vectors, together with the \(z\)-component of the total angular momentum vector. In addition, we can either choose to measure the \(z\)-components of the orbital and spin angular momentum vectors, or the magnitude squared of the total angular momentum vector.
Let \(\psi^{(1)}_{l,s;m,m_s}\) represent a simultaneous eigenstate of \(L^2\), \(S^{\,2}\), \(L_z\), and \(S_z\) corresponding to the following eigenvalues:
\[\begin{aligned} L^2\,\psi^{(1)}_{l,s;m,m_s}&= l\,(l+1)\,\hbar^{\,2}\,\psi^{(1)}_{l,s;m,m_s},\\[0.5ex] S^{\,2}\,\psi^{(1)}_{l,s;m,m_s}&= s\,(s+1)\,\hbar^{\,2}\,\psi^{(1)}_{l,s;m,m_s},\\[0.5ex] L_z\,\psi^{(1)}_{l,s;m,m_s}&= m\,\hbar\,\psi^{(1)}_{l,s;m,m_s},\\[0.5ex] S_z\,\psi^{(1)}_{l,s;m,m_s}&= m_s\,\hbar\,\psi^{(1)}_{l,s;m,m_s}.\end{aligned}\]
It is easily seen that
\[\begin{aligned} J_z\,\psi^{(1)}_{l,s;m,m_s}& = (L_z+S_z)\,\psi^{(1)}_{l,s;m,m_s}= (m+m_s)\,\hbar\,\psi^{(1)}_{l,s;m,m_s}\nonumber\\[0.5ex] &= m_j\,\hbar\,\psi^{(1)}_{l,s;m,m_s}.\end{aligned}\]
Hence,
\[m_j = m+m_s.\label{e11.23}\]
In other words, the quantum numbers controlling the \(z\)-components of the various angular momentum vectors can simply be added algebraically.
Finally, let \(\psi^{(2)}_{l,s;j,m_j}\) represent a simultaneous eigenstate of \(L^2\), \(S^{\,2}\), \(J^{\,2}\), and \(J_z\) corresponding to the following eigenvalues:
\[\begin{aligned} L^2\,\psi^{(2)}_{l,s;j,m_j}&= l\,(l+1)\,\hbar^{\,2}\,\psi^{(2)}_{l,s;j,m_j},\\[0.5ex] S^{\,2}\,\psi^{(2)}_{l,s;j,m_j}&= s\,(s+1)\,\hbar^{\,2}\,\psi^{(2)}_{l,s;j,m_j},\\[0.5ex] J^{\,2}\,\psi^{(2)}_{l,s;j,m_j}&= j\,(j+1)\,\hbar^{\,2}\,\psi^{(2)}_{l,s;j,m_j},\label{e11.26}\\[0.5ex] J_z\,\psi^{(2)}_{l,s;j,m_j}&= m_j\,\hbar\,\psi^{(2)}_{l,s;j,m_j}.\end{aligned}\]
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)
10.2: Angular Momentum in Hydrogen Atom
Richard Fitzpatrick
© Copyright 2021 Physics LibreTexts | CommonCrawl |
\begin{document}
\title{A universal, memory-assisted entropic uncertainty relation} \author{Z.-H. Ma,$^{1,2,+}$ C.-M. Yao,$^{3,4,+}$ Z.-H. Chen,$^{5}$ S. Severini$^{2}$, A. Serafini$^{4}$} \address{$^1$Department of Mathematics, Shanghai Jiao-Tong University, Shanghai, 200240, P. R. China} \address{$^2$Department of Computer Science, University College London, Gower St., WC1E 6BT London, United Kingdom} \address{$^3$Department of Physics and Electronic Science, Hunan University of Arts and Science, Changde, 415000, China} \address{$^4$Department of Physics and Astronomy, University College London, Gower St., WC1E 6BT London, United Kingdom} \address{$^5$Department of Science, Zhijiang College, Zhejiang University of Technology, Hangzhou, 310024, China} \address{$^{+}$ These two authors contributed equally to this work.}
\date{\today}
\begin{abstract} We derive a new memory-assisted entropic uncertainty relation for non-degenerate Hermitian observables where both quantum correlations, in the form of conditional von Neumann entropy, and quantum discord between system and memory play an explicit role. Our relation is `universal', in the sense that it does not depend on the specific observable, but only on properties of the quantum state. We contrast such an uncertainty relation with previously known memory-assisted relations based on entanglement and correlations. Further, we present a detailed comparative study of entanglement- and discord-assisted entropic uncertainty relations for systems of two qubits -- one of which plays the role of the memory -- subject to several forms of independent quantum noise, in both Markovian and non-Markovian regimes. We thus show explicitly that, partly due to the ubiquity and inherent resilience of quantum discord, discord-tightened entropic uncertainty relations often offer a better estimate of the uncertainties in play. \end{abstract}
\pacs{03.67.-a,03.67.Mn,03.65.Yz,03.65.Ud}
\maketitle
\section{Entropic uncertainty relations}
One of the key aspects of quantum theory is that it is fundamentally impossible to know certain things, such as a particle's position and momentum, simultaneously with infinite precision. In fact, quantum mechanical uncertainty principles assert fundamental limits on the precision with which certain pairs of physical properties, such as position and momentum, may be simultaneously known. Originally observed by Heisenberg \cite{Heisenberg}, the uncertainty principle is best known in the Robertson-Schr\"odinger form \cite{Robertson}
$$\Delta X\Delta Y\geq \frac{1}{2}|\langle[X,Y]\rangle|,$$ where $\Delta X$ ( $\Delta Y$) represents the standard deviation of the corresponding observable $X (Y)$, formally represented by a Hermitian operator.
The entropic uncertainty relation for any two general observables was first given by Deutsch in terms of an information-theoretic model \cite{Deutsch}. Afterwards, an improved version was given by Kraus and then proved by Maassen and Uiffink \cite{Kraus}, which strengthen and generalizes Heisenberg's uncertainty relations, and can be written as follows: $$H(X)+H(Y)\geq \log_2\left(\frac{1}{c}\right),$$ where $H$ is the Shannon entropy of the measured observable and $c$ quantifies the `complementarity' between the observables:
$c=\max_{(i,j)} |\langle x_i | y_j\rangle|^2$ if $X$ and $Y$ are non-degenerate
observables ($|x_i\rangle , |y_j\rangle$ being the eigenvectors of $X$ and $Y$, respectively). Recently, the uncertainty principle in terms of entropy has been extended to the case involving entanglement with a quantum memory: It was proven by Berta et al.\ \cite{Berta} that \begin{equation}
S(X|B)+S(Y|B)\geq \log_2\left(\frac{1}{c}\right)+S(A|B), \label{berta} \end{equation}
where $S(X|B)=S\left[\sum_{j}\left(|x_j\rangle\langle x_j|\otimes\mathbbm{1}\right)\varrho_{AB}\left(|x_j\rangle\langle x_j|\otimes\mathbbm{1}\right)\right]$ and
$(S(S|B))=S\left[\sum_{j}\left(|y_j\rangle\langle y_j|\otimes{1}\right)\varrho_{AB}\left(|y_j\rangle\langle y_j|\otimes{1}\right)\right]$ are the average conditional von Neumann entropies representing the uncertainty of the measurement outcomes of $X$ and $Y$ obtained using the information stored in system $B$, given that the initial system plus memory state was $\varrho_{AB}$.
The quantity $S(A|B) = S(\varrho_{AB})-S({\rm Tr}_A(\varrho_{AB}))$ represents instead the conditional von Neumann entropy between system $A$ and $B$, defined in analogy with the classical definition of conditional entropy. The measurement outcomes of two incompatible observables on a particle can be precisely predicted when it is maximally entangled with a quantum memory, as indicated by Eq.(\ref{berta}). This entanglement-assisted entropic uncertainty relation was promptly experimentally tested [6].
Note that the quantum conditional entropy $S(A|B)$, appearing in the lower bound above, is a quantifier of quantum correlations that can be negative for entangled states, and thus may tighten the uncertainty relation without memory. Quantum correlations may also be assessed by quantum discord, which we define in the following. The total correlations between two quantum systems $A$ and $B$ are quantified by the quantum mutual information \begin{equation}\label{Total}
\mathcal{I}(\varrho_{AB})=S(\varrho_{A})+S(\varrho_{B})-S(\varrho_{AB}) \; , \end{equation}
where $\varrho_{A(B)}= \mathrm{Tr}_{B(A)}(\varrho_{AB})$. On the other hand, the classical part of correlations is defined as the maximum information that can be obtained by performing a local measurement, which is defined as $\mathcal{I}(\varrho_{AB}|\{\hat{\Pi}_{A}^{j}\})=S(\varrho_{B})-\sum_j p_j S(\varrho_{B|j})$, where $\varrho_{B|j}=\tr_{A}((\hat{\Pi}_{A}^{j}\otimes \mathbbm{1})\varrho(\hat{\Pi}_{A}^{j}\otimes \mathbbm{1}))/\tr_{AB}[(\hat{\Pi}_{A}^{j}\otimes I)\varrho(\hat{\Pi}_{A}^{j}\otimes \mathbbm{1})]$, $\{\hat{\Pi}_{A}^{j}\}$ are POVM locally performed on subsystem $A$, $p_j$ is the probability of the measurement outcome $j$. Classical correlations are thus quantified by \cite{ved}:
\begin{equation}\label{classical}J_{A}(\varrho_{AB})=\mathrm{sup}_{\{\hat{\Pi}_{A}^{j}\}}\mathcal{I}(\varrho_{AB}|\{\hat{\Pi}_{A}^{j}\})\end{equation} Then quantum $A$-side discord (quantum correlation) is defined as the difference of total correlation and $A-$side classical correlation \cite{Ollivier}: \begin{equation}
D_{A}(\varrho_{AB})=\mathcal{I}(\varrho_{AB})- J_{A}(\varrho_{AB}) \; .\label{Bdiscord} \end{equation}
Recently, by considering the role of quantum discord and classical correlations of the joint system-memory state, Pati et al.\ \cite{Pati} obtained a modified entropic uncertainty relation that tightens the lower bound (\ref{berta}) of Berta et al.\: \begin{equation}
S(R|B)+S(S|B)\geq \log_{2}\frac{1}{c}+S(A|B) +\max\{0, D_{A}(\varrho_{AB})- J_{A}(\varrho_{AB})\} \; .\label{pati} \end{equation}
Entropic uncertainty relations have been shown to hold fundamental consequences for the security of cryptographic protocols \cite{tomamichel,ng,moloktov}, the foundations of thermodynamics \cite{hanggi} and entanglement theory \cite{guhne,niekamp}. See also \cite{wehner} for a quite recent review.
The main finding of the present work is the derivation of a new discord-assisted uncertainty relation, which we shall then contrast with the already known ones (Section 2). We will then move on to consider the behaviour of the different uncertainty relations under decoherence (Section 3), both to shed further light on our newly derived universal entropic uncertainty relation, and to provide the reader with a detailed comparative analysis of the different memory-assisted entropic uncertainty relations in realistic, noisy situations. We shall draw conclusions in Section 4.
\section{An observable-indipendent entropic uncertainty relation}
In the following, we use $X$ and $Z$ to denote two {\em non-degenerate} Hermitian observables described by the POVMs $X=\{X_{j}\}$ and $Z=\{Z_{k}\}$,
where $X_j$ and $Z_{j}$ are orthogonal projectors. Otherwise, we shall apply all the notation introduced in the previous section. Please also bear in mind that, in this paper, we always use $S(\varrho_{AB})$ ($S(X|B)$) to denote the von Neumann entropy (conditional von Neumann entropy) of the quantum state $\varrho_{AB}$ ($(X\otimes I)\varrho_{AB}$), while we use $H(X)$ to denote the Shannon entropy of the discrete probability distribution $P$ of measurement outcomes for $X$. Further, we will make use of the following lemma:
\noindent {\bf Lemma 1.} Let $X:=\{X_{j}\}$ and $Z:=\{Z_{k}\}$ be arbitrary POVMs on $A$, then for all single-partite state $\varrho_{A}$, \begin{equation}
H(X)+ H(Z)\geq \log\frac{1}{c(X)}+\log\frac{1}{c(Z)}\\
+2S(A),\label{Cond2} \end{equation} where $c(X):=\max_{i} \tr(X_{i})$ and $c(Z):=\max_{i} \tr(Z_{i})$, $H(X)$ is the Shannon entropy of the probability $p_i :={\rm Tr}[(X_i ) \varrho_{A}]$, $H(Z)$ is the Shannon entropy of the probability $q_i :={\rm Tr}[(Z_i ) \varrho_{A}]$.
\noindent {\bf Proof.} From Corollary 7 of Ref.\ \cite{Coles}, we know that \begin{equation}
H(X)\geq \log\frac{1}{c(X)}+S(A) \; .\label{Cond2a} \end{equation} By using the above relation twice, we get the result: \begin{equation}
H(X)+ H(Z)\geq \log\frac{1}{c(X)}+\log\frac{1}{c(Z)}\\
+2S(A) \; , \label{Cond2} \end{equation} which proves our lemma. \qed
Clearly, if $X$ and $Z$ are non degenerate Hermitian observables, such that their POVM elements are all one-dimensional projectors, the inequality of Lemma 1 becomes: \begin{equation}
H(X)+ H(Z)\geq
2S(A) \; . \label{Cond3} \end{equation}
We can now move on to our main result:
\noindent {\bf Theorem 1.} Let $X:=\{X_{j}\}$ and $Z:=\{Z_{k}\}$ be non-degenerate Hermitian observables on subsystem $A$, and let $\varrho_{AB}$ be any bipartite state of systems $A$ and $B$. One has \begin{align}
S(X|B)+ S(Z|B)\geq \nonumber\\
+2S(A|B)+ 2D_{A}(\varrho_{AB})\label{bound} \; , \end{align}
where $D_{A}(\varrho_{AB})$ is the quantum discord, $S(X|B)$ and $S(Z|B)$ conditional entropies after measurements on $A$, and
$S(A|B)$ is the von Neumann conditional entropy of state $\varrho_{AB}$, all defined above.
\noindent {\bf Proof.}
Consider a bipartite density operator $\varrho_{AB}$. If Alice performs a measurement of an observable $X$ on subsystem $A$, then the post-measurement state is $\varrho_{AB}^X = \sum_i(X_{i} \otimes {\mathbbm 1}) \varrho_{AB} (X_i \otimes {\mathbbm 1}) = \sum_i p_i X_{i} \otimes \varrho_{B|i}$, where $p_i =
{\rm Tr}[(X_{i} \otimes I) \varrho_{AB}]$ is the probability of obtaining the $i^{\rm th}$ outcome and $\varrho_{B|i}={\rm Tr}_{A}[(X_{i} \otimes I) \varrho_{AB} (X_{i} \otimes {\mathbbm 1})]/p_i$ is the conditional state of the memory $B$ corresponding to this outcome. The conditional von Neumann entropy $S(X|B)$ denotes the ignorance about the measurement outcome $X$ given information stored in a quantum memory held by an observer $B$. Thus, $S(X|B)=S(\varrho_{AB}^X)-S(\varrho_B)$ is the conditional entropy of the state $\varrho_{AB}^X$. This is given by \begin{equation}
S(X|B) = \sum_i p_i S(\varrho_B|i) + H(P) -S(\varrho_B)\label{Cond} \end{equation} Here $P:=(p_i)$ is the probability distribution of the outcomes.
It is worth noticing that, if $B$ is a null system then, of the three terms in (\ref{Cond}), only the term $H(P)$ survives, {\em i.e.}\ $S(X|B) = H(P)$: the conditional von Neumann entropy of the quantum state after the measurement is the same as the Shannon entropy of the discrete probability distribution of measurement outcomes.
Now, denote by $P:=(p_i)$ the probability distribution of the measurement outcomes, so $p_i:={\rm Tr}[(X_i \otimes I) \varrho_{AB}]$; it is clear that ${\rm Tr}[(X_i \otimes I) \varrho_{AB}]={\rm Tr}[(X_i ) \varrho_{A}]$. Hence, the probability distribution $P:=(p_i)$ only depends on the reduced density matrix $\varrho_{A}$. Then the Shannon entropy of $P$ only depends on the single partite state $\varrho_{A}$.
For the bipartite state $\varrho_{AB}$, with Hermitian measurements $X:=\{X_{j}\}$ and $Z:=\{Z_{k}\}$ performed on subsystem $A$, the following holds: \begin{align}
& S\left(X|B\right) +S\left(Z|B\right) \nonumber\\
& =H\left(X\right) -\mathcal{I}(\varrho_{AB}|\{X_{j}\}) +H\left(Z\right) -\mathcal{I}(\varrho_{AB}|\{Z_{k}\}) \nonumber\\
& \geq H\left(X\right) +H\left(Z|\right) -2J_{A}\left(\varrho_{AB}\right)\nonumber\\ & \geq2S(\varrho_{A})-2J_{A}(\varrho_{AB}) \nonumber\\
& =2S(A|B)+ 2D_{A}(\varrho_{AB})\label{proof-of-new-ineq} \, . \end{align} The first identity is a consequence of Eq.~(\ref{Cond}). The first inequality follows from the definition of the classical correlation
$J_{A}\left( \varrho_{AB}\right)$. Since $J_{A}\left( \varrho_{AB}\right)$ is defined as the maximal among all POVMs for $\mathcal{I}(\varrho_{AB}|\{\hat{\Pi}_{A}^{j}\})$,
in general, $\mathcal{I}(\varrho_{AB}|\{X_{j}\})\leq J_{A}\left( \varrho_{AB}\right)$; similarly, $\mathcal{I}(\varrho_{AB}|\{Z_{k}\})\leq J_{A}\left( \varrho_{AB}\right)$. The second inequality comes from Eq.~(\ref{Cond3}), a consequence of the Lemma reported above. Final, the last equality follows by the definition of quantum discord. \qed
We now intend to compare our bound with the inequality (\ref{pati}), which is also related to quantum discord. The term $\frac{1}{c}$ in (\ref{pati}) quantifies the compatibility of the two observables, and thus accounts for specific information concerning the measurements carried out. Thus, one should expect such a bound to be typically tighter than the relation (\ref{bound}). This is in fact often the case. However, aside from the intrinsic value of a universal, observable-independent relation, we find that, in several significant cases which we shall cover in the next section, our bound is more strict than the Inequality (\ref{pati}), and is almost the same as the actual value of the uncertainty. In a sense, in such cases, quantum correlations between the two subsystems make up for the absence of a measurement-specific term like $\frac{1}{c}$ of Ineq.~(\ref{pati}).
In particular, as shown in what follows, our bound turns out to improve quite often on Berta et al.'s lower bound of Inequality (\ref{berta}), based on quantum correlations.
\subsection{Examples}
Let us first illustrate the relevance of our relationship by considering some ad-hoc instances.
\subsubsection{Two-qubit Werner states.}
Consider the two-qubit Werner state
$\varrho_{AB}=\frac{1-f}{4}I_{A}\otimes I_{B}+f|\Psi^{-}\rangle\langle\Psi^{-}|$, where $|\Psi^{-}\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$ is the anti-symmetric Bell state, and $0\leq f\leq 1$. We choose observables $X$ and $Z$ as the two spin observables $\sigma_{x}$ and $\sigma_{z}$. Then the uncertainty can be determined as: \begin{align*}
S(X|B)+ S(Z|B)=2-(1-f)\log_2(1-f)-(1+f)\log_2(1+f) \; . \end{align*} The lower bound in (\ref{bound}) reads: \begin{align*}
2D_{A}(\varrho_{AB}) =2-(1-f)\log_2(1-f)-(1+f)\log_2(1+f) \; , \end{align*}
which is exactly equal to the uncertainty $S(X|B)+ S(Z|B)$. this also coincides with the lower bound (\ref{pati}) of Ref.~\cite{Pati}: \begin{align*}2-(1-f)\log_2(1-f)-(1+f)\log_2(1+f) \; . \end{align*} Instead, the lower bound (\ref{berta}) of Ref.~\cite{Berta} is \begin{align*}2-\frac{1+3f}{4}\log_2((1+3f))-\frac{3(1-f)}{4}\log_2(1-f)\; ,\end{align*} which is smaller, and thus less informative, than the other two.
\subsubsection{Two-qutrit Werner states.} For two qutits, a Werner state can be written as $\varrho_{AB}=\frac{1-f}{6}\Pi^{+}+\frac{f}{3}\Pi^{-}$, where $\Pi^{+}$ is the projector onto the symmetric subspace and $\Pi^{-}$ is the projector onto the antisymmetric subspace. We choose observables $X$ and $Z$ as two generators of $SU(3)$ and define : \begin{align*}
|0\rangle=(1,0,0)^{T},
|1\rangle=(0,1,0)^{T},|2\rangle=(0,0,1)^{T} \, , \end{align*} \begin{align*}
X=|0\rangle\langle 1|+|1\rangle\langle 0|, \quad
Z=|0\rangle\langle 0|-|1\rangle\langle 1| \, , \end{align*} such that \begin{align*}
S(X|B)+ S(Z|B) =f+3-(1-f)\log_2(1-f)-(1+f)\log_2(1+f) \; , \end{align*} \begin{align*} D_{A}(\varrho_{AB})&=2+f\log_2(\frac{f}{2})+(1-f)\log_2(\frac{1-f}{4})\\& -\frac{1-f}{2}\log_2(1-f)-\frac{1+f}{2}\log_2(\frac{1+f}{2}). \end{align*} Our bound (\ref{bound}) then reads: \begin{align*}
2S(A|B)+ 2D_{A}(\varrho_{AB})\\=f+3-(1-f)\log_2(1-f)-(1+f)\log_2(1+f)\; , \end{align*}
which is equal to the uncertainty $S(X|B)+ S(Z|B)$ But the lower bound (\ref{pati}) of Ref.~\cite{Pati} is \begin{align*} -f+1-(1-f)\log_2(1-f)-f\log_2f\\+ \max\{0,2f+f\log_2f-\log_2\frac{3(1+f)}{4}\\-f\log_2(1+f)\}\\ =f+3-\log_23-(1-f)\log_2(1-f)-(1+f)\log_2(1+f)\; , \end{align*} which is smaller than what we obtained. For the lower bound (\ref{berta}) of Ref.~\cite{Berta} one has \begin{align*} -f+1-(1-f)\log_2(1-f)-f\log_2f \; , \end{align*} which is smaller than our lower bound. We have thus identified a situation, with 2-qutrit Werner states, {\em where our bound performs better than the previously known ones}.
\subsubsection{Isotropic states.} Consider a bipartite isotropic state with local Hilbert space dimension $d$, $\varrho=f \phi_{d}+\frac{1-f}{d^2-1}(I-\phi_{d})$. For $d=2$, consider the following observables \begin{align*}
|0\rangle=(1,0)^{T},|1\rangle=(0,1)^{T}, \end{align*} \begin{align*}
X=|0\rangle\langle 1|+|1\rangle\langle 0|,Z=|0\rangle\langle 0|-|1\rangle\langle 1|. \end{align*} Then \begin{align*}
S(X|B)+ S(Z|B)=-\frac{2}{3}(-2f+2(1-f)\log_2(1-f)+\log_2\frac{4(1+2f)}{27}+2f\log_2(1+2f)) \, , \end{align*} \begin{align*} D_{A}(\varrho_{AB})=\frac{1-f}{3}\log_2(\frac{1-f}{3})+f\log_2f-\frac{1+2f}{3}\log_2\frac{1+2f}{6}. \end{align*} The universal lower bound (\ref{bound}) reads: \begin{align*} -\frac{2}{3}(-2f+2(1-f)\log_2(1-f)+\log_2\frac{4(1+2f)}{27}+2f\log_2(1+2f)) \, , \end{align*}
which is exactly equal to the uncertainty $S(X|B)+ S(Z|B)$. The lower bound (\ref{pati}) of Ref.~\cite{Pati} is also equal to the uncertainty $S(X|B)+ S(Z|B)$ in this case. However, the lower bound (\ref{berta}) of Ref.~\cite{Berta} equals \begin{align*} -(1-f)\log_2\frac{1-f}{3}-f\log_2 f \; , \end{align*} which is always smaller than what obtained with discord-assisted bounds.
\begin{figure}
\caption{Plot of the different lower bounds for isotropic states with dimension $d=2$ and observables given by the matrices in Eqs.~(\ref{X1},\ref{Z1}).
The upper blue `+'line and pink solid lines represent the uncertainty $S(X|B)+ S(Z|B)$ and the universal lower bound (\ref{bound}), the lower green `-' line and red `.-' line represent the results of the bounds (\ref{berta}) pof Ref.~\cite{Berta} and (\ref{pati}) of Ref.~\cite{Pati}.}
\label{bound1}
\end{figure}
When $d=3$, instead, by choosing the observables \begin{align*}
|0\rangle=(1,0,0)^{T},|1\rangle=(0,1,0)^{T},|2\rangle=(0,0,1)^{T}, \end{align*} \begin{align*}
X=|0\rangle\langle 1|+|1\rangle\langle 0|,Z=|0\rangle\langle 0|-|1\rangle\langle 1| , \end{align*} one gets \begin{align*}
S(X|B)+ S(Z|B)=-\frac{1}{2}(3(1-f)\log_2\frac{1-f}{8}+\log_2\frac{27(1+3f)}{4}+3f\log_2\frac{1+3f}{12}) , \end{align*} \begin{align*} D_{A}(\varrho_{AB})=\frac{1-f}{4}\log_2(\frac{1-f}{8})+f\log_2f-\frac{1+3f}{4}\log_2\frac{1+3f}{12}. \end{align*} The universal lower bound (\ref{bound}) then reads: \begin{align*} -\frac{1}{2}(3(1-f)\log_2\frac{1-f}{8}+\log_2\frac{27(1+3f)}{4}+3f\log_2\frac{1+3f}{12}), \end{align*}
which is equal to the uncertainty $S(X|B)+ S(Z|B)$. The lower bound (\ref{pati}) of Ref.~\cite{Pati} is instead \begin{align*} -\frac{1}{2}(3(1-f)\log_2\frac{1-f}{8}+\log_2\frac{243(1+3f)}{4}+3f\log_2\frac{1+3f}{12}), \end{align*} which is smaller than ours. The lower bound (\ref{berta}) of Ref.~\cite{Berta} is equal to \begin{align*} -(1-f)\log_2(1-f)-f\log_2 f-(3f-3)-\log_23 \; , \end{align*} which is also smaller than our bound.
\begin{figure}
\caption{Plot of the different lower bounds for isotropic states with dimension $d=3$ and observables given by the matrices in Eqs.~(\ref{X2},\ref{Z2}).
The upper blue `+'line and pink solid lines represent the uncertainty $S(X|B)+ S(Z|B)$ and the universal lower bound (\ref{bound}), the lower green `-' line and red `.-' line represent the results of the bounds (\ref{berta}) pof Ref.~\cite{Berta} and (\ref{pati}) of Ref.~\cite{Pati}. }
\label{bound2}
\end{figure}
For an isotropic state with $d=2$, other cases can be constructed where our lower bound provides one with a substantial advantage, such as \begin{equation} X=\begin{pmatrix} 0.272007 , 0.0483473+0.584816 i \\ 0.0483473-0.584816 i, 0.246297 \end{pmatrix},\label{X1} \end{equation} \begin{equation} Z=\begin{pmatrix} 0.43916 , 0.857154+0.976248 i \\ 0.857154-0.976248 i, 0.515329 \end{pmatrix}. \label{Z1} \end{equation} The different bounds for the observables above are displayed in Fig.~\ref{bound1}. Likewise, cases where our lower bound is tighter can be found for $d=3$, such as \begin{equation} X=\begin{pmatrix} 0.246301 , 0.267394 + 0.627628 i,0.155311 + 0.270053 i \\ 0.267394 - 0.627628 i, 0.752065,0.231887 + 0.500147 i\\ 0.155311 - 0.270053 i,0.231887 - 0.500147 i,0.94377 \end{pmatrix}, \label{X2} \end{equation} \begin{equation} Z=\begin{pmatrix} 0.586665 , 0.146795 + 0.957852 i,0.687252 + 0.677623 i \\ 0.146795 - 0.957852 i, 0.709581,0.405322 + 0.525615 i\\ 0.687252 - 0.677623 i,0.405322 - 0.525615 i,0.901804 \end{pmatrix}, \label{Z2} \end{equation} whose uncertainties and different bounds are depicted in Fig.~\ref{bound2}
\subsubsection{Qubit-qudit states.} Consider the qubit-qutrit state
$\varrho=\alpha (|02\rangle\langle 02|+|12\rangle\langle 12|)+\beta (|\phi^{+}\rangle\langle\phi^{+}|+|\phi^{-}\rangle\langle\phi^{-}|+
|\psi^{+}\rangle\langle\psi^{+}|)+\gamma|\psi^{-}\rangle\langle\psi^{-}|$, where
$\phi_{\pm}=\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle)$ and $\psi_{\pm}=\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle))$, as well as the following observables: \begin{align*}
|0\rangle=(1,0)^{T},|1\rangle=(0,1)^{T}, \end{align*} \begin{align*}
X=|0\rangle\langle 1|+|1\rangle\langle 0|,Z=|0\rangle\langle 0|-|1\rangle\langle 1|. \end{align*} Then \begin{align*}
S(X|B)+ S(Z|B)=4\alpha-4\beta-4\beta \log_2(\beta)-2(\beta+\gamma)\log_2(\beta+\gamma)+2(3\beta+\gamma)\log_2(3\beta+\gamma) \, . \end{align*}
Our bound (\ref{bound}) and the lower bound (\ref{pati}) of Ref.~\cite{Pati} are all equal to the uncertainty $S(X|B)+ S(Z|B)$ in this instance. The bound (\ref{berta}) of Ref.~\cite{Berta} is instead \begin{align*} 4\alpha-3\beta \log_2(\beta)-\gamma \log_2(\gamma)+(3\beta+\gamma)\log_2(3\beta+\gamma) \; , \end{align*} which is always smaller than the previous bounds. Under the following choice of observables: \begin{equation} X=\begin{pmatrix} 0.826411 , 0.443371+0.745704 i \\ 0.443371-0.745704 i, 0.459166 \end{pmatrix}, \label{X3} \end{equation} \begin{equation} Z=\begin{pmatrix} 0.832848 , 0.191194+0.608568 i \\ 0.191194-0.608568 i, 0.509301 \end{pmatrix}, \label{Z3} \end{equation} and parameters $\alpha=0.25$, $0\le\gamma\le0.5$ then, as depicted in Fig.~\ref{bound3}, the universal bound we derived is tighter than any of the previously known ones.
\begin{figure}
\caption{Plot of the different lower bounds for the qubit-qutrit state defined in the text and observables given by the matrices in Eqs.~(\ref{X3},\ref{Z3}).
The upper blue `+'line and pink solid lines represent the uncertainty $S(X|B)+ S(Z|B)$ and the universal lower bound (\ref{bound}), the lower green `-' line and red `.-' line represent the results of the bounds (\ref{berta}) pof Ref.~\cite{Berta} and (\ref{pati}) of Ref.~\cite{Pati}. }
\label{bound3}
\end{figure}
For the following state of one qubit times a four-dimensional quantum system
$\varrho=\alpha (|02\rangle\langle 02|+|03\rangle\langle 03|+|12\rangle\langle 12|+|13\rangle\langle 13|)+\beta (|\phi^{+}\rangle\langle\phi^{+}|+|\phi^{-}\rangle\langle\phi^{-}|+
|\psi^{+}\rangle\langle\psi^{+}|)+\gamma|\psi^{-}\rangle\langle\psi^{-}|$, with observables \begin{align*}
|0\rangle=(1,0,0)^{T},|1\rangle=(0,1,0)^{T},|2\rangle=(0,0,1)^{T}, \end{align*} \begin{align*}
X=|0\rangle\langle 1|+|1\rangle\langle 0|, Z=|0\rangle\langle 0|-|1\rangle\langle 1|, \end{align*} one has \begin{align*}
S(X|B)+ S(Z|B)=8\alpha-4\beta-4\beta \log_2(\beta)-2(\beta+\gamma)\log_2(\beta+\gamma)+2(3\beta+\gamma)\log_2(3\beta+\gamma). \end{align*}
The universal bound (\ref{bound}) and the lower bound (\ref{pati}) of Ref.~\cite{Pati} are all equal to the uncertainty $S(X|B)+ S(Z|B)$, while the bound (\ref{berta}) of Ref.~\cite{Berta} is always smaller and reads \begin{align*} 8\alpha-3\beta \log_2(\beta)-\gamma \log_2(\gamma)+(3\beta+\gamma)\log_2(3\beta+\gamma). \end{align*} Under the following choice of observables: \begin{equation} X=\begin{pmatrix} 0.370786 , 0.344509+0.694499 i \\ 0.344509-0.694499 i, 0.60978 \end{pmatrix},\label{X4} \end{equation} \begin{equation} Z=\begin{pmatrix} 0.303997 , 0.332044+0.448198 i \\ 0.332044-0.448198 i, 0.342387 \end{pmatrix}, \label{Z4} \end{equation} and parameters $\alpha=0.1$, $0\le\gamma\le0.6$ then, as shown in Fig.~\ref{bound4}, the universal bound we derived is tighter than any of the previously known ones.
\begin{figure}
\caption{Plot of the different lower bounds for the qubit by four-dimensional system state defined in the main text and observables given by the matrices in Eqs.~(\ref{X4},\ref{Z4}).
The upper blue `+'line and pink solid lines represent the uncertainty $S(X|B)+ S(Z|B)$ and the universal lower bound (\ref{bound}), the lower green `-' line and red `.-' line represent the results of the bounds (\ref{berta}) pof Ref.~\cite{Berta} and (\ref{pati}) of Ref.~\cite{Pati}. }
\label{bound4}
\end{figure}
\section{Uncertainty relations under decoherence}
In the real world, as already remarked, quantum states are unavoidably disturbed by decoherence induced by the environment. The extent to which the environment affects quantum entanglement or quantum and classical correlations beyond entanglement is a central problem, and much work has been done along these lines \cite{T. Yu,maziero}, in both Markovian \cite{Francesco} and non-Markovian \cite{Wang} regimes. It is thus natural to ask what impact such environmental decoherence has on the quantities entering entropic uncertainty relations in the presence of a quantum memory. Recently, Z. Y. Xu et al.\ \cite{Z. Y.} considered the behaviour of the uncertainty relation under the action of local unital and non-unital noisy channels. Thus, they found out that while unital noise increases the amount of uncertainty, the amplitude- damping nonunital noises may reduce the amount of uncertainty, and bring it closer to its lower bound in the long-time limit. These results shed light on the different competitive mechanisms governing quantum correlations on the one hand and the minimal missing information after local measurements on the other.
In this section, we focus on two quantum bits, with one of the two qubits acting as a memory, and examine the behaviour of different entropic uncertainty relations -- including the newly derived one of Theorem 1 -- with assisting quantum and classical correlations when the two qubits interact with independent environments, in both Markovian and non-Markovian regimes. The most common noise channels (amplitude and phase damping) are analysed. Here, we shall focus on three scenarios in succession: first, we discuss the influence of system-reservoir dynamics of quantum and classical correlations on the entropic uncertainty relation, and compare the differences among three entropic uncertainty relations in Eq. (\ref{berta}), (\ref{pati}) and (\ref{bound}); second, we explore non-Markovian dynamics influence on the entropic uncertainty relation; and third, we discuss two special examples.
\subsection{Uncertainty relations under unital and non-unital local noisy channels}
In order to investigate the behaviour of the uncertainty relation under the influence of independent local noisy channels, in what follows we will consider a system $S$ comprised of qubits $A$ and $B$, each of them interacting independently with its own environment $E_A$ and $E_B$, respectively. The dynamics of two qubits interacting independently with individual environments are described by the solutions of the appropriate Born-Markov Lindblad equations \cite{H. P.}, which can be described conveniently in the Kraus operator formalism \cite{M. A. Nielsen}. Given an initial state for two qubits, its evolution can be written compactly as \begin{equation} \varrho_{AB}(t)=\sum\limits_{uv}M_{uv}\varrho_{AB}(0)M_{uv}^{\dag} , \end{equation} where the Kraus operators $M_{u,v}=M_u\otimes M_v$ \cite{M. A. Nielsen} satisfy the completeness relation $\sum\limits_{u,v}M_{u,v}^{\dag} M_{u,v}=\mathbbm{1}$ at all times. The operators $M_u$ and $M_v$ describe the one-qubit quantum channels.
In the following we shall restrict to two-qubit states with maximally mixed local states that can be written in the form \begin{equation} \varrho_{AB} = \frac14\left(\mathbbm{1}+\sum\limits_{i=1}^3(c_{i}\sigma_i^A\otimes\sigma_i^{B}) \right) , \end{equation} where $\sigma_i^R$ is the standard Pauli matrix in direction $j$ acting on the space of subsystem $R$ for $R=A,B$ and $c_{i}$ is a triple of real coefficients satisfying $0\le c_i \le 1$.
Including environments, the whole initial state will be taken as $\varrho_{AB}\otimes|00\rangle_{E_A E_B}$,
where $|00\rangle_{E_A E_B}$ is the nominal vacuum state of the environments $E_A$ and $E_B$ in which the qubits $A$ and $B$, respectively, are immersed. We present below what happens to the entropic uncertainty relation for some qubit channels of broad interest ({\em i.e.}, amplitude damping and phase damping).
\subsubsection{Amplitude damping channel}
The amplitude-damping channel, which is a classical noise process representing the dissipative interaction between system $S$ and the environment $E$, can be modelled by treating $E$ as a large collection of independent harmonic oscillators interacting weakly with $S$ \cite{H. P.}. The effect of a dissipative channel over one qubit is depicted by the following map \begin{equation} \begin{split} \vert 0 \rangle_{S}\vert 0 \rangle_{E}\mapsto \vert 0 \rangle_{S}\vert 0 \rangle_{E},\\ \vert 1 \rangle_{S}\vert 0 \rangle_{E}\mapsto \sqrt{1-p}\vert 1 \rangle_{S}\vert 0 \rangle_{E}+\sqrt{p}\vert 0 \rangle_{S}\vert 1 \rangle_{E}, \end{split} \end{equation}
where $|0\rangle_S$ is the ground and $|1\rangle_S$ the excited state of the qubit.
The states $|0\rangle_E$ and $|1\rangle_E$ describe the states of the environment with no excitation and one excitation distributed over all its modes ({\em i.e.}, in the normal mode coupled to the qubit). The quantity $p\in[0,1]$ represents a decay probability, which will be a decreasing exponential function of time under Markov approximation. The corresponding Kraus operators describing the amplitude-damping channel acting on the system are given by \cite{M. A. Nielsen} \begin{equation} \begin{split} M_{0}=\left( \begin{array}{cc} 1 & 0 \\ 0 & \sqrt{1-p} \\% \end{array} \right)\otimes \left( \begin{array}{cc} 1 & 0 \\ 0 & \sqrt{1-p} \\% \end{array} \right),\\ M_{1}=\left( \begin{array}{cc} 0 & \sqrt{p} \\ 0 & 0 \\% \end{array} \right)\otimes \left( \begin{array}{cc} 0 & \sqrt{p} \\ 0 & 0 \\% \end{array} \right), \end{split} \end{equation} The total system evolves under the action of the operators in Eq.~(7), obtained by tracing out the degrees of freedom of the reservoir,
in the computational basis ${|00\rangle_{AB} , |01\rangle_{AB} , |10\rangle_{AB} , |11\rangle_{AB}}$ for qubits $A$ and $B$. The density operator after the action of the channel is given by \cite{J. Maziero}
\begin{equation} \hspace*{-2cm} \varrho_{AB}=\frac{1}{4}\left( \begin{array}{cccc} (1+p)^2+(1-p)^2c_3 & 0&0&(1-p)(c_1-c_2) \\ 0 &((1-c_3)+(1+c_3)p)(1-p)&(1-p)(c_1+c_2)&0 \\ 0 &(1-p)(c_1+c_2)&((1-c_3)+(1+c_3)p)(1-p)&0 \\ (1-p)(c_1-c_2) &0 &0&(1-p)^2(1+c_3) \\% \end{array} \right) . \end{equation}
Due to the X structure of the density matrices in Eq.(8), there is a simple closed expression for the concurrence $Con$ present in all bipartitions \cite{S. Luo} \begin{equation} Con(p)=2\max\{0,\lambda_1(p),\lambda_2(p)\}, \end{equation}
with $\lambda_1 (p)=|\varrho_{14} |-\sqrt{\varrho_{22} \varrho_{33} }$ and $\lambda_2 (p)=|\varrho_23 |-\sqrt(\varrho_11 \varrho_44 )$. We can also derive analytical expressions for mutual information and classical correlation: \begin{eqnarray} \fl I[\varrho_{AB}(p)]=-(1-p)\log_2(1-p)-(1+p)\log_2(1+p)\nonumber\\ +\frac{1}{4}(1-p)(1+c_1+c_2-c_3+p+c_3p) \log_2[(1-p)(1+c_1+c_2-c_3+p+c_3p)]\nonumber\\ +\frac{1}{4}(1-p)(1-c_1-c_2-c_3+p+c_3p) \log_2[(1-p)(1-c_1-c_2-c_3+p+c_3p)]\nonumber\\ +\frac{1}{4}(1+p^2+c_3(1-p)^2-\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})\nonumber\\ \log_2[1+p^2+c_3(1-p)^2-\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}]\nonumber\\ +\frac{1}{4}(1+p^2+c_3(1-p)^2+\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})\nonumber\\ \log_2[1+p^2+c_3(1-p)^2+\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}],\nonumber \end{eqnarray}
\begin{eqnarray} \fl C[\varrho_{AB}(p)]=\frac{1}{4}(1+p^2+c_3(1-p)^2-\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}) \log_2[1+p^2+c_3(1-p)^2 \nonumber\\ -\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}]+\frac{1}{4}(1+p^2+c_3(1-p)^2\nonumber\\ +\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}) \log_2[1+p^2+c_3(1-p)^2 +\sqrt{(c_1-c_2)^2(1-p)^2+4p^2}] \nonumber\\- \frac{1}{4}(1+p^2+c_3(1-p)^2+(c_1-c_2)(1-p)) \log_2[1+p^2+c_3(1-p)^2+(c_1-c_2)(1-p)]\nonumber\\- \frac{1}{4}(1+p^2+c_3(1-p)^2-(c_1-c_2)(1-p)) \log_2[1+p^2+c_3(1-p)^2-(c_1-c_2)(1-p)] \nonumber\\ +\frac{1+c}{2}\log_2(1+c)+\frac{1-c}{2}\log_2(1-c),\nonumber \end{eqnarray}
where $c=\max\{|c_1(1-p)|,|c_2(1-p)|,c_3(1-p)^2+p^2\}$.
The quantum discord is then given by [7] \begin{equation} D[\varrho_{AB}(p)]=I[\varrho_{AB}(p)]-C[\varrho_{AB}(p)] . \end{equation}
If one chooses two of the Pauli observables $R=\sigma_i$ and $S=\sigma_j$ ($i, j=1, 2, 3$) as measurements, the left-hand side of Eq.~(\ref{berta}) can be written as \begin{eqnarray} \fl U=2+(1-p)\log_2(1-p)+(1+p)\log_2(1+p) \nonumber\\ -\frac{1}{2}[(1-\sqrt{c_1^2(1-p)^2+p^2})\log_2(1-\sqrt{c_1^2(1-p)^2+p^2})\nonumber\\+ (1+\sqrt{c_1^2(1-p)^2+p^2}) \log_2(1+\sqrt{c_1^2(1-p)^2+p^2})\nonumber\\+ (1-\sqrt{c_2^2(1-p)^2+p^2}) \log_2(1-\sqrt{c_2^2(1-p)^2+p^2})\nonumber\\+ (1+\sqrt{c_2^2(1-p)^2+p^2}) \log_2(1+\sqrt{c_2^2(1-p)^2+p^2}] . \nonumber \end{eqnarray}
On the other hand, the complementarity c of the observables $\sigma_i$ and $\sigma_j$ is always equal to $1/2$, so that the right-hand sides of Eq. (\ref{berta}), Eq. (\ref{pati}) and Eq. (\ref{bound}), which we shall denote by $U_{b1}$, $U_{b2}$ and $U_{b3}$ take the form, respectively, \begin{equation} U_{b1}=1+S(\varrho_{AB})-S(\varrho_{B}) , \end{equation} where $S(\varrho_{AB})=-\frac{1}{4}(1-p)(1+c_1+c_2-c_3+p+c_3p)\log_2[(1-p)(1+c_1+c_2-c_3+p+c_3p)] -\frac{1}{4}(1-p)(1-c_1-c_2-c_3+p+c_3p)\log_2[(1-p)(1-c_1-c_2-c_3+p+c_3p)] -\frac{1}{4}(1+p^2+c_3(1-p)^2-\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})\log_2[(1+p^2+c_3(1-p)^2-\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})] -\frac{1}{4}(1+p^2+c_3(1-p)^2+\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})\log_2[(1+p^2+c_3(1-p)^2+\sqrt{(c_1-c_2)^2(1-p)^2+4p^2})]$ , $S(\varrho_{B})=1-\frac{1-p}{2}\log_2(1-p)-\frac{1+p}{2}\log_2(1+p)$, \begin{equation} U_{b2}=1+S(\varrho_{AB})-S(\varrho_{B})+\max\{0,D[\varrho_{AB}(p)]-C[\varrho_{AB}(p)]\} , \end{equation} \begin{equation} U_{b3}=2S(\varrho_{AB})-2S(\varrho_{B})+2 D[\varrho_{AB}(p)] . \end{equation}
\begin{figure}
\caption{(a) $U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_2$ with initial state $c_1=c_2=c_3=-0.8$ under local amplitude-damping channel with damping probability $p$.
$U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color,
$U_{b3}$-green dot dashed color. (b) Concurrence ($Con$)-yellow solid color, discord (D)- magenta large dashing color,
classical correlation ($C$)-light blue dot color, and mutual information (M)-pink dot dashed color.}
\label{fig:Fig1}
\end{figure}
Let us now choose the initial state $c_1=c_2=c_3=-0.8$. The specifics of the dynamics of the uncertainties clearly depend on the initial state, but these values represent a typical situation. Also, let us denote by $U$ the left hand side of Eqs.~(\ref{berta}), (\ref{pati}) and (\ref{bound}) when two Pauli observables $\sigma_1$ and $\sigma_2$ are chosen. As shown in fig. 1, the quantity $U$ will increase over time due to the gradual decay of correlations between system qubit $A$ and memory qubit $B$, while $U_{b1}$, $U_{b2}$ and $U_{b3}$, corresponding to the lower bounds of the uncertainty in Eq. (\ref{berta}), (\ref{pati}) and (\ref{bound}) respectively, will increase at first, and then decrease to asymptotic values. The dynamics of discord, entanglement and classical correlations present in qubits A and B are shown in the inset of Fig.\ 1. In this case, the non-unital channel induces disentanglement in asymptotic time; the quantum discord first decreases then increases for a short interval, and finally decreases to disappear asymptotically (if one assumes $p$ to fall exponentially in time). This dynamics induces substantial differences among the behaviours of $U_{b1}$, $U_{b2}$ and $U_{b3}$: at very short times, $U_{b3}$ better approximates the evolution of the Shannon entropies, thanks to the permanence of classical correlations at such times, while $U_{b3}$ becomes tighter at intermediate times.
\subsubsection{Phase damping channel}
The phase-damping channel is a unital channel (where a maximally mixed input state is left unchanged) leads to a loss of quantum coherence without loss of energy. The map of this channel on a one-qubit system is given by \begin{equation} \begin{split} \vert 0 \rangle_{S}\vert 0 \rangle_{E}\mapsto \vert 0 \rangle_{S}\vert 0 \rangle_{E},\\ \vert 1 \rangle_{S}\vert 0 \rangle_{E}\mapsto \sqrt{1-p}\vert 1 \rangle_{S}\vert 0 \rangle_{E}+\sqrt{p}\vert 1 \rangle_{S}\vert 1 \rangle_{E}, \end{split} . \end{equation} The corresponding Kraus operators describing the phase-damping channel for the system of qubits A and B can be written as \begin{equation} \begin{split} M_{AB}^0=\left( \begin{array}{cc} 1 & 0 \\ 0 & \sqrt{1-p} \\% \end{array} \right)\otimes \left( \begin{array}{cc} 1 & 0 \\ 0 & \sqrt{1-p} \\% \end{array} \right), \\ M_{AB}^1=\left( \begin{array}{cc} 0 & 0 \\ 0 & \sqrt{p} \\% \end{array} \right)\otimes \left( \begin{array}{cc} 0 & 0 \\ 0 & \sqrt{p} \\% \end{array} \right), \end{split} . \end{equation} For the initial state (5), the evolved density operator of the system $AB$, obtained by tracing out the degrees of freedom of the reservoirs, is given by \begin{equation} \varrho_{AB}=\frac{1}{4}\left( \begin{array}{cccc} \frac{1+c_3}{4} & 0&0&\frac{(1-p)c^{-}}{4} \\ 0 &\frac{1-c_3}{4}&\frac{(1-p)c^{+}}{4}&0 \\ 0 &\frac{(1-p)c^{+}}{4}&\frac{1-c_3}{4}&0 \\ \frac{(1-p)c^{-}}{4} &0 &0&\frac{1+c_3}{4} \\% \end{array} \right) , \end{equation} where $c^{\pm}=c_1\pm c_2$. The mutual information and the classical correlation present in qubits $A$ and $B$ can be computed analytically and are given by \begin{equation} \begin{split} I[\varrho_{AB}(p)]=\frac{1}{4}(1+c_1+c_2-c_3-c_1p-c_2p) \log_2(1+c_1+c_2-c_3-c_1p-c_2p)\\ +\frac{1}{4}(1-c_1+c_2+c_3+c_1p-c_2p) \log_2(1-c_1+c_2+c_3+c_1p-c_2p)\\ +\frac{1}{4}(1+c_1-c_2+c_3-c_1p+c_2p) \log_2(1+c_1-c_2+c_3-c_1p+c_2p)\\ +\frac{1}{4}(1-c_1-c_2-c_3+c_1p+c_2p) \log_2(1-c_1-c_2-c_3+c_1p-c_2p),\\ \end{split} \end{equation} \begin{equation} C[\varrho_{AB}(p)]=\frac{1-c}{2}\log_2(1-c)+\frac{1+c}{2}\log_2(1+c) , \end{equation}
where $c=\max\{|c_1(1-p)|,|c_2(1-p)|,|c_3|\}$ The concurrence of qubits $A$ and $B$ is instead given by \begin{equation} \begin{split} Con(p)=\frac{1}{2} \max\{1+c_3-(c_1-c_2)(1-p),1+c_3\\+(c_1-c_2)(1-p), 1-c_3-(c_1+c_2)(1-p),\\1-c_3+(c_1+c_2)(1-p)\}-1 . \end{split} \end{equation} Thus we can get the following \begin{eqnarray} U&=&2-\frac{1}{2}(1+c_1(1-p))\log_2(1+c_1(1-p))\nonumber\\&&-\frac{1}{2}(1-c_1(1-p))\log_2(1-c_1(1-p)) \nonumber \\ &&-\frac{1}{2}(1+c_2(1-p))\log_2(1+c_2(1-p))\nonumber\\&&-\frac{1}{2}(1-c_2(1-p))\log_2(1-c_2(1-p)), \\ U_{b1}&=&S(\varrho_{AB}), \\ U_{b2}&=&S(\varrho_{AB})+\max\{0,DC\},\\ U_{b3}&=&2S(\varrho_{AB})-2+2(I[\varrho_{AB}(p)]-C[\varrho_{AB}(p)]),\\ \end{eqnarray} where $DC=\frac{1}{4}(1+c_1+c_2-c_3-c_1p-c_2p)\log_2(1+c_1+c_2-c_3-c_1p-c_2p) +\frac{1}{4}(1-c_1+c_2+c_3+c_1p-c_2p)\log_2(1-c_1+c_2+c_3+c_1p-c_2p) +\frac{1}{4}(1+c_1-c_2+c_3-c_1p+c_2p)\log_2(1+c_1-c_2+c_3-c_1p+c_2p) +\frac{1}{4}(1-c_1-c_2-c_3+c_1p+c_2p)\log_2(1-c_1-c_2-c_3+c_1p-c_2p)-(1-c)\log_2(1-c)-(1+c)\log_2(1+c)$.
\begin{figure}
\caption{(a) $U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_3$ with initial state $c_1=c_2=c_3=-0.8$, under the independently local phasedamping channel with p as the damping rate. $U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color, $U_{b3}$-green dot dashed color. (b) Concurrence (Con)-yellow solid color, discord (D)- magenta large dashing color , classical correlation (C)-light blue dot color, and mutual information (M)-pink dot dashed color.
}
\label{fig:Fig2}
\end{figure}
As depicted in Fig.\ 2, choosing $c_1=c_2=c_3=-0.8$, while $U$, $U_{b1}$ and $U_{b2}$ increase all the time while $U_{b3}$ is constant as this unital channel induces the loss of entanglement and discord gradually in asymptotic time, whereas classical correlations remain unchanged. $U_{b1}$ and $U_{b2}$ are almost the same curve during this process, except for a short initial time interval where $U_{b3}$ and $U_{b2}$ coincide. This situation should be contrasted with the analysis carried out in Ref. [11], where the behaviours of memory-assisted entropic uncertainty relations with under noise acting on the system qubit are shown. Obvious differences on the behaviours of the uncertainties are due to the different dynamics of quantum and classical correlations of the joint qubits with respect to single qubit decoherence. Furthermore, here we pay more attention to the action of discord-assited memories, while Ref. [11] mainly focuses on the effect of entanglement assistance.
\subsection{Uncertainty relations in non-Markovian environments}
In this section we study the effect of dissipation on the uncertainty relation by exactly solving a model consisting of two independent qubits subject to two zero-temperature non-Markovian reservoirs. We shall see how the behaviour of the uncertainty relation due to correlation dynamics is affected by the environment being, respectively, `quantum' or `classical', {\em i.e.} with or without back-action on the system.
\subsubsection{Reservoirs with system-environment back-action}
The non-Markovian effects on the dynamics of entanglement and discord presented in a two qubits system have been studied recently [19,20]. Assuming a two qubits system A and B whose dynamics is described by the damped Jaynes-Cummings model, the qubits are coupled to a single cavity mode, which in turn is coupled to a non-Markovian environment. The environments are described by a bath of harmonic oscillators, and the spectral density is written as \cite{H. P.} \begin{equation} J(\omega)=\frac{1}{2\pi}\frac{\gamma_0\tau^2}{(\omega_0-\omega)^2+\tau^2} , \end{equation} where $\tau$ is associated with the reservoir correlation time $t_B$ by the relation $t_B\approx\frac{1}{\tau}$, and $\gamma_0$ is connected to the time scale $t_{R}$ over which the two-qubit system changes, here $t_R\approx\frac{1}{\gamma_0}$, and the strong coupling condition $t_R<2t_B$ is assumed. The two-qubit Hamiltonian under independent amplitude-damping channels can be written as \cite{F. F. Fanchini} \begin{equation} H=\omega_{0}^{j}\sigma_{\dag}^{j}\sigma_{-}^{j}+\sum\limits_{k}\omega_{k}^{j}a_{k}^{(j)\dag}a_{k}^{j}+(\sigma_{\dag}^{j}B^{j}+\sigma_{-}^{j}B^{j\dag}), \end{equation} where $B^((j))=\sum\limits_k g_k^((j))a_k^((j))$ with $g_k^((j))$ being the coupling constant, $\omega_0^((j))$ is the transition frequency of the $j^{\rm th}$ qubit, and $\sigma_{\pm}^((j))$ are the system raising and lowering operators of the $j$th qubit. Here the index $k$ labels the reservoir field modes with frequencies$\omega_k^((j))$, and $a_k^((j)^\dag) (a_k^((j)))$ is their creation (annihilation) operator. Here, and in the following, the Einstein convention sum is used. The initial state of the two qubits is the Bell-like state \begin{equation}
|\psi\rangle=\alpha|00\rangle+\sqrt{1-\alpha^2}|11\rangle . \end{equation}
According to the dynamics of the initial state's density matrix elements given in Ref. \cite{B. Bellomo}, the mutual information, classical correlation and concurrence present in qubits $A$ and $B$ are given by
\begin{eqnarray} \fl I[\varrho_{AB}(t)] = -2a^2p_t\log_2(a^2p_t)-2(1-a^2p_t)\log_2(1-a^2p_t)+2a^2p_t(1-p_t)\log_2(a^2p_t(1-p_t)) \nonumber\\ +[-a^2p_t(1-p_t)\nonumber\\ +\frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t})] \log_2(-a^2p_t(1-p_t)+ \frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t})) \nonumber\\ +[-a^2p_t(1-p_t)\nonumber\\ +\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t})] \log_2(-a^2p_t(1-p_t)+\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t})),\nonumber\\ \fl D[\varrho_{AB}(t)] = \min{D_1,D_2},\nonumber\\ \fl Con(p)= 2\max\{a^2(1-p_t)p_t,\sqrt{2a^2p_t^2+a^4p_t^2(p_t^2-2p_t-1)-2\sqrt{a^4(1-a^2)p_t^4(1-p_ta^2(2-p_t))}}\nonumber\\
\sqrt{2a^2p_t^2+a^4p_t^2(p_t^2-2p_t-1)+2\sqrt{a^4(1-a^2)p_t^4(1-p_ta^2(2-p_t))}}\}\nonumber\\
-2a^2(1-p_t)p_t-\sqrt{2a^2p_t^2+a^4p_t^2(p_t^2-2p_t-1)-2\sqrt{a^4(1-a^2)p_t^4(1-p_ta^2(2-p_t))}}\nonumber\\
-\sqrt{2a^2p_t^2+a^4p_t^2(p_t^2-2p_t-1)+2\sqrt{a^4(1-a^2)p_t^4(1-p_ta^2(2-p_t))}},\nonumber \end{eqnarray}
where \begin{eqnarray} \fl D_1 = a^2p_t(1-p_t)\log_2(a^2p_t(1-p_t)) \nonumber\\ +[-a^2p_t(1-p_t)+\frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t})]\log_2(-a^2p_t(1-p_t) \nonumber\\ +\frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t}))\nonumber\\ +[-a^2p_t(1-p_t)+\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t})]\log_2(-a^2p_t(1-p_t)\nonumber\\ +\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t})) -a^2p_t\log_2(a^2p_t)-(1-a^2p_t)\log_2(1-a^2p_t)\nonumber\\ -a^2p_t\log_2p_t-a^2p_t(1-p_t)\log_2(1-p_t)- a^2(1-p_t)p_t\log_2\frac{a^2p_t(1-p_t)}{1-a^2p_t}-\nonumber\\ (1-2a^2p_t+a^2p_t^2)\log_2\frac{1-2a^2p_t+a^2p_t^2}{1-a^2p_t} , \nonumber \end{eqnarray} \begin{eqnarray} \fl D_2=a^2p_t(1-p_t)\log_2(a^2p_t(1-p_t))\nonumber\\ + [-a^2p_t(1-p_t)+\frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t})]\log_2(-a^2p_t(1-p_t)\nonumber\\ +\frac{1}{2}(1-\sqrt{1-4a^2(1-p_t)p_t}))\nonumber\\ +[-a^2p_t(1-p_t)+\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t})]\log_2(-a^2p_t(1-p_t)\nonumber\\ +\frac{1}{2}(1+\sqrt{1-4a^2(1-p_t)p_t}))\nonumber\\ -\frac{1+\sqrt{1-4a^2p_t(1-p_t)}}{2}\log_2\frac{1+\sqrt{1-4a^2p_t(1-p_t)}}{2}\nonumber\\ -\frac{1-\sqrt{1-4a^2p_t(1-p_t)}}{2}\log_2\frac{1-\sqrt{1-4a^2p_t(1-p_t)}}{2}. \nonumber \end{eqnarray}
\begin{figure}\label{fig:Fig3}
\end{figure}
In Fig.\ 3 we plot the uncertainty $U$, as well as the lower bounds $U_{b1}$, $U_{b2}$ and $U_{b3}$ as functions of the rescaled time $\gamma_0 t$ in the strong coupling regime, with $1/t=0.01\gamma_0$ and $\alpha=\frac{1}{\sqrt{10}}$. $U$, $U_{b1}$, $U_{b2}$ and $U_{b3}$ all oscillate in the long-time limit due to the entanglement and discord between qubits A and B periodically vanishing and reviving. It is apparent that the behaviours of $U_{b1}$ and $U_{b2}$ are the same at short times and tighten the lower bound of the uncertainty with respect of $U_{b3}$. The amplitudes of the oscillations of $U_{b1}$ (or $U_{b2}$) and $U_{b3}$ reduce slowly as the peaks of entanglement and discord dwindle after each revival.
\subsubsection{Reservoirs without system-environment back-action}
We now want to explore how the entropic uncertainty relations are affected by revivals of correlations, including quantum discord and entanglement, occurring in `classical' non-Markovian environments with no back-action. Suppose the pair of non-interacting qubits is in a generic initial Bell-diagonal state: \begin{equation}
\varrho(0)=\sum\limits_{kn}c_{k}^{n}(0)|k^n\rangle\langle k^n|,(k=1,2;n=\pm) , \end{equation}
where $|1^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}$,$|2^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}}$. Each of the two qubits is coupled to a random external field acting as a local environment, so the global dynamical map $\Omega$ applied on the initial state $\varrho(0)$ is of the random external field type \cite{R. Alicki} and yields the state \begin{equation} \varrho(t)=\frac{1}{4}\sum\limits_{j,k=1}^2U_{j}^{A}(t)U_{k}^{B}(t)\varrho(0)U_{j}^{A\dag}(t)U_{k}^{B\dag}(t) \end{equation} where $U_j^R (t)=e^{-iH_j/\hbar} (R=A,B;j=1,2)$ is the time evolution operator with $H_j=i\hbar g(\sigma_+ e^{-i\phi_j }-\sigma_- e^{i\phi_j })$, and $H_j$ is expressed in the rotating frame at the qubit-field resonant frequency
$\omega$. In the basis ${|1\rangle, |0\rangle}$, the time evolution operators $U_j^R (t)$ have the following matrix form \cite{R. Lo Franco} \begin{equation} U_{j}^{R}(t)=\left( \begin{array}{cc} \cos(gt) & e^{-i\phi_j}\sin(gt) \\ e^{-i\phi_j}\sin(gt) &\cos(gt) \\% \end{array} \right), \end{equation} where $j=1, 2, \phi_j$ is the phase of the field at the location of each qubit and is either 0 or$\pi$ with probability $p = \frac{1}{2}$. The interaction between each qubit and its local field mode is assumed to be strong enough so that, for sufficiently long times, the dissipation effects of the vacuum radiation modes on the qubit dynamics can be neglected. From the matrix of Eq.\ (30) we know that the dynamics is cyclic, and the global map $\Omega$ acts within the class of Bell-diagonal states (or states with maximally mixed marginals \cite{S. Luo}). These properties allow us to analytically calculate the correlation quantifiers under the map of Eq.\ (29) for different initial states $\varrho(0)$.
As shown in Fig.\ 4a (right hand side), for an initial Bell-diagonal state with $c_1^+ (0)=0.9$, $c_1^- (0)=0.1$, and $c_2^+ (0)=c_2^- (0)=0$, choosing $p_1=1-p_2=0.025$, under the map of Eq. (29), both entanglement and classical correlations will collapse and revive during the dynamics, while discord keeps approximately constant. In Fig.\ 4a (left hand side) we observe that the lower bounds $U_{b1}$ and $U_{b2}$ coincide all the time, and that $U$, $U_{b1}$, $U_{b2}$ and $U_{b3}$ present periodic oscillations: interestingly, $U$ oscillates is out of phase, periodically saturating the entanglement-assisted uncertainty relation . The amplitudes of oscillating revival do not decay. Also, $U_{b1}$ (or the equivalent $U_{b2}$) provides one with a tighter lower bound for the uncertainty than $U_{b3}$ at all times.
\begin{figure}
\caption{(a) The upper-left figure:$U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_3$ with initial state $c_1^+ (0)=0.9$, $c_1^- (0)=0.1$, $c_2^+ (0)=c_2^- (0)=0$ under the independently local non-Markovianc lassical environments with$p_1=0.025$. $U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color, $U_{b3}$-green dot dashed color. The upper-right figure :concurrence (Con)-yellow solid color, discord (D)- magenta large dashing color , classical correlation (C)-light blue dot color, and mutual information (M)-pink dot dashed color. (b)The lower-left figure:$U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_3$ with initial state as in Fig.4a under the independently local non-Markovian classical environments with $p_1=0.08$. $U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color, $U_{b3}$-green dot dashed color. The lower-right figure:concurrence (Con)-yellow solid color, discord (D)- magenta large dashing color , classical correlation (C)-light blue dot color, and mutual information (M)-pink dot dashed color.
}
\label{fig:Fig4a}
\end{figure}
The choice $p_1=1-p_2=0.08$ for the same initial state, plotted in Fig. 4b, shows a similar behaviour, with the exceptions that the oscillations of $U$ are now in phase with those of the lower bounds (but the periodic saturation of the bounds still occurs), and that $U_{b3}$ provides, at certain points in time, a bound as tight as the other two assisted uncertainty relations. When entanglement and discord increase to their maximum, $U$, $U_{b1}$ (or $U_{b2}$) and $U_{b3}$ will decrease to their minimum, and viceversa. It is worth mentioning that the periodical oscillation of the uncertainties and lower bounds is only a consequence of the non-Markovian character of the independent qubit-reservoir dynamics, whether back-action on the system is present or not. This fact might hence be used to quantify the non-Markovianity of single-qubit dynamics.
\subsection{Special cases}
Let us now discuss two special examples of open system dynamics characterised, respectively, by a particular interplay between quantum discord and classical correlations, and by the presence of discord without entanglement.
\subsubsection{Sudden transition between classical and quantum decoherence}
A sharp transition between `classical' and `quantum' loss of correlations in a composite system characterises certain open quantum systems, when properly parametrized. This kind of behaviour has first been noticed in the case of two qubits locally subject to non-dissipative channels \cite{L. Mazzola}, and then observed in an all-optical experimental setup \cite{Jin-Shi}. An environment-induced sudden change has also been observed in a room temperature nuclear magnetic resonance setup \cite{R. Auccaise}. Moreover sudden change and immunity against some sources of noise were still found when an environment is modelled as classical instead of quantum \cite{R. Lo Franco}, indicating that such a peculiar behavior is in fact quite general.
Here, we adopt the model of Ref. \cite{L. Mazzola}, supposing the initial state is in the class of states of Eq. (5) and consider two independent phase damping channel, so that the time evolution of the whole system is given by \cite{maziero} \begin{equation}
\varrho_{AB}(t)=\sum\limits_{kn}\lambda_{k}^{n}(t)|k^n\rangle\langle k^n|,(k=1,2;n=\pm) , \end{equation} where $\lambda_1^{\pm}(t)=\frac{1}{4}(1\pm c_1(t)\mp c_2(t)+c_3(t))$,$\lambda_2^{\pm}(t)=\frac{1}{4}(1\pm c_1(t)\pm c_2(t)-c_3(t))$, $c_1(t)=c_1 e^{-2\gamma t}$,$c_2(t)=c_2 e^{-2\gamma t}$,$c_3(t)=c_3$, for a damping rate $\gamma$. The parameters chosen for the initial state are $c_1=1, c_3=-c_2=0.6$.
\begin{figure}
\caption{(a) $U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_3$ with initial state $c_1=1, c_3=-c_2=0.6$ under the independently local phase damping channel with¦Ãphase damping rate. $U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color, $U_{b3}$-green dot dashed color. (b) Concurrence (Con)-yellow solid color, discord (D)- magenta large dashing color , classical correlation (C)-light blue dot color, and mutual information (M)-pink dot dashed color.
}
\label{fig:Fig5}
\end{figure}
As shown in Fig. 5, the uncertainty will increase in the long-time limit due to the gradually missing quantum correlations. The dynamics of correlations are shown on the right plot of Fig. 5. In this case, while mutual information and entanglement decrease gradually, classical correlations and discord display two mutually exclusive plateaux (when discord remains constant, classical correlation is decreasing, and vice versa). We can observe that this same phenomenon of sudden transition between discord and classical correlation decoherence occurs in the inset of Fig. 4b. Clearly, this sudden transition influences the behaviour of the corresponding entropic uncertainty relations: as shown in Fig. 5, $U_{b1}$ and $U_{b2}$ here coincide all the time (discord does not tighten the entanglement-assisted uncertainty relation) and $U_{b1}$ (or $U_{b2}$) is always a tighter lower bound for the uncertainty than $U_{b3}$. In particular, $U_{b3}$ is increasing at first, and then has a sudden change as its maximum value coincides with $U_{b1}$ (and $U_{b2}$), {\em i.e.}, $U_{b3}$ keeps constant when discord decays in time.
\subsubsection{Quantum correlations without entanglement}
It is well known that many operations in quantum information processing depend largely on quantum correlations represented by quantum entanglement. However, there are indications that some protocols might display a quantum advantage without the presence of entanglement \cite{animesh}. Besides, correlations quantified by quantum discord can always be ``activated'', even when no quantum entanglement is initially present \cite{marco}. We consider here our two qubits system under a one-sided phase damping channel \cite{Jin-Shi}, in the following initial state \begin{equation} \begin{split}
\varrho(0)=dR |2^{\dag}\rangle\langle 2^{\dag}|+b(1-R)dR |2^{-}\rangle\langle 2^{-}|+bR |1^{\dag}\rangle\langle 1^{\dag}|+d(1-R)dR |1^{-}\rangle\langle 1^{-}|, \end{split} \end{equation}
\begin{figure}
\caption{(a) $U$ and $U_{bi} (i=1, 2, 3)$ of the observables $\sigma_1, \sigma_3$ with initial state $b=0.7, R=0.7, d=03$ under a one-sided phase damping channel with damping rate.$U$-red large dashing color, $U_{b1}$-black solid color, $U_{b2}$-blue dot color, $U_{b3}$-green dot dashed color. (b) concurrence (Con)-yellow solid color, discord (D)- magenta large dashing color , classical correlation (C)-light blue dot color, and mutual information (M)-pink dot dashed color.
}
\label{fig:Fig6}
\end{figure}
where $b=0.7, R=0.7$ and $d=0.3$. As shown in the right plot of Fig.6, entanglement is almost zero during this process, while quantum discord is larger than classical correlations for a certain short time interval. We observe the behaviour of the uncertainty relations from Fig. 6: at first, $U$, $U_{b1}$, $U_{b2}$ and $U_{b3}$ coincide and quickly increase; then, while $U$, $U_{b2}$ and $U_{b3}$ coincide (as discord is bigger than classical correlations), $U_{b1}$ is lower than $U$ (or $U_{b2}$). Furthermore, $U_{b1}$ and $U_{b2}$ will provide a higher lower bound than $U_{b3}$ and coincide with each other when discord becomes smaller than classical correlation.
\section{Conclusions}
We have introduced a new memory-assisted, observable-independent entropic uncertainty relation where quantum discord between system and memory plays an explicit role. We have shown that this uncertainty relation can be tighter than the ones obtained previously by Berta {\em et al.}\ and Pati {\em et al.} Moreover, we have explored the behaviour of these three entropic uncertainty relations with assisting quantum correlations for a two-qubit composite system interacting with two independent environments, in both Markovian and non-Markovian regimes. The most common noise channels (amplitude damping, phase damping) were discussed. The entropic uncertainties (or their lower bounds) will increase under independent local unital Markovian noisy channels, while they may be reduced under the non-unital noise channel. The entropic uncertainties (and their lower bounds) exhibit periodically oscillation due to correlation dynamics under independently non-Markovian reservoirs, whether environment is modeled as quantum or classical. In addition, we have compared the differences among three entropic uncertainty relations in Eq.\ (\ref{berta}), (\ref{pati}) and (\ref{bound}). The lower bound $U_{b2}$ or $U_{b3}$ will tighten the bound on the uncertainty when discord is bigger than classical correlation, which is often the case in practice. The relation between quantum correlations and the uncertainties is subtle, since a certain reduction int he uncertainty may also happen in the presence of small quantum correlations without entanglement. We have also shown that, in essence due to the greater resilience of the nearly ubiquitous quantum discord \cite{ferraro}, situations arise where uncertainty relations tightened by quantum discord offer a better estimate of the actual uncertainties in play. However, the advantage offered in this sense by the quantity $U_{b3}$ of Eq.\ (\ref{bound}) seems to be limited to rather specific circumstances.
\ack This work was supported by the National Natural Science Foundation of China under Grant 61144006, by the Foundation of China Scholarship Council, by the Project Fund of Hunan Provincial Science and Technology Department under Grant 2010FJ3147, and by the Educational Committee of the Hunan Province of China through the Overseas Famous Teachers Programme.
\end{document} | arXiv |
Shelling (topology)
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitting a shelling is called shellable.
Definition
A d-dimensional simplicial complex is called pure if its maximal simplices all have dimension d. Let $\Delta $ be a finite or countably infinite simplicial complex. An ordering $C_{1},C_{2},\ldots $ of the maximal simplices of $\Delta $ is a shelling if the complex
$B_{k}:={\Big (}\bigcup _{i=1}^{k-1}C_{i}{\Big )}\cap C_{k}$
is pure and of dimension $\dim C_{k}-1$ for all $k=2,3,\ldots $. That is, the "new" simplex $C_{k}$ meets the previous simplices along some union $B_{k}$ of top-dimensional simplices of the boundary of $C_{k}$. If $B_{k}$ is the entire boundary of $C_{k}$ then $C_{k}$ is called spanning.
For $\Delta $ not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of $\Delta $ having analogous properties.
Properties
• A shellable complex is homotopy equivalent to a wedge sum of spheres, one for each spanning simplex of corresponding dimension.
• A shellable complex may admit many different shellings, but the number of spanning simplices and their dimensions do not depend on the choice of shelling. This follows from the previous property.
Examples
• Every Coxeter complex, and more generally every building (in the sense of Tits), is shellable.[1]
• The boundary complex of a (convex) polytope is shellable.[2][3] Note that here, shellability is generalized to the case of polyhedral complexes (that are not necessarily simplicial).
• There is an unshellable triangulation of the tetrahedron.[4]
Notes
1. Björner, Anders (1984). "Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings". Advances in Mathematics. 52 (3): 173–212. doi:10.1016/0001-8708(84)90021-5. ISSN 0001-8708.
2. Bruggesser, H.; Mani, P. "Shellable Decompositions of Cells and Spheres". Mathematica Scandinavica. 29: 197–205. doi:10.7146/math.scand.a-11045.
3. Ziegler, Günter M. "8.2. Shelling polytopes". Lectures on polytopes. Springer. pp. 239–246. doi:10.1007/978-1-4613-8431-1_8.
4. Rudin, Mary Ellen (1958). "An unshellable triangulation of a tetrahedron". Bulletin of the American Mathematical Society. 64 (3): 90–91. doi:10.1090/s0002-9904-1958-10168-8. ISSN 1088-9485.
References
• Kozlov, Dmitry (2008). Combinatorial Algebraic Topology. Berlin: Springer. ISBN 978-3-540-71961-8.
| Wikipedia |
The bacteria in a lab dish double in number every four hours. If 500 bacteria cells are in the dish now, in how many hours will there be exactly 32,000 bacteria?
32000 bacteria is $32000/500=64$ times the number currently in the lab dish. Since $64=2^6$, the bacteria had to double 6 times to reach this number. Since the bacteria double every four hours, it takes $4\cdot6=\boxed{24}$ hours. | Math Dataset |
What is so special about spontaneous symmetry breaking? (time reversal example)
I have serious trouble understanding the concept of spontaneous symmetry breaking (in condensed matter specifically). Let's take time reversal in magnetic systems as an example.
Ferromagnetism is said to spontaneously breaks time reversal symmetry. As I understand it, time reversal symmetry can be understood with a time reversal operator $\mathcal{T}$ that reverses the sign of all momentum and spin, so that $\mathcal{T} S_{zi} =- S_{zi} \mathcal{T}$.
We take an hamiltonian that can generate ferromagnetism like Ising $$ H_0 = -J \sum_{<i,j>} S_{zi} \ S_{zj} $$ and notice that $[\mathcal{T}, H_0] = 0$ because there are two spin operators. So no symmetry breaking here.
Apparently, a spontaneous symmetry breaking is manifested in the asymmetry of the ground state rather than that of the hamiltonian. There are two ground states for the hamiltonian one with all spins up ${\left|\left. \uparrow \right>\right.}^{\otimes n}$ and one with all spins down ${\left|\left. \downarrow \right>\right.}^{\otimes n}$. Since $\mathcal{T} {\left|\left. \uparrow \right>\right.}^{\otimes n} = {\left|\left. \downarrow \right>\right.}^{\otimes n}$, time reversing keeps you in the ground state, so no symmetry breaking here.
A point that is often made is that passing below the critical temperature breaks the time reversal symmetry because you will find the system exhibits non vanishing magnetization and thus is in a particular ground state not in a superposition of both. But this is only the case because some noise in the environment (it could also be an irregularity in the system) caused the system to choose a particular direction. We could simply add that noise into the model by saying
$$H= H_0 + \delta H$$ with $[\mathcal{T}, \delta H] \neq 0$. Then the total hamiltonian is not symmetric.
Is that the essence of the so called "spontaneous symmetry breaking"? If it is, what is so special about it?
Couldn't we just say that below the critical temperature the system is greatly susceptible (literally since susceptibilities are discontinuous) to small perturbations?
Is there a rigorous definition of what a spontaneous symmetry breaking is?
quantum-mechanics condensed-matter symmetry-breaking time-reversal-symmetry ferromagnetism
UndeadUndead
Indeed, one of the definitions of spontaneous symmetry breaking is in terms of its susceptibility:
Suppose we add a symmetry breaking perturbation $h \; \delta H$ to our Hamiltonian (as you do), if $$ \lim_{h \to 0} \lim_{N \to \infty} \langle m \rangle \neq 0 $$ then we say our system has spontaneous symmetry breaking.
(Note: $N$ is the number of spins in our system. Indeed, on a mathematical level, non-analyticities can only arise in the thermodynamic limit.)
What is special is that any arbitrarily small perturbation will do. Imagine you have a million spins. If the state is originally in a symmetric state (i.e. not symmetry broken yet), then even if I just apply an arbitrarily small magnetic field on a single spin, the whole system will choose that orientation.
You suggest that the fact one in principle needs the environment to 'make the choice' that this is not really spontaneous. It is true that in that philosophical sense of the word, the direction of magnetization is not 'spontaneous'. But what can be called spontaneous in the universe? If I perfectly balance an egg, then the direction it will eventually roll when it loses its balance is spontaneous (or not spontaneous) in exactly the same sense. And note that once the egg has rolled down (and stopped), the tiny perturbations in the air which influenced its original direction are now no longer sufficient to change its position. I.e.: after the `spontaneous' process, the system is now stable.
The same thing happens in the above magnet: once it has chosen a direction of magnetization, then changing the applied magnetic field on that single spin I mentioned before will not change the total magnetization. So in that sense it is not true that it is so susceptible! One needs to apply an extensive magnetic field (i.e. a field that acts on most of the spins) to change the direction of the magnetization.
That is what is so funny about these systems:
An arbitrarily small perturbation can create a magnetization, but it cannot change it!
On a more quantum-mechanical note, if one has a Hamiltonian whose ground state should display spontaneous symmetry breaking, then if one takes the ground state to be in a symmetric superposition (which one can always do), then this state has ridiculously long entanglement. These are called cat states (in reference to Schrodinger's cat). This is a natural consequence of the above: an interaction with a single spin has to influence all spins at once, which is only possible if every single spin is entangled with every other spin. An example is the state $|\uparrow \uparrow \uparrow \cdots \rangle + |\downarrow \downarrow \downarrow \cdots \rangle$. (Indeed: an interaction with a single spin will collapse this 'cat state' to a product state, and then it is clear that any subsequent single-spin interaction cannot flip the state to the other product state.) Indeed, the way symmetry breaking phases are classified in one spatial dimension is in terms of these entanglement properties [Schuch et al., 2010].
Ruben VerresenRuben Verresen
$\begingroup$ Great answer thanks! So, is this "one-way" sensitivity to perturbations equivalent to the the phenomenon called hy hysteresis? $\endgroup$ – Undead Apr 21 '17 at 0:08
$\begingroup$ Not really. A simple difference is for example that hysteresis is even relevant when you are applying a global (i.e. extensive) magnetic field. $\endgroup$ – Ruben Verresen Apr 21 '17 at 0:14
$\begingroup$ Another question, do you know if there's a way to prove that the definition you give is equivalent to the one given by GaragePhys below? $\endgroup$ – Undead Apr 21 '17 at 14:51
You've already mentioned the exact definition of spontaneous symmmetry breaking:
Spontaneous symmetry breaking for a system that is described by a hamiltonian $H$ with ground state $\left| g \right\rangle$ happens where there is a symmetry transformation of $H$ that doesn't leave the ground state invariant $$[T,H] = 0 \text{ but } T \left| g \right\rangle \neq 0. $$ Much like a stick that's standing on its tip can be rotated around itself but will eventually fall back to a "ground state" that doesn't have this rotational symmetry anymore.
The example you are quoting is a statistical theory, so the fluctuations that drive the system into a state of non-vanishing magnetisation below the critical temperature are already built in from the start.
Spontaneous symmetry breaking is a key ingredient in the Standard Model of particle physics, it is used to explain why the particles that mediate the weak force are massive (W and Z bosons, this is remarkable because you can't achieve that by just putting a mass term in the Lagrangian!).
But it also works the other way around, in that it explains why sometimes there are massless modes in a system (see Goldstone bosons).
GaragePhysGaragePhys
$\begingroup$ Thanks for your answer! I am not sure if it's a typo, do you mean $ T \left g \right \rangle \neq \left g \right \rangle$ instead? Because I can't see any reason why it could give 0. Take the case of an hamiltonian symmetric under parity like an harmonic oscillator. Then applying the parity operator on the ground state gives back the ground state not 0. $\endgroup$ – Undead Apr 21 '17 at 0:20
$\begingroup$ No, that's not a typo, what you mean by the symbol $T$ is the generator of the symmetry, which is an infinitesimal version of it. For example if you have rotations in 3d space acting on the vector: $\left| x \right\rangle = \begin{pmatrix} 0\\ 0\\x\end{pmatrix}$ Let's consider the transformation: $\endgroup$ – GaragePhys Apr 21 '17 at 5:36
$\begingroup$ $$\left| x \right\rangle\rightarrow R \left| x \right\rangle \text{ with } R = \begin{pmatrix} cos(\alpha) & sin(\alpha) & 0 \\ -sin(\alpha) & cos(\alpha) & 0 \\ 0& 0& 1 \end{pmatrix} \approx \alpha \underbrace{\begin{pmatrix}0 & 1 & 0 \\ -1 &0 & 0 \\ 0& 0& 1 \end{pmatrix}}_{ = T} + \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0& 1 \end{pmatrix}$$ Here $\left| x \right\rangle$ is clerly invariant under $R$, which is expressed by $T \left| x \right\rangle= 0$ Here you'd find more about it: \url{en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics}. $\endgroup$ – GaragePhys Apr 21 '17 at 5:36
$\begingroup$ I understand what you mean! Also, do you know where I can find a comprehensive explanation of what Goldstone bosons are (considering I'm from condensed matter and not familiar with field theory formalism) $\endgroup$ – Undead Apr 21 '17 at 18:06
$\begingroup$ The complete title is 'An introduction to quantum field theory', it's the standard reference for QFT. $\endgroup$ – GaragePhys Apr 21 '17 at 19:58
Not the answer you're looking for? Browse other questions tagged quantum-mechanics condensed-matter symmetry-breaking time-reversal-symmetry ferromagnetism or ask your own question.
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What is spontaneous in spontaneous symmetry breaking | CommonCrawl |
Writing in LaTeX: 5 bad habits to avoid
Chad Musick, PhD | Writing
Overall, most academic papers are prepared in Microsoft Word. But did you know that there's another option? It's called LaTeX. For authors new to LaTeX, I'll start with an overview of how it works and the benefits it offers. Then, for authors who are already using LaTeX, I'll give some tips and tricks that I hope will be helpful for preparing your next paper.
What is LaTeX? It's a template system for typesetting documents. First, you prepare your manuscript in a "structured" source file, where you specify a template to produce a typeset version of your paper.
Source + Template → Final Typeset Text
In this system, the content of your document is separated from its format. This means that the same source document can be compiled into different formats by simply specifying a different set of styles. For example, let's look at one document element—the section heading: \section{Introduction}. When you apply the journal template, the section heading will be automatically put in the correct format. For instance.
The formatting of the section headings is just one example. When you apply a LaTeX template, all the other document elements will also be put into the correct format automatically. This includes the main text (e.g., single column or double column), the figure captions, the table titles, and the citations (e.g., superscript numerals1 or bracketed numbers [1]).
Another benefit of LaTeX is the ability to handle mathematical text easily. Instead of searching for a Greek letter in Word and manually italicizing it, LaTeX lets you simply write $\delta$ for δ and $\Delta$ for Δ. You can also easily handle more complicated mathematical expressions and symbols that would be a challenge in Word or its built-in equation editor.
LaTeX is a powerful tool for preparing documents, but a full introduction to the topic is beyond the scope of this article. If you'd like to get started learning LaTeX, we offer a free mini course.
Nowadays, LaTeX is frequently used in many academic fields. Although the use of LaTeX is nearly universal among academics in some subjects (particularly mathematics and computer science), few of us have had a formal class in its use. Because little formal training on LaTeX is given, most authors look to existing texts to see how to produce the desired effects. Such texts are readily available. For example, the arXiv e-print server hosts more than 1,000,000 papers, and LaTeX source can be downloaded for a large proportion of these.
Although this type of imitative learning has allowed widespread adoption of LaTeX, it has also resulted in certain techniques being copied so many times that it is not clear to authors that they are bad habits. In the remainder of this month's article, I'll describe these bad habits and show you what to do instead.
Bad Habit 1: Using "math mode" to produce italics
This is probably the most common bad habit, and rises to the level of mistake. In LaTeX, variables in mathematical expressions are rendered in italics, so $name$ will produce name. This is subtly different than name: the space between the letters in the first reflects that it's being typeset as the product n·a·m·e.
Do This Instead: When you want italicized text, use \textit{text to be italicized}.
Bad Habit 2: Using the wrong kind of quotation mark
To allow complete control over rendering, LaTeX offers a variety of styles for quotation marks. Typically, you'll want the "inverted commas" style (sometimes called "smart quotes"). However, using the inverted commas produced by word processing software is not guaranteed to produce the correct output, and using straight quotes (so-called "dumb quotes") is not either. If you're not careful, it's very easy to end up with right quotation marks on the left side of a word, "like this".
Do This Instead: Use the ` character for a left inverted comma and the ' character for a right inverted comma. To produce double quotes, simply repeat the characters, "like this".
Bad Habit 3: Writing very long mathematical expressions without any spaces
In contrast with the previous types of bad habits, it will often be clear when this one is causing a problem, because the text will extend past the end of the line. By default, LaTeX does not insert line breaks into mathematical expressions. As a consequence, a very long expression will run past the end of the line.
Do This Instead: Switch out of math mode for non-mathematical punctuation (commas are a particularly easy place to overlook doing this). For example, writing "We have 12 variables ($a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $i$, $j$, $k$, $m$)." will result in correct line breaks and correct typesetting of the commas. Writing, instead, "We have 12 variables $(a,b,c,d,e,f,g,h,i,j,k, m).$" will typeset the commas, parentheses, and period as math symbols and prevent the line from breaking when it needs to.
Bad Habit 4: Writing out your citations in full
A disciplined writer can achieve the correct output even when writing out citations in full, but this requires much more work than using one of the citation and reference packages available. When maintaining a reference list by hand, it is necessary to ensure that the references are in the correct order (e.g., order of citation or alphabetically by first author), that all references are cited, and that all citations refer to a source listed as a reference.
Do This Instead: Use a citation package and a reference manager. I recommend the natbib citation package. When using this, you have access to the \citet and \citep commands so that you can distinguish between "ThinkSCIENCE (2015)" and "(ThinkSCIENCE, 2015)" without needing to write names out by hand. There are several popular reference managers (see a comparison) that can output to BiBTeX (the format used by the natbib citation package).
Bad Habit 5: Writing commands to describe the look, rather than the meaning
LaTeX doesn't always have the look that we want. For example, the default is to display vector symbols as having an arrow on top, but some journals specify that vectors should be written as a bold upright letter instead.
Do This Instead: Use the \renewcommand and \newcommand commands to achieve the look you want for a particular type of item. For example, \renewcommand{\vec}[1]{\mathbf{#1}} will cause vectors (written with the \vec command) to appear as bold upright letters, rather than with arrows across the top.* If you decide that you'd rather have arrows, or if the journal requires arrows instead, you can update all of your vectors by changing only one line. As an added benefit, it will be easier to see what you intend if your commands are mostly to express the meaning, rather than the style, of your words.
Special Use for This Tip: You can use commands to automatically switch between "Fig." and "Figure" and to make other changes of this type.
Authors are asked to shoulder an increasing amount of the burden of typesetting and copyediting for academic publishing. On the one hand, this is extra uncompensated work for authors. On the other hand, it gives authors a chance to have more control over the final look and feel of the paper. A typesetting system such as LaTeX can simplify this process. Many users find that once LaTeX is mastered, it is faster than Word for producing beautiful texts.
If you'd like our help in converting a document to LaTeX, as some computer science conferences require, in typesetting a book, or in learning to use LaTeX, please contact us. We're happy to help authors, publishers, and institutions with this.
*This is effective for Roman letters, but Greek letters require special handling. | CommonCrawl |
Sarah peterson and miles
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Dec 10, 2020 · Miles Bonsignore married his girlfriend Sarrah Peterson. 1 (b) - The purpose of home leave is to ensure that employees who live abroad for an extended period undergo reorientation and re-exposure in the United States on a regular basis. Peterson, now 47, was sentenced to death on Jul 19, 2020 · Sarah Peterson Barkema, Facebook पर है. P. Oct 13, 2020 · The annual competition, currently in its 37th year, honors Charles E. The Complete Miles Davis Featuring John Coltrane. The Best Of Miles Davis & John Coltrane (1955-1961) At Newport 1958. The average household income in the Peterson Street area is $70,341. Apr 07, 2011 · Sarah: Mom, let's try and try. But by the time we got to the club, Styles was gone. Now 80, and a conservative Trump May 22, 2013 · In Planning the Home Front, Sarah Jo Peterson offers readers a portrait of the American people—industrialists and labor leaders, federal officials and municipal leaders, social reformers, industrial workers, and their families—that lays bare the foundations of community, the high costs of racism, and the tangled process of negotiation Sarah Dryden-Peterson leads a research program that focuses on the connections between education and community development, specifically the role that education plays in building peaceful and participatory societies. (360) 573-4806. She currently holds a NASA Hubble Fellowship at New York University. Although clearly having the credentials, having graduated in medicine at the national university and a 25-year career in public health including seminal research and policy work in indigenous health and refugee care, nevertheless she was quickly put into the position at the beginning of the COVID-19 Jan 09, 2020 · Sarah is 29 degrees from Colin Powell, 24 degrees from Edwin Bocage, 21 degrees from Chris Calloway, 42 degrees from Miles Davis, 25 degrees from Morgan Freeman, 30 degrees from Lena Horne, 23 degrees from Katherine Johnson, 33 degrees from Amelia King, 27 degrees from Bob Moses, 21 degrees from Barack Obama, 43 degrees from Cicely Tyson and 18 Apr 10, 2013 · Sarah Patterson Farm July, 1863. She . Sarah Peterson, LMT is a Massage Therapist in Vancouver, WA. Sarah Peterson Assistant Principal. Friday, Aug. Feb 09, 2010 · VEACH, John & Sarah PETERSON, 11 Apr 1799 ; WARE, Daniel & Zeruviah HUGHES, 28 Jan 1801 CORSON, Miles & Julian CORSON, 22 Jan 1825 ; CORSON, Peter & Sylvia SMITH Jun 18, 2012 · We finished in 1 hour, 4 minutes, which isn't too shabby for 6. The result is a remarkably compelling narrative that will be of great interest to both historians and planners. was held at 1 p. Jun 11, 2015 · I'm Sarah. May 04, 2017 · Sarah McCammon/NPR. Currently, Sarah Peterson is 132 years, 8 months and 5 days old. Sep 01, 2010 · SARAH DRYDEN-PETERSON is a Social Sciences and Humanities Research Council of Canada postdoctoral fellow at the Ontario Institute for Studies in Education/University of Toronto. It's an external conflict because Ms. Miles Robinson (1821-1908) Roland Oran Roehrman The Rabbit Room fosters Christ-centered community and spiritual formation through music, books, and story. "It stands for heritage Entdecken Sie Die Einhundert: Jazz von Kenny Burrell, Peggy Lee, Sun Ra, Louis Armstrong & Ella Fitzgerald, Quincy Jones & His Orchestra, Wayne Shorter, Charlie Parker, Ornette Coleman, Art Tatum, Dizzy Gillespie & Stuff Smith, Pharoah Sanders, Bill Frisell & Kenny Wheeler & Dave Holland & Lee Konitz, Wes Montgomery, Django Reinhardt, Etta James, Max Roach, Cannonball Adderley, Stan Getz Mar 25, 2013 · Sarah Miles, actor My agent was a man called Robin Fox. Our platform features advanced search and filtering, data insights Jul 28, 2021 · Bob Peterson, who voiced the paperwork-obsessed slug-woman Roz in "Monsters, Inc. She handles all stages of pretrial litigation, including depositions, motions practice, and discovery management. Since 60 minutes is 1 hour, she is running at a speed of 8 miles per hour. Together, they feed 600-head of Holstein steers from 200 pounds to finish on their family farm - Hillside Farms. Sarah attended Russel High School and was voted most likely to succeed and she did succeed in what mattered most: her relationship with God, family and friends. Oct 08, 2005 · In the late 1960s, Vaughan returned to jazz music, performing and making regular recordings. Kind Of Blue Deluxe 50th Anniversary Collector's Edition. Your pace and only as far as you can go. 2 miles. Project oVerview. This historical marker is listed in these topic lists: Agriculture • Science & Medicine • War, US Civil • Women. Lived In West Lebanon NH, Haines City FL, Lebanon NH, Sarasota FL. When 60-year-old Owen Golay talks about the two Confederate flags he flies in his front yard, he sounds like many Southern defenders of such symbols. Sarah Peterson Barkema और आपके अन्य परिचितों से जुड़ने के लिए Facebook में शामिल हों. Actor Leo McKern ,actress Sarah Miles, actor Robert Mitchum on set of the Metro-Goldwyn-Mayer movie "Ryan's Daughter" in 1970. In 1868 he purchased 160 acres of land on Dry Creek, 5 miles east of Sep 25, 2021 · Get all the wedding details for the Miles Bonsignore & Sarah Peterson wedding from their wedding website, including details for the wedding registry, travel, and hotels. Information provided by: Sarah Peterson. I don't plan on being here for the long term but I really appreciate how we work together as a team. Powered by our supporters, we provide free suicide prevention programs to teens and adults at schools, colleges, work places, community centers, gyms, libraries, faith organizations, and other community groups. Mrs. Miles Bonsignore started at 2nd Try LLC in May 2018 as a Production Assistant, and was later promoted to Podcast Producer and Camera Operator. Mar 08, 2018 · It's an internal conflict because Sarah has to decide whether or not to confront her friend. We found 61 addresses and 61 properties on Peterson Street in Alta, IA. ImportKey features trade and import data on over companies world wide, with records updated daily and going back to 2008. com and celebrate them on their big day, Saturday, September 04, 2021. Hans Bidertwas born on 1 Aug 1613 in Baren, Langenbruck, Basel, Switzerland. This, with saxophonist John Apr 04, 2019 · Former cop and convicted wife-killer Drew Peterson thinks federal prison is an upgrade from the confines he was previously in and thinks he should have just stayed a bachelor in a new interview Jun 02, 2021 · In this April 21, 2003 file photo, Sarah Kellison stands in front of a memorial in honor of Laci Peterson outside the house Laci shared with her husband Scott Peterson in Modesto, Calif. Provided to YouTube by Universal Music GroupAutumn Leaves · Cannonball AdderleySomethin' Else℗ 1999 Blue Note RecordsReleased on: 1999-01-01Associated Perfo (Genesis 11:31) Sarah would no doubt have much to do with caring for this elderly parent. She has experience trying cases in state and federal court. Find other locations and directions on Healthgrades. Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 Site Map Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 404-802-8900 The coronavirus disease 2019 (COVID-19) pandemic and the response to the pandemic are combining to produce a tidal wave of need for rehabilitation. ePlay makes connecting with streamers on live cam easier than ever with pocket-play mobile design and one-touch controls for players. O. Peterson has been great so far, and I am grateful to work with loving and kind people. Sarah was born on December 28, 1998 in Slidell, Louisian Sarah Peterson, Actress: Friends. As a locally owned Rapid City company, we offer Real Estate services throughout Western South Dakota and the Black Hills in the communities we know and love. A four-time Grammy Award winner and an NEA Jazz Masters recipient, Vaughan was a defining voice of the genre with a career that spanned more than five decades. He sold this land and rented a farm 1 mile south of Valmont that he ran 3 years. Mom: Sarah, I've told you I don't want to be a runner but, ok. Peterson was a coward and that his performance, under the circumstances, failed to meet the standards of police officers are patently untrue" By Sarah Gray February 26, 2018 ©2020 VOLUNTEER Hampton Roads an affiliate of the Points of Light Global Network, 757-624-2400, P. Programs can be tailored to specific audiences: general, teachers, LGBTQ+, military, and more. Sarah Miles and Robert Bolt at The BAFTA Awards during Sarah Miles and Robert Bolt at 1991 BAFTA awards in London, Great Britain. Among those speaking to the media in the aftermath were members Aug 01, 2019 · Miles buffs refer to his "first and second great quintets". This, with saxophonist John The Salmon SuperHwy Project is an unprecedented community effort that will restore access for fish to almost 180 miles of blocked habitat throughout six major salmon & steelhead rivers of Oregon's North Coast. First Generation. Clear Mourning Jan 26, 2021 · It had been a spectacular year for Sarah Peterson, the country's Chief Medical Officer. I was in a relationship with his son, Willy, an officer in the Coldstream Guards, who later changed his name to James . Sep 25, 2021 · Get all the wedding details for the Miles Bonsignore & Sarah Peterson wedding from their wedding website, including details for the wedding registry, travel, and hotels. If it takes her 15 minutes to run 2 miles, it will take her $4\times15=60$ minutes to run $4\times2=8$ miles at the same pace. Peterson did Mar 01, 2014 · The 1910 Census for Naugatuck, Connecticut (about 5 miles from downtown Waterbury) shows a 16 year old Clara P. 335 likes · 1 talking about this. Freund Aug 05, 2020 · Sarah Cooper has an outstanding résumé. Whether you are buying or selling residential, commercial, business opportunities, or land, our skilled team of agents are The Georgia Department of Corrections, its Employees and Contractors (heretofore known as " GDC ") make no warranty as to the accuracy or completeness of any information obtained through the use of this service. In the past, Sarah has also been known as Sarah E Peterson, Sarah J Rau and Sarah J Peterson. Select an address below to search who owns that property on Peterson Street and uncover many additional details. Laci Peterson's father, Dennis Rocha, died Sunday at age 72. MyRegistry. Before starting work at Second Try, he spent about six months at Buzzfeed in their fellowship program. 13, at Calvary Baptist Church. Like the skin of an apple, Earth's crust is very, very thin in comparison to the overall radius of our planet, Daniel Peterson writes. Join Facebook to connect with Sarah Miles Petersen and others you may know. Throughout the 1970s and '80s she recorded with such jazz notables as Oscar Peterson, Louie Bellson Geri Peterson Sarah Rohman; Lost Springs Cemetery map address, GPS coordinates and phone number. 45. . However, the couple is yet to become parents. Select the best result to find their address, phone number, relatives, and public records. QuiltCon Block Challenge: Pods by QuiltCon Block Challenge. The second was the 1960s group including Wayne Shorter, Herbie Hancock, Ron Carter and Tony Williams. You'll learn how to listen and enjoy this incredible art form by listening to important artists from the past (Miles Davis, Oscar Peterson, Sarah Vaughn) and present (Diana Krall, Brad Mehldau, Pat Metheny) and discover what makes them innovative and unique. It's an internal conflict because Sarah will probably argue with her mom later that night. In addition to her […] Find Sarah Peterson & Miles Bonsignore's Wedding registry at west elm. Easily hire Sarah Peterson for your special event: I'm a singer and accompany myself with acoustic guitar! I love to do covers of all types, and I love to listen to indie, alternative, rock, gospel, wedding, and classical music. The Clear Creek watershed encompasses 3,700 acres with 4. By using this service, you acknowledge that you understand that it is solely your responsibility to verify any information you may Oct 26, 2021 · Girls Aloud singer Sarah Harding, who died last month, was one of the celebrities that he made particularly scathing remarks about. Dec 26, 2020 · 3,828 Likes, 169 Comments - Sarah Peterson (@petesypeterson) on Instagram: "@milesbon and I got engaged! We are now in a Fart War for life. Peterson was born on July 21, 1935 in East Point, GA. 'Go with the Group!'. She conducts research on the role of schools, churches, and nonprofits in the integration of immigrants, the development of communities, and the transformation of society. Related To Joseph Peterson, Kathryn Peterson, Marilyn Peterson, Brian Peterson, Faith Peterson. Model: Sarah Peterson @ Brand Model & Talent. Bio: I'm Sarah. Sarah Peterson is an actress, known for Friends (1994), Jeffrey (1995) and The Sentinel (1996). Call Us at 763-550-1100 for more information. Peterson, FAIA (1906-2004), a founder of the HABS program. Nov 28, 2019 · Gratitude for the dirt beneath our feet. She has extensive experience giving lectures to both scientific peers and to non-scientific audiences Nov 03, 2021 · ShareTweetSharePin0 Shares NASHVILLE, Tenn. com Find a Registry Oct 29, 2019 · As much as anyone, Earlene Branch Peterson, who lost her daughter Nancy and granddaughter Sarah in those murders, is the kind of person Mr. The caravan journeyed first to Haran, some 600 miles (960 km) to the northwest, following the course of the Euphrates. Tired and hungry, we walked next door to Dan Tana's. Sarah Everard, the 33-year-old English woman who was abducted and killed while walking home Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 Site Map Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 404-802-8900 Powered by our supporters, we provide free suicide prevention programs to teens and adults at schools, colleges, work places, community centers, gyms, libraries, faith organizations, and other community groups. We're THE place to get free and instant access to your favorite streamers performing live and sharing content with you 24/7. Jul 05, 2015 · Sunday, May 17, 2015. Maria was born on 15 May 1619 in Switzerland. Jan 23, 2019 · Sarah Brightman is one of the most successful performers of all time and originated the role of Christine in The Phantom of the Opera, one of the most beloved musicals of all time. Apart from her background as a standup comedian, she's worked at Google, written for an animated series and published two humorous how-to books: 100 Tricks to Appear Smart in Meetings and How to Be Successful Without Hurting Men's Feelings. Miles's reported annual income is about $50 - 59,999; with a net worth that tops $10,000 - $24,999. Jan 14, 2021 · Adrian Peterson had been ordered to pay a Pennsylvania loan company $8,268,426. We said goodbye to our Jul 13, 2020 · Victims' relatives most vocal opponents of man's execution. Sarah Pearson is a Danish astrophysicist with a PhD degree from Columbia University in New York. Mar 11, 2021 · First published on Wed 10 Mar 2021 15. 2 miles of upstream habitat. MANN living with her parents Albert Mann and Sarah Mann, and her grandmother Maria Mann. A significant historical month for this entry is July 1863. Fax See products, suppliers and buyers related to SARAH MILES PETERSON. 30 and a mommy to a smart and sassy little girl Cashapp: $Sarahnea09 Welcome to The Real Estate Group. Peterson has given Sarah a detention. He had interests in music, and experimental agricultural and farm techniques. This Clara Mann married Carl Edward Peterson in 1929. Jun 01, 2021 · The abduction and killing of Sarah Everard ignited a movement to make women feel safer on the streets. Even if you're a fan, Brightman may have fallen off of your radar. The talented professionals at PWB offer an array of consulting services including audit and accounting, outsourced bookkeeping, and tax services. Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 Site Map Miles Elementary School • 4215 Bakers Ferry Rd SW Atlanta, GA 30331 404-802-8900 Oct 05, 2021 · Sarah Deer is a citizen of the Muscogee (Creek) Nation of Oklahoma and a University Distinguished Professor at the University of Kansas. In Planning the Home Front, Sarah Jo Peterson offers readers a portrait of the American people—industrialists and labor leaders, federal officials and municipal leaders, social reformers, industrial workers, and their families—that lays bare the foundations of community, the high costs of racism, and the tangled process of negotiation between New Deal visionaries and wartime planners. City of Lincoln/Lancaster County Police Department and Sheriff's Office Warrants Warrants valid as of 02:05 11-15-2021 Jun 18, 2012 · We finished in 1 hour, 4 minutes, which isn't too shabby for 6. The partnership includes federal, state, and local agencies as well as non-governmenta Oct 02, 2009 · Friday, October 16, 2009. 🎂 Sarah Peterson - Age, Bio, Faces and Birthday. PetersonNorfolk - Sarah Michelle Peterson, 15, of Norfolk passed away on Monday, January 20, 2014 at Sentara Norfolk General Hospital. As the American Institute of Certified Planners (AICP) institutes its new Code of Ethics and Professional Conduct, Sarah Jo Peterson, PhD, Assistant Professor of Regional and City Planning at the University of Oklahoma, remembers the old code and its greater capacity to inspire America's planners. She loved God with all her heart, soul and Dr. Now 80, and a conservative Trump Leave a review. Sep 26, 2019 · On June 25, 2017, Peterson was biking with her dad on the Peak to Peak Highway, just six miles away from her Nederland home. Jake Sanders III officiated and the Rev. Her 2015 book, The Beginning and End of Rape: Confronting Sexual Violence in Native America is the culmination of over 25 years of working with survivors and has received several awards, including the Best First Book award from the Native American Indigenous If it takes her 45 minutes to run 6 miles, it will take her $45\div3=15$ minutes to run $6\div3=2$ miles at the same pace. We walked back to the house, stretch and had some protein shakes. Companies all over the world use ImportKey to analyze suppliers, buyers and competitors. Chow to the Troubador, in West Hollywood, is about two miles. In town, Sarah is an odd one, going around in a man's long coat and trying to get rides back home – three point two miles. She pesters the man who fishes aboard her deceased father's skiff. Dec 11, 2020 · Sarah Moore husband: Sarah and husband Pete have been married for 19 years. May 31, 2013 · The drive from Mr. Sarah Peterson handles complex civil litigation and white collar criminal matters. Resides in White River Junction, VT. Uncovering Family Secrets W/ Sarah Peterson - You Can Sit With Us Ep. TryMod. 21 after a defaulted loan in 2016, New York State Supreme Court records show. Sarah E. Rey Del Rio/Getty Images. Our platform features advanced search and filtering, data insights Bio: I'm Sarah. New Podcast. Sarah Allen Peterson died at the age of 85 on May 29, 2021, at her home in Canton, Ga. Explanation: The only sentence that has correct parallel structure is the following one: Sheila Peterson can run 20 miles, swim three miles, and bike 30 miles, all without breaking a sweat. sarah peterson and miles
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\begin{document}
\title{The Inert Drift Atlas Model} \author{Sayan Banerjee, Amarjit Budhiraja, Benjamin Estevez} \maketitle
\begin{abstract} Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle.
The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles.
This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)).
We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z}$ is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.
\noindent\newline
\noindent \textbf{AMS 2010 subject classifications:} 60J60, 60K35, 60J25, 60H10. \newline
\noindent \textbf{Keywords:} Reflecting Brownian motions, degenerate dynamics, Atlas model, singular interaction, exponential ergodicity, product-form stationary distributions. \end{abstract}
\section{Introduction}
\label{sec:intro}
\editc{\subsection{Motivation and Model Description}} In this work we study the long-time behavior of an interacting particle system comprising a massive (inert) particle that moves under the combined influence of an external Newtonian potential (eg. gravitation) and a non-Newtonian `inert drift' resulting from collisions with many microscopic (Brownian) particles. This serves as a simplified model for the motion of a semi-permeable membrane in a fluid having microscopic impurities (see \cite{KnightInert}). The membrane, which allows fluid molecules to pass but is impermeable to the impurities, plays the role of the inert particle.
Mathematically, this model consists of $N$-Brownian particles in ${\mathbb{R}}$, with state processes denoted as $\{X_i(t), t \ge 0\}_{1\le i \le N}$, interacting with the inert particle, with state process $X_0(t)$, according to the following system of equations: For $t \ge 0$, \begin{equation}\label{eq:unrankloc}
\begin{aligned}
X_0(t) &= x_0 + \int_0^t V(s) ds, \;\; V(t) = v_0+ gt - \sum_{i=1}^N \ell_i(t) ,\\
X_i(t) &= x_i + W_i(t) + \ell_i(t), \; 1 \le i \le N.
\end{aligned} \end{equation} Here \editc{$x_0 \le x_1 \le \cdots \le x_N$} denote the initial positions of the $N+1$ particles, $v_0$ the initial velocity of the inert particle, $\{W_i, 1 \le i \le N\}$ are mutually independent standard real Brownian motions, $g \in (0,\infty)$ denotes the gravitation constant and $\ell_i$ is the collision local time between the $i$-th particle and the inert particle which, in particular, satisfies $\ell_i(t) = \int_0^t 1_{\{X_i(s) = X_0(s)\}} d\ell_i(s)$ for $1\le i \le N$ and $t\ge 0$.
The local time interactions model the cumulative transfer of momentum when a Brownian particle collides with the inert particle `infinitely often' on finite time intervals, with each collision resulting in an infinitesimal momentum transfer. Such interactions lie at the heart of this model and interesting long time behavior results from the combined effect of the `soft' gravitational potential and `hard' collisions.
\editc{It follows from \cite{barnes2018strong} (see Theorem 2.5 and Proposition 2.10 therein) that there is a strong solution to the system of equations in \eqref{eq:unrankloc} and the solution satisfies $X_0(t) \le X_i(t)$ for all $t\ge 0$ and $1\le i \le N$ a.s. Using Gronwall's lemma and the Lipschitz property of the Skorohod map it is easy to verify that in fact the system of equations in \eqref{eq:unrankloc} has a unique strong solution. Given this unique solution process $\{X_i(t),\; t \ge 0\}_{0\le i \le N}$ of \eqref{eq:unrankloc} it will be
convenient} to consider the ordered particle system: $$X_{(0)}(t) \le X_{(1)}(t) \le \cdots \le X_{(N)}(t), \; t \ge 0,$$ where $\{X_{(i)}(t) : t \ge 0\}$ denotes the state process of the $i$-th particle from the bottom (\editc{note that the lowest particle, which we call the $0$-th particle from the bottom, is the inert particle, in particular, $X_{(0)}(\cdot) = X_0(\cdot)$}). By an application of Tanaka's formula it is easy to verify that this ranked particle system satisfies the following system of equations: For $t \ge 0$, \begin{equation}\label{eq:rankloc}
\begin{aligned}
X_{(0)}(t) &= x_0 + \int_0^t V(s) ds, \;\; V(t) = v_0+ gt - L_1(t),\\
X_{(1)}(t) &= x_1 + B_1(t) -\frac{1}{2} L_2(t) + L_1(t), \\
X_{(i)}(t) &= x_i + B_i(t) -\frac{1}{2} L_{i+1}(t) + \frac{1}{2} L_{i}(t), \; 2 \le i \le N.
\end{aligned} \end{equation} where \editc{$x_0 \leq x_1 \leq ... \leq x_N$,} $\{B_i, 1 \le i \le N\}$ are standard independent Brownian motions and for $1\le i \le N$, $L_{i}$ denotes the collision local time between the $i$-th and the $(i-1)$-th ranked particle which satisfies $L_i(t) = \int_0^t 1_{\{X_{(i)}(s) = X_{(i-1)}(s)\}} dL_i(s)$ and $L_{N+1}(t)=0$ for all $t\ge 0$ .
We are interested in the time asymptotic behavior of the {\em velocity and gap processes} associated with this system. Namely, denoting $Z_i(t) \doteq X_{(i)}(t) - X_{(i-1)}(t)$, the object of interest is the stochastic process $$(V(t), Z_1(t), \cdots, Z_N(t)).$$ This process is given by the system of equations \begin{equation}\label{eq:gapproc}
\begin{aligned}
V(t) &= v_0+ gt - L_1(t),\\
Z_1(t) &= z_1 + B_1(t) - \int_0^t V(s) ds - \frac{1}{2} L_2(t) + L_1(t), \\
Z_2(t) &= z_2 + B_2(t)- B_1(t) - \frac{1}{2} L_3(t) + L_2(t) - L_1(t),\\
Z_i(t) &= z_i + B_i(t) - B_{i-1}(t) -\frac{1}{2} L_{i+1}(t) + L_{i}(t) - \frac{1}{2} L_{i-1}(t), \; 3 \le i \le N.
\end{aligned} \end{equation}
The model described by equations \eqref{eq:rankloc} (with gaps evolving as in \eqref{eq:gapproc}), which we call the \emph{inert drift Atlas model}, lies at the interface of two well-studied classes of interacting particle systems: \emph{inert drift models} and \emph{rank-based diffusions}, which we summarize below.
\editc{\subsection{Previous Work}}
The case where $N=1$ (namely the two particle system) with $g=0$ was analyzed in \cite{KnightInert}, which initiated the study of inert drift models. It was shown there that the inert particle progressively gains momentum from the local time interactions and eventually escapes the Brownian particle (no further collisions). When $g>0$, \cite{banerjee2019gravitation} showed that the two particles never escape each other. Among other results, the paper showed that the velocity of the inert particle and the gap between the two particles jointly converge in total variation distance to an explicit stationary distribution having a product form density (no rates of convergence were obtained). The two particle model with gravitation and fluid viscosity was investigated in \cite{BanBro}. In \cite{BBCH}, an inert drift model was considered where a particle moves as a diffusion process inside a bounded smooth domain and acquires inert drift when it hits the boundary of the domain. It was shown that the position of the particle and the cumulative inert drift have a product form stationary measure, which is unique under suitable conditions. A variety of related inert drift models have been studied in \cite{BurEtAl,WhiteInert,BurWhi}. When the term $\sum_{i=1}^N \ell_i(t)$ in \eqref{eq:unrankloc} is replaced by $N^{-1}\sum_{i=1}^N \ell_i(t)$ (mean field type interaction), the asymptotic behavior as $N\to \infty$ has been analyzed in \cite{BarHyd,barnes2018strong} where results on hydrodynamic limits and propagation of chaos have been obtained. Recently, unexpected connections have appeared between inert drift models and diffusion limits of load balancing systems like the Join-the-shortest-queue policy in heavy traffic \cite{eschenfeldt2018join},\cite{BanMuk},\cite{banerjee2020join}. More precisely, the joint evolution of the diffusion-scaled number of idle servers and busy servers converges in distribution to a diffusion that resembles the two particle inert drift system with linear drift. Consequently, there are several common themes at the technical level between \cite{banerjee2019gravitation,BanBro} and \cite{BanMuk,banerjee2020join}. Brownian particle systems of the form studied in the current work also arise as diffusion approximations of certain types of queuing systems in which each queue has the same finite capacity which is dynamically controlled in a manner that the increase in capacity is proportional to net job loss due to capacity constraints. In this model, currently under investigation, the individual queues play the role of Brownian particles whereas the dynamically changing queue capacity threshold represents the massive inert particle.
In a somewhat different vein, inspired by problems in mathematical finance, the study of rank-based diffusions \cite{Atlas2},\cite{Atlas1},\cite{Atlas3},\cite{DJO},\cite{sarantsev2017stationary},\cite{AS},\cite{banerjee2021domains} have gained a lot of attention in recent years. These models consist of a collection of particles on the real line which evolve as diffusion processes where the drift and diffusivity of each particle is a function of its relative rank in the system. Closest in spirit to our model is the \emph{Atlas model} where the lowest ranked particle at any time moves as a Brownian motion with constant upward drift while the remaining particles evolve as standard Brownian motions (with zero drift).
\editc{\subsection{Analytical Challenges}}
The Atlas model and the model considered here are examples of particle systems with topological interactions in the terminology of \cite{CDGP}. In such particle systems, interactions between particles are determined by their relative positions. In particular, in both the Atlas model and in the particle system considered here, the lowest particle has different dynamical properties. Specifically, in the Atlas model the lowest particle gets a constant upward drift whereas in the model considered here the lowest particle experiences an {\em inert drift}. However there are some important differences between the two models. Unlike the Atlas model, where the collision local time of the lowest two particles enters directly in the position evolution of the lowest particle, here this local time impacts the velocity of the lowest particle. Indeed, this collision local time is the source of the inert drift of the lowest particle. Furthermore,
there is no Brownian noise in the equation for $X_{(0)}$ in \eqref{eq:rankloc}, unlike in the Atlas model. This results in the deterministic evolution of the velocity process in time periods with no collisions, making the full system, whose long-time behavior is of interest, non-elliptic (in fact, the driving diffusion process in the interior of the domain is not even hypoelliptic). More precisely, the law of $(V(t), Z_1(t), \cdots, Z_N(t))$ for any $t>0$ does not have a density with respect to Lebesgue measure, for general initial conditions. Also, we find that, unlike the Atlas model, the system considered here is not reversible.
Hence, standard techniques for studying ergodicity behavior of elliptic diffusion processes cannot be applied, and one needs new methods.
As noted above, inert two-particle systems have been studied in several previous works, however the current work is the first to study the ergodicity properties of a general $N$-particle system. There are fundamental differences in system behavior
as one goes from $N=1$ to $N>1$ which make the study of ergodicity behavior significantly more demanding. \editc{In particular, as is crucially exploited in \cite{banerjee2019gravitation, BanBro}, in the $N=1$ case, there is a basic \emph{regenerative structure} arising from the fact that at points of decrease of the velocity process, the remaining state coordinate, namely the one corresponding to $Z_1$, is fully determined (in fact equal to $0$). In the general $N$-particle system there is no such simple regenerative structure since, although the first gap coordinate $Z_1$ is once again $0$ at points of decrease of $V$, the remaining coordinates, namely $Z_2, \ldots , Z_N$ can be arbitrary.}
\editc{\subsection{Main Contributions}}
We now briefly describe the main contributions of this work. Since the system is not hypoelliptic, one cannot apply standard existing theory to argue uniqueness of invariant measures. Our first main result says that the Markov proces $(V, \mathbf{Z})= (V, Z_1, \ldots, Z_N)$ admits at most one stationary distribution. We then produce an explicit stationary distribution for the system and together the two results (see Theorems \ref{thm:exisuniq} and \ref{thm:prodform}) prove existence and uniqueness of stationary distributions of $(V, \mathbf{Z})$. We in fact show that the unique stationary distribution takes a product form whose first component (corresponding to the velocity coordinate) is Gaussian and remaining are Exponential (see Theorem \ref{thm:prodform} for the precise form). \editc{In the case $N=1$, a Gaussian-Exponential product form stationary distribution has appeared in previous works \cite{WhiteInert},\cite{BBCH},\cite{banerjee2019gravitation}; however, this is the first work that finds such a product form structure for a general \editc{$N$-particle} system.} This stationary distribution also has striking similarities with the Atlas model where the stationary distribution is a product of exponentials with rates decreasing with the ranks of the particles (see, for example, \cite[Theorem 8]{Atlas1}).
We next study the rate of convergence to stationarity.
In Theorem \ref{thm:geomerg}, we show that the distribution of $(V(t), Z_1(t), \cdots, Z_N(t))$ converges to equilibrium exponentially fast (exponential ergodicity) as $t\to \infty$. \editc{To the best of our knowledge, this is the first result on exponential ergodicity for any type of non-hypoelliptic reflected diffusion in dimensions higher than $2$.}
Finally in Theorem \ref{lln} we establish some law of large numbers type results. In particular, it is shown that the whole system `drifts' to infinity at speed $g/N$. Although this is an intuitive result to expect, our proof crucially hinges on the
rather technical result on exponential moments of return times to certain compact sets that form the basis of the exponential ergodicity proof.
We also find, somewhat surprisingly, that the intensity of collisions when $N \ge 3$ is maximum, in a certain sense, between the first two Brownian particles (rather than between the inert and the first Brownian particle); see Remark \ref{int12}.
\editc{\subsection{Approach}}
A common approach to proving ergodicity or exponential rates of convergence to stationarity for diffusions in domains is by constructing a suitable Lyapunov function by analyzing the interplay between the ``interior drift vector field'' and the reflection vector field (cf. \cite{dupuis1994lyapunov}, \cite{atar2001positive}, \cite{BudLee}). For example, in polyhedral domains with constant (oblique) reflection on each face of the boundary, the key insight in the construction of a Lyapunov function is that the drift vector field for stable systems must lie in the interior of the cone generated by the negatives of the reflection directions.
Note that $\mathbf{Z}$ is a reflected diffusion in the positive orthant $\mathbb{R}^N_+$ with constant oblique reflection at each face. The interior drift of this process is $V(t) \mathbf{e}_1$, where $\mathbf{e}_1$ is the unit vector with $1$ in the first coordinate. Due to the complicated dynamics of $V$, that includes in particular the local time for the first gap process $Z_1$, its behavior in relation to the reflection field seems hard to analyze which makes a direct construction of a explicit form Lyapunov function (as in the above cited works) hard.
In this work we instead take a pathwise approach. The stability in the particle system studied here arises as a result of interplay between the intersection local times for the various particles in the system. This interplay is distilled in Lemma \ref{zless} which identifies a stabilizing `singular' drift that prevents the gaps between the particles from being too large. This key lemma allows us to prove the finiteness of exponential moments of hitting times to certain compact sets by analyzing excursions of the process between suitably chosen stopping times (see Sections \ref{highlev}-\ref{comphit}). In conjunction with results of \cite{DowMeyTwe} (see Proposition \ref{driftcondn} (a)), this analysis furnishes a general abstract form Lyapunov function, given in terms of exponential moments of these hitting times, which is key in the proof of exponential ergodicity.
Another important ingredient in our proofs is establishing a certain minorization estimate (see Proposition \ref{driftcondn} (b)). For hypoelliptic diffusions such an estimate follows readily from the existence of a density for the process at each time $t>0$. However, in our case, establishing a suitable minorization bound involves substantial work and a careful exploitation of the properties of the collision local times of the particles in the system. The proof of this estimate, which uses an intricate and novel
pathwise analysis, is the topic of Section \ref{minosec}.
\editc{\subsection{Future Directions}}
The current work is the first step in our program of analyzing high-dimensional reflected diffusions with inert drift type interactions. The natural next step will be to investigate ergodicity properties of the infinite-dimensional analogue of our model. The corresponding vector of velocity and gap processes is expected to have at least one stationary distribution, given by the $N \rightarrow \infty$ limit of \eqref{eq:statdistn} below. It is unclear if this is the unique stationary distribution. Analogy with the Atlas model suggests infinitely many stationary distributions, each with a non-trivial domain of attraction \cite{sarantsev2017stationary},\cite{DJO},\cite{banerjee2021domains}. Another interesting question concerns the study of hydrodynamic limits of empirical occupation measures of the system and relate them to the path asymptotics of the bottom $k$ particles for $k \in \mathbb{N}$ (see \cite{dembo2017equilibrium} for related results on the Atlas model). Both these directions are currently under investigation.
\subsection{Notation and Preliminaries}
The following notation will be used. For $d \in {\mathbb{N}}$ and $T>0$, we denote by ${\mathcal{C}}([0,T]: {\mathbb{R}}^d)$ (resp. ${\mathcal{C}}([0, \infty): {\mathbb{R}}^d)$) the space of continuous functions on $[0,T]$ (resp. $[0,\infty)$) with values in ${\mathbb{R}}^d$, equipped with the topology of uniform convergence (resp. local uniform convergence). The spaces ${\mathcal{C}}([0,T]: {\mathbb{R}}_+^d)$ (resp. ${\mathcal{C}}([0, \infty): {\mathbb{R}}^d_+)$) of continuous functions with values in the nonnegative orthant ${\mathbb{R}}_+^d$ are defined similarly. For $t \in [0,\infty)$ and $f \in {\mathcal{C}}([0, \infty): \mathbb{R}^d)$, we define
$\|f\|_t \doteq \sup_{0 \leq s \leq t}|f(s)|$, where $|\cdot|$ is the Euclidean norm on $\mathbb{R}^{d}$.
Borel $\sigma$-fields on a metric space $S$ will be denoted as ${\mathcal{B}}(S)$. Inequalities for vectors and vector-valued random variables are understood to be coordinatewise. An open set $G \subset {\mathbb{R}}^d$ is said to have a ${\mathcal{C}}^2$ boundary if each point in $\partial G$ has a neighborhood in which $\partial G$ is the graph of a ${\mathcal{C}}^{2 }$ function of $d-1$ of the coordinates (cf. \cite[Section 6.2]{giltru}). Throughout $\lambda$ will denote the Lebesgue measure on a subset of a Euclidean space whose dimension will be clear from the context.
The following elementary estimate will be used several times. Suppose for $m\in {\mathbb{N}}$, $\tilde B_1, \ldots\editc{,} \tilde B_m$ are mutually independent Brownian motions and $\alpha_1, \ldots\editc{,} \alpha_m \in {\mathbb{R}}_+$. Let $\tilde B_i^* (t)\doteq \sup_{0\le s \le t} |\tilde B_i(s)|$. Then there are $\varrho_1, \varrho_2 \in (0,\infty)$, such that \begin{equation} \label{eq:elemconc}
\mathbb{E}\left( e^{u\sum_{i=1}^m \alpha_i\tilde B_i^* (t)}\right) \le \varrho_1 e^{\varrho_2 u^2 t} \mbox{ for all } t \ge 0 \mbox{ and } u \ge 0.\end{equation} The dependence of the constants $\varrho_1, \varrho_2$ on $m$ and $\alpha_i$ will usually be suppressed from the notation.
\editc{In the next section, we outline our main results. The organization of the paper is summarized at the end of the section.}
\section{Main Results} \label{sec:mainres} Define the $N \times N$ matrix \begin{equation}\label{Rmat} R \doteq \begin{pmatrix}
1 & -\frac{1}{2} & 0 & 0 & \cdots & 0 \\
-1 & 1 & -\frac{1}{2} & 0 & \cdots & 0 \\
0 & -\frac{1}{2} & 1 & -\frac{1}{2} & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & \cdots & \cdots & \cdots & -\frac{1}{2} & 1 \end{pmatrix}. \end{equation} It is easily checked that the matrix $U=I-R$ has the property that $U^T$ is a transient, substochastic matrix and thus has spectral radius strictly less than $1$. Consequently, $R$ is invertible and $W= R^{-1}$ can be written as an infinite sum of matrices with nonnegative entries. The Skorohod problem associated with such matrices $R$ has been well studied and the following result is well known cf. \cite{harrei},\cite{dupish},\cite{dupram}.
We denote by ${\mathcal{C}}_0([0, \infty): \mathbb{R}^N)$ the space of continuous functions $f: [0,\infty) \rightarrow \mathbb{R}^N$ such that $f(0) \geq 0$.
\begin{proposition}\label{skorokhod}
To each $x \in {\mathcal{C}}_0([0, \infty):\mathbb{R}^N)$ there is a unique pair $(\eta,y) \in {\mathcal{C}}([0, \infty):\mathbb{R}_+^N) \times {\mathcal{C}}([0, \infty):\mathbb{R}_+^N)$ such that,
\begin{enumerate}[(i)]
\item for all $t\ge 0$, $y(t) = x(t) + R\eta(t)$,
\item For each $i \in \{1,...,N\}$,
\begin{inparaenum}
\item $\eta_i(0) = 0$,
\item $\eta_i(t)$ is non-decreasing in $t$,
\item $\int_0^{\infty}y_i(t)d\eta_i(t) = 0$.
\end{inparaenum} \end{enumerate} The pair $(\eta,y)$ is called the solution to the Skorokhod problem for $x$ with respect to $R$. The map $\Gamma: {\mathcal{C}}_0([0, \infty):\mathbb{R}^N) \rightarrow {\mathcal{C}}([0, \infty):\mathbb{R}_+^N) \times {\mathcal{C}}([0, \infty):\mathbb{R}_+^N)$ given by \begin{align*}
\Gamma(x) = (\eta,y) = (\Gamma_1(x),\Gamma_2(x)) \end{align*} is Lipschitz in the sense that there is a $c_{\Gamma} \in (0,\infty)$ such that for $x,x' \in {\mathcal{C}}_0([0, \infty):\mathbb{R}^N)$ and $t <\infty$,
$$\|\Gamma_1(x) - \Gamma_1(x')\|_t + \|\Gamma_2(x) - \Gamma_2(x')\|_t \le c_{\Gamma} \|x-x'\|_t.$$ \end{proposition} For $x \in {\mathcal{C}}_0([0, \infty):\mathbb{R}^N)$, we occasionally write $\Gamma_1(x) = (\Gamma_{11}(x), \ldots , \Gamma_{N1}(x))$.
The following result gives strong existence and uniqueness for the system of equations in \eqref{eq:gapproc}. Proof is given in Section \ref{sec:exisuniq}. Let \begin{equation}\label{Amat} A \doteq \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \\ -1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & -1 & 1 & \cdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & 1 & 0 \\ 0 & 0 & 0 & \cdots & -1 & 1 \\ \end{pmatrix}. \end{equation} \begin{theorem}\label{thm:wellposed} Let $(\bar \Omega, \bar{\mathcal{F}}, \{\bar {\mathcal{F}}_t\}_{t\ge 0}, \bar {\mathbb{P}})$ be a filtered probability space on which are given $N$ mutually independent standard real $\bar {\mathcal{F}}_t$-Brownian motions $B_1, \ldots, B_N$ and, $\bar{\mathcal{F}}_0$-measurable random variables $V^0$ and $\mathbf{Z}^0 = (Z_1^0, \ldots, Z_N^0)$ with values in ${\mathbb{R}}$ and ${\mathbb{R}}_+^N$ respectively. Then there is a continuous, $\bar{\mathcal{F}}_t$-adapted, stochastic process $(V(t), Z_1(t), \ldots, Z_N(t))_{0\le t < \infty}$ with values in ${\mathbb{R}} \times {\mathbb{R}}_+^N$ such that, for all $t\ge 0$, \begin{equation}\label{eq:solskor} \begin{aligned}
V(t) &= V^0 + gt - L_1(t), \\
\mathbf{Z}(t) &=\Gamma_2\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V(s) ds + A\mathbf{B}(\cdot)\right)(t),\\
L_1(t) &= \Gamma_{11}\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V(s) ds + A\mathbf{B}(\cdot)\right)(t),\\ \end{aligned} \end{equation} where $\mathbf{B} = (B_1, \ldots, B_N)'$ and $\mathbf{Z} = (Z_1, \ldots , Z_N)'$. Furthermore, if $(\tilde V(t), \tilde Z_1(t), \ldots\editc{,} \tilde Z_N(t))$ is another such process then $$(\tilde V(t), \tilde Z_1(t), \ldots\editc{,} \tilde Z_N(t)) = ( V(t), Z_1(t), \ldots , Z_N(t)) \mbox{ for all } t \ge 0, \mbox{ a.s. } $$ \end{theorem} We remark that, with $\mathbf{Z}$, and $V$ as in the theorem, letting $$\mathbf{L}(t) = (L_1(t) , \ldots , L_N(t))= \Gamma_1\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V(s) ds + A\mathbf{B}(\cdot)\right)(t) \mbox{ and } L_{N+1}(t)=0,\; $$ we have that the following system of equations holds: \begin{equation}\label{eq:gapprocb}
\begin{aligned}
V(t) &= V^0+ gt - L_1(t),\\
Z_1(t) &= Z_1^0 + B_1(t) - \int_0^t V(s) ds - \frac{1}{2} L_2(t) + L_1(t), \\
Z_2(t) &= Z_2^0 + B_2(t)- B_1(t) - \frac{1}{2} L_3(t) + L_2(t) - L_1(t),\\
Z_i(t) &= Z_i^0 + B_i(t) - B_{i-1}(t) -\frac{1}{2} L_{i+1}(t) + L_{i}(t) - \frac{1}{2} L_{i-1}(t), \; 3 \le i \le N.
\end{aligned} \end{equation} Consider the path space $\Omega^* = {\mathcal{C}}([0,\infty): {\mathbb{R}}^N \times {\mathbb{R}} \times {\mathbb{R}}_+^N)$, ${\mathcal{F}}^*$ the corresponding Borel $\sigma$-field on $\Omega^*$. We also consider the space $(\Omega, {\mathcal{F}}) \doteq ({\mathcal{C}}([0,\infty): {\mathbb{R}} \times {\mathbb{R}}_+^N), {\mathcal{B}}({\mathcal{C}}([0,\infty): {\mathbb{R}} \times {\mathbb{R}}_+^N))$. On these two measurable spaces we denote, for $(v, \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, by ${\mathbb{P}}^*_{(v,\mathbf{z})}$ [resp. ${\mathbb{P}}_{(v,\mathbf{z})}$], the probability measures induced by $(\mathbf{B}, V, \mathbf{Z})$ [resp. $(V, \mathbf{Z})$] where $(V, \mathbf{Z})$ is the solution of \eqref{eq:solskor} when $(V^0, \mathbf{Z}^0) = (v, \mathbf{z})$ a.s. Then from the unique solvability in the above theorem it follows that $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ defines a strong Markov family. The next result concerns the stationary distribution of this Markov family. \begin{theorem}\label{thm:exisuniq}
There is a unique stationary distribution for the Markov family $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$. \end{theorem} In fact this unique stationary distribution takes an explicit product form as given by the theorem below. Consider the probability measure $\pi$ on ${\mathbb{R}} \times {\mathbb{R}}_+^N$ given by the formula: \begin{equation}\label{eq:statdistn} \pi(dv,\, dz_1, \, \ldots , dz_N) \doteq c_{\pi}e^{-(v-\frac{g}{N})^2}\prod_{i=1}^Ne^{-2g\editc{\left(\frac{N-i+1}{N}\right)}z_i}\, dv\, dz_1\, \ldots , dz_N, \end{equation} where $c_{\pi}$ is the normalization constant. \begin{theorem}\label{thm:prodform} The probability measure $\pi$ defined in \eqref{eq:statdistn} is the unique stationary distribution of $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$. \end{theorem} \editc{Note that while Theorem \ref{thm:prodform} implies Theorem \ref{thm:exisuniq}, we proceed by first showing that there exists at most one stationary distribution (Theorem \ref{atmostonesd}). The existence and explicit form of the stationary distribution is subsequently exhibited (in Section \ref{sec:prodform}) by proving that the density of $\pi$ solves the partial differential equation (with boundary conditions) arising from the basic adjoint relationship (see \eqref{s1}-\eqref{s4}). We have therefore separated out these results for clarity of exposition.}
Our third result gives exponential ergodicity of the Markov process. Write an element $\omega \in \Omega^*$ [resp. $\omega \in \Omega$] as $\omega = (\beta,\upsilon,\zeta)$ [resp. $\omega = (\upsilon,\zeta)$], where $\beta \in {\mathcal{C}}([0,\infty): {\mathbb{R}}^N)$, $\,\upsilon \in {\mathcal{C}}([0,\infty): {\mathbb{R}})$ and $\,\zeta \in {\mathcal{C}}([0,\infty): {\mathbb{R}}_+^N)$. For $t \in [0,\infty)$, abusing notation, denote the coordinate processes $\mathbf{B}(t), V(t)$ and $\mathbf{Z}(t)$ on $(\Omega^*, {\mathcal{F}}^*)$ [resp. $V(t)$ and $\mathbf{Z}(t)$ on $(\Omega, {\mathcal{F}})$] by the formulae \begin{align*}
\mathbf{B}(t)(\omega) = \beta(t), \;\;
V(t)(\omega) = \upsilon(t), \;\;
\mathbf{Z}(t)(\omega) = \zeta(t), \; \; t\ge 0. \end{align*} Also, we will write $B_i(t)$ and $Z_i(t)$ respectively for the projections of $\mathbf{B}(t)$ and $\mathbf{Z}(t)$ onto their $i^{\textnormal{th}}$ coordinates. Consider the transition probability kernel of the Markov family $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ defined as $${\mathbb{P}}^t((v, \mathbf{z}), A) \doteq {\mathbb{P}}_{(v,\mathbf{z})}((V(t), \mathbf{Z}(t)) \in A), \; t\ge 0, (v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N,\; A \in {\mathcal{B}}({\mathbb{R}} \times {\mathbb{R}}_+^N).$$ Also, for a bounded and measurable $\phi: {\mathbb{R}} \times {\mathbb{R}}_+^N \to {\mathbb{R}}$ we write $${\mathbb{P}}^t((v, \mathbf{z}),\, \phi) \doteq \int_{{\mathbb{R}} \times {\mathbb{R}}_+^N} \phi(\tilde v, \tilde \mathbf{z})\, {\mathbb{P}}^t((v, \mathbf{z}),\, d\tilde v\times d\tilde\mathbf{z}).$$ Similarly, for $\phi$ as above, $\pi(\phi) \doteq \int \phi(\tilde v, \tilde \mathbf{z}) \pi(d\tilde v\times d\tilde\mathbf{z})$. The following theorem shows the convergence of the transition probability kernel to the stationary distribution in the total variation distance at an exponential rate. Denote by $\mbox{{\small BM}}_1$ the class of all measurable
$\phi: {\mathbb{R}} \times {\mathbb{R}}_+^N \to {\mathbb{R}}$ such that $\sup_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N} |\phi(v,\mathbf{z})|\le 1$. \begin{theorem}\label{thm:geomerg}
There is a $\gamma \in (0,1)$ and, for every $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, a $\kappa(v,\mathbf{z}) \in (0,\infty)$, such that for all $t\ge 0$,
$$ \sup_{\phi \in \mbox{{\small BM}}_1} |{\mathbb{P}}^t((v, \mathbf{z}),\, \phi) - \pi(\phi)| \le \kappa(v, \mathbf{z}) \gamma^t .$$ \end{theorem}
\editc{We note here that the proof of exponential ergodicity proceeds through establishing finiteness of exponential moments of certain hitting times. This, in turn, provides the tightness required to furnish an independent proof of existence of a stationary distribution.}
Finally, we prove a strong law of large numbers type result for the system. Recall the ranked particle system $\{X_{(i)}(\cdot)\}_{0 \le i \le N}$ from \eqref{eq:rankloc}. This process can be constructed on $(\Omega^*, {\mathcal{F}}^*, {\mathbb{P}}^*_{(v,\mathbf{z})})$ for any $(v, \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$ by solving the system of equations \editc{in \eqref{eq:solskor} (or equivalently\eqref{eq:gapprocb}),} whose unique pathwise solutions are guaranteed by Theorem \ref{thm:wellposed}, and then defining $X_{(i)}$ by the right side of \eqref{eq:rankloc}.
\begin{theorem}\label{lln} For any $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, the following limits hold ${\mathbb{P}}^*_{(v,\mathbf{z})}$-almost surely: \begin{align} \lim_{t \rightarrow \infty} \frac{X_{(i)}(t)}{t} &= \frac{g}{N}, \ \ 0 \le i \le N,\label{le1}\\ \lim_{t \rightarrow \infty} \frac{L_1(t)}{t} &= g,\label{le2}\\ \lim_{t \rightarrow \infty} \frac{L_i(t)}{t} &= \frac{2(N-i+1)g}{N}, \ \ 2 \le i \le N.\label{le3} \end{align} \end{theorem}
\begin{remark}\label{int12} It is natural to expect that the gaps become larger in some sense as one moves away from the inert particle. This heuristic is quantified in the stochastic monotonicity of the stationary gaps displayed in \eqref{eq:statdistn}. From this, it might appear that the growth rate of the local time $L_i(t)$ (which quantifies the intensity of collisions between the $(i-1)$th and $i$th particle) with $t$ should decrease as $i$ increases from $1$ to $N$. However, Theorem \ref{lln} shows that for $N \ge 3$, $L_2$ grows at a faster rate than $L_1$ and the expected decrease in rates holds from $i=2$ onwards. Hence, perhaps surprisingly, particles indexed $1$ and $2$ collide `more often' than particles $0$ and $1$ as time progresses.
\end{remark} \editc{\subsection{Organization}} \editc{The rest of the paper is organized as follows. In Section \ref{sec:wellposed}, we provide the proof of Theorem \ref{thm:wellposed}. In Section \ref{minosec}, we show a technical estimate which will be integral to the proofs of our main results. In Section \ref{sec:exisuniq}, we show that there is at most one stationary distribution (Theorem \ref{atmostonesd}). In Section \ref{sec:prodform}, we prove Theorem \ref{thm:prodform}. Together, these two results also establish Theorem \ref{thm:exisuniq}. In Section \ref{sec:geomerg}, we give the proof of Theorem \ref{thm:geomerg}. Proofs of several technical results stated in Section \ref{sec:geomerg} (without proof) are provided in Section \ref{sec:techlem}. In Section \ref{sec:lln}, we establish Theorem \ref{lln}.}
\section{Existence and uniqueness of the process}\label{sec:wellposed} In this section, we prove Theorem \ref{thm:wellposed}. The proof uses the Lipschitz property in Proposition \ref{skorokhod}, and a standard Picard iteration scheme. We provide a sketch. Fix $T<\infty$. Let $(V^0, \mathbf{Z}^0)$ be as in the statement of the theorem. Define, for $n \in {\mathbb{N}}_0$, continuous $\bar {\mathcal{F}}_t$-adapted ${\mathbb{R}} \times {\mathbb{R}}_+^N \times {\mathbb{R}}_+^N$ valued processes $\{(V^{(n)}(t), \mathbf{Z}^{(n)}(t)), \mathbf{L}^{(n)}(t))\}_{0\le t \le T}$, recursively, as follows. Let $$V^{(0)}(t) \doteq V^0, \;\; \mathbf{Z}^{(n)}(t)) \doteq \mathbf{Z}^0,\;\; \mathbf{L}^{(n)}(t) = 0, \;\; 0 \le t \le T.$$ Having defined $\{(V^{(k)}(t), \mathbf{Z}^{(k)}(t)), \mathbf{L}^{(k)}(t))\}_{0\le t \le T}$ for $k = 0, \ldots \editc{,} n-1$, define \begin{equation}\label{eq:picit} \begin{aligned} \mathbf{Z}^{(n)}(t) &=\Gamma_2\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V^{(n-1)}(s) ds + A\mathbf{B}(\cdot)\right)(t),\\
\mathbf{L}^{(n)}(t) &= \Gamma_{1}\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V^{(n-1)}(s) ds + A\mathbf{B}(\cdot)\right)(t),\\
V^{(n)}(t) &= V^0 + gt - L_1^{(n)}(t), \end{aligned} \end{equation} where $L_1^{(n)}(t)$ is the first coordinate of $\mathbf{L}^{(n)}(t)$.
From the Lipschitz property in Proposition \ref{skorokhod} it follows that, for any $n \ge 2$, and $t \in [0,T]$, \begin{align*}
\|\mathbf{Z}^{(n)} - \mathbf{Z}^{(n-1)}\|_t + \|\mathbf{L}^{(n)} - \mathbf{L}^{(n-1)}\|_t \le c_{\Gamma}\int_0^t \|V^{(n-1)} - V^{(n-2)}\|_s ds \end{align*} and \begin{align*}
\|V^{(n)} - V^{(n-1)}\|_t = \|L_1^{(n)} - L_1^{(n-1)}\|_t \le \|\mathbf{L}^{(n)} - \mathbf{L}^{(n-1)}\|_t \le c_{\Gamma}\int_0^t \|V^{(n-1)} - V^{(n-2)}\|_s ds. \end{align*} Letting
$$\Delta_n(t) \doteq \|\mathbf{Z}^{(n)} - \mathbf{Z}^{(n-1)}\|_t + \|\mathbf{L}^{(n)} - \mathbf{L}^{(n-1)}\|_t + \|V^{(n)} - V^{(n-1)}\|_t,$$ we have for $n \ge 2$ and $t \in [0,T]$, $\Delta_n(t) \le c_{\Gamma}\int_0^t \Delta_{n-1} (s) ds$. Now a standard argument shows that, a.s., $(V^{(n)}, \mathbf{Z}^{(n)}, \mathbf{L}^{(n)})$ is a Cauchy sequence in ${\mathcal{C}}([0,T]: {\mathbb{R}} \times {\mathbb{R}}_+^N \times {\mathbb{R}}_+^N)$. Let $(V,\mathbf{Z}, \mathbf{L})$ denote the limit. It is easy to verify that this is a $\bar {\mathcal{F}}_t$-adapted process. Furthermore, sending $n\to \infty$ in \eqref{eq:picit} we see that $(V, \mathbf{Z}, \mathbf{L})$ solve, for $0 \le t \le T$, \begin{equation}\label{eq:picitfin} \begin{aligned} \mathbf{Z}(t) &=\Gamma_2\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V(s) ds + A\mathbf{B}(\cdot)\right)(t),\\
\mathbf{L}(t) &= \Gamma_{1}\left(\mathbf{Z}^0 - \mathbf{e}_1 \int_0^{\cdot} V(s) ds + A\mathbf{B}(\cdot)\right)(t),\\
V(t) &= V^0 + gt - L_1(t), \end{aligned} \end{equation} where $L_1(t)$ is the first coordinate of $\mathbf{L}(t)$. In particular, $(V, \mathbf{Z})$ is a solution of \eqref{eq:solskor}. Since $T>0$ is arbitrary this proves the first part of the theorem.
Now suppose that $(V, \mathbf{Z}, \mathbf{L})$ and $(\tilde V, \tilde \mathbf{Z}, \tilde \mathbf{L})$ are two continuous ${\mathbb{R}} \times {\mathbb{R}}_+^N \times {\mathbb{R}}_+^N$ valued $\bar {\mathcal{F}}_t$-adapted processes that solve \eqref{eq:picitfin}. Then, for $t \in [0, T]$, \begin{align*}
\|\mathbf{Z}- \tilde \mathbf{Z}\|_t + \|\mathbf{L}- \tilde \mathbf{L}\|_t &\le c_{\Gamma} \int_0^t \|V-\tilde V\|_s ds =
c_{\Gamma} \int_0^t \|L_1-\tilde L_1\|_s ds\\
& \le c_{\Gamma} \int_0^t (\|\mathbf{Z}- \tilde \mathbf{Z}\|_s + \|\mathbf{L}- \tilde \mathbf{L}\|_s) ds. \end{align*} Using Gr\"onwall's lemma, it then follows that $\mathbf{Z}(t)= \tilde \mathbf{Z}(t)$ and $\mathbf{L}(t) = \tilde \mathbf{L}(t)$ for all $t \in [0,T]$ a.s. which also says that $V(t) = \tilde V(t)$ for all $t \in [0,T]$ a.s. The result follows.
\qed
\section{A Minorization Estimate}\label{minosec} In this section we will establish a minorization estimate for the transition probability kernel ${\mathbb{P}}^t((v, \mathbf{z}), A)$ introduced in the last section. This estimate will be a key ingredient in the proofs of Theorems \ref{thm:exisuniq} and \ref{thm:geomerg}. The deterministic motion of the bottom (inert) particle when $Z_1>0$ results in very singular behavior of our diffusion process manifested, in particular, by the lack of a density of $(V(t),\mathbf{Z}(t))$ with respect to Lebesgue measure for any $t >0$ when the initial condition satisfies $Z_1(0)>0$. Hence, one cannot use standard techniques for establishing a minorization condition for elliptic (or hypoelliptic) diffusions. We take a pathwise approach here by analyzing a suitable collection of driving Brownian paths to obtain a sub-density of the form described in Theorem \ref{minorization}. This is done by first `removing the drift' by applying Girsanov's Theorem and analyzing the simpler system given by gaps between $N$ ordered Brownian motions and the local time at zero of the bottom particle. This, along with an appropriate control of the Radon-Nikodym derivative, yields the desired result.
Let
\begin{equation}
\varsigma \doteq \frac{1}{128}, \;\;\; \varsigma^* \doteq \varsigma + \left(\frac{1}{63}-\frac{1}{64}\right).
\end{equation} \begin{theorem}\label{minorization}
Let $C = [0, \frac{g}{128}] \times [\frac{g}{2},g]^N$.
There exists $D \in {\mathcal{B}}(\mathbb{R} \times \mathbb{R}_+^N)$ such that
$\lambda(D \cap C) > 0$, and such that for each $(v, \mathbf{z}) \in [0, \frac{g}{128}]\times (0, \infty) \times \mathbb{R}_+^{N-1}$, there is a $K_{(v,\mathbf{z})} \in (0,\infty)$ so that \begin{equation}\label{MLminor}
\inf_{t \in [\varsigma, \varsigma^*]}\mathbb{P}^t((v, \mathbf{z}), S) \geq K_{(v,\mathbf{z})}\lambda(S \cap D) \mbox{ for every } S \in {\mathcal{B}}(\mathbb{R} \times \mathbb{R}_+^N). \end{equation}
Moreover, the map $(v,\mathbf{z}) \mapsto K_{(v,\mathbf{z})}$ is measurable and for any $0\le a_i<b_i<\infty$,
$1\le i \le N$, $a_1>0$, with $\bar A = [0, \frac{g}{128}] \times [a_1, b_1]\times \cdots \times [a_N, b_N]$, \begin{equation}\label{MLuniform}
\bar{K}_{\bar A} \doteq \inf_{(v, \mathbf{z}) \in \bar A}K_{(v,\mathbf{z})} > 0. \end{equation}
\end{theorem}
In proving the above it will be convenient to introduce a probability measure $\tilde {\mathbb{P}}^*_{(v, \mathbf{z})}$ that is mutually absolutely continuous to ${\mathbb{P}}^*_{(v, \mathbf{z})}$ and which is somewhat simpler to analyze. This measure corresponds to the law of the processes $(\mathbf{B}, V,\mathbf{Z})$ given as in \eqref{eq:gapprocb} but with $V$ on the right side of equation for $Z_1$ replaced by the $0$ process. Recall the path space $(\Omega^*, {\mathcal{F}}^*)$ and the coordinate processes $(\mathbf{B}, V, \mathbf{Z})$ given on this space. Let $\{{\mathcal{F}}^*_t\}_{t \geq 0}$ be the filtration generated by these coordinate processes. For $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N$
let $\tilde{\mathbb{P}}^*_{(v, \mathbf{z})}$ be the probability measure on $(\Omega^*, {\mathcal{F}}^*)$ such that under
$\tilde{\mathbb{P}}^*_{(v, \mathbf{z})}$ the following hold: \begin{enumerate}[(i)]
\item $\mathbf{B}$ is the standard $N$-dimensional ${\mathcal{F}}^*_t$-Brownian motion.
\item For each $t \in [0,\infty)$, with $L(t) = \Gamma_1(\mathbf{z} + A\mathbf{B}(\cdot))(t)$,
\begin{align}
\begin{split}\label{drifteqn}
\mathbf{Z}(t) = \mathbf{z} + A\mathbf{B}(t) + RL(t), \;\;
V(t) = v+ gt - L_1(t).
\end{split}
\end{align}
\end{enumerate}
\editc{\subsection{Outline of Proof}
The proof of Theorem \ref{minorization} is organized as follows. In Lemma \ref{Novikov}, we establish a version of Novikov's criterion which allows us to relate ${\mathbb{P}}^*_{(v, \mathbf{z})}$ to $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$ via Girsanov's Theorem. In Corollary \ref{girsanov}, we use the preceding lemma to invoke Girsanov's Theorem and make explicit the relation between the two measures.
We next prove a number of technical results in support of Theorem \ref{minorization}. In Lemma \ref{kild}, we establish a minorization condition for a `killed' version of $\mathbf{Z}$ under law $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$, when $\mathbf{Z}(0)$ lies in a certain compact set $F$. In Lemma \ref{subdens}, we prove the existence of a subdensity for the supremum of Brownian motion over a compact time interval under certain constraints on its infimum and final location. This supremum, in turn, is connected to the local time $L_1$ via the Skorohod map. As under law $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$, existence of a subdensity at a fixed time for $(\mathbf{Z}, V)$ is implied by that for $(\mathbf{Z}, L_1)$ (see \eqref{drifteqn}), the above two lemmas are crucial in proving Theorem \ref{minorization}. Lemmas \ref{H2sub1} and \ref{H2sub2} provide a version of the `support theorem' where a tractable event in terms of the driving Brownian motions is constructed under which the gap process $\mathbf{Z}$ at a prescribed time $\varsigma/4$ lies in $F$ almost surely under $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$.
Lastly, we prove Theorem \ref{minorization}. Using the strong Markov property, we analyze the process pathwise between appropriately chosen stopping times. We first let $Z_1$ hit zero at time $\tau_1$ after which, under the event on the driving Brownian motions described in Lemma \ref{H2sub2}, the local time $L_1$ lies in a given Borel set and the gaps $\mathbf{Z}$ lie in the set $F$ introduced in Lemma \ref{kild} at time $\tau_1 + \varsigma/4$. Theorem \ref{minorization} now follows upon combining this and the minorization condition on the killed gap process obtained in Lemma \ref{kild}, which is used in analyzing the subsequent process path.
\subsection{Proof of Theorem \ref{minorization}}}
In order to relate ${\mathbb{P}}^*_{(v, \mathbf{z})}$ with $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$ we establish the following integrability property which will be used to verify a variation of Novikov's criterion. {\color{black} In the following, $\tilde{\mathbb{E}}^*_{(v,\mathbf{z})}$ denotes the expectation under the probability measure $\tilde{\mathbb{P}}^*_{(v, \mathbf{z})}$. Under $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$, the local times $L_i$, $1\le i \le N$ (and with $L_{N+1}=0$) have the following pathwise representation: \begin{equation}\label{locrep} \begin{aligned}
L_1(t) &= \sup_{s \leq t}(-z_1 + \frac{1}{2}L_2(s) - B_1(s))^+,\\
L_2(t) &= \sup_{s \leq t}(-z_2 + \frac{1}{2}L_3(s) + L_1(s) + B_1(s) - B_2(s))^+,\\
L_i(t) &= \sup_{s \leq t}(-z_i + \frac{1}{2}(L_{i+1}(s) + L_{i-1}(s)) + B_{i-1}(s) - B_i(s))^+, \ \ i=3, \ldots, N. \end{aligned} \end{equation}
}
\begin{lemma}\label{Novikov} For every $c\in (0, \infty)$ and $r\in {\mathbb{N}}$, there is a $m \in {\mathbb{N}}$ such that with $t_k = k/m$, $k=0, 1, \ldots \editc{,} rm-1$,
for each $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N$, $${\color{black} \tilde{\mathbb{E}}^*_{(v,\mathbf{z})}\,e^{\frac{c}{2}\int_{t_{k}}^{t_{k+1}} V(s)^2 ds} < \infty.}$$ \end{lemma}
\begin{proof} Fix $c\in (0,\infty)$ and $r\in {\mathbb{N}}$.
Also, fix $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N$. All equalities and inequalities in the proof are almost sure with respect to the measure $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$.
Note that, for $t\ge 0$, by \eqref{locrep}, \begin{align*}
L_1(t) & \leq \frac{1}{2}L_2(t) + \sup_{s \leq t}(-B_1(s)),\\
L_2(t) & \leq \frac{1}{2}L_3(t) + L_1(t) + \sup_{s \leq t}(B_1(s) - B_2(s)),\\
L_i(t) & \leq \frac{1}{2}(L_{i+1}(t) + L_{i-1}(t)) + \sup_{s \leq t}(B_{i-1}(s) - B_i(s)), \ \ i=3, \ldots, N. \end{align*}
Define \begin{equation}\label{bstar} \begin{aligned}
B^*_1(t) &= \sup_{s \leq t}(-B_1(s)) \\
B^*_i(t) &= \sup_{s \leq t}(B_{i-1}(s) - B_i(s)), \textnormal{ for }\, i = 2, \ldots, N \\
\mathbf{B}^*(t) &= (B_1^*(t),...,B_N^*(t)). \end{aligned} \end{equation}
Recall the matrices $U=I-R$ and $W = (I-U)^{-1}$. Then, it is easy to verify that \begin{equation}\label{Umat}
U =
\begin{pmatrix}
0 & \frac{1}{2} & 0 & 0 & 0 & \cdots & 0 & 0 \\
1 & 0 & \frac{1}{2} & 0 & 0 & \cdots & 0 & 0 \\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & \cdots & \cdots & \cdots & \frac{1}{2} & 0
\end{pmatrix}. \end{equation}
and so from the above inequalities we can write, for $t\ge 0$, $$\mathbf{L}(t) \leq U\mathbf{L}(t) +\mathbf{B}^*(t).$$ In particular, recalling that $W$ can be written as an infinite sum of matrices with nonnegative entries, we have that $$L_1(t) \leq (W\mathbf{B}^*(t))_1.$$ Now fix $m\in {\mathbb{N}}$ which will be chosen suitably below. Define $t_k = k/m$, $k=0, 1, \ldots \editc{,} rm-1$. Then, for any $k$ as above, \begin{multline*}
\int_{t_k}^{t_{k+1}}V(s)^2ds \leq m^{-1}\sup_{s \in [t_k, t_{k+1}]}V(s)^2 \\
= m^{-1}\sup_{s \in [t_k, t_{k+1}]}(v+gs-L_1(s))^2
\le 2m^{-1}(|v|+rg)^2 + 2m^{-1} (L_1(r))^2. \end{multline*} It then follows \begin{align*} \max_{0\le k \le rm-1} e^{\frac{c}{2}\int_{t_k}^{t_{k+1}}V(s)^2ds} & \leq c_1 e^{cm^{-1} (W\mathbf{B}^*(r))^2_1} \end{align*} where
$c_1= e^{cm^{-1}(|v|+rg)^2}$.
The expectation of the right side under ${\mathbb{P}}^*_{(v,\mathbf{z})}$ (which is independent of $(v,\mathbf{z}) \in {\mathbb{R}}\times {\mathbb{R}}_+^N$)
is finite for sufficiently large $m$. The result follows. \end{proof} For $(v, \mathbf{z}) \in\mathbb{R} \times \mathbb{R}_+^N$ and $r \in {\mathbb{N}}$, with an abuse of notation, denote the projection of ${\mathbb{P}}^*_{(v, \mathbf{z})}$ [resp. $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$] on $\Omega^r \doteq {\mathcal{C}}([0,r]: {\mathbb{R}}^N\times {\mathbb{R}} \times {\mathbb{R}}_+^N)$ once more as ${\mathbb{P}}^*_{(v, \mathbf{z})}$ [resp. $\tilde{{\mathbb{P}}}^*_{(v, \mathbf{z})}$]. Denote by ${\mathcal{F}}^r$ the Borel $\sigma$-field on $\Omega^r$. The coordinate processes $\mathbf{B}, V, \mathbf{Z}$ on $(\Omega^r, {\mathcal{F}}^r)$ and the canonical filtration $\{{\mathcal{F}}^r_t\}_{0\le t \le r}$ are defined in an analogous manner. Denote by $\mathbf{e}_1$ the unit vector $(1,0, 0,\ldots, 0)'$ in ${\mathbb{R}}^N$.
\begin{corollary}\label{girsanov}
Fix $r\in {\mathbb{N}}$. Define for $t \in [0,r]$, real measurable maps ${\mathcal{E}}(t)$ on $(\Omega^r, {\mathcal{F}}^r)$ as \begin{equation}\label{expmart}
\mathcal{E}(t) \doteq e^{- \sum_{i=1}^N \int_0^t V(s)(A^{-1} \mathbf{e}_1)_i dB_i(s) - \frac{|A^{-1}\mathbf{e}_1|^2}{2}\int_0^ t V(s)^2ds }. \end{equation} Then for every $(v, \mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N$, $\tilde{{\mathbb{E}}}^*_{(v, \mathbf{z})}[{\mathcal{E}}(r)] =1$ and for every $F \in {\mathcal{B}} ({\mathcal{C}}([0,r]: {\mathbb{R}} \times {\mathbb{R}}_+^N))$ \begin{equation}\label{Girs} {\color{black} \mathbb{P}^*_{(v,\mathbf{z})}((V,\mathbf{Z})\in F) = \tilde{\mathbb{E}}^*_{(v,\mathbf{z})}[\textbf{1}_{\{(V,\mathbf{Z}) \in F\}}\mathcal{E}(r)].} \end{equation}
\end{corollary}
\begin{proof}
Fix $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N$ and $r\in {\mathbb{N}}$. For $t \in [0,r]$, define \begin{equation*}
\tilde{\mathbf{B}}(t) \doteq \mathbf{B}(t) + \int_0^t V(s)A^{-1}\mathbf{e}_1 ds . \end{equation*}
By Lemma \ref{Novikov} with $c = |A^{-1}\mathbf{e}_1|^2$ and (a slight modification of) \cite[Corollary 3.5.14]{KarShre}, it follows that $\{{\mathcal{E}}(t)\}_{0\le t \le r}$ is a martingale with respect to the filtration $\{{\mathcal{F}}^r_t\}_{0 \le t \le r}$ under the probability measure $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$. Hence, from Girsanov's theorem, $\{\tilde{\mathbf{B}}(s)\}_{0\le s \le r}$ is a Brownian motion under the probability measure ${\mathbb{Q}}^*_{(v,\mathbf{z})}$ defined by $d{\mathbb{Q}}^*_{(v,\mathbf{z})} \doteq {\mathcal{E}}(r)d\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$. Also, under the measure ${\mathbb{Q}}^*_{(v,\mathbf{z})}$ we have \begin{align*}
\mathbf{Z}(t) & = \Gamma_2(\mathbf{z} + A\mathbf{B}(\cdot))(t)
= \Gamma_2\left(\mathbf{z} + A\left(\tilde{\mathbf{B}}(\cdot) - \int_0^{\cdot} V(s)A^{-1}\mathbf{e}_1ds\right)\right)(t) \\
& = \Gamma_2\left(\mathbf{z} + A\tilde{\mathbf{B}}(\cdot) - \int_0^{\cdot} V(s)ds\,\mathbf{e}_1\right)(t). \end{align*} By the unique solvability given in Theorem \ref{thm:wellposed} and the definition of $\mathbb{P}^*_{(v,\mathbf{z})}$ it now follows that the law of $(V,\mathbf{Z})$ under ${\mathbb{Q}}^*_{(v,\mathbf{z})}$ is same as that under $\mathbb{P}^*_{(v,\mathbf{z})}$. The result follows. \end{proof}
We next prove several technical estimates that will be needed in the proof of Theorem \ref{minorization}.
\begin{lemma}\label{kild} Let
\begin{equation}\label{Fset}
F \doteq [\frac{g}{16},\frac{g}{4}] \times [\frac{g}{10},2g] \times [\frac{3g}{4},2g]^{N-2}. \end{equation} Let $G \subset (\mathbb{R}_+^N)^o$ be an open and bounded domain with ${\mathcal{C}}^{2}$ boundary such that
$$F \subset F_1 \doteq [\frac{g}{16},g] \times [\frac{g}{10},2g] \times [\frac{3g}{4},2g]^{N-2} \subset G.$$ Let $\sigma_F \doteq \inf_{x \in F,\, y \in \partial G}|A^{-1}(x-y)| $ and choose $ \epsilon > 0$ so that $G_{1} \doteq \{x \in G: \inf_{y \in \partial G}|A^{-1}(x-y)| > \epsilon\}$ satisfies $G\supset G_{1} \supset F_1 \supset F$. Define on $(\Omega^*, {\mathcal{F}}^*)$,
$\tau_G = \inf\{t \geq 0: \mathbf{Z}(t) \notin G\}$.
Also, fix a `cemetery point' $\partial^* \in ({\mathbb{R}}_+^N)^c$ and define the `killed process' $\{\mathbf{Z}^*(t)\}$ by
\begin{align}\label{eq:zstar}
\mathbf{Z}^*(t) \doteq
\begin{cases}
\mathbf{Z}(t) & \mbox{ if } t < \tau_G \\
\partial^*& \mbox{ if } t \geq \tau_G,
\end{cases} \end{align} Then, there is a $c_G \in (0,\infty)$ such that for any $J \in \mathcal{B}(\mathbb{R}_+^N)$, \begin{equation}
\inf_{s \in [\frac{\varsigma}{4},\varsigma^*], (v,\mathbf{z}) \in {\mathbb{R}}\times F}\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J) \geq c_G\lambda(J \cap G_{1}). \end{equation}
\end{lemma}
\begin{proof} Fix $s \in [\frac{\varsigma}{4}, \varsigma^*], (v,\mathbf{z}) \in {\mathbb{R}}\times F$ and $J \in \mathcal{B}(\mathbb{R}_+^N)$ with $\lambda(J\cap G_{1}) > 0$.
Since, under $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$, $\mathbf{Z}(t)= \mathbf{z} + A\mathbf{B}(t)$ until the first time it has hit the boundary of the positive orthant, \begin{align*}
\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J) &= \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J \cap G)\\
&= \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{z} + A \mathbf{B}(s) \in J\cap G, \mathbf{z} + A\mathbf{B}(u) \in G \mbox{ for all } u \le s)\\
&= \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(A^{-1}\mathbf{z} + \mathbf{B}(s) \in A^{-1}(J \cap G), \, A^{-1}\mathbf{z} + \mathbf{B}(u) \in A^{-1}(G),\mbox{ for all } u \le s ). \end{align*} Denote the transition probability density at time $t$ of an $N$-dimensional standard Brownian motion in $A^{-1}G$, started from $x$ and killed at the boundary of $A^{-1}G$, by $p_t(x,\cdot)$. Then from the above identities it follows \begin{align}\label{eq:denseq}
\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J) &= \int_{A^{-1}(G\cap J)}p_s(A^{-1}\mathbf{z}, y)dy. \end{align}
From \cite[Theorem 1.1]{ZhangHeatKer} we have that there exists $T > 0$ and $c_1, c_2 \in (0,\infty)$ such that for all $x, y \in A^{-1}G$: \begin{equation}\label{densltT}
p_t(x,y) \geq \editc{\left(\frac{\rho(x)\rho(y)}{t}\wedge 1\right)}\frac{c_1}{t^{N/2}}e^{-c_2|x-y|^2/t},\,\,\,\,\,t \in [0, T] \end{equation} \begin{equation}\label{densgtT}
p_t(x,y) \geq c_1\rho(x)\rho(y)e^{-c_2 t},\,\,\,\,\,\,t \in (T,\infty), \end{equation}
where $\rho(x) = \inf_{r \in \partial G}|x-A^{-1}r|$.
Note that there is a $\eta \in (0,\infty)$ such that
$$\lambda(A^{-1}(C)) = \eta \lambda(C) \mbox{ for all } C \in {\mathcal{B}}({\mathbb{R}}^N).$$ We now estimate the right side of \eqref{eq:denseq}. First suppose that $s > T$. Then since $\mathbf{z} \in F$ and $s \le 1$, \begin{multline*}
\int_{A^{-1}(G\cap J)}p_s(A^{-1}\mathbf{z}, y)dy
\geq \int_{A^{-1}(G\cap J)}c_1\rho(A^{-1}\mathbf{z})\rho(y)e^{-c_2s}dy \\
\geq c_1\sigma_Fe^{-c_2}\int_{A^{-1}( G_{1}\cap J)}\rho(y)dy
\geq c_1\sigma_Fe^{-c_2}\epsilon\eta\lambda( G_{1}\cap J). \end{multline*} Letting $c_{G,1} = c_1\sigma_Fe^{-c_2}\epsilon\eta$, we have from \eqref{eq:denseq}, when $s> T$, \begin{align}\label{KPminor} \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J) \geq c_{G,1}\lambda(J\cap G_1). \end{align} Now consider the case $s \leq T$. Then, a similar estimate shows, \begin{align*} \int_{A^{-1}(J\cap G)}p_s(A^{-1}\mathbf{z},y)dy
& \geq \int_{A^{-1}(J\cap G)}\editc{\left(\frac{\rho(A^{-1}\mathbf{z})\rho(y)}{s}\wedge 1\right)}\frac{c_1}{s^{N/2}}e^{-c_2|A^{-1}\mathbf{z}-y|^2/s}dy \\
& \geq c_{G,2}\,\lambda(J\cap G_1) \\ \end{align*} where $c_{G,2} = \eta c_1((\epsilon\sigma_F)\wedge 1) e^{-4c_2(\varsigma)^{-1}
\sup_{x,y \in A^{-1}G}|x-y|^2}.$ Thus, once more from \eqref{eq:denseq}, when $s\le T$, \begin{align}\label{KPminorb} \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(\mathbf{Z}^*(s) \in J) \geq c_{G,2}\lambda(J\cap G_1). \end{align} Setting $c_G = c_{G,1}\wedge c_{G,2}$, we have the result on combining \eqref{KPminor} and \eqref{KPminorb}. \end{proof}
For $z_1 \in \mathbb{R}$, let $\mathcal{P}_{z_1}$ denote a probability measure on $\Omega^*$ under which the coordinate process $\{B_1(t)\}$ is a standard Brownian motion starting at $z_1$. We will use similar notation for the corresponding expectation. \begin{lemma}\label{subdens} There exists a $\mathscr{K} \in (0, \infty)$ such that for every $I \in \mathcal{B}(\mathbb{R})$, \begin{equation}
\mathcal{P}_{0} \left(\sup_{0\le u \le \frac{\varsigma}{4}} B_1(u) \in I,\, \inf_{0\le u \leq \frac{\varsigma}{4}} B_1(u) > -\frac{6g}{10},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]\right) \geq \mathscr{K}\lambda(I \cap [0,\frac{g}{63}]). \end{equation} \end{lemma} \begin{proof} Let $I \in \mathcal{B}(\mathbb{R})$. We assume without loss of generality that, $I \subset [0,\frac{g}{63}]$ and $I$ is of the form $I = [\beta_1, \beta_2] \subset \mathbb{R}_+$ for $0 \leq \beta_1 \leq \beta_2$ (the choice of $\mathscr{K}$ will be independent of $\beta_1, \beta_2$). Let $\gamma \doteq \frac{g}{2}(-\frac{1}{8}-\frac{1}{16})$ be the midpoint of $ [-\frac{g}{8},-\frac{g}{16}]$. For a level $c \in {\mathbb{R}}$, let $\tau_{c} \doteq \inf\{t \geq 0: B_1(t) = c\}$. Define $\sigma \doteq \tau_{-6g/10}$ and $\tau^{\beta}_i \doteq \tau_{\beta_i}$ for $i=1,2$. Then \begin{align*}
&\mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in I,\, \inf_{u \leq \frac{\varsigma}{4}} B_1(u) > -\frac{6g}{10},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]) \\
& = \mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in I,\, \sigma > \frac{\varsigma}{4},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]). \end{align*} Using the strong Markov property of the Brownian motion, we obtain, \begin{align*}
&\mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in I,\, \sigma > \frac{\varsigma}{4},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]) \\
& \geq \mathcal{P}_{0}(\tau_1^{\beta} \leq \frac{\varsigma}{8} \wedge \sigma,\, \sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in I,\, \sigma > \frac{\varsigma}{4},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]) \\
& = \mathcal{P}_{0}(\textbf{1}_{\{\tau_1^{\beta} \leq \frac{\varsigma}{8} \wedge \sigma\,\}}\Theta(\tau_1^{\beta})), \end{align*} where, for $t \in [0, \frac{\varsigma}{8}]$, \begin{align} \Theta(t) \doteq \mathcal{P}_{\beta_1}(\sup_{u \leq \frac{\varsigma}{4}-t}(B_1(u)) \leq \beta_2,\, \sigma > \frac{\varsigma}{4}-t,\,B_1(\frac{\varsigma}{4}-t) \in [-\frac{g}{8},-\frac{g}{16}]). \end{align} By another application of the strong Markov property, for $t \in [0,\frac{\varsigma}{8}]$, \begin{align*}
\Theta(t) &\geq \mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \tau_2^{\beta} \wedge \frac{\varsigma}{16},\, B_1(s) \in [-\frac{g}{8},-\frac{g}{16}] \mbox{ for all } s \in [\tau_{\gamma}, \frac{\varsigma}{4}-t]) \\
& = \mathcal{P}_{\beta_1}(\textbf{1}_{\{\tau_{\gamma} \leq \tau_2^{\beta} \wedge \frac{\varsigma}{16}\}}\,\Theta'(\tau_{\gamma}))
\end{align*} where for $u \in [0, \frac{\varsigma}{16}]$, \begin{align}
\Theta'(u) \doteq \mathcal{P}_{\gamma}(B_1(s) \in [-\frac{g}{8},-\frac{g}{16}] \mbox{ for all } s \in [0, \frac{\varsigma}{4}-t-u]). \end{align} Thus letting $$\kappa_1 \doteq \mathcal{P}_{\gamma}( B_1(s) \in [-\frac{g}{8},-\frac{g}{16}] \mbox{ for all } s \in [0, \frac{\varsigma}{4}])$$ we have that, for $t \in [0,\frac{\varsigma}{8}]$, \begin{equation} \Theta(t) \ge \kappa_1 \mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \tau_2^{\beta} \wedge \frac{\varsigma}{16}). \end{equation} Also, by an application of the reflection principle, \begin{align*}
\mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \tau_2^{\beta} \wedge \frac{\varsigma}{16}) & = \mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \frac{\varsigma}{16}) - \mathcal{P}_{\beta_1}( \tau_2^{\beta} < \tau_{\gamma} \leq \frac{\varsigma}{16}) \\
& = \mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \frac{\varsigma}{16}) - \mathcal{P}_{\beta_1 + 2(\beta_2-\beta_1)}( \tau_{\gamma} \leq \frac{\varsigma}{16}). \end{align*} From the definition of the stopping times we see, \begin{align*}
\mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \frac{\varsigma}{16}) & = \mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{16}}(B_1(u)) \geq \beta_1 - \gamma ), \\
\mathcal{P}_{\beta_1 + 2(\beta_2-\beta_1)}( \tau_{\gamma} \leq \frac{\varsigma}{16}) & = \mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{16}}(B_1(u)) \geq \beta_1 + 2(\beta_2-\beta_1) - \gamma ). \end{align*} Using the explicit form for the probability density for the law of the maximum of a Brownian motion, we then obtain, \begin{align*}
\mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \frac{\varsigma}{16}) - \mathcal{P}_{\beta_1 + 2(\beta_2-\beta_1)}(\tau_{\gamma} \leq \frac{\varsigma}{16}) & = \int_{\beta_1-\gamma}^{\beta_1 + 2(\beta_2-\beta_1)-\gamma}\frac{4\sqrt{2}}{\sqrt{\pi\varsigma}}e^{-8z^2/\varsigma} \\
& \geq \frac{8\sqrt{2}}{\sqrt{\pi\varsigma}}\inf_{\beta_1-\gamma\leq z\leq \beta_1+2(\beta_2-\beta_1)-\gamma}e^{-8z^2/\varsigma}(\beta_2-\beta_1). \end{align*} Since $$\beta_1 + 2(\beta_2-\beta_1) - \gamma \leq 2\beta_2 - \gamma \leq \frac{2g}{63} + \frac{3g}{32} \le \frac{g}{4},$$ we have $$\inf_{\beta_1-\gamma\leq z\leq 2(\beta_2-\beta_1)-\gamma}e^{-8z^2/\varsigma} \geq e^{-\frac{g^2}{2\varsigma}}.$$ Thus, for $t \in [0, \varsigma/8]$, $$\Theta(t) \ge \kappa_1 \mathcal{P}_{\beta_1}(\tau_{\gamma} \leq \tau_2^{\beta} \wedge \frac{\varsigma}{16}) \ge \kappa_1 \frac{8\sqrt{2}}{\sqrt{\pi\varsigma}} e^{-\frac{g^2}{2\varsigma}} (\beta_2-\beta_1).$$ Finally, observe that, as $I \subset [0, \frac{g}{63}]$, \begin{align*}
\mathcal{P}_{0}(\tau_1^{\beta} \leq \frac{\varsigma}{8} \wedge \sigma\,) \geq \mathcal{P}_{0}(\sup_{u \leq \frac{\varsigma}{8}}(B_1(s)) > \frac{g}{63},\,\,\inf_{u \leq \frac{\varsigma}{8}}(B_1(s)) > -\frac{6g}{10}) \doteq \kappa_2. \end{align*} The result now follows on setting $ \mathscr{K} = \kappa_1\kappa_2\frac{8\sqrt{2}}{\sqrt{\pi\varsigma}}e^{-\frac{g^2}{2\varsigma}}.$
\end{proof}
For $0\le s \le 1$ and $(z_2, z_3, \ldots , z_N) \in {\mathbb{R}}_+^{N-1}$, define \begin{align*}
\hat{L}_1(s) &= \sup_{u \leq s}(-B_1(u)), \;\;\; \hat{Z}_1(s) = B_1(s) + \hat{L}_1(s) \\
\hat{L}_i(s) &= \sup_{u \leq s}(-z_i+B_{i-1}(u)-B_i(u))^+, \ i = 2, \ldots , N .\\
\end{align*}
\begin{lemma}\label{H2sub1} Let $I \in \mathcal{B}(\mathbb{R})$ be such that $I \subset [0,\frac{g}{63}]$ and $(v, {\mathbf{z}}) \in {\mathbb{R}} \times \{0\} \times [g,\frac{3g}{2}]^{N-1}$. Let $H \in {\mathcal{F}}^*$. Then the following are equivalent:
\begin{enumerate}[(a)]
\item On $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s.,
\begin{inparaenum}[(i)]
\item $L_1(\frac{\varsigma}{4}) \in I$,
\item $L_i(\frac{\varsigma}{4}) \leq \frac{g}{6}, \mbox{ for all } i = 2, \ldots ,N$,
\item $\sup_{0 \leq u\leq \frac{\varsigma}{4}}B_1(u) < \frac{6g}{10}$,
\item $\sup_{0 \leq u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8}, \mbox{ for all } i = 2,\ldots, N$,
\item $Z_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}]$.
\end{inparaenum}
\item On $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s.,
\begin{inparaenum}[(i')]
\item $\hat L_1(\frac{\varsigma}{4}) \in I$,
\item $\hat L_i(\frac{\varsigma}{4}) \leq \frac{g}{6}, \mbox{ for all } i = 2, \ldots,N$,
\item $\sup_{0 \leq u\leq \frac{\varsigma}{4}}B_1(u) < \frac{6g}{10}$,
\item $\sup_{0 \leq u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8}, \mbox{ for all } i = 2,\ldots, N$,
\item $\hat Z_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}]$.
\end{inparaenum}
\end{enumerate}
Furthermore, under these equivalent conditions, on $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s., $L_1(\varsigma/4) = \hat L_1(\varsigma/4)$ and
$L_i(\varsigma/4)=0$ for $i= 2, \ldots, N$. \end{lemma}
\begin{proof} Fix $(v,{\mathbf{z}}) \in {\mathbb{R}} \times \{0\} \times [g,\frac{3g}{2}]^{N-1}$. Noting that $z_i \geq g$ for $i = 2, \ldots \editc{,} N$, we see that, when conditions $(i) - (v)$ hold, for all $u \leq \frac{\varsigma}{4}$, \begin{align*}
-z_i+B_{i-1}(u)-B_i(u)+\frac{1}{2}(L_{i-1}(u)+L_{i+1}(u)) &\leq -g + \frac{g}{4} + \frac{g}{6}\leq 0, \; i = 3, \ldots, N, \\
-z_2+B_{1}(u)-B_2(u)+\frac{1}{2}L_{3}(u)+L_{1}(u) &\leq -g + \frac{6g}{10}+\frac{g}{8} + \frac{g}{12} + \frac{g}{63} \leq 0. \end{align*}
Hence, when conditions $(i) - (v)$ hold on $H$, by \eqref{locrep}, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s., $L_i(\frac{\varsigma}{4}) = \hat{L}_i(\frac{\varsigma}{4}) =0$ for $i = 2, \ldots , N$ which in turn says that $L_1(\frac{\varsigma}{4}) = \hat{L}_1(\frac{\varsigma}{4})$ and
$Z_1(\frac{\varsigma}{4}) = \hat{Z}_1(\frac{\varsigma}{4})$. Thus in this case $(i') - (v')$ hold on $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s.
On the other hand, suppose that $(i') - (v')$ hold on $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s. Consider the stopping times, \begin{align*}
\nu_2 & = \inf\{t \geq 0: -z_2 + B_{1}(t) - B_2(t) +\frac{1}{2}L_{3}(t)+L_{1}(t) \geq 0\} \\
\nu_i & = \inf\{t \geq 0: -z_i + B_{i-1}(t) - B_i(t) +\frac{1}{2}(L_{i+1}(t)+L_{i-1}(t)) \geq 0\}, \;\; i = 3, \ldots , N
\end{align*}
and let $\nu = \min_{2 \leq i \leq N}\nu_i$.
Then, for $s \leq \nu$, $L_i(s) =\hat L_i(s) = 0,\, \mbox{ for all } i = 2, ..., N$ and so $L_1(s) = \hat{L}_1(s)$ and $Z_1(s) = \hat{Z}_1(s)$. Thus, if $\nu \leq \frac{\varsigma}{4}$, \begin{align*}
-z_2 + B_1(\nu) - B_2(\nu) +\frac{1}{2} L_3(\nu) + L_1(\nu)
= -z_2 + B_1(\nu) - B_2(\nu) + \hat{L}_1(\nu)
\leq -g +\frac{6g}{10} + \frac{g}{8} + \frac{g}{63} < 0, \end{align*} and, for $i = 3, \ldots , N$, \begin{align*}
-z_i + B_{i-1}(\nu) - B_i(\nu) + \frac{1}{2}(L_{i+1}(\nu)+L_{i-1}(\nu))
= -z_i + B_{i-1}(\nu) - B_i(\nu)
\leq -g +\frac{g}{8} + \frac{g}{8} < 0. \end{align*}
This contradicts the definition of $\nu$ and consequently we must have that $\nu > \frac{\varsigma}{4}$. Thus $(i) - (v)$ hold on $H$, $\tilde{\mathbb{P}}_{(v,{\mathbf{z}})}$ a.s., and the result follows. \end{proof}
Recall the set $F$ introduced in Lemma \ref{kild}. \begin{lemma}\label{H2sub2} Let $I \in \mathcal{B}(\mathbb{R})$ be such that $I \subset [0,\frac{g}{63}]$ and $(v,{\mathbf{z}}) \in {\mathbb{R}} \times \{0\} \times [g,\frac{3g}{2}]^{N-1}$. Let $H \in {\mathcal{F}}^*$ and suppose the equivalent conditions of Lemma \ref{H2sub1} hold on $H$. Then, on $H$, $\tilde{\mathbb{P}}^*_{(v,{\mathbf{z}})}$ a.s., $\mathbf{Z}(\frac{\varsigma}{4}) \in F$.
\end{lemma}
\begin{proof} Under assumptions of the lemma, on $H$, $\tilde{\mathbb{P}}_{(v,{\mathbf{z}})}$ a.s., \begin{align*}
Z_2(\varsigma/4) &= z_2 + B_2(\varsigma/4) - B_1(\varsigma/4) -\frac{1}{2}L_3(\varsigma/4)+L_2(\varsigma/4)-L_1(\varsigma/4) \\
& = z_2 + B_2(\varsigma/4) - B_1(\varsigma/4) - L_1(\varsigma/4)
\geq g -\frac{g}{8}-\frac{6g}{10}-\frac{g}{63} \geq \frac{g}{10}, \\
Z_i(\varsigma/4) &= z_i + B_i(\varsigma/4) - B_{i-1}(\varsigma/4) -\frac{1}{2}(L_{i-1}(\varsigma/4)+L_{i+1}(\varsigma/4))+L_i(\varsigma/4) \\
& = z_i + B_i(\varsigma/4) - B_{i-1}(\varsigma/4) \geq \frac{3g}{4}, \;\;i = 3, \ldots, N.
\end{align*} From Lemma \ref{H2sub1}, under the assumptions of the current lemma, $L_1(\varsigma/4) = \hat L_1(\varsigma/4)$ and so \begin{align*}
\frac{g}{63} & \geq L_1(\varsigma/4) = \hat{L}_1(\varsigma/4)
= \sup_{u \leq \varsigma/4}(-B_1(u)) \geq -B_1(\varsigma/4). \end{align*} Thus we have the upper bound, \begin{align*}
Z_2(\varsigma/4) &= z_2 + B_2(\varsigma/4) - B_1(\varsigma/4) - L_1(\varsigma/4) \leq \frac{3g}{2} + \frac{g}{8} + \frac{g}{63} \leq 2g, \end{align*} Also, for $i = 3, ..., N$, \begin{align*}
Z_i(\varsigma/4) &= z_i + B_i(\varsigma/4) - B_{i-1}(\varsigma/4) \leq \frac{3g}{2} + \frac{g}{8} + \frac{g}{8} \le 2g. \end{align*} Hence, $(Z_2(\varsigma/4), \ldots, Z_N(\varsigma/4)) \in [\frac{g}{10},2g] \times [\frac{3g}{4},2g]^{N-2}$. Also, under the conditions of the lemma $Z_1 \in [\frac{g}{16},\frac{g}{4}]$. Thus $\mathbf{Z}(\varsigma/4) \in F$ and the lemma is proved. \end{proof}
We can now complete the proof of Theorem \ref{minorization}.\\ \ \\
\textbf{Proof of Theorem \ref{minorization}.} Recall $F, G$ and $G_1$ from Lemma \ref{kild}. We will prove the theorem with $D \doteq D_1\times G_1$ where $D_1 = [0, g/128]$. Let $(v,\mathbf{z}) \in [0,\frac{g}{128}] \times (0,\infty) \times {\mathbb{R}}_+^{N-1}$. All equalities and inequalities of random quantities in the proof are under the measure $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$. Let $r \in [\varsigma,\varsigma^*]$ be given. It suffices to establish the estimate in \eqref{MLminor} for $S \in \mathcal{B}(\mathbb{R} \times \mathbb{R}_+^N)$ of the form $S = I \times J,\,\,\,I \in \mathcal{B}(\mathbb{R}),\,\, J \in \mathcal{B}(\mathbb{R}_+^N)$ with {\color{black} $I \subseteq D_1$ and $J \subseteq G_1$, for a choice of the constant $K_{(v,\mathbf{z})}$ independent of $I,J$.}
For such an $S$, letting
$\tilde{B}(t) \doteq \sum_{i=1}^N (A^{-1})_{i,1}B_i(t)$, by Corollary \ref{girsanov}, \begin{align} \begin{split}\label{minorineq1}
\mathbb{P}^r((v, \mathbf{z}), S) & = \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{\{V(r) \in I, \mathbf{Z}(r) \in J\}}\mathcal{E}(1 ) = \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{\{V(r) \in I, \mathbf{Z}(r) \in J\}}\mathcal{E}(r)\\
& = \,\,\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{\{V(r) \in I, \mathbf{Z}(r) \in J\}}e^{ -\int_0^rV(s)d\tilde{B}(s) - \frac{|A^{-1}\mathbf{e}_1|^2}{2}\int_0^rV(s)^2ds }. \end{split} \end{align}
On the set $\{V(r) \in I\}$, $L_1(r) = gr - V(r) + v \leq gr + v \leq gr + \frac{g}{128},$ so that by monotonicity, $L_1(s) \leq g(r+\frac{1}{128})$ for all $s \leq r$. This implies that, on this set, for $s \leq r,$ $-2g \leq -g(r + \frac{1}{128}) \leq V(s) \leq gr + v \leq 2g,$ i.e., $|V(s)| \leq 2g$. Using this estimate in \eqref{minorineq1} we get \begin{equation}\label{eq:mainest}
\mathbb{P}^r((v, \mathbf{z}), S) \ge e^{-2|A^{-1}\mathbf{e}_1|^2g^2}\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{\{V(r) \in I, \mathbf{Z}(r) \in J\}}e^{ -\int_0^rV(s)d\tilde{B}(s) }. \end{equation} By It\^{o}'s formula, \begin{align*}
-\int_0^rV(s)d\tilde{B}(s) & = \int_0^r\tilde{B}(s)dV(s) - V(r)\tilde{B}(r) \\
& = -\int_0^r\tilde{B}(s)dL_1(s) + g\int_0^r\tilde{B}(s)ds - V(r)\tilde{B}(r). \end{align*} Define the stopping time \begin{equation*}
\tau_1 \doteq \inf\{t \geq 0: Z_1(t) = 0\}, \end{equation*} and let \editc{ \begin{align*}
H \doteq & \left\lbrace \tau_1 \leq \frac{\varsigma}{4}, (Z_2(\tau_1), \ldots , Z_N(\tau_1)) \in [g,\frac{3g}{2}]^{N-1}, L_1(\tau_1 + \frac{\varsigma}{4}) \in v+gr - I,\right.\\
&\left.\quad \mathbf{Z}(\tau_1+\frac{\varsigma}{4}) \in F, \mathbf{Z}(s) > 0, \mbox{ for all } s \in [\tau_1+\frac{\varsigma}{4}, r],
\,\, \mathbf{Z}(r) \in J \right\rbrace. \end{align*} } Note that \begin{equation}\label{hcontain} H \subset \{(V(r),\mathbf{Z}(r)) \in I \times J\}. \end{equation} On $H$ we have, \begin{align*}
- V(r)\tilde{B}(r) & \geq- 2g|\tilde{B}(r)|
\geq - 2g|\tilde{B}(r) - \tilde{B}(\tau_1 + \frac{\varsigma}{4})|
-2g|\tilde{B}(\tau_1+\frac{\varsigma}{4}) - \tilde{B}(\tau_1)| - 2g|\tilde{B}(\tau_1)|. \end{align*} In addition, on $H$, \begin{align*}
g\int_0^r\tilde{B}(s)ds & = g\int_0^{\tau_1}\tilde{B}(s)ds + g\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}}(\tilde{B}(s) - \tilde{B}(\tau_1))ds + \frac{g\varsigma}{4}\tilde{B}(\tau_1) \\
& + g\int_{\tau_1+\frac{\varsigma}{4}}^r(\tilde{B}(s) - \tilde{B}(\tau_1+\frac{\varsigma}{4}))ds + g\tilde{B}(\tau_1+\frac{\varsigma}{4})(r-(\tau_1+\frac{\varsigma}{4})) \\
& = g\int_0^{\tau_1}\tilde{B}(s)ds + g\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}}(\tilde{B}(s) - \tilde{B}(\tau_1))ds + g\tilde{B}(\tau_1)(r-\tau_1) \\
& + g\int_{\tau_1+\frac{\varsigma}{4}}^r(\tilde{B}(s) - \tilde{B}(\tau_1+\frac{\varsigma}{4}))ds + g(\tilde{B}(\tau_1+\frac{\varsigma}{4}) - \tilde{B}(\tau_1))(r-(\tau_1+\frac{\varsigma}{4})). \end{align*} Also, by the definition of $\tau_1$, on $H$,
\begin{align*}
-\int_0^r \tilde{B}(s)dL_1(s) & = -\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}} \tilde{B}(s)dL_1(s) \\
& = -\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}} (\tilde{B}(s)-\tilde{B}(\tau_1))dL_1(s) - \tilde{B}(\tau_1)(L_1(\tau_1+\frac{\varsigma}{4})-L_1(\tau_1)) \\
& {\geq -\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}} (\tilde{B}(s)-\tilde{B}(\tau_1))dL_1(s) - g |\tilde{B}(\tau_1)|.} \end{align*}
Now let \begin{align*}
U_1 & \doteq g\int_{\tau_1+\frac{\varsigma}{4}}^r(\tilde{B}(s) - \tilde{B}(\tau_1+\frac{\varsigma}{4}))ds -2g|\tilde{B}(r)-\tilde{B}(\tau_1 + \frac{\varsigma}{4})|,\\
U_2 & \doteq g(\tilde{B}(\tau_1+\frac{\varsigma}{4}) - \tilde{B}(\tau_1))(r-(\tau_1+\frac{\varsigma}{4})) -2g|\tilde{B}(\tau_1+\frac{\varsigma}{4}) - \tilde{B}(\tau_1)|\\
&\,\,\, - \int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}} (\tilde{B}(s)-\tilde{B}(\tau_1))dL_1(s) + g\int_{\tau_1}^{\tau_1+\frac{\varsigma}{4}}(\tilde{B}(s) - \tilde{B}(\tau_1))ds,\\
U_3 & \doteq {- 3g|\tilde{B}(\tau_1)|} + g\int_0^{\tau_1}\tilde{B}(s)ds + g\tilde{B}(\tau_1)(r-\tau_1). \end{align*} Then, by \eqref{hcontain}, we have the lower bound \begin{align}
\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{\{V(r) \in I, \mathbf{Z}(r) \in J\}}e^{ -\int_0^rV(s)d\tilde{B}(s) } \geq \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_He^{ -\int_0^rV(s)d\tilde{B}(s) }
{ \geq } \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_He^{ U_1 + U_2 + U_3}. \label{eq:406} \end{align} Recall the killed process $\mathbf{Z}^*$ from \eqref{eq:zstar}. Define the sets \begin{align*}
H_1(s) & = \{\mathbf{Z}^*(s) \in J\}, \; 0\le s \le 1, \\
H_2(v) &= \left\lbrace L_1(\frac{\varsigma}{4}) \in gr+v-I, L_i(\frac{\varsigma}{4}) \leq \frac{g}{6} \mbox{ for } 2\le i \le N,\,\, \sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10},\,\,\right. \\
&\left. \,\,\,\,\,\,\,\,\,\,\,\,\,\sup_{u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8}\,\,\, i = 2, ..., N,\,\, Z_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}] \right\rbrace, \; \mbox{ where } v\in [0, g/128]. \\
H_3 & = \left\lbrace \tau_1 \leq \frac{\varsigma}{4}, (Z_2(\tau_1), \ldots, Z_N(\tau_1)) \in [g,\frac{3g}{2}]^{N-1}\right\rbrace. \end{align*} Also, set \begin{align*}
U_1'(t) & \doteq g\int_0^{t}\tilde{B}(s)ds -2g|\tilde{B}(t)|,\; 0 \le t \le 1,\\
U_2' & \doteq -3g|\tilde{B}(\frac{\varsigma}{4})|
- \int_{0}^{\frac{\varsigma}{4}} \tilde{B}(s)dL_1(s) + g\int_{0}^{\frac{\varsigma}{4}}\tilde{B}(s)ds.
\end{align*} Applying the Strong Markov Property at $\tau_1+\frac{\varsigma}{4}$ and then $\tau_1$, we have \begin{multline}
\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_He^{ U_1 + U_2 + U_3} \geq \inf_{(\tilde v,\tilde \mathbf{z}) \in {\mathbb{R}}\times F,\, \frac{\varsigma}{4} \leq s \leq r }\tilde{\mathbb{E}}^*_{(\tilde v,\tilde \mathbf{z})} \textbf{1}_{H_1(s)}e^{U'_1(s)} \\
\,\,\,\,\,\,\,\times \inf_{(\hat v,\hat{\mathbf{z}}) \in {\mathbb{R}}\times [g,\frac{3g}{2}]^{N-1}}\tilde{\mathbb{E}}^*_{(\hat v,(0,\hat\mathbf{z}))} \textbf{1}_{\{L_1(\frac{\varsigma}{4}) \in gr+v-I, \mathbf{Z}(\frac{\varsigma}{4}) \in F\}}e^{U_2'} \;\;\times \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{H_3}e^{U_3}. \label{eq409}
\end{multline}
Note that since by assumption $I \subseteq [0, g/128]$, $r \in [\varsigma,\varsigma^*]$, and $v \in [0, g/128]$, \begin{equation}\label{eq:inclu}
\tilde I \doteq gr+v-I \subseteq [0, g/63]. \end{equation} Thus, using Lemma \ref{H2sub2}, we see \begin{multline*}
\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_He^{ U_1 + U_2 + U_3} \geq \inf_{(\tilde v,\tilde \mathbf{z}) \in {\mathbb{R}}\times F,\, \frac{\varsigma}{4} \leq s \leq r }\tilde{\mathbb{E}}^*_{(\tilde v,\tilde \mathbf{z})} \textbf{1}_{H_1(s)}e^{U'_1(s)} \\
\times \inf_{(\hat v,\hat{\mathbf{z}}) \in {\mathbb{R}}\times [g,\frac{3g}{2}]^{N-1}}\tilde{\mathbb{E}}^*_{(\hat v,(0,\hat{\mathbf{z}}))} \textbf{1}_{H_2(v)}e^{U'_2} \times \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{H_3}e^{U_3}. \end{multline*} For the final term, note that, on $H_3$,
{\color{black} $U_3 \ge -5g\sup_{0\le s \le \varsigma^*} |\tilde B(s)|$.} Now, for $M' > 0$, define
\editc{ \begin{align*}
H_3'(M') &= \left\lbrace\sup_{0 \leq s \leq \varsigma^*}|\tilde B(s)| < M',\,\,\, Z_1(s) > 0 \,\,\,\mbox{ for all } s \leq \frac{\varsigma}{8},\,\, \inf_{\frac{\varsigma}{8} \leq s \leq \frac{\varsigma}{4}}Z_1(s) = 0, \,\,\right. \\
& \left. \,\,\,\,\,\,\,\,\,\, (Z_2(s), \ldots \editc{,} Z_N(s)) \in [g,\frac{3g}{2}]^{N-1} \,\,\,\mbox{ for all } s \in [\frac{\varsigma}{8}, \frac{\varsigma}{4}]\right\rbrace. \end{align*} } {\color{black} Clearly $H_3'(M') \subset H_3$. For any $(v, \mathbf{z}) \in [0,\frac{g}{128}] \times (0,\infty) \times {\mathbb{R}}_+^{N-1}$, one can construct suitable Brownian paths to obtain a measurable choice of $M' = M'(v,\mathbf{z})$ such that $$ \kappa_{(v,\mathbf{z})} \doteq \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(H_3'(M'(v,\mathbf{z})))>0. $$ This definition readily implies the measurability of $(v,\mathbf{z}) \mapsto \kappa_{(v,\mathbf{z})}$ through the measurability of the maps $(v,\mathbf{z}) \mapsto M'(v,\mathbf{z})$ and $(v,\mathbf{z}) \mapsto \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(A^\circ)$ for any $A^\circ \in \mathcal{F}$. } {\color{black} Recall the set $\bar A$ from the statement of Theorem \ref{minorization}. Using continuity properties of the transition kernel of Brownian motion in its starting point, the choice of $M'(v,\mathbf{z})$ can be made such that
\begin{equation}\label{eq:unifonc} \sup_{(v,\mathbf{z}) \in \bar A} M'(v,\mathbf{z}) < \infty, \ \ \ \inf_{(v,\mathbf{z}) \in \bar A} \kappa_{(v,\mathbf{z})} >0. \end{equation}
It now follows that, \begin{equation} \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{H_3} e^{U_3} \ge \tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{H_3'(M'(v,\mathbf{z}))} e^{U_3} \ge e^{-5g M'(v,\mathbf{z})} \kappa_{(v,\mathbf{z})}. \label{eq:411}\end{equation}}
Now consider the term involving $H_1(s)$. Note that, on the set $H_1(s)$, $A\mathbf{B}(u) +z \in G$ for all $u \le s$. Since $G$ is bounded, we have that for some $\kappa_G \in (0, \infty)$, under $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}$, for all $(v,\mathbf{z}) \in {\mathbb{R}}\times F$
$$\sup_{0\le u \le s} |\tilde B(u)| \le \kappa_G \mbox{ on } H_1(s), \mbox{ for all } s \in [\varsigma/4, r] \mbox{ and } r \in [\varsigma,\varsigma^*].$$ Thus, from Lemma \ref{kild}, \begin{equation} \inf_{(v,\mathbf{z}) \in {\mathbb{R}}\times F,\, \frac{\varsigma}{4} \leq s \leq r }\tilde{\mathbb{E}}^*_{(v,\mathbf{z})} \textbf{1}_{H_1(s)}e^{U'_1(s)} \ge e^{-3g\kappa_G} \inf_{(v,\mathbf{z}) \in {\mathbb{R}}\times F,\, \frac{\varsigma}{4} \leq s \leq r }\tilde{\mathbb{P}}^*_{(v,\mathbf{z})} (\mathbf{Z}^*(s)\in J) \ge e^{-3g\kappa_G} c_G \lambda(J\cap G_1). \label{eq:413} \end{equation}
Consider finally the term involving $H_2(v)$. From Lemma \ref{H2sub1} (and recalling \eqref{eq:inclu}) it follows that, on $H_2(v)$, for $ v \in [0, g/128]$ and $0\le s \le \varsigma/4$, $$-B_1(s) \le \sup_{u\le \varsigma/4} (-B_1(u)) = L_1(\varsigma/4) \le g/63.$$
Using this and other properties of the set $H_2(v)$, we see that with $c_A \doteq \frac{6g}{10} \sum_{i=1}^N |(A^{-1})_{i1}|$, on $H_2(v)$, \begin{equation}
\sup_{0\le s \le \varsigma/4}|\tilde B(s)| \le c_A. \end{equation} It then follows that, on $H_2(v)$, \begin{equation}
U'_2 \ge -3g c_A - \frac{g \varsigma}{4} c_A - c_A L_1(\varsigma/4) \ge -4gc_A. \end{equation} Thus, we have \begin{equation}\label{eq:424} \inf_{(\hat v,\hat{\mathbf{z}}) \in {\mathbb{R}}\times [g,\frac{3g}{2}]^{N-1}}\tilde{\mathbb{E}}^*_{(\hat v,(0,\hat{\mathbf{z}}))} \textbf{1}_{H_2(v)}e^{U'_2} \ge e^{-4gc_A} \inf_{(\hat v,\hat{\mathbf{z}}) \in {\mathbb{R}}\times [g,\frac{3g}{2}]^{N-1}}\tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(H_2( v)). \end{equation}
Note that the conditions $\sup_{u\le \varsigma/4} (-B_2(u)) \le -13g/30 + \hat z_2$ and $\sup_{u \leq \frac{\varsigma}{4}} B_1(u) < 6g/10$ imply that $\hat L_2(\varsigma/4) \le g/6$. Thus from Lemma \ref{H2sub1}, and using \eqref{eq:inclu} again, \begin{align}
&\tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(H_2(v))\nonumber\\
&\quad= \tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}\left( \hat{L}_1(\frac{\varsigma}{4}) \in \tilde I, \hat{L}_i(\frac{\varsigma}{4}) \leq \frac{g}{6}, \sup_{u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8} \mbox{ for } 2\le i \le N,\,\, \sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10} , \hat{Z}_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}] \right) \nonumber\\
&\quad\geq \tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}\left(\hat{L}_1(\frac{\varsigma}{4}) \in \tilde I,\, \sup_{u \leq \frac{\varsigma}{4}}(-B_2(u)) \leq -\frac{13g}{30}+\hat z_2, \hat{L}_3(\frac{\varsigma}{4}) \leq \frac{g}{6}, ..., \hat{L}_N(\frac{\varsigma}{4}) \leq \frac{g}{6}\right.,\,\, \,\, \\
& \left. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10}, \, \sup_{u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8} \mbox{ for } 2\le i \le N,\,\,\hat{Z}_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}] \right) \nonumber\\
&\quad= K_{\hat{\mathbf{z}}}\,\tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}\left(\hat{L}_1(\frac{\varsigma}{4}) \in \tilde I,\, \sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10},\,\hat{Z}_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}]\right), \label{eq:416} \end{align} where \begin{align}
K_{\hat{\mathbf{z}}}=\tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}\left(\sup_{u \leq \frac{\varsigma}{4}}(-B_2(u)) \leq -\frac{13g}{30}+\hat z_2, \hat{L}_3(\frac{\varsigma}{4}) \leq \frac{g}{6}, ..., \hat{L}_N(\frac{\varsigma}{4}) \leq \frac{g}{6}, \, \sup_{u \leq \frac{\varsigma}{4}}|B_i(u)| < \frac{g}{8} \mbox{ for } 2\le i \le N\,\,\right), \end{align} and in the last step we have used the independence of $B_1$ and $(B_2, \ldots , B_N)$.
Note that $\hat{Z}_1(\frac{\varsigma}{4}) = B_1(\frac{\varsigma}{4}) + \hat{L}_1(\frac{\varsigma}{4})$, so if $\hat{L}_1(\frac{\varsigma}{4}) \in [0, \frac{g}{63}]$ and $\,B_1(\frac{\varsigma}{4}) \in [\frac{g}{16}, \frac{g}{8}]$, then $\hat{Z}_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}]$. Consequently, \begin{align*}
&\tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(\hat{L}_1(\frac{\varsigma}{4}) \in \tilde I,\, \sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10},\,\hat{Z}_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{4}]) \\
& \geq \tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(\sup_{u \leq \frac{\varsigma}{4}}(-B_1(u)) \in \tilde I,\, \sup_{u \leq \frac{\varsigma}{4}} B_1(u) < \frac{6g}{10},\,B_1(\frac{\varsigma}{4}) \in [\frac{g}{16},\frac{g}{8}])\\ &=\, \tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(\sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in \tilde I,\, \inf_{u \leq \frac{\varsigma}{4}} B_1(u) > -\frac{6g}{10},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]), \end{align*} where in the last line we have used the fact that
$\{B(s)\}_{s \leq \frac{\varsigma}{4}}$ is equal in distribution to $\{-B(s)\}_{s\leq \frac{\varsigma}{4}}$.
Applying Lemma \ref{subdens} we have \begin{multline} \tilde{\mathbb{P}}^*_{(\hat v,(0,\hat{\mathbf{z}}))}(\sup_{u \leq \frac{\varsigma}{4}}(B_1(u)) \in \tilde I,\, \inf_{u \leq \frac{\varsigma}{4}} B_1(u) > -\frac{6g}{10},\,B_1(\frac{\varsigma}{4}) \in [-\frac{g}{8},-\frac{g}{16}]) \ge \mathscr{K}\lambda(\tilde I \cap [0,\frac{g}{63}])\\ = \mathscr{K}\lambda(\tilde I ) = \mathscr{K}\lambda( I ) = \mathscr{K}\lambda( I \cap D_1), \label{eq:425} \end{multline} where for the last equality we have used that $I \subseteq [0, g/128]=D_1$. Thus, letting $$\hat K \doteq \inf_{\hat{\mathbf{z}} \in [g,\frac{3g}{2}]^{N-1}} K_{\hat{\mathbf{z}}},$$ we have on combining estimates in \eqref{eq:mainest}, \eqref{eq:406}, \eqref{eq409}, \eqref{eq:411}, \eqref{eq:413}, \eqref{eq:424}, \eqref{eq:416}, \eqref{eq:425}, \begin{align*}
\mathbb{P}^r((v, \mathbf{z}), S) &\ge e^{-2|A^{-1}\mathbf{e}_1|^2g^2} e^{-5gM'(v,\mathbf{z})} \kappa_{(v,\mathbf{z})} e^{-3g\kappa_G} c_G e^{-4gc_A} \hat K \mathscr{K}\lambda(J\cap G_1)\lambda( I \cap D_1)\\
&= K_{(v,\mathbf{z})}\lambda((I\times J)\cap D)
\end{align*} {\color{black} where
$$
K_{(v,\mathbf{z})} = e^{-2|A^{-1}\mathbf{e}_1|^2g^2}e^{-5gM'(v,\mathbf{z})} \kappa_{(v,\mathbf{z})} e^{-3g\kappa_G} c_G e^{-4gc_A} \hat K \mathscr{K}.$$ This proves the first statement in the theorem. The second statement is immediate from the measurability of $(v, \mathbf{z}) \mapsto \kappa_{(v,\mathbf{z})}$ indicated earlier in the proof and \eqref{eq:unifonc}.} \qedsymbol
\section{Stationary Distribution: Uniqueness}\label{sec:exisuniq} In this section, we establish uniqueness of the stationary distribution by using the minorization estimate in Theorem \ref{minorization} in conjunction with the following lemma. This lemma also plays a crucial role in establishing exponential ergodicity of the system. \begin{lemma}\label{uniqlem} For each $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^{N}$, there exists $r_0 \doteq r_0(v,\mathbf{z}) \in \mathbb{N}$ such that \begin{equation*}
\mathbb{P}^{r_0}((v,\mathbf{z}), R) > 0, \end{equation*} where \begin{equation}\label{Rset} R \doteq (0,\frac{g}{128}) \times (0, \infty) \times \mathbb{R}_+^{N-1}. \end{equation} Furthermore, if $v \ge g/128$, we can take $r_0 = 1$. \end{lemma}
\begin{proof} Let $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^{N}$ be given. In view of Corollary \ref{girsanov} it suffices to show that for some $r_0 \in {\mathbb{N}}$ $$ \tilde{\mathbb{P}}^*_{(v,\mathbf{z})}((V(r_0),\mathbf{Z}(r_0))\in R) >0.$$
Consider first the case where $v < g/128$. Define $$ v_1 \doteq \begin{cases} \frac{g}{256} - v, & v < \frac{g}{256},\\ \frac{1}{2}(\frac{g}{128}-v), & v \in [\frac{g}{256}, \frac{g}{128}). \end{cases} $$ Set $v_2 \doteq v+v_1$. Write $v_1 = g(k+t_1)$ with $k \in {\mathbb{N}}_0$ and $t_1\in [0,1)$. Let $r_0 \doteq k+1$ and $t_2 \doteq (k+t_1)/2$. Let $v_3 \doteq gr_0 - v_1$. Fix $\delta \in (0, v_3)$ such that $[v_2-\delta, v_2+\delta] \subset (0, g/128)$. Consider the set $A_1 \in {\mathcal{F}}^*$ defined as $$A_1 \doteq \{L_1(t_2) \in [v_3-\delta, v_3+\delta], Z_1(t)>0 \mbox{ for all } t \in (t_2, r_0]\}.$$ Then on $A_1$, $\mathbf{Z}(r_0) \in (0, \infty)\times {\mathbb{R}}_+^{N-1}$ and \begin{align*}
V(r_0) &= v+ (k+1)g - L_1(r_0) = v+ (k+1)g - L_1(t_2) \in v+ (k+1)g - [v_3-\delta, v_3+\delta]\\
&= [v+ (k+1)g -v_3-\delta, v+ (k+1)g -v_3+\delta]. \end{align*} Also $$v+ (k+1)g -v_3 = v+ (k+1)g -(g(k+1)-v_1) = v+v_1 = v_2$$ Thus on $A_1$, $V(r_0) \in [v_2-\delta, v_2+ \delta] \subset (0, g/128)$ and consequently $A_1 \subset \{(V(r_0), \mathbf{Z}(r_0)) \in R\}$. It is easily verified that $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(A_1) >0$ which proves the result for the case $v < g/128$.
Now consider the case $v\ge g/128$. Let $v_1 \doteq v+g- g/256$ and fix $\delta \in (0, g/256)$. Consider the set $A_2 \in {\mathcal{F}}^*$ defined as $$A_2 \doteq \{L_1(1/2) \in [v_1-\delta, v_1+\delta], Z_1(t)>0 \mbox{ for all } t \in (1/2, 1]\}.$$ Then, with $r_0=1$, we see, on $A_2$, $\mathbf{Z}(r_0) \in (0, \infty)\times {\mathbb{R}}_+^{N-1}$ and \begin{align*}
V(r_0) &= v+ g - L_1(r_0) = v+ g - L_1(1/2) \in v+ g - [v_1-\delta, v_1+\delta]\\
&= [v+ g -v_1-\delta, v+ g -v_1+\delta] = [g/256-\delta, g/256+\delta] = [v_2-\delta, v_2+\delta]\subset (0, g/128). \end{align*} Thus $A_2 \subset \{(V(r_0), \mathbf{Z}(r_0)) \in R\}$. Once again, it is easily verified that $\tilde{\mathbb{P}}^*_{(v,\mathbf{z})}(A_2) >0$ proving the result for the case $v \ge g/128$ with $r_0=1$.
\end{proof} \begin{theorem}\label{atmostonesd} The Markov family $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ has at most one stationary distribution. \end{theorem} \begin{proof} By Birkhoff's ergodic theorem, if there are multiple stationary distributions, then we can find two that are mutually singular \cite[Chapter 4, Theorem 4.4 and Lemma 4.6]{einsiedlerergodic}. Suppose that $\pi, \pi'$ are mutually singular stationary distributions. Then there is a $A \in {\mathcal{B}}(\mathbb{R} \times \mathbb{R}_+^N)$ such that $\pi(A) = \pi'(A^c) = 0$. Recall the set $D$ from Theorem \ref{minorization}. Since $\lambda(D) > 0$, it follows that either $\lambda(D \cap A) > 0$ or $\lambda(D\cap A^c) > 0$. For specificity, suppose $\lambda(D \cap A) > 0$. We will now argue that $\pi(A)>0$, arriving at a contradiction.
By Theorem \ref{minorization}, with $R$ as in Lemma \ref{uniqlem}, for every $(v, \mathbf{z}) \in R$, there is a $K_{(v,\mathbf{z})}>0$ such that \begin{equation}\label{uniqineq}
\mathbb{P}^{\varsigma}((v,\mathbf{z}), A) \geq K_{(v,\mathbf{z})}\lambda(A\cap D). \end{equation} Define the transition probability kernel ${\mathbb{Q}}$ on ${\mathbb{R}} \times {\mathbb{R}}_+^N$ as \begin{equation*}
{\mathbb{Q}}((\tilde v,\tilde \mathbf{z}), S) \doteq \sum_{i=1}^\infty \frac{1}{2^i}\mathbb{P}^{i+\varsigma}((\tilde v,\tilde \mathbf{z}), S), \; (\tilde v,\tilde \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N, \; S \in {\mathcal{B}}({\mathbb{R}} \times {\mathbb{R}}_+^N). \end{equation*} Since $\pi$ is a stationary distribution, we have \begin{equation}\label{eq:statstep}
\pi(A) = \int_{\mathbb{R} \times \mathbb{R}_+^N} {\mathbb{Q}}((\tilde v,\tilde \mathbf{z}), A) d\pi(\tilde v,\tilde \mathbf{z}). \end{equation} Also, for any $(\tilde v,\tilde \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$ and with $r_0 = r_0(\tilde v,\tilde \mathbf{z}) \in {\mathbb{N}}$ as in Lemma \ref{uniqlem}, \begin{align*}
{\mathbb{Q}}((\tilde v,\tilde \mathbf{z}), A) &\ge 2^{-r_0} \mathbb{P}^{r_0+\varsigma}((\tilde v,\tilde \mathbf{z}), A)
\ge 2^{-r_0} \int_{R} \mathbb{P}^{r_0}((\tilde v,\tilde \mathbf{z}), (d v, d\mathbf{z}))
\mathbb{P}^{\varsigma}(( v,\mathbf{z}), A)\\
&\ge 2^{-r_0} \lambda(A\cap D)\int_{R} \mathbb{P}^{r_0}((\tilde v,\tilde \mathbf{z}), (d v, d\mathbf{z}))
K_{(v,\mathbf{z})} >0. \end{align*} From \eqref{eq:statstep} it now follows that $\pi(A)>0$ which gives a contradiction and proves the result. \end{proof}
\section{Product form of stationary density}\label{sec:prodform}
In this section, we prove Theorem \ref{thm:prodform}. The proof relies on `guessing' a product form for the stationary joint density and showing that it satisfies the partial differential equations (along with appropriate boundary conditions) that characterize such stationary densities. This guess is inspired by \cite{banerjee2019gravitation}, where a product form joint density was obtained for the velocity and gap processes of the system comprising one inert particle and one Brownian particle.
\begin{proof}[Proof of Theorem \ref{thm:prodform}] The generator of the process $(V,\mathbf{Z})$ given by \eqref{eq:gapproc} acts on any $f: \mathbb{R} \times \mathbb{R}_+^N \rightarrow \mathbb{R}$ that is continuously differentiable in $v$ and twice continuously differentiable in $(z_1,\dots, z_N)$, and compactly supported in the interior of $\mathbb{R} \times \mathbb{R}_+^N$, by $$ \mathcal{L} f(v,\mathbf{z}) = \frac{1}{2}\sum_{1 \le i,j\le N} h_{ij}\frac{\partial f}{\partial z_i \partial z_j}(v,\mathbf{z}) +g \frac{\partial f}{\partial v}(v,\mathbf{z}) - v\frac{\partial f}{\partial z_1}(v,\mathbf{z}), \ (v,\mathbf{z}) \in \mathbb{R} \times (0,\infty)^N, $$
where $h_{11} = 1$, $h_{ii} = 2$ for $2 \le i \le N$, $h_{ij} = -1$ for $|i-j| = 1$, and $h_{ij}=0$ otherwise. Moreover, from the pathwise existence and uniqueness (Theorem \ref{thm:wellposed}) it readily follows that the associated submartingale problem \cite[Definition 2.1]{KangRam} for our process is well-posed. For $c_0,c_1,\ldots,c_N, { \phi} >0$, consider the function \begin{equation} \pi(v, \mathbf{z}) = c_{\pi}e^{-c_0(v + \phi)^2}\prod_{i=1}^Ne^{-c_iz_i}, \ (v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N, \label{eq:920} \end{equation} where $c_{\pi}$ is the normalization constant ensuring $\int_{\mathbb{R} \times \mathbb{R}_+^N} \pi(v,\mathbf{z}) dv d\mathbf{z} =1$. Translating the conditions (1)-(3) of \cite[Theorem 3]{KangRam}, $\pi$ is the density of a stationary distribution if $\pi$ satisfies the interior condition \begin{align} \mathcal{L}^* \pi(v,\mathbf{z}) \doteq \frac{1}{2}\sum_{1 \le i,j \le N} h_{ij}\frac{\partial \pi}{\partial z_i \partial z_j}(v,\mathbf{z}) -g \frac{\partial \pi}{\partial v}(v,\mathbf{z}) + \frac{\partial (v \pi)}{\partial z_1}(v,\mathbf{z}) = 0, \ (v,\mathbf{z}) \in \mathbb{R} \times (0,\infty)^N, \label{s1} \end{align} and boundary conditions \begin{align}
2v\pi(v,\mathbf{z})+\frac{\partial \pi}{\partial z_1}(v,\mathbf{z})-\frac{\partial \pi}{\partial z_2}(v,\mathbf{z})+\frac{\partial \pi}{\partial v}(v,\mathbf{z}) &= 0 \ \mbox{ if } z_1=0, \label{s2}\\
-\frac{\partial \pi}{\partial z_{i-1}}(v,\mathbf{z}) +2\frac{\partial \pi}{\partial z_i}(v,\mathbf{z}) - \frac{\partial \pi}{\partial z_{i+1}}(v,\mathbf{z})& = 0 \ \mbox{ if } z_i=0, \mbox{ for some } 2 \le i \le N-1, \label{s3}\\
-\frac{\partial \pi}{\partial z_{N-1}}(v,\mathbf{z}) + 2 \frac{\partial \pi}{\partial z_{N}}(v,\mathbf{z})& = 0 \ \mbox{ if } z_N=0. \label{s4}
\end{align}
We will solve for the constants $c_0,c_1,\ldots,c_N, c_{\pi}$ to obtain a $\pi$ satisfying the above conditions.
The conditions \eqref{s3} and \eqref{s4} applied to \eqref{eq:920} yield that \begin{align} c_{i-1} - 2c_i + c_{i+1} &= 0, \,\,\,\,\,i = 2, ..., N-1,\label{s5}\\
c_{N-1} -2c_N &= 0.
\end{align} From these identities, we obtain that \begin{align*}
c_{N-1} &= 2c_N \\
c_{N-2} &= 2c_{N-1} - c_N = 3c_N. \end{align*} Fix $j \in \{2,...,N-1\}$. Suppose that we have $c_i = (N-i+1)c_N$ for all $ j \leq i \leq N$. Then, from \eqref{s5}, \begin{align*}
c_{j-1} = 2c_j - c_{j+1} &= (2(N-j+1) - (N-j))c_N \\
& = (N-(j-1)+1)c_N. \end{align*} Hence, we have by induction that $c_i = (N-i+1)c_N$ for $i = 1,...,N$. Substituting this into \eqref{s2}, we see that $$2v-c_1+c_2-2c_0(v+\phi)=0, \,\,\, \mbox{ for all } \ v \in \mathbb{R}.$$ Since this holds for all $v \in \mathbb{R}$, we must have $c_0 = 1$, and so \begin{align*}
2\phi = c_2 - c_1 = (N-1)c_N - Nc_N = -c_N \end{align*} and thus $c_N = -2\phi$. Next substituting \eqref{eq:920} in \eqref{s1}, \begin{equation}\label{icond} \frac{1}{2}\sum_{1 \leq i,j \leq N}h_{ij}c_ic_j + 2gc_0(v+\phi)-c_1v = 0, \,\,\,\mbox{ for all } \ v \in \mathbb{R}. \end{equation} Again, since this holds for all $v \in \mathbb{R}$, we must have, $$2g = c_1 = Nc_N.$$ From the above relations, we obtain \begin{align}
\label{statdistconst}
c_0 = 1 ,\;\;
c_i = 2\left(\frac{N-i+1}{N}\right)g,\,\,\, \ i = 1,...,N,\;\;
\phi = -\frac{g}{N}.
\end{align} To show that this choice of constants yields a valid density for some stationary distribution, it remains only to demonstrate that \eqref{s1} holds for all $(v,\mathbf{z}) \in \mathbb{R} \times (0,\infty)^N$, or equivalently, from \eqref{icond} and \eqref{statdistconst}, \begin{align}\label{fi1}
\frac{1}{2}\sum_{1 \leq i,j \leq N}h_{ij}c_ic_j - 2\frac{g^2}{N} = 0. \end{align} To see this, note that \begin{align*}
\frac{1}{2}\sum_{1 \leq i,j \leq N}h_{ij}c_ic_j & = \frac{1}{2}\sum_{1 \leq i,j \leq N}h_{ij}\left(\frac{N-i+1}{N}\right)\left(\frac{N-j+1}{N}\right)4g^2 \\
& = \frac{2g^2}{N^2}\sum_{1 \leq i,j \leq N}h_{ij}(N-i+1)(N-j+1) \\
& = \frac{2g^2}{N^2}\sum_{i=1}^N(N-i+1)\sum_{j=1}^N h_{ij}(N-j+1). \end{align*} From the formulae of $\{h_{ij}\}_{1 \le i,j \le N}$, it follows that $$ \sum_{j=1}^N h_{ij}(N-j+1) = \delta_{1,i},\,\,\,\mbox{ for all } \ i = 1,...,N, $$ where $\delta_{1,i}$ is the Kronecker delta function. Hence, \begin{align*}
\frac{1}{2}\sum_{1 \leq i,j \leq N}h_{ij}c_ic_j
&= \frac{2g^2}{N^2}\sum_{i=1}^N(N-i+1)\delta_{1,i}
= \frac{2g^2}{N}, \end{align*} which proves \eqref{fi1}. We have therefore shown that $\pi$ with constants as in \eqref{statdistconst} is indeed the density for a stationary distribution of the process $(V,\mathbf{Z})$. Uniqueness follows from Theorem \ref{thm:exisuniq}. \end{proof}
\section{Exponential Ergodicity}\label{sec:geomerg}
In this section we will prove Theorem \ref{thm:geomerg}. Since the main source of stability in our system is the local time interactions between particles, standard PDE techniques for constructing Lyapunov functions (\cite{khas},\cite{mattingly2002ergodicity},\cite{eberle2019couplings}) for hypoelliptic diffusions are hard to implement. Furthermore, the singular nature of the dynamics arising from the motion of the inert particle, and the spatial dependence of the drift (which contains a $V$ term), make it challenging to adapt the Lyapunov function constructions for reflected Brownian motions, which proceed via an analysis of the associated noiseless system \cite{dupuis1994lyapunov},\cite{atar2001positive}.
\editc{\subsection{Outline of Approach}} Here, we take a different approach to exponential ergodicity by analyzing excursions of the process between appropriately chosen stopping times (see \eqref{eq:sigmatimes}) as the velocity of the inert particle `toggles' between two levels. Control on the exponential moments of these stopping times is established in Sections \ref{highlev} and \ref{sec:lowlev}. In Section \ref{singdrift}, it is shown that the intersection local time between the bottom two particles creates a `singular' drift that results in a reduction of the function $\bar{Z}_2(t) \doteq \sum_{i=1}^{N-1}iZ_{N-i+1}(t)$ of the gaps when observed at successive stopping times. These estimates are combined in Section \ref{comphit} to show that the distribution of return times of the process to an appropriately chosen compact set $C^*$ has exponentially decaying tails. Finally, in Section \ref{ee}, the exponential moments of this return time are used to construct a suitable Lyapunov function. The minorization estimate in Theorem \ref{minorization} is utilized to show that $C^*$ is a `petite' (or small) set in the language of \cite{DowMeyTwe}. These facts together imply exponential ergodicity using the machinery developed in \cite{DowMeyTwe} (see Theorem 6.2 there). \editc{Proofs of some technical lemmas are deferred to Section \ref{sec:techlem} in order to make it easier to see the overall idea.}
\editc{We note here that the connection between finiteness of exponential moments of certain hitting times, Lyapunov functions and exponential ergodicity is not new \cite{meyn2012markov,DowMeyTwe}. The main work in this section is in establishing that exponential moments of associated hitting times are finite through a detailed pathwise analysis of the process. A general treatment of the above connection in the context of diffusion processes has been undertaken in \cite{cattiaux2013poincare,monmarche20192}, among others. However, typically the diffusion processes are assumed to be hypoelliptic and/or reversible with respect to the stationary measure, neither of which apply to our setting.}
\subsection{An Inequality for the Local Time}\label{loctimeineq} In this section we establish an estimate on local times which will be used several times. Recall the matrix $W$ from Section \ref{sec:mainres} and the process $\mathbf{B}^*$ from \eqref{bstar}. \begin{lemma}\label{locin} For any $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^{N}$ and $t \geq 0$, the following inequality holds, $\mathbb{P}^*_{(v,\mathbf{z})}-\textnormal{a.s.}$, for all $i = 1, 2, ..., N$, \begin{equation}\label{LTineq}
L_i(t) \leq W_{i,1}t\sup_{0 \leq s \leq t}(V(s))^+ + \sum_{j=1}^{N}W_{i,j}B^*_j(t). \end{equation} Moreover, with $\bar Y(t) \doteq \sum_{i=2}^N (N-i+1) B_i^*(t) $, \begin{equation}\label{eql2l1}
L_2(t) \le \frac{2(N-1)}{N}L_1(t) + \frac{2}{N} \bar Y(t), \; t \ge 0. \end{equation}
\end{lemma}
\begin{proof} Let $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^{N}$ be given. All inequalities in the proof are a.s. under $\mathbb{P}^*_{(v,\mathbf{z})}$. For $1\le i \le N$, the local times $L_i$ are given as \begin{equation}\label{eq:loctimes} \begin{aligned}
L_1(t) &= \sup_{s \leq t}(-z_1+\frac{1}{2}L_2(s) + \int_0^sV(u)du - B_1(s))^+ \\
L_2(t) &= \sup_{s \leq t}(-z_2 + \frac{1}{2}L_3(s) + L_1(s) + B_1(s) - B_2(s))^+ \\
L_i(t) &= \sup_{s \leq t}(-z_i + \frac{1}{2}(L_{i-1}(s)+L_{i+1}(s))+B_{i-1}(s)-B_{i}(s))^+, \;\; i = 3, \ldots , N. \end{aligned} \end{equation} Using these identities we see that \begin{equation}\label{eq:lbds} \begin{aligned}
L_1(t)
&\leq \sup_{0 \leq s \leq t}(V(s))^{+}t + \frac{1}{2}L_2(t) + \sup_{s \leq t}(-B_1(s)) \\
L_2(t)
&\leq \frac{1}{2}L_3(t) + L_1(t) + \sup_{s \leq t}(B_1(s) - B_2(s)) \\
L_i(t)
&\leq \frac{1}{2}(L_{i+1}(t) + L_{i-1}(t)) + \sup_{s \leq t}(B_{i-1}(s) - B_i(s)), \;\; i = 3, \ldots , N. \end{aligned} \end{equation} Recalling the matrix $U$ from \eqref{Umat} the above inequalities can be written as $$\mathbf{L}(t) \leq \sup_{0 \leq s \leq t}(V(s))^{+}t\mathbf{e}_1 + U\mathbf{L}(t) + \mathbf{B}^*(t),\,\,\,\,\,\, t \geq 0.$$ Using the fact that $W=(I-U)^{-1}$ is a matrix with nonnegative entries, we have, $$\mathbf{L}(t) \leq \sup_{0 \leq s \leq t}(V(s))^{+}tW\mathbf{e}_1 + W\mathbf{B}^*(t),\,\,\,\,\,\, t \geq 0.$$ This proves the first statement in the lemma. For the second inequality, note that from \eqref{eq:lbds} we have $$\sum_{i=3}^N (N-i+1) (L_i(t) - \frac{1}{2} L_{i+1}(t) - \frac{1}{2} L_{i-1}(t)) + (N-1) (L_2(t) - L_1(t) - \frac{1}{2} L_3(t)) \le \sum_{i=2}^N (N-i+1) B_i^*(t) = \bar Y(t).$$ Simplifying the left side, we see, $$\frac{N}{2} L_2(t) - (N-1)L_1(t) \le \bar Y(t)$$ which proves the second statement. \end{proof}
\subsection{Hitting Time of an Upper Velocity Level}\label{highlev}
For $c \in {\mathbb{R}}$, let $\hat\tau_c \doteq \inf\{t \geq 0 : V(t) = c\}$.
The main result of this section is the following control on exponential moments of this hitting time.
\begin{proposition}\label{sigma1} There exists a $\gamma \in (0,\infty)$ such that $$\sup_{ (v,\mathbf{z}) \in [\frac{g}{2N},2g]\times {\mathbb{R}}_+^N}\,\mathbb{E}^*_{(v,\mathbf{z})}\,e^{\gamma\,\hat\tau_{4g}} < \infty.$$ \end{proposition} Proof of the proposition relies on the three lemmas given below. \editc{Proofs of these lemmas are given in Section \ref{props1}. The proposition is proved after the statements of these lemmas.}
\begin{lemma}\label{sec1prop1} There exists a $\beta \in (0,\infty)$ so that $$\sup_{{\mathbf{z}} \in {\mathbb{R}}_+^{N}}\,\mathbb{E}^*_{(0,{\mathbf{z}})}\,e^{\beta\,\hat\tau_{g/(2N)}} < \infty.$$ \end{lemma}
\begin{lemma}\label{sec1prop2} We have \begin{equation} \inf_{(v, \mathbf{z}) \in [\frac{g}{2N},2g]\times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(\hat \tau_{4g} < \hat\tau_{0}) \doteq p >0. \end{equation} \end{lemma}
\begin{lemma}\label{sec1prop3} There exists $\gamma_1 > 0$ so that $$\sup_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\,\mathbb{E}^*_{(v,\mathbf{z})}\,e^{\gamma_1\,(\hat\tau_{4g}\wedge\hat\tau_{0})} < \infty.$$ \end{lemma}
We now prove the main result of the section.\\
\noindent \textbf{Proof of Proposition \ref{sigma1}.} Define $\tau_{-1} = \tau_0 \doteq 0$ and for $i \in {\mathbb{N}}_0$, define \begin{align*}
\tau_{2i+1} \doteq \inf\{t \geq \tau_{2i}: V(t) = 4g\,\,\, or\,\,\, 0\} ,\;\;
\tau_{2i+2} \doteq \inf\{t \geq \tau_{2i+1}: V(t) = \frac{g}{2N}\,\,\, or\,\,\, 4g\}. \end{align*} Define $$\mathscr{N} = \inf\{k \geq 0: V(\tau_{2k+1})=4g\}.$$ From Lemma \ref{sec1prop2} it follows that \begin{equation} \sup_{(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(\mathscr{N} = k) \le (1-p)^{k-1}, \ \ k \ge 0. \end{equation} Note that \begin{align}
\hat\tau_{4g} &\leq \sum_{i=0}^{\mathscr{N}}(\tau_{2i+1}-\tau_{2i-1})
\leq \sum_{i=1}^{\mathscr{N}+1}(\tau_{2i}-\tau_{2i-2}). \label{eq:535} \end{align} By Lemmas \ref{sec1prop1} and \ref{sec1prop3} there are $c_1, c_2 \in (0,\infty)$ such that $$\sup_{(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(\tau_2 \geq t) \leq c_1e^{-c_2t}.$$ It then follows that, for $0 < \alpha < c_2$, \begin{align*}
\sup_{(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N}\mathbb{E}^*_{(v,\mathbf{z})}\,e^{\alpha\,\tau_2} & \le \int_{-\infty}^{\infty}\alpha e^{\alpha s}\sup_{(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(\tau_2 \geq s)ds \\
&\leq 1 + \alpha c_1\int_0^{\infty}e^{(\alpha - c_2)s}ds
= 1 + \frac{\alpha c_1}{c_2-\alpha}. \end{align*} Choose $\delta \in (0,1)$ such that $(1+\delta)(1-p)\doteq \kappa <1$. Now choose $\alpha>0$ sufficiently small such that $$\sup_{(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N}\mathbb{E}^*_{(v,\mathbf{z})}\,e^{2\alpha\,\tau_2} \le (1+\delta).$$
Applying Cauchy-Schwarz and the Strong Markov property we now see that, for any $(v,\mathbf{z}) \in [\frac{g}{2N}, 2g]\times {\mathbb{R}}_+^N$, \begin{align*}
\mathbb{E}^*_{(v,\mathbf{z})}e^{\alpha\sum_{i=1}^{\mathscr{N}+1}(\tau_{2i}-\tau_{2i-2})} & = \sum_{k=0}^{\infty}\mathbb{E}^*_{(v,\mathbf{z})}e^{\alpha\sum_{i=1}^{k+1}(\tau_{2i}-\tau_{2i-2})}\textbf{1}_{\{\mathscr{N} = k\}} \\
& \leq \sum_{k=0}^{\infty}(\mathbb{E}^*_{(v,\mathbf{z})}e^{2\alpha\sum_{i=1}^{k+1}(\tau_{2i}-\tau_{2i-2})})^{\frac{1}{2}}(\mathbb{P}^*_{(v,\mathbf{z})}(\mathscr{N} = k))^{\frac{1}{2}} \\
& \leq \frac{2}{(1-p)^{1/2}}\sum_{k=0}^{\infty}(1+\delta)^{k/2}(1-p)^{\frac{k}{2}}
\leq \sum_{k=0}^{\infty}\kappa^{k/2} < \infty. \end{align*} The result now follows on combining the above estimate with \eqref{eq:535}.
\qedsymbol
\subsection{Hitting Time of a Lower Velocity Level} \label{sec:lowlev} Let $\sigma_1 \doteq \hat \tau_{4g} = \inf\{t \geq 0: V(t) = 4g\}$ and set
$\sigma_2 \doteq \inf\{t \geq \sigma_1: V(t) = 2g\}$.
The main result of the section is the following. \begin{proposition}\label{sigma2} There is $\gamma_2 > 0$ such that $$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\,\,\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{\gamma_2\sigma_2} < \infty.$$ \end{proposition} The proof relies on the following three lemmas. \editc{Proofs of these lemmas are given in Section \ref{sec:pfsigma2}. The proposition is proved after the statements of the lemmas.} \begin{lemma}\label{sec2prop1} There is a $\gamma_3 > 0$ such that $$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\gamma_3\, Z_1(\sigma_1)^{1/2}} < \infty.$$ \end{lemma}
\begin{lemma}\label{sec2prop2} Define $\tau_0^{Z_1} \doteq \inf\{t \geq 0: Z_1(t) = 0\}$. There is a $\gamma_4 > 0$ and
$\kappa_1, \kappa_2 \in (0,\infty)$ such that for any $z_1 \in {\mathbb{R}}_+$ and $\gamma \in (0, \gamma_4]$
\begin{equation*}
\sup_{\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1}}\mathbb{E}^*_{(4g,z_1,\hat{\mathbf{z}})}e^{\gamma \tau_0^{Z_1}} \leq \kappa_1 e^{\kappa_2\gamma z_1^{1/2}}. \end{equation*} \end{lemma}
\begin{lemma}\label{sec2prop3} There exists a $\gamma_5 > 0$ and $\kappa_1', \kappa_2' \in (0, \infty)$ such that for all
$\gamma \in (0, \gamma_5)$ and
${ v \in [2g, \infty)}$, $$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(v, 0,\hat{\mathbf{z}})}e^{\gamma \,\hat \tau_{2g}} \leq \kappa_1' e^{\kappa_2' \gamma v}.$$ \end{lemma}
We now prove the main result of this section. $\,$ \\
\noindent \textbf{Proof of Proposition \ref{sigma2}.} Let $\alpha \in (0,1)$ be such that \begin{equation}\label{eq:alphcon}
\alpha < \gamma_5, \;\; \alpha (1+\kappa_2'g) \le \gamma_4, \;\; 2\alpha (1+\kappa_2'g)\kappa_2 \le \gamma_3, \;\; 2\alpha(1+\kappa_2'g) \le \gamma, \end{equation} where $\gamma_5$ and $\kappa_2'$ are as in Lemma \ref{sec2prop3}, $\kappa_2$ and $\gamma_4$ are as in Lemma \ref{sec2prop2}, $\gamma_3$ is as in Lemma \ref{sec2prop1} and $\gamma$ is as in Proposition \ref{sigma1}.
Fix $\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}$. Define stopping time \begin{align}\label{eq:etastop}
\eta_1 &\doteq \inf\{t \ge \sigma_1: Z_1(t)=0\}. \end{align} Note that $\sigma_2 = \inf\{t \ge \eta_1: V(t) = 2g\}$.
From the strong Markov property, Lemma \ref{sec2prop3}, and recalling the first condition on $\alpha$ from \eqref{eq:alphcon}, \begin{align*} \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{\alpha\sigma_2}
&= \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} \left[ \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}(e^{\alpha\sigma_2} \mid {\mathcal{F}}^*_{\eta_1})\right] \le \kappa_1'\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\kappa_2'\alpha V(\eta_1) + \alpha \eta_1}\\ & \le \kappa_1'\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\kappa_2'\alpha (4g+ g\eta_1) + \alpha \eta_1} \le \kappa_1'e^{4\kappa_2'\alpha g}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\alpha(1+\kappa_2'g) \eta_1}. \end{align*} Thus, with $d_1 = \kappa_1'e^{4\kappa_2' \alpha g}$ and $d_2= (1+\kappa_2'g)$, \begin{equation} \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{\alpha\sigma_2} \le d_1 \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\alpha d_2\eta_1}. \label{eq:541} \end{equation} Using the strong Markov property again, \begin{align*} \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\alpha d_2\eta_1} & = \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} \left[ \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}(e^{\alpha d_2\eta_1} \mid {\mathcal{F}}^*_{\sigma_1})\right]\\ &\le \kappa_1\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} e^{\kappa_2 \alpha d_2Z_1(\sigma_1)^{1/2} + \alpha d_2\sigma_1}\\ &\le \kappa_1\left(\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} e^{2\kappa_2 \alpha d_2Z_1(\sigma_1)^{1/2} }\right)^{1/2} \left(\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} e^{2 \alpha d_2\sigma_1}\right)^{1/2}, \end{align*} where the second inequality is from Lemma \ref{sec2prop2} and on recalling the second condition on $\alpha$ from \eqref{eq:alphcon}, and the last line is from Cauchy-Schwarz inequality. Next, applying Lemma \ref{sec2prop1}, and recalling the third condition on $\alpha$ from \eqref{eq:alphcon}, $$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})} e^{2\kappa_2 \alpha d_2 Z_1(\sigma_1)^{1/2}} \doteq d_3 <\infty.$$ Finally, applying Proposition \ref{sigma1} and recalling the fourth condition on $\alpha$ from \eqref{eq:alphcon} we have $$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{2 \alpha d_2\sigma_1} \doteq d_4<\infty.$$ Combining the above estimates we have $$ \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{\alpha\sigma_2} \le d_1 \kappa_1 d_3^{1/2} d_4^{1/2} <\infty.$$ The result follows. \qedsymbol
\subsection{A Negative Singular Drift Property}\label{singdrift}
For $\mathbf{z} \in {\mathbb{R}}_+^N$, define $\bar{z}_2 \doteq \sum_{i=2}^N(N-i+1)z_i$. Similarly, for ${\mathbb{R}}_+^N$ valued process $\{\mathbf{Z}(t)\}$, we define
for $t\ge 0$, \begin{equation}\label{barz2} \bar{Z}_2(t) \doteq \sum_{i=2}^N(N-i+1)Z_i(t) = \sum_{i=1}^{N-1}iZ_{N-i+1}(t). \end{equation}
The main result of this section is Propositon \ref{neginexp}, where we will show that if $\bar{z}_2$ is large, then the process $\bar{Z}_2(\cdot)$ decreases in expectation in the course of an appropriately large number of excursions of the velocity process between the levels $2g$ and $4g$ (see \eqref{eq:sigmatimes}).
The following lemma gives a key algebraic representation of $\bar{Z}_2(\cdot)$ in terms of $L_1$, $L_{k+1}$ for $k \in \{1,\ldots,N-1\}$, and additional error terms. If $\bar{z}_2$ is large, then there exists $k \in \{1,\ldots,N-1\}$ such that $Z_{k+ 1}(0) = z_{k+1}$ is large. Thus, it takes a long time for this gap to hit zero. Before this time, the lowest (inert) particle `pushes' the bottom $k+1$ particles up and thereby reduces $\bar{Z}_2(\cdot)$, as captured by the $L_1$ term in the lemma. This `singular' drift through local times results in stability and, in turn, exponential ergodicity, of the system. \begin{lemma}\label{zless} Let $Y^{(1)}_{1}(t) = 0$ and $Y^{(1)}_{k}(t) \doteq \sum_{i=2}^{k}(k-i+1) B_i^*(t), \, t \ge 0, \, 2 \le k \le N$. Also define $M(t) \doteq \sum_{i=2}^NB_i(t) - (N-1)B_1(t), \, t \ge 0$. Then for all $(v,\mathbf{z}) \in {\mathbb{R}}\times {\mathbb{R}}_+^N$, ${\mathbb{P}}^*_{(v,\mathbf{z})}$ a.s., \begin{equation}\label{zlesseq}
\bar{Z}_2(t) - \bar{z}_2 { \,\,\le\,\, } M(t) + \frac{N}{k}Y^{(1)}_k(t) + \frac{N}{2k}L_{k+1}(t) - \frac{(N-k)}{k}L_1(t), \ t \ge 0, \ 1 \le k \le N, \end{equation} { with equality for $k = 1$.} \end{lemma}
\begin{proof} Note that, for $(v,\mathbf{z}) \in {\mathbb{R}}\times {\mathbb{R}}_+^N$, under ${\mathbb{P}}^*_{(v,\mathbf{z})}$, for $t\ge 0$,
\begin{align*}
\bar{Z}_2(t) & = -\frac{(N-1)}{2}L_1(t) + \sum_{i=2}^N(N-i+1)\left[(B_i(t)-B_{i-1}(t)-\frac{1}{2}(L_{i+1}(t)+L_{i-1}(t))+L_i(t) + z_i\right] \\
& = -\frac{(N-1)}{2}L_1(t) + \sum_{i=1}^{N-1}i\left[(B_{N-i+1}(t)-B_{N-i}(t)-\frac{1}{2}(L_{N-i+2}(t)+L_{N-i}(t))+L_{N-i+1}(t) + z_{N-i+1}\right] \\
& = \bar z_2 - \frac{(N-1)}{2}L_1(t) + \sum_{i=1}^{N-1}iB_{N-i+1}(t) - \sum_{i=1}^{N-1}iB_{N-i}(t) +\sum_{i=1}^{N-1}i(L_{N-i+1}(t)-\frac{1}{2}(L_{N-i+2}(t)+L_{N-i}(t))). \\ \end{align*} Also, \begin{align*}
\sum_{i=1}^{N-1}iB_{N-i+1}(t) - \sum_{i=1}^{N-1}iB_{N-i}(t) & = \sum_{i=0}^{N-2}(i+1)B_{N-i}(t) - \sum_{i=1}^{N-1}iB_{N-i}(t)
= \sum_{i=2}^NB_i(t) - (N-1)B_1(t). \end{align*} Moreover, \begin{multline*}
\sum_{i=1}^{N-1}i(L_{N-i+1}(t)-\frac{1}{2}(L_{N-i+2}(t)+L_{N-i}(t)))\\
= -\frac{1}{2}\sum_{i=1}^{N-1}i(L_{N-i+2}(t)-L_{N-i+1}(t))
+\frac{1}{2}\sum_{i=1}^{N-1}i(L_{N-i+1}(t)-L_{N-i}(t))) \\
= -\frac{1}{2}\sum_{i=0}^{N-2}(i+1)(L_{N-i+1}(t)-L_{N-i}(t))
+\frac{1}{2}\sum_{i=1}^{N-1}i(L_{N-i+1}(t)-L_{N-i}(t)) \\
= \frac{(N-1)}{2}(L_2(t)-L_1(t))+\frac{1}{2}L_2(t). \end{multline*} Hence, \begin{align*}
\bar{Z}_2(t) & = \bar{z}_2 - \frac{(N-1)}{2}L_1(t) + \sum_{i=2}^NB_i(t) - (N-1)B_1 (t)+ \frac{(N-1)}{2}(L_2(t)-L_1(t))+\frac{1}{2}L_2(t) \\
& = \bar{z}_2 + \sum_{i=2}^NB_i(t) - (N-1)B_1(t) + \frac{N}{2}L_2(t) - (N-1)L_1(t). \end{align*} Consider the martingale $M(t) \doteq \sum_{i=2}^NB_i(t) - (N-1)B_1(t)$. Then \begin{equation}\label{eq:semmart}
\bar{Z}_2(t) = \bar{z}_2 + M(t) + \frac{N}{2}L_2(t) - (N-1)L_1(t).
\end{equation} This proves \eqref{zlesseq} for $k=1$. Now, we will use this along with some local time inequalities to prove \eqref{zlesseq} for $k \ge 2$. Note that, from \eqref{eq:lbds}, \begin{align}
L_2(t) &\leq L_1(t) + B^*_2(t) +\frac{1}{2}L_3(t) \label{eq:eq147} \\
L_i & \leq B^*_i(t) + \frac{1}{2}(L_{i+1}(t) + L_{i-1}(t)),\,\, i = 3, \ldots , N.\nonumber \end{align} From these identities it follows that, for $k \in \{3, \ldots, N\},$ \begin{align*}
\sum_{i=3}^k (k-i+1)\left[(L_i(t) - \frac{1}{2}(L_{i+1}(t) + L_{i-1}(t)))\right] + &(k-1)(L_2(t)-L_1(t)-\frac{1}{2}L_3(t)) \\
&\leq \sum_{i = 2}^k (k-i+1)B^*_i(t) \doteq Y^{(1)}_k(t). \end{align*} On the other hand, \begin{align*}
\sum_{i=3}^k (k-i+1)(L_i(t) - \frac{1}{2}(L_{i+1}(t) + L_{i-1}(t)))& + (k-1)(L_2(t)-L_1(t)-\frac{1}{2}L_3(t)) \\
&= \frac{k}{2}L_2(t) - (k-1)L_1(t) -\frac{1}{2}L_{k+1}(t). \end{align*} Combining the last two displays and multiplying through by $\frac{2}{k}$, \begin{align}\label{eq:1055}
L_2(t) \leq \frac{2(k-1)}{k}L_1(t) + \frac{1}{k}L_{k+1}(t) + \frac{2}{k}Y^{(1)}_k(t), \, k = 3, \ldots , N. \end{align} The last display holds trivially for $k=1$ and also for $k=2$, as can be seen from \eqref{eq:eq147}.
Hence, for all $1\le k \le N$, using \eqref{eq:semmart}, \begin{align}
\bar{Z}_2(t) - \bar{z}_2 & = M(t) + \frac{N}{2}L_2(t) - (N-1)L_1(t) \nonumber\\
& \leq M(t) + \frac{N}{2}(\frac{2}{k}Y^{(1)}_k(t) + \frac{2(k-1)}{k}L_1(t) + \frac{1}{k}L_{k+1}(t)) - (N-1)L_1(t) \nonumber\\
& = M(t) + \frac{N}{k}Y^{(1)}_k(t) + \frac{N}{2k}L_{k+1}(t) - \frac{(N-k)}{k}L_1(t).\label{eq:507} \end{align} This proves the lemma. \end{proof}
Define the sequence of stopping times $\{\sigma_m\}_{m\ge 0}$ as $\sigma_0=0$, and for $i \ge 0$, \begin{equation}
\label{eq:sigmatimes}
\sigma_{2i+1} \doteq \inf\{t \geq \sigma_{2i}: V(t) = 4g\},\;\;
\sigma_{2i+2} \doteq \inf\{t \geq \sigma_{2i+1}: V(t) = 2g\}.
\end{equation} For $\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}$, abusing notation, write $\sum_{i=2}^N(N-i+1)\hat z_i$ as $\bar z_2$. \begin{proposition}\label{neginexp} There exists $\Delta_0 > 0$ so that, for every $\Delta \geq \Delta_0,$ there is a $l \in {\mathbb{N}}$ such that \begin{equation}
\sup_{\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}: \bar z_2 \geq \Delta}\mathbb{E}_{(2g,0,\hat{\mathbf{z}})}(\bar{Z}_2(\sigma_{2l}) - \bar z_2) < 0. \end{equation} \end{proposition}
\editc{This proposition will be proven using the following two lemmas. Proofs of the lemmas are given in Section \ref{sec:pfneginexp}.}
\begin{lemma}\label{zlesslem1} There exists an $l_0 \in \mathbb{N}$ and $c_2 > 0$, such that for all $1\le k<N$, $\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}$, and $l \ge l_0$, \begin{equation}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(\bar{Z}_2(\sigma_{2l}) - \bar{z}_2) \leq -c_2l +\frac{N}{2k}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l}).\label{eq:1140}\end{equation} \end{lemma}
To complete the proof of Proposition \ref{neginexp} we will estimate, in the next lemma, the second term in the bound \eqref{eq:1140}.
Take $\Delta > 0$ and suppose $\hat\mathbf{z} \in {\mathcal{S}}_{\Delta} \doteq \{\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}: \bar{z}_2 \geq \Delta\}$. Then there is a $ k \in \{1, \ldots, N-1\}$ so that \begin{equation}z_{k+1}\geq \frac{\Delta}{N^2}.\label{eq:1132} \end{equation} We will work with this $k$ in the following.
\begin{lemma}\label{zlesslem2} For $\Delta>0$ and $\hat\mathbf{z} \in {\mathcal{S}}_{\Delta}$, let $k = k(\Delta)$ satisfy \eqref{eq:1132}. There exist positive constants $\Delta_1, D_1, D_2,D_3$ such that for any $\Delta \ge \Delta_1$ and $l \in \mathbb{N}$, \begin{align}\label{lkbd}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l})
\le D_1l^{5/2}\editc{\left(\sqrt{l}e^{-D_2\sqrt{\Delta}/l}+e^{-D_3\Delta^{3/2}}\right)}. \end{align} \end{lemma}
\noindent {\bf Proof of Proposition \ref{neginexp}.} With $l_0$ as in Lemma \ref{zlesslem1} and $\Delta_1$ as in Lemma \ref{zlesslem2}, let $\Delta_0' \doteq \max\{\Delta_1, l_0^4\}$. Setting $l = l(\Delta) =\lfloor\Delta^{1/4}\rfloor + 1$, we use Lemma \ref{zlesslem1} and Lemma \ref{zlesslem2} to obtain positive constants $c_2', D_1', D_2'$ such that for all $\Delta \ge \Delta_0'$ and $\hat\mathbf{z} \in {\mathcal{S}}_{\Delta} $, $$\mathbb{E}^*_{(2g,0,\hat\mathbf{z})}(\bar{Z}_{2}(\sigma_{2l}) - \bar{z}_2) \leq -c_2'\Delta^{1/4}+\frac{ND_1'}{4k}\Delta^{3/4}e^{-D_2'\Delta^{1/4}}.$$ The result now follows upon taking $\Delta_0 \geq \Delta_0'$ such that the above bound is negative for all $\Delta \ge \Delta_0$.
\qed
The next proposition shows that $|\bar{Z}_2(\sigma_2)-\bar{Z}_2(0)|$ has a finite exponential moment.
\begin{proposition}\label{z2expmoment} There exists $\gamma_6 > 0$ so that
$$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\gamma_6 |\bar{Z}_2(\sigma_2) - \bar{z}_2|} < \infty.$$ \end{proposition} \begin{proof} From \eqref{eq:semmart} and \eqref{eql2l1}, under $\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}$, \begin{align*}
|\bar{Z}_2(\sigma_2) -\bar{z}_2| &\le |M(\sigma_2)| + \frac{N}{2}L_2(\sigma_2) + (N-1)L_1(\sigma_2)\\
&\le |M(\sigma_2)| + |\bar Y(\sigma_2)| + 2(N-1)L_1(\sigma_2).
\end{align*} Also note that $$L_1(\sigma_2) =2g+ g\sigma_2 - V(\sigma_2) = 2g+ g\sigma_2-2g = g\sigma_2.$$ Thus $$
|\bar{Z}_2(\sigma_2) -\bar{z}_2| \le 2(N-1)g \sigma_2+ |M(\sigma_2)|+ |\bar Y(\sigma_2)|.$$ Hence, writing $Y^\circ(t) := |M(t)| + |\bar Y(t)|$, for any $\gamma >0$, using Cauchy Schwarz inequalty, \begin{equation}\label{pp}
\mathbb{E}^*_{(2g, 0 ,\hat{\mathbf{z}})} e^{\gamma |\bar Z_2(\sigma_2) - \bar z_2|} \le \left(\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{4(N-1)g\gamma \sigma_2}\right)^{1/2} \left(\mathbb{E}^*_{(2g,0, \hat{\mathbf{z}})} e^{2\gamma |Y^\circ(\sigma_2)|}\right)^{1/2}. \end{equation} Recall $\gamma_2$ from Proposition \ref{sigma2}, and write $D := \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\,\,\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{\gamma_2\sigma_2} < \infty$. Proceeding as in the proof of Lemma \ref{sec2prop1} \editc{(see Section \ref{pf:sec2prop1})}, observe using \eqref{eq:elemconc}, Proposition \ref{sigma2} and Markov's inequality that there exist $c, c'>0$ such that for any $\gamma \in (0,\gamma_2/2)$ and any $\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}$, \begin{align*}
\mathbb{E}^*_{(2g,0, \hat{\mathbf{z}})} e^{2\gamma |Y^\circ(\sigma_2)|} \le \sum_{k=0}^{\infty}\left(\mathbb{E}^*_{(2g,0, \hat{\mathbf{z}})} e^{4\gamma\sup_{0 \le s \le k+1} |Y^\circ(s)|}\right)^{1/2}(\mathbb{P}^*_{(2g,0, \hat{\mathbf{z}})}(\sigma_2 \geq k))^{1/2} \le c\sqrt{D}\sum_{k=0}^{\infty}e^{c'\gamma^2(k+1) - \gamma k}. \end{align*} The proposition follows from the above bound, \eqref{pp} and Proposition \ref{sigma2} upon choosing $\gamma \in (0, \min\{\gamma_2/(4(N-1)g), \gamma_2/2\})$ small enough so that the sum on the right side in the above display is finite. \end{proof}
\subsection{Hitting Time of a Compact Set}\label{comphit} Recall the sequence of stopping times $\{\sigma_j\}_{j\in {\mathbb{N}}_0}$ introduced in \eqref{eq:sigmatimes} and the process $\bar Z_2$ defined in \eqref{barz2}. Also fix $\Delta \ge \Delta_0$ where $\Delta_0$ is as in Proposition \ref{neginexp}. Define \begin{align}\label{eq928}
\Gamma' \doteq \inf\{\sigma_{2k} \geq 0: k \in \mathbb{N}, \bar{Z}_2(\sigma_{2k}) \leq \Delta\}, \;\;
\Gamma \doteq \inf\{t \geq 0: Z_1(t) = 0, \bar{Z}_2(t) \leq \Delta, V(t) = 2g\}. \end{align} Recall that for $\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}$, we write $\sum_{i=2}^N(N-i+1)\hat z_i$ as $\bar z_2$.
\begin{proposition}\label{gammabound}
There exist $\gamma_7 > 0$ and $c, c' > 0$ such that for any $\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}$ with $\bar z_2 \ge \Delta$ and any $t \ge c' \bar z_2$,
$$\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}(\Gamma > t) \leq ce^{-\gamma_7 t}.$$ \end{proposition} \begin{proof} From the definition of the stopping times $\{\sigma_j\}_{j\in {\mathbb{N}}_0}$ it follows that, for each $k \in \mathbb{N}$, $\sigma_{2k}$ is a point of decrease of the velocity process and, consequently, $Z_1(\sigma_{2k})=0$. Indeed, if this is not the case, one can produce an open interval containing $\sigma_{2l}$ where the velocity is strictly increasing, leading to a contradiction to the definition of $\sigma_{2k}$.
Since $Z_1(\sigma_{2k})=0$, it follows that $\Gamma' \ge \Gamma$. It therefore suffices to show the result with $\Gamma$ replaced by $\Gamma'$.
By Proposition \ref{neginexp}, we can obtain $l_* \in \mathbb{N}$ and $\mu_*>0$ such that for any $j \in \mathbb{N}$, \begin{equation}\label{nm1}
\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1} } \left\lbrace \mathbb{E}_{(2g,0,\hat{\mathbf{z}})}(\bar{Z}_2(\sigma_{2jl_*}) - \bar{Z}_2(\sigma_{2(j-1)l_*}) \, \vert \, {\mathcal{F}}^*_{\sigma_{2(j-1)l_*}}) + \mu_*\right\rbrace \textbf{1}_{\{\Gamma' > \sigma_{2(j-1)l^*}\}} \le 0, \end{equation} where ${\mathcal{F}}^*_{\sigma_{2(j-1)l_*}}$ denotes the filtration associated with the process stopped at time $\sigma_{2(j-1)l_*}$.
Write $\mathcal{X}_j := \bar{Z}_2(\sigma_{2jl_*}) - \bar{Z}_2(\sigma_{2(j-1)l_*}), \, \tilde{\mathcal{F}}_{j-1} := {\mathcal{F}}^*_{\sigma_{2(j-1)l_*}}, \, j \in \mathbb{N}$. By Proposition \ref{z2expmoment} and the strong Markov property,
$$\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\sup_{j \in \mathbb{N}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\left(e^{\gamma_6|\mathcal{X}_j|} \, \vert \, \tilde{\mathcal{F}}_{j-1}\right) < \infty.$$
This in particular says that $\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\sup_{j \in \mathbb{N}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\left(|\mathcal{X}_j| \, \Big| \, \tilde{\mathcal{F}}_{j-1}\right) < \infty$. From these observations and Markov's inequality, we conclude that there exist positive constants $c_1, c_2$ such that for any $j \in \mathbb{N}$, $$
\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(|\mathcal{X}_j - \mathbb{E}(\mathcal{X}_j \, \vert \, \tilde{\mathcal{F}}_{j-1})| \ge x \, \vert \, \tilde{\mathcal{F}}_{j-1}\right) \le c_1 e^{-c_2 x}, \ x \ge 0. $$ Hence, by \cite[Theorem 2.2]{wainwright2019high} and its proof, there exist non-negative numbers $(\nu,b)$ such that for any $j \in \mathbb{N}$, $$
\sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\left(e^{\lambda (\mathcal{X}_j - \mathbb{E}(\mathcal{X}_j \, \vert \, \tilde{\mathcal{F}}_{j-1}))} \, \vert \, \tilde{\mathcal{F}}_{j-1} \right) \le e^{\nu^2\lambda^2/2}, \ \text{for all} \ |\lambda| < 1/b. $$ Therefore, by \cite[Theorem 2.3]{wainwright2019high}, there exist positive constants $c_3, c_4$ such that for any $\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}$ with $\bar z_2 \ge \Delta$ and any $t > \left(\frac{2}{\mu_*} + \frac{2}{\Delta}\right)\bar z_2$, \begin{align}\label{5.15.1} \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\Gamma' > \sigma_{2l_*\lfloor t \rfloor}\right) &= \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\sum_{j=1}^{\lfloor t \rfloor}\mathcal{X}_j > \Delta - \bar z_2, \, \Gamma' > \sigma_{2l_*\lfloor t \rfloor}\right)\notag\\ &\le \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\sum_{j=1}^{\lfloor t \rfloor}(\mathcal{X}_j - \mathbb{E}(\mathcal{X}_j \, \vert \, \tilde{\mathcal{F}}_{j-1})) > \Delta - \bar z_2 + \mu_*\lfloor t \rfloor, \, \Gamma' > \sigma_{2l_*\lfloor t \rfloor}\right)\notag\\ &\le \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\sum_{j=1}^{\lfloor t \rfloor}(\mathcal{X}_j - \mathbb{E}(\mathcal{X}_j \, \vert \, \tilde{\mathcal{F}}_{j-1})) > \Delta + \mu_*t/4 \right)\notag\\ & \le c_3 e^{-c_4 t}. \end{align} In the above display the first inequality is from \eqref{nm1} while the second inequality is from the facts that due to our condition on $\bar z_2$ and $t$ we have that $\mu_*(t-1) > 2 \bar z_2$ and $t>2$, which says that $$\mu_*\lfloor t \rfloor - \bar z_2 =\frac{1}{2}\mu_*\lfloor t \rfloor - \bar z_2 + \frac{1}{2}\mu_*\lfloor t \rfloor \ge \frac{1}{2}(\mu_*\lfloor t \rfloor- 2 \bar z_2) + \frac{1}{4}\mu_*t \ge \frac{1}{4}\mu_*t. $$ Now, Proposition \ref{sigma2} and the strong Markov property imply that there exists $A \ge 1$ such that $$ \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\left(e^{\gamma_2 \sigma_{2l_*\lfloor t \rfloor }}\right) \le A^{\lfloor t \rfloor}, \ t > 0. $$ Hence, taking $a>0$ such that $e^{\gamma_2a} >A$, we obtain positive constants $c_3', c_4'$ such that for any $t >0$, \begin{equation}\label{5.15.2} \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\sigma_{2l_*\lfloor t \rfloor } > a t\right) \le c_3' e^{-c_4' t}. \end{equation} Using \eqref{5.15.1} and \eqref{5.15.2}, we conclude that there exist positive constants $c_5, c_6$ such that for any $\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}$ with $\bar z_2 \ge \Delta$ and any $t > \left(\frac{2}{\mu_*} + \frac{2}{\Delta}\right)\bar z_2$, $$ \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}(\Gamma' > at) \le \mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\Gamma' > \sigma_{2l_*\lfloor t \rfloor}\right) + \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}\left(\sigma_{2l_*\lfloor t \rfloor } > a t\right) \le c_5 e^{-c_6 t}. $$ The result follows upon taking $c= c_5, \gamma_7 =c_6$ and $c'=a\left(\frac{2}{\mu_*} + \frac{2}{\Delta}\right)$. \end{proof}
\subsection{Completing the Proof of Exponential Ergodicity}\label{ee}
In this section, we will complete the proof of Theorem \ref{thm:geomerg}. We begin with the following proposition the proof of which will be completed in Section \ref{sec:driftcon}. Fix $\Delta \ge \Delta_0$ where $\Delta_0$ is as in Proposition \ref{neginexp}. Define \begin{equation}\label{eq:cstar}
C^* \doteq \{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N: v=2g,\; z_1= 0, \; \bar z_2\le \Delta\}.\end{equation} Let $\tau_{C^*}(1) \doteq \inf\{t \geq 1: (V(t),\mathbf{Z}(t)) \in C^*\}$. \begin{proposition}\label{driftcondn}$\;$ \begin{enumerate}
\item There exists $\eta > 0$ such that \begin{equation}\label{v0}
\tilde{V}_0(v,\mathbf{z}) \doteq \mathbb{E}^*_{(v,\mathbf{z})}e^{\eta\tau_{C^*}(1)} < \infty, \,\,\,\,\,\, \mbox{ for all } (v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N.
\end{equation}
Furthermore, $$\sup_{(v,\mathbf{z}) \in C^*}\tilde V_0(v,\mathbf{z}) \doteq M <\infty.$$
\item There exists a non-zero measure $\nu$ on ${\mathcal{B}}({\mathbb{R}}\times {\mathbb{R}}_+^N)$ and $r_1 \in (0, \infty)$ such that, for all $(v,\mathbf{z}) \in C^*$,
$$\mathbb{P}^{r_1}((v,\mathbf{z}), A) \geq \nu(A) \mbox{ for all } A \in {\mathcal{B}}({\mathbb{R}}\times {\mathbb{R}}_+^N).$$
\end{enumerate} \end{proposition}
Part (2) of the above proposition shows that, in the terminology of Down, Meyn and Tweedie (cf. \cite[Section 3]{DowMeyTwe}, the set $C^*$
is $\nu-$petite (or small) for the Markov family $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$. Together with part (1) of the proposition, this shows that
the conditions of \cite[Theorem 6.2]{DowMeyTwe} are satisfied and consequently, the function $V_0$ defined as
\begin{equation}\label{lyapunov}
V_0(v,\mathbf{z}) \doteq 1 - \frac{1}{\eta} + \frac{1}{\eta}\tilde{V}_0(v,\mathbf{z}), \; (v, \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^{N}, \end{equation} satisfies the drift condition $({\mathcal{D}}_T)$ in \cite[Section 5]{DowMeyTwe}. We will now like to apply \cite[Theorem 5.2]{DowMeyTwe} to conclude the proof of exponential ergodicity. For this we show in the next two results that the Markov process $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ is irreducible and aperiodic.
Recall the set $D$ from Theorem \ref{minorization}. \begin{proposition}\label{irred}
Define the measure $\psi$ on ${\mathcal{B}}({\mathbb{R}}\times {\mathbb{R}}_+^N)$ as $\psi(A)\doteq \lambda(A\cap D)$, $A \in {\mathcal{B}}({\mathbb{R}}\times {\mathbb{R}}_+^N)$. Then the Markov process $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ is
$\psi$-irreducible. \end{proposition}
\begin{proof} Fix $(v,\mathbf{z}) \in \mathbb{R} \times \mathbb{R}_+^N $. Let $B \in \mathcal{B}(\mathbb{R}\times \mathbb{R}_+^N)$ be such that $\lambda(B\cap D) > 0$. To establish $\psi$-irreducibility it suffices to show \begin{equation}
\mathbb{E}^*_{(v,\mathbf{z})} \int_0^{\infty} \textbf{1}_{\{(V(t),\mathbf{Z}(t)) \in B\}}dt >0. \end{equation} From Theorem \ref{minorization}, for each $t \in [\varsigma, \varsigma^*]$ and $(v', \mathbf{z}') \in R= (0,\frac{g}{128}) \times (0, \infty) \times \mathbb{R}_+^{N-1}$, \begin{equation*}
\mathbb{P}^t((v',\mathbf{z}'), B) \ge K_{(v',\mathbf{z}')}\lambda(B \cap D). \end{equation*} Also, from Lemma \ref{uniqlem}, for any $(v, \mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^{N}$, there exists $r_0 \doteq r_0(v,\mathbf{z}) \in \mathbb{N}$ such that \begin{equation}
\mathbb{P}^{r_0}((v,\mathbf{z}), R) > 0. \label{eq:irredpos} \end{equation} Observe that for $t \in [r_0 + \varsigma, r_0 + \varsigma^*],$ \begin{align*}
\mathbb{P}^t((v,\mathbf{z}), B) & = \int_{\mathbb{R}\times\mathbb{R}_+^{N}} \mathbb{P}^{t-r_0}((v',\mathbf{z}'), B) d\mathbb{P}^{r_0}((v,\mathbf{z}), dv',d\mathbf{z}')
\geq \lambda(B\cap D)\int_R K_{(v',\mathbf{z}')} d\mathbb{P}^{r_0}((v,\mathbf{z}), dv',d\mathbf{z}'). \end{align*} The latter expression is strictly positive in view of \eqref{eq:irredpos}, the positivity of $K_{(v,\mathbf{z})}$ for $(v,\mathbf{z}) \in R$ and our assumption concerning $B$. Finally note that \begin{align*}
\mathbb{E}^*_{(v,\mathbf{z})}\int_0^{\infty}\textbf{1}_{\{(V(t),\mathbf{Z}(t)) \in B\}}dt &= \int_0^{\infty}\mathbb{P}^t ((v,\mathbf{z}), B)dt
\geq \int_{r_0+\varsigma}^{r_0+\varsigma^*}\mathbb{P}^t ((v,\mathbf{z}), B)dt > 0. \end{align*} The result follows. \end{proof}
\begin{proposition}\label{aper}
The Markov process $\{{\mathbb{P}}_{(v,\mathbf{z})}\}_{(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N}$ is aperiodic.
\end{proposition}
\begin{proof} Recall the set $C$ and the constant $\bar{K}_{\bar A}$ from Theorem \ref{minorization} and let $\bar{K} \doteq \bar{K}_C$. Define the measure $\nu$ on ${\mathcal{B}}({\mathbb{R}} \times {\mathbb{R}}_+^N)$ as $\nu(B) \doteq \bar{K}\lambda(B \cap D)$, for $B \in {\mathcal{B}}({\mathbb{R}} \times {\mathbb{R}}_+^N)$. From Theorem \ref{minorization} it follows that the set $C$ in the statement of the theorem is
$\nu$-small. Hence, for aperiodicity, it suffices to show that, for some $t_0>0$ \begin{equation}\label{apersuf}
\mathbb{P}^{t}((v,\mathbf{z}),C) > 0, \,\,\,\,\,\mbox{ for all } t \geq t_0, \mbox{ and } (v,\mathbf{z}) \in C. \end{equation} Since $\lambda(C \cap D) > 0$, we have that \eqref{apersuf} holds for $t \in [\varsigma, \varsigma^*]$ and all $(v,\mathbf{z}) \in C$. Let $\delta = \varsigma^*-\varsigma$, We now claim that, for all $m \in {\mathbb{N}}$, $$\mathbb{P}^t((v,\mathbf{z}),C) > 0, \,\,\,\,\,\mbox{ for all } t \in [m\varsigma, m\varsigma + m\delta] \mbox{ and } (v,\mathbf{z}) \in C.$$ Indeed, clearly the result is true with $m=1$, and if the result is true with $m =k$ then it is also true for $m=k+1$ since any $t\in [(k+1)\varsigma, (k+1)\varsigma + (k+1)\delta]$ can be written as $t_1+t_2$ with $t_1 \in [k\varsigma, k\varsigma + k\delta]$ and $t_2 \in [\varsigma, \varsigma+\delta]$, and $$\mathbb{P}^t((v,\mathbf{z}),C) \ge \int_C P^{t_1}((v,\mathbf{z}), (d\tilde v, d\tilde \mathbf{z})) P^{t_2}((\tilde v,\tilde \mathbf{z}),C) >0 \mbox{ for all } (v,\mathbf{z}) \in C. $$ Now choose $k_0\in {\mathbb{N}}$ such that $k_0\delta \ge \varsigma$. Then $\mathbb{P}^t((v,\mathbf{z}),C) > 0$ for all $(v,\mathbf{z}) \in C$ and $t \in [k\varsigma, (k+1)\varsigma]$ for all $k \ge k_0$. We conclude that $\mathbb{P}^t((v,\mathbf{z}),C) > 0$ for all $(v,\mathbf{z}) \in C$ and for all $t\ge k_0\varsigma$. The result follows. \end{proof}
We can now complete the proof of exponential ergodicity.\\
\noindent {\bf Proof of Theorem \ref{thm:geomerg}.} As noted previously, Proposition \ref{driftcondn} shows that
the conditions of \cite[Theorem 6.2]{DowMeyTwe} are satisfied and consequently, the function $V_0$ defined in
\eqref{lyapunov} satisfies the drift condition $({\mathcal{D}}_T)$ in \cite[Section 5]{DowMeyTwe}. Also from Propositions \ref{irred} and \ref{aper} the Markov process is $\psi$-irreducible and aperiodic. The result is now immediate from \cite[Theorem 5.2]{DowMeyTwe}.
\section{Proofs of Some Results from Section \ref{sec:geomerg}.}
\label{sec:techlem}
In this section we present proofs of some technical results stated without proof in Section \ref{sec:geomerg}.
\subsection{Proofs of Lemmas for Proposition \ref{sigma1}.}\label{props1}
In this section we provide the proofs of Lemmas \ref{sec1prop1}, \ref{sec1prop2}, and \ref{sec1prop3} stated in Section \ref{highlev} that were used in the proof of Proposition \ref{sigma1}.
\subsubsection{Proof of Lemma \ref{sec1prop1}.} \label{sec:pfsec1prop1}
Fix ${\mathbf{z}} \in {\mathbb{R}}_+^{N}$. All inequalities in the proof will be a.s. under $\mathbb{P}^*_{(0,{\mathbf{z}})}$.
Using \eqref{LTineq}, we have that for $t \leq \hat\tau_{g/(2N)}$, $$L_1(t) \leq \sum_{i=1}^NW_{1,i}B_i^*(t) + \frac{gW_{1,1}t}{2N}.$$ It can be verified that \begin{equation}
W_{1,1} = N, \mbox{ and } W_{i,1} = 2N - 2(i - 1), \,\, i = 2,...,N. \end{equation} Using this and since $V(t) = gt - L_1(t)$, it follows that \begin{align*}
V(t) &\geq -\sum_{i=1}^NW_{1,i}B_i^*(t) + g(1 - \frac{W_{1,1}}{2N})t
= -\sum_{i=1}^NW_{1,i}B_i^*(t) + \frac{gt}{2} \doteq Q(t). \end{align*}
Define $\hat \sigma_{g/(2N)} \doteq \inf\{t\ge 0: Q(t) = g/(2N)\}$. Then the above inequality implies that $\hat\sigma_{g/(2N)}\geq \hat\tau_{g/(2N)}$. By a standard concentration bound (see \eqref{eq:elemconc})
it follows that there are $\varrho_1, \varrho_2 \in (0, \infty)$ such that \begin{equation}\label{eq:concbm}
\mathbb{E}^*_{(0,{\mathbf{z}})}e^{\theta\sum_{i=1}^NW_{1,i}B_i^*(s)} \le \varrho_1 e^{\varrho_2\theta^2 s} \mbox{ for all } s \ge 0 \mbox{ and } \theta \in (0, \infty). \end{equation} Then, for an arbitrary $\theta, \beta > 0$, we have \begin{align*}
\mathbb{E}^*_{(0,{\mathbf{z}})}e^{\beta \hat\tau_{g/(2N)}} & = \int_0^\infty \mathbb{P}^*_{(0,{\mathbf{z}})}(\hat\tau_{g/(2N)} > \frac{\ln(s)}{\beta})\,\,ds \\
& \leq \int_0^\infty\mathbb{P}^*_{(0,{\mathbf{z}})}(\hat\sigma_{g/(2N)} > \frac{\ln(s)}{\beta})\,\,ds \\
& \leq \int_0^\infty\mathbb{P}^*_{(0,{\mathbf{z}})}(Q(\frac{\ln(s)}{\beta}) < \frac{g}{2N})\,\,ds \\
& \le 1 + \int_1^\infty\mathbb{P}^*_{(0,{\mathbf{z}})}(\frac{g\ln(s)}{2\beta} < \frac{g}{2N}+\sum_{i=1}^NW_{1,i}B_i^*(\frac{\ln(s)}{\beta}))\,\,ds \\
& \leq 1 + e^{\theta g/(2N)}\int_1^\infty e^{-\theta g\ln(s)/2\beta}\mathbb{E}^*_{(0,{\mathbf{z}})}e^{\theta\sum_{i=1}^NW_{1,i}B_i^*(\frac{\ln(s)}{\beta})}\,ds \\
& \leq 1 + \varrho_1e^{\theta g/(2N)}\int_1^\infty s^{-\theta g/2\beta}s^{\theta^2\varrho_2/\beta}\,ds. \end{align*} Now take $$\theta \doteq \frac{g}{4\varrho_2}, \;\; \beta \doteq \frac{\theta g}{8}.$$ Then $$-\theta g/2\beta + \theta^2\varrho_2/\beta = -2.$$ The result follows.
\qed
\subsubsection{Proof of Lemma \ref{sec1prop2}.} We will first show that \begin{equation}\label{step1bb}
\inf_{(v, \mathbf{z}) \in [\frac{g}{4N},4g]\times [1, \infty) \times {\mathbb{R}}_+^{N-1}}\mathbb{P}^*_{(v,\mathbf{z})}(\hat \tau_{4g} < \hat\tau_{0}) \doteq p_1 >0.\end{equation} Note that, for $t>0$, on the set $\{\hat \tau_{4g} >t\}$, for $(v, \mathbf{z}) \in [\frac{g}{4N},4g]\times{\mathbb{R}}_+^N$, under $\mathbb{P}^*_{(v,\mathbf{z})}$, \begin{align*} L_1(t) &\le \sup_{0\leq s\leq t}(-z_1-B_1(s)+\frac{1}{2}L_2(s)+4gs)^+ \le \sup_{0\leq s\leq t}(-z_1-B_1(s)+4gs)^+ + \frac{1}{2}L_2(t)\\ &\le \sup_{0\leq s\leq t}(-z_1-B_1(s)+4gs)^+ + \frac{(N-1)}{N} L_1(t) + \frac{1}{N} \bar Y(t), \end{align*} where the last inequality uses \eqref{eql2l1}. Thus \begin{equation}\label{eq:1033}
L_1(t) \le N \sup_{0\leq s\leq t}(-z_1-B_1(s)+4gs)^+ + \bar Y(t).\end{equation}
Consider the set $A_1 \in {\mathcal{F}}^*$ defined as $$A_1 \doteq \{ -B_1(s)+ 4gs -1 < 0 \mbox{ for all } s \in [0,8] \mbox{ and } \bar Y(8)< g/8N\}.$$ Note that $$\inf_{(v,\mathbf{z})\in {\mathbb{R}} \times {\mathbb{R}}_+^N} {\mathbb{P}}_{(v,\mathbf{z})}^*(A_1) \doteq p_1'>0.$$ Also, for $(v, \mathbf{z}) \in [\frac{g}{4N},4g]\times [1, \infty) \times {\mathbb{R}}_+^{N-1}$, under ${\mathbb{P}}_{(v,\mathbf{z})}^*$, on $A_1$, $$L_1(8\wedge \hat \tau_{4g}) \le N \sup_{0\leq s\leq 8\wedge \hat \tau_{4g}}(-z_1-B_1(s)+4gs )^+ + \bar Y(8) = \bar Y(8) < g/8N < 4g.$$ So, in particular, $$V(8\wedge \hat \tau_{4g}) = V(\hat \tau_{4g})1_{\{\hat \tau_{4g}\le 8\}} + V(8)1_{\{\hat \tau_{4g}> 8\}} \ge 4g 1_{\{\hat \tau_{4g}\le 8\}} + (8g-4g)1_{\{\hat \tau_{4g}> 8\}} = 4g$$ and consequently $\hat \tau_{4g}\le 8$. Also, under the same conditions, for $s<8$, $$V(s\wedge \hat \tau_{4g}) \ge v- L_1(s\wedge \hat \tau_{4g}) \ge v - L_1(8 \wedge \hat \tau_{4g}) > \frac{g}{4N} - \frac{g}{8N} >0.$$ Thus we have $$p_1 = \inf_{(v, \mathbf{z}) \in [\frac{g}{4N},4g]\times [1, \infty) \times {\mathbb{R}}_+^{N-1}}\mathbb{P}^*_{(v,\mathbf{z})}(\hat \tau_{4g} < \hat\tau_{0}) \ge \inf_{(v,\mathbf{z})\in {\mathbb{R}} \times {\mathbb{R}}_+^N} {\mathbb{P}}_{(v,\mathbf{z})}^*(A_1) = p_1'>0.$$ This proves \eqref{step1bb}.
Let $\nu_1 \doteq \inf\{t \ge 0: Z_1(t)\ge 1\}$. In order to complete the proof, from the strong Markov property, it suffices to show that \begin{equation}\label{eq:p2bd}
\inf_{(v, \mathbf{z}) \in [\frac{g}{2N},2g] \times {\mathbb{R}}_+^N} \mathbb{P}^*_{(v,\mathbf{z})}(\nu_1 \wedge \hat \tau_{4g} < \hat\tau_{g/4N}) \doteq p_2>0. \end{equation} Fix $\delta \in (0,1)$ such that $$2gN\delta + \frac{1}{2} gN\delta^2 \le \frac{g}{16N}.$$ Define $A_2 \in {\mathcal{F}}^*$ as $$A_2 \doteq \{B_1(\delta) \ge 1 + 4gN\delta + gN\delta^2 + \frac{3g}{16N}, \; \bar Y(\delta) + N B_1^*(\delta) \le \frac{g}{16N}\}.$$ It is easy to check that $$\inf_{(v,\mathbf{z})\in [\frac{g}{2N},2g]\times {\mathbb{R}}_+^N} {\mathbb{P}}_{(v,\mathbf{z})}^*(A_2) \doteq p_2'>0.$$ Furthermore, as in \eqref{eq:1033}, for $(v, \mathbf{z}) \in [\frac{g}{2N},2g]\times {\mathbb{R}}_+^N$, under ${\mathbb{P}}_{(v,\mathbf{z})}^*$, on $A_2$, $$ L_1(\delta) \le N B_1^*(\delta) + \bar Y(\delta) + N \int_0^{\delta} V^+(s) ds \le N B_1^*(\delta) + \bar Y(\delta) + 2gN\delta + \frac{1}{2} gN\delta^2.$$ Also, under the same conditions, from \eqref{eql2l1}, $$ L_2(\delta) \le 2 L_1(\delta) + \frac{2}{N} \bar Y(\delta) \le 2N B_1^*(\delta) + 2\bar Y(\delta) + 4gN\delta + gN\delta^2 + \frac{2}{N} \bar Y(\delta).$$ Thus \begin{align*} Z_1(\delta) &= z_1 + B_1(\delta) + L_1(\delta) - \frac{1}{2} L_2(\delta) - \int_0^{\delta} V(s) ds \\ &\ge 1 + 4gN\delta + gN\delta^2 + \frac{3g}{16N} - N B_1^*(\delta) - \bar Y(\delta) - 2gN\delta - \frac{1}{2} gN\delta^2 - \frac{1}{N} \bar Y(\delta) - 2g\delta - \frac{1}{2} g\delta^2\\ &\ge 1. \end{align*} Again, under the same conditions, for $0 \le s \le \delta$, \begin{align*}
V(s) &\ge \frac{g}{2N} - L_1(\delta) \ge \frac{g}{2N}
-N B_1^*(\delta) - \bar Y(\delta) - 2gN\delta - \frac{1}{2} gN\delta^2\\
&\ge \frac{g}{2N} - \frac{g}{16N} - 2gN\delta - \frac{1}{2} gN\delta^2 \ge \frac{g}{2N} - \frac{g}{16N} - \frac{g}{16N} > \frac{g}{4N}. \end{align*} It then follows \begin{align*}
p_2 &= \inf_{(v, \mathbf{z}) \in [\frac{g}{2N},2g] \times {\mathbb{R}}_+^N} \mathbb{P}^*_{(v,\mathbf{z})}(\nu_1 \wedge \hat \tau_{4g} < \hat\tau_{g/4N})\\
&\ge \inf_{(v, \mathbf{z}) \in [\frac{g}{2N},2g] \times {\mathbb{R}}_+^N} \mathbb{P}^*_{(v,\mathbf{z})}(\nu_1 < \hat\tau_{g/4N})\\
&\ge \inf_{(v, \mathbf{z}) \in [\frac{g}{2N},2g] \times {\mathbb{R}}_+^N} \mathbb{P}^*_{(v,\mathbf{z})}(A_2) = p_2' >0. \end{align*} This proves \eqref{eq:p2bd} and completes the proof of the lemma.
\qed
\subsubsection{Proof of Lemma \ref{sec1prop3}.} By the strong Markov property, it suffices to show that for some $m \in {\mathbb{N}}$ \begin{equation}\label{eq:mpos}
\inf_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(\hat\tau_{4g}\wedge\hat\tau_{0} \le m) >0. \end{equation} We will prove \eqref{eq:mpos} with $m=5$. We consider two cases:\\ {\bf Case 1:} $z_1\ge 1$. Define $A_1 \in {\mathcal{F}}^*$ as $$A_1 \doteq \{-B_1(s) +4gs -1 \le 0 \mbox{ for all } 0 \le s \le 5, \; \bar Y(5) < g\}.$$ It is easily seen that $$\inf_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(A_1) \doteq \kappa_1 >0.$$ From \eqref{eql2l1} and \eqref{eq:1033} it follows that, for $(v,\mathbf{z}) \in [0,4g] { \times [1,\infty) \times {\mathbb{R}}_+^{N-1}}$, under $\mathbb{P}^*_{(v,\mathbf{z})}$, on $A_1 \cap \{\hat\tau_{4g}\wedge\hat\tau_{0} > 5\}$, $$L_1(5) \le N \sup_{s\le 5} (-1+4gs - B_1(s))^+ + \bar Y(5) < g,$$ and consequently $$V(5) \ge 5g- L_1(5) > 5g-g = 4g.$$ This says that $A_1 \cap \{\hat\tau_{4g}\wedge\hat\tau_{0} > 5\}$ is $\mathbb{P}^*_{(v,\mathbf{z})}$ trivial and so $$\inf_{(v,\mathbf{z}) \in [0,4g] { \times [1,\infty) \times {\mathbb{R}}_+^{N-1}}}\mathbb{P}^*_{(v,\mathbf{z})}(\hat\tau_{4g}\wedge\hat\tau_{0} \le 5) \ge \inf_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(A_1) = \kappa_1 >0.$$ This proves \eqref{eq:mpos} when $z_1 \ge 1$.\\
\noindent{\bf Case 2:} $z_1<1$. Define $A_2 \in {\mathcal{F}}^*$ as $$A_2 \doteq \{B_1(5)<-1-9g\}.$$ Clearly $$\inf_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(A_2) \doteq \kappa_2 >0.$$ Also, for $(v,\mathbf{z}) \in [0,4g] { \times [0,1) \times {\mathbb{R}}_+^{N-1}}$, under $\mathbb{P}^*_{(v,\mathbf{z})}$, on $A_2 \cap \{\hat\tau_{4g}\wedge\hat\tau_{0} > 5\}$, $$ L_1(5) = \sup_{0\le s \le 5} (-z_1 + \frac{1}{2} L_2(s) + \int_0^s V(u) du - B_1(s))^+\ge \sup_{0\le s \le 5}(-1 - B_1(s))^+ > 9g$$ and consequently $$V(5) \le 4g+ 5g - L_1(5) <0.$$ This shows that $A_2 \cap \{\hat\tau_{4g}\wedge\hat\tau_{0} > 5\}$ is $\mathbb{P}^*_{(v,\mathbf{z})}$ trivial and so
$$\inf_{(v,\mathbf{z}) \in [0,4g] { \times [0,1) \times {\mathbb{R}}_+^{N-1}}}\mathbb{P}^*_{(v,\mathbf{z})}(\hat\tau_{4g}\wedge\hat\tau_{0} \le 5 ) \ge \inf_{(v,\mathbf{z}) \in [0,4g] \times {\mathbb{R}}_+^N}\mathbb{P}^*_{(v,\mathbf{z})}(A_2) = \kappa_2 >0.$$ This completes the proof of \eqref{eq:mpos} when $z_1 < 1$. The result follows.
\qed
\subsection{Proofs of Lemmas for Proposition \ref{sigma2}.} \label{sec:pfsigma2}
In this section we provide the proofs of Lemmas \ref{sec2prop1}, \ref{sec2prop2}, and \ref{sec2prop3} stated in Section \ref{sec:lowlev} that were used in the proof of Proposition \ref{sigma2}.
\subsubsection{Proof of Lemma \ref{sec2prop1}.}\label{pf:sec2prop1} Fix $\hat{\mathbf{z}} \in \mathbb{R}^{N-1}$. All inequalities in this proof are $\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}$-a.s. Observe that $4g = V(\sigma_1) = g\sigma_1 - L_1(\sigma_1) + 2g$, so that $L_1(\sigma_1) = g\sigma_1 - 2g$. Then \begin{align*}
Z_1(\sigma_1) &= B_1(\sigma_1) - \frac{1}{2}L_2(\sigma_1) + L_1(\sigma_1) - \int_0^{\sigma_1}V(s)ds \\
&\leq \sup_{0\leq s \leq \sigma_1}(B_1(s)) + g\sigma_1 - \frac{g\sigma_1^2}{2} - 2g\sigma_1 + \int_0^{\sigma_1}L_1(s)ds \\
&\leq \sup_{0\leq s \leq \sigma_1}(B_1(s)) + L_1(\sigma_1)\sigma_1 \\
&\leq \sup_{0\leq s \leq \sigma_1}(B_1(s)) + g \sigma_1^2. \end{align*} Thus, for $\beta > 0$, \begin{align}
e^{\beta(Z_1(\sigma_1))^{1/2}} &\leq e^{\beta(\sup_{0\leq s \leq \sigma_1}(B_1(s)) + g \sigma_1^2)^{1/2}} \\
&\leq e^{\beta(\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2} + \beta \sqrt{g}\sigma_1}
\leq \frac{1}{2}e^{2\beta((\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2}} + \frac{1}{2}e^{2\beta\sqrt{g}\sigma_1},
\label{eq:118} \end{align} where in the final step we use Young's inequality. We now estimate each of the terms in \eqref{eq:118}. We begin by recalling that from Proposition \ref{sigma1}, we can find $\beta_0 \in (0, 1/2)$ such that \begin{equation}
\sup_{\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1} }\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{\beta_0\sigma_1} \doteq c(\beta_0)<\infty. \end{equation} Hence, taking $\beta \in (0, \beta_0/(2\sqrt{g})]$, the second term in \eqref{eq:118} is bounded as \begin{equation} \sup_{\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1} }\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{2\beta\sqrt{g}\sigma_1} \le c(\beta_0). \end{equation} With $\beta \in (0, \beta_0]$ for the first term in \eqref{eq:118}, we have, \begin{align}
\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{2\beta(\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2}}
&\le e^{2\beta} + \mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{2\beta\sup_{0\leq s \leq \sigma_1}B_1(s)}\textbf{1}_{\{\sup_{0\leq s \leq \sigma_1}B_1(s) > 1\}}\label{eq:eq145} \\
& \leq e^{2\beta} + \sum_{k=0}^{\infty}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{2\beta\sup_{0\leq s \leq \sigma_1}B_1(s)}\textbf{1}_{\{k \leq \sigma_1 < k+1\}}\nonumber \\
& \leq e^{2\beta} + \sum_{k=0}^{\infty}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{2\beta\sup_{0\leq s \leq k+1}B_1(s)}\textbf{1}_{\{k \leq \sigma_1 < k+1\}}.\nonumber \end{align} Using Cauchy-Schwarz inequality, \begin{align}
\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}e^{2\beta(\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2}}
& \leq e^{2\beta} + \sum_{k=0}^{\infty}(\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{4\beta\sup_{0\leq s \leq k+1}B_1(s)})^{1/2}(\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}(\sigma_1 \geq k))^{1/2} \nonumber\\
& \leq e^{2\beta} + \varrho_1\sum_{k=0}^{\infty}e^{8\beta^2\varrho_2(k+1)}(\mathbb{P}^*_{(2g,0,\hat{\mathbf{z}})}(\sigma_1 \geq k))^{1/2} \nonumber\\
& \leq e^{2\beta} + \varrho_1 c(\beta_0)^{1/2} e^{8\beta^2\varrho_2} \sum_{k=0}^{\infty}e^{8\beta^2\varrho_2k -\frac{\beta k}{2}} \doteq c_1(\beta) <\infty, \label{eq:907} \end{align} where the finiteness follows on choosing $\beta \in (0, \beta_1]$ for sufficiently small $\beta_1 \in (0, \beta_0]$. The second line above follows from a standard concentration inequality (see \eqref{eq:elemconc}) and the last line from Markov's inequality. Thus for any $\beta \in (0,\beta_1]$, \begin{equation*}
\sup_{\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1}}\mathbb{E}^*_{(2g,0,\hat{\mathbf{z}})}\,e^{2\beta(\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2}}
\doteq c_1(\beta) < \infty. \end{equation*} The result now follows on setting $\gamma_3 = \min\{\beta_0/(2\sqrt{g}),\beta_1\}$.
\qed \subsubsection{Proof of Lemma \ref{sec2prop2}.}
Let $(z_1,\hat{\mathbf{z}}) \in (0,\infty) \times \mathbb{R}_+^{N-1}$. All inequalities of random quantities in this proof are $\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}$-almost sure. For $ t \leq \tau_0^{Z_1}$, we have \begin{align*}
Z_1(t) & = z_1 + B_1(t) - \frac{1}{2}L_2(t) - \int_0^tV(s)ds \\
& = z_1 + B_1(t) - \frac{1}{2}L_2(t) - \int_0^t(gs + 4g)ds \\
& \leq z_1 + \sup_{0 \leq s \leq t}B_1(s) - \frac{gt^2}{2} \doteq H(t). \end{align*} Consequently, $Z_1(t)$ must hit zero before $H(t)$, and so $\tau_0^H \doteq \inf\{t\ge0: H(t) =0\} \geq \tau_0^{Z_1}$. Thus, for arbitrary $\gamma>0$, \begin{align*}
\mathbb{E}^*_{(4g,z_1,\hat{\mathbf{z}})}e^{\gamma\tau_0^{Z_1}} &= 1 + \int_1^{\infty}\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}(\tau_0^{Z_1} > \frac{\ln(s)}{\gamma}) ds \\
& \leq 1 + \int_1^{\infty}\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}(\tau_0^{H} > \frac{\ln(s)}{\gamma}) ds\\
& \leq 1 + \int_1^{\infty}\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}(H(\frac{\ln(s)}{\gamma}) > 0) ds. \end{align*} Thus, using Markov's inequality, for $\theta >0$, \begin{align*} \mathbb{E}^*_{(4g,z_1,\hat{\mathbf{z}})}e^{\gamma \tau_0^{Z_1}} & \le 1 + \int_1^{\infty}\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}(z_1 + \sup_{0 \leq u \leq \frac{\ln(s)}{\gamma}}B_1(s) > \frac{g(\frac{\ln(s)}{\gamma})^2}{2} )ds \\
& \leq 1 + \int_1^{\infty}\mathbb{P}^*_{(4g,z_1,\hat{\mathbf{z}})}(z_1^{1/2} +(\sup_{0 \leq u \leq \frac{\ln(s)}{\gamma}}B_1(s))^{1/2} > \sqrt{\frac{g}{2}}\frac{\ln(s)}{\gamma})ds \\
& \leq 1 + e^{\theta z_1^{1/2}}\int_1^{\infty}s^{-\sqrt{\frac{g}{2}}\frac{\theta}{\gamma}} \mathbb{E}^*_{(4g,z_1,\hat{\mathbf{z}})}e^{\theta(\sup_{0 \leq u \leq \frac{\ln(s)}{\gamma}}B_1(s))^{1/2}}ds \\
& \leq 1 + \varrho_1e^{\theta z_1^{1/2}}\int_1^{\infty}s^{-\sqrt{\frac{g}{2}}\frac{\theta}{\gamma}}s^{\frac{\varrho_2\theta^2}{\gamma}}ds, \end{align*} where in the last line we have used a standard concentration inequality (see \eqref{eq:elemconc}). Now take $\gamma_4 \doteq g/(16\varrho_2)$ and for fixed $\gamma \in (0, \gamma_4]$, take $\theta = 4\sqrt{2}\gamma/\sqrt{g}$. Then it follows that $$-\sqrt{\frac{g}{2}}\frac{\theta}{\gamma} + \frac{\varrho_2\theta^2}{\gamma} { \leq} -2.$$ Thus $$ \sup_{\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1}}\mathbb{E}^*_{(4g,z_1,\hat{\mathbf{z}})}e^{\gamma \tau_0^{Z_1}} \le 1 + \varrho_1e^{4\sqrt{2} \gamma z_1^{1/2}/\sqrt{g}}.$$
The result follows.
\qed
\subsubsection{Proof of Lemma \ref{sec2prop3}.} Let $v \in [2g, \infty),\,\hat{\mathbf{z}} \in \mathbb{R}_+^{N-1}$, and $\gamma > 0$. All inequalities of random quantities in this proof are $\mathbb{P}^*_{(v,0,\hat{\mathbf{z}})}$-almost sure. For $t \leq \hat \tau_{2g}$, $V(t) \geq 2g$, so \begin{align*}
0 \leq Z_1(t) & = B_1(t) + L_1(t) - \frac{1}{2}L_2(t) - \int_0^tV(s)ds
\leq \sup_{0 \leq s \leq t}B_1(s) + L_1(t) - 2gt, \end{align*} from which it follows that $
- L_1(t) \leq \sup_{0 \leq s \leq t}B_1(s) - 2gt. $ Hence, $$V(t) = gt - L_1(t) + v \leq \sup_{0 \leq s \leq t}B_1(s) - gt + v \doteq Q(t).$$ From this inequality we see that $\tau^Q_{2g} \doteq \inf\{t\ge 0: Q(t)=2g\} $ satisfies $\tau^Q_{2g} \geq \hat \tau_{2g}$. Then, for any $\theta > 0$, \begin{align*}
\mathbb{E}^*_{(v,0,\hat{\mathbf{z}})}e^{\gamma\hat \tau_{2g}} &= \int_0^\infty \mathbb{P}^*_{(v,0,\hat{\mathbf{z}})}(\hat \tau_{2g} \geq \frac{1}{\gamma}\ln(s))ds
\leq \int_0^\infty \mathbb{P}^*_{(v,0,\hat{\mathbf{z}})}(\tau^Q_{2g} \geq \frac{1}{\gamma}\ln(s))ds \\
& \leq 1 + \int_1^{\infty}\mathbb{P}^*_{(v,0,\hat{\mathbf{z}})}(Q(\frac{1}{\gamma}\ln(s)) > 2g)ds . \end{align*} Thus by Markov's inequality, \begin{align*}
\mathbb{E}^*_{(v,0,\hat{\mathbf{z}})}e^{\gamma\hat \tau_{2g}} & \leq 1 + e^{-2g\theta}\int_1^{\infty}\mathbb{E}^*_{(v,0,\hat{\mathbf{z}})}e^{\theta Q(\frac{1}{\gamma}\ln(s))}ds \\
& = 1 + e^{\theta(v-2g)}\int_1^{\infty}e^{-\frac{g\theta}{\gamma}\ln(s)}\mathbb{E}^*_{(v,0,\hat{\mathbf{z}})}e^{\theta\sup_{0 \leq t \leq \frac{1}{\gamma}\ln(s)}B_1(t)}ds \\
& \leq 1 + \varrho_1e^{\theta(v-2g)}\int_1^{\infty}s^{-\frac{g\theta}{\gamma}+\frac{\varrho_2\theta^2}{\gamma}}ds, \\ \end{align*} where we have once again used \eqref{eq:elemconc}.
Now let $\gamma_5 \doteq g^2/(8\varrho_2)$ and for fixed $\gamma \in (0, \gamma_5)$, take $\theta = 4 \gamma/g$. Then, for any $\gamma \in (0, \gamma_5)$, $$-\frac{g\theta}{\gamma}+\frac{\varrho_2\theta^2}{\gamma} { \leq} -2.$$ It then follows, for $\gamma \in (0, \gamma_5)$, $$ \sup_{\hat{\mathbf{z}} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(v, 0,\hat{\mathbf{z}})}e^{\gamma \,\hat \tau_{2g}} \le
1 + \varrho_1e^{4 \gamma(v-2g)/g}.
$$
The result follows.
\qed
\subsection{Proofs of Lemmas for Proposition \ref{neginexp}.} \label{sec:pfneginexp} In this section we provide the proofs of Lemmas \ref{zlesslem1} and \ref{zlesslem2} stated in Section \ref{singdrift} that were used in the proof of Proposition \ref{neginexp}.
\subsubsection{Proof of Lemma \ref{zlesslem1}.} Fix $(2g,0, \hat\mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^N$. Since $M(t) = \sum_{i=2}^NB_i(t) - (N-1)B_1(t)$, from Proposition \ref{sigma2} (which implies $\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\sigma_{2l} < \infty$ for any $l \in \mathbb{N}$) and optional sampling theorem (cf. \cite[Section 1.3.C]{KarShre}),
we have from Lemma \ref{zless}, for $l\in {\mathbb{N}}$, \begin{align}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(\bar{Z}_2(\sigma_{2l}) - \bar{z}_2) &\leq \mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\left(M(\sigma_{2l}) + \frac{N}{k}Y^{(1)}_k(\sigma_{2l}) - \frac{(N-k)}{k}L_1(\sigma_{2l}) + \frac{N}{2k}L_{k+1}(\sigma_{2l})\right) \nonumber\\
& = \frac{N}{k}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\,Y^{(1)}_k(\sigma_{2l}) - \frac{(N-k)}{k}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\,L_1(\sigma_{2l}) + \frac{N}{2k}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\,L_{k+1}(\sigma_{2l}). &\ &&\ \label{eq:340} \end{align} Using standard martingale maximal inequalities we have \begin{align*}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}B_i^*(\sigma_{2l}) &\leq
c_0 \sqrt{\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\sigma_{2l}}
= c_0\left(\sum_{i=1}^{l}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(\sigma_{2i}-\sigma_{2(i-1)})\right)^{1/2} \le c_0' \sqrt{l}, \end{align*} where $ c_0, c_0'\in (0, \infty)$ are independent of $\hat\mathbf{z}$ and $l$, and the last inequality once more uses Proposition \ref{sigma2}. Thus, for some $ c_1 \in (0,\infty)$, for all $k = 1, \ldots, N$, $l\in {\mathbb{N}}$, \begin{equation}\label{eq:338}
\sup_{\hat\mathbf{z} \in {\mathbb{R}}_+^{N-1}}\mathbb{E}^*_{(2g,0,\mathbf{z})}Y_k^{(1)}(\sigma_{2l}) \leq c_1 l^{1/2}. \end{equation} Next note that \begin{align}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}\,L_1(\sigma_{2l}) & = \sum_{i=1}^{l}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(L_1(\sigma_{2i}) - L_1(\sigma_{2i-2})) \;\;\ge l\inf_{\tilde\mathbf{z} \in {\mathbb{R}}_+^{N-1}} \mathbb{E}^*_{(2g, 0,\tilde\mathbf{z})}(L_1(\sigma_2))\nonumber\\
&\ge gl\inf_{\tilde\mathbf{z} \in {\mathbb{R}}_+^{N-1}} \mathbb{P}^*_{(2g,0,\tilde\mathbf{z})}(L_1(\sigma_2) > g)\nonumber\\
&= gl\inf_{\tilde\mathbf{z} \in {\mathbb{R}}_+^{N-1}} \mathbb{P}^*_{(2g,0,\tilde\mathbf{z})}(g\sigma_2 - V(\sigma_2) + 2g > g)
= gl\inf_{\tilde\mathbf{z} \in {\mathbb{R}}_+^{N-1}} \mathbb{P}^*_{(2g,0,\tilde\mathbf{z})}(\sigma_2 > 1) = gl, &\ \label{eq:345} \end{align}
where the last equality follows on observing that, under $\mathbb{P}^*_{(2g,0,\tilde\mathbf{z})}$, $\sigma_2> \sigma_1>1$ a.s.
The result follows from \eqref{eq:340}, \eqref{eq:338} and \eqref{eq:345}.
\qed
\subsubsection{Proof of Lemma \ref{zlesslem2}.}
Fix $\Delta>0$ and $(2g,0, \hat\mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^N$ such that $\hat\mathbf{z} \in {\mathcal{S}}_{\Delta}$. All inequalities will be a.s. under ${\mathbb{P}}^*_{(2g,0,\hat\mathbf{z})}$. Let $k=k(\Delta)$ satisfy \eqref{eq:1132}. Define $$\theta_k = \inf\{t \geq 0: Z_{k+1}(t) = 0\}.$$ Then for $t \leq \theta_k$, \begin{align}
Z_{k+1}(t) &= z_{k+1}+B_{k+1}(t)-B_k(t) - \frac{1}{2}(L_k(t) + L_{k+2}(t)) \nonumber\\
& \geq \frac{\Delta}{N^2}+B_{k+1}(t)-B_k(t) - \frac{1}{2}(L_k(t) + L_{k+2}(t)).\label{eq:551} \end{align} To bound $\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l})$, we will obtain an upper bound on the probability that $L_{k+1}(\sigma_{2l}) > 0$, or equivalently, the probability that $Z_{k+1}(\cdot)$ hits zero before time $\sigma_{2l}$, using \eqref{eq:551}. Next, we will estimate $\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l})^2)$. These two will be combined using a Cauchy-Schwarz inequality to obtain an upper bound for $\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l})$.
We will first obtain an upper bound for $L_k(t)$ for $k<N$ and $t \le \theta_k$. When $3 \le k \le N-1$, from \eqref{eq:lbds}, for $t \le \theta_k$, \begin{align}
L_{k}(t) &\leq B^*_k(t) + \frac{1}{2}L_{k-1}(t) \nonumber\\
L_i(t) &\leq B^*_i(t) + \frac{1}{2}(L_{i-1}(t)+L_{i+1}(t)),\,\, 3 \leq i \leq k-1 \ \mbox{ if } \ k \ge 4, \nonumber\\
L_2(t) &\leq L_1(t) + B^*_2(t) + \frac{1}{2}L_3(t).\label{lll} \end{align} Thus, \begin{align*}
\sum_{i=3}^{k-1}&(i-1)(L_i(t)-\frac{1}{2}(L_{i+1}(t)+L_{i-1}(t))) \\
&+ (L_2(t)-L_1(t)-\frac{1}{2}L_3(t)) + (k-1)(L_k(t)-\frac{1}{2}L_{k-1}(t)) \leq \sum_{i=2}^k(i-1)B^*_i(t) \doteq Y^{(2)}_k(t), \end{align*} where the first sum is taken to be zero if $k=3$. The left side in the above inequality equals $
\frac{k}{2}L_k(t) - L_1(t) $ and so we have, whenever $N>k\ge3$, $t \leq \theta_k$, \begin{equation}\label{eq:549}
L_k(t) \leq \frac{2}{k}L_1(t) + \frac{2}{k}Y^{(2)}_k(t).\end{equation} Note that the above inequality holds trivially if $k=1$, and by \eqref{lll} if $k=2$, and so in fact the above holds under $\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}$, with $\hat \mathbf{z} \in {\mathcal{S}}_{\Delta}$, for $k$ satisfying \eqref{eq:1132} and $t \le \theta_k$ .
We now obtain a similar upper bound on $L_{k+2}(t)$ when $k<N$ and $t \le \theta_k$. From \eqref{eq:lbds}, when $k<N-1$, for $t \leq \theta_k$, \begin{align*}
&(N-k-1)(L_{k+2}(t)-\frac{1}{2}L_{k+3}(t))
+ \sum_{i=k+2}^{N-1}(N-i)(L_{i+1}(t)-\frac{1}{2}(L_{i}(t) + L_{i+2}(t))) \\
&\leq \sum_{i=k+1}^{N-1}(N-i)B^*_{i+1}(t) \doteq Y^{(3)}_k(t). \end{align*} The left side equals $\frac{N-k}{2}L_{k+2}(t)$ and so we have, when $k<N-1$, \begin{equation}\label{eq:550}
L_{k+2}(t) \leq \frac{2}{N-k}Y^{(3)}_k(t),\,\,\mbox{ for all } t \leq \theta_k.\end{equation}
Note that when $k=N-1$ the inequality is trivially true. Using \eqref{eq:549} and \eqref{eq:550} in \eqref{eq:551}, we have for $t \leq \theta_k$, under $\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}$
\begin{align*}
Z_{k+1}(t)
&\ge \frac{\Delta}{N^2} - \sum_{i=1}^N B_{i}^*(t) - \frac{2}{k}L_1(t) - \frac{2}{k}Y^{(2)}_k(t)
-\frac{2}{N-k}Y^{(3)}_k(t)\\
&\ge \frac{\Delta}{N^2} - Y^{(4)}_k(t) - \frac{2}{k}L_1(t) , \end{align*} where $Y^{(4)}_k(t) = \sum_{i=1}^N B_{i}^*(t) + \frac{2}{k}Y^{(2)}_k(t) + \frac{2}{N-k}Y^{(3)}_k(t)$. Note that if $L_{k+1}(\sigma_{2l}) > 0$, then $\inf_{0 \leq s \leq \sigma_{2l}}Z_{k+1}(s) = 0$, which in turn implies that $\theta_k \in [0, \sigma_{2l}]$ and so from the above display $$\frac{\Delta}{N^2} \le Y^{(4)}_k(\theta_k) + \frac{2}{k}L_1(\theta_k) \leq Y^{(4)}_k(\sigma_{2l}) + \frac{2}{k}L_1(\sigma_{2l}). $$ As a consequence, \begin{align}
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l}) > 0) & \leq \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\frac{\Delta}{N^2} \leq Y^{(4)}_k(\sigma_{2l}) + \frac{2}{k}L_1(\sigma_{2l})) \\
& \leq \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\frac{\Delta}{2N^2} \leq Y^{(4)}_k(\sigma_{2l}))
+ \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\frac{\Delta}{2N^2} \leq \frac{2}{k}L_1(\sigma_{2l})).\label{eq:631} \end{align} Consider now the first term on the right side. Then, for $T \ge 1$, \begin{align}
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})} (\frac{\Delta}{2N^2} \leq Y^{(4)}_k(\sigma_{2l}))
& \leq \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\sigma_{2l} > T) + \mathbb{P}(\frac{\Delta}{2N^2} \leq Y^{(4)}_k(T)) \nonumber\\
& \leq l { \sup_{\tilde \mathbf{z} \in {\mathbb{R}}_+^{N-1}}}\mathbb{P}^*_{(2g,0, \tilde\mathbf{z})}(\sigma_{2} > T/l) + c_1e^{-c_2\Delta^2/T} \nonumber\\
& \leq c_3le^{-c_4T/l} +c_1e^{-c_2\Delta^2/T}, \label{eq:1020} \end{align} where $c_i \in (0,\infty)$ are constants that do not depend on $\Delta$ or $\hat\mathbf{z} \in {\mathcal{S}}_{\Delta}$, the second inequality uses the strong Markov property and a standard concentration estimate (see \eqref{eq:elemconc}) and the last inequality is a consequence of Proposition \ref{sigma2}.
Now consider the second term on the right side of \eqref{eq:631}. For $T\ge 1$,
\begin{align}
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\frac{\Delta}{2N^2} \leq \frac{2}{k}L_1(\sigma_{2l})) &\le
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\sigma_{2l} > T) + \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\frac{\Delta}{2N^2} \leq \frac{2}{k}L_1(T)) \nonumber\\
&\le c_3le^{-c_4T/l} +\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(L_1(T)\ge \frac{\Delta k}{4N^2}). \label{eq:1019}
\end{align}
Note that under ${\mathbb{P}}^*_{(2g, 0,\hat\mathbf{z})}$, $$\sup_{0 \leq s \leq T}V(s) < 2g + gT.$$ Thus, using \eqref{eql2l1} and \eqref{eq:loctimes}, for any $T\ge 1$, \begin{align*}
L_1(T) &\le \frac{1}{2}L_2(T) + (2g+gT)T + B_1^*(T)\le \frac{(N-1)}{N}L_1(T) + \frac{1}{N} \bar Y(T) + (2g+gT)T + B_1^*(T) \end{align*} and thus, with $\tilde Y(T) = \bar Y(T) + NB_1^*(T)$ and $c_5 = 3gN$ \begin{align}\label{eq:1114}
L_1(T) &\le \bar Y(T) + (2g+gT)TN + NB_1^*(T) { \leq} \tilde Y(T) + c_5T^2. \end{align}
Take $T = T(\Delta) \doteq \frac{1}{2N} \sqrt{\frac{\Delta}{2c_5}}$ and observe that $c_5T^2 \le \Delta k/(8N^2)$ for all $k \in \{1,\ldots,N\}$. Choose $\Delta_1>0$ such that $T(\Delta_1) \ge 1$. Then, it follows by a concentration estimate that, for $\Delta \ge \Delta_1$,
\begin{align}
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(L_1(T)&\ge \frac{\Delta k}{4N^2}) \le
\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\tilde Y(T) \ge \frac{\Delta k}{4N^2} - c_5T^2)\\
&\le \mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(\tilde Y(T) \ge c_5T^2) \le c_6 e^{-c_7T^3}.\label{eq:1017} \end{align} Then, using \eqref{eq:1017}, \eqref{eq:1019} and \eqref{eq:1020} in \eqref{eq:631}, we obtain constants $c_2', c_7', c_8, c_9>0$ such that for all $\Delta \ge \Delta_1$, \begin{equation}\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l}) > 0) \leq 2c_3le^{-c_4\nu \sqrt{\Delta}/l} +c_1e^{-c_2'\Delta^{3/2}} + c_6 e^{-c_7'\Delta^{3/2}} { \leq} 2c_3le^{-c_4\nu \sqrt{\Delta}/l} + c_8e^{-c_9\Delta^{3/2}}. \label{eq:1135} \end{equation}
We will now obtain an upper bound for $\mathbb{E}_{{(2g,0,\hat \mathbf{z})}}(L_{k+1}(\sigma_{2l})^2)$. From \eqref{eq:loctimes} we have that, for $m\ge 2$ and $t\ge 0$, \begin{align*}
&(L_N(t) - \frac{1}{2}L_{N-1}(t)) + \sum_{j=m}^{N-2} (N-j)(L_{j+1}(t) -\frac{1}{2}(L_{j+2}(t)+L_{j}(t))) \\
&\leq \sum_{j=m}^{N-1}(N-j)B_{j+1}^*(t) \doteq Y^{(5)}_{m}(t). \end{align*} The left side above equals $\frac{N-m+1}{2} L_{m+1}(t) - \frac{N-m}{2}L_m(t) $ and so $$\frac{N-m+1}{2} L_{m+1}(t) - \frac{N-m}{2}L_m(t) \le Y^{(5)}_{m}(t).$$ Dividing by $(N-m)(N-m+1)/2$ throughout, we have $$\frac{1}{N-m} L_{m+1}(t)- \frac{1}{N-m+1} L_m(t) \le \frac{2}{(N-m)(N-m+1)}Y^{(5)}_{m}(t),\;\; 2\le m \le N-1.$$ Summing over $m$ from $2$ to $k$, the above yields $$\frac{1}{N-k}L_{k+1}(t)-\frac{1}{N-1}L_2(t) \leq \sum_{m=2}^k\frac{2Y^{(5)}_{m}(t)}{(N-m)(N-m+1)} \doteq Y^{(6)}_k(t).$$ and thus $$L_{k+1}(t) \leq \frac{N-k}{N-1}L_2(t)+(N-k)Y^{(6)}_k(t).$$ From \eqref{eq:1055} (recall it holds for any value of $k\ge 1$) we have $$L_{k+1}(t) \leq \frac{N-k}{N-1}\left(\frac{2(k-1)}{k}L_1(t) + \frac{2}{k}Y^{(1)}_k(t) + \frac{1}{k}L_{k+1}(t)\right)+(N-k)Y^{(6)}_k(t).$$ Thus $$\frac{N(k-1)}{k(N-1)} L_{k+1}(t) \le \frac{2(k-1)(N-k)}{k(N-1)} L_1(t) + \frac{2(N-k)}{k(N-1)}Y^{(1)}_k(t) + (N-k)Y^{(6)}_k(t)$$ and consequently, when $k>1$, \begin{align}
L_{k+1}(t) &\leq \frac{2(N-k)}{N}L_1(t)+\frac{2(N-k)}{N(k-1)}Y^{(1)}_k(t)+\frac{k(N-k)(N-1)}{N(k-1)}Y^{(6)}_k(t) \\
& = \frac{2(N-k)}{N}L_1(t)+Y^{(7)}_k(t),\label{eq:1122} \end{align} where $Y_k^{(7)}(t) = \frac{2(N-k)}{N(k-1)}Y^{(1)}_k(t)+\frac{k(N-k)(N-1)}{N(k-1)}Y^{(6)}_k(t)$. Recalling the inequality \eqref{eql2l1} we see that \eqref{eq:1122} also holds with $k=1$ and $Y^{(7)}_1(t) \doteq \frac{2}{N} \bar Y(t)$.
Using this, we obtain that \begin{align*}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l})^2) & \le 2\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(Y^{(7)}_k(\sigma_{2l}))^2 + \frac{8(N-k)^2}{N^2}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_1^2(\sigma_{2l})\\
& \le 2\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(Y^{(7)}_k(\sigma_{2l}))^2
+ \frac{16(N-k)^2}{N^2}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(\tilde Y(\sigma_{2l}))^2 + \frac{16(N-k)^2c_5^2}{N^2}\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(\sigma_{2l})^4. \end{align*} where the last line is from \eqref{eq:1114}. Thus, using the above bound, the strong Markov property, and Proposition \ref{sigma2}, there is a $b_1 \in (0,\infty)$ such that, for all $l \in {\mathbb{N}}$, $\hat \mathbf{z} \in {\mathcal{S}}_\Delta$, and $k$ satisfying \eqref{eq:1132}, \begin{align*}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l})^2) \le b_1 l^5. \end{align*}
Applying Cauchy-Schwarz inequality and using \eqref{eq:1135}, we obtain positive constants $D_1, D_2, D_3$ such that for all $\Delta \ge \Delta_1$, \begin{align*}
\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l}) & \leq (\mathbb{E}^*_{(2g, 0,\hat\mathbf{z})}L_{k+1}(\sigma_{2l})^2)^{1/2}(\mathbb{P}^*_{(2g, 0,\hat\mathbf{z})}(L_{k+1}(\sigma_{2l}) > 0))^{1/2} \\
& \leq b_1^{1/2} l^{5/2} (2c_3le^{-c_4\nu \sqrt{\Delta}/l} + c_8e^{-c_9\Delta^{3/2}})^{1/2}=
D_1l^{5/2}(\sqrt{l}e^{-D_2\sqrt{\Delta}/l}+e^{-D_3\Delta^{3/2}}), \end{align*} { as desired.}
\qed
\subsection{Proof of Proposition \ref{driftcondn}} \label{sec:driftcon} In this section we give the proof of Proposition \ref{driftcondn}. Proof relies on five preliminary lemmas which extend some estimates derived in Sections \ref{highlev} and \ref{sec:lowlev} to more general starting configurations. The first four are required to verify part (1) of the proposition and the last one is used to check part (2). Proof of the proposition is at the end of the section. Recall the set $C^*$ from \eqref{eq:cstar} and stopping times $\sigma_1, \sigma_2$ from Section \ref{sec:lowlev}. Recall $\Gamma$ from \eqref{eq928}.
\begin{lemma}
\label{lem:800} There exists a $\rho_0 >0$ and $b_1, b_2 >0$ such that, for all $\rho \in (0, \rho_0)$, there is a $b_3(\rho) \in (0,\infty)$ such that for any $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, \begin{equation*}
\mathbb{E}^*_{(v,\mathbf{z})}e^{\rho\Gamma} \leq b_3(\rho)e^{b_1\rho(|v| + z_1 + \bar z_2)}\mathbb{E}^*_{(v,\mathbf{z})}e^{b_2\rho\sigma_1}. \end{equation*} \end{lemma} \begin{proof} Define the stopping time $$\sigma^* \doteq \inf\{t\ge \sigma_2: (V(t), \mathbf{Z}(t)) \in C^*\}.$$ From Proposition \ref{gammabound}, there exist positive constants $d_0, c'$ such that for any $\gamma \in (0, \gamma_7/2)$, where $\gamma_7$ is as in that lemma, \begin{equation}
\mathbb{E}^*_{(2g,0,\hat\mathbf{z})} e^{\gamma \Gamma} \le d_0e^{c'\gamma \bar z_2}, \ \ \hat \mathbf{z} \in {\mathbb{R}}_+^{N-1}. \end{equation} Fix $\rho_0'>0$ such that $ \rho_0' < \min\{\gamma_7, \gamma_5, \gamma_4\}$ and $\rho_0' (1+\kappa_2' g) < \gamma_4$, where $\gamma_5$ and $\kappa_2'$ are as in Lemma \ref{sec2prop3} and $\gamma_4$ is as in Lemma \ref{sec2prop2}. Then, for $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$ and $\rho \in (0, \rho_0'/2)$, \begin{align} \mathbb{E}^*_{(v,\mathbf{z})} e^{\rho\Gamma} \le \mathbb{E}^*_{(v,\mathbf{z})}e^{\rho\sigma^*} &= \mathbb{E}^*_{(v,\mathbf{z})}\left[ \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{\rho\sigma^*}\mid {\mathcal{F}}^*_{\sigma_2}\right]\right] \le d_0 \mathbb{E}^*_{(v,\mathbf{z})} e^{\rho\sigma_2 + c'\rho \bar Z_2(\sigma_2)}\notag\\ & \le d_0\left(\mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\sigma_2}\right)^{1/2}\left(\mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \bar Z_2(\sigma_2)}\right)^{1/2}.\label{eq:649} \end{align} Recall the stopping time $\eta_1 \ge \sigma_1$ defined in \eqref{eq:etastop}. Proceeding as in the proof of Proposition \ref{sigma2} (see \eqref{eq:541}), with $d_1 = 1 + \kappa_1' e^{8\rho \kappa_2' g}$ and $d_2 = (1 + \kappa_2' g)$,
using Lemma \ref{sec2prop2} and Lemma \ref{sec2prop3}, \begin{align} \mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\sigma_2} &\le \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\eta_1 <\sigma_2} \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{2\rho\sigma_2}\mid {\mathcal{F}}^*_{\eta_1}\right]\right] + \mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\eta_1}\\ &\le \kappa_1' \mathbb{E}^*_{(v,\mathbf{z})}e^{2\rho \kappa_2' V(\eta_1) + 2\rho \eta_1} + \mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\eta_1}\\ &\le d_1 \mathbb{E}^*_{(v,\mathbf{z})}e^{2\rho d_2 \eta_1}\\ &= d_1 \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{2\rho d_2 \sigma_1}\mathbb{E}^*_{(v,\mathbf{z})}\left[e^{2\rho d_2 (\eta_1- \sigma_1)} \mid {\mathcal{F}}^*_{\sigma_1} \right]\right]\\ &\le \kappa_1 d_1 \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{2\rho d_2 \sigma_1}e^{2\rho d_2\kappa_2 Z_1(\sigma_1)^{1/2}}\right]\\ & \le \kappa_1 d_1\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{4\rho d_2\kappa_2 Z_1(\sigma_1)^{1/2}}\right)^{1/2}
\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{ 4\rho d_2\sigma_1}\right)^{1/2}, \label{eq:641a} \end{align} where we used Lemma \ref{sec2prop3} for the second inequality, $V(\eta_1) \le 4g + g\eta_1$ in the third inequality, and Lemma \ref{sec2prop2} in the penultimate inequality (and the observation that $\rho'_0 d_2 < \gamma_4$).
Now we estimate exponential moments of $Z_1(\sigma_1)^{1/2}$. Note that, under $\mathbb{P}^*_{(v,\mathbf{z})}$, $L_1(\sigma_1) = g\sigma_1+ v-4g$, from which it follows that
$$Z_1(\sigma_1) \le \sup_{0\le s \le \sigma_1} B_1(s) + z_1 + g \sigma_1^2 + g\sigma_1 + 2|v|\sigma_1 + |v|$$ and so, using $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$ and $\sqrt{ab} \le (a+b)/2$ for $a,b \ge 0$, \begin{align*}
Z_1(\sigma_1)^{1/2} \le \left(\sup_{0\le s \le \sigma_1} B_1(s)\right)^{1/2} + (\sqrt{g} +3/2)\sigma_1 + \frac{1}{2}(z_1 + 3|v| + g + 1).
\end{align*} Using this bound in \eqref{eq:641a} and Young's inequality, we obtain a finite positive constant $d_3$ not depending on $v, \mathbf{z}, \rho$ such that \begin{multline} \mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\sigma_2}\\
\le d_3 e^{2\rho d_2 \kappa_2(z_1 + 3|v|)}\left[\left(\mathbb{E}^*_{(v,\mathbf{z})} e^{8\rho d_2\kappa_2\left(\sup_{0\le s \le \sigma_1} B_1(s)\right)^{1/2}}\right)^{1/2} + \left(\mathbb{E}^*_{(v,\mathbf{z})}e^{8\rho d_2\kappa_2(\sqrt{g} +3/2)\sigma_1}\right)^{1/2}\right]\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{4\rho d_2\sigma_1}\right)^{1/2}.\\\label{i1} \end{multline} The expectation involving $\sup_{0\le s \le \sigma_1} B_1(s)$ above is bounded as in the proof of Lemma \ref{sec2prop1} (see \eqref{eq:907}) to obtain $\rho_0'', d_4,d_5 \in (0,\infty)$ such that $d_5\rho_0''<d_2$, and for any $\rho \in (0, \rho_0'')$ and $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, \begin{align}\label{stopsup}
\mathbb{E}^*_{(v,\mathbf{z})}e^{8\rho d_2\kappa_2(\sup_{0\leq s \leq \sigma_1}B_1(s))^{1/2}}
& \leq e^{8\rho d_2\kappa_2} + d_4 e^{d_5\rho^2}\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{4\rho d_2\sigma_1}\right)^{1/2} \sum_{k=0}^{\infty}e^{d_5\rho^2k - 2\rho d_2 k}\notag\\
&\le e^{8\rho d_2\kappa_2} + d_6(\rho) \left(\mathbb{E}^*_{(v,\mathbf{z})}e^{4\rho d_2\sigma_1}\right)^{1/2}, \end{align} where $d_6(\rho) \doteq d_4 e^{d_5\rho^2}(1-e^{-\rho d_2})^{-1}$.
Using this bound in \eqref{i1}, we conclude that for every $0 < \rho < \min\{\rho_0'/2, \rho_0''\}$, there exists a finite positive constant $d_7(\rho)$ satisfying \begin{equation}\label{ff0}
\mathbb{E}^*_{(v,\mathbf{z})} e^{2\rho\sigma_2} \le e^{2\rho d_2 \kappa_2(z_1 + 3|v|)} d_7(\rho)\mathbb{E}^*_{(v,\mathbf{z})}e^{d_2'\rho\sigma_1}, \end{equation} where $d_2' = \max\{4d_2, 8d_2\kappa_2(\sqrt{g} + 3/2)\}$.
Now, we estimate $\mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \bar Z_2(\sigma_2)}$. From \eqref{eq:semmart} and \eqref{eql2l1}, $\bar Z_2(t) \le \bar z_2 + M(t) + \bar Y(t), t \ge 0$. Hence, writing $\tilde{Y}(t) \doteq M(t) + \bar Y(t)$, $$ \mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \bar Z_2(\sigma_2)} \le e^{2c'\rho\bar z_2}\mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \tilde{Y}(\sigma_2)}. $$ Proceeding exactly as in \eqref{stopsup}, we obtain $\rho_0'''>0$ such that for every $\rho \in (0, \rho_0''')$, there exists a $d_8(\rho) \in (0,\infty)$ such that $$ \mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \tilde{Y}(\sigma_2)} \le \sum_{k=0}^{\infty}\left(\mathbb{E}^*_{(v,\mathbf{z})} e^{4c'\rho\sup_{0 \le s \le k+1} \tilde{Y}(s)}\right)^{1/2}(\mathbb{P}^*_{(v,\mathbf{z})}(\sigma_2 \geq k))^{1/2} \le d_8(\rho)\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{2\rho \sigma_2}\right)^{1/2}, $$ which, along with \eqref{ff0}, gives \begin{equation}\label{ff1}
\mathbb{E}^*_{(v,\mathbf{z})} e^{2c'\rho \bar Z_2(\sigma_2)} \le e^{2c'\rho\bar z_2}d_8(\rho)\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{2\rho \sigma_2}\right)^{1/2} \le e^{2c'\rho\bar z_2}e^{\rho d_2 \kappa_2(z_1 + 3|v|)} d_8(\rho) d_7(\rho)^{1/2}\left(\mathbb{E}^*_{(v,\mathbf{z})}e^{d_2'\rho\sigma_1}\right)^{1/2}. \end{equation} The result, with $\rho_0 \doteq \min\{\rho_0'/2, \rho_0'', \rho_0'''\}$, now follows upon using \eqref{ff0} and \eqref{ff1} in \eqref{eq:649}. \end{proof} \begin{lemma}\label{sec5prop2} Let $\vartheta \doteq \hat \tau_{4g}\wedge \hat \tau_0.$
Then there is a
$\beta_1 > 0$ and $D_1 > 0$ such that, for all $(v,\mathbf{z}) \in {\mathbb{R}}\times {\mathbb{R}}_+^N$,
$$\mathbb{E}^*_{(v,\mathbf{z})}e^{\beta_1\vartheta} \leq D_1 e^{\beta_1(|v|+z_1)}.$$ \end{lemma}
\begin{proof}
Fix $(v,\mathbf{z}) \in {\mathbb{R}}\times {\mathbb{R}}_+^N$. We consider three cases.\\
\noindent {\bf Case 1: $v \in [0,4g]$.} In this case the result is immediate from Lemma \ref{sec1prop3}.
\noindent {\bf Case 2: $v>4g$.} In this case, for all $t \leq \vartheta$, $V(t) > 4g$. Thus, for such $t$, we have \begin{align*}
Z_1(t) & = z_1 + B_1(t) + L_1(t) - \frac{1}{2}L_2(t) - \int_0^tV(s)ds
\leq z_1 + \sup_{0 \leq s \leq t}B_1(s) + L_1(t) - 4gt. \end{align*} Consequently, $- L_1(t) \leq z_1 + \sup_{0 \leq s \leq t}B_1(s) - 4gt$. Thus we have, for $t \leq \vartheta$, $$V(t) = gt - L_1(t) + v \leq z_1 + \sup_{0 \leq s \leq t}B_1(s) - 3gt + v \doteq Q_1(t).$$ Letting $\tau^{Q_1}_{4g} \doteq \inf\{t\ge 0: Q_1(t)= 4g\}$, we have $\tau^{Q_1}_{4g} \geq \vartheta$. Thus, for $\beta_1, \theta>0$, \begin{align*}
\mathbb{E}^*_{(v,\mathbf{z})}e^{\beta_1\vartheta} & \leq 1 + \int_1^{\infty}\mathbb{P}^*_{(v,\mathbf{z})}(Q_1(\frac{1}{\beta_1}\ln(s)) > 4g)ds \\
& \leq 1 + e^{-4g\theta\beta_1}\int_1^{\infty}\mathbb{E}^*_{(v,\mathbf{z})}e^{\theta\beta_1 Q_1(\frac{1}{\beta_1}\ln(s))}ds \\
& = 1 + e^{-4g\theta\beta_1 + \theta\beta_1 (z_1+v)}\int_1^{\infty}e^{-3\theta g\ln(s)}\mathbb{E}^*_{(v,\mathbf{z})}e^{\theta\beta_1\sup_{0 \leq t \leq \frac{1}{\beta_1}\ln(s)}B_1(t)}ds \\
& \leq 1 + \varrho_1e^{-4g\theta\beta_1 + \theta\beta_1 (z_1+v)}\int_1^{\infty}e^{-3\theta g\ln(s)+\varrho_2\theta^2\beta_1\ln(s)}ds \end{align*} where the last line uses the estimate \eqref{eq:elemconc}. Taking $\theta = g^{-1}$ and $\beta_1 = g^2/\varrho_2$, we now see that $$\mathbb{E}^*_{(v,\mathbf{z})}e^{\beta_1\vartheta} \leq 1 +\varrho_1e^{-4g\theta\beta_1 + \theta\beta_1 (z_1+v)}$$ which completes the proof for Case 2.\\
\noindent {\bf Case 3: $v<0$.} In this case, for $t \leq \vartheta$, we have $V(t) < 0$. Thus for such $t$, from \eqref{eq:loctimes} and \eqref{eql2l1}, \begin{align*}
L_1(t) & = \sup_{0 \leq s \leq t}(-z_1 - B_1(s) + \frac{1}{2}L_2(s) + \int_0^sV(u)\,du)^+
\leq B_1^*(t) + \frac{1}{2}L_2(t) \le B_1^*(t) + \frac{N-1}{N} L_1(t) + \frac{1}{N} \bar Y(t). \end{align*} Consequently, $$ L_1(t) \le NB_1^*(t) + \bar Y(t)$$ and so \begin{align*}
V(t) & = gt - L_1(t) + v
\geq gt - NB_1^*(t) - \bar Y(t) + v \doteq Q_2(t). \end{align*} Letting, $\tau_0^{Q_2} \doteq \inf\{t\ge 0: Q_2(t) =0\}$, we then have,
$\tau_0^{Q_2} \geq \vartheta$. Thus, for $\theta, \beta_1 >0$, \begin{align*}
\mathbb{E}^*_{(v,\mathbf{z})}e^{\beta_1\vartheta} & \leq 1 + \int_1^\infty \mathbb{P}^*_{(v,\mathbf{z})}(Q_2(\frac{\ln(s)}{\beta_1}) < 0)\,ds \\
& = 1 + \int_1^\infty \mathbb{P}^*_{(v,\mathbf{z})}(g\frac{\ln(s)}{\beta_1} + v < NB_1^*(\frac{\ln(s)}{\beta_1}) + \bar Y(\frac{\ln(s)}{\beta_1}))\,ds \\
& \leq 1 + e^{-\theta\beta_1 v}\int_1^\infty s^{-\theta g}\mathbb{E}^*_{(v,\mathbf{z})}e^{\theta \beta_1(NB_1^*(\frac{\ln(s)}{\beta_1}) + \bar Y(\frac{\ln(s)}{\beta_1}))}\,ds \\
& \leq 1 + \varrho_1e^{\theta\beta_1 |v|}\int_1^\infty s^{-\theta g}s^{\varrho_2 \beta_1 \theta^2}\,ds, \end{align*} where in the last line we have used the estimate \eqref{eq:elemconc}. Take $\theta = 4g^{-1}$ and $\beta_1 = g^2/(8\varrho_2)$, then
$$ \mathbb{E}^*_{(v,\mathbf{z})}e^{\beta_1\vartheta} \leq 1 + \varrho_1e^{\theta\beta_1 |v|}.$$ This completes the proof for Case 3 and thus the result follows. \end{proof}
\begin{lemma}\label{sigma1est} There is a $\beta_2 > 0$ and $\kappa_1, \kappa_2>0$ such that, for all $(v,\mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^N$,
$$\mathbb{E}^*_{(v,\mathbf{z})}\,e^{\beta_2\sigma_1} \leq \kappa_1 e^{\kappa_2(|v| + z_1)}.$$ \end{lemma}
\begin{proof} From Proposition \ref{sigma1}, with $\gamma$ as in that proposition, \begin{equation}\label{eq:eeq217}
\sup_{\mathbf{z} \in {\mathbb{R}}_+^N} \mathbb{E}^*_{(\frac{g}{2N},\mathbf{z})}e^{\gamma \hat \tau_{4g}} \doteq d_1<\infty. \end{equation} Also, from Lemma \ref{sec1prop1}, with $\beta$ as in that lemma, \begin{equation}\label{eq:222}
\sup_{{\mathbf{z}} \in {\mathbb{R}}_+^{N}}\,\mathbb{E}^*_{(0,{\mathbf{z}})}\,e^{\beta\,\hat\tau_{g/(2N)}} \doteq d_2< \infty. \end{equation} With $\beta_1$ as in Lemma \ref{sec5prop2}, let $\beta_2 \in (0, \min\{\gamma, \beta, \beta_1\})$. Recall the stopping time $\vartheta$ from Lemma \ref{sec5prop2}. Define stopping times $$\vartheta_1 \doteq \inf\{t\ge \vartheta: V(t) = g/(2N)\}, \;\; \vartheta_2 \doteq \inf\{t\ge \vartheta_1: V(t) = 4g\}.$$ Then, for $(v,\mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^N$, \begin{align*} \mathbb{E}^*_{(v,\mathbf{z})}\,e^{\beta_2\sigma_1} \leq \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1=\vartheta\}}\,e^{\beta_2\sigma_1}\right] + \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}\,e^{\beta_2\sigma_1}\right]. \end{align*} From Lemma \ref{sec5prop2}, with $\beta_1$ and $D_1$ as in the lemma, $$\mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1=\vartheta\}}\,e^{\beta_2\sigma_1}\right] \le \mathbb{E}^*_{(v,\mathbf{z})}\left[\,e^{\beta_2\vartheta}\right]
\le D_1 e^{\beta_1(|v|+z_1)}.$$ Next, from \eqref{eq:eeq217}, \begin{align*} \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}\,e^{\beta_2\sigma_1}\right] &\le \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}\,e^{\beta_2\vartheta_2}\right] = \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}\,\mathbb{E}^*_{(v,\mathbf{z})}\left[e^{\beta_2\vartheta_2}\mid {\mathcal{F}}^*_{\vartheta_1}\right]\right]\\ &\le d_1 \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}e^{\beta_2\vartheta_1}\right]. \end{align*} Also, from \eqref{eq:222}, \begin{align*} \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}e^{\beta_2\vartheta_1}\right] &= \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}} \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{\beta_2\vartheta_1}\mid {\mathcal{F}}^*_{\vartheta}\right]\right] \\ &\le d_2 \mathbb{E}^*_{(v,\mathbf{z})}\left[\textbf{1}_{\{\sigma_1 >\vartheta\}}e^{\beta_2\vartheta}\right]
\le d_2 D_1 e^{\beta_1(|v|+z_1)}, \end{align*} where the last line is from Lemma \ref{sec5prop2}. Combining the above estimates, for all $(v,\mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^N$,
$$\mathbb{E}^*_{(v,\mathbf{z})}\,e^{\beta_2\sigma_1} \le D_1 e^{\beta_1(|v|+z_1)}
+ d_1 d_2 D_1 e^{\beta_1(|v|+z_1)}.$$ The result follows. \end{proof}
\begin{lemma}\label{sec5prop3} There is a $\kappa \in (0, \infty)$ such that for every $\alpha > 0$ there is a $s_\alpha > 0$ with
$$\mathbb{E}^*_{(v,\mathbf{z})}e^{\alpha(|V(1)|+Z_1(1) + \bar Z_2(1))} \leq s_\alpha e^{\kappa\alpha(|v|+z_1 + \bar z_2)}, \mbox{ for all } (v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N.$$ \end{lemma}
\begin{proof}
Since $V(t) \leq g + |v|$ for $t \leq 1$, we have from \eqref{LTineq}
$$L_1(1) \le (g+|v|) W_{11} + \sum_{i = 1}^N W_{1,i}B^*_i(1) = N(g+|v|)+ \sum_{i = 1}^N W_{1,i}B^*_i(1).$$ Thus, under $\mathbb{P}^*_{(v,\mathbf{z})}$, \begin{align*}
Z_1(1) + |V(1)| & \leq z_1 + B_1(1) + L_1(1) - \frac{1}{2}L_2(1) - \int_0^1V(s)ds + g + L_1(1) + |v|
\\
&\le 2|v| + \frac{g}{2} + z_1 + \sup_{0 \leq s \leq 1}B_1(s) + 3 L_1(1)\\
& \leq 2|v| + \frac{g}{2} + z_1 + \sup_{0 \leq s \leq 1}B_1(s) +
3(N(g+|v|)+ \sum_{i = 1}^N W_{1,i}B^*_i(1)).
\end{align*} Moreover, from \eqref{eq:semmart} and \eqref{eql2l1}, $\bar Z_2(1) \le \bar z_2 + M(1) + \bar Y(1), \ t \ge 0.$ The result is now immediate from the estimate in \eqref{eq:elemconc}. \end{proof}
For $K \in {\mathbb{N}}$, with $K>256/g$, define $$R_K \doteq [\frac{1}{K}, \frac{g}{128}- \frac{1}{K}] \times [\frac{1}{K}, K] \times [0, K]^{N-1}.$$ \begin{lemma}
\label{lem:lem5}
There is a $K\in {\mathbb{N}}$, $K>256/g$, such that
$$\inf_{(v,\mathbf{z}) \in C^*} \mathbb{P}^1((v,\mathbf{z}), R_K) \doteq c_K >0.$$ \end{lemma} \begin{proof}
Suppose that, for every $K \in {\mathbb{N}}$, $K>256/g$,
$$\inf_{(v,\mathbf{z}) \in C^*} \mathbb{P}^1((v,\mathbf{z}), R_K) =0.$$
Then, we can find a sequence $\{(v_K, \mathbf{z}_K)\}_{K \in {\mathbb{N}}} \subset C^*$ such that
\begin{equation}\label{eq:gtz}
{\mathbb{P}}^1((v_K, \mathbf{z}_K), R_K) \le \frac{1}{K}.
\end{equation}
Since $C^*$ is compact, we can find $(v^*, \mathbf{z}^*) \in C^*$ such that, along a subsequence (labeled again with $K$), $(v_K, \mathbf{z}_K) \to (v^*, \mathbf{z}^*)$.
From the second statement in Lemma \ref{uniqlem},
$${\mathbb{P}}^1((v^*, \mathbf{z}^*), R) >0.$$
Since $R_K$ increase to $R$ as $K\to \infty$, we can find a $K^* \in {\mathbb{N}}$, $K^* > 256/g$, such that
$${\mathbb{P}}^1((v^*, \mathbf{z}^*), R_{K^*}) \doteq a_{K^*}>0.$$
Choose a real, continuous function $f: {\mathbb{R}} \times {\mathbb{R}}_+^N $ such that $0\le f \le 1$, $f=1$ on $R_{K^*}$ and $f = 0$ on $R_{2K^*}^c$.
Then
\begin{align*}
\liminf_{K\to \infty}{\mathbb{P}}^1((v_K, \mathbf{z}_K), R_{2K^*}) &\ge
\liminf_{K\to \infty} \int f(v, \mathbf{z}){\mathbb{P}}^1((v_K, \mathbf{z}_K), (dv, d \mathbf{z}))\\
&= \int f(v, \mathbf{z}){\mathbb{P}}^1((v^*, \mathbf{z}^*), (dv, d\mathbf{z}))
\ge {\mathbb{P}}^1((v^*, \mathbf{z}^*), R_{K^*}) = a_{K^*}>0,
\end{align*}
where the middle equality follows from the Feller property of the transition probability kernel ${\mathbb{P}}^1$. The Feller property can be verified by analyzing two copies of the process \eqref{eq:gapproc} starting from different initial conditions but driven by the same Brownian motion. Using the Lipschitz property of the Skorohod map and Gr\"onwall's lemma, the distance between the coupled processes in sup-norm on any given compact time interval can be made small (in a pathwise sense) if the initial conditions are close enough.
On the other hand, from \eqref{eq:gtz}
$$\limsup_{K\to \infty}{\mathbb{P}}^1((v_K, \mathbf{z}_K), R_{2K^*}) \le \limsup_{K\to \infty}{\mathbb{P}}^1((v_K, \mathbf{z}_K), R_{K}) = 0.$$
This is a contradiction which completes the proof of the lemma. \end{proof}
We can now complete the proof of Proposition \ref{driftcondn}.\\
\noindent \textbf{Proof of Proposition \ref{driftcondn}}. Fix $\eta>0$ such that $\eta < \rho_0$ and $b_2\eta \le \beta_2$, where $\rho_0$ and $b_2$ are as Lemma \ref{lem:800} and $\beta_2$ is as in Lemma \ref{sigma1est}. Combining Lemmas \ref{lem:800} and \ref{sigma1est}, for all $(v, \mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^{N}$,
$$\mathbb{E}^*_{(v,\mathbf{z})}e^{\eta\Gamma} \leq b_3(\eta)e^{b_1 \eta(|v| + z_1 + \bar z_2)}\mathbb{E}^*_{(v,\mathbf{z})}e^{b_2\eta\sigma_1}
\le b_3(\eta)\kappa_1 e^{b_1 \eta(|v| + z_1 + \bar z_2) + \kappa_2(|v| + z_1)}.$$ Thus $$\mathbb{E}^*_{(v,\mathbf{z})}e^{\eta\tau_{C^*}(1)} = \mathbb{E}^*_{(v,\mathbf{z})}\left[ \mathbb{E}^*_{(v,\mathbf{z})}\left[e^{\eta\tau_{C^*}(1)} \mid {\mathcal{F}}^*_1\right]\right]
\le b_3(\eta)\kappa_1 e^{\eta}\mathbb{E}^*_{(v,\mathbf{z})}e^{(b_1\eta + \kappa_2)(|V(1)|+Z_1(1) + \bar Z_2(1))}.
$$ Consequently, with $\alpha = b_1\eta + \kappa_2$, for all $(v, \mathbf{z})\in {\mathbb{R}}\times {\mathbb{R}}_+^{N}$, $$\mathbb{E}^*_{(v,\mathbf{z})}e^{\eta\tau_{C^*}(1)} \le
b_3(\eta)\kappa_1 s_{\alpha} e^{\eta} e^{\kappa \alpha (|v|+z_1 + \bar z_2)},$$ where $s_{\alpha}$ and $\kappa$ are as in Lemma \ref{sec5prop3}. This immediately implies part (1) of the proposition.
We now consider part (2).
Let $K$ be as in Lemma \ref{lem:lem5}. From Theorem \ref{minorization}, with $M_1\doteq \inf_{(v, \mathbf{z}) \in R_K} K_{(v,\mathbf{z})}$ (which is positive) $$\inf_{(v, \mathbf{z}) \in R_K} \mathbb{P}^{\varsigma}(( v,\mathbf{z}), B ) \ge
\lambda(B\cap D)\inf_{(v, \mathbf{z}) \in R_K} K_{(v,\mathbf{z})} = M_1 \lambda(B\cap D).$$ Also, from Lemma \ref{lem:lem5}, with $K$ as in the lemma, for $(v,\mathbf{z}) \in C^*$ and $B \in {\mathcal{B}}({\mathbb{R}} \times {\mathbb{R}}_+^N)$, \begin{align*}
\mathbb{P}^{1+\varsigma}((v,\mathbf{z}), B) &\ge
\int_{R_K} \mathbb{P}^{1}((v,\mathbf{z}), (d\tilde v, d\tilde \mathbf{z}))
\mathbb{P}^{\varsigma}((\tilde v,\tilde \mathbf{z}), B) \\
&\ge M_1 \lambda(B\cap D) \mathbb{P}^{1}((v,\mathbf{z}), R_K)
\ge M_1 c_K\lambda(B\cap D). \end{align*} The result now follows on taking $\nu(\cdot) =M_1 c_K\lambda(\cdot\cap D)$ and $r_1 = 1 + \varsigma$. \qedsymbol\\
\section{Law of large numbers}\label{sec:lln}
In this section, we prove Theorem \ref{lln}. We begin with the following lemma.
\begin{lemma}\label{fl} For any $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^N$, $\mathbb{P}^*_{(v,\mathbf{z})}$-almost every $\omega$, there exists a $m^*(\omega) \in (0, \infty)$ such that $$
|V(t, \omega)| + \sum_{i=1}^N Z_i(t,\omega) \le m^*(\omega) (\log t)^2, \ \mbox{ for all } t \ge 2. $$ \end{lemma}
\begin{proof}
Recall the set $C^*$ from \eqref{eq:cstar} and the stopping time $\tau_{C^*}(1)$ defined just after. Let $\Sigma(t) \doteq |V(t)| + \sum_{i=1}^N Z_i(t), t \ge 0$. We will first show that there exist positive constants $c_1, c_2$ such that \begin{equation}\label{lln1} \sup_{(v,\mathbf{z}) \in C^*} \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le \tau_{C^*}(1)} \Sigma(t) \ge x\right) \le c_1e^{-c_2\sqrt{x}}, \ x >0. \end{equation} Note that, from \eqref{eq:gapproc} and Lemma \ref{locin}, for $(v,\mathbf{z}) \in C^*$ and $t \ge 0$, \begin{align*}
\Sigma(t) &= |{ v}| + \sum_{i=1}^N { z_i} + B_N(t) + g t + L_1(t) + \frac{1}{2}L_N(t)\\ &\le 2g + \Delta + B_N(t) + gt + \sum_{i=1}^NW_{i,1}t\sup_{0 \leq s \leq t}(V(s))^+ + \sum_{i,j=1}^NW_{i,j}B_j^*(t)\\ & \le 2g + \Delta + gt + 2gt\sum_{i=1}^NW_{i,1} + gt^2\sum_{i=1}^NW_{i,1} + B_N(t) + \sum_{i,j=1}^NW_{i,j}B_j^*(t). \end{align*} Using this bound along with part (1) of Proposition \ref{driftcondn} and \eqref{eq:elemconc}, we obtain positive constants $c_1, c_2, x_0$ such that, for any $(v,\mathbf{z}) \in C^*, x \ge x_0$, and choosing $\delta >0$ sufficiently small, \begin{multline*} \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le \tau_{C^*}(1)} \Sigma(t) \ge x\right) \le \mathbb{P}^*_{(v,\mathbf{z})}(\tau_{C^*}(1) \ge \delta \sqrt{x}) + \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le \delta \sqrt{x}} \Sigma(t) \ge x\right)\\ \le \mathbb{P}^*_{(v,\mathbf{z})}(\tau_{C^*}(1) \ge \delta \sqrt{x}) + \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le \delta \sqrt{x}} \left(B_N(t) + \sum_{i,j=1}^NW_{i,j}B_j^*(t)\right) \ge \frac{x}{2}\right) \le c_1 e^{-c_2\sqrt{x}}. \end{multline*} This proves \eqref{lln1}. Define the following stopping times: $$ {\bar{\tau}}_{0} = 0, \ \ {\bar{\tau}}_{i+1} \doteq \inf\{t \ge {\bar{\tau}}_i + 1: (V(t), \mathbf{Z}(t)) \in C^*\}, \ i \ge 0. $$ Using \eqref{lln1}, there exists a positive constant $c_3$ such that for any $(v,\mathbf{z}) \in C^*$, $n \ge 2$ and $m >0$,
\begin{align*} & \mathbb{P}^*_{(v,\mathbf{z})}\left(\Sigma(t) \ge m (\log t)^2 \text{ for some } t \in [n,n+1]\right)\\ & \le \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le n+1}\Sigma(t) \ge c_3 m (\log (n+1))^2\right)
\le \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le {\bar{\tau}}_{n+1}}\Sigma(t) \ge c_3 m (\log (n+1))^2\right)\\
& \le (n+1)\sup_{(v,\mathbf{z}) \in C^*} \mathbb{P}^*_{(v,\mathbf{z})}\left(\sup_{t \le \tau_{C^*}(1)} \Sigma(t) \ge c_3 m (\log (n+1))^2\right)
\le c_1(n+1) e^{-c_2\sqrt{c_3m}\log(n+1)},
\end{align*} where we used the strong Markov property to obtain the third inequality. Choosing $m$ sufficiently large, we see from the Borel-Cantelli Lemma that, for any $(v,\mathbf{z}) \in C^*$, \begin{equation}
\mathbb{P}^*_{(v,\mathbf{z})}\left( \limsup_{t\to \infty} \frac{\Sigma(t)}{(\log t)^2} <\infty\right) = 1. \end{equation} Finally for an arbitrary $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^{N}$, and with $\Gamma$ as in \eqref{eq928} and $A = \{\limsup_{t\to \infty} \frac{\Sigma(t)}{(\log t)^2} <\infty\}$, applying Lemma \ref{lem:800}, \begin{align*}
\mathbb{P}^*_{(v,\mathbf{z})}(A) = \mathbb{P}^*_{(v,\mathbf{z})}(A, \Gamma <\infty)
= \mathbb{E}^*_{(v,\mathbf{z})}\left( \mathbb{P}^*_{(v,\mathbf{z})}\left(A \mid {\mathcal{F}}^*_{\Gamma}\right) 1_{\{\Gamma<\infty\}}\right) = \mathbb{P}^*_{(v,\mathbf{z})}(\Gamma <\infty) = 1. \end{align*} The result follows. \end{proof}
{\bf Proof of Theorem \ref{lln}.} All limits in the proof hold $\mathbb{P}^*_{(v,\mathbf{z})}$-almost surely for arbitrary $(v,\mathbf{z}) \in {\mathbb{R}} \times {\mathbb{R}}_+^{N}$. From \eqref{eq:rankloc}, \begin{equation}\label{ll1} \lim_{t \rightarrow \infty} \frac{\sum_{j=1}^NX_{(j)}(t)}{Nt} = \lim_{t \rightarrow \infty} \frac{V(t) + \sum_{j=1}^NX_{(j)}(t)}{Nt} = \frac{g}{N}. \end{equation} where we used Lemma \ref{fl} in the first equality. Moreover, again using Lemma \ref{fl}, for any $i \in \{0,1,\dots,N\}$, \begin{equation}\label{ll2}
\frac{1}{t}\left|X_{(i)}(t) - \frac{1}{N}\sum_{j=1}^NX_{(j)}(t)\right| \le \frac{1}{t}\left|X_{(N)}(t) - X_{(0)}(t)\right| \le \frac{1}{t} \sum_{i=1}^N Z_i(t) \rightarrow 0 \end{equation} as $t \rightarrow \infty$. The statement in \eqref{le1} now follows from \eqref{ll1} and \eqref{ll2}. Also, \eqref{le2} follows from Lemma \ref{fl} on noting $$ 0 = \lim_{t \rightarrow \infty}\frac{V(t)}{t} = g - \lim_{t \rightarrow \infty}\frac{L_1(t)}{t}. $$ To prove \eqref{le3}, note that from \eqref{eq:rankloc}, \eqref{eq:gapproc} and Lemma \ref{fl}, $$ 0 = \lim_{t \rightarrow \infty}\frac{Z_1(t)}{t} = - \lim_{t \rightarrow \infty}\frac{X_{(0)}(t)}{t} - \frac{1}{2}\lim_{t \rightarrow \infty}\frac{L_2(t)}{t} + \lim_{t \rightarrow \infty}\frac{L_1(t)}{t}, $$ which gives $\lim_{t \rightarrow \infty}\frac{L_2(t)}{t} = \frac{2(N-1)g}{N}$. Again using \eqref{eq:gapproc} and Lemma \ref{fl}, $$ 0 = \lim_{t \rightarrow \infty}\frac{Z_2(t)}{t} = - \frac{1}{2}\lim_{t \rightarrow \infty}\frac{L_3(t)}{t} + \lim_{t \rightarrow \infty}\frac{L_2(t)}{t} - \lim_{t \rightarrow \infty}\frac{L_1(t)}{t}, $$ from which, we obtain $\lim_{t \rightarrow \infty}\frac{L_3(t)}{t} = \frac{2(N-2)g}{N}$. Suppose $N \ge 4$ and, for some $i \in \{3,\dots,N-1\}$, the limit $\lim_{t \rightarrow \infty}\frac{L_j(t)}{t}$ exists and equals $\frac{2(N - j+1)g}{N}$ for all $3 \le j \le i$. Then, using \eqref{eq:gapproc}, $\lim_{t \rightarrow \infty}\frac{L_{i+1}(t)}{t}$ exists and $$ 0 = \lim_{t \rightarrow \infty}\frac{Z_i(t)}{t} = -\frac{1}{2}\lim_{t \rightarrow \infty}\frac{L_{i+1}(t)}{t} + \lim_{t \rightarrow \infty}\frac{L_{i}(t)}{t}-\frac{1}{2}\lim_{t \rightarrow \infty}\frac{L_{i-1}(t)}{t} $$ which implies $\lim_{t \rightarrow \infty}\frac{L_{i+1}(t)}{t} = \frac{2(N-i)g}{N}$. The statement in \eqref{le3} now follows by induction.
\qed \\\\
\noindent \textbf{Acknowledgement:} The research of SB was supported in part by the NSF CAREER award DMS-2141621. The research of AB was supported in part by the NSF awards DMS-1814894 and DMS-1853968. We thank two anonymous referees for their careful suggestions which improved the article. \\\\
\noindent \textbf{Conflict of interest statement: }There are no conflicts of interests.
\noindent{\scriptsize {\textsc{\noindent S. Banerjee, A. Budhiraja, and B. Estevez\newline Department of Statistics and Operations Research\newline University of North Carolina\newline Chapel Hill, NC 27599, USA\newline email: [email protected] \newline email: [email protected] \newline email: [email protected]
} }}
\end{document} | arXiv |
$66,316 in 1993 is worth $68,014.06 in 1994
$66,316 in 1993 has the same purchasing power as $68,014.06 in 1994. Over the 1 year this is a change of $1,698.06.
The average inflation rate of the dollar between 1993 and 1994 was 2.78% per year. The cumulative price increase of the dollar over this time was 2.56%.
So what does this data mean? It means that the prices in 1994 are 680.14 higher than the average prices since 1993. A dollar in 1994 can buy 97.50% of what it could buy in 1993.
The inflation rate for 1993 was 2.99%, while the inflation rate for 1994 was 2.56%. The 1994 inflation rate is higher than the average inflation rate of 2.39% per year between 1994 and 2021.
We can look at the buying power equivalent for $66,316 in 1993 to see how much you would need to adjust for in order to beat inflation. For 1993 to 1994, if you started with $66,316 in 1993, you would need to have $68,014.06 in 1993 to keep up with inflation rates.
So if we are saying that $66,316 is equivalent to $68,014.06 over time, you can see the core concept of inflation in action. The "real value" of a single dollar decreases over time. It will pay for fewer items at the store than it did previously.
In the chart below you can see how the value of the dollar is worth less over 1 year.
If you're interested to see the effect of inflation on various 1950 amounts, the table below shows how much each amount would be worth today based on the price increase of 2.56%.
$1.00 in 1993 $1.03 in 1994
$10.00 in 1993 $10.26 in 1994
$100.00 in 1993 $102.56 in 1994
$1,000.00 in 1993 $1,025.61 in 1994
$10,000.00 in 1993 $10,256.06 in 1994
$100,000.00 in 1993 $102,560.55 in 1994
$1,000,000.00 in 1993 $1,025,605.54 in 1994
We then replace the variables with the historical CPI values. The CPI in 1993 was 144.5 and 148.2 in 1994.
$$\dfrac{ \$66,316 \times 148.2 }{ 144.5 } = \text{ \$68,014.06 } $$
$66,316 in 1993 has the same purchasing power as $68,014.06 in 1994.
To work out the total inflation rate for the 1 year between 1993 and 1994, we can use a different formula:
$$ \dfrac{\text{CPI in 1994 } - \text{ CPI in 1993 } }{\text{CPI in 1993 }} \times 100 = \text{Cumulative rate for 1 year} $$
$$ \dfrac{\text{ 148.2 } - \text{ 144.5 } }{\text{ 144.5 }} \times 100 = \text{ 2.56\% } $$
<a href="https://studyfinance.com/inflation/us/1993/66316/1994/">$66,316 in 1993 is worth $68,014.06 in 1994</a>
"$66,316 in 1993 is worth $68,014.06 in 1994". StudyFinance.com. Accessed on January 19, 2022. https://studyfinance.com/inflation/us/1993/66316/1994/.
"$66,316 in 1993 is worth $68,014.06 in 1994". StudyFinance.com, https://studyfinance.com/inflation/us/1993/66316/1994/. Accessed 19 January, 2022
$66,316 in 1993 is worth $68,014.06 in 1994. StudyFinance.com. Retrieved from https://studyfinance.com/inflation/us/1993/66316/1994/. | CommonCrawl |
\begin{document}
\title{\bf\Large Rooted Hypersequent Calculus for Modal Logic S5
\thanks
{{\it Key Words}
\begin{abstract} We present a rooted hypersequent calculus for modal propositional logic S5. We show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well. \end{abstract} \section{\bf Introduction} The propositional modal logic {\sf S5} is one of the peculiar modal logics in several respects. Most notably from the proof-theoretical point of view, S5 has so far resisted all efforts to provide it with a acceptable cut-free sequent calculus. Whereas the framework of sequent calculi has proven quite successful in providing analytic calculi for a number of normal modal logics such as {\sf K}, {\sf KT} or {\sf S4} \cite{wansing1994sequent}. For some formats of rules it can even be shown that no such calculus can exist \cite{Lellmann-Pattinson}. Perhaps, the easiest way of demonstrating this resistance is Euclideanness axiom: $ {\sf (5)} \,\, \Diamond A\rightarrow \Box\Diamond A $. Sequent calculus systems for S5 have been widely studied for a long time. Several authors have introduced many sequent calculi for S5, including Ohnishi and Matsumoto \cite{ohnishi1959gentzen}, Mints \cite{mints1997indexed}, Sato \cite{sato1980cut}, Fitting \cite{Fitting}, Wansing \cite{wansing1994sequent} and Bra\"uner \cite{brauner2000cut}. The efforts to develop sequent calculus to accommodate cut-free systems for S5 leading to introduce a variety of new sequent framework. Notably, labelled sequent calculus (see e.g. \cite{negri2005proof}), double sequent calculus (see e.g. \cite{indrzejczak1998cut}), display calculus (see e.g. \cite{ belnap1982display, wansing1999predicate}), deep inference system (see e.g. \cite{stouppa2007deep}), nested sequent (see \cite{brunnler2009deep, poggiolesi-Gentzen Calculi}), hypersequent calculus, which was introduced independently in \cite{Avron hyper, mints1974, Pottinger} and finally, grafted hypersequents (\cite{Kuznets Lellmann}), which combines the formalism of nested sequents with that of hypersequents. Hypersequent calculus provided numerous cut-free formulations for the logic S5, including Pottinger \cite{Pottinger}, Avron \cite{Avron hyper}, Restall \cite{restall2005proofnets}, Poggiolesi \cite{poggiolesi2008cut}, Lahav \cite{Lahav}, Kurokawa \cite{kurokawa2013hypersequent}, Bednarska et al \cite{Bednarska Indrzejczak}, and Lellmann \cite{ Lellmann}.
The aim of this paper is to introduce a new sequent-style calculus for S5 by suggesting a framework of rooted hypersequents. A rooted hypersequent is of the form $ \Gamma\Rightarrow\Delta\,||\, P_1\Rightarrow Q_1\,|\cdots|\, P_n\Rightarrow Q_n $, where $ \Gamma $ and $ \Delta $ are multisets of arbitrary formulas and $ P_i $ and $ Q_i $ are multisets of atomic formulas. Precisely, a rooted hypersequent is given by a sequent $ \Gamma\Rightarrow\Delta $, called its root, together with a hypersequent $ \cal H $, called its crown, where all formulas in the components of crown are atomic formulas. The sequents in the crown work as storage for atomic formulas that they might be used to get axioms. This sequent is inspired by the grafted hypersequent in \cite{Kuznets Lellmann}. A difference is that in the grafted hypersequent, all formulas in the crown can be arbitrary formula. Thus, the notion of our calculus is very close to the notion of grafted hypersequent. The main idea for constructing rooted hypersequent is to take an ordinary sequent $ \Gamma\Rightarrow\Delta $ as a root and add sequences of multisets of atomic formulas to it.
Our calculus has the subformula property, and we show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. It is worth pointing out that in order to prove admissibility of cut rule, we make use of a normal form, called Quasi Normal Form, which is based on using modal and negation of modal formulas as literals. Soundness and completeness are established as well.
We proceed as follows. In the next section we recall the modal logic {\sf S5}. In Section \ref{G3S5}, we present rooted hypersequent calculus $ {\cal R}_{\sf S5}$. In Section \ref{section Soundness}, we prove soundness of the system with respect to Kripke models. In Section \ref{Structural properties}, we prove the admissibility of weakening and contraction rules, and some other properties of $ {\cal R}_{\sf S5}$. In Section \ref{sec cut}, we prove admissibility of cut rule, and completeness of the system. Finally, we conclude the paper in Section \ref{conclution}. \section{\bf Modal logic S5} \label{sec 2} In this section, we recall the axiomatic formulation of the modal logic S5.
The language of modal logic S5 is obtained by adding to the language of propositional logic the two modal operators $ \Box $ and $ \Diamond $. Atomic formulas are denoted by $ p,q,r, $ and so on.
Formulas, denoted by $ A,B,C,\ldots $,
are defined by the following grammar:
\[ A:=\bot\,|\top\,|p\,|\neg A|\,A\wedge A|\,A\vee A|\,A\rightarrow A|\,\Diamond A|\,\Box A,\] where $ \bot $ is a constant for falsity, and $ \top $ is a constant for truth.
Modal logic S5 has the following axiom schemes: \begin{align*} & \text{All propositional tautologies,}\\ & \text{(Dual)} \quad \Box A\leftrightarrow \neg \Diamond \neg A,\\ &\text{(K)}\quad\Box(A\rightarrow B)\rightarrow(\Box A\rightarrow \Box B),\\ &\text{(T)}\quad \Box A\rightarrow A,\\ &\text{(5)} \quad \Diamond A\rightarrow \Box\Diamond A. \end{align*} Equivalently, instead of (5) we can use: \begin{align*} & \text{(4)}\quad \Box A\rightarrow \Box\Box A,\\ & \text{(B)} \quad A\rightarrow \Box\Diamond A. \end{align*} The proof rules are Modus Ponens and Necessitation: \begin{center}
\AxiomC{$ A $}
\AxiomC{$ A\rightarrow B $}
\RightLabel{ MP,}
\BinaryInfC{$ B $}
\DisplayProof
$\quad$
\AxiomC{$A$}
\RightLabel{N.}
\UnaryInfC{$\Box A$}
\DisplayProof \end{center} Rule Necessitation can be applied only to theorems (i.e. to formulas derivable from no premise), for a detailed exposition see \cite{Chellas, Blackburn}.
If $ A $ is derivable in S5 from assumption $ \Gamma $, we write $ \Gamma\vdash_{\text{S5}} A $. \section{ Rooted Hypersequent $ \cal{R}_{\sf S5} $}\label{G3S5} Our calculus is based on finite multisets, i.e. on sets counting multiplicities of elements. We use certain categories of letters, possibly with subscripts or primed, as metavariables for certain syntactical categories (locally different conventions may be introduced): \begin{tasks}(2)
\task[$ \bullet $]
$ p$ and $q$ for atomic formulas,
\task[$ \bullet $]
$P$ and $Q$ for multisets of atomic formulas,
\task[$ \bullet $]
$ M$ and $N$ for multisets of modal formulas,
\task[$ \bullet $]
$ \Gamma$ and $\Delta$ for multisets of arbitrary formulas. \end{tasks} In addition, we use the following notations. \begin{itemize} \item The union of multisets $ \Gamma $ and $ \Delta $ is indicated simply by $ \Gamma,\Delta $. The union of a multiset $ \Gamma $ with a singleton multiset $ \{A\} $ is written $ \Gamma, A $. \item
We use $\neg\Gamma$ for multiset of formulas $\neg A$ such that $A\in \Gamma$.
\end{itemize} \begin{Def}
A sequent is a pair of multisets $ \Gamma $ and $ \Delta $, written as $ \Gamma\Rightarrow\Delta $. A hypersequent is a multiset of sequents,
written $ \Gamma_1\Rightarrow\Delta_1\,|\,\cdots,| \Gamma_n\Rightarrow\Delta_n $, where each $ \Gamma_i\Rightarrow\Delta_i $ is called a component. A rooted hypersequent
is given by a sequent $ \Gamma\Rightarrow\Delta $, called its root, together with a hypersequent $ \cal H $, called its crown, where all formulas in the components of crown are atomic formulas,
and is written as $ \Gamma\Rightarrow\Delta\,||\, {\cal H} $. If the crown is the empty hypersequent, the double-line separator can be omitted: a rooted hypersequent $ \Gamma\Rightarrow\Delta $ is understood as $ \Gamma\Rightarrow\Delta\,||\, \emptyset $. Formulas occurring on the
left-hand side of the sequent arrow in the root or a component of the crown are called antecedent formulas; those occurring on the right-hand side succedent formulas. \end{Def} Therefore, the notion of a rooted hypersequent
\[\Gamma\Rightarrow\Delta\, ||\, P_1 \Rightarrow Q_1\,|\, P_2\Rightarrow Q_2\,|\, \ldots \,|\,P_n\Rightarrow Q_n,\]
can be seen as a restriction of the notion of grafted hypersequent as in \cite{Kuznets Lellmann}. \begin{Def}
Let $\Gamma\Rightarrow\Delta\, ||\, P_1 \Rightarrow Q_1\,|\, P_2\Rightarrow Q_2\,|\, \ldots \,|\,P_n\Rightarrow Q_n $ be a rooted hypersequent. Its formula interpretation is the formula $\bigwedge\Gamma\rightarrow \bigvee\Delta\vee\bigvee\limits_{i=1}^n\Box(\bigwedge P_i\rightarrow \bigvee Q_i)$. \end{Def}
The axioms and rules of $ \cal{R}_{\sf S5} $ are given in the following:
\\ \textbf{Initial sequents:} \begin{center}
\begin{tabular}{cccccc}
\AxiomC{}
\RightLabel{\sf Ax}
\UnaryInfC{$p,\Gamma\Rightarrow \Delta,p\,||\, {\cal H} $}
\DisplayProof
&
&
\AxiomC{}
\RightLabel{\sf L$\bot$}
\UnaryInfC{$\bot,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\DisplayProof
&
&
\AxiomC{}
\RightLabel{\sf R$\top$}
\UnaryInfC{$\Gamma\Rightarrow \Delta, \top\,||\, {\cal H} $}
\DisplayProof
\end{tabular} \end{center}
\textbf{Propositional Rules:} \begin{center} \begin{tabular}{ccc}
\AxiomC{$ \Gamma\Rightarrow \Delta,A\,||\, {\cal H} $}
\RightLabel{\sf L$\neg $}
\UnaryInfC{$ \neg A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\DisplayProof
&
&
\AxiomC{$ A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\RightLabel{\sf R$\neg $}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,\neg A\,||\,{\cal H}$}
\DisplayProof
\\ [0.7cm]
\AxiomC{$ A,\Gamma\Rightarrow \Delta \,||\,{\cal H} $}
\AxiomC{$ B,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\RightLabel{\sf L$\vee $}
\BinaryInfC{$ A\vee B,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\DisplayProof
&
&
\AxiomC{$ \Gamma\Rightarrow \Delta, A,B\,||\,{\cal H} $}
\RightLabel{\sf R$\vee $}
\UnaryInfC{$\Gamma\Rightarrow \Delta,A\vee B \,||\,{\cal H}$}
\DisplayProof
\\[0.7cm]
\AxiomC{$ A,B,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\RightLabel{\sf L$\wedge $}
\UnaryInfC{$A\wedge B,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\DisplayProof
&
&
\AxiomC{$ \Gamma\Rightarrow \Delta,A \,||\,{\cal H}$}
\AxiomC{$ \Gamma\Rightarrow \Delta,B\,||\,{\cal H} $}
\RightLabel{\sf R$\wedge $}
\BinaryInfC{$ \Gamma\Rightarrow \Delta,A\wedge B\,||\,{\cal H} $}
\DisplayProof
\\[0.7cm]
\AxiomC{$ \Gamma\Rightarrow \Delta, A \,||\,{\cal H}$}
\AxiomC{$ B,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\RightLabel{\sf L$\rightarrow $}
\BinaryInfC{$ A\rightarrow B,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\DisplayProof
&
&
\AxiomC{$ A,\Gamma\Rightarrow \Delta,B\,||\,{\cal H} $}
\RightLabel{\sf R$\rightarrow $}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,A\rightarrow B\,||\,{\cal H} $}
\DisplayProof \end{tabular} \end{center}
\textbf{Modal Rules:} \begin{center} \begin{tabular}{ccc}
\AxiomC{$ A,M\Rightarrow N\,||\,{\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$\Diamond A,M,P\Rightarrow Q,N\,||\,{\cal H} $}
\DisplayProof
&
&
\AxiomC{$\Gamma\Rightarrow \Delta,\Diamond A,A \,||\,{\cal H}$}
\RightLabel{\sf R$\Diamond $}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,\Diamond A\,||\,{\cal H}$}
\DisplayProof
\\[0.7cm]
\AxiomC{$ A,\Box A,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\RightLabel{\sf L$\Box $}
\UnaryInfC{$\Box A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\DisplayProof
&
&
\AxiomC{$ M\Rightarrow N,A\,||\, {\cal H}\,|\,P\Rightarrow Q$}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$M,P\Rightarrow Q,N,\Box A \,||\,{\cal H}$}
\DisplayProof \end{tabular} \end{center}
\textbf{Structural Rule:} \begin{center} \begin{tabular}{c}
\AxiomC{$M,P_i\Rightarrow Q_i,N\,||\,{\cal H}\,| P\Rightarrow Q\,|{\cal G} $}
\RightLabel{\sf Exch}
\UnaryInfC{$M,P\Rightarrow Q,N\,||\,{\cal H}\,| P_i\Rightarrow Q_i\,|{\cal G} $}
\DisplayProof
\end{tabular} \end{center} Let us make some remarks on the {\sf L$ \Diamond $}, {\sf R$ \Box $} and {\sf Exch} (abbreviates for exchange). All formulas in the conclusions of these rules are atomic or modal formulas. In a backward proof search by applying these rules, atomic formulas in the root of the conclusions move to the crown of the premises as a new sequent. Suppose, $P$ in the antecedent and $Q$ in the succedent of the root sequent of conclusion move to the crown of premise, the formulas in $P$ and $Q$ are saved in the crowns, until applications of the rule {\sf Exch} in a derivation. In other words, The sequents in the crown work as storage for atomic formulas that they might be used to get axioms. By applying the rule {\sf Exch}, the multisets $ P_i$ and $ Q_i $ come out from the crown while $ P $ and $ Q $ move to the crown.
\begin{Examp}\label{exa}{\rm The following sequents are derivable in $ \cal{R}_{\sf S5} $. \begin{enumerate}
\item \label{exa4} $\Rightarrow (r\wedge p)\rightarrow (q\rightarrow \Box(\Diamond (p\wedge q)\wedge \Diamond r)) $ \item \label{exa5} $ \Rightarrow\Box(\Box\neg p\vee p)\rightarrow \Box(\neg p\vee \Box p) $ \end{enumerate}} \end{Examp} \begin{proof}
$ \, $\\
\begin{prooftree}
1.
\AxiomC{}
\RightLabel{Ax}
\UnaryInfC{$ p,q,r\Rightarrow \Diamond(p\wedge q),p $}
\AxiomC{}
\RightLabel{Ax}
\UnaryInfC{$ r,p,q\Rightarrow \Diamond(p\wedge q),q $}
\RightLabel{\sf R$\wedge $}
\BinaryInfC{$ r,p,q\Rightarrow \Diamond(p\wedge q), p\wedge q $}
\RightLabel{\sf R$\Diamond $}
\UnaryInfC{$ r,p,q\Rightarrow \Diamond(p\wedge q) $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Rightarrow \Diamond(p\wedge q)\,||\,r,p,q\Rightarrow $}
\AxiomC{}
\RightLabel{Ax}
\UnaryInfC{$ r,p,q\Rightarrow \Diamond r,r $}
\RightLabel{\sf R$\Diamond $}
\UnaryInfC{$ r,p,q\Rightarrow \Diamond r $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Rightarrow \Diamond r \,||\,r,p,q\Rightarrow$}
\RightLabel{\sf R$\wedge $}
\BinaryInfC{$ \Rightarrow\Diamond (p\wedge q)\wedge \Diamond r\,||\,r,p,q\Rightarrow $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ r,p,q\Rightarrow\Box(\Diamond (p\wedge q)\wedge \Diamond r) $}
\RightLabel{\sf L$\wedge $}
\UnaryInfC{$ r\wedge p,q\Rightarrow \Box(\Diamond (p\wedge q)\wedge \Diamond r) $}
\RightLabel{\sf R$\rightarrow $}
\UnaryInfC{$ r\wedge p\Rightarrow q\rightarrow \Box(\Diamond (p\wedge q)\wedge \Diamond r) $}
\RightLabel{\sf R$\rightarrow $}
\UnaryInfC{$\Rightarrow (r\wedge p)\rightarrow (q\rightarrow \Box(\Diamond (p\wedge q)\wedge \Diamond r)) $} \end{prooftree} \begin{prooftree}
2.
\AxiomC{}
\RightLabel{\sf Ax}
\UnaryInfC{$\Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow p \,||\,\Rightarrow p $}
\RightLabel{\sf L$\neg $}
\UnaryInfC{$\neg p,\Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow \,||\,\Rightarrow p $}
\RightLabel{\sf L$\Box $}
\UnaryInfC{$ \Box\neg p,\Box(\Box\neg p\vee p),p\Rightarrow \,||\,\Rightarrow p $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Box\neg p,\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow $}
\AxiomC{}
\RightLabel{\sf Ax}
\UnaryInfC{$ p,\Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow $}
\RightLabel{\sf L$\vee $}
\BinaryInfC{$ \Box\neg p\vee p,\Box(\Box\neg p\vee p)\Rightarrow p \,||\,p\Rightarrow$}
\RightLabel{{\sf L$\Box$},}
\UnaryInfC{$ \Box(\Box\neg p\vee p)\Rightarrow p\,||\,p\Rightarrow $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ \Box(\Box\neg p\vee p),p\Rightarrow \Box p $}
\RightLabel{\sf R$\neg $}
\UnaryInfC{$ \Box(\Box\neg p\vee p)\Rightarrow \neg p,\Box p $}
\RightLabel{\sf R$\vee $}
\UnaryInfC{$ \Box(\Box\neg p\vee p)\Rightarrow \neg p\vee\Box p $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ \Box(\Box\neg p\vee p)\Rightarrow \Box(\neg p\vee\Box p) $}
\RightLabel{\sf R$\rightarrow $}
\UnaryInfC{$\Rightarrow \Box(\Box\neg p\vee p)\rightarrow \Box(\neg p\vee\Box p) $} \end{prooftree} \end{proof} \section{ Soundness}\label{section Soundness} In this section we prove soundness of the rules with respect to Kripke models.
A Kripke model $\mathcal{M}$ for S5 is a triple $ \mathcal{M}=(W, R, V) $ where $ W $ is a set of states, $ R $ is an equivalence relation on $ W $ and $ V: \varPhi\rightarrow \mathcal{P} (W) $ is a valuation function, where $ \varPhi $ is the set of propositional variables. Suppose that $ w\in W $. We inductively define the notion of a formula $ A $ being satisfied in $\mathcal{M}$ at state $ w $ as follows: \begin{itemize}
\item
$\mathcal{M},w \vDash p \quad \text{iff} \quad w\in V(p),\, \text{where}\,\,
p\in \varPhi $,
\item
$\mathcal{M},w \vDash \neg A \quad \text{iff} \quad \mathcal{M},w \nvDash A,$
\item
$\mathcal{M},w \vDash A\vee B \quad \text{iff} \quad \mathcal{M},w \vDash A \,\,\text{or}\,\, \mathcal{M},w \vDash B,$
\item
$\mathcal{M},w \Rightarrow A\wedge B \quad \text{iff} \quad \mathcal{M},w \vDash A \,\,\text{and}\,\, \mathcal{M},w \vDash B,$
\item
$\mathcal{M},w \vDash A\rightarrow B \quad \text{iff} \quad \mathcal{M},w \nvDash A \,\text{or}\, \mathcal{M},w \vDash B,$
\item
$\mathcal{M},w \vDash \Diamond A \quad \text{iff} \quad \,\,\mathcal{M},v \Rightarrow A\,\,\text{for some}\,\, v\in W \,\,\text{such that}\,\, R(w,v), $
\item
$\mathcal{M},w \vDash \Box A \quad \text{iff} \quad \mathcal{M},v \Rightarrow A \,\,\text{for all}\,\, v\in W \,\text{such that}\, R(w,v). $ \end{itemize} We extend semantical notions to sequents in the following way: \begin{itemize}
\item
$ \mathcal{M},w \vDash \Gamma\Rightarrow \Delta\,||\,P_1\Rightarrow Q_1\,|\cdots P_n\Rightarrow Q_n $ iff
$ \mathcal{M},w \vDash \bigwedge\Gamma\rightarrow \bigvee\Delta\vee\bigvee_{i=1}^n\Box(\bigwedge P_i\rightarrow \bigvee Q_i)$.
\item
$ \mathcal{M} \vDash \Gamma\Rightarrow \Delta \,||\, {\cal H} $ iff $ \mathcal{M},w \Rightarrow \Gamma\Rightarrow \Delta \,||\, {\cal H} $, for all $ w $ in the domain of $ \mathcal{M} $.
\item
$ \vDash \Gamma\Rightarrow \Delta\,||\, {\cal H} $ iff
$ \mathcal{M} \vDash \Gamma\Rightarrow \Delta \,||\, {\cal H} $, for all S5 models $ \mathcal{M} $.
\item
The sequent $ \Gamma\Rightarrow \Delta\,||\, {\cal H} $ is called S5-valid if $ \vDash \Gamma\Rightarrow \Delta \,||\, {\cal H}$. \end{itemize} \begin{Lem}\label{2.1}
Let
$\mathcal{M}=(W,R,V)$
be a {\rm Kripke} model for
$ {\sf S5} $.
\begin{itemize}
\item[{\rm (1)}]
$\mathcal{M},w \Rightarrow \Box A \quad \text{iff} \quad \mathcal{M},w' \Rightarrow \Box A$, for all $ w'\in W $, where $ wRw' $.
\item[{\rm (2)}]
$\mathcal{M},w \Rightarrow \Diamond A \quad \text{iff} \quad \mathcal{M},w' \Rightarrow \Diamond A$, for all $ w'\in W $, where $ wRw' $.
\item[{\rm (3)}]
If
$\mathcal{M},w \Rightarrow A$, then $\mathcal{M},w' \Rightarrow \Diamond A$, for all $ w'\in W $, where $ wRw' $.
\end{itemize} \end{Lem} \begin{proof}
The proof clearly follows from the definition of satisfiability and the fact that
$ R $ is an equivalence relation. \end{proof} \begin{The}[Soundness]\label{Soundness}
If
$ \Gamma\Rightarrow \Delta \,||\, {\cal H}$
is provable in $ \cal{R}_{\sf S5} $, then it is S5-valid. \end{The} \begin{proof}
The proof is by induction on the height of the derivation of $ \Gamma\Rightarrow \Delta\,||\, {\cal H} $. Initial sequents are obviously valid
in every Kripke model for S5. We only check the induction step for rules {\sf R$ \Box $} and {\sf Exch}. The rule {\sf L$ \Diamond $} can be verified similarly.
\begin{itemize}
\item
Rule {\sf R$ \Box $}:
Suppose that the sequent $ \Gamma\Rightarrow \Delta \,||\, {\cal H} $ is $M,P\Rightarrow Q,N,\Box A \,||\,{\cal H} $, the conclusion of rule {\sf R$ \Box $}, with the premise $M\Rightarrow N,A\,||\,P\Rightarrow Q \,|\,{\cal H} $. For convenience, let the hypersequent $ {\cal H} $ be a sequent $ P_1\Rightarrow Q_1 $. Suppose, by induction hypothesis, that the premise is valid, i.e., for every Kripke model $ \cal M $ we have
\begin{equation}\label{3}
\text{If}\,\, \mathcal{M},w\vDash \bigwedge M,\,\, \text{then}\,\, \mathcal{M},w \vDash\bigvee N\vee A \vee \Box(\bigwedge P\rightarrow \bigvee Q)\vee \Box(\bigwedge P_1\rightarrow \bigvee Q_1).
\end{equation}
Assume the conclusion is not S5-valid i.e., there is a model $ \mathcal{M}=(W,R,V) $ and $ w'\in W $ such that
\begin{align}
&\mathcal{M},w'\vDash \bigwedge M\wedge \bigwedge P\label{1} \\
& \mathcal{M},w'\nvDash \bigvee Q\vee \bigvee N\vee\Box A \vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1).\label{2}
\end{align} Thus,
$\mathcal{M},w'\nvDash \bigvee N\vee \Box(\bigwedge P\rightarrow \bigvee Q)\vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1)$. Suppose $ w'Rw $, it follows from \ref{1} and Lemma \ref{2.1} that $ \mathcal{M},w\vDash \bigwedge M $ and
$\mathcal{M},w\nvDash \bigvee N\vee \Box(\bigwedge P\rightarrow \bigvee Q)\vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1)$. Therefore, by (\ref{3}) we have
$ \mathcal{M},w\vDash A $, and so $ \mathcal{M},w'\vDash \Box A $. This leads to a contradiction with (\ref{2}).
\item
Rule {\sf Exch}:
Suppose that the sequent $ \Gamma\Rightarrow \Delta \,||\, {\cal H} $ is the conclusion of the rule {\sf Exch}.
For convenience, let the hypersequent $ \cal{H} $ be a sequent $ P_1\Rightarrow Q_1 $ and let $ \cal{G} $ be a sequent $ P_2\Rightarrow Q_2 $. Suppose, by induction hypothesis, that the premise is valid i.e., for every model $ \cal M $ we have:
\begin{align}\label{6}
&\text{If}\,\, \mathcal{M},w \vDash \bigwedge M\wedge\bigwedge P_i,\,\, \text{then}\nonumber\\
& \mathcal{M},w \vDash \bigvee Q_i\vee \bigvee N\vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1)\vee\Box(\bigwedge P\rightarrow \bigvee Q)\vee \Box(\bigwedge P_2\rightarrow \bigvee Q_2).
\end{align}
Assume the conclusion is not valid i.e., there is a model $ \mathcal{M}=(W,R,V) $ and $ w'\in W $ such that
\begin{align}
&\mathcal{M},w'\vDash \bigwedge M \wedge \bigwedge P\label{4} \\
& \mathcal{M},w'\nvDash \bigvee Q\vee \bigvee N\vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1)\vee \Box(\bigwedge P_i\rightarrow \bigvee Q_i)\vee \Box(\bigwedge P_2\rightarrow \bigvee Q_2) \label{5}
\end{align}
By \ref{5}, we have $\mathcal{M},w\nvDash \bigwedge P_i\rightarrow \bigvee Q_i $ for some $ w$ such that $w'Rw$. Thus, $\mathcal{M},w\vDash\bigwedge P_i$, and $\mathcal{M},w\nvDash\bigvee Q_i$. In addition, because $M$ is a multiset of modal formula, using Lemma \ref{2.1} and (\ref{4}) we can conclude that $\mathcal{M},w\vDash \bigwedge M$. That means, $\mathcal{M},w\vDash \bigwedge M \wedge\bigwedge P_i$ and $\mathcal{M},w\nvDash\bigvee Q_i$ for some $w$ such that $w'Rw$. It follows from (\ref{6}) that
\[\mathcal{M},w \vDash \bigvee N\vee\Box(\bigwedge P_1\rightarrow \bigvee Q_1)\vee\Box(\bigwedge P\rightarrow \bigvee Q)\vee \Box(\bigwedge P_2\rightarrow \bigvee Q_2).\] This, Using Lemma \ref{2.1}, leads to a contradiction with (\ref{5}).
\end{itemize} \end{proof} \section{Structural properties}\label{Structural properties} In this section, we prove the admissibility of weakening and contraction rules, and also some properties of $ \cal{R}_{\sf S5} $, which are used to prove the admissibility of cut rule. Some (parts of) proofs are omitted because they are easy or similar to the proofs in \cite{Buss,negri2008structural,bpt}.
The \textit{height} of a derivation is the greatest number of successive applications of rules in it, where initial sequents have height $0$. The notation
$ \vdash_n\Gamma\Rightarrow \Delta\, ||\, \cal{H} $ means that
$ \Gamma\Rightarrow \Delta\, ||\, \cal{H} $ is derivable with a height of derivation at most $ n $ in the system $ \cal{R}_{\sf S5} $. A rule of $ \cal{R}_{\sf S5} $ is said to be (height-preserving) \textit{admissible} if whenever an instance of its premise(s) is (are) derivable in $ \cal{R}_{\sf S5} $ (with at most height $ n $), then so is the corresponding instance of its conclusion. A rule of $ \cal{R}_{\sf S5} $ is said to be \textit{invertible} if whenever an instance of its conclusion is derivable in $ \cal{R}_{\sf S5} $, then so is the corresponding instance of its premise(s).
The following lemma, shows that the propositional rules are height-preserving invertible. The proof is by induction on the height of derivations. \begin{Lem}\label{invertibility of propositional rules }
All propositional rules are height-preserving invertible in $ \cal{R}_{\sf S5} $. \end{Lem} In order to prove the admissibility of cut rule, we introduce the following structural rules, called {\sf Merge} and $ {\sf Merge}^{\sf c} $. Rule $ {\sf Merge}^{\sf c} $ allows us to merge two crown components in the crown, and Rule {\sf Merge} allows us to merge a component in the crown with the root component. \begin{Lem}\label{Merge}
The following rules are height-preserving admissible
\begin{center}
\AxiomC{$ \Gamma\Rightarrow \Delta\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I} $}
\RightLabel{$ {\sf Merge}^{\sf c} $}
\UnaryInfC{$\Gamma\Rightarrow \Delta\,||\, {\cal H} \,|\,P_i,P_j\Rightarrow Q_i,Q_j\,|\, {\cal G}\,|\,{\cal I}$}
\DisplayProof
$ \quad $
\AxiomC{$ \Gamma\Rightarrow \Delta\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G} $}
\RightLabel{\sf Merge}
\UnaryInfC{$ \Gamma,P_i\Rightarrow Q_i, \Delta\,||\, {\cal H} \,|\, {\cal G} $}
\DisplayProof
\end{center} \end{Lem} \begin{proof}
Both rules are proved simultaneously by induction on the height of the derivations of the premises. In both rules, if the premise is an initial sequent, then the conclusion is an initial sequent too.
For the induction step, we consider only cases where the last rule is {\sf L$\Diamond $} or {\sf Exch}; since the rule {\sf R$\Box $} is treated symmetrically and for the remaining
rules it suffices to apply the induction hypothesis to the premise and
then use the same rule to obtain deduction of the conclusion.\\
For the rule {\sf Merge} we consider the following cases.
Case 1. Let $ \Gamma=\Diamond A,M,P $ and
$ \Delta=Q,N $
and let $ \Diamond A $ be the principal formula:
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ \vdash_n A,M \Rightarrow N\,||\, {\cal H} \,|P_i\Rightarrow Q_i\,|\, {\cal G}\,|\, P\Rightarrow Q$}
\RightLabel{{\sf L$\Diamond $},}
\UnaryInfC{$\vdash_{n+1} \Diamond A,M,P \Rightarrow Q,N\,||\, {\cal H} \,|P_i\Rightarrow Q_i\,|\, {\cal G} $}
\end{prooftree}
By induction hypothesis (IH), we have:
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ \vdash_n A,M \Rightarrow N\,||\, {\cal H} \,|P_i\Rightarrow Q_i\,|\, {\cal G}\,|\, P\Rightarrow Q $}
\RightLabel{IH ($ {\sf Merge}^{\sf c} $)}
\UnaryInfC{$ \vdash_n A,M \Rightarrow N\,||\, {\cal H} \,|\, {\cal G}\,|\, P,P_i\Rightarrow Q_i,Q$}
\RightLabel{{\sf L$\Diamond $}.}
\UnaryInfC{$ \vdash_{n+1} \Diamond A,M,P,P_i\Rightarrow Q_i,Q,N \,||\,{\cal H} \,|\, {\cal G} $}
\end{prooftree}
Case 2. Let $ \Gamma=M,P$ and $ \Delta=Q,N $, and let the last rule be as:
\begin{prooftree}
\AxiomC{$\vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H} \,|\,P\Rightarrow Q\,|\, {\cal G} $}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$\vdash_{n+1} M,P\Rightarrow Q,N\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G} $}
\end{prooftree}
In this case, by induction hypothesis, the sequent $\vdash_n M,P,P_i\Rightarrow Q_i,Q,N\,||\, {\cal H} \,|\, {\cal G} $ is obtained.\\
For the rule $ {\sf Merge}^{\sf c} $ we consider the following cases.
Case 1. The premise is derived by {\sf Exch}. If this rule does not apply to $ P_i\Rightarrow Q_i $ and $ P_j\Rightarrow Q_j $, then the conclusion is obtained by applying induction hypothesis and then applying the same rule {\sf Exch}. Therefore, let the last rule be as follows in which the rule {\sf Exch} apply to the component $ P_i \Rightarrow Q_i $:
\begin{prooftree}
\AxiomC{$ \vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H} \,|\,P\Rightarrow Q\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I} $}
\RightLabel{{\sf Exch},}
\UnaryInfC{$\vdash_{n+1} M,P\Rightarrow Q,N\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I} $}
\end{prooftree} where $ \Gamma=M,P $ and $ \Delta=Q,N $. Then the conclusion is derived as follows
\begin{prooftree}
\AxiomC{$ \vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H} \,|\,P\Rightarrow Q\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I} $}
\RightLabel{{\sf Merge} (IH)}
\UnaryInfC{$\vdash_n M,P_i,P_j\Rightarrow Q_j,Q_i,N\,||\, {\cal H} \,|\,P\Rightarrow Q\,|\, {\cal G}\,|\,{\cal I} $}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$\vdash_{n+1} M,P\Rightarrow Q,N\,||\, {\cal H} \,|\,P_i,P_j\Rightarrow Q_j,Q_i\,|\, {\cal G}\,|\,{\cal I} $}
\end{prooftree}
Case 2. The premise is derived by {\sf L$ \Diamond $}.
\begin{prooftree}
\AxiomC{$ \vdash_n A,M\Rightarrow N\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I}\,|\,P\Rightarrow Q$}
\RightLabel{{\sf L$ \Diamond $},}
\UnaryInfC{$\vdash_{n+1} \Diamond A,M,P\Rightarrow Q,N\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I}$} \end{prooftree} where $ \Gamma=\Diamond A,M,P $ and $ \Delta= Q,N $. Then the conclusion is obtained as follows: \begin{prooftree}
\AxiomC{$ \vdash_n A,M\Rightarrow N\,||\, {\cal H} \,|\,P_i\Rightarrow Q_i\,|\, {\cal G}\,|\,P_j\Rightarrow Q_j\,|\,{\cal I}\,|\,P\Rightarrow Q$}
\RightLabel{IH}
\UnaryInfC{$\vdash_n A,M\Rightarrow N\,||\, {\cal H} \,|\,P_i,P_j\Rightarrow Q_i,Q_j\,|\, {\cal G}\,|\,{\cal I}\,|\,P\Rightarrow Q$}
\RightLabel{{\sf L$ \Diamond $}.}
\UnaryInfC{$\vdash_{n+1} \Diamond A,M,P\Rightarrow Q,N\,|| {\cal H} \,|\,P_i,P_j\Rightarrow Q_i,Q_j\,|\, {\cal G}\,|\,{\cal I}$} \end{prooftree} \end{proof} \subsection{Admissibility of weakening }
In this subsection, we prove the admissibility of structural rules of external weakening $ {\sf EW} $, crown weakening $ {\sf W}^{\sf c} $ , left rooted weakening $ {\sf LW} $, right rooted weakening $ {\sf RW} $.
\begin{Lem}\label{external weakening}
The rule of external weakening:
\begin{prooftree}
\AxiomC{$ \Gamma\Rightarrow \Delta\,||\, {\cal H}\,|\, \cal{G} $}
\RightLabel{$ \sf EW $}
\UnaryInfC{$ \Gamma\Rightarrow \Delta\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\, \cal{G} $}
\end{prooftree}
is height-preserving admissible in $ {\cal R}_{\sf S5} $.
\end{Lem}
\begin{proof}
By straightforward induction on the height of the derivation of the premise.
\end{proof}
\begin{Lem}\label{modal and atomic weakening}
The rule of crown weakening:
\begin{prooftree}
\AxiomC{$ \Gamma\Rightarrow\Delta\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i\,|\, {\cal G} $}
\RightLabel{$ {\sf W}^{\sf c} $}
\UnaryInfC{$ \Gamma\Rightarrow\Delta\,||\, {\cal H}\,|\,P,P_i\Rightarrow Q_i,Q\,|\, {\cal G} $}
\end{prooftree} is height-preserving admissible in $ {\cal R}_{\sf S5} $.
\end{Lem} \begin{proof}
{\rm
It follows by the height-preserving admissibility of the two rules $ \sf EW $ and $ {\sf Merge}^{\sf c} $. } \end{proof} \begin{Lem} \label{W2}
The rules of left and right weakening:
\begin{center}
\AxiomC{$ \Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{\sf LW}
\UnaryInfC{$ A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\DisplayProof
$\qquad$
\AxiomC{$ \Gamma\Rightarrow \Delta \,||\, {\cal H}$}
\RightLabel{\sf RW,}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,A\,||\, {\cal H} $}
\DisplayProof
\end{center}
are admissible, where $ A $ is an arbitrary formula. \end{Lem} \begin{proof}
{\rm
Both rules are proved simultaneously by induction on the complexity of $ A $ with subinduction on the height of the derivations. The admissibility of the rules for atomic formula follows by admissibility of the two rules {\sf EW} and {\sf Merge}. The admissibility of the rules for modal formula are straightforward by induction on the height of the derivation. The other cases are proved easily.
} \end{proof} \subsection{Invertibility } In this subsection, first we introduce a normal form called Quasi Normal Form, which are used to prove the admissibility of the contraction and cut rules. Then we show that the structural and modal rules are invertible. \begin{Def}[Quasi-literal]
A quasi-literal is an atomic formula, a negation of atomic formula, a modal formula, or a negation of modal formula.
\end{Def} \begin{Def}[Quasi-clause and quasi-phrase] A quasi-clause (quasi-phrase) is either a single quasi-literal or a conjunction (disjunction) of quasi-literals.
\end{Def} \begin{Def}
A formula is in conjunctive (disjunctive) quasi-normal form, abbreviated CQNF (DQNF), iff it is a conjunction (disjunction) of quasi-clauses.
\end{Def} \begin{Examp}
The formula $(\neg\Box(A\rightarrow B)\vee p \vee \Diamond C)\wedge(\neg q)\wedge(\Box\Diamond A\vee \neg \Diamond(A\wedge B)\vee\neg r)$ is in CQNF, which $(\neg\Box(A\rightarrow B)\vee p \vee \Diamond C)$, $(\neg q)$, and $(\Box\Diamond A\vee \neg \Diamond(A\wedge B)\vee\neg r)$ are quasi-clauses. \end{Examp} Without loss of generality we can assume that every formula in CQNF is as follows \[\bigwedge_{i=1}^k(\bigvee P_i\vee\bigvee \neg Q_i\vee\bigvee M_i\vee\bigvee \neg N_i),\] and every formula in DQNF is as follows \[\bigvee_{i=1}^l(\bigwedge P_i\vee\bigwedge \neg Q_i\vee\bigwedge M_i\vee\bigwedge \neg N_i),\] where $P_i$ and $Q_i$ are multisets of atomic formulae and $M_i$ and $N_i$ are multisets of modal formulae.
Similar to the propositional logic, Every formula can be transformed into an equivalent formula in CQNF, and an equivalent formula in DQNF.
For these quasi-normal forms we have the following lemmas. \begin{Lem}\label{NFcut1}
Let $ A $ be an arbitrary formula, and let $\bigwedge_{i=1}^k(\bigvee \neg P_i\vee\bigvee Q_i\vee\bigvee \neg M_i\vee\bigvee N_i)$ and $\bigvee_{i=1}^{l}(\bigwedge P'_i\vee\bigwedge \neg Q'_i\vee\bigwedge M'_i\vee\bigwedge \neg N'_i)$ be equivalent formulae in CQNF and DQNF, respectively, for $A$. Then
\begin{itemize}
\item[{\rm (i)}]
$ \vdash\Gamma\Rightarrow \Delta, A\,||\, {\cal H} $
iff
$\vdash M_i,P_i,\Gamma\Rightarrow \Delta, Q_i, N_i\,||\, {\cal H}$,
for every $i=1,\ldots,k$.
\item[{\rm (ii)}]
$\vdash A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $
iff
$\vdash M'_i, P'_i,\Gamma\Rightarrow \Delta,Q'_i, N'_i\,||\, {\cal H}$,
for every $i=1,\ldots,l$.
\end{itemize} \end{Lem} \begin{proof}
We consider the first assertion; the second is treated symmetrically.\\
$ (\Leftarrow) $. Suppose that $ M_i,P_i,\Gamma\Rightarrow \Delta, Q_i, N_i\,||\, {\cal H}$ is derivable
for every $i=1,\ldots,k$. By applying propositional rules, where non-modal subformulae of $ A $ are principal formulae in a backward proof search with $ \Gamma\Rightarrow \Delta, A\,||\, {\cal H} $ in the bottom.
We obtain a derivation, in which the topsequents are $ M_i,P_i,\Gamma\Rightarrow \Delta, Q_i, N_i\,||\, {\cal H}$, $i=1,\ldots,k$.\\
$ (\Rightarrow) $. Suppose $ \Gamma\Rightarrow \Delta, A \,||\, {\cal H}$ is derivable. By applying the invertibility of the propositional rules, we reach a derivation for $ M_i,P_i,\Gamma\Rightarrow \Delta, Q_i, N_i\,||\, {\cal H}$, for every $i=1,\ldots,k$. \end{proof} \begin{Cor}\label{left to right1}
Both rules in the above lemma from left to right are height-preserving admissible. In other words:
\item[{\rm (i)}]
If $ \vdash_n\Gamma\Rightarrow \Delta, A\,||\, {\cal H} $,
then
$\vdash_n M_i,P_i,\Gamma\Rightarrow \Delta, Q_i, N_i\,||\, {\cal H}$,
for every $i=1,\ldots,k$.
\item[{\rm (ii)}]
If $\vdash_n A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $,
then
$\vdash_n M'_i, P'_i,\Gamma\Rightarrow \Delta,Q'_i, N'_i\,||\, {\cal H}$,
for every $i=1,\ldots,l$. \end{Cor} \begin{Lem}\label{NFcut2}
Let $ A $ be an arbitrary formula, and let $\bigwedge_{i=1}^k(\bigvee\neg P_i\vee\bigvee Q_i\vee\bigvee\neg M_i\vee\bigvee N_i)$ and $\bigvee_{i=1}^{l}(\bigwedge P'_i\vee\bigwedge \neg Q'_i\vee\bigwedge M'_i\vee\bigwedge \neg N'_i)$ be equivalent formulae in CQNF and DQNF, respectively, for $A$. Then
\begin{itemize}
\item[{\rm (i)}]
$\vdash \Gamma\Rightarrow \Delta,\Box A\,||\, {\cal H} $
iff
$\vdash M_i,\Gamma\Rightarrow \Delta, N_i\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i$,
for every $i=1,\ldots,k$.
\item[{\rm (ii)}]
$ \vdash\Diamond A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $
iff
$\vdash M'_i, \Gamma\Rightarrow \Delta, N'_i\,||\, {\cal H}\,|\,P'_i\Rightarrow Q'_i$,
for every $i=1,\ldots,l$.
\end{itemize} \end{Lem} \begin{proof}
We prove the first assertion; the second is proved by a similar argument. For the direction from right to left observe that by Lemma \ref{NFcut1}, $ \vdash M_i,\Gamma\Rightarrow \Delta, N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i $ implies
$ \vdash M_i,M,P\Rightarrow N,Q, N_i\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i$, where $ M,P$ and $N,Q $ are multisets of
atomic and modal formulae corresponding to formulae in $ \Gamma $ and $ \Delta$. Then by applying rule {\sf Exch} we have:
\begin{prooftree}
\AxiomC{$ \vdash M_i,M,P\Rightarrow N,Q, N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i$}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$ \vdash M_i,M,P_i\Rightarrow N,Q_i, N_i\,||\, {\cal H} \,|\,P\Rightarrow Q$}
\end{prooftree}
Therefore, $ \vdash M_i,M,P_i\Rightarrow N,Q_i, N_i\,||\, {\cal H}\,|\,P\Rightarrow Q $, for $ i=1,\ldots,k $, and then by Lemma \ref{NFcut1} we have\\
$\vdash M\Rightarrow N,A\,||\, {\cal H}\,|\,P\Rightarrow Q $. Hence, by applying rule {\sf R$ \Box $} we have: \begin{prooftree}
\AxiomC{$\vdash M\Rightarrow N,A\,||\, {\cal H}\,|\,P\Rightarrow Q$}
\RightLabel{\sf R$ \Box $}
\UnaryInfC{$\vdash M,P\Rightarrow N,Q,\Box A\,||\, {\cal H} $}
\end{prooftree}
Again by applying Lemma \ref{NFcut1} we have $\vdash \Gamma\Rightarrow \Delta,\Box A \,||\, {\cal H}$.\\
For the opposite direction, by induction on $ n $, we prove that if $ \vdash_n\Gamma\Rightarrow \Delta,\Box A\,||\, {\cal H} $, then\\
$\vdash_n M_i,\Gamma\Rightarrow \Delta, N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i$, for $ i=1,\ldots,k $. For $ n=0 $, if $ \Gamma\Rightarrow \Delta,\Box A\,||\, {\cal H} $ is an initial sequent, then $ \Box A $ is not principal, and so $ M_i,\Gamma\Rightarrow \Delta, N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i$ is an initial sequent too, for $ i=1,\ldots,k $. Assume height-preserving admissible up to height n, and let
$$ \vdash_{n+1} \Gamma\Rightarrow \Delta,\Box A\,||\, {\cal H} $$ If $ \Box A $ is not principal, we apply induction hypothesis to the premise(s) and then use the same rule to obtain deductions of $ M_i,\Gamma\Rightarrow \Delta, N_i\,||\,{\cal H} \,|\, P_i\Rightarrow Q_i$, for $ i=1,\ldots,k $. If on the other hand $ \Box A $ is principal, the derivation ends with
\begin{prooftree}
\AxiomC{$\vdash_n M\Rightarrow N, A\,||\,{\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{{\sf R$ \Box $},}
\UnaryInfC{$\vdash_{n+1}M,P\Rightarrow Q,N,\Box A\,||\,{\cal H} $}
\end{prooftree}
where $ \Gamma= M,P $ and $ \Delta= Q,N$. Therefore, by applying Corollary \ref{left to right1}, we have
$$\vdash_n M_i,P_i,M\Rightarrow N, N_i,Q_i\,||\,{\cal H}\,|\,P\Rightarrow Q, $$
for $ i=1,\ldots,k $. Hence, by applying rule {\sf Exch} the conclusion is obtained as follows:
\begin{prooftree}
\AxiomC{$\vdash_n M_i,P_i,M\Rightarrow N, N_i,Q_i\,||\,{\cal H} \,|\,P\Rightarrow Q$ }
\RightLabel{{\sf Exch}.}
\UnaryInfC{$\vdash_{n+1} M_i,P,M\Rightarrow N, N_i,Q\,||\,{\cal H}\,|\,P_i\Rightarrow Q_i $ }
\end{prooftree} \end{proof} \begin{Cor}\label{left to right2}
Both rules in the above lemma from left to right are height-preserving admissible. In other words:
\begin{itemize}
\item[{\rm (i)}]
If $\vdash_n \Gamma\Rightarrow \Delta,\Box A\,||\, {\cal H} $,
then
$\vdash_n M_i,\Gamma\Rightarrow \Delta, N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i$,
for every $i=1,\ldots,k$.
\item[{\rm (ii)}]
If $ \vdash_n\Diamond A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $,
then
$\vdash_n M'_i, \Gamma\Rightarrow \Delta, N'_i\,||\, {\cal H}\,|\,P'_i\Rightarrow Q'_i$,
for every $i=1,\ldots,l$.
\end{itemize} \end{Cor} \begin{Lem} \label{inversion for modal rules}
The structural and modal rules are invertible in $ \cal{R}_{\sf S5} $. \end{Lem} \begin{proof}
Rules {\sf L$ \Box $} and {\sf R$ \Diamond $} have repetition of the principle formula in the premises; so that we can obtain a derivation of premises of these rules by weakening their conclusions.
We prove that the rules {\sf R$ \Box $} and {\sf Exch} are invertible; the rule {\sf L$ \Diamond $} is treated similarly.\\
Let $\vdash M,P\Rightarrow Q,N,\Box A\,||\, {\cal H}$, and let $\bigwedge_{i=1}^n(\bigvee P_i\vee\bigvee \neg Q_i\vee\bigvee M_i\vee\bigvee \neg N_i)$ be an equivalent formula in CQNF for $ A $. Then, by applying Lemma \ref{NFcut2} we have
$$\vdash M_i,M,P\Rightarrow Q, N,N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i, $$ for $ i=1,\ldots,n $. Thus, by applying rule {\sf Exch} we have
\begin{prooftree}
\AxiomC{$M_i,M,P\Rightarrow Q, N,N_i\,||\, {\cal H}\,|\,P_i\Rightarrow Q_i $}
\RightLabel{\sf Exch}
\UnaryInfC{$M_i,M,P_i\Rightarrow Q_i, N,N_i\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\end{prooftree}
Therefore, $ \vdash M_i,M,P_i\Rightarrow Q_i, N,N_i\,||\, {\cal H}\,|\, P\Rightarrow Q$, for $ i=1,\ldots,n $, and then again by applying Lemma \ref{NFcut1} we have $\vdash M\Rightarrow N,A\,||\, {\cal H}\,|\, P\Rightarrow Q $.
Finally, we prove that the rule {\sf Exch}
is invertible. Let $ \mathcal{D} $ be a derivation of
$$ M,P\Rightarrow Q,N\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\,{\cal G}. $$ If the last rule applied in $ \mathcal{D} $ is propositional rules or {\sf Exch}, where the rule {\sf Exch} does not apply to $ P_i \Rightarrow Q_i $, apply induction hypothesis to the premise and then apply the last rule with the same principal formula to obtain deduction of $M,P_i\Rightarrow Q_i,N\,||\,{\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $. Therefore, let the last rule be {\sf R$ \Box $} as follows:
\begin{prooftree}
\AxiomC{$M\Rightarrow N', A\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\,{\cal G}\,|\,P\Rightarrow Q $}
\RightLabel{{\sf R$ \Box $},}
\UnaryInfC{$ M,P\Rightarrow Q,N',\Box A\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\,{\cal G} $}
\end{prooftree}
where $ N=N',\Box A $, we have
\begin{prooftree}
\AxiomC{$M\Rightarrow N', A\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\,{\cal G}\,|\,P\Rightarrow Q $}
\RightLabel{{\sf R$ \Box $}.}
\UnaryInfC{$ M,P_i\Rightarrow Q_i,N',\Box A\,||\,{\cal H}\,|\,{\cal G}\,|\,P\Rightarrow Q $ }
\end{prooftree} If the last rule is {\sf L$ \Diamond $}, the proof is similarly. If the last rule is {\sf Exch} which is applied to $ P_i \Rightarrow Q_i $, the conclusion is obtained.
\end{proof} \subsection{Admissibility of contraction } In order to prove the admissibility of the contraction rule, we need the following lemma. \begin{Lem}\label{*} The following rules are admissible.
\begin{center}
\AxiomC{$ \Diamond A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\LeftLabel{{\rm (\rom{1})}}
\UnaryInfC{$ A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\DisplayProof
$ \qquad $
\AxiomC{$ \Gamma\Rightarrow \Delta,\Box A\,||\,{\cal H} $}
\LeftLabel{\rm (\rom{2})}
\UnaryInfC{$\Gamma\Rightarrow \Delta, A \,||\,{\cal H}$}
\DisplayProof
\end{center} \end{Lem} \begin{proof}
We only consider the part $ {\rm (\rom{1})} $; the other is proved similarly. Let $ \vdash \Diamond A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $, and let $\bigvee_{i=1}^{k}(\bigwedge P'_i\vee\bigwedge \neg Q'_i\vee\bigwedge M'_i\vee\bigwedge \neg N'_i)$ be an equivalent formula in DQNF for $ A $. Then we have \begin{prooftree}
\AxiomC{$\vdash \Diamond A,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\RightLabel{Lemma \ref{NFcut2}}
\UnaryInfC{$\vdash M'_i,\Gamma\Rightarrow \Delta, N'_i\,||\,{\cal H}\,|\, P'_i\Rightarrow Q'_i$}
\RightLabel{\sf Merge}
\UnaryInfC{$\vdash M'_i,\Gamma,P'_i\Rightarrow Q'_i,\Delta, N'_i \,||\,{\cal H}$}
\RightLabel{Lemma \ref{NFcut1}}
\UnaryInfC{$ \vdash A,\Gamma\Rightarrow \Delta \,||\,{\cal H}$}
\end{prooftree} \end{proof} \begin{Cor}\label{**}
The following rule is admissible.
\begin{prooftree}
\AxiomC{$ \Diamond A,\Gamma\Rightarrow \Delta,\Box B\,||\,{\cal H} $}
\UnaryInfC{$ A,\Gamma\Rightarrow \Delta,B\,||\,{\cal H} $}
\end{prooftree} \end{Cor}
We now prove the admissibility of rules left rooted contraction {\sf LC}, right rooted contraction {\sf RC}, left crown contraction $ {\sf LC}^{\sf c} $, right crown contraction $ {\sf RC}^{\sf c} $ and finally, external contraction {\sf EC}, which are required for the proof of the admissibility of cut rule. First we consider the rules {\sf LC} and {\sf RC} for atomic formula, these rules and rules $ {\sf LC}^{\sf c} $ and $ {\sf RC}^{\sf c} $ are proved simultaneously. \begin{Lem} \label{atomic contraction}
The following rules are height-preserving admissible,
\[\begin{array}{cc}
\AxiomC{$ p,p,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\RightLabel{\sf LC}
\UnaryInfC{$ p,\Gamma\Rightarrow \Delta\,||\,{\cal H} $}
\DisplayProof
&
\AxiomC{$ \Gamma\Rightarrow \Delta,q,q \,||\,{\cal H} $}
\RightLabel{\sf RC,}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,q\,||\,{\cal H} $}
\DisplayProof
\\[0.5cm]
\AxiomC{$ \Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\, p,p,P_i\Rightarrow Q_i\,|\,{\cal G} $} \RightLabel{$ {\sf LC} ^{\sf c}$}
\UnaryInfC{$ \Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\, p,P_i\Rightarrow Q_i\,|\,{\cal G} $} \DisplayProof &
\AxiomC{$ \Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i,q,q\,|\,{\cal G} $} \RightLabel{$ {\sf RC}^{\sf c} $}
\UnaryInfC{$ \Gamma\Rightarrow\Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i,q\,|\,{\cal G} $} \DisplayProof \end{array} \] where $ p $ and $ q $ are atomic formulae. \end{Lem} \begin{proof}
{\rm
All rules are proved simultaneously by induction on the height of derivation of the premises. In all cases, if the premise is an initial sequent, then the conclusion is an initial sequent too. We consider some cases; the other cases are proved by a similar argument.
For the rule $ {\sf LC} ^{\sf c}$, let $ \Gamma=M,P$ and $ \Delta=Q,N$, and let the last rule be \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash_n M,p,p,P_i\Rightarrow Q_i,N\,||\,{\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\RightLabel{{\sf Exch},}
\UnaryInfC{$\vdash_{n+1} M,P\Rightarrow Q,N\,||\,{\cal H}\,|\, p,p,P_i\Rightarrow Q_i\,|\,{\cal G} $} \end{prooftree}
then we have \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ \vdash_n M,p,p,P_i\Rightarrow Q_i,N\,||\,{\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\RightLabel{IH ({\sf LC})}
\UnaryInfC{$\vdash_n M,p,P_i\Rightarrow Q_i,N\,||\,{\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$\vdash_{n+1}M,P\Rightarrow Q,N\,||\,{\cal H}\,|\, p,P_i\Rightarrow Q_i\,|\,{\cal G} $} \end{prooftree} For the other last rules $\sf R $, use induction hypothesis on the premise, and then apply the rule $ \sf R $.\\
For the rule $\sf LC $, let $ \Gamma=M,P$ and $ \Delta=Q,N$, and let the last rule be as follows \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H}\,|\,p,p,P \Rightarrow Q\,|\, {\cal G}$}
\RightLabel{{\sf Exch},}
\UnaryInfC{$\vdash_{n+1} p,p,M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i\,|\, {\cal G} $} \end{prooftree} then we have \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H}\,|\,p,p,P \Rightarrow Q\,|\, {\cal G}$}
\RightLabel{IH ($ {\sf LC} ^{\sf c}$)}
\UnaryInfC{$\vdash_n M,P_i\Rightarrow Q_i,N\,||\, {\cal H}\,|\,p,P \Rightarrow Q\,|\, {\cal G}$}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$\vdash_{n+1} M,p,P\Rightarrow Q,N\,||\, {\cal H}\,|\,P_i \Rightarrow Q_i\,|\, {\cal G} $} \end{prooftree} Let the last rule be {\sf L$\Diamond $} as: \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash_n B,M\Rightarrow N\,||\, {\cal H}\,|\,p,p,P\Rightarrow Q$}
\RightLabel{{\sf L$\Diamond $},}
\UnaryInfC{$\vdash_{n+1} p,p,\Diamond B,M,P\Rightarrow Q,N\,||\, {\cal H} $} \end{prooftree} where $ \Gamma=\Diamond B,M,P$ and $ \Delta=Q,N$. Then we have \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash_n B,M\Rightarrow N\,||\, {\cal H}\,|\,p,p,P\Rightarrow Q $}
\RightLabel{IH ($ {\sf LC} ^{\sf c}$)}
\UnaryInfC{$\vdash_n B,M\Rightarrow N\,||\,{\cal H}\,|\,p,P\Rightarrow Q $}
\RightLabel{{\sf L$\Diamond $},}
\UnaryInfC{$\vdash_{n+1} p,P,\Diamond B,M\Rightarrow N,Q\,||\, {\cal H} $} \end{prooftree} The rule {\sf R$ \Box $} is treated similarly; for the other rules use induction hypothesis and then use the same rule. \begin{Lem}
The rules of contraction,
\begin{center}
\AxiomC{$ A,A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{\sf LC}
\UnaryInfC{$ A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\DisplayProof
$\qquad$
\AxiomC{$ \Gamma\Rightarrow \Delta,A,A\,||\, {\cal H} $}
\RightLabel{{\sf RC},}
\UnaryInfC{$ \Gamma\Rightarrow \Delta,A\,||\, {\cal H} $}
\DisplayProof
\end{center} are admissible. \end{Lem} \begin{proof}
Both rules are proved simultaneously by induction on the complexity of $ A $ with subinduction on the height of the derivations. The lemma holds for atomic formula $ A $, by Lemma \ref{atomic contraction}. For the other cases, if $ A $ is not principal in the last rule (either modal or propositional), apply inductive
hypothesis to the premises and then apply the last rule. Here we only consider the rule $ {\sf LC} $; the admissibility of the rule {\sf RC} is proved similarly. Suppose
$ \vdash A,A,\Gamma\Rightarrow \Delta\,||\, {\cal H} $, we consider some cases, in which $ A $ is principal formula in the last rule:
\end{proof}
Case 1. $ A= \Diamond B $:
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash B,\Diamond B,M\Rightarrow N\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{{\sf L$\Diamond $},}
\UnaryInfC{$\vdash \Diamond B,\Diamond B,M,P\Rightarrow N,Q\,||\, {\cal H} $}
\end{prooftree}
where $ \Gamma=M,P$ and $ \Delta=N,Q$. Then we have
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash B,\Diamond B,M\Rightarrow N\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{ Lemma \ref{*} }
\UnaryInfC{$\vdash B, B,M\Rightarrow N\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{ IH }
\UnaryInfC{$\vdash B,M\Rightarrow N\,||\, {\cal H}\,|\,P\Rightarrow Q$}
\RightLabel{{\sf L$\Diamond $}.}
\UnaryInfC{$\vdash \Diamond B,M,P\Rightarrow N,Q\,||\, {\cal H} $}
\end{prooftree}
Case 2. $ A=\Box B$: \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash B,\Box B,\Box B,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{{\sf L$\Box $},}
\UnaryInfC{$\vdash \Box B,\Box B,\Gamma\Rightarrow \Delta\,||\, {\cal H} $} \end{prooftree} then \begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$\vdash B,\Box B,\Box B,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{IH}
\UnaryInfC{$\vdash B,\Box B,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{{\sf L$\Box $}.}
\UnaryInfC{$\vdash \Box B,\Gamma\Rightarrow \Delta\,||\, {\cal H} $} \end{prooftree}
Case 3. Let $ A=B\rightarrow C $ and let the last rule as \begin{prooftree}
\AxiomC{$ \mathcal{D}_1 $}
\noLine
\UnaryInfC{$\vdash B\rightarrow C,\Gamma\Rightarrow \Delta,B\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}_2 $}
\noLine
\UnaryInfC{$\vdash C,B\rightarrow C,\Gamma\Rightarrow \Delta\,||\, {\cal H} $}
\RightLabel{{\sf L$\rightarrow $},}
\BinaryInfC{$\vdash B\rightarrow C,B\rightarrow C,\Gamma\Rightarrow \Delta\,||\, {\cal H} $} \end{prooftree} By applying the invertibility of the rule {\sf L$ \rightarrow $}, see Lemma \ref{invertibility of propositional rules }, to the first premise, we get
$$ \vdash \Gamma\Rightarrow \Delta,B,B\,||\, {\cal H} $$ and applying to the second premise, we get $\vdash C,C,\Gamma\Rightarrow \Delta\,||\, {\cal H} $. We then use the induction hypothesis and obtain $\vdash \Gamma\Rightarrow \Delta,B\,||\, {\cal H} $ and $\vdash C,\Gamma\Rightarrow \Delta\,||\, {\cal H} $. Thus by {\sf L$\rightarrow $}, we get $\vdash B\rightarrow C,\Gamma\Rightarrow \Delta \,||\, {\cal H}$.\\
} \end{proof} \begin{Lem}
The rule of external contraction,
\begin{center}
\AxiomC{$ \Gamma\Rightarrow \Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\, P_i\Rightarrow Q_i\,|\, {\cal G} $}
\RightLabel{$ \sf EC $}
\UnaryInfC{$ \Gamma\Rightarrow \Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\, {\cal G} $}
\DisplayProof
\end{center}
is height-preserving admissible.
\end{Lem} \begin{proof} If the premise is derivable, then the conclusion is derived as follows
\begin{prooftree}
\AxiomC{$\vdash_n \Gamma\Rightarrow \Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\, P_i\Rightarrow Q_i\,|\, {\cal G} $}
\RightLabel{$ {\sf Merge}^{\sf c} $}
\UnaryInfC{$\vdash_n \Gamma\Rightarrow \Delta\,||\,{\cal H}\,|\, P_i,P_i\Rightarrow Q_i,Q_i\,|\, {\cal G} $}
\RightLabel{$ {\sf RC}^{\sf c} $, $ {\sf LC} ^{\sf c}$}
\UnaryInfC{$\vdash_n \Gamma\Rightarrow \Delta\,||\,{\cal H}\,|\, P_i\Rightarrow Q_i\,|\, {\cal G} $}
\end{prooftree} this rule is height-preserving admissible because the rules $ {\sf Merge}^{\sf c} $, $ {\sf RC}^{\sf c} $, and $ {\sf LC} ^{\sf c}$ are height-preserving admissible. \end{proof} \section{Admissibility of cut }\label{sec cut} In this section, we prove the admissibility of cut rule and completeness theorem.
In cut rule, \begin{prooftree}
\AxiomC{$ \Gamma\Rightarrow \Delta,D\,||\, {\cal H} $}
\AxiomC{$ D,\Gamma'\Rightarrow \Delta'\,||\, {\cal H'} $}
\RightLabel{\sf Cut,}
\BinaryInfC{$ \Gamma,\Gamma'\Rightarrow \Delta,\Delta'\,||\, {\cal H}\,|\, {\cal H'} $}
\end{prooftree} the crown of the conclusion is the union of crowns of the premises.
If cut formula $ D $ is atomic, then the admissibility of the cut rule is proved simultaneously with the following rule which is called crown cut: \begin{prooftree}
\AxiomC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,{\cal G} $}
\RightLabel{$ {\sf Cut}^{\sf c} $}
\AxiomC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $} \end{prooftree}
\textbf{Note}: The following exchanges between root and crown parts should be noted in the crown cut rule $ {\sf Cut}^{\sf c} $.
\begin{tasks}(4)
\task[$ \bullet $] $ P $ and $ P_i $
\task [$ \bullet $] $ Q $ and $ Q_i $
\task [$ \bullet $] $P' $ and $ P'_i $
\task [$ \bullet $] $ Q' $ and $ Q'_i $
\end{tasks}
\begin{Lem}\label{atomic cut}
The following rules are admissible, where $ p$ is an atomic formula.
\begin{prooftree}
\AxiomC{$ \Gamma\Rightarrow \Delta,p\,||\, {\cal H} $}
\AxiomC{$ p,\Gamma'\Rightarrow \Delta'\,||\, {\cal H'} $}
\RightLabel{\sf Cut,}
\BinaryInfC{$ \Gamma,\Gamma'\Rightarrow \Delta,\Delta'\,||\, {\cal H}\,|\, {\cal H'} $}
\end{prooftree}
\begin{prooftree}
\AxiomC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,{\cal G} $}
\RightLabel{$ {\sf Cut}^{\sf c} $}
\AxiomC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree}
\end{Lem}
\begin{proof}
Both rules are proved simultaneously by induction on the sum of heights of derivations of the two premises, which is called cut-height. First, we consider the rule $ {\sf Cut} $. If both of the premises are initial sequents, then the conclusion is an initial sequent too, and if only one of the premises is an initial sequent, then the conclusion is obtained by weakening. \\
If one of the last rules in the derivations of the premises is not
{\sf L$\Diamond $},
{\sf R$\Box $}, or {\sf Exch},
then the cut rule can be transformed into cut(s) with lower
cut-height as usual. Thus we consider cases which the last rules are {\sf L$\Diamond $},
{\sf R$\Box $}, and {\sf Exch}.\\
\textbf{Case 1. The left premise is derived by {\sf R$ \Box $}.} Let $\Gamma=M,P $ and $\Delta=Q,N,\Box A $, and let $ \Box A $ be the principal formula. We have three subcases according to the last rule in the derivation of the right premise. \\
\textbf{Subcase 1.1} The right premise is derived by {\sf L$ \Diamond $}. Let $\Gamma'=\Diamond B,M',P'$
and $ \Delta '=Q',N'$, and let the last rules be as follows
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N,A\,||\,{\cal H}\,|\,P\Rightarrow Q,p $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M,P\Rightarrow Q,N,\Box A,p\,||\, {\cal H}$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ B,M'\Rightarrow N' \,||\, {\cal H}'\,|\,p,P'\Rightarrow Q' $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$ p,\Diamond B,M',P'\Rightarrow Q',N'\,||\, {\cal H}' $}
\RightLabel{{\sf Cut}.}
\BinaryInfC{$ M,P, \Diamond B,M',P'\Rightarrow Q,N,\Box A,Q',N'\,||\, {\cal H}\,|\,{\cal H}' $}
\end{prooftree}
For this cut, let
$ \bigwedge_{i=1}^{k}(\bigvee\neg P_i\vee \bigvee Q_i\vee \bigvee\neg M_i\vee \bigvee N_i) $
be an equivalent formula in CQNF of $ A $. Then, by Corollary \ref{left to right1}, which is height-preserving admissible, we get the following derivation from $ \mathcal{D} $
\begin{prooftree}
\AxiomC{$ \mathcal{D}_1 $}
\noLine
\UnaryInfC{$ M_i,P_i,M\Rightarrow N,Q_i, N_i \,||\, {\cal H}\,|\,P\Rightarrow Q,p, $}
\end{prooftree} for every clause $ (\bigvee\neg P_i\vee \bigvee Q_i\vee \bigvee\neg M_i\vee \bigvee N_i) $.
Then by applying crown cut rule ${\sf Cut}^{\sf c} $, we get the following derivation for every clauses in CQNF of $ A $
\begin{prooftree}
\AxiomC{$ \mathcal{D}_1 $}
\noLine
\UnaryInfC{$ M_i,P_i,M\Rightarrow N,Q_i, N_i \,||\,{\cal H}\,|\,P\Rightarrow Q,p $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ B,M'\Rightarrow N' \,||\, {\cal H}' \,|\,p,P'\Rightarrow Q' $}
\RightLabel{{\sf L$\Diamond $}}
\UnaryInfC{$ \Diamond B,M'\Rightarrow N' \,||\, {\cal H}' \,|\,p,P'\Rightarrow Q' $}
\RightLabel{$ {\sf Cut}^{\sf c} $}
\BinaryInfC{$ M_i,P,M,\Diamond B,M',P'\Rightarrow Q,N,N_i,Q',N'\,||\,{\cal H}\,|\,P_i\Rightarrow Q_i\,|\, {\cal H}'\,|\,P'_i\Rightarrow Q'_i $}
\end{prooftree}
and then using Lemma \ref{NFcut2}, the conclusion is obtained.\\
\textbf{Subcase 1.2.} The right premise is derived by {\sf R$ \Box $}. Similar to Case 1.1.\\
\textbf{Subcase 1.3.} The right premise is derived by {\sf Exch}. Let $ {\cal H}'={\cal G}'\,|\,P'_i\Rightarrow Q'_i\,|\,{\cal I}' $, and let the last rules be as follows:
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N,A\,||\, {\cal H}\,|\,P\Rightarrow Q,p $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M,P\Rightarrow Q,N,\Box A,p\,||\, {\cal H}$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\,p,P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$p,M',P'\Rightarrow Q',N'\,||\,{\cal G}'\,|\,P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\RightLabel{{\sf Cut}.}
\BinaryInfC{$M,P,M',P' \Rightarrow Q,N,\Box A,Q',N'\,||\, {\cal H}\,|\,{\cal G}'\,|\,P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree}
This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$M\Rightarrow N,A\,||\, {\cal H}\,|\, P\Rightarrow Q,p $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M\Rightarrow N,\Box A\,||\, {\cal H}\,|\,P\Rightarrow Q,p$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\,p,P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{$ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$M,P,M',P' \Rightarrow Q,N,\Box A,Q',N'\,||\, {\cal H}\,|\,{\cal G}'\,|\,P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree} \textbf{Case 2. The left premise is derived by {\sf L$ \Diamond $}.} Similar to the case 1.\\
\textbf{Case 3. The left premise is derived by {\sf Exch}.} Let the hypersequent $ {\cal H}$ be as ${\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I} $. According to the last rule applied in the right premise, we have the following subcases:\\ \textbf{Subcase 3.1.} The right premise is derived by {\sf R$ \Box $} or {\sf L$ \Diamond $}. Similar to the case 1.2.\\
\textbf{Subcase 3.2.} The right premise is derived by {\sf Exch}. Let the hypersequent $ {\cal H}'$ be as ${\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\, {\cal I}' $ and let the last rules be as
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$M,P_i\Rightarrow Q_i,N\,||\,{\cal G}\,|\, P\Rightarrow Q,p\,|\, {\cal I} $}
\RightLabel{\sf Exch}
\UnaryInfC{$M,P\Rightarrow Q,N,p\,||\,{\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\, p,P'\Rightarrow Q'\,|\, {\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$p,M',P'\Rightarrow Q',N'\,||\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\, {\cal I}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$M,P,M',P'\Rightarrow Q,N,Q',N'\,||\,{\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I}\,|\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\, {\cal I}' $}
\end{prooftree}
where $ \Gamma=M,P $, $ \,\Delta=Q,N $, $\, \Gamma'=M',P' $, and $ \Delta'=Q',N'\, $. This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$M,P_i\Rightarrow Q_i,N\,||\,{\cal G}\,|\, P\Rightarrow Q,p\,|\, {\cal I} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\, p,P'\Rightarrow Q'\,|\, {\cal I}' $}
\RightLabel{$ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$M,P,M',P'\Rightarrow Q,N,Q',N'\,||\,{\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I}\,|\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\, {\cal I}' $}
\end{prooftree} \textbf{Case 4. The right premise is derived by {\sf L$ \Diamond $}, {\sf R$ \Box $}, or {\sf Exch}.} Similar to the above cases for the left premise.\\ Now we prove the admissibility of the crown cut rule $ {\sf Cut}^{\sf c} $. If the left premise is an instance of initial sequent $ {\sf L \bot} $ or $ {\sf R \top} $, then so is the conclusion. If the left premise is an instance of initial sequent {\sf Ax}, we have two cases as follows. If $ M $ and $ N $ contain a common atomic formula, then so is the conclusion. If $ P $ and $ Q $ contain a common atomic formula, then apply the rule {\sf Exch} and then apply the rules weakening to obtain deduction of the conclusion. If the right premise is an initial sequent or both of the premises are initial sequent, the conclusion is obtained by the same argument.\\
The last rule applied in the premises of the crown cute rule can only be modal rules since all formulae in this rule are modal or atomic. Similar to the proof of the cut rule, we only consider the rules {\sf L$ \Diamond $}, {\sf R$ \Box $}, and {\sf Exch}.\\
\textbf{Case 1. The left premise is derived by L$ \Diamond $.}
Let $ M=\Diamond B,M_1 $ and the derivation be as follows
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ B,M_1\Rightarrow N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\, {\cal G}\,|\,P\Rightarrow Q $}
\RightLabel{L$ \Diamond $}
\UnaryInfC{$ \Diamond B,M_1,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\, {\cal G} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{$ {\sf Cut}^{\sf c} $,}
\BinaryInfC{$\Diamond B,M_1,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\, {\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree}
For this case, first we transform $ B $ into $\bigvee_{j=1}^{m}(\bigwedge P''_j\vee\bigwedge \neg Q''_j\vee\bigwedge M''_j\vee\bigwedge \neg N''_j)$, an equivalent formula in DQNF for $ B $. Then by Corollary \ref{left to right1} we get $ \mathcal{D}_1 $ from $ \mathcal{D} $ as:
\begin{prooftree}
\AxiomC{$ \mathcal{D}_1 $}
\noLine
\UnaryInfC{$ P''_j,M''_j,M_1\Rightarrow N,Q''_j,N''_j\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\, {\cal G}\,|\,P\Rightarrow Q $}
\end{prooftree} for every phrases in DQNF. Thus since the rules in Corollary \ref{left to right1} are height-preserving admissible by induction hypothesis we have
\begin{prooftree}
\AxiomC{$ \mathcal{D}_1 $}
\noLine
\UnaryInfC{$ P''_j,M''_j,M_1\Rightarrow N,Q''_j,N''_j\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\, {\cal G} \,|\,P\Rightarrow Q$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{$ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$ M''_j, M_1,P_i,M',P'_i\Rightarrow N,Q_i,N''_j,Q'_i,N'\,||\,{\cal H}\,|\,P''_j\Rightarrow Q''_j \,|\,{\cal G} \,|\,P\Rightarrow Q \,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}'$}
\end{prooftree}
Therefore, by applying Lemma \ref{NFcut2}, the conclusion is obtained.\\
\textbf{Case 2. The left premise is derived by {\sf R$ \Box $}.} Similar to Case 1.\\
\textbf{Case 3. The left premise is derived by {\sf Exch}.} We have two subcases according to the principal formula; the cut formula $ p $ is principal formula or not. If the cut formula is not principal, then the derivation is transformed into a derivation with lower cut-height as follows:\\
Let the hypersequent $ {\cal G} $ be as $ P''\Rightarrow Q''\,|\,{\cal I} $ and the last rule {\sf Exch} be as follows:
\begin{prooftree}
\AxiomC{$\mathcal{D} $}
\noLine
\UnaryInfC{$ M,P''\Rightarrow Q'',N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,P\Rightarrow Q\,|\,{\cal I} $}
\RightLabel{\sf Exch}
\UnaryInfC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,P''\Rightarrow Q''\,|\,{\cal I} $}
\AxiomC{$\mathcal{D}' $}
\noLine
\UnaryInfC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{ $ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,P''\Rightarrow Q''\,|\,{\cal I}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree} This cut is transformed into \begin{prooftree}
\AxiomC{$\mathcal{D} $}
\noLine
\UnaryInfC{$ M,P''\Rightarrow Q'',N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,P\Rightarrow Q\,|\,{\cal I} $}
\AxiomC{$\mathcal{D}' $}
\noLine
\UnaryInfC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{ $ {\sf Cut}^{\sf c} $}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P''\Rightarrow Q''\,|\,P\Rightarrow Q\,|\,{\cal I}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $} \end{prooftree} Therefore, let the left premise is derive by {\sf Exch} where the cut formula $ p $ is principal. We have three subcases according to the last rule applied in the right premise.\\
\textbf{Subcase 3.1.} The right premise is derived by {\sf Exch}.
\begin{prooftree}
\AxiomC{$\mathcal{D} $}
\noLine
\UnaryInfC{$ M,P_i\Rightarrow Q_i,p,N\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\RightLabel{\sf Exch}
\UnaryInfC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,{\cal G} $}
\AxiomC{$\mathcal{D}'$}
\noLine
\UnaryInfC{$ M',p,P'_i\Rightarrow Q'_i,N'\,||\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$ M',P'\Rightarrow Q',N'\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{ $ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree}
This cut is transformed into the first cut as follows:
\begin{prooftree}
\AxiomC{$\mathcal{D} $}
\noLine
\UnaryInfC{$ M,P_i\Rightarrow Q_i,p,N\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\AxiomC{$\mathcal{D}'$}
\noLine
\UnaryInfC{$ M',p,P'_i\Rightarrow Q'_i,N'\,||\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\RightLabel{\sf Cut}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree}
\textbf{Subcase 3.2.} The right premise is derived by {\sf R$ \Box $}. Let $ N'=N'',\Box A $ and $ \Box A $ be the principal formula:
\begin{prooftree}
\AxiomC{$\mathcal{D} $}
\noLine
\UnaryInfC{$ M,P_i\Rightarrow Q_i,p,N\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G} $}
\RightLabel{\sf Exch}
\UnaryInfC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,{\cal G} $}
\AxiomC{$\mathcal{D}'$}
\noLine
\UnaryInfC{$ M'\Rightarrow N'', A\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}'\,|\,P'\Rightarrow Q' $}
\RightLabel{\sf R$ \Box $}
\UnaryInfC{$ M',P'\Rightarrow Q',N'',\Box A\,||\, {\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}' $}
\RightLabel{ $ {\sf Cut}^{\sf c} $.}
\BinaryInfC{$ M,P_i,M',P'_i\Rightarrow Q_i,N,Q'_i,N'',\Box A\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P'\Rightarrow Q'\,|\,{\cal G}' $}
\end{prooftree} For this cut, let $ \bigwedge_{j=1}^{k}(\bigvee\neg P''_j\vee \bigvee Q''_j\vee \bigvee\neg M''_j\vee \bigvee N''_j) $ be an equivalent formula in CQNF of $ A $. Then, by Corollary \ref{left to right1}, which is height-preserving admissible, we get the following derivation from $ \mathcal{D}' $ \begin{prooftree}
\AxiomC{$ \mathcal{D}'_1 $}
\noLine
\UnaryInfC{$ M''_j,P''_j,M'\Rightarrow N'',Q''_j, N''_j\,||\,{\cal H}'\,|\,p, P'_i\Rightarrow Q'_i\,|\,{\cal G}'\,|\,P'\Rightarrow Q' , $} \end{prooftree} for every clause $ (\bigvee\neg P''_j\vee \bigvee Q''_j\vee \bigvee\neg M''_j\vee \bigvee N''_j) $. Then by applying the rule $ {\sf Cut}^{\sf c} $, we get the following derivation for every clauses in CQNF of $ A $
\begin{prooftree}
\AxiomC{$ M,P\Rightarrow Q,N\,||\, {\cal H}\,|\, P_i\Rightarrow Q_i,p\,|\,{\cal G}$}
\AxiomC{$ \mathcal{D}'_1 $}
\RightLabel{ $ {\sf Cut}^{\sf c} $}
\BinaryInfC{$ M,P_i,M''_j,P'_i, M'\Rightarrow Q_i,N,N'',Q'_i,N''_j\,||\, {\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}\,|\, {\cal H}'\,|\, P''_j\Rightarrow Q''_j\,|\,{\cal G}'\,|\, P'\Rightarrow Q'$} \end{prooftree}
Then, by applying Lemma \ref{NFcut2}, the conclusion is obtained.\\ \textbf{Subcase 3.3.} The right premise is derived by {\sf L$ \Diamond $}. Similar to Subcase 3.2.
\end{proof} Now, we prove the admissibility of the cut rule for arbitrary formula. \begin{The}\label{cut}
The rule of cut \begin{prooftree}
\AxiomC{$ \Gamma\Rightarrow \Delta,D\,||\, {\cal H} $}
\AxiomC{$ D,\Gamma'\Rightarrow \Delta'\,||\, {\cal H'} $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ \Gamma,\Gamma'\Rightarrow \Delta,\Delta'\,||\, {\cal H}\,|\, {\cal H'} $} \end{prooftree} where $ D $ is an arbitrary formula, is admissible in $ {\cal R}_{\sf S5}$. \end{The} \begin{proof}
The proof proceeds by induction on the structure of the cut formula
$ D $ with subinduction on the cut-height i.e., the sum of the heights of the
derivations of the premises. The admissibility of the rule for atomic cut formula follows from Lemma \ref{atomic cut}, therefore we consider cases where $ D $ is not atomic formula.
If the cut formula is of the form
$ \neg A $,
$ A\wedge B $,
$ A\vee B $, or
$ A\rightarrow B $,
then using invertibility of the propositional rules, \Cref{invertibility of propositional rules }, the cut rule can be transformed into cut rules where cut formula is reduced, i.e, cut formula is
$ A $ or
$ B $. Thus it remains to consider cases, where cut formula is of the form $ \Diamond A $ or $ \Box A $. We only consider the case where the cut formula is of the form $ \Box A $; the other case is proved similarly. We distinguish the following cases:\\
\textbf{1. Cut formula $ \Box A $ is principal in the left premise only.}
we consider the last rule applied to the right premise of cut. If the last rule applied is a propositional rule, then the derivation is transformed into a derivation of lower cut-height as usual. Thus we will consider modal rules.\\
\textbf{Subcase 1.1.} The right premise is derived by {\sf R$ \Box $}. Let $ \Gamma'=M',P' $
and
$ \Delta'=Q',N',\Box B $, and let cut rule be as follows
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N, A\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,M'\Rightarrow N', B \,||\, {\cal H}'\,|\,P'\Rightarrow Q'$}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ \Box A,M',P'\Rightarrow Q',N', \Box B \,||\, {\cal H}'$}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ M,P,M',P'\Rightarrow Q,N,Q',N',\Box B ||\, {\cal H}\,|\, {\cal H}' $}
\end{prooftree}
where $ \Gamma=M,P $ and $ \Delta=Q,N $. This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N, A\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M\Rightarrow N, \Box A\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,M'\Rightarrow N', B \,||\, {\cal H}'\,|\,P'\Rightarrow Q' $}
\RightLabel{\sf Cut}
\BinaryInfC{$ M,M'\Rightarrow N, N', B\,||\,{\cal H}\,|\,P\Rightarrow Q \,|\,{\cal H}'\,|\, P'\Rightarrow Q' $}
\RightLabel{$ {\sf Merge}^{\sf c} $}
\UnaryInfC{$ M,M'\Rightarrow N, N', B\,||\, {\cal H}\,|\, {\cal H}'\,|\,P,P'\Rightarrow Q,Q' $}
\RightLabel{{\sf R$\Box $}.}
\UnaryInfC{$M,M',P,P'\Rightarrow Q,Q', N, N', \Box B\,||\, {\cal H}\,|\, {\cal H}' $}
\end{prooftree}
\textbf{Case 1.2.} The right premise is derived by {\sf L$ \Diamond $}. This case is treated similar to the above case.\\
\textbf{Case 1.2.} The right premise is derived by {\sf R$ \Diamond $}. Let
$ \Delta'=\Delta'',\Diamond B $ and cut rule be as follows
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,\Gamma'\Rightarrow \Delta'',\Diamond B,B\,||\, {\cal H}' $}
\RightLabel{\sf R$\Diamond $}
\UnaryInfC{$ \Box A,\Gamma'\Rightarrow \Delta'', \Diamond B\,||\, {\cal H}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ M,P,\Gamma'\Rightarrow Q,N,\Delta'',\Diamond B\,\,||\, {\cal H}\,|\, {\cal H}' $}
\end{prooftree}
where $ \Gamma=M,P $ and $ \Delta=Q,N $. This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,\Gamma'\Rightarrow \Delta'',\Diamond B,B\,||\, {\cal H}' $}
\RightLabel{\sf Cut}
\BinaryInfC{$ M,P,\Gamma'\Rightarrow Q,N, \Delta'', \Diamond B,B\,||\, {\cal H}\,|\, {\cal H}' $}
\RightLabel{{\sf R$\Diamond $}.}
\UnaryInfC{$ M,P,\Gamma'\Rightarrow Q,N,\Delta'',\Diamond B\,\,||\, {\cal H}\,|\, {\cal H}' $}
\end{prooftree}
\textbf{Case 1.3.} The right premise is derived by {\sf L$ \Box $}. This case is treated similar to the above case.\\
\textbf{Case 1.4.} The right premise is derived by {\sf Exch}. Let $ \Gamma'=M',P' $ and
$ \Delta'=Q',N' $, and let the hypersequent $ {\cal H}' $ be as $ {\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $:
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N, A\,||\,{\cal H}\,|\, P\Rightarrow Q $}
\RightLabel{\sf R$ \Box $}
\UnaryInfC{$ M,P\Rightarrow Q,N, \Box A\,||\,{\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Box A,M',P'\Rightarrow Q',N'\,||\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ M,P,M',P'\Rightarrow Q,N,Q',N'\,||\,{\cal H}\,|\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree}
where $ \Gamma=M,P $ and $ \Delta=Q,N $. This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N, A\,||\,{\cal H}\,|\, P\Rightarrow Q $}
\RightLabel{\sf R$ \Box $}
\UnaryInfC{$ M\Rightarrow N, \Box A\,||\,{\cal H}\,|\, P\Rightarrow Q$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Box A,M',P'_i\Rightarrow Q'_i,N'\,||\,{\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{\sf Cut}
\BinaryInfC{$ M,M',P'_i\Rightarrow N,Q'_i, N'\,||\,{\cal H}\,|\, P\Rightarrow Q\,|\,{\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{$ {\sf Merge}^{\sf c} $}
\UnaryInfC{$M,M',P'_i\Rightarrow N,Q'_i, N'\,||\,{\cal H}\,|\,{\cal G}'\,|\, P,P'\Rightarrow Q,Q'\,|\,{\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$ M,M',P,P'\Rightarrow N,N',Q,Q'\,||\,{\cal H}\,|\,{\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree} \textbf{ 2. Cut formula $ \Box A $ is principal in both premises.}
Let $ \Gamma=M,P $
and
$ \Delta=Q,N $, and let cut rule be as follows
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ M\Rightarrow N, A\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ A,\Box A,\Gamma'\Rightarrow \Delta'\,||\, {\cal H}' $}
\RightLabel{\sf L$\Box $}
\UnaryInfC{$ \Box A,\Gamma'\Rightarrow \Delta' \,||\, {\cal H}'$}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ M,P,\Gamma'\Rightarrow Q,N, \Delta'\,||\, {\cal H}\,|\,{\cal H}' $}
\end{prooftree}
This cut is transformed into
\begin{prooftree}
\AxiomC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H} $}
\RightLabel{ \ref{*}}
\UnaryInfC{$ M,P\Rightarrow Q,N, A\,||\, {\cal H} $}
\AxiomC{$ M,P\Rightarrow Q,N, \Box A\,||\, {\cal H}$}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ A,\Box A,\Gamma'\Rightarrow \Delta'\,||\, {\cal H}' $}
\RightLabel{\sf Cut }
\BinaryInfC{$ M,P,A,\Gamma'\Rightarrow Q,N,\Delta'\,||\, {\cal H}\,|\,{\cal H}' $}
\RightLabel{\sf Cut}
\BinaryInfC{$ M,P,M,P,\Gamma'\Rightarrow Q,N,Q,N,\Delta'\,||\, {\cal H}\,|\, {\cal H}\,|\,{\cal H}' $}
\RightLabel{{\sf LC}, {\sf RC}, {\sf EC}}
\UnaryInfC{$ M,P,\Gamma'\Rightarrow Q,N, \Delta'\,||\, {\cal H}\,|\,{\cal H}' $}
\end{prooftree}
\textbf{ 3. Cut formula $ \Box A $ is not principal in the left premise.}\\
According to the last rule in the derivation of the left premise, we have subcases. For propositional rules the cut rule can be transformed into a derivation with cut(s) of lower cut-height.\\
\textbf{Case 3.1.} The left premise is derived by {\sf L$ \Diamond $}. Suppose that $ \Gamma=\Diamond C,M,P$ and $ \Delta=N,Q $ where $ \Diamond C $ is the principal formula. Again, we have the following subcases according to the last rule in derivation of the right premise, and we only consider the modal rules.\\
\textbf{Subcase 3.1.1} The right premise is derived by {\sf L$ \Diamond $}. Let
$ \Gamma'=\Diamond B,M',P'\,$ and $\Delta'=Q',N' $.
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ C,M\Rightarrow N, \Diamond A\,||\, {\cal H} \,|\,P\Rightarrow Q $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$ \Diamond C,M,P\Rightarrow Q,N, \Diamond A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Diamond A, B,M'\Rightarrow N' \,||\, {\cal H}'\,|\, P'\Rightarrow Q' $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$ \Diamond A,\Diamond B,M',P'\Rightarrow Q',N'\,||\, {\cal H}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ \Diamond C,M,P, \Diamond B,M',P'\Rightarrow Q,N,Q',N'\,||\, {\cal H}\,|\, {\cal H}' $}
\end{prooftree}
This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ C,M\Rightarrow N, \Diamond A\,||\, {\cal H} \,|\,P\Rightarrow Q $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Diamond A, B,M'\Rightarrow N' \,||\,{\cal H}'\,|\, P'\Rightarrow Q' $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$ \Diamond A, \Diamond B,M'\Rightarrow N' \,||\,{\cal H}'\,|\, P'\Rightarrow Q'$}
\RightLabel{ \sf Cut }
\BinaryInfC{$ C,M, \Diamond B,M'\Rightarrow N,N'\,||\, {\cal H} \,|\,P\Rightarrow Q\,|\,{\cal H}'\,|\, P'\Rightarrow Q' $}
\RightLabel{$ {\sf Merge}^{\sf c} $}
\UnaryInfC{$ C,M, \Diamond B,M'\Rightarrow N,N'\,||\, {\cal H}\,|\, {\cal H}'\,|\,P,P'\Rightarrow Q,Q' $}
\RightLabel{\sf L$\Diamond $.}
\UnaryInfC{$ \Diamond C,M,P, \Diamond B,M',P'\Rightarrow Q,N,Q',N' \,||\, {\cal H}\,|\, {\cal H}'$}
\end{prooftree}
\textbf{Subcase 3.1.2} The right premise is derived by {\sf R$ \Box $} or {\sf Exch}. Similar to Subcase 3.1.1.\\
\textbf{Subcase 3.1.3} The right premise is derived by {\sf L$ \Box $}.
Let
$ \Gamma'=\Box B,M',P'\,$ and $\Delta'=Q',N' $.
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$ C,M\Rightarrow N, \Diamond A\,||\, {\cal H}\,|\,P\Rightarrow Q $}
\RightLabel{\sf L$\Diamond $}
\UnaryInfC{$ \Diamond C,M,P\Rightarrow Q,N, \Diamond A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Diamond A, B, \Box B,M',P'\Rightarrow Q',N'\,||\, {\cal H}' $}
\RightLabel{\sf L$\Box $}
\UnaryInfC{$ \Diamond A,\Box B,M',P'\Rightarrow Q',N'\,||\, {\cal H}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ \Diamond C,M,P, \Diamond B,M',P'\Rightarrow Q,N,Q',N'\,||\, {\cal H}\,|\,{\cal H}' $}
\end{prooftree}
This cut is transformed into
\begin{prooftree}
\AxiomC{$ \Diamond C,M,P\Rightarrow Q,N, \Diamond A\,||\, {\cal H} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Diamond A, B, \Box B,M',P'\Rightarrow Q',N'\,||\, {\cal H}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$ \Diamond C,M,P, B,\Box B,M',P'\Rightarrow Q,N,Q',N'\,||\, {\cal H}\,|\,{\cal H}' $}
\RightLabel{\sf L$ \Box $}
\UnaryInfC{$ \Diamond C,M,P, \Box B,M',P'\Rightarrow Q,N,Q',N' \,||\, {\cal H}\,|\,{\cal H}'$}
\end{prooftree}
\textbf{Subcase 3.1.3} The right premise is derived by {\sf R$ \Diamond $}. Similar to Subcase 3.1.2.\\
\textbf{Case 3.2.} The left premise is derived by {\sf R$ \Box $}. Similar to Case 3.1.\\
\textbf{Case 3.2.} The left premise is derived by {\sf Exch}. Suppose the hypersequent ${\cal H}$ be as $ {\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I} $. We have the following subcases according to the last rule in derivation of the right premise, and we only consider modal rules here. \\
\textbf{Subcase 3.2.1} The right premise is derived by {\sf Exch}. Let the hypersequent $ {\cal H}' $ be as ${\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $, and let the last rules be as
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$M,P_i\Rightarrow Q_i,N, \Diamond A\,||\, {\cal G}\,|\, P\Rightarrow Q\,|\, {\cal I} $}
\RightLabel{\sf Exch}
\UnaryInfC{$M,P\Rightarrow Q,N,\Diamond A\,||\, {\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$\Diamond A,M',P'_i\Rightarrow Q'_i,N'\,||\, {\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$\Diamond A,M',P'\Rightarrow Q',N'\,||\, {\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$M,P,M',P' \Rightarrow Q,N,Q',N'\,||\, {\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I}\,|\, {\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree}
where $ \Gamma=M,P $, $ \,\Delta=Q,N $, $\, \Gamma'=M',P' $, and $ \Delta'=Q',N'$. This cut is transformed into
\begin{prooftree}
\AxiomC{$ \mathcal{D} $}
\noLine
\UnaryInfC{$M,P_i\Rightarrow Q_i,N, \Diamond A\,||\, {\cal G}\,|\, P\Rightarrow Q\,|\, {\cal I} $}
\AxiomC{$ \mathcal{D}' $}
\noLine
\UnaryInfC{$ \Diamond A,M',P'_i\Rightarrow Q'_i,N'\,||\, {\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Diamond A,M'\Rightarrow N'\,||\,P'_i\Rightarrow Q'_i\,|\, {\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{{\sf Cut},}
\BinaryInfC{$M,P_i,M'\Rightarrow Q_i,N,N',||\, {\cal G}\,|\, P\Rightarrow Q\,|\, {\cal I}\,|\,P'_i\Rightarrow Q'_i\,|\, {\cal G}'\,|\, P'\Rightarrow Q'\,|\,{\cal I}' $}
\RightLabel{${\sf Merge}^{\sf c}$}
\UnaryInfC{$ M,P_i,M'\Rightarrow Q_i,N,N',||\, {\cal G}\,|\, P,P'\Rightarrow Q,Q'\,|\, {\cal I}\,|\,P'_i\Rightarrow Q'_i\,|\, {\cal G}'\,|\,{\cal I}' $}
\RightLabel{{\sf Exch}.}
\UnaryInfC{$ M,P,M',P' \Rightarrow Q,N,Q',N'\,||\, {\cal G}\,|\, P_i\Rightarrow Q_i\,|\, {\cal I}\,|\, {\cal G}'\,|\, P'_i\Rightarrow Q'_i\,|\,{\cal I}' $}
\end{prooftree}
\textbf{Subcase 3.2.2} The right premise is derived by {\sf R$ \Box $}. Similar to Subcase 3.2.1.\\
\textbf{Subcase 3.2.3} The right premise is derived by {\sf R$ \Diamond $} or {\sf L$ \Box $}. Similar to Subcase 3.1.3.\\ \end{proof} \begin{The}\label{complete}
The following are equivalent.
\begin{itemize}
\item[{\rm (1)}]
The sequent
$ \Gamma\Rightarrow A $ is S5-valid.
\item[{\rm (2)}]
$ \Gamma\vdash_{\text{S5}} A $.
\item[{\rm (3)}]
The sequent
$ \Gamma\Rightarrow A $
is provable in $ \cal{R}_{\sf S5} $. \end{itemize} \end{The} \begin{proof}
(1) implies (2) by completeness of S5.
(3) implies (1) by soundness of $ \cal{R}_{\sf S5} $. We show that (2) implies (3).
Suppose
$ A_1,\ldots,A_n $
is an S5-proof of $ A $ from $ \Gamma $. This means that $ A_n $ is $ A $ and that each $ A_i $ is in $ \Gamma $, is an axiom, or is inferred by modus ponens or necessitation. It is straightforward to
prove, by induction on $ i $, that $\vdash \Gamma\Rightarrow A_i $ for each $ A_i $.
Case 1. $ A_i\in \Gamma $: Let $ \Gamma= A_i,\Gamma' $. It can be easily proved that $ A_i,\Gamma'\Rightarrow A_i $ is derivable in $ \cal{R}_{\sf S5} $ by induction on the complexity of $ A_i $ and using weakening rules.
Case 2. $ A_i $ is an axiom of S5: All axioms of S5 are easily proved in $ \cal{R}_{\sf S5} $. As a typical example, in the following we prove the axiom 5:
\begin{prooftree}
\AxiomC{$ A\Rightarrow A,\Diamond A $}
\RightLabel{$ \sf R\Diamond $}
\UnaryInfC{$ A\Rightarrow \Diamond A $}
\RightLabel{$ \sf L\Diamond $}
\UnaryInfC{$ \Diamond A\Rightarrow \Diamond A $}
\RightLabel{\sf R$\Box $}
\UnaryInfC{$ \Diamond A\Rightarrow \Box\Diamond A $}
\RightLabel{{\sf R$\rightarrow $}.}
\UnaryInfC{$ \Rightarrow \Diamond A\rightarrow \Box\Diamond A $}
\end{prooftree}
Case 3. $ A_i $ is inferred by modus ponens: Suppose $ A_i $ is inferred from $ A_j $ and $ A_j\rightarrow A_i $, $ j<i $, by use of the cut rule we prove $ \Gamma\Rightarrow A_i $:
\begin{prooftree}
\AxiomC{}
\RightLabel{IH}
\UnaryInfC{$ \Gamma\Rightarrow A_j $}
\AxiomC{}
\RightLabel{ IH }
\UnaryInfC{$ \Gamma\Rightarrow A_j\rightarrow A_i $}
\AxiomC{$ A_j\Rightarrow A_i, A_j $}
\AxiomC{$ A_i\Rightarrow A_i $}
\RightLabel{\sf L$\rightarrow $}
\BinaryInfC{$ A_j\rightarrow A_i, A_j\Rightarrow A_i $}
\RightLabel{\sf Cut}
\BinaryInfC{$A_j,\Gamma\Rightarrow A_i $}
\RightLabel{\sf Cut}
\BinaryInfC{$ \Gamma,\Gamma\Rightarrow A_i $}
\RightLabel{{\sf LC}. }
\UnaryInfC{$ \Gamma\Rightarrow A_i $}
\end{prooftree}
Case 4. $ A_i $ is inferred by necessitation: Suppose $ A_i=\Box A_j $ is inferred from $ A_j $ by necessitation. In this case, $ \vdash_{S5}A_j $ (since the rule necessitation can be applied only to premises which are derivable in the axiomatic system) and so we have: \begin{prooftree} \AxiomC{} \RightLabel{IH} \UnaryInfC{$ \Rightarrow A_j $} \RightLabel{\sf R$ \Box $} \UnaryInfC{$ \Rightarrow\Box A_j $} \RightLabel{{\sf LW}.} \UnaryInfC{$ \Gamma\Rightarrow A_i $} \end{prooftree}
\end{proof} \begin{Cor} $ \cal{R}_{\sf S5} $ is sound and complete with respect to the S5-{\rm Kripke} frames. \end{Cor} \section{Concluding Remarks}\label{conclution} We have presented the system $ \cal{R}_{\sf S5} $, using rooted hypersequent, a sequent-style calculus for S5 which enjoys the subformula property. We have proved the soundness and completeness theorems, and the admissibility of the weakening, contraction and cut rules in the system. In the first draft of this paper, we wrote the rules {\sf L$ \Diamond $} and {\sf R$ \Box $} as follows: \begin{center}
\AxiomC{$ M,A\Rightarrow\Box\bigvee(\neg P,Q),N $}
\RightLabel{\sf L$ \Diamond $}
\UnaryInfC{$ \Diamond A,M,P\Rightarrow Q,N $}
\DisplayProof
$ \qquad $
\AxiomC{$ M\Rightarrow\Box\bigvee(\neg P,Q),N,A $}
\RightLabel{{\sf R$ \Box $}.}
\UnaryInfC{$ M, P\Rightarrow Q,N,\Box A $}
\DisplayProof
\end{center} In these rules we can use $ \Diamond\bigwedge(P,\neg Q) $ in the antecedent instead of $ \Box\bigvee(\neg P,Q) $ in the succedent of the premises, since these formulae are equivalent and have the same role in derivations as storages; they equivalently can be exchanged, or be taken both of them. Taking each of them, one can prove the admissibility of the others. By applying these rules in a backward proof search, although the formulae in the premises are constructed from atomic formulae in the conclusions, the subformula property does not hold. Then, we decided to use semicolon in the middle part of the sequents. Using extra connective semicolon ($ ; $) we introduced a new sequent-style calculus for S5, and called it $ \text{G3{\scriptsize S5}}^; $. Sequents in $ \text{G3{\scriptsize S5}}^; $ are of the form $ \Gamma;P_1;\ldots;P_n\Rightarrow Q_n;\ldots;Q_1;\Delta $, where $ \Gamma $ and $ \Delta $ are multisets of arbitrary formulae, and $ P_i $ and $ Q_i $ are multisets of atomic formulae which serve as storages. For convenience, we used $ H $ and $ G $ to denote the sequence of multisets $P_1;\ldots;P_n$ and $ Q_n;\ldots;Q_1$, respectively. Thus, we used $ \Gamma;H\Rightarrow G;\Delta $ to denote the sequents. The main idea for constructing this sequent is to take an ordinary sequent $ \Gamma\Rightarrow\Delta $ as a root and add two sequences of multisets of atomic formulae to it. The system $ \text{G3{\scriptsize S5}}^; $ is obtained by extending G3c for propositional logic with the following rules: \begin{align*} &\AxiomC{$ A,\Box A,\Gamma;H\Rightarrow G; \Delta $} \RightLabel{L$\Box $} \UnaryInfC{$ \Box A,\Gamma;H\Rightarrow G;\Delta $} \DisplayProof && \AxiomC{$ M;P;H\Rightarrow G;Q;N,A$} \RightLabel{\sf R$\Box $} \UnaryInfC{$M,P;H\Rightarrow G;Q,N,\Box A $} \DisplayProof \\[0.15cm] &\AxiomC{$ A,M;P;H\Rightarrow G;Q;N $} \RightLabel{L$\Diamond $} \UnaryInfC{$\Diamond A,M,P;H\Rightarrow G;Q,N $} \DisplayProof && \AxiomC{$ \Gamma;H\Rightarrow G; \Delta,\Diamond A,A $} \RightLabel{\sf R$\Diamond $} \UnaryInfC{$ \Gamma;H\Rightarrow G;\Delta,\Diamond A $} \DisplayProof \end{align*} \begin{prooftree}
\AxiomC{$M,P_i;H_1;P;H_2\Rightarrow G_2;Q;G_1;Q_i,N $}
\RightLabel{\sf Exch}
\UnaryInfC{$M,P;H_1;P_i;H_2\Rightarrow G_2;Q_i;G_1;Q,N $} \end{prooftree} where $M$ and $N $ are multisets of modal formulae. In backward proof search, by applying the rules L$ \Diamond $, {\sf R$ \Box $}, and {\sf Exch} atomic formulae in $ P $ and $ Q $ in the conclusions move to the middle (storage) parts (between two semicolons) in the premises. These formulae, which are stored in the middle parts of sequents until applications of the rule {\sf Exch} in a derivation, are called related formulae, and $ (P,Q) $ is called a related pair of multisets. By applying the rule {\sf Exch}, related formulae in $ P_i $ and $ Q_i $ come out from the middle part. In other words, multisets $ P_i $ and $ Q_i $ exist together from storages by applications of the rule {\sf Exch}, if they have entered them together previously by applications of the rules {\sf L$ \Diamond $}, {\sf R$ \Box $}, or {\sf Exch}.
In the following, we prove a simple sequent to show details of this system. Below, $ H=G=\emptyset $ and so for convenience we omit their semicolons, also $ P=\emptyset $ and $ Q=p $ in the rules {\sf R$ \Box $} and {\sf Exch}: \begin{prooftree}
\AxiomC{$ p,\Box p\Rightarrow p$}
\RightLabel{\sf L$ \Box $}
\UnaryInfC{$ \Box p\Rightarrow p $}
\RightLabel{\sf Exch}
\UnaryInfC{$ \Box p;\Rightarrow p; $}
\RightLabel{\sf R$ \neg $}
\UnaryInfC{$ ;\Rightarrow p; \neg \Box p $}
\RightLabel{\sf R$ \Box $}
\UnaryInfC{$ \Rightarrow p,\Box\neg \Box p $} \end{prooftree} This system has the subformula property, and we showed that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well.
The intended interpretation of the sequent $ \Gamma;P_1;\ldots;P_n\Rightarrow Q_n;\ldots;Q_1;\Delta $ is defined as follows \[\bigwedge\Gamma\rightarrow \bigvee\Delta\vee\bigvee_{i=1}^n\Box(\bigwedge P_i\rightarrow \bigvee Q_i).\] This interpretation is similar to the standard formula interpretation of both nested sequents (\cite{brunnler2009deep}) and grafted hypersequents (\cite{Kuznets Lellmann}). A nested sequent is a structure $ \Gamma\Rightarrow\Delta, [\mathcal{N}_1],\ldots,[\mathcal{N}_n] $, where $ \Gamma\Rightarrow\Delta $ is an ordinary sequent and each $ \mathcal{N}_i $ is again a nested sequent. A grafted hypersequent is a structure
$ \Gamma\Rightarrow \Delta\parallel \Gamma_1\Rightarrow \Delta_1\,|\cdots|\Gamma_n\Rightarrow \Delta_n $, where $ \Gamma\Rightarrow \Delta $ is called root or trunk, and each $ \Gamma_i\Rightarrow \Delta_i $ is called a component. The standard formula interpretation of nested sequent is given recursively as \[ (\Gamma\Rightarrow\Delta, [\mathcal{N}_1],\ldots,[\mathcal{N}_n])^{tr}:=\bigwedge \Gamma\rightarrow\bigvee\Delta \vee \Box(\mathcal{N}_1)^{tr}\vee\ldots\vee \Box(\mathcal{N}_n)^{tr},\] where $(\mathcal{N}_i)^{tr}$ is the standard formula interpretation of the nested sequent $ \mathcal{N}_i $. A grafted hypersequents is essentially the same as the nested sequent $ \Gamma\Rightarrow\Delta,[\Gamma_1\Rightarrow \Delta_1],\ldots, [\Gamma_n\Rightarrow \Delta_n] $, and the interpretation of grafted hypersequents is adapted from the nested sequent setting as well. Therefore, the sequent $ \Gamma;P_1;\ldots;P_n\Rightarrow Q_n;\ldots;Q_1;\Delta $, can also be transformed into rooted hypersequent
$$ \Gamma\Rightarrow \Delta\parallel P_1\Rightarrow Q_1\,|\cdots|P_n\Rightarrow Q_n, $$ where all formulae in each component $ P_i\Rightarrow Q_i $ are atomic formulae. Therefore, we decided to rewrite the sequent in the framework of grafted hypersequent and we call it rooted hypersequent.\\
\textbf{Acknowledgement}. The authors would like to thank Meghdad Ghari for useful suggestions.
\newlength{\bibitemsep}\setlength{\bibitemsep}{.2\baselineskip plus .05\baselineskip minus .05\baselineskip} \newlength{\bibparskip}\setlength{\bibparskip}{0pt} \let\oldthebibliography\thebibliography \renewcommand\thebibliography[1]{
\oldthebibliography{#1}
\setlength{\parskip}{\bibitemsep}
\setlength{\itemsep}{\bibparskip} } \addcontentsline{toc}{section}{References}
\end{document} | arXiv |
\begin{document}
\title{The Poincar\'e-Hopf Theorem for relative braid classes}
\author{S. Muna\`{o}, R.C. Vandervorst\footnote{Department of Mathematics, VU University Amsterdam, The Netherlands.}}
\maketitle
{\abstract
\emph{Braid Floer homology} is an invariant of proper relative braid classes \cite{GVVW}. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid classes is non-trivial,
then additional closed integral curves of the Hamiltonian equations are forced
via a Morse type theory. In this article we show that certain information contained in the braid Floer homology --- the Euler-Floer characteristic --- also forces closed integral curves and periodic points of arbitrary vector fields and diffeomorphisms and leads to a Poincar\'e-Hopf type Theorem. The Euler-Floer characteristic for any proper relative braid class can be computed via a finite cube complex that serves as a model for the given braid class.
The results in this paper are restricted to the 2-disc, but can be extend to two-dimensional surfaces (with or without boundary). } \section{Introduction} \label{intro} Let ${ \mathbb{D}^2}\subset { \mathbb{R}}^2$ denote the standard (closed) 2-disc in the plane with coordinates $x=(p,q),$ i.e. ${ \mathbb{D}^2}:=\{(p,q)\in{ \mathbb{R}}^2: p^2+q^2\leq 1\},$ and let $X(x,t)$ be a smooth 1-periodic vector field on ${ \mathbb{D}^2}$, i.e. $X(x,t+1) = X(x,t)$ for all $x\in { \mathbb{D}^2}$ and $t\in { \mathbb{R}}$.
The vector field $X$ is tangent to the boundary $\partial { \mathbb{D}^2}$, i.e $X(x,t) \cdot \nu =0$ for all $x\in \partial{ \mathbb{D}^2}$, where $\nu$ the outward unit normal on $\partial{ \mathbb{D}^2}$. The set of vector fields satisfying these hypotheses is denoted by $\mathcal{F}_{\parallel}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$. \emph{Closed integral curves} $x(t)$ of $X$ are integral curves\footnote{Integral curves of $X$ are smooth functions $x: { \mathbb{R}} \to { \mathbb{D}^2}\subset { \mathbb{R}}^{2}$ that satisfy the differential equation $x' = X(x,t)$.} of $X$ for which $x(t+\ell) = x(t)$ for some $\ell\in { \mathbb{N}}$. Every integral curve of $X$ with minimal period $\ell$ defines a closed loop in the configuration space ${ \mathbf{C}}_{\ell}({ \mathbb{D}^2})$ of $\ell$ unordered distinct points. A collection of distinct closed integral curves with periods $\ell_j$ defines a closed loop in ${ \mathbf{C}}_{m}({ \mathbb{D}^2})$, with $m = \sum_{j} \ell_{j}$. As curves in the cylinder ${ \mathbb{D}^2}\times [0,1]$ such a collection of integral curves
represents a geometric braid which corresponds to a unique word $\beta_y \in { \mathbf{B}}_m$, modulo conjugacy and full twists: \begin{equation} \label{eqn:cong} \beta_y \sim \beta_y \Delta^{2k} \sim \Delta^{2k} \beta_y, \end{equation} where $\Delta^2$ is a full positive twist and ${ \mathbf{B}}_m$ is the Artin Braid group on $m$ strands.
Let $y$ be a geometric braid consisting of closed integral curves of $X$, which will be referred to as a \emph{skeleton}. The curves $y^i(t)$, $i=1,\cdots, m$ satisfy the periodicity condition $y(0) = y(1)$ as point sets, i.e. $ y^i(0)=y^{\sigma(i)}(1)$ for some permutation $\sigma \in S_m$.
In the configuration space ${ \mathbf{C}}_{n+m}({ \mathbb{D}^2})$\footnote{The space of continuous mapping ${ \mathbb{R}}/{ \mathbb{Z}} \to X$, with $X$ a topological space, is called the free loop space of $X$ and is denoted by ${\mathcal{L}} X$.} we consider closed loops of the form $x \rel y := \bigl\{x^1(t),\cdots x^n(t),y^1(t),\cdots, y^m(t)\bigr\}$. The path component of $x\rel y$ of closed loops in ${\mathcal{L}} { \mathbf{C}}_{n+m}({ \mathbb{D}^2})$
is denoted by $[x\rel y]$ and is called a \emph{relative braid class}. The loops $x'\rel y' \in [x\rel y]$,
keeping $y'$ fixed, is denoted by $[x']\rel y'$ and is called a \emph{fiber}.
Relative braid classes are path components
of braids which have at least two components and the components are labeled into two groups: $x$ and $y$. The intertwining of $x$ and $y$ defines various different braid classes. A relative braid class $[x\rel y]$ in ${ \mathbb{D}^2}$ is \emph{proper} if components $x_c\subset x$ cannot be deformed onto (i) the boundary $\partial{ \mathbb{D}^2}$, (ii) itself,\footnote{This condition is separated into two cases: (i) a component in $x$ cannot be not deformed into a single strand, or (ii) if a component in $x$ can be deformed into a single strand, then the latter
necessarily intersects $y$ or a different component in $x$.} or other components $x_c' \subset x$, or (iii) components in $y_c \subset y$, see \cite{GVVW} for details. In this paper we are mainly concerned with relative braids for which $x$ has only one strand. To proper relative braid classes $[x\rel y]$ one can assign the invariants ${ \mathrm{HB}}_*([x\rel y])$, with coefficients in ${ \mathbb{Z}}_2$, called \emph{Braid Floer homology}. In the following subsection we will briefly explain the construction of the invariants ${ \mathrm{HB}}_*([x\rel y])$ in case that $x$ consists of one single strand. See \cite{GVVW} for more details on Braid Floer homology.
\subsection{A brief summary of Braid Floer homology}
Fix a Hamiltonian vector field $X_H$ in $\mathcal{F}_{||}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$ of the form $X_H(x,t)=J\nabla H(x,t),$ where $$ J=\left( \begin{array}{cc} 0 & -1\\ 1 & 0 \end{array} \right) $$ and $H$ is a Hamiltonian function with the properties: \begin{enumerate} \item [(i)]$H\in C^{\infty}({ \mathbb{D}^2} \times{ \mathbb{R}}/{ \mathbb{Z}};{ \mathbb{R}})$;
\item [(ii)]$H(x,t)|_{x\in\partial{ \mathbb{D}^2}}=0$, for all $t\in { \mathbb{R}}/{ \mathbb{Z}}$. \end{enumerate} For closed integral curves of $X_H$ of period 1 we define the Hamilton action
$$ {\mathscr{A}}_{H}(x)=\int_0^1\tfrac{1}{2}Jx\cdot x_t-H(x,t)\ dt, $$ Critical points of the action functional ${\mathscr{A}}_H$ are in one-to-one correspondence with closed integral curves of period 1. Assume that
$y=\{y^j(t)\}$ is a collection of closed integral curves of the Hamilton vector field $X_H$, i.e. periodic solutions of the $y_t^j=X_H(y^j,t)$. Consider a proper relative braid class $[x]\rel y$, with $x$ 1-periodic and seek closed integral curves $x\rel y$ in $[x]\rel y$. The set of critical points of ${\mathscr{A}}_H$ in $[x]\rel y$ is denoted by $\Crit_{{\mathscr{A}}_H}([x]\rel y)$.
In order to understand the set $\Crit_{{\mathscr{A}}_H}([x]\rel y)$ we consider the negative $L^2$-gradient flow of ${\mathscr{A}}_H$.
The $L^2$-gradient flow $ u_s=-\nabla_{L^2}{\mathscr{A}}_H(u) $ yields the Cauchy-Riemann equations $$ u_s(s,t)+Ju_t(s,t)+\nabla H(u(s,t),t)=0, $$ for which the stationary solutions $u(s,t)=x(t)$ are the critical points of ${\mathscr{A}}_{H}$.
To a braid $y$ one can assign an integer ${\mathrm{Cross}}(y)$ which counts the number of crossings (with sign) of strands in the standard planar projection. In the case of a relative braid $x \rel y$ the number ${\mathrm{Cross}}(x\rel y)$ is an invariant of the relative braid class $[x\rel y]$. In
\cite{GVVW}
a monotonicity lemma is proved, which states
that along solutions $u(s,t)$ of the nonlinear Cauchy-Riemann equations, the number ${\mathrm{Cross}}(u(s,\cdot)\rel y)$ is non-increasing (the jumps correspond to `singular braids', i.e. `braids' for which intersections occur). As a consequence an isolation property for proper relative braid classes exists: the set bounded solutions of the Cauchy-Riemann equations in a proper braid class fiber $[x]\rel y$, denoted by
${\mathscr{M}}([x]\rel y;H)$, is compact and isolated with respect to the topology of uniform convergence on compact subsets of ${ \mathbb{R}}^2$. These facts provide all the ingredients to follows Floer's approach towards Morse Theory for the Hamiltonian action \cite{Floer1}. For generic Hamiltonians which satisfy (i) and (ii) above and for which $y$ is a skeleton, the critical points in $[x]\rel y$ of the action ${\mathscr{A}}_H$ are non-degenerate and the set of connecting orbits ${\mathscr{M}}_{x_{-},x_{+}}([x]\rel y;H)$ are smooth finite dimensional manifolds. To critical in $\Crit_{{\mathscr{A}}_H}([x]\rel y)$ we assign a relative index $\mu^{CZ}(x)$ (the Conley-Zehnder index) and $$ \dim {\mathscr{M}}_{x_{-},x_{+}}([x]\rel y;H)=\mu^{CZ}(x_{-})-\mu^{CZ}(x_{+}). $$ Define the free abelian groups $C_k$ over the critical points of index $k$, with coefficients in ${ \mathbb{Z}}_{2}$, i.e. $$ C_{k}([x]\rel y;H):=\bigoplus_{x\in \Crit_{{\mathscr{A}}_H}([x]\rel y), \atop \mu(x)=k}{ \mathbb{Z}}_{2}\langle x\rangle, $$ and the boundary operator $$ \partial_{k} = \partial_k([x]\rel y;H): C_{k}\to C_{k-1}, $$ which counts the number of orbits (modulo 2) between critical points of index $k$ and $k-1$ respectively. Analysis of the spaces ${\mathscr{M}}_{x_{-},x_{+}}([x]\rel y;H)$ reveals that $(C_*,\partial_{*})$ is a chain complex, and its (Floer) homology is denoted by $ { \mathrm{HB}}_{*}([x]\rel y;H). $
Different choices of $H$ yields isomorphic Floer homologies and $$ { \mathrm{HB}}_{*}([x]\rel y) = \varprojlim { \mathrm{HB}}_{*}([x]\rel y;H), $$ where the inverse limit is defined with respect to the canonical isomorphisms $a_k(H,H'): { \mathrm{HB}}_k([x]\rel y,H) \to { \mathrm{HB}}_k([x]\rel y,H')$. Some properties are: \begin{enumerate} \item [(i)] the groups ${ \mathrm{HB}}_k([x]\rel y)$ are defined for all $k\in { \mathbb{Z}}$ and are finite, i.e. ${ \mathbb{Z}}_2^d$ for some $d\ge 0$; \item [(ii)] the groups ${ \mathrm{HB}}_k([x]\rel y)$ are invariants for the fibers in the same relative braid class $[x\rel y]$, i.e. if $x\rel y\sim x'\rel y'$, then ${ \mathrm{HB}}_k([x]\rel y)\cong { \mathrm{HB}}_k([x']\rel y')$. For this reason we will write ${ \mathrm{HB}}_{*}{([x\rel y])}$; \item [(iii)] if $(x\rel y)\cdot \Delta^{2\ell}$ denotes composition with $\ell$ full twists, then ${ \mathrm{HB}}_k([(x\rel y)\cdot\Delta^{2\ell}])\cong { \mathrm{HB}}_{k-2\ell}([x\rel y])$. \end{enumerate}
\subsection{The Euler-Floer characteristic and the Poincar\'{e}-Hopf Formula} Braid Floer homology is an invariant of conjugacy classes in ${ \mathbf{B}}_{n+m}$ and can be computed from purely topological data. The \emph{Euler-Floer characteristic} of ${ \mathrm{HB}}_*\bigl([x\rel y]\bigr)$ is defined as follows: \begin{equation} \label{EulerFloer-1} \chi\bigl(x\rel y\bigr) = \sum_{k\in { \mathbb{Z}}} (-1)^{k}{\dim { \mathrm{HB}}_k([x\rel y])}. \end{equation}
In Section \ref{comp} we show that the Euler-Floer characteristic of ${ \mathrm{HB}}_*\bigl([x\rel y]\bigr)$ can be computed from a finite cube complex which serves as a model for the braid class.
A 1-periodic function $x \in C^{1}({ \mathbb{R}}/{ \mathbb{Z}})$ is an \emph{isolated} closed integral curve of $X$ if there exists an $\epsilon>0$ such that $x$ is the only solution of the differential equation \begin{equation} \label{eqn:E} {\mathscr{E}}\bigl(x(t)\bigr) =
\frac{dx}{dt}(t)- X\bigl(x(t),t\bigr), \end{equation}
in $B_{\epsilon}(x) \subset C^{1}({ \mathbb{R}}/{ \mathbb{Z}})$. For isolated, and in particular non-degenerate closed integral curves we can define an \emph{index} as follows. Let $\Theta \in {\rm M}_{2\times 2}({ \mathbb{R}})$ be any matrix satisfying $\sigma(\Theta) \cap 2\pi k i{ \mathbb{R}} = \varnothing$, for all $k\in { \mathbb{Z}}$ and let $\eta\mapsto R(t;\eta)$ be a curve in $C^\infty\bigl({ \mathbb{R}}/{ \mathbb{Z}};{\rm M}_{2\times 2}({ \mathbb{R}})\bigr)$, with $R(t;0) = \Theta$ and $R(t;1) = D_xX(x(t),t)$ --- the linearization of $X$ at $x(t)$. Then $\eta \mapsto F(\eta) = \frac{d}{dt} - R(t;\eta)$ defines a curve in $\fred_0(C^1,C^0)$, where we denote by $\fred_0(C^1,C^0)$ the space of Fredholm operators of index $0$ between $C^1$ and $C^0.$ Denote by $\Sigma \subset \fred_0(C^1,C^0)$ the set of non-invertible operators and by $\Sigma_1 \subset \Sigma$ the non-invertible operators with a 1-dimensional kernel. If the end points of $F$ are invertible one can choose the path $\eta\mapsto R(t;\eta)$ such that $F(\eta)$ intersects $\Sigma$ in $\Sigma_1$ and all intersections are transverse. If $\gamma = \# \hbox{~intersections of}~ F(\eta)~\hbox{with}~\Sigma_1$, then \begin{equation} \label{eqn:index0} \iota(x) = -\sgn(\det(\Theta)) (-1)^\gamma. \end{equation} This definition is independent of the choice of $\Theta$, see Section \ref{sec:proof}.
The above definition can be expressed in terms of the Leray-Schauder degree. Let $M\in {\rm GL}(C^0,C^1)$ be any isomorphism such that $\Phi_M(x) := M {\mathscr{E}}(x)$ is of the form `identity + compact'.
Then the index of an isolated closed integral curve is given by \begin{equation} \label{eqn:index} \iota(x) = - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \deg_{ LS}(\Phi_M,B_{\epsilon}(x),0).
\end{equation} where $\beta_M(\Theta)$ is the number of negative eigenvalues of $M\frac{d}{dt} - M\Theta$ counted with multiplicity. The latter definition holds for both non-degenerate and isolated 1-periodic closed integral curves of $X$. In Section \ref{sec:proof} we show that the two expressions for the index are the same and we show that they are independent of the choices of $M$ and $\Theta$.
\begin{theorem}[Poincar\'e-Hopf Formula] \label{thm:PH1} Let $y$ be a skeleton of closed integral curves of a vector field $X\in \mathcal{F}_{\parallel}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$ and let $[x\rel y]$ be a proper relative braid class. Suppose that all 1-periodic
closed integral curves of $X$ are isolated, then for all closed integral curves $x_0\rel y$ in $[x_0]\rel y$ it holds that \begin{equation} \label{eqn:PH2} \sum_{x_{0}} \iota(x_{0}) = \chi\bigl(x\rel y\bigr). \end{equation}
\end{theorem}
The index formula can be used to obtain existence results for closed integral curves in proper relative braid classes. \begin{theorem} \label{thm:exist} Let $y$ be a skeleton of closed integral curves of a vector field $X\in \mathcal{F}_{\parallel}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$ and let $[x\rel y]$ be a proper relative braid class. If $\chi\bigl(x\rel y\bigr) \not = 0$, then there exist closed integral curves $x_0\rel y$ in $[x]\rel y$. \end{theorem}
The analogue of Theorem \ref{thm:PH1} can also be proved for relative braid class $[x\rel y]$ in ${ \mathbf{C}}_{n+m}({ \mathbb{D}^2})$. Our theory also provides detailed information about the linking of solutions. In Section \ref{sec:examples} we give various examples and compute the Euler-Floer characteristic.
This does not provide a procedure for computing the braid Floer homology.
\begin{remark} \label{rmk:fitz} {\em In this paper Theorem \ref{thm:PH1} is proved using the standard Leray-Schauder degree theory in combination with the theory of spectral flow and parity for operators on Hilbert spaces. The Leray-Schauder degree is related to the Euler characteristic of Braid Floer homology. An other approach is the use the degree theory developed by Fitzpatrick et al. \cite{Fitzpatrick:1992vv}. } \end{remark}
\subsection{Discretization and computability}
The second part of the paper deals with the computability of the Euler-Floer characteristic. This is obtained through a finite dimensional model. A model is constructed in three steps: \begin{enumerate} \item[(i)] compose $x\rel y$ with $\ell\ge 0$ full twists $\Delta^2$, such that $(x\rel y)\cdot \Delta^{2\ell}$ is isotopic to a positive braid $x^+\rel y^+$; \item[(ii)] relative braids $x^+\rel y^+$ are isotopic to Legendrian braids $x_L\rel y_L$ on ${ \mathbb{R}}^2$, i.e. braids which have the form $x_L=(q_t,q)$ and $y_L=(Q_t,Q),$ where $q=\pi_2 x$ and $Q=\pi_2 y,$ and $\pi_2$ the projection onto the $q-$coordinate; \item[(iii)] discretize $q$ and $Q = \{Q^j\}$ to $q_d = \{q_i\}$, with $q_i=q(i/d), i=0,\dots,d$ and $Q_D = \{ Q_D^j\}$, with $Q_D^j = \{Q^j_i\}$ and $Q^j_i = Q^j(i/d)$ respectively, and consider the piecewise linear interpolations connecting the \emph{anchor} points $q_i$ and $Q^j_i$ for $i=0,\dots,d$. A discretization $q_D\rel Q_D$ is \emph{admissible} if the linear interpolation is isotopic to $q\rel Q$. All such discretization form the discrete relative braid class $[q_D\rel Q_D]$, for which each fiber is a finite cube complex.
\end{enumerate}
\begin{remark} {\em If the number of
discretization points is not large enough, then the discretization may not be admissible and therefore not capture the topology of the braid. See \cite{GVV} and Section \ref{subsec:discbraid} for more details. } \end{remark}
For $d>0$ large enough there exists an admissible discretization $q_D\rel Q_D$ for any Legendrian representative $x_L\rel y_L \in [x\rel y]$ and thus an associated discrete relative braid class $[q_D\rel Q_D]$. In \cite{GVV} an invariant for discrete braid classes was introduced. Let $[q_D]\rel Q_D$ denote a fiber in $[q_D\rel Q_D]$, which is a cube complex with a finite number of connected components and their closures are denoted by $N_j$. The faces of the hypercubes $N_j$ can be co-oriented in direction of decreasing the number of crossing in $q_D\rel Q_D$, and we define $N_j^-$ as the closure of the set of faces with outward pointing co-orientation. The sets $N_j^-$ are called \emph{exit sets}. The invariant for a fiber is given by $$ {\mathrm{HC}}_*([q_D]\rel Q_D) = \bigoplus_{j} H_*(N_j, N_j^-). $$ This discrete braid invariant is well-defined for any $d>0$ for which there exist admissible discretizations and is independent of both the particular fiber and the discretization size $d$. For the associated Euler characteristic we therefore write $\chi\bigl(q_D\rel Q_D\bigr)$. The latter is an Euler characteristic of a topological pair.
The Euler characteristic of the Braid Floer homology $\chi(x\rel y)$ can be related to the Euler characteristic of the associated discrete braid class.
\begin{theorem} \label{thm:discrete} Let $[x\rel y]$ a proper relative braid class and $\ell\ge 0$ is an integer such that $(x\rel y)\cdot \Delta^{2\ell}$ is isotopic to a positive braid $x^+\rel y^+$. Let
$q_D\rel Q_D$ be an admissible discretization, for some $d>0$, of a Legendrian representative $x_L\rel y_L \in [x^+\rel y^+]$.
Then $$ \chi(x\rel y)=\chi(q_D\rel Q^*_D), $$ where $Q_D^*$ is the augmentation of $Q_D$ by adding the constant strands $ \pm 1$ to $Q_D$. \end{theorem}
The idea behind the proof of Theorem \ref{thm:discrete} is to first relate $\chi(x\rel y)$ to mechanical Lagrangian systems and then use a discretization approach based on the method of broken geodesics. Theorem \ref{thm:discrete} is proved in Section \ref{comp}.
In Section \ref{sec:examples} we use the latter to compute the Euler-Floer characteristic for various examples of proper relative braid classes.
\subsection{Additional topological properties} \label{subset:add}
In this paper we do not address the question whether the closed integral curves $x\rel y$ are non-constant, i.e. are not
equilibrium points.
By considering relative braid classes
where $x$ consists of more than one strand one can study non-constant closed integral curves. Braid Floer homology for relative braids with $x$ consisting of $n$ strands is defined in \cite{GVVW}. The ideas in this paper extend to relative braid classes with multi-strand braids $x$. In Section \ref{sec:examples} we give an example of a multi-strand $x$ in $x\rel y$ and explain how this yields the existence of non-trivial closed integral curves.
The invariant $\chi\bigl(q_D\rel Q_D\bigr)$ is a true Euler characteristic and $$ \chi\bigl(q_D\rel Q_D\bigr) = \chi\bigl([q_D]\rel Q_D,[q_D]^-\rel Q_D\bigr), $$ where $[q_D]^-\rel Q_D$ is the exit. A similar characterization does not a priori exist for $[x]\rel y$.
This problem is circumvented by considering Hamiltonian systems and carrying out Floer's approach towards Morse theory (see \cite{Floer1}), by using the isolation property of $[x]\rel y$. The fact that the Euler characteristic of Floer homology is related to the Euler characteristic of a topological pair indicates that Floer homology is a good substitute for a suitable (co)-homology theory. For more details see Section \ref{comp} and Remark \ref{eqn:realEC}.
Braid Floer homology developed for the 2-disc ${ \mathbb{D}^2}$ can be extended to more general 2-dimensional manifolds. This generalization of Braid Floer homology for 2-dimensional manifolds can then be used to
extend the results in this paper to more general surfaces. \vskip.4cm
\noindent {\bf Acknowledgment.} The authors wish to thank J.B. van den Berg for the many stimulating discussions on the subject of Braid Floer homology.
\section{Closed integral curves} Let $X\in \mathcal{F}_{\parallel}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$, then closed
integral curves of $X$ of period 1 satisfy the differential equation \begin{equation}\label{general vf} \left\{ \begin{array}{ll} \displaystyle \frac{dx}{dt}=X(x,t), \quad x\in { \mathbb{D}^2}, ~t\in { \mathbb{R}}/{ \mathbb{Z}},\\ x(0)=x(1). \end{array} \right. \end{equation}
Consider the unbounded operator
$L_{\mu}:C^1({ \mathbb{R}}/{ \mathbb{Z}}) \subset C^0({ \mathbb{R}}/{ \mathbb{Z}}) \to C^0({ \mathbb{R}}/{ \mathbb{Z}})$, defined by
$$
L_{\mu}:= - J\frac{d}{dt}+\mu, \quad \mu \in { \mathbb{R}}.
$$
The operator
is invertible for $\mu\not= 2\pi k, k\in \mathbb{Z}$ and
the inverse $L_{\mu}^{-1}: C^0({ \mathbb{R}}/{ \mathbb{Z}}) \to C^0({ \mathbb{R}}/{ \mathbb{Z}})$ is compact.
Transforming Equation (\ref{general vf}), using $L_\mu^{-1}$, yields
the equation $\Phi_{\mu}(x) =0$, where
$$ \Phi_{\mu}(x):=x-L_{\mu}^{-1}\bigl(-JX(x,t)+\mu x\bigr). $$ If we set $$ K_{\mu}(x):=L_{\mu}^{-1}\bigl(-JX(x,t)+\mu x\bigr), $$ then $\Phi_{\mu}$ is of the form $\Phi_\mu(x) = x -K_{\mu}(x)$, where $K_\mu$ is a (non-linear) compact operator on $C^0({ \mathbb{R}}/{ \mathbb{Z}})$. Since $X$ is a smooth vector field the mapping $\Phi_{\mu}$ is a smooth mapping on $C^0({ \mathbb{R}}/{ \mathbb{Z}})$.
\begin{proposition} \label{prop:equiv1}
A function $x\in C^0({ \mathbb{R}}/{ \mathbb{Z}})$, with $|x(t)|\le 1$ for all $t$, is a solution of $\Phi_{\mu}(x) = 0$ if and only if $x\in C^1({ \mathbb{R}}/{ \mathbb{Z}})$ and $x$ satisfies Equation (\ref{general vf}). \end{proposition}
\begin{proof} If $x\in C^{1}({ \mathbb{R}}/{ \mathbb{Z}};{ \mathbb{D}^2})$ is a solution of Equation (\ref{general vf}), then $\Phi_{\mu}(x) =0$ is obviously satisfied. On the other hand, if $x\in C^{0}({ \mathbb{R}}/{ \mathbb{Z}};{ \mathbb{D}^2})$ is a zero of $\Phi_{\mu}$, then $x = K_{\mu}(x) \in C^{1}({ \mathbb{R}}/{ \mathbb{Z}})$, since $R(L_{\mu}^{-1}) \subset C^{1}({ \mathbb{R}}/{ \mathbb{Z}})$. Applying $L_{\mu}$ to both sides
shows that $x$ satisfies Equation (\ref{general vf}). \end{proof}
Note that the zero set $\Phi_{\mu}^{-1}(0)$ does not depend on the parameter $\mu$. In order to apply the Leray-Schauder degree theory we consider appropriate bounded, open subsets $\Omega \subset C^0({ \mathbb{R}}/{ \mathbb{Z}})$, which have the property that $\Phi_{\mu}^{-1}(0) \cap \partial \Omega =\varnothing$. Let $\Omega = [x]\rel y$, where $[x]\rel y$ is a proper relative braid fiber, and $y = \{y^1,\cdots,y^m\}$ is a skeleton of closed integral curves for the vector field $X$.
\begin{proposition} \label{lemma isolation} Let $[x\rel y]$ be a proper relative braid class and let $\Omega = [x]\rel y$ be the fiber given by $y$. Then, there exists an $0<r<1$ such that $$
|x(t)| < r, ~~\hbox{and~~}~~|x(t) - y^{j}(t)| >1-r,\quad \forall ~j=1,\cdots,m,\quad \forall ~t\in { \mathbb{R}}, $$
and for all $x\in \Phi_{\mu}^{-1}(0) \cap \Omega = \{x\in \Omega~|~x=K_\mu(x)\}$.
\end{proposition}
\begin{proof} Since $\Omega\subset C^0({ \mathbb{R}}/{ \mathbb{Z}})$ is a bounded set and $K_\mu$ is compact, the solution set $\Phi^{-1}_{\mu}(0)\cap\Omega$ is compact. Indeed, let $x_n= K_\mu(x_n)$ be a sequence in $\Phi^{-1}_{\mu}(0)\cap\Omega$, then $K_\mu(x_{n_k}) \to x$, and thus $x_{n_k} \to x$, which, by continuity, implies that $K_\mu(x_{n_k}) \to K_\mu(x)$, and thus $x\in \Phi^{-1}_{\mu}(0)\cap\Omega$.
Let $x_{n} \in \Phi^{-1}_{\mu}(0)\cap\Omega$ and assume that such an $0<r<1$ does not exist. Then, by the compactness of $\Phi_{\mu}^{-1}(0)\cap \Omega$, there is a subsequence $x_{n_{k}}\to x$ such that one, or both of the following two possibilities hold:
(i) $|x(t_0)|=1$ for some $t_0$. By the uniqueness of solutions of Equation (\ref{general vf}) and the invariance of the boundary $\partial { \mathbb{D}^2}$ ($X(x,t)$ is tangent to the boundary),
$|x(t)| =1$ for all $t$, which is impossible since $[x] \rel y$ is proper;
(ii) $x(t_0) = y^j(t_0)$ for some $t_0$ and some $j$. As before, by the uniqueness of solutions of Equation (\ref{general vf}), then $x(t)=y^j(t)$ for all $t$, which again contradicts the fact that $[x]\rel y$ is proper.
\end{proof}
By Proposition \ref{lemma isolation} the Leray-Schauder degree $\deg_{LS}(\Phi_{\mu},\Omega,0)$ is well-defined. Consider the
Hamiltonian vector field \begin{equation}\label{hamiltonian vector} X_H=J \nabla H, \quad J=\left(
\begin{array}{cc}
0 & -1\\
1 & ~~0
\end{array}
\right), \end{equation} where $H(x,t)$ is a smooth Hamiltonian such that $X_H \in \mathcal{F}_{\parallel}({ \mathbb{D}^2}\times { \mathbb{R}}/{ \mathbb{Z}})$ and $y$ is a skeleton for $X_H$. Such a Hamiltonian can always be constructed, see \cite{GVVW}, and the class of such Hamiltonians will be denote by $\mathcal{H}_{\parallel}(y)$. Since $y$ is a skeleton for both $X$ and $X_H$, it is a skeleton for the linear homotopy $X_{\alpha}=(1-\alpha)X+\alpha X_{H}$, $\alpha\in [0,1]$. Associated with the homotopy $X_\alpha$ of vector fields we define the homotopy $$ \Phi_{\mu,\alpha}(x):=x-L_{\mu}^{-1}\bigl(-JX_\alpha(x,t)+\mu x\bigr) = x-K_{\mu,\alpha}(x),\quad \alpha \in [0,1], $$ with $K_{\mu,\alpha}(x) = L_{\mu}^{-1}\bigl(-JX_\alpha(x,t)+\mu x\bigr)$. Proposition \ref{lemma isolation} applies for all $\alpha\in [0,1]$, i.e. by compactness there exists a uniform $0<r<1$ such that $$
|x(t)| < r, ~~\hbox{and~~}~~|x(t) - y^{j}(t)| >1-r, $$
for all $t\in { \mathbb{R}}$, for all $j$ and for all $x\in \Phi_{\mu,\alpha}^{-1}(0) \cap \Omega = \{x\in \Omega~|~x=K_{\mu,\alpha}(x) \}$ and all $\alpha \in [0,1]$.
By the homotopy invariance of the Leray-Schauder degree we have \begin{equation} \label{homtop} \deg_{LS}(\Phi_{\mu},\Omega,0)=\deg_{LS}(\Phi_{\mu,\alpha},\Omega,0)=\deg_{LS}(\Phi_{\mu,H},\Omega,0), \end{equation} where $\Phi_{\mu,0} = \Phi_{\mu}$ and $\Phi_{\mu,1} = \Phi_{\mu,H}$. Note that the zeroes of $\Phi_{\mu,H}$
correspond to critical point of the functional \begin{equation}\label{action} {\mathscr{A}}_H(x)=\int_0^1 \tfrac{1}{2} J x\cdot x_t - H(x,t) dt, \end{equation} and are denoted by $\Crit_{{\mathscr{A}}_H}([x]\rel y)$. In \cite{GVVW} invariants are defined which provide information about $\Phi_{\mu,H}^{-1}(0)\cap \Omega = \Crit_{{\mathscr{A}}_H}([x]\rel y)$ and thus $\deg_{LS}(\Phi_{\mu,H},\Omega,0)$. These invariants are the Braid Floer homology groups ${ \mathrm{HB}}_*\bigl([x]\rel y\bigr)$ as explained in the introduction. In the next section we examine spectral properties of the solutions of $\Phi_{\mu,\alpha}^{-1}(0)\cap \Omega$ in order to compute $\deg_{LS}(\Phi_{\mu,H},\Omega,0)$ and thus $\deg_{LS}(\Phi_{\mu},\Omega,0)$.
\begin{remark} \label{rmk:other} {\em There is obviously more room for choosing appropriate operators $L_{\mu}$ and therefore functions $\Phi_{\mu}$. In Section \ref{sec:proof} this issue will be discussed in more detail. } \end{remark}
\section{Parity, Spectral flow and the Leray-Schauder degree}
The
Leray-Schauder degree of an isolated zero $x$ of $\Phi_{\mu}(x) =0$ is called the local degree.
A zero $x \in \Phi_{\mu}^{-1}(0)$ is non-degenerate if $1\not\in \sigma(D_xK_{\mu}(x))$, where
$D_x K_{\mu}(x): C^0({ \mathbb{R}}/{ \mathbb{Z}}) \to C^0({ \mathbb{R}}/{ \mathbb{Z}})$ is the (compact) linearization at $x$ and is given by $D_x K_\mu(x) = L^{-1}_{\mu}(-JD_x X(x,t)+\mu)$.
If $x$ is a non-degenerate zero, then it is an isolated zero and
the degree can be determined from spectral information.
\begin{proposition}\label{beta} Let $x\in C^0({ \mathbb{R}}/{ \mathbb{Z}})$ be a non-degenerate zero of $\Phi_{\mu}$ and let $\epsilon>0$ be sufficiently small such that
$B_\epsilon(x) = \bigl\{\tilde x\in C^0({ \mathbb{R}}/{ \mathbb{Z}})~|~|\tilde x(t) - x(t)|<\epsilon,\forall t\bigr\}$ is a neighborhood in which $x$ is the only zero.
Then $$ \deg_{LS}\bigl(\Phi_{\mu},B_\epsilon(x),0\bigr)=\deg_{LS}\bigl(\Id-D_x K_{\mu}(x),B_\epsilon(x),0\bigr) =(-1)^{\beta_{\mu}(x)} $$ where $$ \beta_{\mu}(x)=\sum_{\sigma_{j}>1,\ \sigma_{j}\in\sigma(D_x K_{\mu}(x))}\beta_{j},\quad \beta_{j}=\dim\left(\bigcup_{i=1}^{\infty}\ker\bigl(\sigma_{j}\Id-D_x K_{\mu}(x)\bigr)^i\right), $$ which will be referred to as the Morse index of $x$, or alternatively the Morse index of linearized operator $D_x\Phi_\mu(x)$.
\end{proposition} \begin{proof} See \cite{Lloyd}. \end{proof}
The functions $\Phi_{\mu,\alpha}(x) = x - K_{\mu,\alpha}(x)$ are of the form `identity + compact' and Proposition \ref{beta} can be applied to non-degenerate zeroes of $\Phi_{\mu,\alpha}(x) = 0$. If we choose the Hamiltonian $H \in \mathcal{H}_{\parallel}^{{\rm reg}}(y)$ `generically', then the zeroes of $\Phi_{\mu,H}$ are non-degenerate, i.e. $1 \not \in \sigma(D_x K_{\mu,H}(x))$, where $D_x K_{\mu,H}(x) = D_x K_{\mu,1}(x)$. By compactness there are only finitely many zeroes in a fiber $\Omega = [x]\rel y$.
\begin{lemma} \label{nondeg} Let $x\in \Phi_{\mu,H}^{-1}(0)\cap \Omega$. Then following criteria for non-degeneracy are equivalent: \begin{enumerate} \item [(i)] $1 \not \in \sigma(D_x K_{\mu,H}(x))$; \item [(ii)] the operator $B=-J\frac{d}{dt} -D^2_xH(x(t),t)$ is invertible; \item [(iii)] let $\Psi(t)$ be defined by $B\Psi(t) =0$, $\Psi(0) = \Id$, then $\det(\Psi(1) -\Id)\neq 0$. \end{enumerate} \end{lemma}
\begin{proof} A function $\psi$ satisfies $D_x K_{\mu,H}(x) \psi = \psi$ if and only if $B\psi =0$, which shows the equivalence between (i) and (ii). The equivalence between (ii) and (iii) is proved in \cite{GVVW}. \end{proof}
The generic choice of $H$ follows from Proposition 7.1 in \cite{GVVW} based on criterion (iii). Hamiltonians for which the zeroes of $\Phi_{\mu,H}$ are non-degenerate are denoted by $\mathcal{H}_{\parallel}^{{\rm reg}}(y)$. Note that \emph{no} genericity is needed for $\alpha \in [0,1)$! For the Leray-Schauder degree this yields \begin{equation} \label{formula1} \deg_{LS}(\Phi_{\mu,\alpha},\Omega,0)=\deg_{LS}(\Phi_{\mu,H},\Omega,0)=\sum_{x\in \Crit_{{\mathscr{A}}_H}([x]\rel y)}(-1)^{\beta_{\mu,H}(x)}, \end{equation} for all $ \alpha \in [0,1]$ and where $\beta_{\mu,H}(x)$ is the Morse index of $\Id - D_x K_{\mu,H}(x)$.
The goal is to determine the Leray-Schauder degree $\deg_{LS}(\Phi_{\mu},\Omega,0)$ from information contained in the Braid Floer homology groups ${ \mathrm{HB}}_*([x]\rel y)$. In order to do so we examen the Hamiltonian case.
In the Hamiltonian case the linearized operator $D_x\Phi_{\mu,H}(x)$ is given by $$ A := D_x\Phi_{\mu,H}(x) = \Id - D_x K_{\mu,H}(x) = \Id - L_\mu^{-1} \bigl( D^2_x H(x(t),t) + \mu\bigr), $$ which is a bounded operator on $C^0({ \mathbb{R}}/{ \mathbb{Z}})$. The operator $A$ extends to a bounded operator on $L^2({ \mathbb{R}}/{ \mathbb{Z}})$.
Consider the path $\eta\mapsto A(\eta)$, $\eta\in I=[0,1]$, given by \begin{equation} \label{choice1} A(\eta)=\Id-L_{\mu}^{-1}(S(t;\eta)+\mu) = \Id - T_\mu(\eta), \end{equation} where $S(t;\eta)$ a smooth family of symmetric matrices and $T_{\mu}(\eta) = L_{\mu}^{-1}(S(t;\eta)+\mu)$.
The endpoints satisfy $$ S(t;0)=\theta \Id, \quad S(t;1)=D^{2}_xH(x(t),t), $$ with $\theta\not=2\pi k$, for some $k\in \mathbb{Z}$ and $D^{2}_xH(x(t),t)$ is the Hessian of $H$ at a critical point in
$\Crit_{{\mathscr{A}}_H}([x]\rel y)$. The path of $\eta\mapsto A(\eta)$ is a path bounded linear Fredholm operators on $L^2({ \mathbb{R}}/{ \mathbb{Z}})$ of Fredholm index 0, which are
compact perturbations of the identity and
whose endpoints are
invertible. \begin{lemma} \label{lem:all-s} The path $\eta\mapsto A(\eta)$ defined in (\ref{choice1}) is a smooth path of
bounded linear Fredholm operators in $H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$ of index $0$, with invertible endpoints.
\end{lemma} \begin{proof} By the smoothness of $S(t;\eta)$ we have that
$\Vert S(t;\eta)x\Vert_{H^{m}}\le C\Vert x\Vert_{H^m}$, for any $x\in H^{m}({ \mathbb{R}}/{ \mathbb{Z}})$ and any $m\in { \mathbb{N}} \cup \{0\}$. By interpolation the same holds for all $x\in H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$ and the claim follows from the fact that $L_{\mu}^{-1}: H^{s}({ \mathbb{R}}/{ \mathbb{Z}}) \to H^{s+1}({ \mathbb{R}}/{ \mathbb{Z}}) \hookrightarrow H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$ is compact. \end{proof}
\subsection{Parity of paths of linear Fredholm operators} \label{subsec:parity1} Let $\eta \mapsto \Lambda(\eta)$ be a smooth path of bounded linear Fredholm operators of index $0$ on a Hilbert space $\mathscr{H}$. A crossing $\eta_{0}\in I$ is a number for which the operator $\Lambda(\eta_{0})$ is not invertible. A crossing is simple if $\dim \ker \Lambda(\eta_0) =1$. A path $\eta\mapsto \Lambda(\eta)$ between invertible ends can always be perturbed to have only simple crossings. Such paths are called generic.
Following \cite{FitzPej,Fitzpatrick3,Fitzpatrick:1992vv,Fitzpatrick:1999ur}, we define the \emph{parity} of a generic path $\eta\mapsto \Lambda(\eta)$ by \begin{equation} \label{eqn:parity} \parity(\Lambda(\eta),I):= \prod_{\ker \Lambda(\eta_{0}) \not = 0}(-1) = (-1)^{\displaystyle{\cross(\Lambda(\eta),I)}}, \end{equation} where $\cross(\Lambda(\eta),I) = \# \{\eta_{0}\in I~:~\ker A(\eta_{0}) \not = 0\}$. The parity is a homotopy invariant with values in ${ \mathbb{Z}}_{2}$. In \cite{FitzPej,Fitzpatrick3,Fitzpatrick:1992vv,Fitzpatrick:1999ur} an alternative characterization of parity is given via the Leray-Schauder degree. For any Fredholm path $\eta \mapsto \Lambda(\eta)$ there exists a path $\eta \mapsto M(\eta)$, called a \emph{parametrix}, such that $\eta \mapsto M(\eta)\Lambda(\eta)$ is of the form `identity + compact'. For parity this gives: $$ \parity(\Lambda(\eta),I) = \deg_{LS}\bigl(M(0)\Lambda(0)\bigr) \cdot \deg_{LS}\bigl(M(1)\Lambda(1)\bigr), $$ where $ \deg_{LS}\bigl(M(\eta)\Lambda(\eta)\bigr) = \deg_{LS}\bigl(M(\eta)\Lambda(\eta),\mathscr{H},0\bigr)$, for $\eta = 0,1$, and the expression is independent of the choice of parametrix. The latter extends the above definition to arbitrary paths with invertible endpoints. For a list of properties of parity see \cite{FitzPej,Fitzpatrick3,Fitzpatrick:1992vv,Fitzpatrick:1999ur}.
\begin{proposition}\label{prop:morsespectral} Let $\eta\mapsto A(\eta)$ be the path of bounded linear Fredholm operators on $H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$ defined by (\ref{choice1}). Then \begin{equation}\label{morsespectral-par} \parity(A(\eta),I) = (-1)^{\beta_{A(0)}} \cdot (-1)^{\beta_{A(1)}} = (-1)^{\beta_{A(0)}-\beta_{A(1)}}.
\end{equation} where $\beta_{A(0)}$ and $\beta_{A(1)}$ are the Morse indices of $A(0)$ and $A(1)$ respectively. \end{proposition}
\begin{proof} For $\eta\mapsto A(\eta)$ the parametrix is the constant path $\eta\mapsto M(\eta) = \Id$. From Proposition \ref{beta} we derive that $$ \deg_{LS}\bigl(A(0)\bigr) = (-1)^{\beta_{A(0)}},\quad{\rm and}\quad \deg_{LS}\bigl(A(1)\bigr) = (-1)^{\beta_{A(1)}}, $$ which proves the first part of the formula. Since $\beta(A(0)) - \beta(A(1)) = \bigl[ \beta(A(0)) + \beta(A(1)) \bigr] \mod 2$, the second identity follows.
\end{proof}
\begin{lemma} \label{eigenA0} For $\theta>0$, the Morse index for $A(0)$ is given by $\beta_{A(0)} = 2\left\lceil\frac{\mu+\theta}{2\pi}\right\rceil$. \end{lemma}
\begin{proof}
The eigenvalues of the operator $A(0)$ are given by $ \lambda=\frac{-\theta+2k\pi}{\mu+2k\pi} $ and all have multiplicity 2. Therefore number of integers $k$ for which $\lambda<0$ is equal to $ \left\lceil\frac{\mu+\theta}{2\pi}\right\rceil$ and consequently $\beta_{A(0)}=2\left\lceil\frac{\mu+\theta}{2\pi}\right\rceil$.
\end{proof}
If $x\in \Phi_{\mu,H}^{-1}(0)$ is a non-degenerate zero, then its local degree can be expressed in terms of the parity of $A(\eta)$.
\begin{proposition} \label{LSspecflow1} Let $x\in \Phi_{\mu,H}^{-1}(0)$ be a non-degenerate zero, then \begin{equation} \label{LSspecflow2} \deg_{LS}\bigl(\Phi_{\mu,H},B_\epsilon(x),0\bigr) = \parity(A(\eta), I), \end{equation} where $\eta\mapsto A(\eta)$ is given by (\ref{choice1}). \end{proposition}
\begin{proof} From Proposition \ref{beta} we have that $\deg_{LS}\bigl(\Phi_{\mu,H},B_\epsilon(x),0\bigr) = (-1)^{\beta_{A(1)}}$ and by Equation (\ref{morsespectral-par}),
$ \parity(A(\eta),I) = (-1)^{\beta_{A(0)}}\cdot (-1)^{\beta_{A(1)}} = (-1)^{\beta_{A(1)}},
$
which completes the proof. \end{proof}
\subsection{Parity and spectral flow} \label{subsec:parity2} The spectral flow is a more refined invariant for paths of selfadjoint operators. For $x\in H^s({ \mathbb{R}}/{ \mathbb{Z}})$ we use the Fourier expansion
$x = \sum_{k\in { \mathbb{Z}}} e^{2\pi Jkt}x_{k}$ and $\sum_{k\in { \mathbb{Z}}} |k|^{2s} |x_k|^2 < \infty$. From the functional calculus of the selfadjoint operator $$ -J \frac{d}{dt} x = \sum_{k\in { \mathbb{Z}}} \bigl(2\pi k \bigr)e^{2\pi Jkt}x_{k}, $$ we define the selfadjoint operators \begin{equation} \label{eqn:extra-self}
N_\mu x = \sum_{k\in { \mathbb{Z}}} \bigl(2\pi |k| + \mu\bigr) e^{2\pi Jkt}x_{k}, \quad\hbox{and}\quad P_{\mu} x = \sum_{k\in { \mathbb{Z}}} \frac{2\pi k +\mu}{2\pi |k| + \mu} e^{2\pi Jkt}x_{k}. \end{equation} For $\mu>0$ and $\mu \not = 2\pi k$, $k\in { \mathbb{Z}}$, the operator $P_{\mu}$ is an isomorphism on $H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$, for all $s\ge 0$.\footnote{As before $\Vert P_{\mu} x \Vert_{H^{s}} \le \Vert x\Vert_{H^{s}}$ and $\Vert P_{\mu}^{-1} x \Vert_{H^{1/2}} \le C(\mu) \Vert x\Vert_{H^{1/2}}$, $\mu>0$ and $\mu\not = 2\pi k$.} Consider the path \begin{equation} \label{eqn:adjusted} C(\eta) = P_{\mu}A(\eta) = P_{\mu} - N_\mu^{-1}(S(t;\eta)+\mu), \end{equation} which is a path of operators of Fredholm index 0. The constant path $\eta \mapsto M_\mu(\eta) = P_{\mu}^{-1}$ is a parametrix for $\eta\mapsto C(\eta)$ (see \cite{Fitzpatrick:1992vv,Fitzpatrick:1999ur}) and since $M_\mu C(\eta) = A(\eta)$, the parity of $C(\eta)$ is given by \begin{equation} \label{eqn:parity2} \parity(C(\eta),I) = \parity(A(\eta),I). \end{equation}
Using $N_\mu$, with $\mu>0$ and $\mu \not = 2\pi k$, we define an equivalent norm on the Sobolev spaces $H^{s}({ \mathbb{R}}/{ \mathbb{Z}})$: $$ (x,y)_{H^{s}} := \bigl( N_\mu^s x, N_\mu^{s}y\bigr)_{L^{2}},\quad \forall x,y\in H^{s}({ \mathbb{R}}/{ \mathbb{Z}}). $$ \begin{lemma} \label{lem:selfadjoint} The operators $C(\eta)$ are selfadjoint on $\Bigl(H^{1/2}({ \mathbb{R}}/{ \mathbb{Z}}),(\cdot,\cdot)_{H^{1/2}}\Bigr)$ for all $\eta \in I$, and $\eta \mapsto C(\eta)$ is a path of selfadjoint operators on $H^{1/2}({ \mathbb{R}}/{ \mathbb{Z}})$. \end{lemma}
\begin{proof} From the functional calculus we derive that $$ (P_{\mu}x,y)_{H^{s}} = \sum_{k\in { \mathbb{Z}}} p_\mu(k) n^{2s}_\mu(k) x_k y_k = (x,P_\mu y)_{H^{s}}, $$
where $n_\mu(k) = 2\pi |k| +\mu$ and $p_\mu(k) = \frac{2\pi k+\mu}{2\pi |k|+\mu}$.
For $s=1/2$ we have that \begin{align*} \bigl(N_\mu^{-1}(S(t;\eta)+\mu)x,y\bigr)_{H^{1/2}} &= \bigl((S(t;\eta)+\mu)x,y\bigr)_{L^{2}} = \bigl(x,(S(t;\eta)+\mu)y\bigr)_{L^{2}}\\
&= \bigl(x,N_\mu^{-1}(S(t;\eta)+\mu)y\bigr)_{H^{1/2}}, \end{align*} which completes the proof. \end{proof}
For a path $\eta \mapsto \Lambda(\eta)$ of \emph{selfadjoint} operators on a Hilbert space $\mathscr{H}$, which is continuously differentiable in the (strong) operator topology we define the crossing operator
${ {\Gamma}}(\Lambda,\eta) = \pi \frac{d}{d\eta} \Lambda(\eta) \pi|_{\ker \Lambda(\eta)}$, where $\pi$ is the orthogonal projection onto $\ker \Lambda(\eta)$. A crossing $\eta_{0}\in I$ is a number for which the operator $\Lambda(\eta_{0})$ is not invertible. A crossing is regular if ${ {\Gamma}}(\Lambda,\eta_{0})$ is non-singular. A point $\eta_0$ for which $\dim \ker \Lambda(\eta_0) =1$, is called a simple crossing. A path $\eta\mapsto \lambda(\eta)$ is called generic if all crossings are simple.
A path $\eta\mapsto \Lambda(\eta)$ with invertible endpoints can always be chosen to be generic by a small perturbation.
At a simple crossing $\eta_0$, there exists a $C^1$-curve $\lambda(\eta)$, for $\eta$ near $\eta_0$, and $\lambda(\eta)$ is an eigenvalue of $\Lambda(\eta)$, with $\lambda(\eta_0) =0$ and $\lambda'(\eta_0) \neq 0$, see \cite{RS1,robbinsalamonmaslovindex}. The spectral flow for a generic path is defined by \begin{equation}\label{specflowf} \specflow(\Lambda(\eta),I)=\sum_{ \lambda(\eta_0)=0}\sgn (\lambda'(\eta_0)). \end{equation}
For a simple crossing $\eta_0$ the crossing operator is simply multiplication by $\lambda'(\eta_0)$ and \begin{equation}\label{innerprodeigenvalues} { {\Gamma}}(\Lambda,\eta)\psi(\eta_0) = \Bigl(\frac{d}{d\eta} \Lambda(\eta_0)\psi(\eta_0),\psi(\eta_0)\Bigr)_{\mathscr{H}}\psi(\eta_0)=\lambda'(\eta_0)\psi(\eta_0), \end{equation} where $\psi(\eta_0)$ is normalized in $\mathscr{H}$, and \begin{equation} \label{eqn:eig2} \lambda'(\eta_0) = \Bigl( \frac{d}{d\eta} \Lambda(\eta_0) \psi(\eta_0)\psi(\eta_0)\Bigr)_\mathscr{H}. \end{equation}
The spectral flow is defined for any continuously differentiable path $\eta \mapsto \Lambda(\eta)$ with invertible endpoints.
From the theory in \cite{Fitzpatrick:1999ur} there is a connection between the spectral flow of $\Lambda(\eta)$ and its parity: \begin{equation} \label{eqn:par-spec} \parity(\Lambda(\eta),I) = (-1)^{\displaystyle{\specflow(\Lambda(\eta),I)}}, \end{equation} which in view of Equation (\ref{eqn:parity}) follows from the fact that $\cross(\Lambda(\eta),I) = \specflow(\Lambda(\eta),\eta) \mod 2$ in the generic case.
The path $\eta \mapsto C(\eta)$ defined in (\ref{eqn:adjusted}) is a continuously differentiable path of operators on $H = H^{1/2}({ \mathbb{R}}/{ \mathbb{Z}})$ with invertible endpoints, and therefore both parity and spectral flow are well-defined. If we combine Equations (\ref{LSspecflow2}) and (\ref{eqn:parity2}) with Equation (\ref{eqn:par-spec}) we obtain \begin{equation} \label{eqn:connect} \deg_{LS}(\Phi_{\mu,H},B_{\epsilon}(x),0) = \parity(A(\eta),I) = (-1)^{\displaystyle{\specflow(C(\eta),I)}}. \end{equation} In the next section we link the spectral flow of $C(\eta)$ to the Conley-Zehnder indices of non-degenerate zeroes and therefore to the Euler-Floer characteristic.
\section{The Conley-Zehnder index} \label{sec:CZ} We discuss the Conley-Zehnder index for Hamiltonian systems and mechanical systems, and explain the relation with the local degree and the Morse index for mechanical systems.
\subsection{Hamiltonian systems} \label{subsec:CZ1} For a non-degenerate 1-periodic solution $x(t)$ of the Hamilton equations the Conley-Zehnder index can be defined as follows. The linearized flow $\Psi$ is given by $$ \left\{ \begin{array}{ll} \displaystyle-J\frac{d\Psi}{dt}-D^2_xH(x,t)\Psi=0\\ \Psi(0)=\Id, \end{array} \right. $$ By Lemma \ref{nondeg}(iii), a 1-periodic solution is non-degenerate if $\Psi(1)$ has no eigenvalues equal to 1. The Conley-Zehnder index is defined using the symplectic path $\Psi(t)$. Following \cite{robbinsalamonmaslovindex}, consider the crossing form ${ {\Gamma}}(\Psi,t)$, defined for vectors $\xi \in \ker(\Psi(t)-\Id)$,
\begin{equation} \label{czsegn} { {\Gamma}}(\Psi,t)\xi = \omega\bigl(\xi,\frac{d}{dt}\Psi(t)\xi\bigr) = (\xi,D^2_xH(x(t),t)\xi).
\end{equation}
A crossing $t_0>0$ is defined by $\det(\Psi(t_0)-\Id) =0$. A crossing is regular if the crossing form is non-singular. A path $t\mapsto \Psi(t)$ is regular if all crossings are regular. Any path can be approximated by a regular path with the same endpoints and which is homotopic to the initial path, see \cite{RS1} for details. For a regular path $t\mapsto \Psi(t)$ the Conley-Zehnder index is given by \begin{equation}\label{maslovsegn} \mu^{CZ}(\Psi)= \frac{1}{2}\sgn D^2_xH(x(0),0)) + \sum_{t_0>0,\atop \det(\Psi(t_0)-\Id)=0}\sgn { {\Gamma}}(\Psi,t_0).
\end{equation} For a non-degenerate 1-periodic solution $x(t)$ we define the Conley-Zehnder index as
$ \mu^{CZ}(x) := \mu^{CZ}(\Psi), $ and the index is integer valued.
Let $x$ be a 1-periodic solution and consider the path $\eta \mapsto B(\eta;x) = -J\frac{d}{dt}-S(t;\eta)$, where, as before, $S(t;\eta)$ is a smooth path of symmetric matrices with endpoints $S(t;0)=\theta \Id$ and $S(t;1)=D^{2}_xH(x(t),t)$ with $\theta\not=2\pi k, k\in \mathbb{Z}$. The operators $B(\eta)= B(\eta;x)$ are unbounded operators on $L^2({ \mathbb{R}}/{ \mathbb{Z}})$, with domain $H^1({ \mathbb{R}}/{ \mathbb{Z}})$. A path $\eta\mapsto B(\eta)$ is continuously differentiable in the (weak) operator topology of ${\mathcal{B}}(H^1,L^2)$
and Hypotheses (A1)-(A3) in \cite{robbinsalamonmaslovindex} are satisfied. We now repeat the definition of spectral flow for a path of unbounded operators as developed in \cite{robbinsalamonmaslovindex}. The crossing operator for a path $\eta\mapsto B(\eta)$ is given by ${ {\Gamma}}(B,\eta) = \pi \frac{d}{d\eta}B(\eta) \pi|_{\ker B(\eta)}$, where $\pi$ is the orthogonal projection onto $\ker B(\eta)$.
A crossing $\eta_{0}\in I$ is a number for which the operator $B(\eta_{0})$ is not invertible. A crossing is regular if ${ {\Gamma}}(B,\eta_{0})$ is non-singular. A point $\eta_0$ for which $\dim \ker B(\eta_0) =1$, is called a simple crossing. A path $\eta\mapsto B(\eta)$ is called generic if all crossing are simple.
A path $\eta\mapsto B(\eta)$ can always be chosen to be generic.
At a simple crossing $\eta_0$ there exists a $C^1$-curve $\ell(\eta)$, for $\eta$ near $\eta_0$, and $\ell(\eta)$ is an eigenvalue of $B(\eta)$ with $\ell(\eta_0) =0$ and $\ell'(\eta_0) \neq 0$. The spectral flow for a generic path is defined by \begin{equation}\label{specflowfB} \specflow(B(\eta),I)=\sum_{\ell(\eta_0)=0}\sgn (\ell'(\eta_0)), \end{equation} and at simple crossings $\eta_0$, \begin{equation} \label{innerprodeigenvalues} { {\Gamma}}(B,\eta)\phi(\eta_0) = \Bigl(\frac{d}{d\eta}B(\eta_0)\phi(\eta_0),\phi(\eta_0)\Bigr)_{L^2}\phi(\eta_0)=\ell'(\eta_0)\phi(\eta_0), \end{equation} after normalizing $\phi(\eta_0)$ in $L^2({ \mathbb{R}}/{ \mathbb{Z}})$. As before the derivative of $\ell$ at $\eta_0$ is given by
\begin{equation}\label{czinnerprod} \ell'(\eta_0) =- \bigl(\partial_{\eta}S(t;\eta_0)\phi(\eta_0),\phi(\eta_0)\bigr)_{L^2}. \end{equation}
\begin{proposition} \label{second} Let $\eta\mapsto B(\eta),{\eta\in I}$, as defined above, be a generic path of unbounded self-adjoint operators with invertible endpoints, and let $\eta\mapsto\Psi(\eta;t)$ be the associated path of symplectic matrices defined by $$ \left\{ \begin{array}{ll} \displaystyle-J\frac{d\Psi}{dt}(t;\eta)-S(t;\eta)\Psi(t;\eta)=0\\ \Psi(0;\eta)=\Id, \end{array} \right. $$
Then \begin{equation} \label{czspectral} \specflow(B(\eta),I)= \mu^{CZ}_{B(0)}-\mu^{CZ}_{B(1)} \end{equation} where $\mu^{CZ}_{B(0)}=\mu^{CZ}(\Psi(t;0))$, $\mu^{CZ}_{B(1)}=\mu^{CZ}(\Psi(t;1))$. \end{proposition} \begin{proof} The expression for the spectral flow follows from \cite{robbinsalamonmaslovindex} and \cite{GVVW}. \end{proof}
In the case $\eta =0$, the Conley-Zehnder index $\mu^{CZ}_{B(0)}$ can be computed explicitly. Recall that $B(0)=-J\frac{d}{dt}-S(0)=-J\frac{d}{dt}-\theta\Id$. \begin{lemma} \label{CZB0} Let $\theta>0$ (fixed) and $\theta\not=2\pi k$, then $\mu^{CZ}_{B(0)}=1+ 2{\left\lfloor \frac{\theta}{2 \pi}\right\rfloor}$. \end{lemma}
\begin{proof} The solution to $B(0)\Psi(t) = 0$ is given by $\Psi(t) = e^{\theta J t}$ and $\det(\Psi(1)- \Id) =0$
exactly when $t=t_0=\frac{2\pi k}{\theta}$. By (\ref{czsegn}) and (\ref{maslovsegn}) we have that ${ {\Gamma}}(\Psi,t) \xi = \theta |\xi|^2$ and therefore
$\mu^{CZ}_{B(0)}=1+ 2{\left\lfloor \frac{\theta}{2 \pi}\right\rfloor}$,
which proves the lemma. \end{proof}
The zeroes $x\in \Phi_{\mu,H}^{-1}(0)$ in $\Omega = [x]\rel y$ can estimated by Braid Floer homology ${ \mathrm{HB}}_*([x]\rel y)$ of $\Omega = [x]\rel y$.
The \emph{Euler-Floer} characteristic of ${ \mathrm{HB}}_*([x]\rel y)$ is defined as \begin{equation} \label{EulerFloer} \chi\bigl( { \mathrm{HB}}_*([x]\rel y)\bigr) := \sum_{k\in { \mathbb{Z}}} (-1)^{k}{\dim { \mathrm{HB}}_k([x]\rel y)}. \end{equation} In \cite{GVVW} the following analogue of the Poincar\'e-Hopf formula is proved. \begin{proposition} \label{PHF} For a proper braid class $[x]\rel y$ and a generic Hamiltonian $H\in \mathcal{H}_{\parallel}^{{\rm reg}}(y)$, it holds that $$ \chi\bigl( { \mathrm{HB}}_*([x]\rel y)\bigr)= \sum_{x\in\Phi_{\mu,H}^{-1}(0)} (-1)^{\mu^{CZ}(x)}. $$ \end{proposition}
It remains to show that $\chi\bigl( { \mathrm{HB}}_*([x]\rel y)\bigr)$ and $\deg_{LS}(\Phi_{\mu,H},\Omega,0)$ are related.
\begin{proposition} \label{CZspecfl} For a proper braid class $[x]\rel y$ and a generic Hamiltonian $H \in \mathcal{H}_{\parallel}^{{\rm reg}}(y)$, we have that \begin{equation} \label{LSspecflow3} \chi\bigl( { \mathrm{HB}}_*([x]\rel y)\bigr) = -\sum_{x_i\in \Phi_{\mu,H}^{-1}(0)}(-1)^{-\displaystyle\specflow(B(\eta;x), I)}, \end{equation} where $\eta\mapsto B(\eta;x)$ is given above for $x\in \Phi_{\mu,H}^{-1}(0)$. \end{proposition}
\begin{proof}
By Proposition \ref{second} and Lemma \ref{CZB0} the spectral flow satisfies, \begin{align*} \mu^{CZ}(x) &= \mu^{CZ}_{B(1;x)} = \mu^{CZ}_{B(0)} - \specflow(B(\eta;x), I)\\ &= 1+ 2{\left\lfloor \tfrac{\theta}{2 \pi}\right\rfloor} - \specflow(B(\eta;x), I). \end{align*} This implies $$ (-1)^{\displaystyle\mu^{CZ}(x)} = -(-1)^{ - \displaystyle\specflow(B(\eta;x), I)}, $$
which completes the proof. \end{proof}
\subsection{Mechanical systems} \label{subsec:CZ2} A mechanical system is defined as the Euler-Lagrange equations of the Lagrangian density $L(q,t) = \frac{1}{2} q_t^2 - V(q,t)$. The linearization at a critical points $q(t)$ of the Lagrangian action is given by the unbounded opeartor $$ -\frac{d^2}{dt^2} - D^2_qV(q(t),t): H^2({ \mathbb{R}}/{ \mathbb{Z}}) \subset L^2({ \mathbb{R}}/{ \mathbb{Z}}) \to L^2({ \mathbb{R}}/{ \mathbb{Z}}). $$ Consider a path of unbounded self-adjoint operators on $L^2({ \mathbb{R}}/{ \mathbb{Z}})$ given by $\eta \mapsto D(\eta) = -\frac{d^2}{dt^2} - Q(t;\eta)$, with $Q(t;\eta)$ smooth. If $D(0)$ and $D(1)$ are invertible, then the spectral flow is well-defined.
\begin{proposition} \label{prop:morsespectral11} Assume that the endpoints of $\eta\mapsto D(\eta)$ are invertible. Then \begin{equation} \label{morsespectral} \specflow(D(\eta), I)=\beta_{D(0)}-\beta_{D(1)}, \end{equation} where $\beta_{D(0)}$ and $\beta_{D(1)}$ are the Morse indices of $D(0)$ and $D(1)$ respectively. \end{proposition}
\begin{proof} In \cite{robbinsalamonmaslovindex} the concatenation property of the spectral flow is proved. We use concatenation as follows. Let $c>0$ be a sufficiently large constant such that $D(0)+c\Id$ and $D(1)+c\Id$ are positive definite self-adjoint operators on $L^2({ \mathbb{R}}/{ \mathbb{Z}})$. Consider the paths $\eta\mapsto D_1(\eta) = D(0) + \eta c\Id$ and $\eta\mapsto D_2(\eta) = D(1) + (1-\eta) c\Id$. Their concatenation $D_1\#D_2$ is a path from $D(0)$ to $D(1)$ and $\eta \mapsto D_1\#D_2$ is homotopic to $\eta\mapsto D(\eta)$. Using the homotopy invariance and the concatenation property of the spectral flow we obtain $$ \specflow(D(\eta), I) = \specflow(D_1\#D_2, I) = \specflow(D_1,I) + \specflow(D_2,I). $$ Since $D(0)$ is invertible, the regular crossings of $D_1(\eta)$ are given by $\eta^1_i = -\frac{\lambda_i}{c}$, where $\lambda_i$ are negative eigenvalues of $D(0)$. By the positive definiteness of $D(0)+c\Id$, the negative eigenvalues of $D(0)$ satisfy $0> \lambda_i > -c$. For the crossing $\eta_i$ this implies $$ 0 < \eta_i = -\frac{\lambda_i}{c} < 1, $$ and therefore the number of crossings equals the number of negative eigenvalues of $D(0)$ counted with multiplicity. By the choice of $c$, we also have that $\frac{d}{d\eta} D_1(\eta) = c\Id$ is positive definite and therefore the signature of the crossing operator of $D_1(\eta)$ is exactly the number of negative eigenvalues of $D(0)$, i.e. $\specflow(D_1,I) =\beta_{D(0)}$. For $D_2(\eta)$ we obtain, $\specflow(D_2,I) = - \beta_{D(1)}$. This proves that $\specflow(D(\eta), I)=\beta_{D(0)}-\beta_{D(1)}$.
\end{proof}
For a mechanical system we have the Hamiltonian $H(x,t) = \frac{1}{2} p^2 + V(q,t)$. As such the Conley-Zenhder index of a critical point $q$ can be defined as the Conley-Zehnder index of $x = (q_t,q)$ using the mechanical Hamiltonian, see also \cite{Abbondandolo:2003fy} and \cite{Duistermaat:1976vq}.
\begin{lemma} \label{index} Let $q$ be a critical point of the mechanical Lagrangian action, then the associated Conley-Zehnder index $\mu^{CZ}(x)$ is well-defined, and
$\mu^{CZ}(x) =\beta(q)$, where $\beta(q)$ is the Morse index of $q$. \end{lemma}
\begin{proof} As before, consider the curves $\eta\mapsto B(\eta)$ and $\eta\mapsto D(\eta)$, $\eta\in I=[0,1]$ given by $$ B(\eta) = -J \frac{d}{dt} - \left(\begin{array}{cc}1 & 0 \\0 & Q(t;\eta)\end{array}\right),\quad D(\eta) = -\frac{d^2}{dt^2} - Q(t;\eta). $$ The crossing forms of the curves are the same --- ${ {\Gamma}}(B,\eta) = { {\Gamma}}(D,\eta)$ --- and therefore also the crossings $\eta_0$ are identical. Indeed, $B(\eta_0)$ is non-invertible if and only if $D(\eta_0)$ is non-invertible. Consequently, $\specflow\bigl(B(\eta),I\bigr) = \specflow\bigl(D(\eta),I\bigr)$ and the Propositions \ref{second} and \ref{prop:morsespectral11} then imply that \begin{equation} \label{funid2} \beta_{D(0)} - \beta_{D(1)} = \mu^{CZ}_{B(0)} - \mu^{CZ}_{B(1)}. \end{equation} Now choose $Q(t;\eta)$ such that $Q(t;0) = d^2 V(q(t),t) +c$ and $Q(t;1) = D^2_q V(q(t),t)$ and such that $\eta\mapsto B(\eta)$ and $\eta\mapsto D(\eta)$ are regular curves. If $c \ll 0$, then $\beta_{D(0)} = 0$. In order to compute $ \mu^{CZ}_{B(0)}$ we invoke the crossing from ${ {\Gamma}}(\Psi,t)$ for the associated symplectic path $\Psi(t)$ as explained in Section \ref{sec:CZ}. Crossings at $t_0\in (0,1]$ correspond to non-trivial solutions of the equation $D(0)\psi =0$ on $[0,t_0]$, with periodic boundary conditions. To be more precise, let $\Psi = (\phi,\psi)$, then $B(0)\Psi = 0$ is equivalent to $\psi_t = \phi$ and $-\phi_t - \bigl(D^2_qV(q(t),t) + c\bigr) \psi= 0$, which yields the equation $D(0)\psi =0$.
For the latter the kernel is trivial for any $t_0 \in (0,1]$. Indeed, if $\psi$ is a solution, then $\int_0^{t_0} |\psi_t|^2 = \int_0^{t_0} (D^2_q V(q,t) + c) \psi^2 <0$, which is a contradiction. Therefore, there are no crossing $t_0 \in (0,1]$. As for $t_0=0$ we have that $ \bigl(D^2_qV(q(0),0) + c\bigr) < 0$, which implies that $\sgn S(0;0) = 0$ and therefore $\mu^{CZ}_{B(0)} =0$, which proves the lemma. \end{proof}
\section{The spectral flows are the same} \label{sec:flowsame} In order to show that the spectral flows are the same we use the fact that the paths $\eta\mapsto C(\eta)$ and $\eta \mapsto B(\eta)$ for a non-degenerate zero $x_i\in \Phi_{\mu,H}^{-1}(0)$ are chosen to have only simple crossings for their crossing operators, i.e.
zero eigenvalues are simple.
In this case the spectral flows are determined by the signs of the derivatives of the eigenvalues at the crossings.
For $\eta \mapsto B(\eta)$ the expression given by Equation (\ref{czinnerprod}) and from Equation (\ref{eqn:eig2}) a similar expression for $\eta\mapsto C(\eta)$ can be derived and is given by \begin{equation} \label{morseinnerprod} \lambda'(\eta_0) = - \bigl(N_{\mu}^{-1}\partial_{\eta}S(t;\eta_0)\psi(\eta_0),\psi(\eta_0)\bigr)_{H^{1/2}}
= -\bigl(\partial_{\eta}S(t;\eta_0)\psi(\eta_0),\psi(\eta_0)\bigr)_{L^{2}} \end{equation}
\begin{lemma}\label{labeloperatorAB} The sets $\{\eta\in [0,1]: C(\eta)\psi(\eta)=0\}$ and $\{\eta\in [0,1]: B(\eta)\phi(\eta)=0\}$ are the same, and the operators $C(\eta)$ and $B(\eta)$ have the same eigenfunctions at crossings $\eta_0$. In particular, $\eta \mapsto B(\eta)$ is generic if and only if $\eta \mapsto C(\eta)$ is generic. \end{lemma}
\begin{proof} Given $\eta_0\in [0,1]$ such that $C(\eta_0)\psi(\eta_0)=0$, then $$ P_{\mu}\psi(\eta_0)-N_\mu^{-1}(S(\eta_0;t)+\mu)\psi(\eta_0)=0, $$ and thus
$\psi(\eta_0)-L_{\mu}^{-1}(S(\eta_0;t)+\mu)\psi(\eta_0)=0$, which is equivalent to the equation $\left(-J\frac{d}{dt}-S(t;\eta_0)\right)\psi(\eta_0)=0,$ i.e. $B(\eta_0)\psi(\eta_0)=0.$
\end{proof}
\begin{lemma} \label{sign1} For all $\mu >0$, with $\mu\not = 2\pi k$, $k\in { \mathbb{Z}}$, $\sgn \lambda'(\eta_0)=\sgn \ell'(\eta_0)$ for all crossings at $\eta_0$. \end{lemma}
\begin{proof} The eigenfunctions $\psi(\eta_{0})$ in Equation (\ref{morseinnerprod}) for $\lambda'(\eta_{0})$ are normalized in $H^{1/2}({ \mathbb{R}}/{ \mathbb{Z}})$ and therefore they relate to the eigenfunctions $\phi(\eta_{0})$ in Equation (\ref{czinnerprod}) for $\ell'(\eta_{0})$ as follows: $$ \psi(\eta_{0}) = \frac{\phi(\eta_{0})}{\Vert \phi(\eta_{0})\Vert_{H^{1/2}}}, \quad \Vert \phi(\eta_{0})\Vert_{L^{2}} =1. $$ Combining Equations (\ref{czinnerprod}) and (\ref{morseinnerprod}) then gives \begin{align*} \lambda'(\eta_{0}) &= -\bigl(\partial_{\eta}S(t;\eta_0)\psi(\eta_0),\psi(\eta_0)\bigr)_{L^{2}}\\ &= - \frac{1}{\Vert\phi(\eta_{0})\Vert^{2}_{H^{1/2}}} \bigl(\partial_{\eta}S(t;\eta_0)\phi(\eta_0),\phi(\eta_0)\bigr)_{L^{2}} = \frac{\ell'(\eta_{0})}{\Vert\phi(\eta_{0})\Vert^{2}_{H^{1/2}}}, \end{align*} which proves the lemma.
\end{proof}
Lemma \ref{sign1} implies that for any non-degenerate $x\in \Phi_{\mu,H}^{-1}(0)\cap \Omega$ \begin{equation} \label{eqn:specsame1} \specflow(C(\eta;x),I) = \specflow(B(\eta;x),I), \end{equation} where $B(\eta;x)$ and $C(\eta;x)$ are the above described path associated with $x$. Therefore \begin{equation} \label{eqn:link}
\parity(A(\eta;x),I) = (-1)^{\displaystyle{\specflow(C(\eta;x),I)}} =(-1)^{\displaystyle{\specflow(B(\eta;x),I)}}, \end{equation}
which yields the following proposition.
\begin{proposition} \label{eqspfl}
The Leray-Schauder degree satisfies $$ \deg_{LS}(\Phi_{\mu,H},\Omega,0) = -\chi\bigl({ \mathrm{HB}}_*([x]\rel y)\bigr). $$ \end{proposition}
\begin{proof} For any Hamiltonian $H\in \mathcal{H}_{\parallel}(y)$ there exists a generic Hamiltonian $\tilde H \in \mathcal{H}_{\parallel}^{{\rm reg}}(y)$ such all zeroes
$x_i\in \Phi_{\mu,\tilde H}^{-1}(0)\cap \Omega$ are non-degenerate.
Since $\Omega = [x]\rel y$ is isolating for all Hamiltonians in $\mathcal{H}_{\parallel}(y)$, the invariance if the Leray-Schauder degree
yields $\deg_{LS}\bigl(\Phi_{\mu,H},\Omega,0\bigr) = \deg_{LS}\bigl(\Phi_{\mu,\tilde H},\Omega,0\bigr)$.
From the Propositions \ref{LSspecflow1} and \ref{CZspecfl} and Equation (\ref{eqn:link}) we conclude that \begin{align*} \deg&_{LS}\bigl(\Phi_{\mu,H},\Omega,0\bigr) = \deg_{LS}\bigl(\Phi_{\mu,\tilde H},\Omega,0\bigr)\\ &= \sum_{x\in \Phi_{\mu,\tilde H}^{-1}(0) }\deg_{LS}\bigl(\Phi_{\mu,\tilde H},B_{\epsilon}(x),0\bigr) = \sum_{x\in \Phi_{\mu,\tilde H}^{-1}(0) } \parity(A(\eta;x), I)\\ &= \sum_{x\in \Phi_{\mu,\tilde H}^{-1}(0) } (-1)^{\displaystyle\specflow(B(\eta;x), I)} = \sum_{x\in \Phi_{\mu,\tilde H}^{-1}(0) } (-1)^{-\displaystyle\specflow(B(\eta;x), I)}\\ &= -\chi\bigl({ \mathrm{HB}}_*([x]\rel y)\bigr), \end{align*} which completes the proof. \end{proof}
\begin{remark}
{\em As $\mu\gg 1$ it holds that $\ell'(\eta_{0}) \sim \mu \lambda'(\eta_{0})$. Indeed, $\Vert \phi(\eta_{0})\Vert_{H^{1/2}}^{2} = \sum_{k} (2\pi |k| +\mu) a_{k}^{2}$, where $a_{k}$ are the Fourier coefficients of $\phi(\eta_{0})$ and $\sum_{k}a_{k}^{2} =1$. Since $\phi(\eta_0)$ are smooth functions the Fourier coefficients satisfy the following properties. For any $\delta>0$ and any $s>0$, there exists $N_{\delta,s}>0$ such that $
\sum_{|k|\geq N }| k|^{2s} |a_k |^2\leq \delta$, for all $ N\ge N_{\delta,s}$. From the latter it follows that $\sum_{k} 2\pi |k| a_{k}^{2} \le C$, with $C>0$ independent of $\eta_{0}$ and $\mu$. We derive that $\mu \le \Vert \phi(\eta_{0})\Vert_{H^{1/2}}^{2} \le C +\mu$ and $$ 1\leftarrow\frac{\mu}{C+\mu} \le \frac{\mu\lambda'(\eta_0)}{\ell'(\eta_0)} = \frac{\mu}{\Vert\phi(\eta_0)\Vert_{H^{1/2}}^2} \le \frac{\mu}{\mu} =1, $$
as $\mu \to \infty$, which proves our statement. } \end{remark}
\section{The proof of Theorems \ref{thm:PH1} and \ref{thm:exist}} \label{sec:proof} We start with
the proof of Theorem \ref{thm:exist}. Since ${ \mathrm{HB}}_*([x]\rel y)$ is an invariant of the proper braid class $[x\rel y]$ it does not depend on a particular fiber $[x]\rel y$. Therefore we denote the Euler-Floer characteristic by $\chi(x\rel y) := \chi\bigl({ \mathrm{HB}}_*([x]\rel y)\bigr)$.
Recall the homotopy invariance of the Leray-Schauder degree as expressed in Equation (\ref{homtop}) \begin{equation*} \deg_{LS}(\Phi_{\mu},\Omega,0)=\deg_{LS}(\Phi_{\mu,\alpha},\Omega,0)=\deg_{LS}(\Phi_{\mu,H},\Omega,0). \end{equation*} By Proposition \ref{eqspfl} we have that $$ \deg_{LS}(\Phi_{\mu},\Omega,0) = \deg_{LS}(\Phi_{\mu,H},\Omega,0) = - \chi(x\rel y), $$ and $ \chi(x\rel y) \not = 0$, then
implies that $\Phi_{\mu}^{-1}(0)\cap \Omega \neq \varnothing$. Therefore there exists a closed integral curves in any relative braid class fiber of $[x\rel y]$, whenever $ \chi(x\rel y) \not = 0$, and this completes the proof of Theorem \ref{thm:exist}.
The remainder of this section is to prove the Poincar\'e-Hopf Formula in Theorem \ref{thm:PH1} for closed integral curves in proper braid fibers. The mapping $$ {\mathscr{E}}: C^1({ \mathbb{R}}/{ \mathbb{Z}}) \to C^0({ \mathbb{R}}/{ \mathbb{Z}}),\quad {\mathscr{E}}(x) = \frac{dx}{dt} - X(x,t), $$ is smooth (nonlinear) Fredholm mapping of index $0$. Let $M \in {\rm GL}(C^0,C^1)$ be an isomorphism such that $M{\mathscr{E}}(x)$ is of the form $M{\mathscr{E}}(x) = \Phi_M(x) = x - K_M(x)$, with $K_M: C^1({ \mathbb{R}}/{ \mathbb{Z}}) \to C^1({ \mathbb{R}}/{ \mathbb{Z}})$ compact. Such isomorphisms $M$ (constant parametrices) obviously exist. For example $M= \Bigl( \frac{d}{dt} + 1\Bigr)^{-1}$, or $M = - J L_\mu^{-1}$. The mappings $\Phi_M: C^1({ \mathbb{R}}/{ \mathbb{Z}}) \to C^1({ \mathbb{R}}/{ \mathbb{Z}})$ are Fredholm mappings of index $0$.
Let $x\in C^1({ \mathbb{R}}/{ \mathbb{Z}})$ be a non-degenerate zero of ${\mathscr{E}}$ and recall the index $\iota(x)$:
$$ \iota(x) = - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \deg_{LS}\bigl(\Phi_M,B_\epsilon(x),0\bigr), $$ where $\Theta \in {\rm M}_{2\times 2}({ \mathbb{R}})$, with $\sigma(\Theta) \cap 2\pi k i{ \mathbb{R}}=\varnothing$, $k\in { \mathbb{Z}}$ and $\beta_M(\Theta)$ is the Morse index of $\Id - K_M(0)$.
\begin{lemma} \label{lem:index1} The index $\iota(x)$ for a non-degenerate zero of ${\mathscr{E}}$ is well-defined, i.e. independent of the choices of $M\in {\rm GL}(C^0,C^1)$
and $\Theta \in {\rm M}_{2\times 2}({ \mathbb{R}})$. \end{lemma}
\begin{proof}
Consider smooth paths $\eta \mapsto F_\Theta(\eta)$, defined by $F_\Theta(\eta) = \frac{d}{dt} - R(t;\eta)$, where $R(t;0) = \Theta$ and $R(t;1) = D_x X(x(t),t)$. The path $$ F_\Theta: [0,1] \to \fred_0(C^1,C^0) $$
has invertible end points, and by the theory in \cite{FitzPej,Fitzpatrick3} we have that the parity of $\eta\mapsto F_\Theta(\eta)$ is well-defined and independent of $M$, i.e. \begin{align*}
\parity(F_\Theta(\eta),I) &= \parity(D_{M,\Theta}(\eta),I) = (-1)^{\beta_M(\Theta)}(-1)^{\beta_M(x)}\\
&= (-1)^{\beta_M(\Theta)} \deg_{LS}\bigl(\Phi_M,B_\epsilon(x),0\bigr), \end{align*} where $D_{M,\Theta}(\eta) = MF_\Theta(\eta)$ and $ \beta_M(x)$ is the Morse index of $D_{M,\Theta}(1) = \Id - K_M(1)$. It remains to show that the index $\iota(x)$ is independent with respect to $\Theta$. Let $\Theta$ and $\Theta'$ be admissible matrices and let $\eta \mapsto G(\eta)$ be a path connecting $G(0) = \frac{d}{dt} - \Theta$ and $G(1) = \frac{d}{dt} - \Theta'$. For the parities it holds that $$
\parity(F_{\Theta}(\eta),I) = \parity(G(\eta),I)\cdot \parity(F_{\Theta'}(\eta),I).
$$ To compute $ \parity(G(\eta),I)$ we consider a special parametrix $M_\mu$, given by $M_\mu = \Bigl( \frac{d}{dt} + \mu\Bigr)^{-1}$, $\mu>0$. From the definition of parity we have that $$ \parity(G(\eta),I) = \parity(M_\mu G(\eta),I) = \deg_{LS}\bigl(M_\mu G(0)\bigr) \cdot \deg_{LS}\bigl(M_\mu G(1)\bigr). $$ We now compute the Leray-Schauder degrees of $M_\mu G(0)$ and $M_\mu G(1)$. We start with $\Theta$ and in order to compute the degree we determine the Morse index. Consider the eigenvalue problem $$ M_\mu G(0) \psi = \lambda \psi, \quad \lambda \in { \mathbb{R}}, $$ which is equivalent to $(1-\lambda) \frac{d\psi}{dt} = \bigl( \Theta + \lambda \mu\bigr) \psi$. Non-trivial solutions are given by $\psi(t) = \exp{\Bigl(\frac{ \Theta + \lambda \mu}{1-\lambda}t\Bigr)} \psi_0$, which yields the condition $\frac{\theta+\lambda \mu}{1-\lambda} = 2\pi k i$, $k\in { \mathbb{Z}}$, where $\theta$ is an eigenvalues of $\Theta$. We now consider three cases:
(i) $\theta_\pm = a \pm ib$. In case of a negative eigenvalue $\lambda$ we have $\frac{a+\lambda\mu}{1-\lambda} = 0$ and $\frac{b}{1-\lambda} = 2\pi k$. The same $\lambda<0$ also suffices for the conjugate eigenvalue via $\frac{-b}{1-\lambda} = -2\pi k$. This implies that any eigenvalue $\lambda<0$ has multiplicity 2, and thus $\deg_{LS}\bigl(M_\mu G(0)\bigr) =1$.
(ii) $\theta_\pm\in{ \mathbb{R}}$, $\theta_-\cdot \theta_+>0$. In case of a negative eigenvalue $\lambda$ we have
$\frac{\theta_\pm+\lambda\mu}{1-\lambda} = 0$ and thus $\lambda_\pm = -\frac{\theta_\pm}{\mu}$, which yields two negative or two positive eigenvalues. As before $\deg_{LS}\bigl(M_\mu G(0)\bigr) =1$.
(iii) $\theta_\pm\in{ \mathbb{R}}$, $\theta_-\cdot \theta_+<0$. From case (ii) we easily derive that there exist two eigenvalues $\lambda_\pm$, one positive and one negative, and therefore $\deg_{LS}\bigl(M_\mu G(0)\bigr) = -1$.
These cases combined impliy that $\deg_{LS}\bigl(M_\mu G(0)\bigr) = \sgn(\det(\Theta))$ and $$ \parity(G(\eta),I) = \sgn(\det(\Theta)) \cdot \sgn(\det(\Theta')). $$ From the latter we derive: \begin{align*} \sgn(\det(\Theta)) &\cdot \parity(F_{\Theta}(\eta),I) \\ &=\sgn(\det(\Theta))\cdot \sgn(\det(\Theta))\cdot \sgn(\det(\Theta')) \cdot \parity(F_{\Theta'}(\eta),I) \\ &=\sgn(\det(\Theta')) \cdot \parity(F_{\Theta'}(\eta),I), \end{align*} which proves the independence of $\Theta$. \end{proof}
Lemmas \ref{lem:index1} shows that the index of a non-degenerate zero of ${\mathscr{E}}$ is well-defined. We now show that the same holds for isolated zeroes.
\begin{lemma} \label{lem:indep} The index $\iota(x)$ for an isolated zero of ${\mathscr{E}}$ is well-defined and for a fixed choice of $M$ and $\Theta$ the index is given by $$ \iota(x) = - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \deg_{LS}\bigl(\Phi_M,B_\epsilon(x),0\bigr), $$ where $\epsilon>0$ is small enough such that $x$ is the only zero of ${\mathscr{E}}$ in $B_\epsilon(x)$.
\end{lemma}
\begin{proof}
By the Sard-Smale Theorem one can choose an arbitrarily small $h\in C^{0}({ \mathbb{R}}/{ \mathbb{Z}})$, $\Vert h\Vert_{C^{0}} <\epsilon'$, such that $h$ is a regular value of ${\mathscr{E}}$ and ${\mathscr{E}}^{-1}(h)\cap B_{\epsilon}(x)$ consists of finitely many non-degenerate zeroes in $x_{h}$. Set $\widetilde {\mathscr{E}}(x) = {\mathscr{E}}(x) -h$ and define \begin{equation} \label{eqn:sum1} \iota(x) = \sum_{x_{h}\in \widetilde{\mathscr{E}}^{-1}(0)\cap B_{\epsilon}(x)} \iota(x_{h}).
\end{equation} We now show that $\iota(x)$ is well-defined. Choose a fixed parametrix $M$ (for ${\mathscr{E}}$) and fixed $\Theta \in {\rm M}_{2\times 2}({ \mathbb{R}})$, and let $\widetilde \Phi_M = M\widetilde {\mathscr{E}}$, then $$ \sum_{x_{h} } \iota(x_{h}) = - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \sum_{x_{h} }\deg_{LS}(\widetilde\Phi_{M},B_{\epsilon_h}(x_h),0), $$ where $B_{\epsilon_h}(x_h)$ are sufficiently small neighborhoods containing only one zero. From Leray-Schauder degree theory we derive that $$
\sum_{x_{h} }\deg_{LS}(\widetilde\Phi_{M},B_{\epsilon_h}(x_h),0) = \deg_{LS}(\widetilde\Phi_{M},B_{\epsilon }(x),0) = \deg_{LS}(\Phi_{M},B_{\epsilon }(x),0),
$$
which proves the lemma.
\end{proof}
Theorem \ref{thm:PH1} now follows from the Leray-Schauder degree. Suppose all zeroes of ${\mathscr{E}}$ in $\Omega = [x]\rel y$ are isolated, then Lemma \ref{lem:indep} implies that \begin{align*} \sum_{x\in {\mathscr{E}}^{-1}(0)\cap \Omega} \iota(x) &= - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \sum_{x} \deg_{LS}\bigl(\Phi_M,B_\epsilon(x),0\bigr)\\ &= - \sgn(\det(\Theta)) (-1)^{\beta_M(\Theta)} \deg_{LS}\bigl(\Phi_M,\Omega,0\bigr) \end{align*} Since the latter expression is independent of $M$ and $\Theta$ we choose $M = L_\mu^{-1}$ and $\Theta = \theta J$. Then, $\Phi_M = \Phi_\mu$, and for the indices we have $\sgn(\det(\theta J)) = 1$ and by Lemma \ref{eigenA0}, $(-1)^{\beta_{L_\mu^{-1}}(\theta J)} = 1$. By Proposition \ref{eqspfl}, $\deg_{LS}(\Phi_\mu,\Omega,0) = -\chi\bigl(x\rel y\bigr)$, which, by substitution of these choices into the index formula, yields $$ \sum_{x\in {\mathscr{E}}^{-1}(0)\cap \Omega} \iota(x) = \chi\bigl(x\rel y\bigr), $$ completing the proof Theorem \ref{thm:PH1}.
\section{Computing the Euler-Floer characteristic} \label{comp} In section we prove Theorem \ref{thm:discrete} and show that the Euler-Floer characteristic can be determined via a discrete topological invariant.
\subsection{Hyperbolic Hamiltonians on ${ \mathbb{R}}^2$} \label{subsec:hypham} Consider Hamiltonians of the form \begin{equation} \label{eqn:special1} H(x,t) = \frac{1}{2} p^2 - \frac{1}{2} q^2 + h(x,t), \end{equation} where $h$ satisfies the following hypotheses:
\begin{enumerate} \item [(h1)] $h\in C^\infty({ \mathbb{R}}^2\times { \mathbb{R}}/{ \mathbb{Z}})$; \item [(h2)] ${\rm supp}(h) \subset { \mathbb{R}}\times [-R,R]\times { \mathbb{R}}/{ \mathbb{Z}}$, for some $R>0$; \item [(h3)] $\Vert h\Vert_{C_b^2({ \mathbb{R}}^2\times { \mathbb{R}}/{ \mathbb{Z}})} \le c$. \end{enumerate}
\begin{lemma} \label{lem:special2} Let $H$ be given by (\ref{eqn:special1}), with $h$ satisfying (h1)-(h3). Then, there exists a constant $R'\ge R>0$, such any 1-periodic solution of $x$ of $x' = X_H(x,t)$ satisfies the estimate $$
|x(t)| \le R',\quad \hbox{for all}~~t\in { \mathbb{R}}/{ \mathbb{Z}}. $$ \end{lemma}
\begin{proof} The Hamilton equation in local coordinates are given by $$ p_t = q- h_q(p,q,t),\quad q_t = p + h_p(p,q,t). $$ Since $h$ is smooth we can rewrite the equations as \begin{equation} \label{eqn:special3} q_{tt} = h_{pq}(p,q,t) q_t + \bigl( 1+ h_{pp}(p,q,t)\bigr) \bigl(q-h_q(p,q,t)\bigr) + h_{pt}(p,q,t). \end{equation} If $x(t)$ is a 1-periodic solution to the Hamilton equations, and suppose there exists an interval $I= [t_0,t_1]\subset [0,1]$ such that
$|q(t)|>R$ on $\Int(I)$ and $|q(t)|\bigr|_{\partial I} = R$. The function $q|_{I}$ satisfies the equation $q_{tt} -q = 0$, and obviously such solutions do not exist. Indeed, if $q|_I\ge R$, then $q_t(t_0)\ge 0$ and $q_t(t_1) \le 0$
and thus $0\ge q_t|_{\partial I} = \int_I q \ge R|I|>0$, a contradiction. The same holds for $q|_I\le -R$. We conclude that $$
|q(t)| < R,\quad \hbox{for all}~~t\in { \mathbb{R}}/{ \mathbb{Z}}. $$ We now use the a priori $q$-estimate in combination with Equation (\ref{eqn:special3}) and Hypothesis (h3). Multiplying Equation (\ref{eqn:special3}) by $q$ and integrating over $[0,1]$ gives: \begin{align*} \int_0^1 q_t^2 &= - \int_0^1 h_{pq} q_t q - \int_0^1 \bigl( 1+ h_{pp}\bigr) \bigl(q-h_q\bigr)q - \int_0^1 h_{pt} q\\
&\le C\int_0^1 |q_t| + C \le \epsilon \int_0^1 q_t^2 + C_\epsilon, \end{align*} which implies that $\int_0^1 q_t^2 \le C(R)$. The $L^2$-norm of the right hand side in (\ref{eqn:special3}) can be estimated using the $L^\infty$ estimate on $q$ and the $L^2$-estimate on $q_t$, which yields $\int_0^1 q_{tt}^2 \le C(R)$. Combining these estimates we have
that $\Vert q \Vert_{H^2({ \mathbb{R}}/{ \mathbb{Z}})} \le C(R)$ and thus $|q_t(t)| \le C(R)$, for all $t\in { \mathbb{R}}/{ \mathbb{Z}}$.
From the Hamilton equations it follows that $|p(t)|\le |q_t(t)| + C$, which proves the lemma. \end{proof}
\begin{lemma} \label{lem:special4}
If $H(x,t;\alpha)$, $\alpha \in [0,1]$ is a (smooth) homotopy of Hamiltonians satisfying (h1)-(h3) with uniform constants $R>0$ and $c>0$, then $|x_\alpha(t)|\le R'$, for all 1-periodic solutions and for all $\alpha\in [0,1]$. \end{lemma}
\begin{proof} The a priori $H^2$-estimates in Lemma \ref{lem:special2} hold with uniform constants with respect to $\alpha \in [0,1]$. This then proves the lemma. \end{proof}
\subsection{Braids on ${ \mathbb{R}}^2$ and Legendrian braids} \label{subset:legen} In Section \ref{intro} we defined braid classes as path components of closed loops in ${\mathcal{L}}{ \mathbf{C}}_n({ \mathbb{D}^2})$, denoted by $[x]$. If we consider closed loops in ${ \mathbf{C}}_n({ \mathbb{R}}^2)$, then the braid classes will be denoted by $[x]_{{ \mathbb{R}}^2}$. The same notation applies to relative braid classes $[x\rel y]_{{ \mathbb{R}}^2}$. A relative braid class is proper if components $x_c \subset x$ cannot be deformed onto
(i) itself, or other components $x_c' \subset x$, or (ii) components $y_c \subset y$. A fiber $[x]_{{ \mathbb{R}}^2}\rel y$ is \emph{not} bounded!
In order to compute the Euler-Floer characteristic of $[x\rel y]$ we assume without loss of generality that $x\rel y$ is a positive representative. If not we compose $x\rel y$ with a sufficient number of positive full twists such that the resulting braid is positive, i.e. only positive crossings, see \cite{GVVW} for more details. The Euler-Floer characteristic remains unchanged. We denote a positive representative $x^+\rel y^+$ again by $x\rel y$.
Define an augmented skeleton $y^*$ by adding the constant strands $y_-(t) = (0,-1)$ and $y_+(t) = (0,1)$. For proper braid classes it holds that $[x\rel y] = [x\rel y^*]$. For notational simplicity we denote the augmented skeleton again by $y$. We also choose the representative $x\rel y$ with the additional the property that $\pi_2 x \rel \pi_2 y$ is a relative braid diagram, i.e. there are no tangencies between the strands, where $\pi_2$ the projection onto the $q$-coordinate. We denote the projection by $q\rel Q$, where $q=\pi_2 x$ and $Q=\pi_2 y$. Special braids on ${ \mathbb{R}}^2$ can be constructed from (smooth) positive braids. Define
$x_L = (q_t,q)$ and $y_L = (Q_t,Q)$, where the subscript $t$ denotes differentiating with respect to $t$. These are called \emph{Legendrian braids} with respect to $\theta = pdt - dq$.
\begin{lemma} \label{repres1} For positive braid $x \rel y$ with only transverse, positive crossings, the braids $x_L \rel y_L$ and $x\rel y$ are isotopic as braids on ${ \mathbb{R}}^2$. Moreover, if $x_L\rel y_L$ and $x'_L\rel y'_L$ are isotopic Legrendrian braids, then they are isotopic via a Legendrian isotopy. \end{lemma}
\begin{proof} By assumption
$x\rel y $ is a representative for which the braid diagram
$q\rel Q$ has only
positive transverse crossings.
Due to the transversality of intersections
the associated Legendrian braid $x_L\rel y_L$ is a braid $[x\rel y]_{{ \mathbb{R}}^2}$. Consider the homotopy $$ \zeta^j(t,\tau) = \tau p^j(t) + (1-\tau)q_t^j, $$ for every strand $q^j$. At $q$-intersections, i.e. times $t_0$ such that $q^j(t_0) = q^{j'}(t_0)$ for some $j\not =j'$, it holds that $p^j(t_0) - p^{j'}(t_0)$ and $q^j_t(t_0) - q_t^{j'}(t_0)$ are non-zero and have the same sign since all crossings in $x\rel y$ are positive! Therefore, $\zeta^j(t_0,\tau) \not = \zeta^{j'}(t_0,\tau)$ for any intersection $t_0$ and any $\tau\in [0,1]$, which shows that $x\rel y$ and $x_L\rel y_L$ are isotopic.
Since $x_L\rel y_L$ and $x'_L\rel y'_L$ have only positive crossings, a smooth Legendrian isotopy exists. \end{proof}
The associated equivalence class of Legendrian braid diagrams is denoted by $[q\rel Q]$ and its fibers by $[q]\rel Q$.
\subsection{Lagrangian systems} \label{subsec:lagr} Legendrian braids can be described with Lagrangian systems and Hamiltonians of the form $H_L(x,t) = \frac{1}{2} p^2 -\frac{1}{2} q^2 + g(q,t)$. On the potential functions $g$ we impose the following hypotheses: \begin{enumerate} \item [(g1)] $g \in C^\infty({ \mathbb{R}}\times { \mathbb{R}}/{ \mathbb{Z}})$; \item [(g2)] ${\rm supp}(g) \subset [-R,R]\times { \mathbb{R}}/{ \mathbb{Z}}$, for some $R>1$. \end{enumerate}
In order to have a straightforward construction of a mechanical Lagrangian we may consider a special representation of $y$. The Euler-Floer characteristic $\chi\bigl(x\rel y\bigr)$ does not depend on the choice of the fiber $[x]\rel y$ and therefore also not on the skeleton $y$. We assume that $y$ has linear crossings in $y_L$. Let $t=t_0$ be a crossing and let $I(t_0)$ be the set of labels defined by: $i,j\in I(t_0)$, if $i\not = j$ and $Q^i(t_0) = Q^j(t_0)$. A crossing at $t=t_0$ is \emph{linear} if $$ Q_t^i(t)= {\rm constant},\quad \forall i\in I(t_0),~{\rm and~}\quad\forall t\in (-\epsilon+t_0, \epsilon+t_0), $$ for some $\epsilon = \epsilon(t_0)>0$. Every skeleton $Q$ with transverse crossings is isotopic to a skeleton with linear crossings via a small
local deformation at crossings.
For Legendrian braids $x_L\rel y_L \in [x\rel y]_{{ \mathbb{R}}^2}$ with linear crossings the following result holds: \begin{lemma} \label{specialHam} Let $y_L$ be a Legendrian skeleton with linear crossings. Then, there exists a Hamiltonian of the form $H_L(x,t) = \frac{1}{2} p^2 -\frac{1}{2} q^2 + g(q,t)$, with
$g$ satisfying Hypotheses (g1)-(g2), and $R>0$ sufficiently large,
such that $y_L$ is a skeleton for $X_{H_L}(x,t)$. \end{lemma}
\begin{proof} Due to the linear crossings in $y_L$ we can follow the construction
in \cite{GVVW}.
For each strand $Q^i$ we define the potentials $g^i(t,x) = - Q_{tt}^i(t) q$.
By construction $Q^i$ is a solution of the equation $Q^i_{tt} = - g^i_q(t,Q^i)$.
Now choose small tubular neighborhoods of the strands $Q^i$ and cut-off functions $\omega^i$ that are equal to 1
near $Q^i$ and are supported in the tubular neighborhoods.
If the tubular neighborhoods are narrow enough, then ${\rm supp}(\omega^i g^i) \cap {\rm supp}(\omega^j g^j) = \varnothing$, for all $i\not = j$, due to the fact that at crossings the functions $g^i$ in question are zero. This implies that all strands $Q^i$ satisfy the differential equation $Q^i_{tt} = - \sum_i\omega^j(t) g^j_q(Q^i,t)$ and
on $[-1 ,1 ]\times { \mathbb{R}}/{ \mathbb{Z}}$, the function is $\sum_i\omega^i(t) g^i(q,t)$ is compactly supported. The latter follows from the fact that for the constant strands $Q^i=\pm 1$, the potentials $g^i$ vanish. Let $R>1$ and define $$ \tilde g^i(t,q) = \begin{cases}
g^i(t,q) & \text{for ~ } |q|\le 1, ~t\in { \mathbb{R}}/{ \mathbb{Z}},\\
-\frac{1}{2m} q^2 & \text{for~} |q|\ge R, ~t\in { \mathbb{R}}/{ \mathbb{Z}}. \end{cases} $$ where $m = \# Q$, which yields smooth functions $\tilde g^i$ on ${ \mathbb{R}}\times { \mathbb{R}}/{ \mathbb{Z}}$. Now define $$ g(q,t) = \frac{1}{2} q^2 + \sum_{i=1}^m \tilde g^i(q,t). $$ By construction ${\rm supp}(g) \subset [-R,R]\times { \mathbb{R}}/{ \mathbb{Z}}$, for some $R>1$ and the strands $Q^i$ all satisfy the Euler-Lagrange equations $Q^i_{tt} = Q^i - g_{qq}(Q^i,t)$, which completes the proof.
\end{proof}
The Hamiltonian $H_L$ given by Lemma \ref{specialHam} gives rise to a Lagrangian system with the Lagrangian action given by \begin{equation} \label{eqn:lagr} {\mathscr{L}}_g = \int_0^1 \frac{1}{2} q_t^2 +\frac{1}{2} q^2 -g(q,t) dt. \end{equation} The braid class $[q]\rel Q$ is bounded due to the special strands $\pm 1$ and all free strands $q$ satisfy $-1 \le q(t) \le 1$. Therefore, the set of critical points of ${\mathscr{L}}_g$ in $[q]\rel Q$ is a compact set. The critical points of ${\mathscr{L}}_g$ in $[q]\rel Q$ are in one-to-one correspondence with the zeroes of the equation $$ \Phi_{\mu,H_L}(x) =x-L_{\mu}^{-1}\bigl(\nabla H_L(x,t)+\mu x\bigr) =0, $$ in the set $\Omega_{{ \mathbb{R}}^2} = [x_L]_{{ \mathbb{R}}^2} \rel y_L$, which implies that $\Phi_{\mu,H_L}$ is a proper mapping on $\Omega_{{ \mathbb{R}}^2}$. From Lemma \ref{lem:special2} we derive that the zeroes of $\Phi_{\mu,H_L}$ are contained in ball in ${ \mathbb{R}}^2$ with radius $R'>1$, and thus $\Phi_{\mu,H_L}^{-1}(0) \cap \Omega_{{ \mathbb{R}}^2} \subset B_{R'}(0)\subset C^1({ \mathbb{R}}/{ \mathbb{Z}})$. Therefore the Leray-Schauder degree is well-defined and in the generic case Lemma \ref{index} and Equations (\ref{eqn:connect}), (\ref{czspectral}) and (\ref{eqn:specsame1}) yield \begin{equation} \label{eqn:finaldegree1} \deg_{LS}(\Phi_{\mu,H_L},\Omega_{{ \mathbb{R}}^2},0) = - \sum_{x\in \Phi_{\mu,H_L}^{-1}(0) \cap \Omega_{{ \mathbb{R}}^2}} (-1)^{\mu^{CZ}(x)} = - \sum_{q \in \Crit({\mathscr{L}}_g) \cap ([q]\rel Q)} (-1)^{\beta(q)}. \end{equation}
We are now in a position to use a homotopy argument. We can scale $y$ to a braid $\rho y$ such that the rescaled Legendrian braid $\rho y_L$ is supported in ${ \mathbb{D}^2}$. By Lemma \ref{repres1}, $y$ is isotopic to $y_L$ and scaling defines an isotopy
between $y_L$ and $\rho y_L$. Denote the isotopy from $y$ to $\rho y_L$ by $y_\alpha$. By Proposition \ref{eqspfl} we obtain that for both skeletons $y$ and $\rho y_L$ it holds that $$ \deg_{LS}(\Phi_{\mu,H},\Omega,0) = -\chi\bigl(x\rel y\bigr) = \deg_{LS}(\Phi_{\mu,H_\rho},\Omega_\rho,0), $$ where $\Omega_\rho = [\rho x_L]\rel \rho y_L \subset [x\rel y]$ and $H_\rho \in \mathcal{H}_{\parallel}(\rho y_L)$. Now extend $H_\rho$ to ${ \mathbb{R}}^2\times { \mathbb{R}}/{ \mathbb{Z}}$, such that Hypotheses (h1)-(h3) are satisfied for some $R>1$. We denote the Hamiltonian again by $H_\rho$. By construction all zeroes of $\Phi_{\mu,H_\rho}$ in $[\rho x_L]\rel \rho y_L$ are supported in ${ \mathbb{D}^2}$ and therefore the zeroes of $\Phi_{\mu,H_\rho}$ in $[\rho x_L]_{{ \mathbb{R}}^2}\rel \rho y_L$ are also supported in ${ \mathbb{D}^2}$. Indeed, any zero intersects ${ \mathbb{D}^2}$, since the braid class is proper and since $\partial { \mathbb{D}^2}$ is invariant for the Hamiltonian vector field, a zero is either inside or outside ${ \mathbb{D}^2}$. Combining these facts implies that a zero lies inside ${ \mathbb{D}^2}$. This yields $$
\deg_{LS}(\Phi_{\mu,H_\rho},\Omega_{\rho,{ \mathbb{R}}^2},0) = \deg_{LS}(\Phi_{\mu,H_\rho},\Omega_\rho,0) = -\chi\bigl(x\rel y\bigr) , $$ where $\Omega_{\rho,{ \mathbb{R}}^2} = [\rho x_L]_{{ \mathbb{R}}^2} \rel \rho y_L$.
For the next homotopy we keep the skeleton $\rho y_L$ fixed as well as the domain $\Omega_{\rho,{ \mathbb{R}}^2}$. Consider the linear homotopy of Hamiltonians $$ H_1(x,t;\alpha) = \frac{1}{2} p^2 - \frac{1}{2} q^2 + (1-\alpha) h_\rho(x,t) + \alpha g_\rho(q,t), $$ where $H_{\rho,L}(t,x) = \frac{1}{2} p^2 - \frac{1}{2} q^2 + g_\rho(q,t)$ given by Lemma \ref{specialHam}. This defines an admissible homotopy since $\rho y_L$ is a skeleton for all $\alpha\in [0,1]$. The uniform estimates are obtained, as before, by Lemma \ref{lem:special4}, which allows application of the Leray-Schauder degree: $$ \deg_{LS}(\Phi_{\mu, H_{\rho,L}},\Omega_{\rho,{ \mathbb{R}}^2},0) = \deg_{LS}(\Phi_{\mu, H_\rho},\Omega_{\rho,{ \mathbb{R}}^2},0) = -\chi\bigl(x\rel y\bigr). $$ Finally, we scale $\rho y_L$ to $y_L$ via $y_{\alpha,L} = (1-\alpha) \rho y_L + \alpha y_L$ and we consider the
homotopy $$ H_2(x,t;\alpha) = \frac{1}{2} p^2 - \frac{1}{2} q^2 + g(q,t;\alpha), $$ between $H_L$ and $H_{\rho,L}$, where $g(q,t;\alpha)$ is found by applying Lemma \ref{specialHam} to $y_{\alpha,L}$. The uniform estimates from Lemma \ref{lem:special4} allows us to apply
the Leray-Schauder degree: $$ \deg_{LS}(\Phi_{\mu,H_L},\Omega_{{ \mathbb{R}}^2},0) = \deg_{LS}(\Phi_{ \mu,H_{\rho,L}},\Omega_{\rho,{ \mathbb{R}}^2},0) = -\chi\bigl(x\rel y\bigr). $$ Combining the equalities for the various Leray-Schauder degrees with (\ref{eqn:finaldegree1}) yields: \begin{equation} \label{eqn:finaldegree2} - \deg_{LS}(\Phi_{H_L},\Omega_{{ \mathbb{R}}^2},0) = \chi\bigl(x\rel y\bigr) = \sum_{q \in \Crit({\mathscr{L}}_g) \cap ([q]\rel Q)} (-1)^{\beta(q)}. \end{equation}
\subsection{Discretized braid classes} \label{subsec:discbraid} The Lagrangian problem (\ref{eqn:lagr}) can be treated by using a variation on the method of broken geodesics.
If we choose $1/d>0$ sufficiently small, the integral
\begin{equation}
S_i(q_{i},q_{i+1}) = \min_{q(t)\in E_i(q_{i},q_{i+1})\atop |q(t)|\le 1} \int_{\tau_{i}}^{\tau_{i+1}} \frac{1}{2} q_t^2 + \frac{1}{2} q^2- g(q,t) dt, \end{equation}
has a unique minimizer $q^{i}$, where $E_i(q_{i},q_{i+1}) = \bigl\{ q\in H^1(\tau_{i},\tau_{i+1})~|~ q(\tau_{i})=q_{i},~q(\tau_{i+1}) = q_{i+1}\bigr\}$, and $\tau_{i} = i/d$. Moreover, if $1/d$ is small, then the minimizers are non-degenerate and $S_i$ is a smooth function of $q_{i}$ and $q_{i+1}$. Critical points $q$ of ${\mathscr{L}}_g$ with $|q(t)|\le 1$ correspond to sequences $q_D = (q_0,\cdots,q_d)$, with $q_0 = q_d$, which are critical points of the discrete action \begin{equation} {\mathscr{W}}(q_D) = \sum_{i=0}^{d-1} S_i(q_i,q_{i+1}). \end{equation} A concatenation $\#_{i} q^{i}$ of minimizers $q^{i}$ is continuous and is an element in the function space
$H^{1}({ \mathbb{R}}/{ \mathbb{Z}})$, and is referred to as a \emph{broken geodesic}. The set of broken geodesics $\#_{i}q^{i}$ is denoted by $E(q_{D})$ and standard arguments using the non-degeneracy of minimizers $q^{i}$ show that $E(q_{D}) \hookrightarrow H^{1}({ \mathbb{R}}/{ \mathbb{Z}})$ is a smooth, $d$-dimensional submanifold in $H^{1}({ \mathbb{R}}/{ \mathbb{Z}})$. The submanifold $E(q_{D})$ is parametrized by sequences $D_{d}=\{q_{D}\in { \mathbb{R}}^{d}~|~|q_{i}|\le 1 \}$ and yields the following commuting diagram: $$ \begin{diagram} \node{E(q_{D})}\arrow{e,l}{{\mathscr{L}}_g}\node{{ \mathbb{R}}}\\ \node{D_{n}}\arrow{n,l}{\#_{i}}\arrow{ne,r}{{\mathscr{W}}} \end{diagram} $$ In the above diagram $\#_{i}$ is regarded as a mapping $q_{D}\mapsto \#_{i}q^{i}$, where the minimizers $q_{i}$ are determined by $q_{D}$. The tangent space to $E(q_{D})$ at a broken geodesic $\#_{i}q^{i}$ is identified by \begin{align*}
T_{\#_{i}q^{i}}E(q_{D}) = \bigl\{ &\psi\in H^{1}({ \mathbb{R}}/{ \mathbb{Z}})~|~-\psi_{tt} + \psi-g_{qq}(q^{i}(t),t)\psi =0,\\ &\psi(\tau_{i}) = \delta q_{i},~~\psi(\tau_{i+1}) = \delta q_{i+1},~~\delta q_{i}\in { \mathbb{R}}, \forall i\bigr\}, \end{align*} and $\#_{i}q^{i} + T_{\#_{i}q^{i}}E(q_{D})$ is the tangent hyperplane at $\#_{i}q^{i}$. For $H^{1}({ \mathbb{R}}/{ \mathbb{Z}})$ we have the following decomposition for any broken geodesic $\#_{i}q^{i}\in E(q_{D})$: \begin{equation} \label{eqn:decomp} H^{1}({ \mathbb{R}}/{ \mathbb{Z}}) = E' \oplus T_{\#_{i}q^{i}}E(q_{D}), \end{equation}
where $E' = \{\eta \in H^{1}({ \mathbb{R}}/{ \mathbb{Z}})~|~\eta(\tau_{i}) = 0,~~\forall i\}$. To be more specific the decomposition is orthogonal with respect to the quadratic form $$ D^2{\mathscr{L}}_g(q)\phi\widetilde \phi = \int_{0}^{1} \phi_{t}\widetilde\phi_{t} + \phi\widetilde\phi - g_{qq}(q(t),t)\phi\widetilde\phi dt,\quad \phi,\widetilde\phi\in H^{1}({ \mathbb{R}}/{ \mathbb{Z}}). $$ Indeed, let $\eta \in E'$ and $\psi \in T_{\#_{i}q^{i}}E(q_{D})$, then \begin{align*} D^2{\mathscr{L}}_g(\#_{i}q^{i})\eta\psi &= \sum_{i}\int_{\tau_{i}}^{\tau_{i+1}} \eta_{t}\psi_{t} + \eta \psi - g_{qq}(q^{i}(t),t)\xi\eta dt\\
&= \sum_{i}\psi_{t}\eta\bigl|_{\tau_{i}}^{\tau_{i+1}} - \sum_{i}\int_{\tau_{i}}^{\tau_{i+1}} \bigl[-\psi_{tt}+\psi +g_{qq}(q^{i}(t),t)\psi\bigr] \eta dt =0. \end{align*} Let $\phi = \eta + \psi$, then $$ D^2{\mathscr{L}}_g(\#_{i}q^{i})\phi\widetilde\phi = D^2{\mathscr{L}}_g(\#_{i}q^{i})\eta\widetilde\eta + D^2{\mathscr{L}}_g(\#_{i}q^{i})\psi\widetilde\psi, $$
by the above orthogonality. By construction the minimizers $q^{i}$ are non-degenerate and therefore $D^2{\mathscr{L}}_g|_{E'}$ is positive definite. This implies that the Morse index of a (stationary) broken geodesic is determined by
$D^2{\mathscr{L}}_g|_{T_{\#_{i}q^{i}}E(q_{D})}$. By the commuting diagram for ${\mathscr{W}}$ this implies that the Morse index is given by quadratic form $D^{2}{\mathscr{W}}(q_{D})$. We have now proved the following lemma that relates the Morse index of critical points of the discrete action ${\mathscr{W}}$ to Morse index of the `full' action ${\mathscr{L}}_g$.
\begin{lemma} \label{lem:same-index} Let $q$ be a critical point of ${\mathscr{L}}_g$ and $q_D$ the corresponding critical point of ${\mathscr{W}}$, then the Morse indices are the same i.e. $\beta(q) = \beta(q_D )$. \end{lemma}
For a 1-periodic function $q(t)$ we define the mapping $$ q \xrightarrow{D_d} q_D = (q_0,\cdots,q_d),\quad q_i = q(i/d),~~i=0,\cdots,d, $$ and $q_D$ is called the discretization of $q$. The linear interpolation $$ q_D \mapsto \ell_{q_D}(t) = \#_i \Bigl[ q_i + \frac{q_{i+1} - q_i}{d} t\Bigr], $$ reconstructs a piecewise linear 1-periodic function. For a relative braid diagram $q\rel Q$, let $q_D\rel Q_D$ be its discretization, where $Q_D$ is obtained by applying $D_d$ to every strand in $Q$. A discretization $q_D\rel Q_D$ is \emph{admissible} if $\ell_{q_D}\rel \ell_{Q_D}$ is homotopic to $q\rel Q$, i.e. $\ell_{q_D}\rel \ell_{Q_D} \in [q\rel Q]$. Define the \emph{discrete} relative braid class $[q_D\rel Q_D]$ as the set of `discrete relative braids' $q_D'\rel Q_D'$, such that $\ell_{q_D'}\rel \ell_{Q_D'} \in [q\rel Q]$. The associated fibers are denoted by $[q_D]\rel Q_D$. It follows from \cite{GVV}, Proposition 27, that $[q_D\rel Q_D]$ is guaranteed to be connected when
$$ d> \#\{\hbox{~crossings in}~q\rel Q\}, $$ i.e.
for any two discrete relative braids $q_D\rel Q_D$ and $q_D'\rel Q_D'$, there exists a homotopy $q_D^\alpha\rel Q_D^\alpha$ (discrete homotopy) such that $\ell_{q_D^\alpha}\rel \ell_{Q_D^\alpha}$ is a path in $[q\rel Q]$. Note that fibers are not necessarily connected! For a braid classes $[q\rel Q]$ the associated discrete braid class $[q_D\rel Q_D]$ may be connected for a smaller choice of $d$.
We showed above that if $1/d>0$ is sufficiently small, then the critical points of ${\mathscr{L}}_g$, with $|q|\le 1$, are in one-to-one correspondence with the critical points of ${\mathscr{W}}$, and their Morse indices coincide by Lemma \ref{lem:same-index}. Moreover, if $1/d>0$ is small enough, then for all critical points of ${\mathscr{L}}_g$ in $[q]\rel Q$, the associated discretizations
are admissible
and $[q_D\rel Q_D]$ is a connected set. The discretizations of the critical points of ${\mathscr{L}}_g$ in $[q]\rel Q$
are critical points of ${\mathscr{W}}$ in the discrete braid class fiber $[q_D]\rel Q_D$.
Now combine the index identity with (\ref{eqn:finaldegree2}), which yields \begin{equation} \label{eqn:to-discr} \chi(x\rel y) = \sum_{q \in \Crit({\mathscr{L}}_g) \cap ([q]\rel Q)} (-1)^{\beta(q)} = \sum_{q_D \in \Crit({\mathscr{W}}) \cap ([q_D]\rel Q_D)} (-1)^{\beta(q_D)}. \end{equation}
\subsection{The Conley index for discrete braids} \label{sec:con-br}
In \cite{GVV} an invariant for discrete braid classes $[q_D\rel Q_D]$ is defined based on the Conley index. The invariant ${\mathrm{HC}}_*([q_D]\rel Q_D)$ is independent of the fiber and can be described as follows. A fiber $[q_D]\rel Q_D$ is a finite dimensional cube complex with a finite number of connected components. Denote the closures of the connected components by $N_j$. The faces of the hypercubes $N_j$ can be co-oriented in direction of decreasing the number of crossing in $q_D\rel Q_D$, and define $N_j^-$ as the closure of the set of faces with outward pointing co-orientation. The sets $N_j^-$ are called \emph{exit sets}. The invariant is given by $$ {\mathrm{HC}}_*([q_D]\rel Q_D) = \bigoplus_{j} H_*(N_j, N_j^-). $$ The invariant is well-defined for any $d>0$ for which there exist admissible discretizations and is independent of both the fiber and the discretization size. From \cite{GVV} we have for any Morse function ${\mathscr{W}}$ on a proper braid class fiber $[q_D]\rel Q_D$, \begin{equation} \label{eqn:from GVV} \sum_{q_D \in \Crit({\mathscr{W}}) \cap ([q_D]\rel Q_D)} (-1)^{\beta(q_D)}= \chi\bigl( {\mathrm{HC}}_*([q_D]\rel Q_D)\bigr)=: \chi\bigl(q_D\rel Q_D\bigr). \end{equation} The latter can be computed for any admissible discretization and is an invariant for $[q\rel Q]$. Combining \ref{eqn:to-discr} and \ref{eqn:from GVV} gives \begin{equation} \label{eqn:charss} \chi\bigl(x\rel y\bigr) = \chi\bigl(q_D\rel Q_D\bigr). \end{equation} In this section we assumed without loss of generality that $x\rel y$ is augmented and since the Euler-Floer characteristic is a braid class invariant, an admissible discretization is construction for an appropriate augmented, Legendrian representative $x_L\rel y_L$. Summarizing $$ \chi\bigl(x\rel y\bigr) =\chi\bigl(x_L\rel y_L^*\bigr) = \chi\bigl(q_D\rel Q_D^*\bigr). $$ Since $\chi\bigl(q_D\rel Q_D^*\bigr)$ is the same for any admissible discretization, the Euler-Floer characteristic can be computed using any admissible discretization, which proves Theorem \ref{thm:discrete}.
\begin{remark} \label{eqn:realEC} {\em The invariant $\chi\bigl(q_D\rel Q_D\bigr)$ is a true Euler characteristic of a topological pair. To be more precise $$ \chi\bigl(q_D\rel Q_D\bigr) = \chi\bigl([q_D]\rel Q_D,[q_D]^-\rel Q_D\bigr), $$ where $[q_D]^-\rel Q_D$ is the exit set a described above. A similar characterization does not a priori exist for $[x]\rel y$. Firstly, it is more complicated to designate the equivalent of an exit set $[x]^-\rel y$ for $[x]\rel y$, and secondly it is not straightforward to develop a (co)-homology theory that is able to provide meaningful information about the topological pair $\bigl([x]\rel y,[x]^-\rel y\bigr)$. This problem is circumvented by considering Hamiltonian systems and carrying out Floer's approach towards Morse theory (see \cite{Floer1}), by using the isolation property of $[x]\rel y$. The fact that the Euler characteristic of Floer homology is related to the Euler characteristic of topological pair indicates that Floer homology is a good substitute for a suitable (co)-homology theory. } \end{remark}
\section{Examples} \label{sec:examples}
We will illustrate by means of two examples that the Euler-Floer characteristic is computable and can be used to find closed integral curves of vector fields on the 2-disc.
\subsection{Example} Figure \ref{cont discr}[left] shows the braid diagram $q\rel Q$ of a positive relative braid
$x\rel y$.
The discretization with $q_D\rel Q_D$, with $d=2$, is shown in Figure \ref{cont discr}[right]. The chosen discretization is admissible and defines the relative braid class $[q_D\rel Q_D]$.
There are five strands, one is free and four are fixed. We denote the points on the free strand by $q_D=(q_0,q_1)$ and on the skeleton by $Q_D= \{Q^1,\cdots, Q^4\}$, with $Q^i = (Q^i_0,Q^i_1)$, $i=1,\cdots,4$.
\begin{figure}
\caption{A positive braid diagram [left] and an admissible discretization [right].}
\label{cont discr}
\end{figure}
In Figure \ref{fig:config1}[left] the braid class fiber $[q_D]\rel Q_D$ is depicted. The coordinate $q_0$ is allowed to move between $Q_0^3$ and $Q_0^2$ and $q_1$ remains in the same braid class if it varies between $Q_1^1$ and $Q_1^4$. For the values $q_0=Q_0^3$ and $q_0=Q_0^2$ the relative braid becomes singular and if $q_0$ crosses these values two intersections are created. If $q_1$ crosses the values $Q_1^1$ or $Q_1^4$ two intersections are destroyed. This provides the desired co-orientation, see Figure \ref{fig:config1}[middle]. The braid class fiber $[q_D]\rel Q_D$ consists of 1 component and
we have that
$$
N={\mathrm{cl}}([q_D\rel Q_D])=\{(q_0,q_1): Q_0^3\leq q_0\leq Q_0^2,Q_1^1\leq q_1\leq Q_1^4 \},
$$
and the exit set is
$$
N^-=\{(q_0,q_1): q_1=Q_1^1, ~{\rm or~} q_1=Q_1^4\}.
$$
For the Conley index this gives: $$ {\mathrm{HC}}_k([q_D]\rel Q_D)=H_k(N,N^-;{ \mathbb{Z}})\cong\left\{ \begin{array}{ll} {{ \mathbb{Z}}} & {k=1}\\ 0 & {\rm otherwise} \end{array} \right. $$ \begin{figure}\label{fig:config1}
\end{figure}
The Euler characteristic of $\bigl([q_D]\rel Q_D,[q_D]^-\rel Q_D\bigr)$ can be computed now and the Euler-Floer characteristic
$\bigl(x\rel y\bigr)$ is given by $$ \chi (x\rel y)= \chi\bigl([q_D]\rel Q_D,[q_D]^-\rel Q_D\bigr) = -1\not=0 $$ From Theorem \ref{thm:exist} we derive that any vector field for which $y$ is a skeleton has at least 1
closed integral curve $x_0\rel y \in [x]\rel y$. Theorem \ref{thm:exist} also implies that any
orientation preserving diffeomorphism $f$ on the 2-disc which fixes the set of four points $A_4$, whose mapping class $[f;A_4]$ is represented by the braid $y$ has an additional fixed point.
\subsection{Example} The theory can also be used to find additional closed integral curves by concatenating the skeleton $y$. As in the previous example $y$ is given by Figure \ref{cont discr}. Glue $\ell$ copies of the skeleton $y$ to its $\ell$-fold concatenation and a reparametrize time by $t\mapsto \ell\cdot t$. Denote the rescaled $\ell$-fold concatenation of $y$ by $\#_\ell y$. Choose $d= 2\ell$ and discretize $\#_\ell y$ as in the previous example. \begin{figure}
\caption{A discretization of a braid class with a 5-fold concatenation of the skeleton $y$. The number of odd anchor points in middle position is $\mu = 3$. }
\label{fig:entropy}
\label{ciao}
\end{figure} For a given braid class $[x\rel \#_\ell y]$, Figure \ref{ciao} below shows a discretized representative $q_D \rel \#_\ell Q_D$, which is admissible. For the skeleton $\#_\ell Q_D$ we can construct
$3^\ell-2$ proper relative braid classes in the following way: the even anchor points of the free strand $q_D$ are always in the middle and for the odd anchor points we have 3 possible choices: bottom, middle, top (2 braids are not proper). We now compute the Conley index of the $3^\ell-2$ different proper discrete relative braid classes and show that the Euler-Floer characteristic is non-trivial for these relative braid classes.
The configuration space $N = {\mathrm{cl}}\bigl( [q_D]\rel \#_\ell Q_D\bigr)$ in this case is given by a cartesian product of $2\ell$ closed intervals, and therefore a $2\ell$-dimensional hypercube. We now proceed by determining the exit set $N^-$. As in the previous example
the co-orientation is found by a union of faces with an outward pointing co-orientation.
Due to the simple product structure of $N$, the set $N^-$ is determined by the odd
anchor points in the middle position.
Denote the number of middle positions at odd anchor points by $\mu.$ In this way $N^-$ consists of opposite faces at at odd anchor points in middle position, see Figure \ref{fig:entropy}. Therefore
$$ {\mathrm{HC}}_k([q_D]\rel \#_\ell Q_D)=H_k(N,N^-)=\left\{ \begin{array}{ll} {{ \mathbb{Z}}}_2 & {k=\mu}\\ 0 & {k\not = \mu,} \end{array} \right. $$ and the Euler-Floer characterisc is given by $$ \chi\bigl(x\rel\#_\ell y\bigr) = (-1)^\mu \not = 0. $$ Let $X(x,t)$ be a vector field for which $y$ is a skeleton of closed integral curves, then $\#_\ell y$ is a skeleton for the vector field $X^\ell(x,t) := \ell X_\ell(x,\ell t)$. From Theorem \ref{thm:exist} we derive that there exists a closed integral curve in each of the $3^\ell -2$ proper relative classes $[x]\rel y$ described above. For the original vector field $X$ this yields $3^\ell -2$ distinct closed integral curves.
Using the arguments in \cite{VVW} one can find a compact invariant set for $X$ with positive topological entropy, which proves that the associated flow is `chaotic' whenever $y$ is a skeleton of given integral curves
\subsection{Example} So far we have not addressed the question whether the closed integral curves $x\rel y$ are non-trivial, i.e.\!\! not equilibrium points of $X$. The theory can also be extended in order to find non-trivial closed integral curves. This paper restricts to relative braids where $x$ consists of just one strand. Braid Floer homology for relative braids with $x$ consisting of $n$ strands is defined in \cite{GVVW}. To illustrate the importance of multi-strand braids we consider the discrete braid class in Figure \ref{ciao-bene}.
\begin{figure}
\caption{A discretization of a braid class with a 3-fold concatenation of the skeleton $y$. The number of odd anchor points in middle position is $\mu = 2$ [right]. If we represent all translates of $x$ we obtain a proper relative braid class where $x$ is a 3-strand braid [left]. The latter provides additional linking information. }
\label{ciao-bene}
\end{figure}
The braid class depicted in Figure \ref{ciao-bene}[right] is discussed in the previous example and the Euler-Floer characteristic is equal to 1. By considering all translates of $x$ on the circle ${ \mathbb{R}}/{ \mathbb{Z}}$, we obtain the braids in Figure \ref{ciao-bene}[left]. The latter braid class is proper and encodes extra information about $q_D$ relative to $Q_D$. The braid class fiber is a 6-dimensional cube with the same Conley index as the braid class in Figure \ref{ciao-bene}[right]. Therefore, $$ \chi(q_D\rel Q_D) = (-1)^2 = 1. $$ As in the 1-strand case, the discrete Euler characteristic
can used to compute the associated Euler-Floer characteristic of $x\rel y$ and $\chi(x\rel y) = 1$. The skeleton $y$ thus forces solutions $x\rel y$ of the above described type. The additional information we obtain this way is that for braid classes $[x\rel y]$, the associated closed integral curves for $X$ cannot be constant and therefore represent non-trivial closed integral curves.
\end{document} | arXiv |
\begin{document}
\title{Trace Inclusion for One-Counter Nets Revisited}
\begin{abstract}
\MCOL{One-counter nets}\ (\MCOL{OCN}) consist of a nondeterministic finite control and a single integer counter that cannot be fully tested for zero. They form a natural subclass of both One-Counter Automata, which allow zero-tests and Petri Nets/VASS, which allow multiple such weak counters.
The \MCOL{trace inclusion problem}\ has recently been shown to be undecidable for \MCOL{OCN}.
In this paper, we contrast the complexity of two natural restrictions which imply decidability.
First, we show that
\MCOL{trace inclusion}\ between an \MCOL{OCN}\ and a \emph{deterministic} \MCOL{OCN}\ is NL-complete,
even with arbitrary binary-encoded initial counter-values as part of the input.
Secondly, we show Ackermannian completeness of for the trace \emph{universality} problem of nondeterministic \MCOL{OCN}. This problem is equivalent to checking \MCOL{trace inclusion}\ between a finite and a \MCOL{OCN}-process.
\end{abstract}
\section{Introduction}
A fundamental question
in formal verification is
if the behaviour of one process can be reproduced by -- or equals that of -- another given process.
These inclusion and equivalence problems, respectively
have been studied for various notions of behavioural preorders and equivalences
and for many
computational models.
\MCOL{Trace inclusion}/equivalence asks if the set of \emph{traces}, all emittable sequences of actions, of one process is contained in/equal to that of another. Other than for instance Simulation preorder, \MCOL{trace inclusion}\ lacks a strong locality of failures, which makes this problem intractable or even undecidable already for very limited models of computation.
We consider \MCOL{one-counter nets}, which consist of a finite control and a single integer counter that cannot be fully tested for zero, in the sense that an empty counter can only restrict possible moves. They are subsumed by \MCOL{One-counter automata}\ (\MCOL{OCA}) and thus Pushdown Systems, which allow explicit zero-tests by reading a bottom marker on the stack.
At the same time, \MCOL{OCN}\ are a subclass of Petri Nets or Vector Addition Systems with states (VASS): they are exactly the one-dimensional VASS and thus equivalent to Petri Nets with at most one unbounded place.
\emph{Related work}. \Textcite{VP1975} show the decidability of the trace equivalence problem for \emph{deterministic} \MCOL{one-counter automata}\ (\MCOL{DOCA}). This problem has recently been shown to be NL-complete by \textcite{BGJ2013}, assuming fixed initial counter-values.
The equivalence of \MCOL{deterministic pushdown automata}\ is known to be decidable \cite{Sen1997} and
primitive recursive \cite{Sti2002}, but the exact complexity is still open.
\Textcite{Val1973} proves the undecidability of both \MCOL{trace inclusion}\ for
\MCOL{DOCA}\ and universality for nondeterministic \MCOL{OCA}.
\Textcite{JEM1999} consider \MCOL{trace inclusion}\ between Petri Nets and finite systems and prove decidability in both directions.
\Textcite{Jan1995} showed that \MCOL{trace inclusion}\ becomes undecidable if one compares processes of Petri Nets with at least two unbounded places. In \cite{HMT2013}, the authors show that \MCOL{trace inclusion}\ is undecidable already for (nondeterministic) \MCOL{one-counter nets}. Simulation preorder however, is known to be decidable and PSPACE-complete for
this model \cite{AC1998,JKM2000,HLMT2013},
which implies a PSPACE\ upper bound for \MCOL{trace inclusion}\ on \MCOL{DOCN}\ as \MCOL{trace inclusion}\ and simulation coincide for deterministic systems.
\Textcite{HWT1996} compare the classes of \emph{languages}
defined by \MCOL{DOCN}\ with various acceptance modes and in a series of papers consider the respective inclusion problems.
They derive procedures that exhaustively search for a bounded witness that work in time and space polynomial in the size of the automata if the initial counter-values are fixed.
We show that for monotone relations like \MCOL{trace inclusion}\ or the inclusion of languages defined by acceptance with final states, one can speed up the search for suitable witnesses.
\emph{Our contribution}. We fix the complexity of two well-known decidable decision problems regarding the traces of one-counter processes.
First, we show that \MCOL{trace inclusion}\ between \emph{deterministic} \MCOL{one-counter net}\ is NL-complete. Our upper bound holds even if only the supposedly larger process is deterministic and if (binary encoded) initial counter-values are part of the input.
This matches
the trivial NL lower bound derived from DFA universality. Our technique uses short certificates for the existence of (possibly long) distinguishing traces. The sizes of certificates are polynomial in the number of states of the finite control and they can be verified in space logarithmic in the binary representation of the initial counter-values.
Our second result is that \MCOL{trace universality}\ of \emph{nondeterministic} \MCOL{OCN}\ is Ackermann-complete. This problem can be easily seen to be (logspace) inter-reducible with checking \MCOL{trace inclusion}\ between a finite process and a process of a \MCOL{OCN}.
\section{Background}
We write $\mathbb{N}$ for the set of non-negative integers.
For any set $A$, let $A^*$ denote the set of finite strings over $A$ and $\varepsilon\in A^*$ the empty string.
\input{definitions/ocn}
An important property of \MCOL{one-counter nets}\ is that the step relation and therefore also \MCOL{trace inclusion}\ is monotone with respect to the counter: \begin{lemma}[Monotonicity]\label{lem:monotonicity} If $pm\step{a}p'm'$ then $p(m+1)\step{a}p'(m'+1)$. This in particular means that $T(pm)\subseteq T(p(m+1))$ holds for any \MCOL{OCN}-process $pm$. \end{lemma}
The next lemma justifies our focus on processes of complete \MCOL{OCN}. The proof is a simple construction and can be found in \cref{app:reduction}. The idea is to first determinize $\NN{A}$ by consistently relabelling all transitions of $\NN{A}$ and $\NN{B}$, and then complete the net $\NN{B}$ by introducing a sink state.
\begin{restatable}[Normal Form Assumption]{lemma}{lemreduction} \label{lem:reduction}
\MCOL{Trace inclusion}\ for \MCOL{OCN}\ is logspace-reducible to \MCOL{trace inclusion}\ between a determinisic and
a complete \MCOL{OCN}.
More precisely, given \MCOL{OCNs}\ $\NN{A}$ and $\NN{B}$
with state sets $N$ and $M$,
one can construct a
\MCOL{DOCN}\
$\NN{A'}$ with states $N$
and a complete \MCOL{OCN}\ $\NN{B'}$
with states $M'\supseteq M$
such
that the following holds for
any two processes $pm$ and $qn$ of $\NN{A}$ and $\NN{B}$,
respectively:
\begin{equation}
T_\NN{A}(pm)\subseteq T_\NN{B}(qn)
\iff T_\NN{A'}(pm)\subseteq T_\NN{B'}(qn).
\end{equation}
Moreover,
the constructed net $\NN{B'}$ is deterministic if the original net $\NN{B}$ is.
\end{restatable} Due to the undecidability of \MCOL{trace inclusion}\ for \MCOL{OCN}\ \cite{HMT2013}, a direct consequence of \cref{lem:reduction} is that \MCOL{trace inclusion}\ $T_\NN{A}(pm)\subseteq T_\NN{B}(qn)$ is already undecidable if we allow the net $\NN{B}$ to be nondeterministic. Unless otherwise stated, we will from now on assume a \MCOL{DOCN}\ $\NN{A}=(Q_A,\mathit{Act},\delta_A)$ and a complete \MCOL{DOCN}\ $\NN{B}=(Q_B,\mathit{Act},\delta_B)$.
\section{Trace Inclusion for Deterministic One-Counter Nets}
We characterize witnesses for non-inclusion $T_\NN{A}(pm)\not\subseteq T_\NN{B}(qn)$, starting with
some notation to express paths and their effects.
\input{definitions/net_paths} Notice that
the absolute values of the effect and guard of a path are bounded by its length. We consider the synchronous product of the control graphs of two given \MCOL{deterministic one-counter nets}. \input{definitions/product_graph}
Since both $\NN{A}$ and $\NN{B}$ are deterministic and $\NN{B}$ is complete,
a trace $w\in T_\NN{A}(pm)$
uniquely determines a path from state $(p,q)$ in their product. We therefore identify witnesses for non-inclusion with the paths they induce in the product.
\input{definitions/witnesses}
Every witness $\pi$ for $(pm,qn)$ completely exhausts the counter in the process of $\NN{B}$: $(pm,qn)\Step{\pi}{}{}(p'm',q'0)$. This is because a process of a complete net can only \emph{not} make an $a$-step in case the counter is empty.
\begin{example}
Consider two nets
given by
self-loops
$p\step{a,0}p$
and $q\step{a,-1}q,$ respectively.
Their product is the cycle $L=(p,q)\xrightarrow{a,0,-1}(p,q)$
with effects $\effect{A}(L)=0$ and $\effect{B}(L)=-1$.
The only witness for $(pm,qn)$ for initial counter-values $m,n\in\mathbb{N}$ is
$L^{n}$, which has length
polynomial in the sizes of the nets
\emph{and} the initial counter-values, but not in the sizes
of the nets alone. \end{example} The previous example shows that if binary-encoded initial counter-values are part of the input, we can only bound the length of shortest witnesses exponentially. However, we will see that it suffices to consider witnesses of a certain regular form only. This leads to
small certificates for non-inclusion, which can be stepwise guessed and verified in space logarithmic in the size of the nets.
A crucial ingredient for our characterization is the monotonicity of witnesses, a direct consequence of the monotonicity of the steps in \MCOL{OCNs}\ (\cref{lem:monotonicity}):
\begin{lemma}\label{lem:witness-monotonicity}
If $\pi$ is a witness for $(pm,qn)$
then for all $m'\ge m$ and $n'\le n$ some prefix of $\pi$
is a witness for $(pm',qn')$. \end{lemma}
The intuition behind the further characterization of witnesses is that in order to show non-inclusion, one looks for a path
that is enabled
in the process of $\NN{A}$ and moreover exhausts the counter in the process of $\NN{B}$. Since any sufficiently long path will revisit control-states in the product, we can compare such paths with respect to their effect on the counters and see that some are ``better'' than others. For instance, a cycle that only increments the counter in $\NN{B}$ and decrements the one in $\NN{A}$ is surely suboptimal considering our goal to find a (shortest) witness. The characterization \cref{thm:form} essentially states that if a witness exists, then also one that, apart from short paths, combines only the most productive cycles.
\input{definitions/loops}
Note that no loop is longer than $|V|$ because it visits exactly one node twice.
\begin{example}
\label{ex:witness}
Consider two \MCOL{DOCN}\ such that their product is the graph depicted
below,
where we identify transitions with their action labels for simplicity and
\noindent \begin{minipage}{\linewidth}
\begin{wrapfigure}[9]{r}{0.430\linewidth}
\includegraphics[width=0.97\linewidth]{pictures/example-product.pdf}
\end{wrapfigure}
let $v_0=(p,p')\in V$.
The paths $t_0t_1t_2$,
$t_3t_4$ and $t_6$ are loops with
slopes $3/1$, $2/1$ and $1/1$
and types $\TypeUU$, $\TypeUU$ and $\TypeDD,$
respectively.
The path $(t_0t_1t_2)(t_3t_4)^9t_5(t_6)^{20}$ is a witness for $(p0,p'10)$
of length $42$.
By replacing $8$ occurrences of the loop $(t_3t_4)$ with
$(t_0t_1t_2)^8$ we derive the longer witness
$(t_0t_1t_2)^9(t_3t_4)t_5(t_6)^{20}$, which has
essentially the same structure but is more efficient in the sense
that for the same effect
on $\NN{B}$ it achieves a higher counter-effect on $\NN{A}$.
\end{minipage} \end{example}
\begin{restatable}[]{theorem}{thmform} \label{thm:form}
Fix a \MCOL{DOCN}\ $\NN{A}$,
a complete \MCOL{DOCN}\ $\NN{B}$,
and
let $K\in\mathbb{N}$ be the number of nodes in their product.
There is a bound $c\in\mathbb{N}$ that depends polynomially on $K$,
such that
the following holds
for any two processes $pm$ and $qn$ of $\NN{A}$ and $\NN{B}$.
If $T(pm)\not\subseteq T(qn)$,
then there is a witness for $(pm,qn)$
that is either no longer than $c$ or has
one of the following forms:
\begin{enumerate}
\item $\pi_0L_0^{l_0}\pi_1$, where
$L_0$ is a loop of type $\TypeUD$ and $\pi_0,\pi_1$ are no longer than $c$,
\item $\pi_0L_0^{l_0}\pi_1L_1^{l_1}\pi_2$, where
$L_0$ and $L_1$ are loops of type $\TypeUU$ and $\TypeDD$ with $S(L_0) > S(L_1)$
and $\pi_0,\pi_1,\pi_2$ are no longer than $c$,
\item $\pi_0L_0^{l_0}\pi_1$, where
$L_0$ is a loop of type $\TypeDD$
and $\pi_0,\pi_1$ are no longer than $c$,
\end{enumerate}
where in all cases, the number of iterations $l_0,l_1\in\mathbb{N}$
are polynomial in $K$ and the initial
counter-values $m$ and $n$ of the given processes.
\end{restatable} \begin{proofsketch}
The overall idea of the proof is to explicitly rewrite
witnesses into one of the canonical forms.
More specifically,
we introduce a system of path-rewriting rules
which simplify witnesses by removing, reducing or changing some
loops as in \cref{ex:witness}.
We show that the rules preserve witnesses and
any sequence of successive rule applications must eventually terminate
with a normalized path, to which none of the rules is applicable.
Such a witness can be decomposed as
\begin{equation}
\label{eq:form:sketch}
\pi=\pi_0L_0^{l_0}\pi_{1}L_{1}^{l_1}\dots\pi_kL_k^{l_k}\pi_{k+1}
\end{equation}
where the $L_i$ are (pairwise different) loops and the
$\pi_i$ are short, i.e.~polynomially bounded.
Moreover the rules are designed in such a way that almost all $l_i$ are polynomially bounded. By almost all we mean except one in the first and third form of the witness or two in the witness of the second form. This means that unravelling of those loops with polynomially bounded $l_i$ and glueing them with surrounding $\pi_i$ to get paths $\pi_0,\pi_1,\pi_2$ does not blow up of the length of $\pi_0,\pi_1, \pi_2$ above polynomial
bound $c$.
\qed \end{proofsketch}
Notice that the bound $c$ in the claim of \cref{thm:form} depends only on the number of states. We now derive a decision procedure for \MCOL{trace inclusion}\ that works in logarithmic space.
\begin{theorem}
Let $pm$ and $qn$ be processes of \MCOL{OCN}\ $\NN{A}$
and \MCOL{DOCN}\ $\NN{B},$
respectively, where $m,n$ are given in binary.
There is a nondeterministic algorithm that
decides $T(pm)\subseteq T(qn)$ in
logarithmic space.
\end{theorem} \input{proofs/main}
\section{Universality of Nondeterministic One-Counter Nets}
\label{sec:universality}
To contrast the result of the previous section we now turn to the problem of checking \MCOL{trace inclusion}\ between a finite process and a nondeterministic \MCOL{OCN}. This problem is known to be decidable, even for general Petri nets \cite{JEM1999} and it
can be easily seen to be (logspace) inter-reducible with the trace universality problem, because \MCOL{OCNs}\ are closed under products with finite systems.
For \MCOL{OCN}, trace universality can be decided using a simple well-quasi-order based saturation method that determinizes the net on the fly.
We will see that this procedure is optimal: The problem is Ackermannian, i.e.~it is non-primitive recursive and lies exactly at level $\omega$ of the Fast Growing Hierarchy \cite{FFSS2011}.
\newcommand{\N_{\bottom}}{\mathbb{N}_{\ensuremath{\bot}}} Let $\N_{\bottom}$ be the set of non-negative integers plus a special least element $\ensuremath{\bot}$ and let $\max$ be the total function that returns the maximal element of any nonempty finite subset and $\ensuremath{\bot}$ otherwise.
Consider a set $S\subseteq Q\times\mathbb{N}$ of processes of an \MCOL{OCN}\ $\NN{N}=(Q,\mathit{Act},\delta)$. We lift the definition of traces to sets of processes in the natural way: the \emph{traces} of $S$ are $T(S)=\bigcup_{qn\in S} T(qn)$.
By the monotonicity of \MCOL{trace inclusion}\ (\cref{lem:monotonicity}), the traces of a finite set of processes are determined
only by the traces of its maximal elements.
\begin{definition}
Let $Q=\{q_1,q_2,\dots,q_k\}$ be the states-set of some \MCOL{OCN}.
For a finite set $S\subseteq Q\times\mathbb{N}$
define the \emph{macrostate} as the vector
$M_S
\in \N_{\bottom}^k$
where for each $0< i\le k$,
$M_S(i)= M_S(q_i) = \max\{n\;|\; q_in \in S\}$.
In particular, the macrostate for a singleton set $S=\{q_in\}$
is the vector with value $n$ at the $i$-th coordinate
and $\bot$ on all others.
The \emph{norm} of a macrostate $M \in \N_{\bottom}^k$
is $\norminf{M} = \max\{M(i) \;|\; 0<i\leq k\}.$
We define a step relation $\WStep{a}{}{}$
for all $a\in\mathit{Act}$ on the set of macrostates as follows:
\begin{equation}
(n_1,n_2,\dots,n_k)\WStep{a}{}{} (m_1,m_2,\dots,m_k)
\end{equation}
iff
$m_i = \max\{n\:|\: \exists n_j\neq\ensuremath{\bot}.\: q_jn_j\step{a}q_in \}$
for all $0< i\le k$.
The \emph{traces} of macrostate $M$ are
$T(M) = \bigcup_{0< i\le k} T(q_{i\, } M(i))$,
where $T(q\bot)=\emptyset$.
For two macrostates $M,N$ we say $M$ is \emph{covered} by $N$ and write
$M\sqsubseteq N$,
if it is pointwise smaller, i.e., $M(i)\le N(i)$ for all
$0< i\le k$.
For convenience,
we will write $\{q_1=n_1, q_2=n_2,\dots,q_l=n_l\}$
for the macrostate with value $M(i)=n_i$
whenever $q_i=n_i$ is listed
and $\bot$ otherwise. \end{definition}
Steps on macrostates correspond to the classical powerset construction and each macrostate represents the finite set of possible processes the \MCOL{OCN}\ can be in, where all non-maximal ones (w.r.t.~their counter-value) are pruned out.
\begin{example}{Macrostate}
\begin{minipage}[b]{0.35\textwidth}
\begin{tikzpicture}[node distance=2cm]
\node (P) at (0,0) {$q2$};
\node (Q) at (0, 3) {$q1$};
\node (R) at (2,1.5) {$q3$};
\node (nic) at (0,-2) {};
\path[->]
(P) edge node[left] {$a, 1$} (Q)
(Q) edge node[right] {$a, 0$} (R)
(R) edge node[right] {$a, -1$} (P);
\path[->]
(R) edge [loop right] node[right] {$a,1$} (R);
\end{tikzpicture}
\end{minipage} \begin{minipage}[b]{0.6\textwidth} Consider automaton $\NN{A}$ like on the picture, state $q_3$ and a counter value $4$; we analyse traces, $T(q_3 4)$. If we go via an edges labelled by $a$ once we can see that $T(q_3 4)=\{\varepsilon\}\, \cup\, aT(q_2 3)\, \cup\, aT(q_35)$. This implies that $T(q_3 4)$ is universal iff $T(q_3, 5)\cup T(q_2, 3)$ is universal, i.e. contains $\mathit{Act}^*$. Making similar analysis after using two more $a$ we get that $T(q_3 4)$ is universal iff $T(q_37)\, \cup \, T(q_2 5)\, \cup \, T(q_1 5)\, \cup \, T(q_3 4)$ is universal. But we know that $T(q_3 4)\subseteq T(q_3 7)$ which implies that $T(q_37)\, \cup \, T(q_2 5)\, \cup \, T(q_1 5)\, \cup \, T(q_3 4)=T(q_37)\, \cup \, T(q_2 5)\, \cup \, T(q_1 5)$. This immediately lead to introduce macrostates $M_{\{q_3 7,q_2 5,q_1 5,q_ 34\} }= (5,5,7).$ The norm $\norminf{M_{\{q_3 7,q_2 5,q_1 5,q_ 34\} }}= 7.$ On the other hand $M_{\{q_3 4\} }=(\ensuremath{\bot}, \ensuremath{\bot}, 4)$ which means that states $q_1$ and $q_2$ are not present and in this case $M(1)=M(q_1)=\ensuremath{\bot}$. Moreover we can write that $M_{\{q_ 34\} }\sqsubseteq M_{\{q_3 7,q_2 5,q_1 5,q_ 34\} }.$
\end{minipage}
\end{example}
The next lemma directly follows from these definitions and monotonicity (\cref{lem:monotonicity}). \begin{lemma}\label{lem:macroprops}\ \begin{enumerate}
\item The covering-order $\sqsubseteq$ is a well-quasi-order on $\N_{\bottom}^k$, the set
of all macrostates. Moreover, $M\sqsubseteq N$ implies $T(M)\subseteq T(N)$.
\label{lem:macroprops:wqo}
\item If $M\wstep{a}N$ then $\norminf{N} \le \norminf{M}+1$.
\label{lem:macroprops:norminc}
\item For any finite set $S\subseteq Q\times\mathbb{N}$ it holds that $T(S) = T(M_S)$.
\label{lem:macroprops:traces} \end{enumerate} \end{lemma}
Dealing with macrostates allows us to treat universality as a reachability problem:
By point~\ref{lem:macroprops:traces} of \cref{lem:macroprops} we see that a process $qn$ is \emph{not} trace universal, $\mathit{Act}^* \neq T(qn)$, if and only if $M_{\{qn\}} \WStep{}{*}{} (\ensuremath{\bot},\bot,\dots,\bot)$.
We take the perspective of a pathfinder, whose goal it is to reach $(\bot)^k$.
\input{univ-upper}
\input{univ-lower}
For the rest of this section, we recall a recent result from \textcite{FFSS2011}, that allows us to provide the exact complexity of the \MCOL{OCN}\ trace universality problem in terms of its level in the Fast-Growing Hierarchy.
\begin{definition}[Fast-Growing Hierarchy] \label{def:fastgrowing-hierarchy}
Consider the family of functions $F_n:\mathbb{N}\to\mathbb{N}$
where for $x,k\in\mathbb{N}$,
\begin{align*}
F_0(x) = x+1 \text{\quad and }
&&
F_{k+1}(x) = F_k^{x+1}(x).
\end{align*}
Here, $F^k$ denotes the $k$-fold application of $F$.
Moreover, define $F_{\omega}(x)=F_x(x)$ for the first limit ordinal $\omega$.
For $k\le \omega$,
$\mathfrak{F}_k$ denotes
the least class of functions
that contains all constants and is closed under
substitution, sum, projections, limited recursion and applications
of functions $F_n$ for $n\le k$.
Already $\mathfrak{F}_2$ contains all elementary functions
and the union $\bigcup_{k\in\mathbb{N}}\mathfrak{F}_k$ of all finite levels
contains exactly the primitive-recursive functions.
A function is called \emph{Ackermannian} if it is in
$\mathfrak{F}_{\omega}\setminus\bigcup_{k\in\mathbb{N}}\mathfrak{F}_k$.
\end{definition}
A sequence $x_0, x_1,\dots,x_l$ of macrostates is called \emph{good} if there are indices $0\le i<j\le l$ such that $x_i\sqsubseteq x_j$ and \emph{bad} otherwise. The sequence is \emph{$t$-controlled} by $f:\mathbb{N}\to\mathbb{N}$ if $\norminf{x_i} < f(i+t)$ for every index $0\le i\le l$. \begin{theorem}[\cite{FFSS2011}]\label{thm:badseq}
Let $f:\mathbb{N}\to\mathbb{N}$ be a monotone function in $\mathfrak{F}_\gamma$
such that $f(x)\ge\max\{1,x\}$ for some $\gamma\ge 1$.
There is a function $L_{k,f}(t)$ in $\mathfrak{F}_{k+\gamma-1}$
that computes a bound on the maximal length of
bad sequences in $\N_{\bottom}^k$ that are $t$-controlled by $f$.
\end{theorem}
\begin{corollary}
\MCOL{Trace universality}\ of \MCOL{OCN}\ is Ackermannian.
\end{corollary} \begin{proof}
By \cref{thm:univ-ack}, it suffices to show that the problem is
in $\mathfrak{F}_\omega$.
Recall the procedure that, for a given process $pm$ of a net with
$k$ control-states, guesses a shortest terminating path from the initial
macrostate
(a witness for non-universality),
and stops unsuccessfully if a macrostate covers
one that has been seen before.
The time and space requirements of this procedure
are bounded in terms of the longest non-increasing (w.r.t.~covering) sequence
of $k$-dimensional macrostates.
These are bad sequences where the norm of the initial macrostate is $m$, the
counter-value of the process to check for universality.
By point~\ref{lem:macroprops:norminc} of \cref{lem:macroprops}, such sequences are $m$-controlled by the successor function $f(x)=x+1$, which is in $\mathfrak{F}_1$. By \cref{thm:badseq}, computing the bound and running the procedure above
is in $\mathfrak{F}_k$.
As $k$ is part of the input, this yields a procedure in $\mathfrak{F}_\omega$. \qed \end{proof}
\section{Conclusion}
We have shown NL-completeness of the general \MCOL{trace inclusion}\ problem for deterministic \MCOL{one-counter nets}, where initial counter-values are part of the input.
Our proof is based on a characterization of the shape of possible witnesses in terms of a small number of polynomially-sized templates.
Realizability of such templates can be verified in space logarithmic only in the size of the underlying state space.
Our procedure is therefore independent of the number of action symbols and transitions in the input nets.
To prove the characterization theorem we use witness rewriting rules, the correctness of which crucially depends on the monotonicity of \MCOL{trace inclusion}\ w.r.t.~counter-values. In fact, we only make use of this property in the net on the left but similarly one can define rules that exploit only the monotonicity in the process on the right. With some additional effort one can extend this argument also for trace inclusion between \MCOL{DOCN}\ and \MCOL{DOCA}\ or vice versa (see \cite{thesis}).
The second part of the paper explores the complexity of the universality problem for nondeterministic \MCOL{OCN}, and \MCOL{trace inclusion}\ between finite systems and \MCOL{OCN}\ that easily reduces to \MCOL{OCN}\ universality. Here we show that the simplest known algorithm which uses a well-quasi-order based saturation technique has already optimal complexity: The problem is Ackermannian, i.e., not primitive recursive.
\paragraph*{Acknowledgement.}
We thank Mary Cryan, Diego Figueira and Sylvain Schmitz for helpful discussions and the anonymous reviewers of an earlier draft for their constructive feedback. Piotr Hofman acknowledges a partial support by the Polish NCN grant 2013/09/B/ST6/01575.
\printbibliography[heading=bibintoc]
\appendix
\section{Normal-Form Assumption}
\label{app:reduction}
We consider here what is sometimes called \emph{realtime} automata, in which no silent ($\varepsilon$-labelled) transitions are present. In the absence of zero-tests, the usual syntactic restriction for deterministic Pushdown Automata, (no state with outgoing $\varepsilon$-transition may have outgoing transitions labelled by $a\neq\varepsilon$) and the lack of an explicit zero-test in our model implies that all states on $\varepsilon$-cycles are essentially deadlocks. A process in such a state can either silently exhaust the counter and deadlock or divert into an infinite $\varepsilon$ loop. With respect to their traces, those processes are equivalent. This means one can eliminate $\varepsilon$-transitions by removing $\varepsilon$-cycles and replacing the remaining short paths by direct steps
(and normalize the effects of single transitions back to $\{-1,0,1\}$). Such a reduction works in $\mathcal{O}(\log n)$ space.
Allowing $\varepsilon$-transitions thus leaves the complexity of \MCOL{trace inclusion}\ invariant.
\input{reduction} \input{proofs/reduction}
\section{Checking Weighted Inequalities in Logspace}
\label{app:inequalities}
\begin{lemma}\label{lem:complexity}
Inequalities
of the form $m \cdot A + B \ge n\cdot C + D$
where all coefficients are non-negative integers given in binary
can be verified in $\mathcal{O}(\log (A+B+C+D))$ deterministic space.
\end{lemma} \begin{proof} Assume w.l.o.g.~that the bit-representations of $m$ and $n$ are of the same length, as are those of $A,B,C$ and $D$, and we have the least significant bit on the right.
To check $m\ge n$, we can stepwise read their binary representation from right to left, flipping an ``output'' bit $\textit{Out}$ on the way: Initially, $\textit{Out}:=1$; in every step set $\textit{Out}:=0$ if the current bit in $m$ is strictly smaller than that in $n$; set $\textit{Out}:=1$ if the current bit in $m$ is strictly bigger than that in $n$ and otherwise proceed without touching $\textit{Out}$. The inequality holds iff $\textit{Out}=1$ after completely reading the input.
To check the weighted variant, we use the same algorithm but multiply $m\cdot A$, and $n\cdot C$ on the fly, using standard long binary multiplication. We use a scratchpad to store the intermediate sums, starting with values $B$ and $D$.
In a step that reads the $i$th bit $m[i]$ of $m$, we want to add $A\cdot 2^i$ to the intermediate sum if $m[i]=1$. We can do that by shifting the binary representation of $A$ left $i$ times and adding the result to the current scratchpad. We see that none of the bits up to $i-1$ in the scratchpad are affected by this operation. We can therefore discard (and use for the comparison in our simple algorithm above) the rightmost bit of the scratchpad in every step.
The claim now follows from the observation that the necessary size of the scratchpad is bounded by $B+A+1$. \qed \end{proof}
\section{Proof of Theorem \ref{thm:form}}
\label{sec:form_proof}
We show that it is safe to consider only witnesses in a reduced form, and derive bounds on the length of certain subpaths.
For this, we introduce path rewriting rules that exchange occurrences of some loops by others. We then show (in \cref{lem:witness_preservation}) that these rules preserve witnesses and (\cref{lem:rules_wqo}) cannot be applied indefinitely. For \emph{reduced} witnesses, those to which no rules are applicable, we derive (\cref{lem:bounds}) bounds on the multiplicities of loops that are less productive than others, which will enable us to prove \cref{thm:form}.
For the rest of this section let $V$ and $E$ be the sets of nodes and edges in the product of $\NN{A}$ and $\NN{B}$.
We start with an easy observation:
Because no loop $L$ is longer than $|V|$, we conclude that $(\effect{\NN{A}}(L),\effect{\NN{B}}(L))\in \{-V\ldots V\}\times \{-V\ldots V\},$ so there are $F_0 := (2\cdot|V|+1)^2$ different values the pair $\effect{\NN{A}}(L),\effect{\NN{B}}(L)$ can have.
Moreover, if a witness exists, then also one that does not contain different loops with the same effects: if
$\pi_0L_0\pi_1L_1\pi_2$ is a witness where $|\pi_1|>0$ and $L_0,L_1$ are two loops with $\effect{}(L_0)=\effect{}(L_1)$, then either some prefix of $\pi_0L_0^2\pi_1\pi_2$ (if $\effect{A}(L_0)\ge 0$)
or some prefix of $\pi_0\pi_1L_1^2\pi_2$ (if $\effect{A}(L_0)<0$) must also be a witness by \cref{lem:witness-monotonicity}. We can therefore consider w.l.o.g.~only \emph{sane} paths, which are of the form
\begin{equation}\label{eq:form:sane}
\pi=\pi_0L_0^{l_0}\pi_{1}L_{1}^{l_1}\dots\pi_rL_r^{l_r}\pi_{r+1}
\end{equation} where $r\le F_0$, all $\pi_i$ are acyclic and all loops have pairwise different effects.
\input{definitions/rules}
\begin{example} Consider \cref{ex:witness} again: The substitution suggested there is an application of the rule {\textit{\textbf{UUL}}}\ to the path $\pi=(t_0t_1t_2)(t_3t_4)^9t_5(t_6)^{20}$, where $L_0=(t_0t_1t_2)$, $L_1=(t_3t_4)$ and $x=y=8$. The result is a reduced witness for $(p0,p'10)$ of length $50$. Shorter reduced witnesses for $(p0,p'10)$ exist, for example $(t_0t_1t_2)^6t_5t_6^{16}$, but because of their different loop structure, these cannot be obtained from $\pi$ by applying rewriting rules, as these do not change the structure, i.e., which loops occur and in which order, of a path. This means that our rules do not necessarily preserve minimality of witnesses. \end{example}
In the next two \cref{lem:witness_preservation,lem:rules_wqo}, we show that the rewriting rules preserve witnesses
and that continuous rule application must eventually terminate.
\begin{restatable}[]{lemma}{lemwitnesspreservation} \label{lem:witness_preservation}
If $\pi$ is a sane witness for $(pm,p'm')$
and $\rho$ is the result of applying one of the rules
to $\pi$, then $\rho$ is also a sane witness for $(pm,p'm')$. \end{restatable}
\begin{proof} \input{proofs/lem_witness_preservation} \qed \end{proof}
\begin{restatable}[]{lemma}{lemruleswqo} \label{lem:rules_wqo}
Any sequence of successive applications of rules to a given
path $\pi$ must eventually terminate. \end{restatable}
\input{proofs/lem_rules_wqo}
\Cref{lem:witness_preservation,lem:rules_wqo} allow us to focus on witnesses that are \emph{reduced}, i.e., which are sane and to which none of the rewriting rules is applicable.
We can now derive bounds on the multiplicities of loops in reduced paths.
\begin{restatable}[]{lemma}{lembounds}
\label{lem:bounds}
Let
$\pi=\pi_0L_0^{l_0}\pi_1L_1^{l_1}\pi_2$
be a reduced path
where
$L_0,L_1$ are loops occurring with multiplicities
$l_0>0$ and $l_1>0$.
\begin{enumerate}
\item\label{lem:bounds:PL}
If $Type(L_0)=Type(L_1)=\TypeUU$ and $S(L_0)\ge S(L_1)$
then $l_1\le |V|$
\item\label{lem:bounds:PR}
If $Type(L_0)=Type(L_1)=\TypeUU$ and $S(L_0) < S(L_1)$
then $l_0\le |\pi_1| + 2|V|$
\item\label{lem:bounds:NL}
If $Type(L_0)=Type(L_1)=\TypeDD$ and $S(L_0)<S(L_1)$
then $l_1< |V|^2 + 2|\pi_1|$
\item\label{lem:bounds:NR}
If $Type(L_0)=Type(L_1)=\TypeDD$ and $S(L_0)\ge S(L_1)$
then $l_0< |V|$
\item\label{lem:bounds:PN}
If $Type(L_0)=\TypeUU$, $Type(L_1)=\TypeDD$ and $S(L_0) \le S(L_1)$
then $l_0\le |\pi_1|+|V|$ or $l_1\le |V|$.
\end{enumerate}
\end{restatable} \input{proofs/lem_bounds}
Finally, we are ready to prove \cref{thm:form}. \thmform* \begin{proof} \input{proofs/form} \end{proof}
a process $pm$ of $\NN{N}$, we can speak about macrostates of $\NN{N}$. Observe that if there is a word $w\in\mathit{Act}^*$ which is not a traces of $pm$ then it has to start from $\sharp$ letter. After reading the first letter $\sharp$, the macrostate of the $\NN{N}$ corresponds to the initial configuration of the \MCOL{ICM}. The main invariant that justifies the contruction is as follows. Let $f$ and $g$ be corresponding configuration of \MCOL{ICM}\ and a macrostate of $\NN{N},$ respectively; \begin{enumerate} \item for any valid move of \MCOL{ICM}\ from $f$ to $f'$ there is a letter $a$ such that it forces a move from $g$ to $g'$ and $f'$ corresponds to $g'$. \item for any move via letter $a$ from $g$ to $g'$ such that $g'$ does not contain $U$ the universal state there is a move of \MCOL{ICM}\ from $f$ to $f'$ such that $f'$ corresponds to $g$. \end{enumerate} Both properties can be easily proven by analysis case by case possible moves and responses for them.
Second important fact is that any word $w$ which does not belong to traces of $\NN{N}$ must have three properties: \begin{enumerate} \item any prefix of $w$ can not lead to a macrostate that contains $U$ the universal state, \item $w$ must contain letter $\$$. \item if $w$ is of a form $\sharp v \$ v'$ and word $\sharp v \$ $ does not lead to a macrostate that contains $U$ the universal state then $w$ does not belong to traces of $\NN{N}$. \end{enumerate}
Combining these five facts it is easy to see that any accepting run of \MCOL{ICM}\ can be translated into a word that do not belong to traces of $\NN{N}$ and vice-versa.
Namely, for any accepting run $r$ of \MCOL{ICM}\ there is a word $w$ which is not a trace of $\NN{N}.$ The word $w$ starts form $\sharp$ then there is a word that corresponds to $r$ and finally symbol $\$$ twice. Observe that rules according to which $\NN{N}$ is build guarantee that after first $\$$ the macrostate equals $\{\bot\cdots \bot \}$, so second $\$$ goes outside of set of traces.
\input{proofs/lem_witness_preservation}
\lemruleswqo* \input{proofs/lem_rules_wqo}
\lembounds* \input{proofs/lem_bounds}
\thmform* \begin{proof} \input{proofs/form} \end{proof}
\end{document} | arXiv |
\begin{document}
\title{The Bose-Chowla argument for Sidon sets}
\begin{abstract} Let $h \geq 2$ and let ${ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\ldots, c_h$, consider the linear form $\varphi = c_1 x_1 + c_2x_2 + \cdots + c_h x_h$. The \emph{representation function} $R_{ \mathcal{A},\varphi}(n)$
counts the number of $h$-tuples $(a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h$ such that $\varphi(a_1,\ldots, a_h) = n$. The $h$-tuple $\mathcal{A}$ is a \emph{$\varphi$-Sidon system of multiplicity $g$} if $R_{\mathcal A,\varphi}(n) \leq g$ for all $n \in \ensuremath{\mathbf Z}$. For every positive integer $g$, let $F_{\varphi,g}(n)$ denote the largest integer $q$ such that there exists a $\varphi$-Sidon system $\mathcal {A} = (A_1,\ldots, A_h)$ of multiplicity $g$ with \[
A_i \subseteq [1,n] \qqand |A_i| = q \] for all $i =1,\ldots, h$. It is proved that, for all linear forms $\varphi$, \[ \limsup_{n\rightarrow \infty} \frac{F_{\varphi,g}(n)}{n^{1/h}} < \infty \] and, for linear forms $\varphi$ whose coefficients $c_i$ satisfy a certain divisibility condition, \[ \liminf_{n\rightarrow\infty} \frac{F_{\varphi,h!}(n)}{n^{1/h}} \geq 1. \] \end{abstract}
\section{Classical Sidon sets} Let $A$ be a subset of an additive abelian group or semigroup $\Lambda$. For every positive integer $h$, let $A^h$ be the set of all $h$-tuples of elements of $A$. The symmetric group $S_h$ acts on the set $A^h$ by permutation of coordinates: For all $\sigma \in S_h$ and $(a_1,\ldots, a_h) \in A^h$, we define \begin{equation} \label{BC-Sidon:action} \sigma \left( a_1,\ldots, a_h \right) = \left( a_{\sigma(1)}, \ldots, a_{\sigma(h)} \right). \end{equation} The orbits of this action define an equivalence relation $\sim$ on $A^h$: For $(a_1,\ldots, a_h) \in A^h$ and $(a'_1,\ldots, a'_h) \in A^h $, we have \[ (a_1,\ldots, a_h) \sim (a'_1,\ldots, a'_h) \] if and only if there is a permutation $\sigma \in S_h$ such that \[ a'_i = a_{\sigma(i)} \qquad \text{for all $i \in \{1,\ldots, h\}$.} \] Let $[ a_1,\ldots, a_h ]$ denote the orbit of the $h$-tuple $(a_1,\ldots, a_h)$ and let \[ A^h/S_h = \left\{ [a_1,\ldots, a_h] : (a_1,\ldots, a_h) \in A^h \right\}. \] be the set of equivalence classes of $A^h$. The number of $h$-tuples in the orbit $[a_1,\ldots, a_h ]$ is at most $h!$ and is equal to $h!$ if and only if the coordinates of the $h$-tuple are distinct.
Consider the linear form \[ \varphi = \varphi (x_1, \ldots, x_h) = x_1 + \cdots + x_h. \] If $(a_1,\ldots, a_h) \sim (a'_1,\ldots, a'_h)$, then for some permutation $\sigma \in S_h$ we have \[ \varphi(a_1,\ldots, a_h) = a_1+\cdots + a_h = a_{\sigma(1)} + \cdots + a_{\sigma(h)} = \varphi(a'_1,\ldots, a'_h). \] Thus, the function $[\varphi]$ defined by \[ [\varphi]( [a_1,\ldots, a_h] ) = \varphi(a_1,\ldots, a_h) = a_1+\cdots + a_h \]
is a well-defined function on the orbit space $A^h /S_h$.
The \emph{$h$-fold sumset} of $A$ is the set $hA$ of all sums of $h$ not necessarily distinct elements of $A$: \begin{align*} hA & = \left\{ a_1+\cdots + a_h: (a_1,\ldots, a_h) \in A^h \right\} \\ & = \left\{ \varphi(a_1,\ldots, a_h): (a_1,\ldots, a_h) \in A^h \right\} \\ & = \left\{ [\varphi] ( [a_1,\ldots, a_h] ): [a_1,\ldots, a_h] \in A^h/S_h \right\} \end{align*}
We define two representation functions for the $h$-fold sumset $hA$: For $w \in \Lambda$, \begin{align*} r_{A,h}(w) & = \card \left\{ [a_1,\ldots, a_h] \in A^h/S_h : a_1 + \cdots + a_h = w \right\} \end{align*} and \begin{align*} R_{A,h}(w) & = \card\left\{ (a_1,\ldots, a_h) \in A^h: a_1 + \cdots + a_h = w \right\}. \end{align*}
Because the coordinates of the $h$-tuple $(a_1,\ldots, a_h)$ are not necessarily distinct, we have \[ R_{A,h}(w) \leq h! \ r_{A,h}(w) \] for all $w \in \Lambda$.
The set $A$ is a \emph{classical Sidon set of order $h$} or a \emph{$B_h$-set} if $r_{A,h}(w) \leq 1$ for all $w \in \Lambda$. The set $A$ is a \emph{classical Sidon set of order $h$ and multiplicity $g$ }
or, simply, a \emph{$B_h[g]$-set} if $r_{A,h}(w) \leq g$ for all $w \in \Lambda$.
For every integer $m \geq 2$ we also define the \emph{modular representation function} \[ r^{(m)}_{A,h}(w) = \card \left\{ [a_1,\ldots, a_h] \in A^h/S_h : a_1 + \cdots + a_h \equiv w \pmod{m} \right\}. \] The set $A$ is a \emph{classical Sidon set of order $h$ modulo $m$} if $r^{(m)}_{A,h}(w) \leq 1$ for all $w \in \Lambda$.
If $A$ is a classical Sidon set of order $h$ modulo $m$ for some $m \geq 2$, then
$A$ is a classical Sidon set of order $h$.
Halberstam and Roth~\cite{halb-roth66} and O'Bryant~\cite{obry04} are excellent surveys of results on Sidon sets obtained before 2004.
\section{The Bose-Chowla argument} Consider classical Sidon sets for the group \ensuremath{\mathbf Z}\ of integers.
Let $F_h(n)$ denote the largest Sidon set of order $h$ contained in the set of consecutive integers $\{1,2,\ldots, n\}$. A simple counting argument shows that $F_h(n) \ll n^{1/h}$ and so \[ \limsup_{n\rightarrow\infty} \frac{F_h(n)}{n^{1/h}} < \infty. \] Bose and Chowla~\cite{bose-chow62} proved that \[ \liminf_{n\rightarrow\infty} \frac{F_h(n)}{n^{1/h}} > 0. \] Their main result is the following.
\begin{theorem}[Bose-Chowla] \label{BC-Sidon:theorem:BC-1} Let $h \geq 2$ and let $q$ be a prime power. Let $\ensuremath{{\mathbf F}_q} = \{ \lambda_1,\ldots, \lambda_{q}\}$ be a finite field with $q$ elements and let $\ensuremath{\mathbf F }_{q^h}$ be an extension field of \ensuremath{{\mathbf F}_q}\ of degree $h$. Let $\theta$ be a generator of the cyclic group $\ensuremath{\mathbf F }_{q^h}^{\times}$. For all $\lambda_j \in \ensuremath{{\mathbf F}_q}$, there is a unique integer $a_j \in \{1,2,\ldots, q^h - 2\}$ such that \[ \theta^{a_j} = \theta - \lambda_j. \] The set \[ A = \left\{ a_j : j \in \{1,\ldots, q\} \right\} \subseteq \{1,2,\ldots, q^h - 2\} \] is a classical Sidon set of order $h$ modulo $q^h -1$ and cardinality $q$. \end{theorem}
\begin{proof} The element $\theta$ generates the field extention $\ensuremath{ \mathbf F}_{q^h}/\ensuremath{{\mathbf F}_q}$, and so the minimal polynomial of $\theta$ has degree $h$. Because $\theta$ has order $^{q^h-1}$ in the cyclic group $\ensuremath{ \mathbf F}_{q^h}^{\times}$, for all integers $u$ and $v$ we have $\theta^u = \theta^v$ if and only if $u\equiv v \pmod{q^h-1}$.
Note that $\theta \notin \ensuremath{{\mathbf F}_q}$ because $h \geq 2$ and $\theta$ generate $\ensuremath{\mathbf F }_{q^h}^{\times}$. Thus, $\theta - \lambda_j \neq 0$ for all $\lambda_j \in \ensuremath{{\mathbf F}_q}$.
It follows that for all $j \in \{1,\ldots, q\}$ there is a unique integer $a_j \in \{0,1,2,\ldots, q^h - 2\}$ such that \[ \theta^{a_j} = \theta - \lambda_j. \] If $a_j=0$, then $1 = \theta^0= \theta - \lambda_j$ and $\theta = \lambda_j +1 \in \ensuremath{{\mathbf F}_q}$, which is absurd. Therefore, $a_j \in \{1,2,\ldots, q^h - 2\}$.
We have $a_j = a_k$
if and only if $\theta-\lambda_j = \theta^{a_j} = \theta^{a_k} = \theta-\lambda_k$ if and only if $j = k$. It follows that the set \[ A = \left\{ a_j : j \in \{1,\ldots, q \} \right\} \] has cardinality $q$.
Let $(a_1,\ldots, a_h)$ and $(a'_1,\ldots, a'_h)$ be $h$-tuples of elements of $A$ such that \[ a_1 \leq a_2 \leq \cdots \leq a_h \qqand a'_1 \leq a'_2 \leq \cdots \leq a'_h. \] There exist unique elements $\lambda_j \in \ensuremath{{\mathbf F}_q}$ and $\lambda'_j \in \ensuremath{{\mathbf F}_q}$ such that \[ \theta^{a_j} = \theta - \lambda_j \qqand \theta^{a'_j} = \theta - \lambda'_j \] for all $j \in \{1,2,\ldots, h\}$. If \[ a_1 + a_2 + \cdots + a_h \equiv a'_1 + a'_2 + \cdots + a'_h \pmod{q^h-1} \] then \begin{align*} \prod_{j =1}^h (\theta - \lambda_j) & = \prod_{j =1}^h \theta^{a_j} = \theta^{a_1 + a_2 + \cdots + a_h}
= \theta^{a'_1 + a'_2 + \cdots + a'_h} = \prod_{j=1}^h \theta^{a'_j} \\
& = \prod_{j =1}^h (\theta - \lambda'_j). \end{align*}
The polynomial \[ f(t) = \prod_{j =1}^h (t - \lambda_j) - \prod_{j =1}^h (t - \lambda'_j) \] is either the zero polynomial or a nonzero polynomial of degree at most $h-1$ with coefficients in \ensuremath{{\mathbf F}_q}. Moreover, \[ f(\theta) = 0. \] The minimal polynomial of $\theta$ has degree $h$, and so $\theta$ is not a root of a nonzero polynomial of degree less than $h$. Therefore, $f(t)$ must be the zero polynomial, and so the polynomials $\prod_{j =1}^h (t - \lambda_j)$ and $\prod_{j =1}^h (t - \lambda'_j)$ have the same roots with the same multiplicities. Thus, $(\lambda_1,\ldots, \lambda_h)$ is a permutation of $(\lambda'_1,\ldots, \lambda'_h)$, and so $(a_1,\ldots, a_h) = (a'_1,\ldots, a'_h)$. Equivalently, if $(a_1,\ldots, a_h) \in A^h$ and $(a'_1,\ldots, a'_h) \in A^h$ and $(a_1,\ldots, a_h) \neq (a'_1,\ldots, a'_h)$, then \[ a_1 + a_2 + \cdots + a_h \not\equiv a'_1 + a'_2 + \cdots + a'_h \pmod{q^h -1} \] and so $A$ is a classical Sidon set of order $h$ modulo $q^h-1$. This completes the proof. \end{proof}
\begin{corollary} \label{BC-Sidon:corollary:BC-1} For every prime power $q$ and every integer $h \geq 2$, \[ F_h(q^h -2 ) \geq q. \] \end{corollary}
\begin{proof} A classical Sidon set of order $h$ modulo $q^h-1$ is a classical Sidon set of order $h$. \end{proof}
\begin{theorem}[Bose-Chowla] \label{BC-Sidon:theorem:BC-2} For every integer $h \geq 2$, \[ \liminf_{n\rightarrow \infty} \frac{F_h(n )}{n^{1/h}} \geq 1. \] \end{theorem}
\begin{proof} It has been known since Hoheisel~\cite{hohe30} (and, more recently, Heath-Brown~\cite{heat88b}) that there is a real number $\alpha$ with $0 < \alpha < 1$ such that if $p$ and $p'$ are consecutive primes, then $p' -p < p^{\alpha}$.
For every integer $n \geq 2^h$, let $p$ be the largest prime such that $p \leq n^{1/h}$ and let $p'$ be the smallest prime such that $p' > n^{1/h}$. The primes $p$ and $p'$ are consecutive, and so, by Hoheisel, \[ p \leq n^{1/h} < p' \leq p+p^{\alpha} \leq p+ n^{\alpha/h} \] The function $F_h(n)$ is increasing. Applying Corollary~\ref{BC-Sidon:corollary:BC-1} with $q = p$, we obtain \[ F_h(n) \geq F_h(p^h) \geq F_h(p^h-2) \geq p \geq n^{1/h} - n^{\alpha /h}. \] Therefore, \[ \liminf_{n\rightarrow \infty} \frac{F_h(n )}{n^{1/h}} \geq \liminf_{n\rightarrow \infty} \frac{ n^{1/h} - n^{\alpha /h}}{n^{1/h}} = \liminf_{n\rightarrow \infty} \left( 1 - \frac{ 1}{ n^{(1-\alpha) /h}} \right) = 1. \] This completes the proof. \end{proof}
\section{Sidon systems for linear forms}
Fix an integer $h \geq 2$. Let $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers and let \[ A_1 \times \cdots \times A_h = \left\{ (a_1,\ldots, a_h) : a_i \in A_i \text{ for } i = 1,\ldots, h \right\}. \] For nonzero integers $c_1,\ldots, c_h$, we consider the linear form \[ \varphi = c_1 x_1 + c_2x_2 + \cdots + c_h x_h \] and the set \[ \varphi(\ensuremath{ \mathcal A})
= \left\{ \varphi(a_1,\ldots, a_h) : (a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h \right\}. \]
For every integer $n$, the \emph{representation function} $R_{\ensuremath{ \mathcal A},\varphi}(n)$
counts the number of $h$-tuples $(a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h$ such that $\varphi(a_1,\ldots, a_h) = n$.
Let $W$ be a finite or infinite set of integers. The $h$-tuple \ensuremath{ \mathcal A}\ is a \emph{$\varphi$-Sidon system for $W$} if $R_{\ensuremath{ \mathcal A},\varphi}(w) \leq 1$ for all $w \in W$. If \ensuremath{ \mathcal A}\ is a $\varphi$-Sidon system for $W$, then the statements \[ (a_1,\ldots, a_h) \in A_1\times \cdots \times A_h, \qquad (a'_1,\ldots, a'_h) \in A_1\times \cdots \times A_h \]
and \[
\varphi(a_1,\ldots, a_h) = \varphi(a'_1,\ldots, a'_h) = w \in W \] imply \[ (a_1,\ldots, a_h) = (a'_1,\ldots, a'_h). \] More generally, the $h$-tuple \ensuremath{ \mathcal A}\ is a \emph{$\varphi$-Sidon system of multiplicity $g$ for $W$} if $R_{\ensuremath{ \mathcal A},\varphi}(w) \leq g$ for all $w \in W$.
The $h$-tuple \ensuremath{ \mathcal A}\ is a \emph{$\varphi$-Sidon system} if \ensuremath{ \mathcal A}\ is a \emph{$\varphi$-Sidon system} for \ensuremath{\mathbf Z}, that is, if $R_{\ensuremath{ \mathcal A},\varphi}(n) \leq 1$ for all integers $n$. The $h$-tuple \ensuremath{ \mathcal A}\ is a \emph{$\varphi$-Sidon system of multiplicity $g$} if $R_{\ensuremath{ \mathcal A},\varphi}(n) \leq g$ for all integers $n$.
Let $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers. We have \[
\prod_{i=1}^h |A_i| = \sum_{w\in \varphi(\ensuremath{ \mathcal A})} R_{\ensuremath{ \mathcal A},\varphi}(w) \] If \ensuremath{ \mathcal A}\ is a $\varphi$-Sidon system, then \[
\sum_{w\in \varphi(\ensuremath{ \mathcal A})} R_{\ensuremath{ \mathcal A},\varphi}(w) = |\varphi(\ensuremath{ \mathcal A})|. \] If \ensuremath{ \mathcal A}\ is a $\varphi$-Sidon system of mutiplicity $g$, then \[
\sum_{w\in \varphi(\ensuremath{ \mathcal A})} R_{\ensuremath{ \mathcal A},\varphi}(w) \leq g|\varphi(\ensuremath{ \mathcal A})|. \]
For example, let $d_1,\ldots, d_h$ be integers such that $d_i \geq 2$ for all $i = 1,\ldots, h$ and let $d = d_1d_2\cdots d_h$. Let $W = [0, d-1]$. Consider the finite sets \[ A_i = [0, d_i-1] \qquad \text{for $i = 1,\ldots h$ } \] and the linear form \[ \varphi = x_1 + d_1x_2 + d_1d_2 x_3 + \cdots + d_1d_2\cdots d_{h-1} x_h. \] We have $R_{\ensuremath{ \mathcal A},\varphi}(w) = 1$ for all $w \in W$ and $R_{\ensuremath{ \mathcal A},\varphi}(w) = 0$ for all integers $w \not\in W$. The $h$-tuple $\ensuremath{ \mathcal A} = (A_1, \ldots, A_h)$ is a Sidon system.
\section{The size of $\varphi$-Sidon systems}
Let $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers and let $(t_1,\ldots, t_h)$ be an $h$-tuple of integers. For $i \in \{1,\ldots, h\}$, the translate of the set $A_i$ by the integer $t_i$ is the set \[ A_i + t_i = \{a_i+t: a_i \in A_i\} \] and the translate of \ensuremath{ \mathcal A}\ by $(t_1,\ldots, t_h)$ is the $h$-tuple \[ \ensuremath{ \mathcal A} + \ensuremath{\mathbf t} = \ensuremath{ \mathcal A}+ (t_1,\ldots, t_h) = (A_1+t_1,\ldots, A_h + t_h). \]
Let $\varphi = \sum_{i=1}^h c_ix_i$, where the coefficients $c_i$ are nonzero integers and let \[ t^* = \varphi(t_1,\ldots, t_h). \]
For all $(a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h$, we have \begin{align*} \varphi(a_1+t_1,\ldots, a_h+t_h) & = \sum_{i=1}^h c_i (a_i+t_i) = \sum_{i=1}^h c_i a_i + \sum_{i=1}^h c_i t_i \\ & = \varphi(a_1,\ldots, a_h) + \varphi(t_1,\ldots, t_h) \\
& = \varphi(a_1,\ldots, a_h) + t^*. \end{align*} Thus, $\varphi(a_1,\ldots, a_h) = b$ if and only if $ \varphi(a_1+t_1,\ldots, a_h+t_h) = b + t^*$, and so \[ R_{\ensuremath{ \mathcal A},\varphi}(b) = R_{\ensuremath{ \mathcal A}+ \ensuremath{\mathbf t},\varphi}(b+t^*). \] It follows that for $h$-tuples of nonempty finite sets of integers or $h$-tuples of nonempty sets of integers that are bounded below, it suffices to consider only $h$-tuples of sets of nonnegative integers $(A_1,\ldots, A_h)$ with $0 \in A_i$ for all $i \in \{1,\ldots, h\}$.
Let $\varphi = c_1 x_1 + \cdots + c_h x_h$, where the coefficients $c_i$ are nonzero integers, and let \[
C = \sum_{i=1}^h |c_i|. \] Let $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers
and let $n$ be a positive integer such that $|a_i| \leq n$ for all $a_i \in \bigcup_{i=1}^h A_i$. For all $(a_1, \ldots, a_h) \in A_1 \times \cdots \times A_h$ we have \[
\left| \varphi(a_1, \ldots, a_h) \right| = \left| \sum_{i=1}^h c_ia_i\right|
\leq \sum_{i=1}^h \left|c_i\right| |a_i| \leq C n \] and so \[ \varphi(A_1,\ldots, A_h) \subseteq [ -C n, Cn ]. \] Therefore, \[
\left| \varphi(A_1,\ldots, A_h) \right| \leq 2C n+1. \] If $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ is a $\varphi$-Sidon system of multiplicity $g$, then \[
g\left| \varphi(A_1,\ldots, A_h) \right| \geq \prod_{i=1}^h |A_i|. \]
If $|A_i| = q$ for all $i \in \{1,\ldots, h\}$, then \begin{equation} \label{BC-Sidon:qCn}
q^h = \prod_{i=1}^h |A_i| \leq g(2Cn+1). \end{equation}
For every positive integer $g$, let $F_{\varphi,g}(n)$ denote the largest integer $q$ such that there exists a $\varphi$-Sidon system $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ of multiplicity $g$ such that \[
A_i \subseteq [1,n] \qqand |A_i| = q \] for all $i =1,\ldots, h$. Inequality~\eqref{BC-Sidon:qCn} implies that \begin{equation} \label{BC-Sidon:FgCn} \limsup_{n\rightarrow \infty} \frac{F_{\varphi,g}(n)}{n^{1/h}} \leq \left(2gC\right)^{1/h} < \infty. \end{equation}
\section{Constructing large $\varphi$-Sidon systems}
\begin{lemma} \label{BC-Sidon:lemma:uh-1} Let $p$ be a prime number and let $h \geq 2$. There exists an integer $u_p$ such that \begin{equation} \label{BC-Sidon:uh-1} \gcd(u_p,p) = \gcd(u_p^h -1,p) = 1 \end{equation} if and only if $p-1$ does not divide $h$. \end{lemma}
\begin{proof} Because $p$ is prime we have $\gcd(u,p) = 1 \text{ or } p$ for all integers $u$. If $p-1$ divides $h$, then $h = (p-1)t$ for some integer $t$. For all integers $u \not\equiv 0 \pmod{p}$ we have $u^{p-1} \equiv 1 \pmod{p}$ and so \[ u^h \equiv \left(u^{p-1}\right)^t \equiv 1 \pmod{p} \] and \[ \gcd(u^h -1,p) = p. \] Thus, if $p-1$ divides $h$, then no integer satisfies~\eqref{BC-Sidon:uh-1}.
Suppose that $p-1$ does not divide $h$. For every primitive root $u_p$ modulo $p$ we have
$u_p \not\equiv 0 \pmod{p}$ and $u_p^h \not\equiv 1 \pmod{p}$ and so $u_p$ satisfies~\eqref{BC-Sidon:uh-1}. This completes the proof. \end{proof}
\begin{lemma} \label{BC-Sidon:lemma:qh-1} Let $h \geq 2$ and let $\ensuremath{ \mathcal P}(h)$ be the finite set of primes $p$ such that $p-1$ divides $h$. Let $\varphi = c_1x_1+\cdots + c_hx_h$ be a linear form whose coefficients $c_1,\ldots, c_h$ are nonzero integers such that $\gcd(c_i, p)=1$ for all $i \in \{1,\ldots, h\}$ and $p \in \ensuremath{ \mathcal P}(h)$. There exist positive integers $u$ and $Q$ with $\gcd(u,Q) = 1$ such that every prime $q$ in the infinite arithmetic progression $u \pmod{Q}$ satisfies \[ \gcd(q^h -1,c_i) = 1 \] for all $i \in \{1,\ldots, h\}$. \end{lemma}
\begin{proof} Let $\ensuremath{ \mathcal P}(c_1,\ldots, c_h)$ be the finite set of primes $p$ such that $p$ divides $c_i$ for some $i \in \{1,\ldots, h\}$ and let \[ Q = \prod_{p \in \ensuremath{ \mathcal P}(c_1,\ldots, c_h)} p. \] If $p \in \ensuremath{ \mathcal P}(c_1,\ldots, c_h)$, then $p \notin \ensuremath{ \mathcal P}(h)$ and so $p-1$ does not divide $h$, and so, by Lemma~\ref{BC-Sidon:lemma:uh-1}, there is an integer $u_p$ that satisfies~\eqref{BC-Sidon:uh-1}. By the Chinese remainder theorem, there is an integer $u$ such that \[ u \equiv u_p \pmod{p} \] for all $p \in \ensuremath{ \mathcal P}(c_1,\ldots, c_h)$. We have $\gcd(u,p) = \gcd(u_p,p) = 1$ for all $p \in \ensuremath{ \mathcal P}(c_1,\ldots, c_h)$ and so $\gcd(u,Q)=1$. By Dirichlet's theorem, there are infinitely many primes $q$ such that \[ q \equiv u \pmod{Q}. \] For all $p \in \ensuremath{ \mathcal P}(c_1,\ldots, c_h)$ we have \[ \gcd(q^h -1,p) = \gcd(u^h -1,p) = \gcd(u_p^h -1,p) = 1 \] and so \[ \gcd(q^h -1,c_i) = 1 \] for all $i \in \{1,\ldots, h\}$. This completes the proof. \end{proof}
\begin{theorem} \label{BC-Sidon:theorem:BCN-1} Consider the linear form \[ \varphi = c_1 x_1 + \cdots + c_h x_h \] where $h \geq 2$ and $c_1,\ldots, c_h$ are nonzero integers. Let $q$ be a prime such that \[ \gcd(q^h-1,c_i) = 1 \] for all $i \in \{1,\ldots, h\}$. There is an $h$-tuple $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ of sets of integers with \[
A_i \subseteq \{1,\ldots, q^h -2\} \qqand |A_i| = q \] for all $i \in \{1,\ldots, h\}$ such that $\ensuremath{ \mathcal A}$ is a $\varphi$-Sidon system of multiplicity at most $h!$. Equivalently, \begin{equation} \label{BC-Sidon:BCN} F_{\varphi,h!} (q^h -2) \geq q. \end{equation} \end{theorem}
\begin{proof} Let $\ensuremath{{\mathbf F}_q}$ be the finite field with $q$ elements and let $\ensuremath{\mathbf F }_{q^h}$ be an extension field of \ensuremath{{\mathbf F}_q}\ of degree $h$. The multiplicative group $\ensuremath{\mathbf F }_{q^h}^{\times}$ is cyclic of order $q^h - 1$. Let $\theta$ be a generator of $\ensuremath{\mathbf F }_{q^h}^{\times}$. For all $i \in \{1,\ldots, h\}$ we have $\gcd(q^h-1,c_i) = 1$ and so $\theta^{c_i}$ is also a generator of the cyclic group $\ensuremath{\mathbf F }_{q^h}^{\times}$.
Let $\ensuremath{{\mathbf F}_q} = \{\lambda_1,\ldots, \lambda_q \}$. The inequality $h \geq 2$ implies that $\theta \notin \ensuremath{\mathbf F }_q$ and so, for all $j \in \{1,\ldots, q\}$, we have $\theta \neq \lambda_j$. Equivalently, $\theta - \lambda_j \in \ensuremath{\mathbf F }_{q^h}^{\times}$. For all $i \in \{1,\ldots, h\}$, the element $\theta^{c_i}$ generates $\ensuremath{\mathbf F }_{q^h}^{\times}$. It follows that there is a unique integer \[ a_{i,j} \in \{0,1,\ldots, q^h -2\} \] such that \[ \theta^{c_i a_{i,j}} = \theta - \lambda_j. \] If $a_{i,j} = 0$, then $\theta = \lambda_j + 1 \in \ensuremath{{\mathbf F}_q}$, which is absurd. Therefore, \[ a_{i,j} \in \{1,\ldots, q^h -2\} \] for all $i \in \{1,\ldots,h\}$ and $j \in \{1,\ldots, q \}$. Moreover, if $1 \leq j < k \leq q$, then $\lambda_j \neq \lambda_k$ and so $a_{i,j} \neq a_{i,k}$ . The set \[ A_i = \left\{ a_{i,j}: j = 1,\ldots, q \right\} \] is a subset of $\{1,\ldots, q^h -2\}$ of cardinality $q$ for all $i =1,\ldots,h$.
Let $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$. For all $(a_{1,j_1},\ldots, a_{h,j_h}) \in A_1 \times \cdots \times A_h$ we have \begin{align*} \theta^{\varphi \left( a_{1,j_1},\ldots, a_{h,j_h} \right) } & = \theta^{\sum_{i=1}^h c_ia_{i,j_i} }
= \prod_{i=1}^h \theta^{c_ia_{i,j_i} } = \prod_{i=1}^h (\theta - \lambda_{j_i}) = f(\theta) \end{align*} where \[ f(t) = \prod_{i=1}^h (t- \lambda_{j_i}) = t^h - \left(\sum_{i=1}^h \lambda_{j_i} \right) t^{h-1} + \cdots + (-1)^h \prod_{i=1}^h\lambda_{j_i} \] is a monic polynomial of degree $h$ with coefficients in \ensuremath{{\mathbf F}_q}.
Similarly, for $(a_{1,j'_1},\ldots, a_{h,j'_h}) \in A_1 \times \cdots \times A_h$ we have \begin{align*} \theta^{\varphi\left( a_{1,j'_1},\ldots, a_{h,j'_h} \right) }
& = \prod_{i=1}^h (\theta - \lambda_{j'_i}) = g(\theta) \end{align*} where \[ g(t) = \prod_{i=1}^h (t- \lambda_{j'_i}) = t^h - \left(\sum_{i=1}^h \lambda_{j'_i} \right) t^{h-1} + \cdots + (-1)^h \prod_{i=1}^h\lambda_{j'_i} \] is also a monic polynomial of degree $h$ with coefficients in \ensuremath{{\mathbf F}_q}.
The relation $\varphi\left( a_{1,j_1},\ldots, a_{h,j_h} \right) = \varphi\left( a_{1,j'_1},\ldots, a_{h,j'_h} \right)$ implies \[ f(\theta) = \theta^{\varphi \left( a_{1,j_1},\ldots, a_{h,j_h} \right) } = \theta^{\varphi\left( a_{1,j'_1},\ldots, a_{h,j'_h} \right) } = g(\theta) \] and so $\theta$ is a root of the polynomial $f(t)-g(t)$. If $f(t) \neq g(t)$, then $f(t) - g(t)$ is a nonzero polynomial of degree at most $h-1$. This is impossible because the minimal polynomial of $\theta$ has degree $h$. Therefore, \[
\prod_{i=1}^h (t- \lambda_{j_i}) = f(t) = g(t) = \prod_{i=1}^h (t- \lambda_{j'_i})
\] and so $(\lambda_{j'_1},\ldots, \lambda_{j'_h})$ is a permutation of $(\lambda_{j_1},\ldots, \lambda_{j_h})$ and $(j'_1,\ldots, j'_h)$ is a permutation of $(j_1,\ldots, j_h)$. There are at most $h!$ such permutations. It follows that for every integer $w$ there are at most $h!$ elements $(a_1,\ldots, a_h) \in A_1\times \cdots \times A_h$ such that $\varphi(a_1,\ldots, a_h) = w$. Therefore, $R_{\ensuremath{ \mathcal A},\varphi}(w) \leq h!$ and $\ensuremath{ \mathcal A} = (A_1,\ldots, A_h)$ is a $\varphi$-Sidon system of multiplicity at most $h!$. This completes the proof. \end{proof}
\begin{theorem} \label{BC-Sidon:theorem:BCN} Let $h \geq 2$ and let $\ensuremath{ \mathcal P}(h)$ be the finite set of primes $p$ such that $p-1$ divides $h$. Let $\varphi = c_1x_1+\cdots + c_hx_h$ be a linear form whose coefficients $c_1,\ldots, c_h$ are nonzero integers such that for all $i \in \{1,\ldots, h\}$ and $p \in \ensuremath{ \mathcal P}(h)$. Then \[ \liminf_{n\rightarrow\infty} \frac{F_{\varphi,h!}(n)}{n^{1/h}} \geq 1. \] \end{theorem}
\begin{proof} There is an analog of Hoheisel's theorem for sufficiently large primes in arithmetic progressions. Let $u$ and $Q$ be relatively prime positive integers. Baker, Harman, and Pintz~\cite{bake-harm-pint97} proved that there is a real number $\alpha = \alpha(u,Q)$ with $0 < \alpha < 1$ such that if $p$ and $p'$ are sufficiently large consecutive primes in the arithmetic progression $u \pmod{Q}$, then $p' - p < p^{\alpha}$.
By Lemma~\ref{BC-Sidon:lemma:qh-1},there exist positive integers $u$ and $Q$ with $\gcd(u,Q) = 1$ such that every prime $q$ in the infinite arithmetic progression $u \pmod{Q}$ satisfies \[ \gcd(q^h -1,c_i) = 1 \] for all $i \in \{1,\ldots, h\}$.
For every sufficiently large integer $n$, let $p$ be the largest prime such that $p\equiv u \pmod{Q}$ and $p \leq n^{1/h}$. Let $p'$ be the smallest prime such that $p\equiv u \pmod{Q}$ and $p' > n^{1/h}$. Then $p$ and $p'$ are consecutive primes in the arithmetic progression $ u \pmod{Q}$, and so \[ p \leq n^{1/h} < p' \leq p+p^{\alpha} \leq p+ n^{\alpha/h}. \] Applying inequality~\eqref{BC-Sidon:BCN} from Theorem~\ref{BC-Sidon:theorem:BCN-1} with $q = p$, we obtain \[ F_{\varphi,h!}(n) \geq F_{\varphi,h!}(p^h) \geq F_{\varphi,h!}(p^h-2) \geq p \geq n^{1/h} - n^{\alpha /h}. \] Therefore, \[ \liminf_{n\rightarrow \infty} \frac{F_{\varphi,h!}(n )}{n^{1/h}} \geq \liminf_{n\rightarrow \infty} \frac{ n^{1/h} - n^{\alpha /h}}{n^{1/h}} = \liminf_{n\rightarrow \infty} \left( 1 - \frac{ 1}{ n^{(1-\alpha) /h}} \right) = 1. \] This completes the proof. \end{proof}
\section{Open problems} Let $h \geq 2$ and let $\varphi = \sum_{i=1}^h c_i x_i$ be a linear form with nonzero integer coefficients. \begin{enumerate} \item[(i)] Is it true that \[ \liminf_{n\rightarrow\infty} \frac{F_{\varphi,h!}(n)}{n^{1/h}} > 0 \] with no condition on the primes that divide the coefficients $c_i$? \item[(ii)] Is it true that there is a finite set $\ensuremath{ \mathcal P}(h)$ of prime numbers such that if none of the coefficients of $\varphi$ is divisible by a prime in $\ensuremath{ \mathcal P}(h)$, then \[ \liminf_{N\rightarrow \infty} \frac{F_{\varphi}(N)}{N^{1/h}} > 0? \] \item[(iii)] Is it true that \[ \liminf_{N\rightarrow \infty} \frac{F_{\varphi}(N)}{N^{1/h}} > 0 \] with no condition on the primes that divide the coefficients $c_i$? This would be the analog of the Bose-Chowla theorem for classical Sidon sets. \end{enumerate}
\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
\end{document} | arXiv |
\begin{document}
\title{Local Antimagic Coloring of Some Graphs} \begin{abstract}
Given a graph $G =(V,E)$, a bijection $f: E \rightarrow \{1, 2, \dots,|E|\}$ is called a local antimagic labeling of $G$ if the vertex weight $w(u) = \sum_{uv \in E} f(uv)$ is distinct for all adjacent vertices. The vertex weights under the local antimagic labeling of $G$ induce a proper vertex coloring of a graph $G$. The \textit{local antimagic chromatic number} of $G$ denoted by $\chi_{la}(G)$ is the minimum number of weights taken over all such local antimagic labelings of $G$. In this paper, we investigate the local antimagic chromatic numbers of the union of some families of graphs, corona product of graphs, and necklace graph and we construct infinitely many graphs satisfying $\chi_{la}(G) = \chi(G)$.\\
{\setlength{\parindent}{0cm}\textbf{Keywords:} Antimagic Graph, Local Antimagic Graph, Local Antimagic Chromatic Number.}\\
{\setlength{\parindent}{0cm}\textbf{AMS Subject Classification 2021: 05C 78.}}
\end{abstract}
\section{Introduction}
The coloring problems in Graph Theory are one of the oldest, most widely known and unsolved problems in Mathematics. They have been the central research topic for centuries among graph theorists. Recently Arumugam et al.\cite{lac_arumugam}, and Bensmail et al.\cite{lac_bensmail} independently defined the notion of local antimagic labeling of a graph. Arumugam et al.~\cite{lac_arumugam} studied the vertex coloring induced by local antimagic labeling. Throughout this paper, we assume that $G \not \cong K_2$. For graph theoretic terminology and notations, we refer to West~\cite{west}.\\
Hartsfield and Ringel \cite{pearls_in_graph_theory} introduced the concept of antimagic labeling of a graph. Given a graph $G = (V,E)$, let $f : E \rightarrow \{1, 2, \dots, |E|\}$ be a bijection. For each vertex $u \in V$, the weight of $u$ induced by $f$ is $w(u) = \displaystyle\sum_{uv \in E} f(uv)$. If the induced weights under $f$ of any two vertices of $G$ are distinct, then $f$ is called antimagic labeling of $G$, and the graph $G$, which admits such a labeling, is called an antimagic graph.\\
An antimagic labeling $f$ of a graph $G$ is said to be \textit{local antimagic} if weights induced by $f$ of adjacent vertices are distinct. Local antimagic labeling naturally induces a proper vertex coloring of a graph $G$. The \textit {local antimagic chromatic number} $\chi_{la}(G)$ of a graph $G$ is the minimum number of colors used over all colorings of $G$ induced by local antimagic labeling of $G$.\\
In \cite{lac_arumugam}, authors calculated the local antimagic chromatic number of a few families of graphs \textit{viz} path, cycle, wheel, etc. Further, they conjectured that {\it every graph other than $K_2$ is local antimagic}. Haslegrave \cite{haslegrave_proof} proved this conjecture.\\
We will use a magic rectangle and magic rectangle sets to obtain local antimagic labelings of some graphs. A \textit{magic rectangle} $MR(a,b)$ is an array whose entries are $\{1,2, \dots, ab\}$, each appearing once, with all its row sums equal to a constant $\rho = \frac{b(ab+1)}{2}$ and all its column sums equal to a constant $\sigma = \frac{a(ab+1)}{2}$. Froncek \cite{mrs1, mrs2} generalized the idea of magic squares to magic rectangle sets. A \textit{magic rectangle set} $\mathcal{M} = MRS(a, b; c)$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of $\{1, 2, \dots, abc\}$, each appearing once, with all row sums in every rectangle equals to a constant $\rho = \frac{b(abc+1)}{2}$ and all column sums in every rectangle equal to a constant $\sigma = \frac{a(abc+1)}{2}$.\\
In this paper, we investigated the local antimagic chromatic numbers of the union of some families of graphs, the corona product of graphs, necklace graph and we construct infinitely many graphs satisfying $\chi_{la}(G) = \chi(G)$.
\section{Known Results}
The following century-old existence result was given by Harmuth \cite{mr1,mr2}, which gives the necessary and sufficient conditions for the existence of a magic rectangle of a given order.
\begin{theorem}\cite{mr1, mr2} \label{mr} A magic rectangle $MR(a,b)$ exists if and only if $a,b > 1$, $ab>4$, and $a \equiv b\pmod{2}$. \end{theorem} Froncek \cite{mrs1, mrs2} proved the existence of MRS$(a,b;c)$. \begin{theorem} \cite{mrs2} \label{mrs2.3} Let $a, b, c$ be positive integers such that $1 < a \le b$. Then a magic rectangle set MRS$(a, b; c)$ exists if and only if either $a, b, c$ are all odd, or $a$ and $b$ are both even, $c$ is arbitrary, and $(a,b) \ne (2,2)$. \end{theorem} \begin{theorem}\cite{mrs1}\label{mrs1.2} If $b \equiv 0 \pmod{2}$ and $b \ge 4$, then a magic rectangle set MRS$(2, b; c)$ exists for every $c$. \end{theorem} \begin{theorem}\cite{mrs1}\label{mrs1.1} If $a \equiv b \equiv 0\pmod{2}, a \ge 2$ and $b \ge 4$, then a magic rectangle set MRS$(a, b; c)$ exists for every $c$. \end{theorem}
\begin{theorem}\cite{lac_pendent_lau}\label{th:lac_bound} Let $G$ be a graph having $k$ pendants. If $G$ is not $K_2$, then $\chi_{la}(G) \ge k + 1$ and the bound is sharp. \end{theorem} \begin{theorem}\cite{lac_union_baca}\label{baca1} Let $G$ be a $4r$-regular graph, $r \ge 1$. Then for every positive integer $m, \chi_{la}(mG) \le \chi_{la}(G)$. \end{theorem} \begin{theorem}\cite{lac_union_baca}\label{baca2} Let $G$ be a $(4r+2)$-regular graph, $r \ge 0$. Then for every positive integer $m, \chi_{la}(mG) \le \chi_{la}(G)$. \end{theorem} \begin{theorem} \cite{lac_cycle_lau} \begin{align*} \chi_{la}(K_{p,q}) = \begin{cases}
q+1 \quad & \text{if } q>p=1\\
2 \quad & \text{if } q > p \ge 2 \text{ and } p \equiv q\pmod{2}\\
3 \quad & \text{otherwise.}
\end{cases} \end{align*} \end{theorem}
\section{Local Antimagic Labeling of Union of Graphs}
With the knowledge of local antimagic chromatic numbers of various classes of well-known graphs, researchers started calculating local antimagic chromatic numbers for graphs obtained from known graphs \cite{lac_corona_arumugam, lac_join_lau, lac_lau, lac_corona_sugeng}. Handa et. al. \cite{adarsht} started by investing the local antimagic chromatic number of the union of paths, cycles, and complete bipartite graphs.\\
BaÄa et al.~\cite{lac_union_baca} investigated independently the local antimagic chromatic number and upper bounds for the union of paths, the union of cycles, the union of trees and other graphs and their proof techniques are different from the proof techniques given in this paper.\\
The union of two graphs $G_1 = (V_1,\ E_1)$ and $G_2 = (V_2,\ E_2)$ is the graph $G = (V,\ E)$ with vertex set $V = V_1 \cup V_2$ and edge set $E = E_1 \cup E_2$.\\
Note that, if $G_1, G_2, \dots , G_n$ are graphs such that $\chi(G_i)=\chi_{i}$ then $\chi(\bigcup_i G_i)= \mbox{max}\{ \chi_{i}: 1 \leq i \leq n\}$. We have the following observation for the local antimagic chromatic number. \begin{observation} \label{bound} For the graphs $G_1, G_2, \dots , G_n$, $\chi_{la} (G_j) \le \chi_{la}(\bigcup_{1 \le i \le n} G_i)$ for each $j$. \end{observation}
\begin{theorem} The graph $rP_n$ is local antimagic with $3 \le \chi_{la}(rP_n)\leq 2r+2$. \end{theorem}
\begin{proof} Let the vertex set of $rP_n$ be $ V(rP_n) = \{v_i^j : 0\leq j\leq r-1,~~ 0\leq i\leq n-1\}$, where for each $j$, $v_0^j, v_1^j,\dots,v_{n-1}^j$ is a path of length $n$. The lower bound is obvious from Observation \ref{bound}. For upper bound, we consider the following two cases.\\
{\bf Case 1:} $n$ is even.\\
We define edge labeling $ f:E\longrightarrow \{1,2,\dots,nr-r\}$ as follow: $$f(v_i^jv_{i+1}^j) =\begin{cases}
(r-\frac{j}{2})(n-1)- \frac{i}{2} &\mbox{if } i,j\equiv 0\pmod{2}\\
\frac{j}{2}(n-1) +\frac{i+1}{2} & \mbox{if } i\equiv 1\pmod{2}, j\equiv 0\pmod{2}\\
\frac{(j-1)}{2})(n-1)+ \frac{i+1}{2}+ \frac{n-1}{2} &\mbox{if } i\equiv 0\pmod{2}, j\equiv 1\pmod{2}\\
(r-\frac{j-1}{2})(n-1)- \frac{i-1}{2}
- \frac{n}{2} &\mbox{if } i,j\equiv 1\pmod{2}.
\end{cases}$$
Then the induced vertex weights are as follows:
For $i\neq 0 \; \mbox{and}\; n-1$, $$ w(v_i^j) =\begin{cases}
r(n-1) & \mbox{if } i\equiv j\pmod{2} \\ r(n-1)+1 & \mbox{if } i\not\equiv j\pmod{2}. \end{cases}$$
For $i= 0$ $$ w(v_0^j) =\begin{cases} (r-\frac{j}{2})(n-1) & \mbox{if } j\equiv 0\pmod{2}\\
(n-1)(\frac{j-1}{2})+\frac{n}{2} & \mbox{if } j\equiv 1\pmod{2}. \end{cases}$$
For $i=n-1$ $$ w(v_{n-1}^j) =\begin{cases}
\frac{j}{2}(n-1)-\frac{n-2}{2} & \mbox{if } j\equiv 0\pmod{2}\\
(n-1)(\frac{j+1}{2}) & \mbox{if } j\equiv 1\pmod{2}. \end{cases}$$
{\bf Case 2:} $n$ is odd.\\
We define edge labeling $ f:E\longrightarrow \{1,2,\dots,nr-r\}$ as follow: $$f(v_i^jv_{i+1}^j) =\begin{cases}
r(n-1)- \frac{j(n-1)}{2}-\frac{i}{2} &\mbox{if } i\equiv 0\pmod{2}\\
\frac{j(n-1)}{2} +\frac{i+1}{2} & \mbox{if } i\equiv 1\pmod{2}.
\end{cases}$$
The induced vertex weights are as follows:
$i\neq 0 \; \mbox{and}\; n-1$, $$ w(v_i^j) =\begin{cases}
r(n-1) & \mbox{if } i\equiv 0\pmod{2} \\ r(n-1)+1 & \mbox{if } i\equiv 1\pmod{2}. \end{cases}$$
$$ w(v_i^j) =\begin{cases} r(n-1) - \frac{j(n-1)}{2}& \mbox{if } i = 0 \\
(\frac{n-1}{2})(j+1) & \mbox{if } i= n-1. \end{cases}$$
Since we have $2r+2$ distinct vertex weights, we conclude that $\chi_{la}(rP_n)\leq 2r+2$. \end{proof}
Next, we investigate the local antimagic chromatic number for the union of cycles. Let the vertex set of $rC_n$ be $ V(rC_n) = \{v_i^j : 1\leq j\leq r,~~ 0\leq i\leq n-1\}$ where for each $j$, $v_0^j, v_1^j,\dots,v_{n-1}^j$ is a cycle of length $n$.
\begin{lemma}{\label{cycle1}} If $n$ is even then the graph $rC_n$ is local antimagic with $\chi_{la}(rC_n)=3$. \end{lemma} \begin{proof} By Observation \ref{bound}, $\chi_{la}(C_n) = 3 \le \chi_{la}(rC_n)$. So it is sufficient to give a local antimagic labeling which induces exactly $3$ distinct weights. Consider the following edge labeling $f$ as
\[f(v_i^jv_{i+1}^j) = \begin{cases}
\frac{(j-1)n}{2}+ \frac{i}{2} +1 & \mbox{if } i\equiv 0\pmod{2}\\
rn-\frac{(j-1)n}{2}- \frac{i-1}{2} & \mbox{if } i\equiv 1\pmod{2}
\end{cases}\] Then the induced vertex weights are as follows \[ w(v_i^j) = \begin{cases}
rn+1 & \mbox{if } i\equiv 1\pmod{2}, i\neq 0\\
rn+2 & \mbox{if } i\equiv 0\pmod{2}, i\neq 0\\
rn+\frac{4-n}{2} & \mbox{if } i=0
\end{cases}\] Hence $f$ is local antimagic labeling, and it induces $3$ weights as required. \end{proof} The illustration of local antimagic labeling of $2C_6$ is shown in Figure \ref{2C6}.
\begin{figure}
\caption{Local antimagic labeling of $2C_6$.}
\label{2C6}
\end{figure}
We give an upper bound on the local antimagic chromatic number for the union of odd length cycles. \begin{lemma}{\label{cycle2}}
If $n$ is odd then the graph $rC_n$ is local antimagic with $\chi_{la}(rC_n)\leq r+2$. \end{lemma} \begin{proof} We define a local antimagic labeling $f:E\longrightarrow \{1,2,\dots,nr\}$ as $$f(v_i^jv_{i+1}^j) =\begin{cases} \frac{jn}{2}+ \frac{i}{2} +1&\mbox{if } i\equiv 0\pmod{2}, j\equiv 0\pmod{2}\\
(r- \frac{j}{2})n- \frac{i-1}{2} & \mbox{if } i\equiv 1\pmod{2}, j\equiv 0\pmod{2}\\
(r- \frac{(j-1)}{2})n- \frac{i}{2}- \frac{(n-1)}{2} &\mbox{if } i\equiv 0\pmod{2}, j\equiv 1\pmod{2}\\
(\frac{j-1}{2})n+ \frac{i+1}{2} + \frac{n+1}{2} &\mbox{if } i\equiv 1\pmod{2}, j\equiv 1\pmod{2}.
\end{cases}$$
Then the induced vertex weights are as follows:
for $i\neq 0$ $$ w(v_i^j) =\begin{cases} rn+2 & \mbox{if } i\equiv j\pmod{2} \\ rn+1 & \mbox{if } i\not\equiv j\pmod{2}. \end{cases}$$
and for $i= 0$ $$ w(v_i^j) =\begin{cases} nj+\frac{n+3}{2} & \mbox{if } j\equiv 0\pmod{2}.\\
2n(r-\frac{j-1}{2})-\frac{3}{2}(n-1) & \mbox{if } j\equiv 1\pmod{2}. \end{cases}$$
Since we have $r+2$ distinct vertex weights, we conclude that $\chi_{la}(rC_n)\leq r+2$. \end{proof} From the Lemmas \ref{cycle1} and \ref{cycle2}, we have the following theorem. \begin{theorem} For $n \geq 3$ the local antimagic chromatic number $3 \le \chi_{la}(rC_n) \leq r+2$. \end{theorem}
Let $mK_{1,n}$ denotes $m$ copies of a star $K_{1,n}$. Let $u_1,u_2,\dots,u_m$ be the $m$ central vertices of $mK_{1,n}$. Let $v_{i,1},v_{i,2},\dots,v_{i,n}$ be $n$ pendant vertices adjacent to the central vertex $u_i$. Note that $deg(u_i)= n$ and $|E(mK_{1,n})| = mn$. \begin{lemma} \label{star1} The $\chi_{la}(mK_{1,n}) = mn + 1$ if $m \ne n$ and satisfies one of the following conditions: \begin{enumerate}
\item $1 < m \le n$, $m$ and $n$ are both even and $(m,n) \ne (2,2)$.
\item $n \equiv 0 \pmod{2},\ n \ge 4,\ m = 2$.
\item $m \equiv n \equiv 0\pmod{2},\ m \ge 2, n \ge 4$.
\end{enumerate} \end{lemma} \begin{proof} Let $m$ and $n$ be as stated. Then by the existence of a magic rectangle set, we have a magic rectangle set $MR(n,m;1)$. For each $i\ (1 \le i \le m)$ we label the edges in $i$th copy of $K_{1,n}$ by entries in $i$th column of $MR(n,m;1)$. This gives $w(u_i)= \frac{n(nm+1)}{2}$ for all $i\ (1\leq i\leq m)$ and $w(v_{i,j}) = f(u_iv_{i,j})$.
Since pendant vertices contribute $mn$ distinct colors and each $u_i$ has the same weight, the total number of distinct weights is $mn+1$. This proves the theorem. \end{proof}
\begin{lemma}{\label{star2}}
For $m > 3,\ n > 2$ and $n(m-1) > 4$, $\chi_{la}(mK_{1,n}) = mn+2$, if $m\equiv 0\pmod{2}$ and $n\equiv 1\pmod{2}$. \end{lemma} \begin{proof}
Since $n\equiv 1\pmod{2}$ and $m \equiv 0\pmod{2}$, then $n\equiv m-1 \equiv 1\pmod{2}$, therefore by Theorem \ref{mr}, there exists $ n\times (m-1)$ magic rectangle $M(n, m-1)$. Let $ C_1, C_2,\dots, C_{m-1}$ be the columns of the $M(n, m-1)$. We label the edges in the first $m-1$ copies of $ K_{1,n}$ using respective columns $ C_1, C_2,\dots, C_{m-1}$. Label the edges in the $m$th copy of $ K_{1,n}$ using the remaining set of labels $\{n(m-1)+1,\ n(m-1)+2, \dots,mn\}$ where sum of all these labels is $n^2(m-1)+\frac{n(n+1)}{2}$. The pendant vertices of $G$ induce $mn$ different weights. For the support vertices $\{u_1,u_2,\dots, u_m\}$, the weights are as follow: \begin{equation*}
w(u_i) = \begin{cases}
\frac{n(mn-n+1)}{2}& \mbox{if } i = 1,2,\dots, m-1\\
n^2(m-1) +\frac{n(n+1)}{2}& \mbox{if } i = m.
\end{cases} \end{equation*}
The total number of distinct weights under this labeling is $mn+2$. Hence we conclude that $\chi_{la}(mK_{1,n}) = mn+2$. \end{proof}
\begin{lemma} \label{star3} For $m \geq 1$ and for even $n$ the $\chi_{la}(mK_{1,n}) = mn+1$. \end{lemma} \begin{proof} Now we define an edge labeling $f$ as follows: \begin{equation*} f(u_iv_{i,j}) = \begin{cases}
(i-1)\frac{n}{2}+j & \mbox{if } 1\leq j\leq \frac{n}{2}\\
mn-(n-j)-(i-1)\frac{n}{2} & \mbox{if } \frac{n}{2}+1\leq j\leq n.
\end{cases} \end{equation*} The induced vertex weights are: $w(v_{i,j})= f(u_iv_{i,j}))$ and $w(u_i)= \frac{n}{2}(mn+1)$.
Since pendant vertices contribute $mn$ distinct weights and each $u_i (1\leq i\leq m)$ has the same weight, the total number of distinct weights is $mn+1$. \end{proof}
The illustration of the local antimagic coloring of $3K_{1,3}$ is shown in Figure \ref{3K_{1,3}}. \begin{figure}
\caption{Local antimagic labeling of $3K_{1,3}$.}
\label{3K_{1,3}}
\end{figure}
The following theorem is evident from the above Lemmas \ref{star1}, \ref{star2} and \ref{star3}. \begin{theorem} $mn+1 \le \chi_{la} (m K_{1,n}) \leq mn+2.$ \end{theorem} The chromatic number of a complete graph on $n$ vertices is $n$. It is easy to observe and prove that $\chi_{la}(K_n) = n = \chi(K_n)$. Moreover, with some conditions on $n$, we have proved that $\chi_{la}(mK_n) = n$. \begin{proposition} \label{prop:1} For $n \ge 3, \chi_{la}(K_n) = n$. \end{proposition} \begin{proof} Let $V(K_n) = \{ v_1, v_2, \dots, v_n\}$ and for each $1 \le j \le n$ and $j < i \le n$ let $e_{i-1} = v_jv_i$. Define edge labeling $f$ by $f(e_i) = (j-1)n - \frac{j(j-1)}{2}+1$. It is easy to observe that the weights, $w(v_1), \dots, w(v_n)$ are in increasing order. Hence, $\chi_{la}(K_n) \le n = \chi(K_n)$. This proves the proposition. \end{proof} \begin{proposition} For $n \ge 3, n \equiv 1 \text{ or } 2\pmod{4}, \chi_{la}(mK_{n}) = n$. \end{proposition} \begin{proof} Let $n \ge 3$. If $n \equiv 1 \pmod{4}$ then $K_n$ is $4r$-regular for some $r \ge 0$ and proof follows by Theorem~\ref{baca1}. If $n \equiv 2\pmod{4}$ then $K_n$ is $(4r+2)$-regular for some $r \ge 1$ and it contains $(n-1)$ even spanning cycles. Hence proof follows by Theorem~\ref{baca2}.\\ \end{proof}
\begin{theorem} Let $\chi_{la}(rK_{m,n}) = 2$ if positive integers $m,\ n,\ r$ with $m \ne n$ satisfies one of the following conditions: \begin{enumerate}
\item $1 < m \le n$ and $m$ and $n$ are both even, $r \ge 1$, and $(m,n) \ne (2,2)$.
\item $1 < m \le n$ and $m, n, r$ are all odd.
\item $m \equiv 0 \pmod{2},\ m \ge 4,\ n = 2$ and $r \ge 1$.
\item $m \equiv n \equiv 0\pmod{2},\ m \ge 2, n \ge 4$ and $r \ge 1$. \end{enumerate} \end{theorem} \begin{proof} By Theorems \ref{mrs2.3} to \ref{mrs1.1} on the existence of magic rectangle sets, we have the existence of MRS$(m,n;r)$ for each of the above cases. Suppose there is a magic rectangle set $MRS(m,n;r)$. Let $M_1,M_2,\dots, M_r$ denotes the $r$ magic rectangles in $MRS(m,n;r)$. For $1\leq k\leq r$, define the vertex set of $k^{th}$ copy of $K_{m,n}$ as $V =\{v_i^k, w_j^k : 1\leq i\leq n, 1\leq j\leq m \}$, where $\{v_i^k : 1 \le i \le m\}$ and $\{w_i^k : 1 \le i \le n\}$ form the respective partite sets of $k^{th}$ copy of $K_{m,n}$.\\
Now for each $k\ (1\leq k\leq r)$ and each $i\ (1 \le i \le n) $ we label the edge set $\{ v_i^k w_j^k : 1\leq j\leq m \}$ with the numbers in the $i$th row of $M_k$. Since the sum of elements in any row or column in the magic rectangle set is equal, the resulting labelling is a local antimagic that induces $2$ different colors. Therefore, $\chi_{la}(rK_{m,n}) \le 2$. Also, $\chi(rK_{m,n}) = 2$. We conclude that $\chi_{la}(rK_{m,n}) = 2$ whenever there exists a MRS$(m,n;r)$. \end{proof} Figure \ref{fig:3k_2,4} illustrates the local antimagic labeling of $3K_{2,4}$. \begin{figure}
\caption{Local antimagic labeling of $3K_{2,4}$.}
\label{fig:3k_2,4}
\end{figure}\\
Next, we obtain $\chi_{la}$ for complete bipartite graphs as a corollary to the above theorem. More on $\chi_{la}(K_{m,n})$ can be found in \cite{lac_arumugam, lac_lau}.
\begin{corollary} Let $\chi_{la}(K_{m,n}) = 2$ if positive integers $m \ne n$ satisfies one of the following conditions: \begin{enumerate}
\item $1 < m \le n$ and $m$ and $n$ are both even, $r \ge 1$, and $(m,n) \ne (2,2)$.
\item $1 < m \le n$ and $m, n, r$ are all odd.
\item $m \equiv 0 \pmod{2},\ m \ge 4,\ n = 2$ and $r \ge 1$.
\item $m \equiv n \equiv 0\pmod{2},\ m \ge 2, n \ge 4$ and $r \ge 1$. \end{enumerate} \end{corollary}
An interesting class of graphs is called a {\it necklace graph} having common vertices. A $u,v$ \textit{necklace} is a list of cycles $C_1, C_2, \dots, C_t$ such that $u \in C_1$, $v \in C_t$, consecutive cycles share exactly one vertex, and non-consecutive cycles are disjoint (see Figure \ref{fig:uv-necklace}). The number of edges in all the cycles is known as the \textit{length of the necklace}. We provide an upper bound for the local antimagic chromatic number for this class of graph.
\begin{figure}
\caption{A $u,v$-necklace.}
\label{fig:uv-necklace}
\end{figure}
\begin{theorem} Let $G$ be an $u,v$-necklace on $t \ge 2$ cycles such that $G$ has no adjacent vertices of degree $4$. Then $\chi_{la}(G) \le t + 2$.
\end{theorem} \begin{proof} Let $G$ be an $u,v$-necklace of length $n$, where $C_i$ be a cycle of length $n_i$ for $1 \le i \le t$. By the definition of $G$, we have an Eulerian tour traversed clockwise starting and ending at $u$. We enumerate the edges of $G$ as we follow the Eulerian tour as shown in Figure \ref{fig:uv-necklace}. Now we label the edges as
\begin{equation*}
f(e_i) =\begin{cases}
\frac{i+1}{2} & \text{if $i$ is odd} \\
n - (\frac{i}{2} - 1) & \text{if $i$ is even}.
\end{cases} \end{equation*} Then for each vertex $x$ of degree $2$ other than $u$, it is easy to see that $w(x) = n+1 \mbox{ or } n+2$ and \begin{align*} w(u) = \begin{cases}
1 + \frac{n+1}{2} \text{ if } n \text{ is even}\\
1 + \frac{n+2}{2} \text{ if } n \text{ is odd} \end{cases} \end{align*} For a vertex $y$ of degree $4$, $w(y) > n+2$ and there are $t-1$ such vertices. This proves that $f$ induces $t + 2$ colors as required. \end{proof} \section{Some Other Results} Arumugam et al.\cite{lac_corona_arumugam} and Premalatha et al.\cite{lac_tree_premalatha} studied the local antimagic chromatic number of corona product $P_n \circ \overline{K}_m$, $C_n \circ \overline{K}_m$ and Setiawan et al.\cite{lac_corona_sugeng} have studied it for corona product $P_m \circ P_k$. The following theorem gives the bounds on the local antimagic chromatic number of corona product $G \circ \overline{K}_m$ for any graph $G$.
\begin{theorem} Let $G$ be a graph with $p$ vertices and $q$ edges such $\chi_{la}(G) = r$, if $m\equiv p\pmod{2}$ then $mp\leq \chi_{la}(G\circ \overline{K}_m)\leq mp+r$. \end{theorem}
\begin{proof} Since $\chi_{la}(G) = r$, there is a local antimagic bijection $f:E\longrightarrow\{1,2,\dots,q \}$ with $r$ distinct weights. Further since $m\equiv p\pmod{2}$, there exits a magic rectangle $MR(m,p)$ of order $m\times p$. Let $C_1, C_2,\dots,C_p$ be the $p$ columns of the magic rectangle $MR(m,p)$. Let $ u_1,u_2,\dots, u_p$ be the vertices and $e_1, e_2,\dots,e_q$ are t edges of the graph $G$. Let $v_i^{j}$ be the pendent vertices adjacent to the vertex $u_i$, $1\leq j\leq m$, $1\leq i\leq p$.\\ We define an edge labeling $g:G\circ \overline{K}_m\longrightarrow \{1,2,\dots, q+mp \}$ by $g(e_i)= f(e_i)$ and $g(u_iv_i^{j})= q + c_{ij}$, where $c_{i,j}$ is the $(i,j){th}$ entry of $MR(m,p)$.
Now weights of the vertices under $g$ are $w_g(u_i)= w_f(u_i)+ \frac{m(mp+1)}{2}+mq$, $w_g(v_i^{j}) = g(u_iv_i^{j})$.
Thus, we have $r+mp$ distinct weights. Hence $\chi_{la}(G\circ \overline{K_m})\leq r + mp$. Since there are $mp$ pendent vertices, by Theorem \ref{th:lac_bound} $\chi_{la}(G\circ \overline{K_m})\geq mp$. This proves the theorem. \end{proof}
We know that the order of a clique $G'$ of a graph $G$ is the lower bound of $\chi(G)$. A similar result holds for local antimagic chromatic number, as illustrated in the following lemma. \begin{lemma} \label{clique} If a graph $G$ contain a $k$-clique then $\chi_{la}(G)\geq k$. \end{lemma} \begin{proof} Let $G^\prime$ be a k- clique in $G$ and let $f $ be a local antimagic labeling of $G$. Since every vertex $v_i \in G^\prime$ is adjacent to $k-1$ other vertices and $f$ is local antimagic labeling of $G$, it follows that for every vertex pair $v_i, v_j \in G^\prime$, $ w(v_i)\neq w(v_j)$. Therefore the weights of the vertices of $G^\prime$ under $f$ are distinct, hence $\chi_{la}(G)\geq k$. \end{proof} \begin{lemma} \label{sub} Let $G$ be a graph with vertex $v$ such that $deg(v) = \Delta(G) \ge 2$. Then, there is a subgraph $H$ of $G$ such that $\chi_{la}(H) = \Delta(G)+1$. \end{lemma} \begin{proof} Let $G$ be a graph with vertex $v$ such that $deg(v) = \Delta(G)$. We consider a subgraph $H$ with vertex set $ \{v, v_i: vv_i \in E(G)\}$ and edge set $\{vv_i : vv_i \in E(G)\}$. Since $H$ is a star, $\chi_{la}(H) = \Delta(G)+1$ (see \cite{lac_lau}). \end{proof} We know that for a given subgraph $H$ of a graph $G$, $\chi(H) \le \chi(G)$. But this need not be the case in local antimagic chromatic number. Using Lemma \ref{sub}, we give some explicit examples where the inequality does not hold. Lau et. al \cite{lac_lau} calculated the local antimgaic chromatic number of bipartite graph and wheel: \begin{align*} \chi_{la}(K_{p,q}) &= \begin{cases} q+1 & \text{if } q>p=1\\ 2 \quad & \text{if } q > p \ge 2 \text{ and } p \equiv q\pmod{2}\\ 3 \quad & \mbox{otherwise}\\ \end{cases}\\
\chi_{la}(W_{n}) &= \begin{cases} 4 \quad & \text{if } n \equiv 0 \text{ or } 1 \text{ or } 3 \pmod{4}\\ 3 \quad & \mbox{otherwise}. \end{cases} \end{align*} For $q > p \ge 2$ using construction given in Lemma~\ref{sub} with $G \cong K_{p,q}$ we obtain subgraph $H$ of $K_{p,q}$ with $\chi_{la}(H) = q + 1 > 3 \ge \chi_{la}(K_{p,q})$. Similarly for $G \cong W_n$, where $n \ge 5$, we obtain subgraph $H$ of $W_n$ such that $\chi_{la}(H) = n + 1 > 4 \ge \chi_{la}(W_n)$.\\
We pose the following problem.\\ \begin{problem} Characterise graphs $G$ for which $\chi_{la}(H) \le \chi_{la}(G)$, for all subgraphs $H$ of $G$. \end{problem} \section{Construction}
Lau et al.\cite{lac_cycle_lau} have constructed graph $G$ from cycles for which $\chi(G) = \chi_{la}(G)$. In this section we give a recursive method to construct infinitely many graphs $\{G_i\}$ such that $\chi(G_i) = \chi_{la}(G_i)$ from the given graph $G$ such that $\chi(G) = \chi_{la}(G)$. \\ \\ \noindent \textbf{Construction:}\\
Let $G$ be a local antimagic graph with local antimagic labeling $f_0$ such that $|E(G)| = m_0$ satisfying $\chi_{la} (G)= \chi(G)$. Let $|V(G)| = n \ge 3$.\\
\textbf{Case i:} $n \ge 4$ is even.\\
Let $q$ be an even positive integer and let $G_0 = G$. For each $i \ge 1$ consider $G_{i} = G_{i-1} + \overline{K}_q$, where $V(\overline{K}_q) = \{u_1, u_2, \dots, u_q\}$. Observe that $|E(G_i)| = m + \sum_{j=1}^{i} (n+(j-1)q)q = m_i$ (say) and that \begin{equation} \label{eq:1} \chi(G_i) = \chi(G_{i-1}) + 1. \end{equation} First we show that $\chi_{la}(G_{i}) \le \chi_{la}(G_{i-1}) + 1$. Since, $n \equiv q \pmod{2}$, $n+(i-1)q \equiv q\pmod{2}$ for each $i$, there exists a magic rectangle $MR(n+(i-1)q, q)$. Add $m_{i-1}$ to each entry of $MR(n+(i-1)q, q)$ to obtain a new rectangle $MR'$ of the same size in which row sum $(\rho)$, and column sum $(\sigma)$ are constant. Label the edges from $u_j$ to $V(G_{i-1})$ by $i$th column of $MR'$. Then $w(u_j) = \sigma$ for each $j$ and $w_{G_{i}}(x) = w_{G_{i-1}}(x) + \rho, \; \forall x \in V(G_{i-1})$. Since $q$ even was arbitrary, we can choose $q$ so that $w_{G_{i}}(x) > w(u_{j})$ for all $i$ and $j$. This proves that $\chi_{la}(G_{i}) \le \chi_{la}(G_{i-1}) + 1$. Now we show that $\chi(G_i) = \chi_{la}(G_i)$ for each $i$ by induction on $i$. For $i = 0$, the result is trivial. Suppose result is true for $i = t$ i.e. $\chi(G_t) = \chi_{la}(G_t)$. Then \begin{align*} \chi(G_{t+1}) &= \chi(G_{t} + \overline{K}_q)\\
&= \chi(G_t)+1\\
&= \chi_{la}(G_t)+1\\
&\ge \chi_{la}(G_{t+1}) \end{align*} and $\chi(G_{t+1}) = \chi(G_{t}) + 1 = \chi(G_{t} + \overline{K}_q) \le \chi_{la}(G_{t} + \overline{K}_q)$. This proves that $\chi(G_{t+1}) = \chi_{la}(G_{t+1})$. Hence by induction the result is true for all $i \ge 0$. Thus, $\{G_i\}_i$ is the required sequence of graphs satisfying the property $\chi(G_i) = \chi_{la}(G_{i})$ for each $i$.\\
\textbf{Case ii:} $n \ge 3$ is odd. \\ Let $q \ge 3$ be an odd integer. Then for each odd $i \ge 1$, $n+(i-1)q \equiv q\pmod{2}$. Hence there is a magic rectangle $MR(n+(i-1)q,q)$ and repeated application of the above procedure; we obtain the graphs $\{G_{2i-1}\}_{i}$ satisfying the property $\chi(G_i) = \chi_{la}(G_{i})$.\\
Hence given a graph $G$ such that $\chi(G) = \chi_{la}(G)$ we construct infinitely many graphs $\{G_i\}$ such that $\chi(G_i) = \chi_{la}(G_i)$. \\
From the above construction, proof of the following theorem is evident.\\
\begin{theorem} \label{multipartite} If $r_1, r_2, \dots, r_t$, where $t \ge 3$ are positive integers such that $a = r_1 + r_2 + \dots + r_{t-1}, r_{t} >1, ar_{t} > 4, a\ne r_{t}$ and $a \equiv r_{t}\pmod{2}$ then for the complete $t$-partite graph $K_{r_1, r_2, \dots, r_t}$, $\chi_{la}(K_{r_1, r_2, \dots, r_t}) = \chi_{la}(K_{r_1, r_2, \dots, r_{t-1}}) + 1$. \end{theorem} As an illustration, consider the complete tripartite graph $K_{p,q,r}$ where $p+q, r>1, (p+q)r > 4 p+q \ne r$ and $p+q \equiv r \pmod{2}$. Then \begin{align*} \chi_{la}(K_{p,q,r}) = \begin{cases} q+2 \quad & \text{if } q>p=1\\ 3 \quad & \text{if } q > p \ge 2 \text{ and } p\equiv q\pmod{2}\\ 4 \quad & \text{otherwise.} \end{cases} \end{align*} \section{Conclusion and Scope} In this paper, we obtained the local antimagic chromatic number for unions of some graphs and a few others. There are infinitely many graphs $G$ for which $\chi_{la}(G) = \chi(G)$, but a complete characterization has yet to be discovered.
\end{document} | arXiv |
Electromagnetic vector fields are composed of elaborate mathematical structures described with vector differential operators, and physicists cannot intuitively understand and distinguish of them. If one intends to learn about the changes in the electric displacement D in medium of polarization P, or the magnetic intensity H in the medium of magnetization M, knowing the features of the fields is impossible unless one calculates and visualizes them. Here, a vector fields platform visualizing the most basic vector fields: E = $-\nabla\varphi$, B = $\nabla\times$ A, D = $\varepsilon_0$E+P, H = $-\nabla\varphi*$, and B = $\mu_0$(H+M), present the fields with its norms in the vector differential mode completely in Mathematica.
The essentiality of metabolic reactions is estimated for hundreds of species by the flux balance analysis for the growth rate after removal of individual and pairs of reactions. About 10% of reactions are essential, i.e., growth stops without them. This large-scale and cross-species study allows us to determine ad hoc ages of each reaction and species. We find that when a reaction is older and in younger species, the reaction is more likely to be essential. Such correlations may be attributable to the recruitment of alternative pathways during evolution to ensure the stability of important reactions. | CommonCrawl |
binomial distribution probability calculator
The calculator can also solve for the number of trials required. The binomial coefficient, $ \binom{n}{X} $ is defined by Binomial Distribution is expressed as BinomialDistribution[n, p] and is defined as; the probability of number of successes in a sequence of n number of experiments (known as Bernoulli Experiments), each of the experiment with a success of probability p. The below given binomial calculator helps you to estimate the binomial distribution based on number of events and probability of success. Binomial Probability Calculator More about the binomial distribution probability so you can better use this binomial calculator: The binomial probability is a type of discrete probability distribution that can take random values on the range of [0, n] [0,n], where n n is the sample size. Binomial Distribution Calculator The calculator will find the binomial and cumulative probabilities, as well as the mean, variance and standard deviation of the binomial distribution. where $n$ is the number of trials, $p$ is the probability of success on a single trial, and $X$ is the number of successes. $$ $$ P(X) = \binom{n}{X} \cdot p^X \cdot (1-p)^{n-X} $$ This tutorial explains how to use the following functions on a TI-84 calculator to find binomial probabilities: binompdf (n, p, x) returns the probability associated with the binomial pdf. The full binomial probability formula with the binomial coefficient is The number of trials (n) is 10. \cdot p^X \cdot (1-p)^{n-X} $$ If we apply the binomial probability formula, or a calculator's binomial probability distribution (PDF) function, to all possible values of X for 5 trials, we can construct a complete binomial distribution table. You will also get a step by step solution to follow. Trials, n, must be a whole number greater than 0. Binomial distribution is a discrete distribution, whereas normal distribution is a continuous distribution. Using the Binomial Probability Calculator Enter the trials, probability, successes, and probability type. \cdot 0.65^3 \cdot (1-0.65)^{5-3} $$ For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems. Binomial Distribution is expressed as BinomialDistribution [n, p] and is defined as; the probability of number of successes in a sequence of n number of experiments (known as Bernoulli Experiments), each of the experiment with a success of probability p. This is the number of times the event will occur. $$ P(3) = \frac{5!}{3!(5-3)!} Enter the number of trials in the $n$ box. The probability type can either be a single success ("exactly"), or an accumulation of successes ("less than", "at most", "more than", "at least"). $$ P(X) = \frac{n!}{X!(n-X)!} To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution. Use this binomial probability calculator to easily calculate binomial cumulative distribution function and probability mass given the probability on a single trial, the number of trials and events. Successes, X, must be a number less than or equal to the number of trials. The binomial probability calculator will calculate a probability based on the binomial probability formula. A binomial distribution is one of the probability distribution methods. How to Calculate Binomial Probabilities on a TI-84 Calculator The binomial distribution is one of the most commonly used distributions in all of statistics. If doing this by hand, apply the binomial probability formula: Trials, n, must be a whole number greater than 0. The sum of the probabilities in this table will always be 1. $$ \binom{n}{X} = \frac{n!}{X!(n-X)!} Why We Use Them and What They Mean, How to Find a Z-Score with the Z-Score Formula, How To Use the Z-Table to Find Area and Z-Scores. Do the calculation of binomial distribution to calculate the probability of getting exactly 6 successes.Solution:Use the following data for the calculation of binomial distribution.Calculation of binomial distribution can be done as follows,P(x=6) = 10C6*(0.5)6(1-0.5)10-6 = (10!/6!(10-6)! Binomial Probability Calculator Use the Binomial Calculator to compute individual and cumulative binomial probabilities. A binomial distribution is one of the probability distribution methods.
Ffxiv Master Recipes 8 Alchemist, Deutsche Kuche Bread, Examples Of Critical Thinking In Mental Health Nursing, Thai Square Vegan Menu, How To Use A Scanner Windows 10, Facebook Logo Png Transparent Background Black, Grixis Delver Legacy 2020, Lice Meaning In Malayalam, Jolly Onion Reviews, | CommonCrawl |
Estimating risk ratio from any standard epidemiological design by doubling the cases
Yilin Ning1,2,
Anastasia Lam3,4 &
Marie Reilly5
Despite the ease of interpretation and communication of a risk ratio (RR), and several other advantages in specific settings, the odds ratio (OR) is more commonly reported in epidemiological and clinical research. This is due to the familiarity of the logistic regression model for estimating adjusted ORs from data gathered in a cross-sectional, cohort or case-control design. The preservation of the OR (but not RR) in case-control samples has contributed to the perception that it is the only valid measure of relative risk from case-control samples. For cohort or cross-sectional data, a method known as 'doubling-the-cases' provides valid estimates of RR and an expression for a robust standard error has been derived, but is not available in statistical software packages.
In this paper, we first describe the doubling-of-cases approach in the cohort setting and then extend its application to case-control studies by incorporating sampling weights and deriving an expression for a robust standard error. The performance of the estimator is evaluated using simulated data, and its application illustrated in a study of neonatal jaundice. We provide an R package that implements the method for any standard design.
Our work illustrates that the doubling-of-cases approach for estimating an adjusted RR from cross-sectional or cohort data can also yield valid RR estimates from case-control data. The approach is straightforward to apply, involving simple modification of the data followed by logistic regression analysis. The method performed well for case-control data from simulated cohorts with a range of prevalence rates. In the application to neonatal jaundice, the RR estimates were similar to those from relative risk regression, whereas the OR from naive logistic regression overestimated the RR despite the low prevalence of the outcome.
By providing an R package that estimates an adjusted RR from cohort, cross-sectional or case-control studies, we have enabled the method to be easily implemented with familiar software, so that investigators are not limited to reporting an OR and can examine the RR when it is of interest.
The familiarity and wide adoption of logistic regression analysis for binary outcomes has resulted in the independent effect of a risk factor being most commonly reported as an adjusted odds ratio (OR) from logistic regression. The ease of communication and interpretation of a risk ratio (RR, also known as relative risk) is well recognized [1] and it is common for investigators to present and discuss the OR as an approximation to a RR for a rare outcome. However, each of these estimators comes with some consequences [2] and their advantages and disadvantages have been discussed extensively in the epidemiological literature. An important limitation of the OR, that is not shared by the RR, is the noncollapsibility that is the subject of ongoing discussion [3]. As a result of this property, the OR can vary across sub-groups defined by a variable unrelated to the exposure, which imposes limitations on its interpretation. Another disadvantage of the OR that is not shared by the RR is that it is sensitive to the choice of scale [4]. Thus there are situations where an adjusted RR can provide a better understanding of the data and research findings [5] and overcome the limitations of only reporting an OR [6]. Of particular concern in global public health is the misinterpretation of the OR as a RR, supporting exaggerated claims of the magnitude of associations [7].
If the underlying disease process follows a relative risk model, and not a logistic model, methods have long been available for estimating the RR from cohort or cross-sectional data: using log-binomial regression [8], or if this has convergence issues, Poisson regression [9] or Cox regression [10]. In the early years of case-control studies, a simple "correction" to the OR was proposed to yield a less biased estimate of the RR [11], but this was later shown to be biased in the presence of confounding [9]. A paper discussing eight methods of estimating the RR [12] from cohort or cross-sectional data presented an intriguing approach referred to as "doubling-the-cases", motivated in the early 1980's by Miettinen [13], where manipulation of the data enables the RR to be estimated using standard logistic regression. Assuming the outcome in the data is coded as 1 for cases and 0 for non-cases, the data set is expanded with an additional record for each case, in which the outcome is changed to 0, and a logistic regression analysis of this expanded data set provides an unbiased estimate of the RR. However, the naive standard error reported by the logistic regression is only valid for low incidence rates, and is otherwise biased upwards, representing the additional uncertainty that has been added to the data by having the same individual covariate profile associated with being both a case and a non-case. A robust sandwich estimator, first proposed in the early 1990s [14], corrects for the doubling of cases in the modified data, and has since been shown to perform well in simulation studies [12, 15]. However, statistical software packages do not provide an estimate of this standard error, so that a valid measure of precision is not easily available for the RR estimate. As a result of this computational challenge for cohort and cross-sectional studies, and the lack of methodology and software for case-control sampling, the simple and intuitive doubling-of-cases approach is absent from the standard tool-box of health researchers.
The early work that developed the robust standard error [14] demonstrated that the doubling-of-cases approach can also be applied to case-cohort data. Since the subcohort is a random sample of the whole cohort, it can be easily shown that the logistic regression of the expanded case-cohort data provides a valid estimate of the RR, and the prevalence can be recovered from the intercept using the subcohort sampling fraction. Unlike the subcohort in a case-cohort study, when a case-control sample is drawn from a cohort, these data are not representative of the larger cohort, resulting in the distortion of the estimate of RR (but not of the OR). However, if the sampling fractions are known, the cohort can be represented by up-weighting the observed data using sampling weights [16]. Since the doubling-of-cases approach uses the standard logistic regression model, it is straightforward to accommodate such sampling weights for valid estimation of the RR from case-control samples. However, additional work is required to incorporate the weights when correcting for the overestimation of variability due to the doubling of cases.
In this paper, we describe the doubling-of-cases approach in the cohort setting and then extend its application to the estimation of adjusted RR from case-control data, where the controls are selected either by random or stratified sampling. We derive an expression for the robust standard error and facilitate the use of the method by implementing it as an R package. We evaluate the performance of the approach using simulated data, and illustrate its application in the analysis of the effect of preterm birth on the risk of neonatal jaundice.
Doubling of cases in cohort studies
To introduce the doubling-of-cases approach for estimating the RR, first consider a crude analysis using a cohort of N subjects, with a binary disease indicator Y (1 for cases and 0 for non-cases) and a binary exposure X (e for exposed and \(\bar {e}\) for unexposed). As illustrated in Fig. 1, the doubling-of-cases approach involves expanding the cohort, by including each case twice, where the outcome on the second record is coded as a non-case. Such modification does not change the number of cases in the expanded cohort (where the outcome is denoted by Y∗), but increases the number of non-cases to N (see Fig. 1 and details in Table 1). Hence, the crude OR computed from the expanded cohort is identical to the RR from the original cohort.
Doubling the cases in a cohort of N subjects, where N.1 subjects are cases. The first subscript indicates the exposure status (e for exposed and \(\bar {e}\) for unexposed) and the second subscript indicates the outcome (1 for cases and 0 for non-cases). A dot (".") for either suffix denotes the total (i.e., no stratification on that variable)
Table 1 Equivalence of the crude RR computed from a cohort of N subjects and the crude OR computed from the expanded cohort with N+N.1 records, where \(N_{.1} = N_{e1} + N_{\bar {e}1}\) is the total number of cases in the original cohort
Mantel-Haenszel OR from expanded cohort
In the presence of an additional categorical confounder, Z, the adjusted RR can be computed from the cohort using the Mantel-Haenszel approach, which is a weighted average of the RRs within each of the strata defined by Z [17]. Similarly, the Mantel-Haenszel OR from the expanded cohort is a weighted average of the ORs within each of the expanded strata (which are shown in Table 1 to be identical to the RRs in the original strata) using weights \(w^{*k} = \left (N^{k}_{\bar {e}1} N^{k}_{e.}\right) / \left (N^{k} + N^{k}_{.1}\right)\) for the OR from the k-th expanded stratum, which differ from the weights used to compute the Mantel-Haenszel RR [17]: \(w^{k} = \left (N^{k}_{\bar {e}1} N^{k}_{e.}\right) / N^{k}\) for the RR from the k-th stratum. It will be shown below that both the Mantel-Haenszel RR and the expanded data Mantel-Haenszel OR are estimating the same underlying parameter, the true adjusted RR.
Logistic regression of expanded cohort
The doubling-of-cases approach in regression analysis of cohort and case-cohort studies was first described in 1993 [14], and more recently was referred to as expanded data logistic regression [15]. Here we will briefly describe the approach by generalising the expanded data Mantel-Haenszel OR introduced above.
Assume the following relative risk log-binomial regression model for the probability of being a case for an individual with exposure X in stratum Z:
$$ \ln Pr(Y = 1 \mid X, Z) = \alpha + \beta X + \gamma Z, $$
where expβ represents the adjusted RR (with adjustment for Z) [18–21]. When the cohort is expanded by doubling the cases, the prevalence in each exposure group in the original cohort becomes the odds in that exposure group in the expanded cohort (see Table 2). Hence, a log-linear model for the prevalence in the original cohort gives rise to a log-linear model for the odds, i.e., a logistic regression model, in the expanded cohort:
$$ \ln \frac{Pr(Y^{*} = 1 \mid X, Z)}{1 - Pr(Y^{*} = 1 \mid X, Z)} = \alpha + \beta X + \gamma Z, $$
Table 2 Equivalence of the adjusted RR assessed in a log-binomial regression model of the original cohort with N subjects and the adjusted OR assessed in a logistic regression model of the expanded cohort with N+N.1 records, where \(N_{.1} = N_{e1} + N_{\bar {e}1}\) is the total number of cases in the original cohort
which estimates the same regression coefficients as the log-binomial regression model in Eq. (1). The robust sandwich-type standard error (SE), derived by the same authors [14] to correct for this overestimation, is described in the next section.
Robust Sandwich-type SE for expanded data logistic regression
It can be readily seen from Table 2 that the probability of the modified outcome being 1 in the expanded cohort is:
$$ p^{*} = Pr(Y^{*} = 1 \mid X, Z) = \frac{Pr(Y = 1 \mid X, Z)}{1 + Pr(Y = 1 \mid X, Z)}. $$
For the relative risk regression model defined in Eq. (1), the following pseudo log-likelihood was used [14] for estimating the regression coefficient, β, and its variability:
$$ {}\begin{aligned} l &= \sum_{i = 1}^{N} \{Y_{i} \ln(p^{*}_{i}) + \ln(1 - p^{*}_{i})\} \\ &= \sum_{i = 1}^{N} \{[Y_{i} \ln(p^{*}_{i}) + (1 - Y_{i}) \ln(1 - p^{*}_{i})] + Y_{i} \ln(1 - p^{*}_{i})\}, \end{aligned} $$
where the subscript i indicates the i-th subject in the original cohort of N subjects. This pseudo log-likelihood is exactly the log-likelihood of the logistic regression of the expanded cohort, where the first component (in the square brackets) represents the regular log-likelihood contribution from the N subjects in the cohort, and the second component corresponds to the additional 'non-cases' created by doubling the cases. Hence, the regular maximum likelihood estimate from logistic regression analysis of the expanded data provides a valid estimate for β= ln(RR).
To describe the robust sandwich-type SE that was proposed [14] for the estimated ln(RR), it is useful to introduce a column vector to collectively denote the covariates observed from the i-th subject in the original cohort: xi=(1,Xi,Zi)T, where the first element corresponds to the intercept term in Eq. (1). The components in constructing the sandwich-type SE are derived from the following first-order derivative of the pseudo log-likelihood, l:
$$ U = \sum_{i = 1}^{N} U_{i} = \sum_{i = 1}^{N} \left\{Y_{i}\left(1 - p^{*}_{i}\right) - p^{*}_{i}\right\} \boldsymbol{x}_{i}^{T} = \sum_{i = 1}^{N} r^{*}_{i} \boldsymbol{x}_{i}^{T}, $$
where \(r^{*}_{i} = Y_{i}(1 - p^{*}_{i}) - p^{*}_{i}\) is derived from the error terms (i.e., the difference between the observed outcome and the estimated probability) from the logistic regression of the expanded cohort. For a case in the original cohort, where \(Y_{i} = 1, r^{*}_{i} = (1 - p^{*}_{i}) + (- p^{*}_{i})\) is the summation of the error terms corresponding to the two records in the expanded cohort, one as a case and the other coded as a non-case but with the same covariates (and hence the same probability \(p^{*}_{i}\)). For a non-case where \(Y_{i} = 0, r^{*}_{i} = - p^{*}_{i}\) is the error term corresponding to the single record in the expanded data for this subject.
The proposed robust covariance matrix for the regression coefficients, (β,γ)T is then:
$$ V = H_{1}^{-1} H_{2} H_{1}^{-1}, $$
where \(H_{1}^{-1}\) is the inverse of the hessian matrix of l, estimated by the naive covariance matrix from the logistic regression of the expanded cohort, and H2 is the covariance matrix of U, estimated by:
$$ \hat{H}_{2} = \sum_{i = 1}^{N} \hat{U}_{i} \hat{U}_{i}^{T} = \sum_{i = 1}^{N} \hat{r}^{*2}_{i} \boldsymbol{x}_{i}^{T} \boldsymbol{x}_{i}, $$
and the \(\hat {r}^{*}_{i}\) terms are computed from the residuals of the expanded data logistic regression as described above.
Doubling of cases in case-control studies
When a case-control sample is drawn from a cohort, the sample prevalence is solely dependent on the case:control ratio. However, a case-control sample can be regarded as "intentionally missing" data, and provided the sampling fractions are known, valid cohort estimates (including the RR) can be obtained by up-weighting the sample observations using inverse probability weights to "reconstruct" the cohort. It is common for all cases in the cohort to be sampled into the case-control study, and for controls to be matched to cases on one or more characteristics. In such studies, the weight is 1 for the cases and the weights for controls are calculated as the inverse of the sampling fraction of the non-cases within the matching strata. If controls are selected by simple random sampling, the weights are simply the inverse of the overall sampling fraction of non-cases in the cohort.
Weighted logistic regression of expanded case-control data
As a direct extension of expanded data logistic regression for estimating the RR in cohort studies, we propose a weighted logistic regression of expanded data from a case-control study. As before, each case in the case-control sample is doubled, but the analysis of the expanded data is conducted with a weighted logistic regression, where the weight of each individual in the expanded data is inherited from the sampling fractions that yielded the original case-control sample. Note that doubling of cases is a part of the analytical approach and does not affect the sampling of case-control data or the calculation of sampling fractions. Using similar arguments as for cohort data [14], we propose a robust sandwich-type SE for the estimate of the β parameter in the logistic regression model, i.e., the estimated ln(RR), and describe it in the next section.
Robust SE for expanded data weighted logistic regression of case-control data
Consider the analysis of a case-control sample of n subjects drawn from a cohort of size N. Assuming all cases and a simple random sample of controls are included, the sampling weight (denoted by w) of each case in this case-control sample is 1, and for each control it is the number of controls in the cohort divided by the number of sampled controls. For matched case-control samples, the sampling weights for controls are the ratios of available controls to sampled controls within each stratum defined by the matching factors. An unbiased estimate of ln(RR) can be obtained from this case-control sample by using the doubling-of-cases approach, provided the individual sampling weights are incorporated in the analysis. More specifically, the pseudo log-likelihood becomes a weighted pseudo log-likelihood:
$$ l_{w} = \sum_{i = 1}^{n} \{w_{i} Y_{i} \ln(p^{*}_{i}) + w_{i} \ln(1 - p^{*}_{i})\}, $$
which is the log-likelihood corresponding to the weighted logistic regression analysis of the expanded case-control sample. The first order derivative of lw is:
$$ {}U_{w} = \sum_{i = 1}^{N} w_{i} U_{i} = \sum_{i = 1}^{n} w_{i} \{Y_{i}(1 - p^{*}_{i}) - p^{*}_{i}\} \boldsymbol{x}_{i}^{T} = \sum_{i = 1}^{n} w_{i} r^{*}_{i} \boldsymbol{x}_{i}^{T}. $$
Following the derivation of Eq. (6) for cohort designs, we propose the following as a robust covariance matrix for the estimates from a weighted analysis:
$$ V_{w} = H_{w1}^{-1} H_{w2} H_{w1}^{-1}, $$
where \(H_{w1}^{-1}\) denotes the inverse of the Hessian matrix of lw and is estimated by the naive covariance matrix from the (weighted) logistic regression of the expanded case-control data, and Hw2 is the covariance matrix of Uw, estimated by:
$$ \hat{H}_{w2} = \sum_{i = 1}^{n} w_{i}^{2} \hat{U}_{i} \hat{U}_{i}^{T} = \sum_{i = 1}^{n} (w_{i} r^{*}_{i})^{2} \boldsymbol{x}_{i}^{T} \boldsymbol{x}_{i}. $$
Simulation study
To evaluate our proposed estimator and robust SE for the RR from case-control data, we simulated a cohort consisting of N=1000 subjects, where 400 subjects were male (Z=1) and the remainder were female. To generate a confounding effect of sex, the probability of being exposed (X=1) was 0.4 for males and 0.2 for females and the outcome generated from the following log-binomial model:
$$ \ln P(Y = 1 \mid X, Z) = \alpha + \ln(RR) X + \ln(1.5) Z. $$
The intercept term was assigned values corresponding to prevalence rates of approximately 10%, 20%, 30% and 40%. We considered true values of RR=1,1.25,1.5,2. For each simulated cohort, we implemented four designs: a 1:1 and 1:2 case-control ratio, each with controls selected randomly or matched on sex.
For the simulated cohort data, we estimated the RR using the log-binomial regression model (the true data-generating model), the expanded data logistic regression model, and other simple/naive estimators: the Mantel-Haenszel RR, expanded data Mantel-Haenszel OR, and the naive logistic regression model (where the estimated OR is viewed as an approximation for the RR). The case-control data was analysed by weighted logistic regression of the expanded data and by logistic regression of the original case-control sample. Although an unweighted logistic regression analysis with adjustment for matching factors is valid for estimating the OR of other covariates, we chose to perform a weighted analysis of matched case-control data to also enable valid estimation of the coefficients of the matching factors. The distributions of the estimates from the doubling-of-cases approaches over 2000 simulation cycles under each scenario were examined on boxplots, where they were compared to the estimates from the correct analysis (Mantel-Haenszel RR or log-binomial model) and the naive estimates. The performance of the method was evaluated by averaging the bias, empirical SE and robust SE, and computing the coverage of the (robust) 95% confidence interval, the type I error rate (when the true RR was 1) and power (when the true RR was not 1).
Illustrative example
We analysed risk factors for neonatal jaundice in infants born to Swedish women between 1992 and 2002 [22]. From the singleton livebirths recorded by the Swedish Medical Birth Register during this calendar period, we excluded infants at risk of neonatal jaundice due to known maternal alloimmunisation or potential alloimmunisation due to a history of transfusion, resulting in 657,264 infants for analysis. In addition to the sex and prematurity of the infant, information was available for maternal age, body mass index (BMI), parity (nulliparous or multiparous) and smoking status. After excluding births with missing information on maternal BMI or smoking status, the final cohort consisted of 547,466 births. Maternal BMI was dichotomised at 25, and maternal age was dichotomised at 35 years. We assessed the association of neonatal jaundice with the six factors described above and the presence of an interaction between preterm birth and parity by analysing the full cohort and a 1:2 case-control sample matched on maternal age and the sex of the infant. The cohort data was analysed using naive logistic regression, log-binomial regression and expanded data logistic regression models. The matched case-control sample was analysed using weighted logistic regression and expanded data weighted logistic regression.
All analyses were performed using R (version 4.0.1). We implemented the expanded data (weighted) logistic regression model as an R package named DoublingOfCases (available from: https://github.com/nyilin/DoublingOfCases). The naive logistic regression model was implemented by the glm function with family = binomial(link = "logit"), and for the weighted logistic regression, the inverse sampling weights were specified via the weights option. The log-binomial regression model was implemented by the glm function with family = binomial(link = "log").
For the simulated cohorts, the expanded data Mantel-Haenszel OR and expanded data logistic regression OR performed well, providing estimates similar to the Mantel-Haenszel RR and the log-binomial RR respectively, regardless of the prevalence in the cohort or the true value of the RR (see Figs. 2 and 3A). The bias in the naive OR increased as expected with larger values of RR and prevalence. Simulation scenarios with a prevalence rate of 40% and true RR of 1.5 or 2 approached the boundary of the parameter space of relative risk models, with the maximum event probability close to 0.80 and 0.95 respectively. The log-binomial regression model failed to converge in 2 and 1432 of the 2000 simulation cycles in these two scenarios respectively, but in the cycles where it converged, it provided valid estimates of the RR (see Appendix Table 5 for detailed simulation results).
Estimated RR across 2000 simulations using different levels of prevalence and true RR values. Estimates were computed using the Mantel-Haenszel (M-H) RR (clear boxes) and expanded data M-H OR methods (shaded boxes)
Estimated RR from expanded data logistic regression across 2000 simulations scenarios with different levels of prevalence and true RR values, compared to estimates from a naive logistic regression and the (true) log-binomial model. Estimates are presented for the full cohort data (panel A) and for matched case-control samples (panel B) with 1:1 (clear boxes) and 1:2 (shaded boxes) case:control ratio
The "Cohort" column in Fig. 4 summarises the good performance of the expanded data logistic regression estimator in all simulation scenarios, with estimated RR close to the true value, coverage close to 95%, type I error close to 5% and power comparable to that of the log-binomial regression model. The robust SE of the estimated RR from the expanded data logistic regression model was similar to the empirical SE, and similar to the variability from the log-binomial regression model when the latter converged (see Appendix Table 5). The naive logistic regression model had a type I error close to 5% and power comparable with the expanded data logistic regression model in all scenarios, as might be expected. Although the estimated OR was a reasonable approximation to the RR (with small bias and coverage close to 95%) when the exposure had no effect (i.e., when RR = 1) or when the prevalence was low (10%), there was an increase in bias and decrease in coverage with increasing prevalence, especially when estimating a larger RR.
Ratio of the estimated and true RR (panel A) and the coverage (panel B), type I error (panel C) and power (panel D) of the RR estimated by naive and expanded data logistic regression analysis of case-control samples in simulation studies. 2000 simulation iterations were repeated in each simulation scenario. The results from log-binomial regression of the simulated cohort data are displayed for comparison: for a prevalence of 40%, this model failed to converge for 2 simulation cycles for RR=1.5 and 1432 cycles for RR=2, and these were excluded. Panel B excludes scenarios with RR > 1 where the naive/weighted logistic regression model had coverage lower than 50%
A similar performance was observed for the weighted logistic regression and expanded data weighted logistic regression models when applied to case-control data. Figure 3B presents the distributions of the RR estimates from 1:1 and 1:2 matched case-control studies, and the performance in terms of bias, coverage, Type I error and power are illustrated for random and matched 1:1 sampling in the second and third columns of Fig. 4 (details in Appendix Table 6).
A total of 21,441 (3.9%) of the infants in the cohort of 547,466 births were diagnosed with neonatal jaundice. The majority of these cases were firstborn infants, with only 3148 born to multiparous mothers. The crude OR associated with preterm birth was 28.0 but the crude RR was only 16.6, and the stronger association among multiparous mothers (crude OR=32.2 and crude RR=20.4) compared to nulliparous mothers (crude OR=23.4 and crude RR=13.1) suggested a possible interaction effect between these two factors. The large difference between the crude OR and RR suggests that the adjusted OR estimated from a naive logistic regression analysis would not be a reasonable approximation to the adjusted RR.
The log-binomial and expanded data logistic regression models provided similar estimates for the association of neonatal jaundice with each of the factors studied, except for a somewhat larger estimate for the association with overweight from the expanded data logistic regression model. Both models identified premature delivery as a strong risk factor for neonatal jaundice, with an estimated relative risk of approximately 13-fold among nulliparous mothers and 20-fold among multiparous mothers (see Table 3). Compared to infants of mothers with maternal BMI below 25, infants of overweight mothers (BMI ≥ 25) had an approximate 20%-26% higher risk of neonatal jaundice. Multiparity was associated with a decreased risk. Despite the low prevalence of the outcome in this population, the OR from a naive logistic regression model considerably overestimated the RR for preterm birth, almost by a factor of 2 for nulliparous mothers and 1.5 for multiparous mothers. Similar estimates were obtained by analysing the matched case-control sample using the weighted logistic and expanded data weighted logistic regression models, by incorporating the sampling weights (see Table 4).
Table 3 Adjusted ORs estimated using naive logistic regression, and adjusted RRs from log-binomial and expanded data logistic regression analysis, using data from a cohort study of neonatal jaundice. In addition to covariates in the table, estimates are adjusted for sex of infant, maternal age and smoking status
Table 4 Adjusted ORs estimated from weighted logistic regression and adjusted RRs estimated from expanded data weighted logistic regression models, using data sampled in a 1:2 case-control design from the infant cohort, matched on infant sex and maternal age. In addition to covariates in the table, estimates are adjusted for smoking status
Despite the attractive properties of the RR, there has been wide adoption of methods for estimating the OR, due in part to its mathematical and statistical properties, such as the reciprocity with respect to the choice of reference group [23] and the avoidance of predicted probabilities greater than 1. But the OR also has some unattractive properties not shared by the RR. Although the Mantel-Haenszel RR can be computed for simple tabular data, the more general log-binomial regression model for estimating an adjusted RR is not as widely known as the corresponding logistic regression model for estimating an adjusted OR. As a result of this familiarity, and the straightforward interpretation and ease of communication of the RR, investigators often present an adjusted OR as an approximation to the adjusted RR for rare outcomes. This has been further encouraged by articles in the medical literature that continue to present the OR as a feature of the case-control study design [24, 25], although there are methods of sampling that offer estimates of RR [26]. In addition to non-rare outcomes, there are other situations where the RR estimate may be useful or more appropriate, and a recent tutorial article on best practice encourages researchers to examine their results in more than one way [5] when there are valid alternatives. We have provided such an alternative, the doubling-of-cases approach, that is intuitively appealing and utilises the familiar logistic regression model after a simple modification to the data. Although it has been known for some time that this method provides a valid adjusted RR from cohort or cross-sectional data, the standard error of the estimate is not available from statistical software packages. In this paper, we first provided an introduction to this method in the context of cohort or cross-sectional data, and then extended the approach to data collected in a case-control design, deriving a robust estimate of the standard error. In contrast to the optional use of weighted logistic regression to improve precision or enable estimation of coefficients of matching factors [16, 27], a weighted analysis is required for valid estimation of a RR from case-control data. Where the case-control study has been implemented in a well-defined population or cohort, these weights are easily available from simple frequency distributions. To make the method accessible to data analysts, we have implemented it as an R package (available from https://github.com/nyilin/DoublingOfCases) that seamlessly estimates adjusted RRs from cohort, cross-sectional and case-control studies.
Using simulated data, we demonstrated that the expanded data weighted logistic regression of a case-control sample, with or without matching, produced similar estimates to the adjusted RR estimated from the full cohort. Our simulation studies also demonstrated the overestimation of a RR by the OR from a simple logistic regression model, even when the outcome is rare, especially for strong effects. In contrast, the weighted logistic regression model of the expanded data generated valid estimates for the RR, even for common outcomes. Our proposed robust SE for the RR estimated from case-control data performed well in estimating the variability of the adjusted RR.
In an application to neonatal jaundice, we found a positive association with preterm birth (which was stronger among multiparous mothers) and maternal overweight, and a negative association with multiparity, consistent with the literature [22, 28]. Although it is often assumed that the OR is a reasonable approximation for the RR when studying a rare outcome, this example demonstrated that the OR can considerably overestimate the RR of a rare event when assessing a very strong association: although the prevalence of the outcome (neonatal jaundice) in the cohort was only 3.9%, the adjusted ORs for preterm (23.5 and 32.5 among nulliparous and multiparous mothers, respectively) were considerably larger than the adjusted RRs estimated from log-binomial regression (12.9 and 20.1) or expanded data logistic regression (13.0 and 20.4).
In our simulation study, we encountered a practical difficulty that is known in the implementation of log-binomial regression models in statistical software packages: the algorithm may fail to converge. In our simulation scenario of moderate effect (true RR=2) and high prevalence (40%), the log-binomial regression failed to identify valid starting values for the coefficients in more than 70% of the iterations. While this could be resolved by using crude RR estimates as starting values (data not shown), such issues may not be so easily overcome in practice. For example, Deddens and Petersen [29] created a simple numerical example with outcome Y=(0,0,0,0,1,0,1,1,1,1) and a single exposure X=(1,2,3,4,5,6,7,8,9,10), where the R implementation (via the glm function) failed to converge even when the true estimates were used as starting values. This difficulty, and sometimes inability, to reach convergence in maximising the likelihood of the log-binomial regression model, has been widely discussed in the literature [12, 15], and a computationally expensive approach to alleviate the problem has been made available in SAS [30]. An alternative approach that avoids convergence issues when estimating the RR is the Poisson regression model (with robust SE), which has a similar good performance to that of expanded data logistic regression when applied to cohort data [12, 15], or to case-control data that incorporates sampling weights (see Fig. 5 in Appendix). The Poisson regression model approximates the binomial distribution of the binary outcome using a Poisson distribution, whose statistical properties may not be familiar to many applied data analysts, making them reluctant to embark on such an analysis. In contrast, the doubling-of-cases approach is easily accessible as it leverages on the simple equivalence between the RR from the original data and the OR from the expanded data that is common to crude and adjusted analyses, and uses one of the most common analytical tools in epidemiology, the logistic regression model.
A potential practical limitation of the doubling-of-cases method for matched case-control data is that it is necessary to know the sampling fractions within the matching strata, as these are needed to enable the analysis to 'reconstruct' the background population/cohort from the case-control sample. The availability of this information will depend on whether the case-control study was conducted within a well-defined population, the nature and extent of the matching factors and the available data resources. Where a study is conducted using national or regional health registers, and cases and controls matched on basic demographic data (such as sex and age category), then the necessary information will be available from population statistics offices. The sampling fractions will also be known for studies that identify cases and controls from electronic medical records. However, the necessary data may be difficult or impossible to obtain for case-control studies that are implemented in the course of clinical work in low-resource settings with limited data infrastructure.
Another limitation of the doubling-of-cases approach, in common with Poisson regression, is the potential bias in the estimated ln(RR) when some subjects have estimated probabilities greater than or equal to 1. In the small numerical example from Deddens and Petersen [29] mentioned above, both the Poisson regression and the expanded data logistic regression had estimated probabilities larger than 1 for the 9th and 10th observations and both methods overestimated the RR to some extent: compared to the correct estimate (with 95% CI) of 1.23 (1.01 - 1.51), the Poisson regression with robust SE estimated the RR as 1.38 (1.13 - 1.70) and the expanded data logistic regression estimate was 1.44 (1.14 - 1.82). Although our illustrative example did not have large estimated probabilities (maximum 0.71), RR estimates are also known to be potentially biased when estimating a strong association with exposure [15], as occurred in the expanded data logistic regression analysis of the very strong association of prematurity with neonatal jaundice. Although the doubling-of-cases approach may result in some bias in the estimates of the RR in such settings, it can still be used by data analysts as a simple first approach. Large estimated probabilities may suggest that the log-linear assumption is inadequate, in which case the regression analysis should consider transformations of continuous covariates and/or interactions between covariates to more appropriately model the underlying data-generating mechanism.
As a result of the method presented in this paper and the provision of a software package for its implementation, investigators can choose whether to report an adjusted OR or RR, or both, regardless of the study design. The method offers a simple and formal way of justifying the reporting of an adjusted OR as an approximate RR, regardless of the prevalence. Another important advantage is that it facilitates the comparison of findings to published RRs and the inclusion of estimates in meta-analyses that may be challenged by the mixed reporting of OR and RR.
Tables 5 and 6 present the detailed simulation results that were visualised in Fig. 4.
Figure 5 presents the results of a supplemental simulation study, in which each simulated cohort was analysed using Poisson regression, and each simple and matched case-control sample using weighted Poisson regression with inverse probability weighting. As illustrated in the Figure, the performance of the (weighted) Poisson regression was comparable with the doubling-of-cases approach in all scenarios investigated.
Table 5 Bias, empirical SE (Emp. SE), mean SE, coverage, type I error and power of lnRR from log-binomial and expanded data logistic regression analysis of simulated cohort, with 2000 simulation iterations in each scenario
Table 6 Bias, empirical SE (Emp. SE), mean SE, coverage, type I error and power of lnRR from expanded data weighted logistic regression analysis of simulated case-control data, with 2000 simulation iterations in each scenario
Ratio of the estimated to true RR (panel A) and the coverage (panel B), type I error (panel C) and power (panel D) of (weighted) Poisson regression of original data and (weighted) logistic regression of expanded data, from 2000 simulation iterations of each scenario
Data sharing is not applicable to this article as no new data were created or analyzed in this study. The R package created for this application, named DoublingOfCases, is available for download from: https://github.com/nyilin/DoublingOfCases.
RR:
Relative Risk
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We thank Jay Achar for fruitful discussions of this work.
This work was supported by Cancerfonden (the Swedish Cancer Society) contract number 16 0497. The funding body had no role in the study design, data analysis and interpretation, or writing the manuscript. Open access funding provided by Karolinska Institute.
Saw Swee Hock School of Public Health, National University of Singapore and National University Health System, Singapore, Singapore
Yilin Ning
Duke-National University of Singapore Medical School, Singapore, Singapore
School of Geography and Sustainable Development, University of St Andrews, St Andrews, United Kingdom
Anastasia Lam
Max Planck Institute for Demographic Research, Rostock, Germany
Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
Marie Reilly
The study was conceived by MR, simulations and data analyses conducted by AL and YN and R code developed by YN. YN drafted the manuscript, which was reviewed and revised by all authors. All authors approved the final version of the manuscript for submission.
N/A, no new data were used for this study.
N/A.
Ning, Y., Lam, A. & Reilly, M. Estimating risk ratio from any standard epidemiological design by doubling the cases. BMC Med Res Methodol 22, 157 (2022). https://doi.org/10.1186/s12874-022-01636-3
Accepted: 12 May 2022
Doubling-of-cases
Expanded data logistic regression
Log-binomial regression
Poisson regression
Weighted analysis | CommonCrawl |
Basics of Geometric Algebra
Grade Projection
Multivector Products
Reverse of Multivector
Reciprocal Frames
Manifolds and Submanifolds
Geometric Derivative
Geometric Derivative on a Manifold
Normalizing Basis for Derivatives
Linear Differential Operators
Split Differential Operator
Linear Transformations/Outermorphisms
Multilinear Functions
Algebraic Operations
Covariant, Contravariant, and Mixed Representations
Contraction and Differentiation
From Vector to Tensor
Parallel Transport and Covariant Derivatives
What is Geometric Algebra?¶
Basics of Geometric Algebra¶
Geometric algebra is the Clifford algebra of a real finite dimensional vector space or the algebra that results when the vector space is extended with a product of vectors (geometric product) that is associative, left and right distributive, and yields a real number for the square (geometric product) of any vector [HS84], [DL03]. The elements of the geometric algebra are called multivectors and consist of the linear combination of scalars, vectors, and the geometric product of two or more vectors. The additional axioms for the geometric algebra are that for any vectors \(a\), \(b\), and \(c\) in the base vector space ([DL03],p85):
\[\begin{split}\begin{array}{c} a\lp bc \rp = \lp ab \rp c \\ a\lp b+c \rp = ab+ac \\ \lp a + b \rp c = ac+bc \\ aa = a^{2} \in \Re. \end{array}\end{split}\]
If the dot (inner) product of two vectors is defined by ([DL03],p86)
\[\be a\cdot b \equiv (ab+ba)/2, \ee\]
then we have
\[\begin{split}\begin{aligned} c &= a+b \\ c^{2} &= (a+b)^{2} \\ c^{2} &= a^{2}+ab+ba+b^{2} \\ a\cdot b &= (c^{2}-a^{2}-b^{2})/2 \in \Re \end{aligned}\end{split}\]
Thus \(a\cdot b\) is real. The objects generated from linear combinations of the geometric products of vectors are called multivectors. If a basis for the underlying vector space are the vectors \({\left \{{{{\eb}}_{1},\dots,{{\eb}}_{n}} \rbrc}\) (we use boldface \(\eb\)'s to denote basis vectors) a complete basis for the geometric algebra is given by the scalar \(1\), the vectors \({{\eb}}_{1},\dots,{{\eb}}_{n}\) and all geometric products of vectors
\[\be {{\eb}}_{i_{1}}{{\eb}}_{i_{2}}\dots {{\eb}}_{i_{r}} \mbox{ where } 0\le r \le n\mbox{, }0 \le i_{j} \le n \mbox{ and } i_{1}<i_{2}<\dots<i_{r} \ee\]
Each base of the complete basis is represented by a non-commutative symbol (except for the scalar 1) with name \({{\eb}}_{i_{1}}\dots {{\eb}}_{i_{r}}\) so that the general multivector \({\boldsymbol{A}}\) is represented by (\(A\) is the scalar part of the multivector and the \(A^{i_{1},\dots,i_{r}}\) are scalars)
\[\begin{split}\be {\boldsymbol{A}} = A + \sum_{r=1}^{n}\sum_{\substack{i_{1},\dots,i_{r}\\ 0\le i_{j}<i_{j+1} \le n}} A^{i_{1},\dots,i_{r}}{{\eb}}_{i_{1}}{{\eb}}_{i_{2}}\dots {{\eb}}_{r} \ee\end{split}\]
The critical operation in setting up the geometric algebra is reducing the geometric product of any two bases to a linear combination of bases so that we can calculate a multiplication table for the bases. Since the geometric product is associative we can use the operation (by definition for two vectors \(a\cdot b \equiv (ab+ba)/2\) which is a scalar)
\[\be \label{reduce} {{\eb}}_{i_{j+1}}{{\eb}}_{i_{j}} = 2{{\eb}}_{i_{j+1}}\cdot {{\eb}}_{i_{j}} - {{\eb}}_{i_{j}}{{\eb}}_{i_{j+1}} \ee\]
These processes are repeated until every basis list in \({\boldsymbol{A}}\) is in normal (ascending) order with no repeated elements. As an example consider the following
\[\begin{split}\begin{aligned} {{\eb}}_{3}{{\eb}}_{2}{{\eb}}_{1} &= (2({{\eb}}_{2}\cdot {{\eb}}_{3}) - {{\eb}}_{2}{{\eb}}_{3}){{\eb}}_{1} \\ &= 2{\lp {{{\eb}}_{2}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{1} - {{\eb}}_{2}{{\eb}}_{3}{{\eb}}_{1} \\ &= 2{\lp {{{\eb}}_{2}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{1} - {{\eb}}_{2}{\lp {2{\lp {{{\eb}}_{1}\cdot {{\eb}}_{3}} \rp }-{{\eb}}_{1}{{\eb}}_{3}} \rp } \\ &= 2{\lp {{\lp {{{\eb}}_{2}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{1}-{\lp {{{\eb}}_{1}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{2}} \rp }+{{\eb}}_{2}{{\eb}}_{1}{{\eb}}_{3} \\ &= 2{\lp {{\lp {{{\eb}}_{2}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{1}-{\lp {{{\eb}}_{1}\cdot {{\eb}}_{3}} \rp }{{\eb}}_{2}+ {\lp {{{\eb}}_{1}\cdot {{\eb}}_{2}} \rp }{{\eb}}_{3}} \rp }-{{\eb}}_{1}{{\eb}}_{2}{{\eb}}_{3} \end{aligned}\end{split}\]
which results from repeated application of eq. (\(\ref{reduce}\)). If the product of basis vectors contains repeated factors eq. (\(\ref{reduce}\)) can be used to bring the repeated factors next to one another so that if \({{\eb}}_{i_{j}} = {{\eb}}_{i_{j+1}}\) then \({{\eb}}_{i_{j}}{{\eb}}_{i_{j+1}} = {{\eb}}_{i_{j}}\cdot {{\eb}}_{i_{j+1}}\) which is a scalar that commutes with all the terms in the product and can be brought to the front of the product. Since every repeated pair of vectors in a geometric product of \(r\) factors reduces the number of non-commutative factors in the product by \(r-2\). The number of bases in the multivector algebra is \(2^{n}\) and the number containing \(r\) factors is \({n\choose r}\) which is the number of combinations or \(n\) things taken \(r\) at a time (binomial coefficient).
The other construction required for formulating the geometric algebra is the outer or wedge product (symbol \({\wedge}\)) of \(r\) vectors denoted by \(a_{1}{\wedge}\dots{\wedge}a_{r}\). The wedge product of \(r\) vectors is called an \(r\)-blade and is defined by ([DL03],p86)
\[\be a_{1}{\wedge}\dots{\wedge}a_{r} \equiv \sum_{i_{j_{1}}\dots i_{j_{r}}} \epsilon^{i_{j_{1}}\dots i_{j_{r}}}a_{i_{j_{1}}}\dots a_{i_{j_{1}}} \ee\]
where \(\epsilon^{i_{j_{1}}\dots i_{j_{r}}}\) is the contravariant permutation symbol which is \(+1\) for an even permutation of the superscripts, \(0\) if any superscripts are repeated, and \(-1\) for an odd permutation of the superscripts. From the definition \(a_{1}{\wedge}\dots{\wedge}a_{r}\) is antisymmetric in all its arguments and the following relation for the wedge product of a vector \(a\) and an \(r\)-blade \(B_{r}\) can be derived
\[\be \label{wedge} a{\wedge}B_{r} = (aB_{r}+(-1)^{r}B_{r}a)/2 \ee\]
Using eq. (\(\ref{wedge}\)) one can represent the wedge product of all the basis vectors in terms of the geometric product of all the basis vectors so that one can solve (the system of equations is lower diagonal) for the geometric product of all the basis vectors in terms of the wedge product of all the basis vectors. Thus a general multivector \({\boldsymbol{B}}\) can be represented as a linear combination of a scalar and the basis blades.
\[\be {\boldsymbol{B}} = B + \sum_{r=1}^{n}\sum_{i_{1},\dots,i_{r},\;\forall\; 0\le i_{j} \le n} B^{i_{1},\dots,i_{r}}{{\eb}}_{i_{1}}{\wedge}{{\eb}}_{i_{2}}{\wedge}\dots{\wedge}{{\eb}}_{r} \ee\]
Using the blades \({{\eb}}_{i_{1}}{\wedge}{{\eb}}_{i_{2}}{\wedge}\dots{\wedge}{{\eb}}_{r}\) creates a graded algebra where \(r\) is the grade of the basis blades. The grade-\(r\) part of \({\boldsymbol{B}}\) is the linear combination of all terms with grade \(r\) basis blades.
Grade Projection¶
The scalar part of \({\boldsymbol{B}}\) is defined to be grade-\(0\). Now that the blade expansion of \({\boldsymbol{B}}\) is defined we can also define the grade projection operator \({\left <{{\boldsymbol{B}}} \right >_{r}}\) by
\[\be {\left <{{\boldsymbol{B}}} \right >_{r}} = \sum_{i_{1},\dots,i_{r},\;\forall\; 0\le i_{j} \le n} B^{i_{1},\dots,i_{r}}{{\eb}}_{i_{1}}{\wedge}{{\eb}}_{i_{2}}{\wedge}\dots{\wedge}{{\eb}}_{r} \ee\]
\[\be {\left <{{\boldsymbol{B}}} \right >_{}} \equiv {\left <{{\boldsymbol{B}}} \right >_{0}} = B \ee\]
Multivector Products¶
Then if \({\boldsymbol{A}}_{r}\) is an \(r\)-grade multivector and \({\boldsymbol{B}}_{s}\) is an \(s\)-grade multivector we have
\[\be {\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s} = {\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{{\left |{r-s}\right |}}}+{\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{{\left |{r-s}\right |}+2}}+\cdots {\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{r+s}} \ee\]
and define ([HS84],p6)
\[\begin{split}\begin{aligned} {\boldsymbol{A}}_{r}{\wedge}{\boldsymbol{B}}_{s} &\equiv {\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{r+s}} \\ {\boldsymbol{A}}_{r}\cdot{\boldsymbol{B}}_{s} &\equiv {\left \{ { \begin{array}{cc} r\mbox{ and }s \ne 0: & {\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{{\left |{r-s}\right |}}} \\ r\mbox{ or }s = 0: & 0 \end{array}} \right \}} \end{aligned}\end{split}\]
where \({\boldsymbol{A}}_{r}\cdot{\boldsymbol{B}}_{s}\) is called the dot or inner product of two pure grade multivectors. For the case of two non-pure grade multivectors
\[\begin{split}\begin{aligned} {\boldsymbol{A}}{\wedge}{\boldsymbol{B}} &= \sum_{r,s}{\left <{{\boldsymbol{A}}} \right >_{r}}{\wedge}{\left <{{\boldsymbol{B}}} \right >_{{s}}} \\ {\boldsymbol{A}}\cdot{\boldsymbol{B}} &= \sum_{r,s\ne 0}{\left <{{\boldsymbol{A}}} \right >_{r}}\cdot{\left <{{\boldsymbol{B}}} \right >_{{s}}} \end{aligned}\end{split}\]
Two other products, the left (\(\rfloor\)) and right (\(\lfloor\)) contractions, are defined by
\[\begin{split}\begin{aligned} {\boldsymbol{A}}\lfloor{\boldsymbol{B}} &\equiv \sum_{r,s}{\left \{ {\begin{array}{cc} {\left <{{\boldsymbol{A}}_r{\boldsymbol{B}}_{s}} \right >_{r-s}} & r \ge s \\ 0 & r < s \end{array}} \right \}} \\ {\boldsymbol{A}}\rfloor{\boldsymbol{B}} &\equiv \sum_{r,s}{\left \{ {\begin{array}{cc} {\left <{{\boldsymbol{A}}_{r}{\boldsymbol{B}}_{s}} \right >_{s-r}} & s \ge r \\ 0 & s < r\end{array}} \right \}} \end{aligned}\end{split}\]
Reverse of Multivector¶
A final operation for multivectors is the reverse. If a multivector \({\boldsymbol{A}}\) is the geometric product of \(r\) vectors (versor) so that \({\boldsymbol{A}} = a_{1}\dots a_{r}\) the reverse is defined by
\[\begin{aligned} {\boldsymbol{A}}^{{\dagger}} \equiv a_{r}\dots a_{1} \end{aligned}\]
where for a general multivector we have (the the sum of the reverse of versors)
\[\be {\boldsymbol{A}}^{{\dagger}} = A + \sum_{r=1}^{n}(-1)^{r(r-1)/2}\sum_{i_{1},\dots,i_{r},\;\forall\; 0\le i_{j} \le n} A^{i_{1},\dots,i_{r}}{{\eb}}_{i_{1}}{\wedge}{{\eb}}_{i_{2}}{\wedge}\dots{\wedge}{{\eb}}_{r} \ee\]
note that if \({\boldsymbol{A}}\) is a versor then \({\boldsymbol{A}}{\boldsymbol{A}}^{{\dagger}}\in\Re\) and (\(AA^{{\dagger}} \ne 0\))
\[\be {\boldsymbol{A}}^{-1} = {\displaystyle\frac{{\boldsymbol{A}}^{{\dagger}}}{{\boldsymbol{AA}}^{{\dagger}}}} \ee\]
The reverse is important in the theory of rotations in \(n\)-dimensions. If \(R\) is the product of an even number of vectors and \(RR^{{\dagger}} = 1\) then \(RaR^{{\dagger}}\) is a composition of rotations of the vector \(a\). If \(R\) is the product of two vectors then the plane that \(R\) defines is the plane of the rotation. That is to say that \(RaR^{{\dagger}}\) rotates the component of \(a\) that is projected into the plane defined by \(a\) and \(b\) where \(R=ab\). \(R\) may be written \(R = e^{\frac{\theta}{2}U}\), where \(\theta\) is the angle of rotation and \(U\) is a unit blade \(\lp U^{2} = \pm 1\rp\) that defines the plane of rotation.
Reciprocal Frames¶
If we have \(M\) linearly independent vectors (a frame), \(a_{1},\dots,a_{M}\), then the reciprocal frame is \(a^{1},\dots,a^{M}\) where \(a_{i}\cdot a^{j} = \delta_{i}^{j}\), \(\delta_{i}^{j}\) is the Kronecker delta (zero if \(i \ne j\) and one if \(i = j\)). The reciprocal frame is constructed as follows:
\[\be E_{M} = a_{1}{\wedge}\dots{\wedge}a_{M} \ee\]
\[\be E_{M}^{-1} = {\displaystyle\frac{E_{M}}{E_{M}^{2}}} \ee\]
\[\be a^{i} = \lp -1\rp ^{i-1}\lp a_{1}{\wedge}\dots{\wedge}\breve{a}_{i} {\wedge}\dots{\wedge}a_{M}\rp E_{M}^{-1} \ee\]
where \(\breve{a}_{i}\) indicates that \(a_{i}\) is to be deleted from the product. In the standard notation if a vector is denoted with a subscript the reciprocal vector is denoted with a superscript. The set of reciprocal vectors will be calculated if a coordinate set is given when a geometric algebra is instantiated since they are required for geometric differentiation when the Ga member function Ga.mvr() is called to return the reciprocal basis in terms of the basis vectors.
Manifolds and Submanifolds¶
A \(m\)-dimensional vector manifold4, \(\mathcal{M}\), is defined by a coordinate tuple (tuples are indicated by the vector accent "\(\vec{\;\;\;}\)")
\[\be \vec{x} = \paren{x^{1},\dots,x^{m}}, \ee\]
and the differentiable mapping (\(U^{m}\) is an \(m\)-dimensional subset of \(\Re^{m}\))
\[\be \f{\bm{e}^{\mathcal{M}}}{\vec{x}}\colon U^{m}\subseteq\Re^{m}\rightarrow \mathcal{V}, \ee\]
where \(\mathcal{V}\) is a vector space with an inner product5 (\(\cdot\)) and is of \({{\dim}\lp {\mathcal{V}} \rp } \ge m\).
Then a set of basis vectors for the tangent space of \(\mathcal{M}\) at \(\vec{x}\), \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\), are
\[\be \bm{e}_{i}^{\mathcal{M}} = \pdiff{\bm{e}^{\mathcal{M}}}{x^{i}} \ee\]
\[\be \f{g_{ij}^{\mathcal{M}}}{\vec{x}} = \bm{e}_{i}^{\mathcal{M}}\cdot\bm{e}_{j}^{\mathcal{M}}. \ee\]
A \(n\)-dimensional (\(n\le m\)) submanifold \(\mathcal{N}\) of \(\mathcal{M}\) is defined by a coordinate tuple
\[\be \vec{u} = \paren{u^{1},\dots,u^{n}}, \ee\]
and a differentiable mapping
\[\be \label{eq_79} \f{\vec{x}}{\vec{u}}\colon U^{n}\subseteq\Re^{n}\rightarrow U^{m}\subseteq\Re^{m}, \ee\]
Then the basis vectors for the tangent space \({{{\mathcal{T}_{\vec{u}}}\lp {\mathcal{N}} \rp }}\) are (using \({{{{\eb}}^{\mathcal{N}}}\lp {\vec{u}} \rp } = {{{{\eb}}^{\mathcal{M}}}\lp {{{\vec{x}}\lp {\vec{u}} \rp }} \rp }\) and the chain rule)6
\[\be \f{\bm{e}_{i}^{\mathcal{N}}}{\vec{u}} = \pdiff{\f{\bm{e}^{\mathcal{N}}}{\vec{u}}}{u^{i}} = \pdiff{\f{\bm{e}^{\mathcal{M}}}{\vec{x}}}{x^{j}}\pdiff{x^{j}}{u^{i}} = \f{\bm{e}_{j}^{\mathcal{M}}}{\f{\vec{x}}{\vec{u}}}\pdiff{x^{j}}{u^{i}}, \ee\]
\[\be \label{eq_81} \f{g_{ij}^{\mathcal{N}}}{\vec{u}} = \pdiff{x^{k}}{u^{i}}\pdiff{x^{l}}{u^{j}} \f{g_{kl}^{\mathcal{M}}}{\f{\vec{x}}{\vec{u}}}. \ee\]
Going back to the base manifold, \(\mathcal{M}\), note that the mapping \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{n}\subseteq\Re^{n}\rightarrow \mathcal{V}\) allows us to calculate an unnormalized pseudo-scalar for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\),
\[\be \f{I^{\mathcal{M}}}{\vec{x}} = \f{\bm{e}_{1}^{\mathcal{M}}}{\vec{x}} \W\dots\W\f{\bm{e}_{m}^{\mathcal{M}}}{\vec{x}}. \ee\]
With the pseudo-scalar we can define a projection operator from \(\mathcal{V}\) to the tangent space of \(\mathcal{M}\) by
\[\be \f{P_{\vec{x}}}{\bm{v}} = (\bm{v}\cdot \f{I^{\mathcal{M}}}{\vec{x}}) \paren{\f{I^{\mathcal{M}}}{\vec{x}}}^{-1} \;\forall\; \bm{v}\in\mathcal{V}. \ee\]
In fact for each tangent space \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) we can define a geometric algebra \({{\mathcal{G}}\lp {{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}} \rp }\) with pseudo-scalar \(I^{\mathcal{M}}\) so that if \(A \in {{\mathcal{G}}\lp {\mathcal{V}} \rp }\) then
\[\be \f{P_{\vec{x}}}{A} = \paren{A\cdot \f{I^{\mathcal{M}}}{\vec{x}}} \paren{\f{I^{\mathcal{M}}}{\vec{x}}}^{-1} \in \f{\mathcal{G}}{\Tn{\mathcal{M}}{\vec{x}}}\;\forall\; A \in \f{\mathcal{G}}{\mathcal{V}} \ee\]
and similarly for the submanifold \(\mathcal{N}\).
If the embedding \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{n}\subseteq\Re^{n}\rightarrow \mathcal{V}\) is not given, but the metric tensor \({{g_{ij}^{\mathcal{M}}}\lp {\vec{x}} \rp }\) is given the geometric algebra of the tangent space can be constructed. Also the derivatives of the basis vectors of the tangent space can be calculated from the metric tensor using the Christoffel symbols, \({{\Gamma_{ij}^{k}}\lp {\vec{u}} \rp }\), where the derivatives of the basis vectors are given by
\[\be \pdiff{\bm{e}_{j}^{\mathcal{M}}}{x^{i}} =\f{\Gamma_{ij}^{k}}{\vec{u}}\bm{e}_{k}^{\mathcal{M}}. \ee\]
If we have a submanifold, \(\mathcal{N}\), defined by eq. (\(\ref{eq_79}\)) we can calculate the metric of \(\mathcal{N}\) from eq. (\(\ref{eq_81}\)) and hence construct the geometric algebra and calculus of the tangent space, \({{{\mathcal{T}_{\vec{u}}}\lp {\mathcal{N}} \rp }}\subseteq {{{\mathcal{T}_{{{\vec{x}}\lp {\vec{u}} \rp }}}\lp {\mathcal{M}} \rp }}\).
If the base manifold is normalized (use the hat symbol to denote normalized tangent vectors, \(\hat{{\eb}}_{i}^{\mathcal{M}}\), and the resulting metric tensor, \(\hat{g}_{ij}^{\mathcal{M}}\)) we have \(\hat{{\eb}}_{i}^{\mathcal{M}}\cdot\hat{{\eb}}_{i}^{\mathcal{M}} = \pm 1\) and \(\hat{g}_{ij}^{\mathcal{M}}\) does not posses enough information to calculate \(g_{ij}^{\mathcal{N}}\). In that case we need to know \(g_{ij}^{\mathcal{M}}\), the metric tensor of the base manifold before normalization. Likewise, for the case of a vector manifold unless the mapping, \({{{\eb}^{\mathcal{M}}}\lp {\vec{x}} \rp }\colon U^{m}\subseteq\Re^{m}\rightarrow \mathcal{V}\), is constant the tangent vectors and metric tensor can only be normalized after the fact (one cannot have a mapping that automatically normalizes all the tangent vectors).
Geometric Derivative¶
The directional derivative of a multivector field \({{F}\lp {x} \rp }\) is defined by (\(a\) is a vector and \(h\) is a scalar)
\[\be \paren{a\cdot\nabla_{x}}F \equiv \lim_{h\rightarrow 0}\bfrac{\f{F}{x+ah}-\f{F}{x}}{h}. \label{eq_50} \ee\]
Note that \(a\cdot\nabla_{x}\) is a scalar operator. It will give a result containing only those grades that are already in \(F\). \({\lp {a\cdot\nabla_{x}} \rp }F\) is the best linear approximation of \({{F}\lp {x} \rp }\) in the direction \(a\). Equation (\(\ref{eq_50}\)) also defines the operator \(\nabla_{x}\) which for the basis vectors, \({\left \{{{\eb}_{i}} \rbrc}\), has the representation (note that the \({\left \{{{\eb}^{j}} \rbrc}\) are reciprocal basis vectors)
\[\be \nabla_{x} F = {\eb}^{j}{\displaystyle\frac{\partial F}{\partial x^{j}}} \ee\]
If \(F_{r}\) is a \(r\)-grade multivector (if the independent vector, \(x\), is obvious we suppress it in the notation and just write \(\nabla\)) and \(F_{r} = F_{r}^{i_{1}\dots i_{r}}{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}\) then
\[\be \nabla F_{r} = {\displaystyle\frac{\partial F_{r}^{i_{1}\dots i_{r}}}{\partial x^{j}}}{\eb}^{j}\lp {\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}} \rp \ee\]
Note that \({\eb}^{j}\lp {\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}} \rp\) can only contain grades \(r-1\) and \(r+1\) so that \(\nabla F_{r}\) also can only contain those grades. For a grade-\(r\) multivector \(F_{r}\) the inner (div) and outer (curl) derivatives are
\[\be \nabla\cdot F_{r} = \left < \nabla F_{r}\right >_{r-1} = {\eb}^{j}\cdot {{\displaystyle\frac{\partial {F_{r}}}{\partial {x^{j}}}}} \ee\]
\[\be \nabla{\wedge}F_{r} = \left < \nabla F_{r}\right >_{r+1} = {\eb}^{j}{\wedge}{{\displaystyle\frac{\partial {F_{r}}}{\partial {x^{j}}}}} \ee\]
For a general multivector function \(F\) the inner and outer derivatives are just the sum of the inner and outer derivatives of each grade of the multivector function.
Geometric Derivative on a Manifold¶
In the case of a manifold the derivatives of the \({\eb}_{i}\)'s are functions of the coordinates, \({\left \{{x^{i}} \rbrc}\), so that the geometric derivative of a \(r\)-grade multivector field is
\[\begin{split}\begin{aligned} \nabla F_{r} &= {\eb}^{i}{{\displaystyle\frac{\partial {F_{r}}}{\partial {x^{i}}}}} = {\eb}^{i}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}} {\lp {F_{r}^{i_{1}\dots i_{r}}{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp } \nonumber \\ &= {{\displaystyle\frac{\partial {F_{r}^{i_{1}\dots i_{r}}}}{\partial {x^{i}}}}}{\eb}^{i}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp } +F_{r}^{i_{1}\dots i_{r}}{\eb}^{i}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp }\end{aligned}\end{split}\]
where the multivector functions \({\eb}^{i}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp }\) are the connection for the manifold7.
The directional (material/convective) derivative, \({\lp {v\cdot\nabla} \rp }F_{r}\) is given by
\[\begin{split}\begin{aligned} {\lp {v\cdot\nabla} \rp } F_{r} &= v^{i}{{\displaystyle\frac{\partial {F_{r}}}{\partial {x^{i}}}}} = v^{i}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}} {\lp {F_{r}^{i_{1}\dots i_{r}}{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp } \nonumber \\ &= v^{i}{{\displaystyle\frac{\partial {F_{r}^{i_{1}\dots i_{r}}}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp } +v^{i}F_{r}^{i_{1}\dots i_{r}}{{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp },\end{aligned}\end{split}\]
so that the multivector connection functions for the directional derivative are \({{\displaystyle\frac{\partial {}}{\partial {x^{i}}}}}{\lp {{\eb}_{i_{1}}{\wedge}\dots{\wedge}{\eb}_{i_{r}}} \rp }\). Be careful and note that \({\lp {v\cdot\nabla} \rp } F_{r} \ne v\cdot {\lp {\nabla F_{r}} \rp }\) since the dot and geometric products are not associative with respect to one another (\(v\cdot\nabla\) is a scalar operator).
Normalizing Basis for Derivatives¶
The basis vector set, \({\left \{ {{\eb}_{i}} \rbrc}\), is not in general normalized. We define a normalized set of basis vectors, \({\left \{{{\boldsymbol{\hat{e}}}_{i}} \rbrc}\), by
\[\be {\boldsymbol{\hat{e}}}_{i} = {\displaystyle\frac{{\eb}_{i}}{\sqrt{{\left |{{\eb}_{i}^{2}}\right |}}}} = {\displaystyle\frac{{\eb}_{i}}{{\left |{{\eb}_{i}}\right |}}}. \ee\]
This works for all \({\eb}_{i}^{2} \neq 0\). Note that \({\boldsymbol{\hat{e}}}_{i}^{2} = \pm 1\).
Thus the geometric derivative for a set of normalized basis vectors is (where \(F_{r} = F_{r}^{i_{1}\dots i_{r}} \bm{\hat{e}}_{i_{1}}\W\dots\W\bm{\hat{e}}_{i_{r}}\) and [no summation] \(\hat{F}_{r}^{i_{1}\dots i_{r}} = F_{r}^{i_{1}\dots i_{r}} \abs{\bm{\hat{e}}_{i_{1}}}\dots\abs{\bm{\hat{e}}_{i_{r}}}\)).
\[\be \nabla F_{r} = \eb^{i}\pdiff{F_{r}}{x^{i}} = \pdiff{F_{r}^{i_{1}\dots i_{r}}}{x^{i}}\bm{e}^{i} \paren{\bm{\hat{e}}_{i_{1}}\W\dots\W\bm{\hat{e}}_{i_{r}}} +F_{r}^{i_{1}\dots i_{r}}\bm{e}^{i}\pdiff{}{x^{i}} \paren{\bm{\hat{e}}_{i_{1}}\W\dots\W\bm{\hat{e}}_{i_{r}}}. \ee\]
To calculate \({\eb}^{i}\) in terms of the \({\boldsymbol{\hat{e}}}_{i}\)'s we have
\[\begin{split}\begin{aligned} {\eb}^{i} &= g^{ij}{\eb}_{j} \nonumber \\ {\eb}^{i} &= g^{ij}{\left |{{\eb}_{j}}\right |}{\boldsymbol{\hat{e}}}_{j}.\end{aligned}\end{split}\]
This is the general (non-orthogonal) formula. If the basis vectors are orthogonal then (no summation over repeated indexes)
\[\begin{split}\begin{aligned} {\eb}^{i} &= g^{ii}{\left |{{\eb}_{i}}\right |}{\boldsymbol{\hat{e}}}_{i} \nonumber \\ {\eb}^{i} &= {\displaystyle\frac{{\left |{{\eb}_{i}}\right |}}{g_{ii}}}{\boldsymbol{\hat{e}}}_{i} = {\displaystyle\frac{{\left |{{\boldsymbol{\hat{e}}}_{i}}\right |}}{{\eb}_{i}^{2}}}{\boldsymbol{\hat{e}}}_{i}.\end{aligned}\end{split}\]
Additionally, one can calculate the connection of the normalized basis as follows
\[\begin{split}\begin{aligned} {{\displaystyle\frac{\partial {{\lp {{\left |{{\eb}_{i}}\right |}{\boldsymbol{\hat{e}}}_{i}} \rp }}}{\partial {x^{j}}}}} =& {{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}}, \nonumber \\ {{\displaystyle\frac{\partial {{\left |{{\eb}_{i}}\right |}}}{\partial {x^{j}}}}}{\boldsymbol{\hat{e}}}_{i} +{\left |{{\eb}_{i}}\right |}{{\displaystyle\frac{\partial {{\boldsymbol{\hat{e}}}_{i}}}{\partial {x^{j}}}}} =& {{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}}, \nonumber \\ {{\displaystyle\frac{\partial {{\boldsymbol{\hat{e}}}_{i}}}{\partial {x^{j}}}}} =& {\displaystyle\frac{1}{{\left |{{\eb}_{i}}\right |}}}{\lp {{{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}} -{{\displaystyle\frac{\partial {{\left |{{\eb}_{i}}\right |}}}{\partial {x^{j}}}}}{\boldsymbol{\hat{e}}}_{i}} \rp },\nonumber \\ =& {\displaystyle\frac{1}{{\left |{{\eb}_{i}}\right |}}}{{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}} -{\displaystyle\frac{1}{{\left |{{\eb}_{i}}\right |}}}{{\displaystyle\frac{\partial {{\left |{{\eb}_{i}}\right |}}}{\partial {x^{j}}}}}{\boldsymbol{\hat{e}}}_{i},\nonumber \\ =& {\displaystyle\frac{1}{{\left |{{\eb}_{i}}\right |}}}{{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}} -{\displaystyle\frac{1}{2g_{ii}}}{{\displaystyle\frac{\partial {g_{ii}}}{\partial {x^{j}}}}}{\boldsymbol{\hat{e}}}_{i},\end{aligned}\end{split}\]
where \({{\displaystyle\frac{\partial {{\eb}_{i}}}{\partial {x^{j}}}}}\) is expanded in terms of the \({\boldsymbol{\hat{e}}}_{i}\)'s.
Linear Differential Operators¶
First a note on partial derivative notation. We shall use the following notation for a partial derivative where the manifold coordinates are \(x_{1},\dots,x_{n}\):
\[\be\label{eq_66a} \bfrac{\partial^{j_{1}+\cdots+j_{n}}}{\partial x_{1}^{j_{1}}\dots\partial x_{n}^{j_{n}}} = \partial_{j_{1}\dots j_{n}}. \ee\]
If \(j_{k}=0\) the partial derivative with respect to the \(k^{th}\) coordinate is not taken. If \(j_{k} = 0\) for all \(1 \le k \le n\) then the partial derivative operator is the scalar one. If we consider a partial derivative where the \(x\)'s are not in normal order such as
\[\be {\displaystyle\frac{\partial^{j_{1}+\cdots+j_{n}}}{\partial x_{i_{1}}^{j_{1}}\dots\partial x_{i_{n}}^{j_{n}}}}, \ee\]
and the \(i_{k}\)'s are not in ascending order. The derivative can always be put in the form in eq (\(\ref{eq_66a}\)) since the order of differentiation does not change the value of the partial derivative (for the smooth functions we are considering). Additionally, using our notation the product of two partial derivative operations is given by
\[\be \partial_{i_{1}\dots i_{n}}\partial_{j_{1}\dots j_{n}} = \partial_{i_{1}+j_{1},\dots, i_{n}+j_{n}}. \ee\]
A general general multivector linear differential operator is a linear combination of multivectors and partial derivative operators denoted by
\[\be\label{eq_66b} D \equiv D^{i_{1}\dots i_{n}}\partial_{i_{1}\dots i_{n}}. \ee\]
Equation (\(\ref{eq_66b}\)) is the normal form of the differential operator in that the partial derivative operators are written to the right of the multivector coefficients and do not operate upon the multivector coefficients. The operator of eq (\(\ref{eq_66b}\)) can operate on mulitvector functions, returning a multivector function via the following definitions.
\(F\) as
\[\be D\circ F = D^{j_{1}\dots j_{n}}\circ\partial_{j_{1}\dots j_{n}}F,\label{eq_67a} \ee\]
\[\be F\circ D = \partial_{j_{1}\dots j_{n}}F\circ D^{j_{1}\dots j_{n}},\label{eq_68a} \ee\]
where the \(D^{j_{1}\dots j_{n}}\) are multivector functions and \(\circ\) is any of the multivector multiplicative operations.
Equations (\(\ref{eq_67a}\)) and (\(\ref{eq_68a}\)) are not the most general multivector linear differential operators, the most general would be
\[\be D \left( F \right) = {D^{j_{1}\dots j_{n}}}\left({\partial_{j_{1}\dots j_{n}}F}\right), \ee\]
where \({{D^{j_{1}\dots j_{n}}}\lp {} \rp }\) are linear multivector functionals.
The definition of the sum of two differential operators is obvious since any multivector operator, \(\circ\), is a bilinear operator \({\lp {{\lp {D_{A}+D_{B}} \rp }\circ F = D_{A}\circ F+D_{B}\circ F} \rp }\), the product of two differential operators \(D_{A}\) and \(D_{B}\) operating on a multivector function \(F\) is defined to be (\(\circ_{1}\) and \(\circ_{2}\) are any two multivector multiplicative operations)
\[\begin{split}\begin{aligned} {\lp {D_{A}\circ_{1}D_{B}} \rp }\circ_{2}F &\equiv {\lp {D_{A}^{i_{1}\dots i_{n}}\circ_{1} \partial_{i_{1}\dots i_{n}}{\lp {D_{B}^{j_{1}\dots j_{n}} \partial_{j_{1}\dots j_{n}}} \rp }} \rp }\circ_{2}F \nonumber \\ &= {\lp {D_{A}^{i_{1}\dots i_{n}}\circ_{1} {\lp {{\lp {\partial_{i_{1}\dots i_{n}}D_{B}^{j_{1}\dots j_{n}}} \rp } \partial_{j_{1}\dots j_{n}}+ D_{B}^{j_{1}\dots j_{n}}} \rp } \partial_{i_{1}+j_{1},\dots, i_{n}+j_{n}}} \rp }\circ_{2}F \nonumber \\ &= {\lp {D_{A}^{i_{1}\dots i_{n}}\circ_{1}{\lp {\partial_{i_{1}\dots i_{n}}D_{B}^{j_{1}\dots j_{n}}} \rp }} \rp } \circ_{2}\partial_{j_{1}\dots j_{n}}F+ {\lp {D_{A}^{i_{1}\dots i_{n}}\circ_{1}D_{B}^{j_{1}\dots j_{n}}} \rp } \circ_{2}\partial_{i_{1}+j_{1},\dots, i_{n}+j_{n}}F,\end{aligned}\end{split}\]
where we have used the fact that the \(\partial\) operator is a scalar operator and commutes with \(\circ_{1}\) and \(\circ_{2}\).
Thus for a pure operator product \(D_{A}\circ D_{B}\) we have
\[\be D_{A}\circ D_{B} = \paren{D_{A}^{i_{1}\dots i_{n}}\circ\paren{\partial_{i_{1}\dots i_{n}}D_{B}^{j_{1}\dots j_{n}}}} \partial_{j_{1}\dots j_{n}}+ \paren{D_{A}^{i_{1}\dots i_{n}}\circ_{1}D_{B}^{j_{1}\dots j_{n}}} \partial_{i_{1}+j_{1},\dots, i_{n}+j_{n}} \label{eq_71a} \ee\]
and the form of eq (\(\ref{eq_71a}\)) is the same as eq (\(\ref{eq_67a}\)). The basis of eq (\(\ref{eq_71a}\)) is that the \(\partial\) operator operates on all object to the right of it as products so that the product rule must be used in all differentiations. Since eq (\(\ref{eq_71a}\)) puts the product of two differential operators in standard form we also evaluate \(F\circ_{2}{\lp {D_{A}\circ_{1}D_{B}} \rp }\).
We now must distinguish between the following cases. If \(D\) is a differential operator and \(F\) a multivector function should \(D\circ F\) and \(F\circ D\) return a differential operator or a multivector. In order to be consistent with the standard vector analysis we have \(D\circ F\) return a multivector and \(F\circ D\) return a differential operator. Then we define the complementary differential operator \(\bar{D}\) which is identical to \(D\) except that \(\bar{D}\circ F\) returns a differential operator according to eq (\(\ref{eq_71a}\))8 and \(F\circ\bar{D}\) returns a multivector according to eq (\(\ref{eq_68a}\)).
A general differential operator is built from repeated applications of the basic operator building blocks \({\lp {\bar{\nabla}\circ A} \rp }\), \({\lp {A\circ\bar{\nabla}} \rp }\), \({\lp {\bar{\nabla}\circ\bar{\nabla}} \rp }\), and \({\lp {A\pm \bar{\nabla}} \rp }\). Both \(\nabla\) and \(\bar{\nabla}\) are represented by the operator
\[\be \nabla = \bar{\nabla} = e^{i}\pdiff{}{x^{i}}, \ee\]
but are flagged to produce the appropriate result.
In the our notation the directional derivative operator is \(a\cdot\nabla\), the Laplacian \(\nabla\cdot\nabla\) and the expression for the Riemann tensor, \(R^{i}_{jkl}\), is
\[\be \paren{\nabla\W\nabla}\eb^{i} = \half R^{i}_{jkl}\paren{\eb^{j}\W\eb^{k}}\eb^{l}. \ee\]
We would use the complement if we wish a quantum mechanical type commutator defining
\[\be \com{x,\nabla} \equiv x\nabla - \bar{\nabla}x, \ee\]
, or if we wish to simulate the dot notation (Doran and Lasenby)
\[\be \dot{F}\dot{\nabla} = F\bar{\nabla}. \ee\]
Split Differential Operator¶
To implement the general "dot" notation for differential operators in python is not possible. Another type of symbolic notation is required. I propose what one could call the "split differential operator." For \(\nabla\) denote the corresponding split operator by two operators \({{\nabla}_{\mathcal{G}}}\) and \({{\nabla}_{\mathcal{D}}}\) where in practice \({{\nabla}_{\mathcal{G}}}\) is a tuple of vectors and \({{\nabla}_{\mathcal{D}}}\) is a tuple of corresponding partial derivatives. Then the equivalent of the "dot" notation would be
\[\be \dot{\nabla}{\lp {A\dot{B}C} \rp } = {{\nabla}_{\mathcal{G}}}{\lp {A{\lp {{{\nabla}_{\mathcal{D}}}B} \rp }C} \rp }.\label{splitopV} \ee\]
We are using the \(\mathcal{G}\) subscript to indicate the geometric algebra parts of the multivector differential operator and the \(\mathcal{D}\) subscript to indicate the scalar differential operator parts of the multivector differential operator. An example of this notation in 3D Euclidean space is
\[\begin{split}\begin{aligned} {{\nabla}_{\mathcal{G}}} &= {\lp {{{\eb}}_{x},{{\eb}}_{y},{{\eb}}_{z}} \rp }, \\ {{\nabla}_{\mathcal{D}}} &= {\lp {{{\displaystyle\frac{\partial {}}{\partial {x}}}},{{\displaystyle\frac{\partial {}}{\partial {y}}}},{{\displaystyle\frac{\partial {}}{\partial {x}}}}} \rp },\end{aligned}\end{split}\]
To implement \({{\nabla}_{\mathcal{G}}}\) and \({{\nabla}_{\mathcal{D}}}\) we have in the example
\[\begin{split}\begin{aligned} {{\nabla}_{\mathcal{D}}}B &= {\lp {{{\displaystyle\frac{\partial {B}}{\partial {x}}}},{{\displaystyle\frac{\partial {B}}{\partial {y}}}},{{\displaystyle\frac{\partial {B}}{\partial {z}}}}} \rp } \\ {\lp {{{\nabla}_{\mathcal{D}}}B} \rp }C &= {\lp {{{\displaystyle\frac{\partial {B}}{\partial {x}}}}C,{{\displaystyle\frac{\partial {B}}{\partial {y}}}}C,{{\displaystyle\frac{\partial {B}}{\partial {z}}}}C} \rp } \\ A{\lp {{{\nabla}_{\mathcal{D}}}B} \rp }C &= {\lp {A{{\displaystyle\frac{\partial {B}}{\partial {x}}}}C,A{{\displaystyle\frac{\partial {B}}{\partial {y}}}}C,A{{\displaystyle\frac{\partial {B}}{\partial {z}}}}C} \rp }.\end{aligned}\end{split}\]
Then the final evaluation is
\[\be {{\nabla}_{\mathcal{G}}}{\lp {A{\lp {{{\nabla}_{\mathcal{D}}}B} \rp }C} \rp } = {{\eb}}_{x}A{{\displaystyle\frac{\partial {B}}{\partial {x}}}}C+{{\eb}}_{y}A{{\displaystyle\frac{\partial {B}}{\partial {y}}}}C+{{\eb}}_{z}A{{\displaystyle\frac{\partial {B}}{\partial {z}}}}C, \ee\]
which could be called the "dot" product of two tuples. Note that \(\nabla = {{\nabla}_{\mathcal{G}}}{{\nabla}_{\mathcal{D}}}\) and \(\dot{F}\dot{\nabla} = F\bar{\nabla} = {\lp {{{\nabla}_{\mathcal{D}}}F} \rp }{{\nabla}_{\mathcal{G}}}\).
For the general multivector differential operator, \(D\), the split operator parts are \({{D}_{\mathcal{G}}}\), a tuple of basis blade multivectors and \({{D}_{\mathcal{D}}}\), a tuple of scalar differential operators that correspond to the coefficients of the basis-blades in the total operator \(D\) so that
\[\be \dot{D}{\lp {A\dot{B}C} \rp } = {{D}_{\mathcal{G}}}{\lp {A{\lp {{{D}_{\mathcal{D}}}B} \rp }C} \rp }. \label{splitopM} \ee\]
If the index set for the basis blades of a geometric algebra is denoted by \({\left \{{n} \rbrc}\) where \({\left \{{n} \rbrc}\) contains \(2^{n}\) indices for an \(n\) dimensional geometric algebra then the most general multivector differential operator can be written9
\[\be D = {{\displaystyle}\sum_{l\in{\left \{ {n} \rbrc}}{{\eb}}^{l}D_{{\left \{ {l} \rbrc}}} \ee\]
\[\be \dot{D}{\lp {A\dot{B}C} \rp } = {{D}_{\mathcal{G}}}{\lp {A{\lp {{{D}_{\mathcal{D}}}B} \rp }C} \rp } = {{\displaystyle}\sum_{l\in{\left \{ {n} \rbrc}}{{\eb}}^{l}{\lp {A{\lp {D_{l}B} \rp }C} \rp }} \ee\]
\[\be {\lp {A\dot{B}C} \rp }\dot{D} = {\lp {A{\lp {{{D}_{\mathcal{D}}}B} \rp }C} \rp }{{D}_{\mathcal{G}}} = {{\displaystyle}\sum_{l\in{\left \{ {n} \rbrc}}{\lp {A{\lp {D_{l}B} \rp }C} \rp }{{\eb}}^{l}}. \ee\]
The implementation of equations \(\ref{splitopV}\) and \(\ref{splitopM}\) is described in sections Instantiating a Multivector and Multivector Derivatives.
Linear Transformations/Outermorphisms¶
In the tangent space of a manifold, \(\mathcal{M}\), (which is a vector space) a linear transformation is the mapping \(\underline{T}\colon{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\rightarrow{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) (we use an underline to indicate a linear transformation) where for all \(x,y\in {{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) and \(\alpha\in\Re\) we have
\[\begin{split}\begin{aligned} {{\underline{T}}\lp {x+y} \rp } =& {{\underline{T}}\lp {x} \rp } + {{\underline{T}}\lp {y} \rp } \\ {{\underline{T}}\lp {\alpha x} \rp } =& \alpha{{\underline{T}}\lp {x} \rp }\end{aligned}\end{split}\]
The outermorphism induced by \(\underline{T}\) is defined for \(x_{1},\dots,x_{r}\in{{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) where \(\newcommand{\f}[2]{{#1}\lp {#2} \rp } \newcommand{\Tn}[2]{\f{\mathcal{T}_{#2}}{#1}} r\le\f{\dim}{\Tn{\mathcal{M}}{\vec{x}}}\)
\[\be \newcommand{\f}[2]{{#1}\lp {#2} \rp } \newcommand{\W}{\wedge} \f{\underline{T}}{x_{1}\W\dots\W x_{r}} \equiv \f{\underline{T}}{x_{1}}\W\dots\W\f{\underline{T}}{x_{r}} \ee\]
If \(I\) is the pseudo scalar for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) we also have the following definitions for determinate, trace, and adjoint (\(\overline{T}\)) of \(\underline{T}\)
\[\begin{split}\begin{align} \f{\underline{T}}{I} \equiv&\; \f{\det}{\underline{T}}I\text{,} \label{eq_82}\\ \f{\tr}{\underline{T}} \equiv&\; \nabla_{y}\cdot\f{\underline{T}}{y}\text{,} \label{eq_83}\\ x\cdot \f{\overline{T}}{y} \equiv&\; y\cdot \f{\underline{T}}{x}.\ \label{eq_84}\\ \end{align}\end{split}\]
If \({\left \{{{{\eb}}_{i}} \rbrc}\) is a basis for \({{{\mathcal{T}_{\vec{x}}}\lp {\mathcal{M}} \rp }}\) then we can represent \(\underline{T}\) with the matrix \(\underline{T}_{i}^{j}\) used as follows (Einstein summation convention as usual) -
\[\be \f{\underline{T}}{\eb_{i}} = \underline{T}_{i}^{j}\eb_{j}, \label{eq_85} \ee\]
The let \({\lp {\underline{T}^{-1}} \rp }_{m}^{n}\) be the inverse matrix of \(\underline{T}_{i}^{j}\) so that \({\lp {\underline{T}^{-1}} \rp }_{m}^{k}\underline{T}_{k}^{j} = \delta^{j}_{m}\) and
\[\be \underline{T}^{-1}{\lp {a^{i}{{\eb}}_{i}} \rp } = a^{i}{\lp {\underline{T}^{-1}} \rp }_{i}^{j}{{\eb}}_{j} \label{eq_85a} \ee\]
and calculate
\[\begin{split}\begin{aligned} \underline{T}^{-1}{\lp {\underline{T}{\lp {a} \rp }} \rp } &= \underline{T}^{-1}{\lp {\underline{T}{\lp {a^{i}{{\eb}}_{i}} \rp }} \rp } \nonumber \\ &= \underline{T}^{-1}{\lp {a^{i}\underline{T}_{i}^{j}{{\eb}}_{j}} \rp } \nonumber \\ &= a^{i}{\lp {\underline{T}^{-1}} \rp }_{i}^{j} \underline{T}_{j}^{k}{{\eb}}_{k} \nonumber \\ &= a^{i}\delta_{i}^{j}{{\eb}}_{j} = a^{i}{{\eb}}_{i} = a.\end{aligned}\end{split}\]
Thus if eq \(\ref{eq_85a}\) is used to define the \(\underline{T}_{i}^{j}\) then the linear transformation defined by the matrix \({\lp {\underline{T}^{-1}} \rp }_{m}^{n}\) is the inverse of \(\underline{T}\).
In eq. (\(\ref{eq_85}\)) the matrix, \(\underline{T}_{i}^{j}\), only has it's usual meaning if the \({\left \{{{{\eb}}_{i}} \rbrc}\) form an orthonormal Euclidean basis (Minkowski spaces not allowed). Equations (\(\ref{eq_82}\)) through (\(\ref{eq_84}\)) become
\[\begin{split}\begin{aligned} {{\det}\lp {\underline{T}} \rp } =&\; {{\underline{T}}\lp {{{\eb}}_{1}{\wedge}\dots{\wedge}{{\eb}}_{n}} \rp }{\lp {{{\eb}}_{1}{\wedge}\dots{\wedge}{{\eb}}_{n}} \rp }^{-1},\\ {{{\mbox{tr}}}\lp {\underline{T}} \rp } =&\; \underline{T}_{i}^{i},\\ \overline{T}_{j}^{i} =&\; g^{il}g_{jp}\underline{T}_{l}^{p}.\end{aligned}\end{split}\]
A important form of linear transformation with a simple representation is the spinor transformation. If \(S\) is an even multivector we have \(SS^{{\dagger}} = \rho^{2}\), where \(\rho^{2}\) is a scalar. Then \(S\) is a spinor transformation is given by (\(v\) is a vector)
\[\be {{S}\lp {v} \rp } = SvS^{{\dagger}} \ee\]
if \({{S}\lp {v} \rp }\) is a vector and
\[\be {{S^{-1}}\lp {v} \rp } = \frac{S^{{\dagger}}vS}{\rho^{4}}. \ee\]
\[\begin{split}\begin{aligned} {{S^{-1}}\lp {{{S}\lp {v} \rp }} \rp } &= \frac{S^{{\dagger}}SvS^{{\dagger}}S}{\rho^{4}} \nonumber \\ &= \frac{\rho^{2}v\rho^{2}}{\rho^{4}} \nonumber \\ &= v. \end{aligned}\end{split}\]
One more topic to consider is whether or not \(T^{i}_{j}\) should be called the matrix representation of \(T\) ? The reason that this is a question is that for a general metric \(g_{ij}\) is that because of the dependence of the dot product on the metric \(T^{i}_{j}\) does not necessarily show the symmetries of the underlying transformation \(T\). Consider the expression
\[\begin{split}\begin{aligned} a\cdot{{T}\lp {b} \rp } &= a^{i}{{\eb}}_{i}\cdot{{T}\lp {b^{j}{{\eb}}_{j}} \rp } \nonumber \\ &= a^{i}{{\eb}}_{i}\cdot {{T}\lp {{{\eb}}_{j}} \rp }b^{j} \nonumber \\ &= a^{i}{{\eb}}_{i}\cdot{{\eb}}_{k} T_{j}^{k}b^{j} \nonumber \\ &= a^{i}g_{ik}T_{j}^{k}b^{j}.\end{aligned}\end{split}\]
\[\be T_{ij} = g_{ik}T_{j}^{k} \ee\]
that has the proper symmetry for self adjoint transformations \((a\cdot{{T}\lp {b} \rp } = b\cdot{{T}\lp {a} \rp })\) in the sense that if \(T = \overline{T}\) then \(T_{ij} = T_{ji}\). Of course if we are dealing with a manifold where the \(g_{ij}\)'s are functions of the coordinates then the matrix representation of a linear transformation will also be a function of the coordinates. Assuming we use \(T_{ij}\) for the matrix representation of the linear transformation, \(T\), then if we given the matrix representation, \(T_{ij}\), we can construct the linear transformation given by \(T^{i}_{j}\) as follows
\[\begin{split}\begin{aligned} T_{ij} &= g_{ik}T_{j}^{k} \nonumber \\ g^{li}T_{ij} &= g^{li}g_{ik}T_{j}^{k} \nonumber \\ g^{li}T_{ij} &= \delta_{k}^{l}T_{j}^{k} \nonumber \\ g^{li}T_{ij} &= T_{j}^{l}.\end{aligned}\end{split}\]
Any program/code that represents \(T\) should allow one to define \(T\) in terms of \(T_{ij}\) or \(T_{j}^{l}\) and likewise given a linear transformation \(T\) obtain both \(T_{ij}\) and \(T_{j}^{l}\) from it. Please note that these considerations come into play for any non-Euclidean metric with respect to the trace and adjoint of a linear transformation since calculating either requires a dot product.
Multilinear Functions¶
A multivector multilinear function10 is a multivector function \({{T}\lp {A_{1},\dots,A_{r}} \rp }\) that is linear in each of it arguments11 (it could be implicitly non-linearly dependent on a set of additional arguments such as the position coordinates, but we only consider the linear arguments). \(T\) is a tensor of degree \(r\) if each variable \(A_{j}\) is restricted to the vector space \(\mathcal{V}_{n}\). More generally if each \(A_{j}\in{{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\) (the geometric algebra of \(\mathcal{V}_{n}\)), we call \(T\) an extensor of degree-\(r\) on \({{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\).
If the values of \({{T} \lp {a_{1},\dots,a_{r}} \rp }\) \(\lp a_{j}\in\mathcal{V}_{n}\;\forall\; 1\le j \le r \rp\) are \(s\)-vectors (pure grade \(s\) multivectors in \({{\mathcal{G}}\lp {\mathcal{V}_{n}} \rp }\)) we say that \(T\) has grade \(s\) and rank \(r+s\). A tensor of grade zero is called a multilinear form.
In the normal definition of tensors as multilinear functions the tensor is defined as a mapping
\[T:{\huge \times}_{i=1}^{r}\mathcal{V}_{i}\rightarrow\Re,\]
so that the standard tensor definition is an example of a grade zero degree/rank$ r $ tensor in our definition.
Algebraic Operations¶
The properties of tensors are (\(\alpha\in\Re\), \(a_{j},b\in\mathcal{V}_{n}\), \(T\) and \(S\) are tensors of rank \(r\), and \(\circ\) is any multivector multiplicative operation)
\[\begin{split}\begin{aligned} {{T}\lp {a_{1},\dots,\alpha a_{j},\dots,a_{r}} \rp } =& \alpha{{T}\lp {a_{1},\dots,a_{j},\dots,a_{r}} \rp }, \\ {{T}\lp {a_{1},\dots,a_{j}+b,\dots,a_{r}} \rp } =& {{T}\lp {a_{1},\dots,a_{j},\dots,a_{r}} \rp }+ {{T}\lp {a_{1},\dots,a_{j-1},b,a_{j+1},\dots,a_{r}} \rp }, \\ {{\lp T\pm S\rp }\lp {a_{1},\dots,a_{r}} \rp } \equiv& {{T}\lp {a_{1},\dots,a_{r}} \rp }\pm{{S}\lp {a_{1},\dots,a_{r}} \rp }.\end{aligned}\end{split}\]
Now let \(T\) be of rank \(r\) and \(S\) of rank \(s\) then the product of the two tensors is
\[\be \f{\lp T\circ S\rp}{a_{1},\dots,a_{r+s}} \equiv \f{T}{a_{1},\dots,a_{r}}\circ\f{S}{a_{r+1},\dots,a_{r+s}}, \ee\]
where "\(\circ\)" is any multivector multiplicative operation.
Covariant, Contravariant, and Mixed Representations¶
The arguments (vectors) of the multilinear function can be represented in terms of the basis vectors or the reciprocal basis vectors
\[\begin{split}\begin{aligned} a_{j} =& a^{i_{j}}{{\eb}}_{i_{j}}, \label{vrep}\\ =& a_{i_{j}}{{\eb}}^{i_{j}}. \label{rvrep}\end{aligned}\end{split}\]
Equation (\(\ref{vrep}\)) gives \(a_{j}\) in terms of the basis vectors and eq (\(\ref{rvrep}\)) in terms of the reciprocal basis vectors. The index \(j\) refers to the argument slot and the indices \(i_{j}\) the components of the vector in terms of the basis. The covariant representation of the tensor is defined by
\(\newcommand{\indices}[1]{#1}\begin{aligned} T\indices{_{i_{1}\dots i_{r}}} \equiv& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{r}}} \rp } \\ {{T}\lp {a_{1},\dots,a_{r}} \rp } =& {{T}\lp {a^{i_{1}}{{\eb}}_{i_{1}},\dots,a^{i_{r}}{{\eb}}_{i_{r}}} \rp } \nonumber \\ =& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{r}}} \rp }a^{i_{1}}\dots a^{i_{r}} \nonumber \\ =& T\indices{_{i_{1}\dots i_{r}}}a^{i_{1}}\dots a^{i_{r}}.\end{aligned}\)$
Likewise for the contravariant representation
\[\begin{split}\begin{aligned} T\indices{^{i_{1}\dots i_{r}}} \equiv& {{T}\lp {{{\eb}}^{i_{1}},\dots,{{\eb}}^{i_{r}}} \rp } \\ {{T}\lp {a_{1},\dots,a_{r}} \rp } =& {{T}\lp {a_{i_{1}}{{\eb}}^{i_{1}},\dots,a_{i_{r}}{{\eb}}^{i_{r}}} \rp } \nonumber \\ =& {{T}\lp {{{\eb}}^{i_{1}},\dots,{{\eb}}^{i_{r}}} \rp }a_{i_{1}}\dots a_{i_{r}} \nonumber \\ =& T\indices{^{i_{1}\dots i_{r}}}a_{i_{1}}\dots a_{i_{r}}.\end{aligned}\end{split}\]
One could also have a mixed representation
\[\begin{split}\begin{aligned} T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}} \equiv& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{s}},{{\eb}}^{i_{s+1}}\dots{{\eb}}^{i_{r}}} \rp } \\ {{T}\lp {a_{1},\dots,a_{r}} \rp } =& {{T}\lp {a^{i_{1}}{{\eb}}_{i_{1}},\dots,a^{i_{s}}{{\eb}}_{i_{s}}, a_{i_{s+1}}{{\eb}}^{i_{s}}\dots,a_{i_{r}}{{\eb}}^{i_{r}}} \rp } \nonumber \\ =& {{T}\lp {{{\eb}}_{i_{1}},\dots,{{\eb}}_{i_{s}},{{\eb}}^{i_{s+1}},\dots,{{\eb}}^{i_{r}}} \rp } a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}},\dots a^{i_{r}} \nonumber \\ =& T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots a^{i_{r}}.\end{aligned}\end{split}\]
In the representation of \(T\) one could have any combination of covariant (lower) and contravariant (upper) indexes.
To convert a covariant index to a contravariant index simply consider
\[\begin{split}\begin{aligned} \f{T}{\eb_{i_{1}},\dots,\eb^{i_{j}},\dots,\eb_{i_{r}}} =& \f{T}{\eb_{i_{1}},\dots,g^{i_{j}k_{j}}\eb_{k_{j}},\dots,\eb_{i_{r}}} \nonumber \\ =& g^{i_{j}k_{j}}\f{T}{\eb_{i_{1}},\dots,\eb_{k_{j}},\dots,\eb_{i_{r}}} \nonumber \\ T_{i_{1}\dots}{}^{i_{j}}{}_{\dots i_{r}} =& g^{i_{j}k_{j}}T\indices{_{i_{1}\dots i_{j}\dots i_{r}}}. \end{aligned}\end{split}\]
Similarly one could lower an upper index with \(g_{i_{j}k_{j}}\).
Contraction and Differentiation¶
The contraction of a tensor between the \(j^{th}\) and \(k^{th}\) variables (slots) is
\[\be \f{T}{a_{i},\dots,a_{j-1},\nabla_{a_{k}},a_{j+1},\dots,a_{r}} = \nabla_{a_{j}}\cdot\lp \nabla_{a_{k}}\f{T}{a_{1},\dots,a_{r}}\rp . \ee\]
This operation reduces the rank of the tensor by two. This definition gives the standard results for metric contraction which is proved as follows for a rank \(r\) grade zero tensor (the circumflex "\(\breve{\:\:}\)" indicates that a term is to be deleted from the product).
\[\begin{split}\begin{align} \f{T}{a_{1},\dots,a_{r}} =& a^{i_{1}}\dots a^{i_{r}}T_{i_{1}\dots i_{r}} \\ \nabla_{a_{j}}T =& \eb^{l_{j}} a^{i_{1}}\dots\lp\partial_{a^{l_j}}a^{i_{j}}\rp\dots a_{i_{r}}T_{i_{1}\dots i_{r}} \nonumber \\ =& \eb^{l_{j}}\delta_{l_{j}}^{i_{j}} a^{i_{1}}\dots \breve{a}^{i_{j}}\dots a^{i_{r}}T_{i_{1}\dots i_{r}} \\ \nabla_{a_{m}}\cdot\lp\nabla_{a_{j}}T\rp =& \eb^{k_{m}}\cdot\eb^{l_{j}}\delta_{l_{j}}^{i_{j}} a^{i_{1}}\dots \breve{a}^{i_{j}}\dots\lp\partial_{a^{k_m}}a^{i_{m}}\rp \dots a^{i_{r}}T_{i_{1}\dots i_{r}} \nonumber \\ =& g^{k_{m}l_{j}}\delta_{l_{j}}^{i_{j}}\delta_{k_{m}}^{i_{m}} a^{i_{1}}\dots \breve{a}^{i_{j}}\dots\breve{a}^{i_{m}} \dots a^{i_{r}}T_{i_{1}\dots i_{r}} \nonumber \\ =& g^{i_{m}i_{j}}a^{i_{1}}\dots \breve{a}^{i_{j}}\dots\breve{a}^{i_{m}} \dots a^{i_{r}}T_{i_{1}\dots i_{j}\dots i_{m}\dots i_{r}} \nonumber \\ =& g^{i_{j}i_{m}}a^{i_{1}}\dots \breve{a}^{i_{j}}\dots\breve{a}^{i_{m}} \dots a^{i_{r}}T_{i_{1}\dots i_{j}\dots i_{m}\dots i_{r}} \nonumber \\ =& \lp g^{i_{j}i_{m}}T_{i_{1}\dots i_{j}\dots i_{m}\dots i_{r}}\rp a^{i_{1}}\dots \breve{a}^{i_{j}}\dots\breve{a}^{i_{m}}\dots a^{i_{r}} \label{eq108} \end{align}\end{split}\]
Equation (\(\ref{eq108}\)) is the correct formula for the metric contraction of a tensor.
If we have a mixed representation of a tensor, \(T\indices{_{i_{1}\dots}{}^{i_{j}}{}_{\dots i_{k}\dots i_{r}}}\), and wish to contract between an upper and lower index (\(i_{j}\) and \(i_{k}\)) first lower the upper index and then use eq (\(\ref{eq108}\)) to contract the result. Remember lowering the index does not change the tensor, only the representation of the tensor, while contraction results in a new tensor. First lower index
\[\be T\indices{_{i_{1}\dots}{}^{i_{j}}{}_{\dots i_{k}\dots i_{r}}} \xRightarrow{\small Lower Index} g_{i_{j}k_{j}}T\indices{_{i_{1}\dots}{}^{k_{j}}{}_{\dots i_{k}\dots i_{r}}} \ee\]
Now contract between \(i_{j}\) and \(i_{k}\) and use the properties of the metric tensor.
\[\begin{split}\begin{aligned} g_{i_{j}k_{j}}T\indices{_{i_{1}\dots}{}^{k_{j}}{}_{\dots i_{k}\dots i_{r}}} \xRightarrow{\small Contract}& g^{i_{j}i_{k}}g_{i_{j}k_{j}}T\indices{_{i_{1}\dots}{}^{k_{j}}{}_{\dots i_{k}\dots i_{r}}} \nonumber \\ =& \delta_{k_{j}}^{i_{k}}T\indices{_{i_{1}\dots}{}^{k_{j}}{}_{\dots i_{k}\dots i_{r}}}. \label{114a}\end{aligned}\end{split}\]
Equation (\(\ref{114a}\)) is the standard formula for contraction between upper and lower indexes of a mixed tensor.
Finally if \({{T}\lp {a_{1},\dots,a_{r}} \rp }\) is a tensor field (implicitly a function of position) the tensor derivative is defined as
\[\begin{aligned} {{T}\lp {a_{1},\dots,a_{r};a_{r+1}} \rp } \equiv \lp a_{r+1}\cdot\nabla\rp {{T}\lp {a_{1},\dots,a_{r}} \rp },\end{aligned}\]
assuming the \(a^{i_{j}}\) coefficients are not a function of the coordinates.
This gives for a grade zero rank \(r\) tensor
\[\begin{split}\begin{aligned} \lp a_{r+1}\cdot\nabla\rp {{T}\lp {a_{1},\dots,a_{r}} \rp } =& a^{i_{r+1}}\partial_{x^{i_{r+1}}}a^{i_{1}}\dots a^{i_{r}} T_{i_{1}\dots i_{r}}, \nonumber \\ =& a^{i_{1}}\dots a^{i_{r}}a^{i_{r+1}} \partial_{x^{i_{r+1}}}T_{i_{1}\dots i_{r}}.\end{aligned}\end{split}\]
From Vector to Tensor¶
A rank one tensor is a vector since it satisfies all the axioms for a vector space, but a vector in not necessarily a tensor since not all vectors are multilinear (actually in the case of vectors a linear function) functions. However, there is a simple isomorphism between vectors and rank one tensors defined by the mapping \({{v}\lp {a} \rp }:\mathcal{V}\rightarrow\Re\) such that if \(v,a \in\mathcal{V}\)
\[\be \f{v}{a} \equiv v\cdot a. \ee\]
So that if \(v = v^{i}{{\eb}}_{i} = v_{i}{{\eb}}^{i}\) the covariant and contravariant representations of \(v\) are (using \({{\eb}}^{i}\cdot{{\eb}}_{j} = \delta^{i}_{j}\))
\[\be \f{v}{a} = v_{i}a^{i} = v^{i}a_{i}. \ee\]
Parallel Transport and Covariant Derivatives¶
The covariant derivative of a tensor field \({{T}\lp {a_{1},\dots,a_{r};x} \rp }\) (\(x\) is the coordinate vector of which \(T\) can be a non-linear function) in the direction \(a_{r+1}\) is (remember \(a_{j} = a_{j}^{k}{{\eb}}_{k}\) and the \({{\eb}}_{k}\) can be functions of \(x\)) the directional derivative of \({{T}\lp {a_{1},\dots,a_{r};x} \rp }\) where all the arguments of \(T\) are parallel transported. The definition of parallel transport is if \(a\) and \(b\) are tangent vectors in the tangent spaced of the manifold then
\[\be \paren{a\cdot\nabla_{x}}b = 0 \label{eq108a} \ee\]
if \(b\) is parallel transported. Since \(b = b^{i}{{\eb}}_{i}\) and the derivatives of \({{\eb}}_{i}\) are functions of the \(x^{i}\)'s then the \(b^{i}\)'s are also functions of the \(x^{i}\)'s so that in order for eq (\(\ref{eq108a}\)) to be satisfied we have
\[\begin{split}\begin{aligned} {\lp {a\cdot\nabla_{x}} \rp }b =& a^{i}\partial_{x^{i}}{\lp {b^{j}{{\eb}}_{j}} \rp } \nonumber \\ =& a^{i}{\lp {{\lp {\partial_{x^{i}}b^{j}} \rp }{{\eb}}_{j} + b^{j}\partial_{x^{i}}{{\eb}}_{j}} \rp } \nonumber \\ =& a^{i}{\lp {{\lp {\partial_{x^{i}}b^{j}} \rp }{{\eb}}_{j} + b^{j}\Gamma_{ij}^{k}{{\eb}}_{k}} \rp } \nonumber \\ =& a^{i}{\lp {{\lp {\partial_{x^{i}}b^{j}} \rp }{{\eb}}_{j} + b^{k}\Gamma_{ik}^{j}{{\eb}}_{j}} \rp }\nonumber \\ =& a^{i}{\lp {{\lp {\partial_{x^{i}}b^{j}} \rp } + b^{k}\Gamma_{ik}^{j}} \rp }{{\eb}}_{j} = 0.\end{aligned}\end{split}\]
Thus for \(b\) to be parallel transported we must have
\[\be \partial_{x^{i}}b^{j} = -b^{k}\Gamma_{ik}^{j}. \label{eq121a} \ee\]
The geometric meaning of parallel transport is that for an infinitesimal rotation and dilation of the basis vectors (cause by infinitesimal changes in the \(x^{i}\)'s) the direction and magnitude of the vector \(b\) does not change.
If we apply eq (\(\ref{eq121a}\)) along a parametric curve defined by \({{x^{j}}\lp {s} \rp }\) we have
\[\begin{split}\begin{align} \deriv{b^{j}}{s}{} =& \deriv{x^{i}}{s}{}\pdiff{b^{j}}{x^{i}} \nonumber \\ =& -b^{k}\deriv{x^{i}}{s}{}\Gamma_{ik}^{j}, \label{eq122a} \end{align}\end{split}\]
and if we define the initial conditions \({{b^{j}}\lp {0} \rp }{{\eb}}_{j}\). Then eq (\(\ref{eq122a}\)) is a system of first order linear differential equations with initial conditions and the solution, \({{b^{j}}\lp {s} \rp }{{\eb}}_{j}\), is the parallel transport of the vector \({{b^{j}}\lp {0} \rp }{{\eb}}_{j}\).
An equivalent formulation for the parallel transport equation is to let \({{\gamma}\lp {s} \rp }\) be a parametric curve in the manifold defined by the tuple \({{\gamma}\lp {s} \rp } = {\lp {{{x^{1}}\lp {s} \rp },\dots,{{x^{n}}\lp {s} \rp }} \rp }\). Then the tangent to \({{\gamma}\lp {s} \rp }\) is given by
\[\be \deriv{\gamma}{s}{} \equiv \deriv{x^{i}}{s}{}\eb_{i} \ee\]
and if \({{v}\lp {x} \rp }\) is a vector field on the manifold then
\[\begin{split}\begin{align} \paren{\deriv{\gamma}{s}{}\cdot\nabla_{x}}v =& \deriv{x^{i}}{s}{}\pdiff{}{x^{i}}\paren{v^{j}\eb_{j}} \nonumber \\ =&\deriv{x^{i}}{s}{}\paren{\pdiff{v^{j}}{x^{i}}\eb_{j}+v^{j}\pdiff{\eb_{j}}{x^{i}}} \nonumber \\ =&\deriv{x^{i}}{s}{}\paren{\pdiff{v^{j}}{x^{i}}\eb_{j}+v^{j}\Gamma^{k}_{ij}\eb_{k}} \nonumber \\ =&\deriv{x^{i}}{s}{}\pdiff{v^{j}}{x^{i}}\eb_{j}+\deriv{x^{i}}{s}{}v^{k}\Gamma^{j}_{ik}\eb_{j} \nonumber \\ =&\paren{\deriv{v^{j}}{s}{}+\deriv{x^{i}}{s}{}v^{k}\Gamma^{j}_{ik}}\eb_{j} \nonumber \\ =& 0. \label{eq124a} \end{align}\end{split}\]
Thus eq (\(\ref{eq124a}\)) is equivalent to eq (\(\ref{eq122a}\)) and parallel transport of a vector field along a curve is equivalent to the directional derivative of the vector field in the direction of the tangent to the curve being zero.
If the tensor component representation is contra-variant (superscripts instead of subscripts) we must use the covariant component representation of the vector arguments of the tensor, \(a = a_{i}{{\eb}}^{i}\). Then the definition of parallel transport gives
\[\begin{split}\begin{aligned} {\lp {a\cdot\nabla_{x}} \rp }b =& a^{i}\partial_{x^{i}}{\lp {b_{j}{{\eb}}^{j}} \rp } \nonumber \\ =& a^{i}{\lp {{\lp {\partial_{x^{i}}b_{j}} \rp }{{\eb}}^{j} + b_{j}\partial_{x^{i}}{{\eb}}^{j}} \rp },\end{aligned}\end{split}\]
and we need
\[\be \paren{\partial_{x^{i}}b_{j}}\eb^{j} + b_{j}\partial_{x^{i}}\eb^{j} = 0. \label{eq111a} \ee\]
To satisfy equation (\(\ref{eq111a}\)) consider the following
\[\begin{split}\begin{aligned} \partial_{x^{i}}{\lp {{{\eb}}^{j}\cdot{{\eb}}_{k}} \rp } =& 0 \nonumber \\ {\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} + {{\eb}}^{j}\cdot{\lp {\partial_{x^{i}}{{\eb}}_{k}} \rp } =& 0 \nonumber \\ {\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} + {{\eb}}^{j}\cdot{{\eb}}_{l}\Gamma_{ik}^{l} =& 0 \nonumber \\ {\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} + \delta_{l}^{j}\Gamma_{ik}^{l} =& 0 \nonumber \\ {\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} + \Gamma_{ik}^{j} =& 0 \nonumber \\ {\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} =& -\Gamma_{ik}^{j}\end{aligned}\end{split}\]
Now dot eq (\(\ref{eq111a}\)) into \({{\eb}}_{k}\) giving
\[\begin{split}\begin{aligned} {\lp {\partial_{x^{i}}b_{j}} \rp }{{\eb}}^{j}\cdot{{\eb}}_{k} + b_{j}{\lp {\partial_{x^{i}}{{\eb}}^{j}} \rp }\cdot{{\eb}}_{k} =& 0 \nonumber \\ {\lp {\partial_{x^{i}}b_{j}} \rp }\delta_{j}^{k} - b_{j}\Gamma_{ik}^{j} =& 0 \nonumber \\ {\lp {\partial_{x^{i}}b_{k}} \rp } = b_{j}\Gamma_{ik}^{j}.\end{aligned}\end{split}\]
Thus if we have a mixed representation of a tensor
\[\be \f{T}{a_{1},\dots,a_{r};x} = \f{T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}}{x}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots a_{i_{r}}, \ee\]
the covariant derivative of the tensor is
\[\begin{split}\begin{align} {\lp {a_{r+1}\cdot D} \rp } {{T}\lp {a_{1},\dots,a_{r};x} \rp } =& {{\displaystyle\frac{\partial {T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}}}{\partial {x^{r+1}}}}}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots a^{r}_{i_{r}} a^{i_{r+1}} \nonumber \\ &\hspace{-0.5in}+ \sum_{p=1}^{s}{{\displaystyle\frac{\partial {a^{i_{p}}}}{\partial {x^{i_{r+1}}}}}}T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}a^{i_{1}}\dots \breve{a}^{i_{p}}\dots a^{i_{s}}a_{i_{s+1}}\dots a_{i_{r}}a^{i_{r+1}} \nonumber \\ &\hspace{-0.5in}+ \sum_{q=s+1}^{r}{{\displaystyle\frac{\partial {a_{i_{p}}}}{\partial {x^{i_{r+1}}}}}}T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots\breve{a}_{i_{q}}\dots a_{i_{r}}a^{i_{r+1}} \nonumber \\ =& {{\displaystyle\frac{\partial {T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}}}{\partial {x^{r+1}}}}}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots a^{r}_{i_{r}} a^{i_{r+1}} \nonumber \\ &\hspace{-0.5in}- \sum_{p=1}^{s}\Gamma_{i_{r+1}l_{p}}^{i_{p}}T\indices{_{i_{1}\dots i_{p}\dots i_{s}}^{i_{s+1} \dots i_{r}}}a^{i_{1}}\dots a^{l_{p}}\dots a^{i_{s}}a_{i_{s+1}}\dots a_{i_{r}}a^{i_{r+1}} \nonumber \\ &\hspace{-0.5in}+ \sum_{q=s+1}^{r}\Gamma_{i_{r+1}i_{q}}^{l_{q}}T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{q} \dots i_{r}}}a^{i_{1}}\dots a^{i_{s}}a_{i_{s+1}}\dots a_{l_{q}}\dots a_{i_{r}}a^{i_{r+1}} . \label{eq126a} \\ \end{align}\end{split}\]
From eq (\(\ref{eq126a}\)) we obtain the components of the covariant derivative to be
\[\begin{aligned} {{\displaystyle\frac{\partial {T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{r}}}}}{\partial {x^{r+1}}}}} - \sum_{p=1}^{s}\Gamma_{i_{r+1}l_{p}}^{i_{p}}T\indices{_{i_{1}\dots i_{p}\dots i_{s}}^{i_{s+1}\dots i_{r}}} + \sum_{q=s+1}^{r}\Gamma_{i_{r+1}i_{q}}^{l_{q}}T\indices{_{i_{1}\dots i_{s}}^{i_{s+1}\dots i_{q}\dots i_{r}}}.\end{aligned}\]
The component free form of the covariant derivative (the one used to calculate it in the code) is
\[\be \mathcal{D}_{a_{r+1}} {{T}\lp {a_{1},\dots,a_{r};x} \rp } \equiv \nabla T - \sum_{k=1}^{r}{{T}\lp {a_{1},\dots,{\lp {a_{r+1}\cdot\nabla} \rp } a_{k},\dots,a_{r};x} \rp }. \ee\]
By the manifold embedding theorem any \(m\)-dimensional manifold is isomorphic to a \(m\)-dimensional vector manifold
This product in not necessarily positive definite.
In this section and all following sections we are using the Einstein summation convention unless otherwise stated.
We use the Christoffel symbols of the first kind to calculate the derivatives of the basis vectors and the product rule to calculate the derivatives of the basis blades where (http://en.wikipedia.org/wiki/Christoffel_symbols)
\[\be \Gamma_{ijk} = {\frac{1}{2}}{\lp {{{\displaystyle\frac{\partial {g_{jk}}}{\partial {x^{i}}}}}+{{\displaystyle\frac{\partial {g_{ik}}}{\partial {x^{j}}}}}-{{\displaystyle\frac{\partial {g_{ij}}}{\partial {x^{k}}}}}} \rp }, \ee\]
\[\be {{\displaystyle\frac{\partial {{{\eb}}_{j}}}{\partial {x^{i}}}}} = \Gamma_{ijk}{{\eb}}^{k}. \ee\]
The Christoffel symbols of the second kind,
\[\be \Gamma_{ij}^{k} = {\frac{1}{2}}g^{kl}{\lp {{{\displaystyle\frac{\partial {g_{li}}}{\partial {x^{j}}}}}+{{\displaystyle\frac{\partial {g_{lj}}}{\partial {x^{i}}}}}-{{\displaystyle\frac{\partial {g_{ij}}}{\partial {x^{l}}}}}} \rp }, \ee\]
could also be used to calculate the derivatives in term of the original basis vectors, but since we need to calculate the reciprocal basis vectors for the geometric derivative it is more efficient to use the symbols of the first kind.
In this case \(D_{B}^{j_{1}\dots j_{n}} = F\) and \(\partial_{j_{1}\dots j_{n}} = 1\).
For example in three dimensions \({\left \{{3} \rbrc} = (0,1,2,3,(1,2),(2,3),(1,3),(1,2,3))\) and as an example of how the superscript would work with each grade \({{\eb}}^{0}=1\), \({{\eb}}^{1}={{\eb}}^{1}\), \({{\eb}}^{{\lp {1,2} \rp }}={{\eb}}^{1}{\wedge}{{\eb}}^{2}\), and \({{\eb}}^{{\lp {1,2,3} \rp }}={{\eb}}^{1}{\wedge}{{\eb}}^{2}{\wedge}{{\eb}}^{3}\).
We are following the treatment of Tensors in section 3–10 of [HS84].
We assume that the arguments are elements of a vector space or more generally a geometric algebra so that the concept of linearity is meaningful. | CommonCrawl |
EZ-AG: structure-free data aggregation in MANETs using push-assisted self-repelling random walks
V. Kulathumani1,
M. Nakagawa1 &
A. Arora2
Journal of Internet Services and Applications volume 9, Article number: 5 (2018) Cite this article
This paper describes EZ-AG, a structure-free protocol for duplicate insensitive data aggregation in MANETs. The key idea in EZ-AG is to introduce a token that performs a self-repelling random walk in the network and aggregates information from nodes when they are visited for the first time. A self-repelling random walk of a token on a graph is one in which at each step, the token moves to a neighbor that has been visited least often. While self-repelling random walks visit all nodes in the network much faster than plain random walks, they tend to slow down when most of the nodes are already visited. In this paper, we show that a single step push phase at each node can significantly speed up the aggregation and eliminate this slow down. By doing so, EZ-AG achieves aggregation in only O(N) time and messages. In terms of overhead, EZ-AG outperforms existing structure-free data aggregation by a factor of at least log(N) and achieves the lower bound for aggregation message overhead. We demonstrate the scalability and robustness of EZ-AG using ns-3 simulations in networks ranging from 100 to 4000 nodes under different mobility models and node speeds. We also describe a hierarchical extension for EZ-AG that can produce multi-resolution aggregates at each node using only O(NlogN) messages, which is a poly-logarithmic factor improvement over existing techniques.
The focus of this paper is on computing order and duplicate insensitive data aggregates (also referred to as ODI-synopsis) and delivering them to every node in a mobile ad-hoc network (MANET) [1–4]. We are specifically motivated by data aggregation requirements in extremely large scale mobile sensor networks [5] such as networks of UAVs, military networks, network of mobile robots and dense vehicular networks, where the number of nodes are often several thousands.
In an order and duplicate insensitive (ODI) synopsis, the same data can be aggregated multiple times but the result is unaffected. MAX, MIN and BOOLEAN OR are natural examples of such duplicate insensitive data aggregation. These queries by themselves are quite common in many applications and some examples are provided below.
As one specific example, consider the application domain of intelligent transportation systems using dense vehicular ad-hoc networks (VANETs) [6]. VANETs are mobile networks supported by both vehicle to vehicle (V2V) and vehicle to road-side infrastructure (V2I) communication, which are in turn enabled by Dedicated Short Range Communication units (DSRC) on board each vehicle [7]. VANETs can be used for improving vehicular safety as well as efficiency by dynamically updating traffic maps and providing efficient re-routes [8]. For such applications, EZ-AG can be used to generate duplicate insensitive aggregates such as the maximum speed or minimum speed in a given area (that are indicative of congestion). It can be used to answer queries such as is there any vehicle that exceeded a certain speed?. It can be also used to answer V2I network management queries such as is there at least one active infrastructure unit within a given area?. VANETs are also often augmented with environmental sensors for tasks such as pollution monitoring [9]. In such applications, aggregation queries related to the sensors can be answered using EZ-AG.
EZ-AG can also be used for data aggregation in networks of drones, UAVs [10] and underwater robotic swarms [11]. For instance, EZ-AG can be used to answer queries such as which is the drone with minimum or maximum battery level? or which robotic fish detects maximum pollution? Aggregation queries resolved by EZ-AG can also be used for consensus driven control applications. For example, EZ-AG can be used to dynamically navigate networks of aerial vehicles towards the area with minimum turbulence [12] or to dynamically navigate a swarm of robotic fish [13] towards regions of higher vegetation.
Other duplicate sensitive statistical aggregates such as COUNT and AVERAGE can also be implemented with ODI synopsis using probabilistic techniques [4, 14]. Using these extensions, EZ-AG can be used to generate duplicate sensitive aggregates such as the number of vehicles in a road segment or average speed of vehicles in a road segment.
In static sensor networks and networks with stable links, data aggregation can be performed by routing along fixed structures such as trees or network backbones [15–18]. However, in MANETs, routing has proven to be quite challenging beyond scales of a few hundred nodes primarily because topology driven structures are unstable and are likely to incur a high communication overhead for maintenance in the presence of node mobility [19]. Therefore, structure-free techniques are more appropriate for data aggregation in MANETs. However, a simple technique like all to all flooding which involves dissemination of data from each node to every other node in the network is not scalable as it incurs an overall cost of O(N2), where N is the number of nodes in the network. Therefore, in this paper we explore the use of self-repelling random walks as a structure free method for data aggregation.
Overview of approach
Random walks are appropriate for data aggregation in mobile networks because they are inherently unaffected by node mobility. The idea is to introduce a token in the network that successively visits all nodes in the network using a random walk traversal and computes the overall aggregate. We say that a node is visited by a token when the node gets exclusive access to the token; the visitation period can be used by the node to add node-specific information into the token, resulting in data aggregation. Note that the concept of visiting all nodes individually differs from that of token dissemination [20, 21] over the entire network where it suffices for every node to simply hear at least one token, as opposed to getting exclusive access to a token.
Note, however that traditional random walks may be too slow in visiting all nodes in the network because they may get stuck in regions of already visited nodes. Hence, in this paper we consider self-repelling random walks [22]. A self-repelling random walk is one in which at each step the walk moves towards one of the neighbors that has been least visited [22] (with ties broken randomly). Self-repelling random walks were introduced in the 1980s and have been studied extensively in the physics literature. One of the striking properties of self-repelling random walks is the remarkable uniformity with which they visit nodes in a graph, i.e., without getting stuck in already visited regions.
Indeed, our results in this paper confirm that until about 85% coverage, duplicate visits are very rare with self-repelling random walks highlighting the efficiency with which a majority of nodes in the network can be visited without extra overhead. However, we observe a slow down when going towards 100% coverage because when most of the nodes are already visited, the token executing self-repelling random walk has to explore the graph to find the next unvisited node. To correct this shortcoming, we introduce a complementary push phase that speeds up the convergence of the random walk. The push phase consists of just one message from each node: before the random walk is started, each node announces its own state to all its neighbors. Note that the push consists of only a single hop broadcast from a node to its neighbors as opposed to a flood which consists of disseminating a node's state to all the nodes in the entire network. Thus, after the push phase, each node now carries information about all its neighbors. As a result, when the random walk executes, it does not have to visit all nodes to finish the aggregation. In fact, we show that the aggregation can finish before the slow down starts for the self-repelling random walk. As a result both the aggregation time and number of messages are now bounded by O(N), as shown in our analysis.
Summary of contributions
We introduce a novel structure-free technique for data aggregation in MANETs that exploits properties of self-repelling random walks and complements it with a push phase. We find that a little push goes a long way in speeding up aggregation and reducing message overhead. In fact, the push phase consists of just a single message from each node to its neighbors. By adding this push phase, we show that both the aggregation time and number of messages are bounded in EZ-AG by O(N). In fact, we show that aggregation is completed in significantly less than N token transfers. The protocol is thus extremely simple, requires very little state maintenance (each nodes only remembers the number of times it has been visited), requires no network structures or clustering.
We compare our results with structure-free techniques for ODI data aggregation such as gossiping and show a log(N) factor improvement in messages compared to existing gossip based techniques. We evaluate our protocol using simulations in ns-3 on networks ranging from 100 to 4000 nodes under various mobility models and node speeds. We also evaluate and compare our protocol with a prototype tree-based technique for data aggregation (i.e., structure based) and show that our protocol is better suited for MANETs and remains scalable under high mobility. In fact, the performance of EZ-AG improves as node mobility increases.
Finally, we also provide an extension to EZ-AG which supplies multi-resolution aggregates to each node. In networks that are quite large, providing each node with only a single aggregate may not be sufficient. On the other hand, providing each node with information about every other node is not scalable. Hierarchical EZ-AG addresses this issue by providing each node with multiple aggregates of neighborhoods of increasing size around itself. Each node can thus have information from all parts of the network, but with a resolution that decays exponentially with distance. This idea is motivated by the fact that in many systems information about nearby regions is more relevant and important than far away regions with progressively increasing importance as distance decreases. Moreover, we also show that aggregates of nearby regions can be obtained at a progressively faster rate than farther regions. Hierarchical EZ-AG uses only O(NlogN) messages and outperforms existing techniques for multi-resolution data aggregation by a factor of log4.4N.
Outline of the paper
In Section 2, we describe related work and specifically compare our contributions with existing work in structured protocols, structure free protocols and random walks. In Section 3, we state the system model. In Section 4, we describe the EZ-AG protocol. In Section 5, we analytically characterize the bound on messages and time for EZ-AG. In Section 6, we describe a hierarchical extension for EZ-AG. In Section 7, we describe the results of our evaluation using ns-3 and compare EZ-AG with a prototype tree-based protocol for data aggregation. We conclude in Section 8.
Structure-based protocols
The problem of data aggregation and one-shot querying has been well studied in the context of static sensor networks. It has been shown that in-network aggregation techniques using spanning trees and network backbones are efficient and reliable solutions for the problem [15–18]. However, in the context of a mobile network, such fixed routing structures are likely to be unstable and could potentially incur a high communication overhead for maintenance [19]. In this paper, we have systematically compared EZ-AG with a prototype tree-based technique for data aggregation and have shown that it outperforms the tree-based idea in mobile networks. We notice that the improvement gets progressively more significant as the average node speed increases.
Structure-free protocols
Flooding, neighborhood gossip and spatial gossip are three structure-free techniques that can be used for data aggregation. Note that flooding data from all nodes to every other node has a messaging cost of O(N2). Alternatively, one could use multiple rounds of neighborhood gossip where in each round a node averages the current state of all its neighbors and this procedure is repeated until convergence [23, 24]. However, this method requires several iterations and has also been shown to have a communication cost and completion time of O(N2) for convergence in grids or random geometric graphs, where connectivity is based on locality [25].
In [1, 2], a spatial gossip technique is described where each node chooses another node in the network (not just neighbors) at random and gossips its state. When this is repeated O(log1+εN) times, all nodes in the network learn about the aggregate state. Note that this scheme requires O(N.polylog(N)) messages. Our random walk based protocol, EZ-AG, requires only O(N) messages. Note also that while all this prior work is on static networks, we demonstrate our results on mobile ad-hoc networks.
Random walks
Random walks and their cover times (time taken to visit all nodes) have been studied extensively for different types of static graphs [26, 27]. In this paper, we are specifically interested in time varying graphs that are relevant in the context of mobile networks.
Self-avoiding and self-repelling random walks are variants of random walks which bias the walk towards unvisited nodes [22]. The unformity in coverage of such random walks in 2-d lattices has been pointed out in [28]. Our paper extends the analysis of self-repelling random walks presented in [28] for application in mobile ad-hoc networks that are modeled as time varying random geometric graphs. Further, we show that by complementing self-repelling random walks with a push phase, we can complete aggergation in O(N) time and messages. The idea of locally biasing random walks and its impact in speeding up coverage has been pointed out in [29] for static networks. Self-repelling random walks are different than the local bias technique presented in [29]. Moreover, we show how to improve the convergence of self-repelling random walks using a complementary push-phase and demonstrate our results on mobile networks.
In a recent paper [30], we have addressed the problem of duplicate-sensitive aggregation using self-repelling random walks and in that solution we have used a gradient technique to speed up self-repelling random walks. The short temporary gradients introduced in [30] are used to pull the token towards unvisited nodes so that each node is visited at least once. The solution in [30] requires O(N.log(N)) messages. In this paper, we address duplicate insensitive aggregation and show that it can be achieved using self-repelling random walks with just O(N) messages.
We consider a mobile network of N nodes modeled as a geometric Markovian evolving graph [31]. Each node has a communication range R. We assume that the N nodes are independently and uniformly deployed over a square region of sides \(\sqrt {A}\) resulting in a network density ρ=N/A of the deployed nodes. Consider the region to be divided into square cells of sides \(R/\sqrt {2}\). Thus the diagonal of each such cell is the communication range R. Let R2>2clog(N)/ρ. It has been shown that there exists a constant c>1 such that each such cell has θ(logN) nodes whp, i.e., the degree of each node is θ(logN) whp. Such graphs are referred to as geo-dense geometric graphs [29]. Denote d=θ(logN) as the degree of connectivity.
The objective of the protocol is to compute a duplicate insensitive aggregate of the state of nodes in a MANET. The aggregate could be initiated by any of the nodes in the MANET or by a special static node such as a base station that is connected to the rest of the nodes. The aggregate needs to be disseminated to all nodes in the network. The protocol could be invoked in a one-shot or periodic aggregation mode.
Mobility model
We consider 3 different mobility models for our evaluations.
The first is a random direction mobility model (with reflection) [32, 33] for the nodes. This is a special case of the random walk mobility model [34]. In this mobility model, at each interval a node picks a random direction uniformly in the range [0,2π] and moves with a constant speed that is randomly chosen in the range [v l ,v h ]. At the end of each interval, a new direction and speed are calculated. If the node hits a boundary, the direction is reversed. Motion of the nodes is independent of each other. An important characteristic of this mobility model is that it preserves the uniformity of node distribution: given that at time t=0 the position and orientation of users are independent and uniform, they remain uniformly distributed for all times t>0 provided the users move independently of each other [31, 33].
The second is random waypoint mobility model. Here, each mobile node randomly selects one location in the simulation area and then travels towards this destination with constant velocity chosen randomly from [v l ,v h ] [34]. Upon reaching the destination, the node stops for a duration defined by the pause time. After this duration, it again chooses another random destination and the process is repeated. We set the pause time to 2 s between successive changes.
The third is Gauss Markov mobility model. In this model, the velocity of mobile node is assumed to be correlated over time and modeled as a Gauss-Markov stochastic process [34]. We set the temporal dependence parameter α=0.75. Velocity and direction are changed every 1 s in the Gauss Markov Model.
We consider node speeds in the range of 3 to 21 m/s. For the deployment density that we have chosen, a mapping between node speed and the average link changes per node per second is listed in Table 1. This table quantifies the link instability caused by node mobility at different node speeds. As seen in Table 1, because of high network density, the network structure is rapidly changing at the speeds chosen for evaluation.
Table 1 Mapping between speed and link changes per node per second (rounded off to integer)
While we have chosen the above mobility models for evaluation, we expect the results to hold even under other models such as motion on a Manhattan grid (suitable for vehicular networks). The crucial aspect of mobility that we capture in our evaluations is the high rate at which links change per second which is quantified in Table 1. Our results highlight that performance of EZ-AG actually improves with higher mobility speeds.
A key metric that we are interested in is the number of times the token is transferred to already visited nodes. We present this in the form of exploration overhead which is defined as the ratio of the number of token transfers to the number of unique nodes whose data has been aggregated into the token. We compute exploration overhead at different stages of coverage as the random walk progresses.
Typically, random walks are evaluated in terms of their cover times, which is defined as the time required to visit all nodes. For a standard random walk, the notion of physical time, messages and the number of steps are all equivalent. However, for the push assisted self-repelling random walks these are somewhat different. The total number of messages required to complete the data aggregation includes the push messages, the messages involved in the self-repelling random walk and the messages involved in disseminating the result to all the nodes using a flood. Moreover, each token transfer step itself consists of announcement, token request and token transfer messages. Thus, although proportional, the number of messages is different than the number of token transfer steps. Hence we separately characterize the number of messages during empirical evaluation.
Finally we note that since we study random walks on mobile networks, the notion of time is also related to node speed. Moreover, when dealing with wireless networks, time also involves messaging delays. Therefore, during empirical evaluation we separately characterize the actual convergence time (in seconds) along with the number of steps (i.e., number of token transfers).
EZ-AG consists of 4 phases as shown in Fig. 1a. These phases are described below. The steps involved in the self-repelling random walk phase are shown in Fig. 1b. The communication cost in each of these phases is analyzed in Section 5.
Summary of EZ-AG protocol: EZ-AG consists of 4 phases as shown in part (a). The steps involved in the random walk phase are shown in part (b)
Aggregation request phase: The node requesting the aggregate first initiates a flood in the network to notify all nodes about the interest in the aggregate. Note that each node broadcasts this flood message exactly once. This results in N messages.
Push phase: Once a node receives this request, it pushes its state to its neighbors. Each node uses the data received from its neighbors to compute an aggregate of the state of all its neighbors. Note that the push consists of only a single hop broadcast from a node to all its neighbors. In contrast, a flood consists of disseminating a node's data to the entire network. Thus, the push phase also requires exactly N messages because each node broadcasts its data once.
Self-repelling random walk phase: Soon after the initiator sends out an aggregate request, it also initiates a token to perform a self-repelling random walk. A node that has the token broadcasts an announce message. Nodes that receive the announce message reply back with a token request message and include the number of times they have been visited by the token in this request. The node that holds the token selects the requesting node which has been visited least number of times (with ties broken randomly) and transfers the token to that node. This token transfer is repeated successively. Note that nodes which hear a token announcement schedule a token request at a random time t r within a bounded interval, where t r is proportional to the number of times that they have been visited. Thus nodes that have not been visited or visited fewer times send a request message earlier. When a node hears a request from a node that has been visited fewer or same number of times, it suppresses its request. Thus, the number of requests received for a token announcement remains fairly constant and irrespective of network density.
We note specifically that tokens do not grow in size when they visit successive nodes because they only carry the aggregated state. Determination of the next node to visit is done with the help of individual nodes which maintain a count of the number of times they have been visited so far. This information is conveyed to the token holder after the announce message, which is then used to determine the next node to be visited. Thus, even at individual nodes, the state maintenance is minimal (each node only remembers the number of times it has been visited).
In the following section, we prove analytically that the aggregate can be computed from all nodes in the network whp in O(N) token transfers. In the empirical evaluation, we show that the median number of token transfers is actually only kn, where 0<k<1, and k is unaffected by network size. Thus, the median exploration overhead is less than 1. One can use this observation to terminate the self-repelling random walk after exactly N steps and whp one can expect that data from all the nodes has been aggregated.
Result dissemination phase: Once the aggregate has been computed, the result can simply be flooded back to all the nodes by the node that holds the result. This requires O(N) messages. Another potential solution (when aggregate is only required at a base station) is to transmit the aggregated tokens using a long distance transmission link (such as cellular or satellite links) in hybrid MANETs where the long links are used for infrequent, high priority data.
The protocol is thus extremely simple, requires very little state maintenance, and requires no network structures or clustering.
Reliability of token transfer
The reliable transfer of tokens from one node to another is important for successful operation of EZ-AG. If a token is released by a node, but the intended recipient did not receive the token reply message, the token is lost. Reliability of token transfer can be imposed by requiring an acknowledgement from the node receiving the token and re-sending the token if an acknowledgement was not received. However, it is possible that the token was transferred correctly to a neighbor but the acknowledgement was lost or the recipient of the token moved away from the communication range of a sender. In this case, a duplicate token may be created by this process. But, since EZ-AG computes duplicate insensitive aggregates, the addition of a duplicate token will not impact the accuracy.
In this section, we first show that the aggregation time and message overhead for push assisted self-repelling random walks is O(N). We consider a static network for our analysis. In Section 7, we evaluate the protocol under different mobility models and verify that the results hold even in the presence of mobility.
First, we state the following claim regarding the uniformity in the distribution of visited nodes during the progression of a self-repelling random walk.
The distribution of visited nodes (and unvisited nodes) remains spatially uniform during the progression of a self-repelling random walk.
Argument: Our claim is based on the analysis of uniformity in coverage of self-repelling random walks in [28] and in [35]. In [28], the variance in the number of visits per node of self-repelling random walks is shown to be tightly bounded, resulting in a uniform distribution of visited nodes across the network. More precisely, let n i (t,x) be the number of times a node i has been visited, starting from a node x. The quantity studied in [28] is the variance \((1/N)\left (\sum _{i} (n_{i}(t,x) - \mu)^{2}\right)\), where \(\mu = (1/N)\left (\sum _{i} n_{i}(t,x)\right)\). It is seen that this variance is bounded by values less than 1 even in lattices of dimensions 2048×2048. A detailed extension of this analysis for mobile networks is presented in Section 7.1 which shows the uniformity with which nodes are visited during a self-repelling random walk. We use this to infer that even after the walk started, the distribution of visited nodes (and by that token, unvisited nodes) remains uniform. The result shows that the self-repelling random walk is not stuck in regions of already visited nodes - instead, it spreads towards unvisited areas.
Theorem 1
The required number of messages for data aggregation by EZ-AG in a connected, static network of N nodes with uniform distribution of node locations is O(N).
We note that the aggregation request flood and the result dissemination flood require O(N) messages. During the push phase, each node broadcasts its state once and this also requires only N messages. Now, we analyze the self-repelling random walk phase.
Consider the region to be divided into square cells of sides \(R/\sqrt {2}\) (see Fig. 2). Thus the diagonal of each such cell is the communication range R. Recall from our system model that each such cell has θ(logN) nodes whp at all times and there are O(N/log(N)) such cells. Therefore, at the end of the push phase, each node has aggregated information about its θ(logN) cell neighbors. Also note that the network can be divided into θ(N/log(N)) sets of nodes that each contain information about θ(log(N)) nodes within their cell. Therefore, the self-repelling random walk has to visit at least one node in each cell to finish aggregating information from all nodes.
Proof synopsis: Consider the region divided into square cells with diagonal size R. At the end of single step push phase, each node has information about all nodes in its cell. So it is sufficient for the token (performing a self-repelling random walk) to visit one node in each cell to finish aggregation
To analyze the number of token transfers required to visit at least one node in each cell, we use the analogous coupon collector problem (also known as the double dixie cup problem) which studies the expected number of coupons to be drawn from B categories so that at least 1 coupon is drawn from each category [36]. To ensure that at least 1 coupon is drawn from each category whp, the required number of draws is O(B.log(B)). Using this result and the fact that a self-repelling random walk traverses a network uniformly, we infer that O((N/logN)∗log(N/logN)) token transfers are needed to visit at least 1 node in each of the θ(N/logN) cells.
Note that log(N)>log(N/log(N)). Hence, the required number of messages for the push assisted self-repelling random walk based aggregation protocol is O(N/log(N)∗log(N)), i.e., O(N). □
Note that in the presence of mobility, the node locations with respect to cells may not be preserved during the push phase. Therefore the generation of θ(N/log(N)) identical partitions of network state as described in the above analysis may not exactly hold. However, in Section 7 we empirically ascertain that kN token transfers (where k<1) are still sufficient to aggregate data from all nodes even in the presence of mobility. In fact, we observe that the required token transfers actually decrease with increasing speed, indicating that data aggregation using self-repelling random walks is actually helped by mobility.
It follows from the above result that the total time for aggregation is also O(N). The impact of network effects such as collisions on the message overhead and aggregation time (if any) will be evaluated in Section 7.
In terms of communication, for data aggregation to complete, we note that each node has to at least transmit its own data once. Thus, O(N) is an absolute lower bound in terms of communication messages for data aggregation. We have thus shown that EZ-AG achieves this lower bound of O(N) for data aggregation and therefore is indeed quite efficient in terms of communication. Moreover, we also show that the random walk phase terminates in exactly N token passes. Also during each transfer of the token, the number of requests for the token remain fairly constant and low (See Fig. 12). Thus, it is not the case that the constants of proportionality are high either.
By way of contrast, in a pure flooding based approach, each node will have to flood the data to every other node resulting in O(N2) cost. Instead, EZ-AG first aggregates the data using O(N) cost and then floods the result in O(N) cost, thus resulting in a total of only O(N) communication cost. The impact of this order efficiency becomes increasingly significant as network size increases.
Extension for hierarchical aggregation
When a network is quite large, providing each node with only a single aggregate for the entire network may not be sufficient. On the other hand, providing each node with information about every other node is not scalable. We therefore pursue an extension to EZ-AG where each node can receive multi-resolution aggregates of neighborhoods with exponentially increasing sizes around itself. This way, each node can have information from all parts of the network but with a resolution that decays exponentially with distance. This idea is motivated by the fact that in many systems information about nearby regions is more relevant and important than far away regions with progressively increasing importance as distance decreases. In this section, we describe how EZ-AG can be extended to provide such multi-resolution synopsis of nodes in a network with only O(NlogN) messages.
Existing techniques for such hierarchical aggregation require O(Nlog5.4N) messages [1]. Thus, EZ-AG offers a poly-logarithmic factor improvement in terms of number of messages for hierarchical aggregation. Moreover, EZ-AG can also be used to generate hierarchical aggregates that are distance-sensitive in refresh rate, where aggregates of nearby regions are supplied at a faster rate than farther neighborhoods.
We divide the network into square cells at different levels (0, 1,.. P) of exponentially increasing sizes (shown in Fig. 3). At the lowest level (level 0), each cell is of sides \(R / \sqrt {2}\). Recall from our system model that each such cell has θ(log(N)) nodes whp. For simplicity, let us denote θ(log(N) by the symbol δ. Thus, there are N/δ cells at level 0. Note that 4 adjoining cells of level i constitute a cell of level i+1. Thus, each cell at level j has δ4j nodes whp. At the highest level P, there is only one cell with all the N nodes. Note that P=log4(N/δ). At any given time, a node belongs to one cell at each level.
Extension of EZ-AG to deliver multi-resolution aggregates: The network is partitioned into cells of increasing hierarchy where the cell at smallest level is of diagonal R. The node y shown in the figure would receive an aggregate corresponding to one cell at each level that it belongs to. In this case, it would receive aggregates for cells A, B, C and D. The largest cell D consists of the entire network
To deliver multi-resolution aggregates, we introduce a token and execute EZ-AG at each cell at every level. A token for a given cell is only transferred to nodes within that cell and floods its aggregate to nodes within that cell. Thus, there are N/δ instances of EZ-AG at level 0 and each instance computes aggregates for δ nodes, i.e., θ(logN) nodes.
The computation and dissemination of aggregates by different instances of EZ-AG are not synchronized. Thus, a node may receive aggregates of different levels at different times. Also, since the nodes are mobile, an aggregate at level l received by a node at any given time corresponds to the cell of the same level l in which it resides at that instant.
An ODI aggregate at level j can be computed using hierarchical EZ-AG in O(4jδ) time and messages.
Note that each cell at level j contains θ(4jδ) nodes whp. Therefore, using Theorem 1, EZ-AG only requires O(4jδ) time and messages to compute aggregate within the cell. □
We note from the above theorem that aggregates at level 0 can be published every O(δ) time, aggregates at level 1 can be published every O(4δ) time and so on. Thus, aggregates for cells at smaller levels can be published exponentially faster than those for larger cells. Thus, if the tokens repeatedly compute an aggregate and disseminate within their respective cells, EZ-AG can generate hierarchical aggregates that are distance-sensitive in refresh rate, where aggregates of nearby regions are supplied at a faster rate than farther neighborhoods.
Hierarchical EZ-AG can compute an ODI aggregate for all cells at all levels using O(NlogN) messages.
Note that a cell at level 0 contains δ nodes and there are N/δ such cells. The aggregate for cells at level 0 can be computed using O(δ) messages.
In general, there are N/4jδ cells at level j and aggregates for these cells can be computed using O(4jδ) messages. Summing up from levels 0 to P, the total aggregation message cost (M) for hierarchical EZ-AG can be computed as follows.
$$\begin{array}{@{}rcl@{}} M &=& \sum_{j=0}^{P} 4^{j} \delta \frac{N}{4^{j} \delta} \\ &=& \sum_{j=0}^{P} N \\ &=& O(N log N) \end{array} $$
Thus, hierarchical EZ-AG can compute an ODI aggregate for all cells at all levels using O(NlogN) messages. □
Comparison of hierarchical EZ-AG with gossip techniques
In [1, 2], a spatial gossip technique is described where each node chooses another node in the network (not just neighbors) at random and gossips its state. When this is repeated O(log1+ε(N)) times (where ε>1), all nodes in the network learn about the aggregate state. Note that this scheme requires O(N.polylog(N)) messages. EZ-AG requires only O(N) messages.
In [1], an extension to the spatial gossip technique is described which provides a multi-resolution synopsis of the network state at each node. The technique described in [1] requires O(Nlog5.4(N)) messages. The hierarchical extension of EZ-AG only requires O(NlogN) messages.
In this section, we systematically evaluate the performance of EZ-AG using simulations in ns-3. We set up MANETs ranging from 100 to 4000 nodes using the network model described in Section 3. Nodes are deployed uniformly in the network with a deployment area and communication range such that R2=4log(N)/ρ. Thus, the network is geo-dense with c=2, i.e., each node has on average 2log(N) neighbors whp and the network is connected whp. We test such networks in our simulations with the following mobility models: 2-d random walk, random waypoint and Gauss-Markov (described in Section 3). The average node speeds range from 3 to 21 m/s. We also consider static networks as a special case.
First, we analyze the convergence characteristics of the push-assisted self-repelling random walk phase in EZ-AG and compare that with self-repelling random walks and plain random walks. Next, we analyze the total messages and time taken by EZ-AG. Finally, we compare EZ-AG with a prototype tree based protocol and with gossip based techniques.
Coverage uniformity
First, in Fig. 4a, b and c, we show the number of times each node is visited when the self-repelling random walk has finished visiting 50% of the nodes, 75% of the nodes and 85% of the nodes. We observe that most of the nodes are just visited once and this result holds even at 1000 nodes. These graphs highlight the uniformity with which nodes are visited as self-repelling random walks progress. The self-repelling random walk is not stuck in regions of already visited nodes - instead, it spreads towards unvisited areas. Otherwise, one would have observed more duplicate visits to the previously visited nodes.In Fig. 4d, we analyze the distribution of number of visits at each node when 100% coverage is attained. Here, we see that most nodes are visited 2 or 3 times and the distribution falls off rapidly after that.
Distribution of number of visits at each node at different stages of exploration of a self-repelling random walk (network size 100,400 and 1000 nodes). a 50% coverage, b 75% coverage, c 85% coverage, d 100% coverage
We then compare the uniformity in coverage with that of pure random walks. In Fig. 5, we plot the number of visits to each node until all nodes are visited at least once for a 500 node network. In comparison with self-repelling random walks (Fig. 5b), we observe that the tail of the distribution is much longer and the number of duplicate visits is much higher for pure random walks.
Analysis of coverage uniformity: Distribution of number of visits at each node in a pure random walk and self-repelling random walk. a Visits distribution: Pure random walk, b Visits distribution: Self-repelling walk
Convergence characteristics
Next, in Fig. 6, we show the exploration overhead of self-repelling random walk during different stages of coverage. As seen in Fig. 6, until about 85% coverage, self-repelling random walks have an exploration overhead of around 1 (irrespective of network size) but then the overhead starts to rise sharply. This is because, until this point self-repelling enables a token to find an unvisited node directly and there are very few wasted explorations. A slowdown for self-repelling random walk is noticed after this point. As a result, the exploration overhead at 100% coverage is close to 2 and moreover it increases with network size. This is what we aim to address using EZ-AG.
Exploration overhead as a function of percentage of nodes visits for self-repelling random walks
The exploration overhead at 100% coverage is shown in Fig. 7 for self-repelling random walks and EZ-AG (i.e., push-assisted self-repelling random walks). As seen in the figure, the exploration overhead for self-repelling random walks grows with a logarithmic trend due to the wasted explorations towards the tail end of the random walk phase when most of the nodes are already visited. The push assisted self-repelling random walks remove these wasted explorations and as a result the median exploration overhead stays constant at all network sizes and is actually less than 1 (approximately 0.75 as seen in Fig. 7).
Exploration overhead at 100% coverage as a function of network size
Impact of mobility and speed
In Fig. 8a and b, we evaluate the impact of mobility model and network speed on the exploration overhead of push assisted self-repelling random walks. We observe that even though random waypoint and Gauss Markov models do not preserve the uniform distribution of node locations, the exploration overhead exhibits a similar trend. As seen in Table 1, the network structure is rapidly changing at the speeds chosen for evaluation. Despite this, in Fig. 8b, we observe that the exploration overhead actually starts decreasing with node speed (this is shown more clearly in Fig. 9 for networks with different sizes).
Analysis of exploration overhead of EZ-AG for different mobility models and network speeds. a Impact of mobility models, b Impact of node speed
Exploration overhead as a function of node speed (network size = 500 nodes)
Variance and terminating condition
In Fig. 10, we show the variation in exploration overhead for EZ-AG over 50 different trials at different network sizes. We observe that irrespective of network size, for 97.5% of the trials, the exploration overhead is smaller than 1. We can use this to design a terminating condition for the random walk phase of the protocol. For example, we could terminate the random walk phase after exactly N steps, and then start the dissemination of the aggregate.
Variance in exploration overhead
Messages and time
In Fig. 11a and b, we show the total number of messages and the total aggregation time as a function of network size for the aggregation protocol based on push-assisted self-repelling random walks. The total number of messages required to complete the data aggregation includes the push messages, the messages involved in the self-repelling random walk phase and the messages involved in disseminating the result to all the nodes using a flood. Note that, each token transfer step itself consists of announcement, token request and token transfer messages. These are all included in Fig. 11a which shows that the messages grow linearly with network size.
Analysis of time and messages for EZ-AG. a Total messages, b Aggregation time
Number of token requests generated per token transfer
An interesting aspect of the token transfer procedure is the number of requests generated for a token during each iteration. Note that the average number of neighbors increases as θ(logN) when the network size increases. However, from Fig. 12, the number of token requests per transfer is seen to be independent of the number of neighbors. From the box plot of Fig. 12, we observe that the average number of token requests in each trial is in the range of 1−3. This is because nodes that are visited less often send a request earlier than those that are visited more times. And, if a node hears a request from a node that has been visited less often than itself, it suppresses its request. Thus, irrespective of the neighborhood density, the number of token requests per node stay constant.
As seen in Fig. 11b, the total aggregation time also exhibits a linear trend. Note that the measurement of time is quite implementation specific and incorporates messaging latency in the wireless network. For instance, in our implementation each transaction (i.e., each iteration of token announcement, token requests and token passing) took on average 25 ms. But this number could be much smaller using methods such as [37] that use collaborative communication for estimating neighborhood sizes that satisfy given predicates.
Comparison with structured tree based protocol
In this section, we compare the performance of our protocol with a structured approach for one-shot duplicate insensitive data aggregation that involves maintaining network structures such as spanning trees. For our comparison, we use a prototype tree-based protocol that we describe briefly. The idea is very similar to other tree-based aggregation protocols developed for static sensor networks [16, 17], but the key difference is that the tree is periodically refreshed to handle mobility as described below.
The initiating node maintains a tree structure rooted at itself by flooding a request message in the network. Each node maintains a parent variable. When a node hears a flood message for the first time, it marks the sending node as its parent. It then schedules a data transmission for its parent at a random time chosen within the next 25 ms. The message is successively forwarded through the tree structure until it reaches the root. During this process, a node also opportunistically aggregates multiple messages in its transmission queue before forwarding data to its parent. A message could be lost because a node's parent has moved away or due to collisions. To handle message losses, a node repeats its data transmission to its parent until an acknowledgement is received from its parent. While this basic protocol is sufficient for a static network, the network structure is constantly evolving in a mobile network. Hence, the initiating node periodically refreshes the tree by broadcasting a new request every 2 s (with a monotonically increasing sequence number to allow nodes to reset their parents). The refreshing of the tree is stopped when data from all nodes has been received at the initiating node.
In Fig. 13a, we compare the total messages required for the tree-based protocol and the random walk based protocol at different node speeds. As seen in this figure, for static networks the tree based protocol is more efficient. However as the mobility increases, the random walk based protocol starts increasing in efficiency. In Fig. 13b we compare the total aggregation time which also exhibits a similar trend.
Comparison of time and messages for EZ-AG and tree-based protocol at different node speeds with a network size of 500 nodes. a Total messages, b Aggregation time
In Fig. 14 we compare the total number messages as a function of network size at an average speed of 9 m/s. Here we observe that the self-repelling random walk based protocol exhibits a linear trend while the tree based protocol exhibits a super-linear trend. This is due to the potentially large number of re-transmissions experienced by the tree-based protocol in a mobile network. This graph also shows that EZ-AG is far more scalable with network size under mobility than structure-based techniques for data aggregation.
Total messages as a function of network size for EZ-AG and tree based protocol (node speed = 9 m/s)
In this paper, we have presented a scalable, robust and lightweight protocol for duplicate insensitive data aggregation in MANETs that exploits the simplicity and efficiency of self-repelling random walks. We showed that by complementing self-repelling random walks with a single step push phase, our protocol can achieve data aggregation in O(N) time and messages. In terms of message overhead, our protocol outperforms existing structure free gossip protocols by a factor of log(N). We quantified the performance of our protocol using ns-3 simulations under different network sizes and mobility models. We also showed that our protocol outperforms structure based protocols in mobile networks and the improvement gets increasingly significant as average node speed increases.
We have shown that EZ-AG meets the lower bound of O(N) in terms of communication requirements for aggregation. Also, each node only needs to store the number of times it has been visited. Thus, EZ-AG is lightweight in terms of both communication requirements and memory utilization. It also makes rather minimal assumptions of the underlying network. In particular, it does not assume knowledge of node addresses or locations, require a neighborhood discovery service or network topology information, or depend upon any particular routing or transport protocols such as TCP/IP.
We also described a hierarchical extension to EZ-AG that provides multi-resolution aggregates of the network state to each node. It outperforms existing technique by a factor of O(log4.4N) in terms of number of messages.
Note that EZ-AG uses only a single step push phase, i.e. a one hop broadcast from every node to its neighbors. Extending the push phase beyond a single hop may improve the speed of convergence, but at increased complexity. Requiring each neighbor to further push the data (i.e., a 2 hop push) essentially increases the communication cost by a factor equal to the degree of connectivity d. Pushing across the network diameter is essentially flooding with a cost of O(N2). A single step push, on the other hand, maintains the communication cost at O(N), while significantly speeding up the aggregation.
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This research was not supported by any external funding source.
Department of Computer Science and Electrical Engineering, West Virginia University, Morgantown, WV 26505, USA
V. Kulathumani & M. Nakagawa
The Samraksh Company, Dublin, 43017, OH, USA
A. Arora
V. Kulathumani
M. Nakagawa
VK conceived the idea of introducing a push phase to speed up self=repelling random walks and worked on the analytic proofs. MN designed the algorithm and carried out experimental evaluations. AA contributed in the design and troubleshooting of the idea and also helped draft the manuscript. All authors read and approved the final manuscript.
Correspondence to V. Kulathumani.
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.
Kulathumani, V., Nakagawa, M. & Arora, A. EZ-AG: structure-free data aggregation in MANETs using push-assisted self-repelling random walks. J Internet Serv Appl 9, 5 (2018). https://doi.org/10.1186/s13174-018-0077-4
Mobile ad-hoc
Scalable and robust data aggregation
Multi-resolution synopsis | CommonCrawl |
Why do some people write the gravitational force as proportional to $\mathbf{r}/|\mathbf{r}|^3$?
I'm reading Mathematical Aspects of Classical and Celestial Mechanics, Second Edition by Arnold, Kozlov, and Neishtadt. It occurred to me that many people like to use third power when mention the law of universal gravitation.
$$F_{ij}=\frac{\gamma m_i m_j}{|r_{ij}|^3}r_{ij}$$ where $r_{ij}=r_i-r_j, \gamma =\text{const}>0$
Is this just some conversion/tradition/habit, or does it carry any significance? i.e. the absolute value changed sign and derivative?
newtonian-gravity vectors notation
edited Jun 19 at 15:27
knzhou
$\begingroup$ One advantage is that you don't need to introduce a unit vector for $\vec{r}_{ij}$. This is the same as writing $\frac{\gamma m_i m_j}{|\vec{r}_{ij}|^2} \frac{\vec{r}_{ij}}{|\vec{r}_{ij}|} = \frac{\gamma m_i m_j}{|\vec{r}_{ij}|^2} \hat{r}_{ij}$. $\endgroup$ – Tob Ernack Jun 19 at 3:53
$\begingroup$ I think this is a fairly common convention. I've seen it on Biot-Savart as well. $\endgroup$ – mathysics Jun 19 at 5:01
$\begingroup$ Notice that it is not an absolute value but vector norm. $\endgroup$ – Matt Jun 19 at 12:02
$\begingroup$ It's a convenient form of typing it to avoid the use of unit vectors. Imagine typing an entire book using $\hat{hats}$. $\endgroup$ – J. Manuel Jun 19 at 14:41
$\begingroup$ Because all the cool guys are doing it. $\endgroup$ – Paracosmiste Jun 19 at 14:56
This is not about gravity but about making maths easier; the same idea comes up whenever you have a force in the radial direction. The reason that ${\bf r}/r^3$ is a little easier to work with than $\hat{\bf r}/r^2$ is the following.
First, one can write it out in rectangular coordinates easily: $$ \frac{\bf r}{r^3} = \frac{1}{r^3} \left( \begin{array}{c} x\\y\\z \end{array} \right) $$
Secondly, because $\hat{\bf r}$ has a square root hidden in it which makes some manipulations trickier: $$ \hat{\bf r} = (x^2 + y^2 + z^2)^{-1/2}(x {\bf i} + y {\bf j} + z {\bf k}) $$ Indeed, when doing things like differentiation, the first step is often to write $\hat{\bf r}$ as ${\bf r}/r$.
Overall, the move to replace $\hat{\bf r}$ by ${\bf r}/r$ reduces the number of different symbols in the mathematical expression and that is the main reason why it is regularly done. Having completed some algebra, one may then choose to present the result in terms of $\hat{\bf r}$ so as to draw attention to the overall scaling of the result, for example in order to make it clear that the gravitational law is an inverse square not an inverse cube law.
edited Jun 19 at 9:13
answered Jun 19 at 7:25
Andrew SteaneAndrew Steane
\begin{equation} \mathbf{F}_{ij} \boldsymbol{=}\mathrm k\dfrac{m_i m_j}{\Vert \mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\Vert^2 } \mathbf{n}_{ij} \tag{1}\label{1} \end{equation} where $\:\mathbf{n}_{ij}\:$ the unit vector along the vector $\:\left(\mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\right)$. But \begin{equation} \mathbf{n}_{ij} \boldsymbol{=}\dfrac{\left(\mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\right)}{\Vert \mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\Vert } \tag{2}\label{2} \end{equation} so the third power \begin{equation} \mathbf{F}_{ij} \boldsymbol{=}\mathrm k\dfrac{m_i m_j}{\Vert \mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\Vert^3 } \left(\mathbf{r}_{i}\boldsymbol{-} \mathbf{r}_{j}\right) \tag{3}\label{3} \end{equation}
FrobeniusFrobenius
$\begingroup$ So the authors presumably stated early on that, in their text, bold face quantities are all vectors. The only scalar in the OP's equation are the masses? $\endgroup$ – DJohnM Jun 19 at 6:31
$\begingroup$ @DJohnM ...the masses $\:m_i,m_j\:$ and the constant $\:\mathrm k$... $\endgroup$ – Frobenius Jun 19 at 6:48
$\begingroup$ @DJohnM I don't think I've seen boldface used for anything other than vectors. $\endgroup$ – David Richerby Jun 19 at 13:19
$\begingroup$ But why (3) and not (1). Is it a tradition\habit? $\endgroup$ – J. Manuel Jun 19 at 15:59
$\begingroup$ @J.Manuel You see both in the literature, and I would wager (1) is actually more common as it makes manifest the inverse square relationship. (3) can often be easier to work with however. $\endgroup$ – gabe Jun 19 at 17:58
It is a matter of convenience:
Let $\vec{r_{ij}}$ be a distance vector with magnitude $r_{ij}$ along the line connecting the masses $m_i$ and $m_j$. Then:
(I): $\vec{r_{ij}}$ squared is a scalar whose value equals its magnitude squared. Proof: $\vec{r_{ij}}^2=\vec{r_{ij}} \cdot \vec{r_{ij}}=r_{ij} r_{ij} \cos{0}=r_{ij}^2$.
(II): $\vec{r_{ij}}$ can be written as $\vec{r_{ij}}=r_{ij} \hat{r_{ij}}$, with $\hat{r_{ij}}$ being a unit vector having the same direction as $\vec{r_{ij}}$. Proof: This goes as an axiom ;-)
If ones writes Newton's law of gravity as $$F_{ij}=\frac{γm_im_j}{\vec{r_{ij}}^2}=\frac{γm_im_j}{r_{ij}^2} \tag{1}$$
Then (1) is an incomplete description of the gravitational force. Equation (1) only represents the magnitude of the gravitational force, as can be noticed by the fact that the right side of it is just a scalar.
It is a well-established observation that gravity is a force whose direction is along the line connecting the masses $m_i$ and $m_j$. As a force, gravity must be written as a vector, and therefore the right form would be
$$\vec{F_{ij}}=\frac{γm_im_j}{r_{ij}^2} \hat{r_{ij}} \tag{2}$$
From (II), equation (2) can be written as
$$\vec{F_{ij}}=\frac{γm_im_j}{r_{ij}^2} \frac{\vec{r_{ij}}}{r_{ij}}=\frac{γm_im_j}{r_{ij}^3} \vec{r_{ij}} \tag{3}$$
Typing vectors and unit vectors with arrows and "hats" atop (as in current case) is a little cumbersome. To solve for typing vectors with arrows, one can select other options as boldfacing them for example. To avoid unit vectors one uses the convention $\frac{\vec{r}}{r}$, i.e., the choice of (3) is a convenient form of writing these sort of equations, especially for works containing a huge amount of text.
It is not only easier (if one uses a WYSIWYG text editor of typing machine) to type equation (3) using these conventions, but visually cleaner as well. Just check:
\begin{equation} \mathbf{F}_{ij} \boldsymbol{=}\mathrm γ\dfrac{m_i m_j}{\mathbf{r}_{ij}^3 } \mathbf{r}_{ij} \tag{4} \end{equation}
As was pointed out by @AndrewSteane, there are even mathematical advantages by using this notation, hence, it is just more convenient to use the $\frac{\bf r}{r^3}$ notation.
The problem with this, however, is that people (almost) always is introduced to Newton's law of gravity in its non-vector form (equation 1) and is expecting to find the inverse square law in a first glance, therefore, it may look odd for beginners until one gets used to it some point in future ;-)
answered Jun 19 at 14:30
J. ManuelJ. Manuel
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Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula
Rashad M. Asharabi 1, and Jürgen Prestin 2,,
Department of Mathematics, College of Arts and Sciences, Najran University, Najran, Saudi Arabia
Institute of Mathematics, University of Lübeck, D-23562 Lübeck, Germany
Received September 2019 Revised February 2020 Published May 2020
Fund Project: The first author gratefully acknowledges the support by the Alexander von Humboldt Foundation under grant 3.4-YEM/1142916
Figure(4) / Table(6)
The use of sampling methods in computing eigenpairs of two-parameter boundary value problems is extremely rare. As far as we know, there are only two studies up to now using the bivariate version of the classical and regularized sampling series. These series have a slow convergence rate. In this paper, we use the bivariate sinc-Gauss sampling formula that was proposed in [6] to construct a new sampling method to compute eigenpairs of a two-parameter Sturm-Liouville system. The convergence rate of this method will be of exponential order, i.e. $ O(\mathrm{e}^{-\delta N}/\sqrt{N}) $ where $ \delta $ is a positive number and $ N $ is the number of terms in the bivariate sinc-Gaussian formula. We estimate the amplitude error associated to this formula, which gives us the possibility to establish the rigorous error analysis of this method. Numerical illustrative examples are presented to demonstrate our method in comparison with the results of the bivariate classical sampling method.
Keywords: Sinc approximation, multiparameter spectral theory, eigencurve, Sturm-Liouville systems, Bernstein space.
Mathematics Subject Classification: Primary: 34B05, 34B09; Secondary: 65F18.
Citation: Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185
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Figure 1. The eigencurves in Example 1
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Figure 2. (a) The logarithm of the norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,20}, \mu_{k,0,20})\|_{\mathbb{R}^{2}} $ for $ k = 1,\ldots,6 $ in Example 1. (b) The logarithm of the norm error $ \|(\lambda_{3}^{*},\mu_{3}^{*})-(\lambda_{3,0,N},\mu_{3,0,N})\|_{\mathbb{R}^{2}} $ for $ N = 10,15,20,25 $ in Example 1
Table 1. Comparisons
Methods Region of approximation Convergence rate
WKS sampling $ [-N,N]^{2} $ $ \ln N/\sqrt{N} $
Regularized sampling $ [-N,N]^{2} $ $ \ln N/N^{m+1/2} $
Sinc-Gaussian sampling $ \prod_{j=1}^{2}[(n_{j}-1/2)h_{j},(n_{j}+1/2)h_{j}] $ $ \mathrm{e}^{-\delta N}/\sqrt{N} $
Table 2. Approximation of eigenpairs with $ h = 1 $
$ k $ $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Bivariate WKS sampling
1 1.813797507802172 1.513239555736101
6 10.88280054723645 10.836602869705539
Bivariate sinc-Gauss sampling
6 10.882796185506988 10.83675471755794
Table 3. The norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,15},\mu_{k,0,15})\|_{\mathbb{R}^{2}} $
$ k $ Bivariate WKS sampling Bivariate sinc-Gauss sampling
$ h=1 $ $ h=0.5 $
1 8.73082$ \times 10^{-6} $ 1.00333$ \times 10^{-9} $ 6.94810$ \times 10^{-12} $
2 3.30572$ \times 10^{-5} $ 5.91952$ \times 10^{-10} $ 1.23606$ \times 10^{-11} $
{$ k $} $ \lambda_{k,0,15} $ $ \mu_{k,0,15} $
Table 5. The norm error $ \|(\lambda_{k}^{*},\mu_{k}^{*})-(\lambda_{k,0,15},\mu_{k,0,15})\|_{\mathbb{R}^{2}} $ with $ h = 1 $
1 1.77230$ \times 10^{-5} $ 6.16607$ \times 10^{-9} $
Table 6. Approximation of eigenpairs with $ h = 1 $ and $ \varepsilon = 10^{-8} $
{$ k $} $ \lambda_{k,\varepsilon,15} $ $ \mu_{k,\varepsilon,15} $
[1ex] Bivariate WKS sampling
8 10.525522861700562 10.538515646451458
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Rashad M. Asharabi Jürgen Prestin | CommonCrawl |
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently.[1]
Differences between low-dimensional and high-dimensional topology
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists.
The distinction is because surgery theory works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work. Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a homotopy of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory.
A modification of the Whitney trick can work in 4 dimensions, and is called Casson handles – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
Important tools in geometric topology
Main article: List of geometric topology topics
Fundamental group
Main article: Fundamental group
In all dimensions, the fundamental group of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
Orientability
Main article: Orientability
A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
Handle decompositions
Main article: Handle decomposition
A handle decomposition of an m-manifold M is a union
$\emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M$
where each $M_{i}$ is obtained from $M_{i-1}$ by the attaching of $i$-handles. A handle decomposition is to a manifold what a CW-decomposition is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory.
Local flatness
Main article: Local flatness
Local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds.
Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If $x\in N,$ we say N is locally flat at x if there is a neighborhood $U\subset M$ of x such that the topological pair $(U,U\cap N)$ is homeomorphic to the pair $(\mathbb {R} ^{n},\mathbb {R} ^{d})$, with a standard inclusion of $\mathbb {R} ^{d}$ as a subspace of $\mathbb {R} ^{n}$. That is, there exists a homeomorphism $U\to R^{n}$ such that the image of $U\cap N$ coincides with $\mathbb {R} ^{d}$.
Schönflies theorems
Main article: Jordan-Schönflies theorem
The generalized Schoenflies theorem states that, if an (n − 1)-dimensional sphere S is embedded into the n-dimensional sphere Sn in a locally flat way (that is, the embedding extends to that of a thickened sphere), then the pair (Sn, S) is homeomorphic to the pair (Sn, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their independent proofs[2][3] of this theorem.
Branches of geometric topology
Low-dimensional topology
Main article: Low-dimensional topology
Low-dimensional topology includes:
• Surfaces (2-manifolds)
• 3-manifolds
• 4-manifolds
each have their own theory, where there are some connections.
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.
2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
Knot theory
Main article: Knot theory
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.
High-dimensional geometric topology
In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.
A characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not. In other words, characteristic classes are global invariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry.
Surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by Milnor (1961). Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 3.
More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M ′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other interesting invariants of the manifold are known.
The classification of exotic spheres by Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.
See also
• Category:Maps of manifolds
• List of geometric topology topics
• Plumbing (mathematics)
References
1. "What is geometric topology?". math.meta.stackexchange.com. Retrieved May 30, 2018.
2. Brown, Morton (1960), A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76. MR0117695
3. Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 1959 59–65. MR0117693
• R. B. Sher and R. J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN 0-444-82432-4.
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| Wikipedia |
\begin{document}
\title{Scenery Reconstruction on Finite Abelian Groups}
\author{Hilary Finucane\footnote{Supported by an ERC grant.}, Omer
Tamuz\footnote{Supported by ISF grant 1300/08. Omer Tamuz is a
recipient of the Google Europe Fellowship in Social Computing, and
this research is supported in part by this Google Fellowship.} and
Yariv Yaari\footnote{Weizmann Institute, Rehovot 76100, Israel}}
\maketitle
\begin{abstract}
We consider the question of when a random walk on a finite abelian
group with a given step distribution can be used to reconstruct a
binary labeling of the elements of the group, up to a
shift. Matzinger and Lember (2006) give a sufficient condition for
reconstructibility on cycles. While, as we show, this condition is
not in general necessary, our main result is that it is necessary
when the length of the cycle is prime and larger than 5, and the
step distribution has only rational probabilities. We extend this
result to other abelian groups.
\end{abstract}
\section{Introduction} Benjamini and Kesten~\cite{BenjKesten:96} consider the following model: Let $G=(V,E)$ be a graph and let $f_1,f_2:V \to \{0,1\}$ be binary labelings of the vertices, or ``sceneries". Let $v(t)$, for $t \in \N$, be the position of a particle performing a random walk on $G$. Given an observation of one of the sequences $\{f_1(v(t))\}$ or $\{f_2(v(t))\}$, is it possible to decide which of the two sequences was observed? This is the problem of distinguishing sceneries. Benjamini and Kesten give some conditions under which one can distinguish correctly with probability one, and show that in some cases this cannot be done. Lindenstrauss~\cite{Lindenstrauss:1999} showed that when $G=\Z$ there exist uncountably many functions $f$ that cannot be distinguished given $\{f(v(t))\}$.
The problem of scenery reconstruction is that of learning a completely unknown scenery $f:V \to \{0,1\}$ by observing $\{f(v(t))\}$. L\"owe, Matzinger and Merkel~\cite{lowe2004reconstructing} showed that when $G=\Z$ and the values of $f$ are chosen i.i.d.\ uniformly (from a large enough set), then almost all functions $f$ can be reconstructed. Furthermore, Matzinger and Rolles~\cite{Matzinger:2003} showed that $f$ can be reconstructed in the interval $[-n,n]$ with high probability from a polynomial sample. Further related work has been pursued by Howard~\cite{Howard:96, howard1996orthogonality,howard1997distinguishing}, and a good overview is given by Kesten~\cite{kesten1998distinguishing}.
We focus on the case when the graph $G$ is an undirected cycle of size $n$. One may think of this graph as having $\Z_n = \Z/n\Z$ as its vertex set, with $(k,\ell) \in E$ whenever $k-\ell \in \{-1,1\}$; equivalently, $G$ is the Cayley graph of $\Z_n$ with generating set $\{-1,1\}$. We characterize a random walk on this graph by a step distribution $\gamma$ on $\Z_n$ such that at each turn with probability $\gamma(k)$ the particle jumps $k$ steps: $\gamma(k) = \P{v(t+1)-v(t)=k}$. Indeed, this is simply a random walk on the group $\Z_n$. We choose $v(1)$ uniformly from $\Z_n$.
Let the functions $f_1:\Z_n \to \{0,1\}$ and $f_2:\Z_n \to \{0,1\}$ be two sceneries. Fixing the random walk $\gamma$, we say that $f_1$ and $f_2$ belong to the same equivalence class if the distribution of $\{f_1(v(t))\}$ is equal to $\{f_2(v(t))\}$. Note that for the finite case, unlike the infinite case, these two distributions are different if and only if they are mutually singular, if and only if $f_1$ and $f_2$ can be distinguished with probability one from a single realization of $\{f(v(t))\}$. Thus, our equivalence relation can also be thought of as indistinguishability.
A first observation is that if $f_1$ and $f_2$ differ only by a cyclic shift, i.e.\ there exists an $\ell$ such that for all $k$ it holds that $f_1(k)=f_2(k+\ell)$ (again using addition in $\Z_n$), then $f_1$ and $f_2$ are in the same equivalence class. Hence the equivalence classes of functions that cannot be distinguished contain, at the least, all the cyclic shifts of their members.
The trivial random walk that jumps a single step to the right w.p.\ $1$ clearly induces minimal equivalence classes; that is, classes of functions related by cyclic shifts and nothing more. Another trivial random walk - the one that jumps to any $k \in \Z_n$ with uniform probability - induces equivalence classes that contain all the functions with a given number of ones.
For some random walks the classes of indistinguishable functions can be surprising. Consider the following example, illustrated in Fig.~\ref{fig:hilary-walk} below. The random walk is on the cycle of length $n$, for $n$ divisible by $6$ ($n=12$ in the figure). Its step function is uniform over $\{-2,-1,1,2\}$, so that it jumps either two steps to the left, one to the left, one to the right or two to the right, all with probability $1/4$.
Let $f_1(k)$ be $0$ for even $k$ and $1$ otherwise. Let $f_2(k)$ be $0$ for $k \mod 6 \in \{0,1,2\}$ and $1$ otherwise. It is easy to see that these two functions are indistinguishable, since the sequence of observed labels will be a sequence of uniform i.i.d.\ bits in both cases.
\begin{figure}
\caption{ The random walk depicted here has a
step distribution that is uniform over $\{-2,-1,1,2\}$. It cannot be
used to distinguish the two very different sceneries $f_1$ and
$f_2$ above.}
\label{fig:f_1}
\label{fig:f_2}
\label{fig:hilary-walk}
\end{figure}
The question that we tackle is the following: which random walks induce minimal equivalence classes? In other words, for which random walks can any two sceneries that differ by more than a shift be distinguished? In the finite case, when any two sceneries differing by more than a shift can be distinguished, it is also possible to reconstruct any scenery up to a shift. So we call a random walk that induces minimal equivalence classes {\bf reconstructive}: \begin{definition}
Let $\gamma:H \to \R$ be the step distribution of a random walk $v(t)$
on a finite group $H$, so that $v(1)$ is picked uniformly from $H$
and $\gamma(k) = \P{v(t+1)-v(t) = k}$. Then $v(t)$ is {\bf
reconstructive} if the distributions of
$\{f_1(v(t))\}_{t=1}^\infty$ and $\{f_2(v(t))\}_{t=1}^\infty$ are
identical only if $f_1$ is a shift of $f_2$. \end{definition}
We are interested in exploring the conditions under which $v(t)$ is reconstructive. Howard~\cite{Howard:96} answers this question for $\gamma$ with support on $\{-1,0,1\}$. For symmetric walks in which $\gamma(-1) = \gamma(1) \neq 0$ he shows that $f$ can be reconstructed up to a shift and a mirror image flip (that is, $f_1$ and $f_2$ cannot be distinguished when $f_1(k)=f_2(-k)$). In all other cases (except the trivial $\gamma(0)=1$) he shows that the equivalence classes are minimal.
Matzinger and Lember~\cite{matzinger2006reconstruction} introduce the use of the Fourier transform to the study of this question. They prove the following theorem\footnote{Matzinger and Lember's notion of
a reconstructive random walk is slightly different, in that they
only require reconstruction up to a shift {\em and flip}, where the
{\em flip} of $f(k)$ is $f(-k)$. Thus Theorem~\ref{thm:distinct-f}
differs in this point from theirs. They also require reconstruction
from a single sequence of observations, which, as we point out
above, is equivalent to our notion in the case of finite groups.} : \begin{theorem}[Matzinger and
Lember~\cite{matzinger2006reconstruction}, Theorem 3.2]
\label{thm:distinct-f}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $\Z_n$. Let
$\hat{\gamma}$ be the Fourier Transform of $\gamma$. Then $v(t)$ is
reconstructive if the Fourier coefficients $\{\hat{\gamma}(x)\}_{x \in
\Z_n}$ are distinct. \end{theorem} We provide the proof for this theorem in the appendix for the reader's convenience, extending it (straightforwardly) to random walks on any abelian group\footnote{Matzinger and Lember state it
for ``periodic sceneries on $\Z$'', which are equivalent to
sceneries on cycles.}. We henceforth use $\hat{\gamma}$ to denote the Fourier transform of $\gamma$.
This condition is not necessary, as we show in Theorem~\ref{thm:counterexample}. Our main result is that this condition {\em is} necessary for random walks on $\Z_n$ when $n$ is prime and larger than 5, and $\gamma$ is rational. \begin{theorem}
\label{thm:prime-distinct}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $\Z_n$, for
$n$ prime and larger than five, and let $\gamma(k)$ be rational for all
$k$. Then $v(t)$ is reconstructive only if the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \Z_n}$ are distinct. \end{theorem}
When a step distribution $\gamma$ is rational (i.e., $\gamma(k)$ is rational for all $k$) then it can be viewed as a uniform distribution over a finite multiset $\Gamma$. Here a finite multiset is a finite collection of elements with repetitions.
Using this representation we make the following definition: \begin{definition}
\label{def:drift}
The {\em drift} of a random walk on $\Z_n$ with step distribution
uniform over the multiset $\Gamma$ is $D(\Gamma) = \sum_{k \in
\Gamma} k$ (with addition in $\Z_n$). \end{definition} We show that a walk with non-zero drift is reconstructive on prime-length cycles. \begin{theorem}
\label{cor:drift}
Suppose $n$ is prime and greater than 5, and suppose
$v(t)$ is a random walk over $\Z_n$, with step distribution uniform
over the multiset $\Gamma$. Then if $D(\Gamma) \neq 0$ then $v(t)$
is reconstructive. \end{theorem}
We show that any fixed, bounded rational random walk on a large enough prime cycle is either symmetric or reconstructive. Symmetric random walks are those for which $\gamma(k)=\gamma(-k)$ for all $k$. They are not reconstructive because they cannot distinguish between $f(k)$ and $f(-k)$.
Given a step distribution $\gamma : \mathbb{Z} \rightarrow [0,1]$ on $\Z$, let $\gamma_n$ denote the step distribution on $\mathbb{Z}_n$ induced by $\gamma$ via $$\gamma_n(k)= \sum_{a = k\, mod\, n} \gamma(a).$$ \begin{theorem}\label{cor:bounded}
Let $\gamma : \Z \rightarrow [0,1]$ be a distribution over $\Z$ with
bounded support, and assume $\gamma(a) \in \mathbb{Q}$ for all
$a$. Then either $\gamma$ is symmetric, or there exists an $N$ such
that for all prime $n > N$, $\gamma_n$ is reconstructive. \end{theorem}
Finally, we extend the result of Theorem~\ref{thm:prime-distinct} to random walks on any abelian group of the form $Z_{p_1}^{d_1} \times \cdots \times Z_{p_m}^{d_m}$ where $p_1, \ldots, p_m$ are primes larger than $5$. \begin{theorem}\label{thm:general}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on
$\mathbb{Z}_{p_1}^{d_1} \times \cdots \times
\mathbb{Z}_{p_m}^{d_m}$, such that $\gamma(k_1, \ldots, k_m) \in \mathbb Q$
for all $(k_1, \ldots , k_m)$, and suppose that $p_i>5$ is prime for
all $i$. Then $v(t)$ is reconstructive only if the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \Z_n}$ are distinct. \end{theorem} This result applies, in particular, to walks on $\Z_n$ where $n$ is square-free (i.e., is not divisible by a square of a prime) and not divisible by $2$, $3$ or $5$, since such groups are isomorphic to groups of the form considered in this theorem.
We show that the rationality condition of our main theorem is tight by presenting an example of a random walk with prime $n$, irrational probabilities and non-distinct Fourier coefficients where reconstruction is possible. It remains open, however, whether the result holds for rational random walks on general cycles of composite length.
This paper will proceed as follows. In Section~\ref{sec:preliminaries}, we will review some useful algebraic facts. In Section~\ref{sec:mainproof}, we will prove Theorem~\ref{thm:prime-distinct}. In Section~\ref{sec:corollaries} we will prove Theorem~\ref{cor:drift}, on walks with non-zero drift, and Theorem~\ref{cor:bounded}, on walks with bounded support. In Sections~\ref{sec:tightness}, and \ref{sec:extensions} we show the tightness of the main theorem, and extend it to products of prime-length cycles, respectively. In Section~\ref{sec:openproblems}, we will present some open problems.
\section{Preliminaries} \label{sec:preliminaries}
Before presenting the proofs of our theorems, we would like to refresh the reader's memory of abelian (commutative) groups and their Fourier transforms, as well as introduce our notation. For the cyclic group $\Z_n$, the Fourier transform $\hat{\gamma}: \Z_n \to \mathbb C$ of $\gamma :\Z_n \to \mathbb C$ is defined by \begin{align*}
\hat{\gamma}(x) = \sum_{k \in \Z_n}\omega_n^{kx}\gamma(k). \end{align*} where $\omega_n = e^{-\frac{2\pi}{n}i}$.
Any finite abelian group can be written as the Cartesian product of cycles with prime power lengths: if $H$ is abelian then there exist $\{n_i\}$ such that $H$ is isomorphic to the {\em torus} $\Z_{n_1}\times \cdots \times \Z_{n_m}$. Thus an element $h \in H$ can be thought of as a vector so that $h_i$ is in $\Z_{n_i}$.
When all the $n_i$'s are the same then $H=\Z_n^d$ is a {\em regular
torus}, and when $n$ is prime then it is a {\em prime regular
torus}. Here a natural dot product exists for $h,k \in H$: $h \cdot k=\sum_{i=1}^dh_ik_i$, where both the multiplication and summation are over $\Z_n$. The Fourier transform for $\Z_n^d$ is thus \begin{align*}
\hat{\gamma}(x) = \sum_{k \in \Z_n^d}\omega_n^{k \cdot x}\gamma(k). \end{align*}
The representation of abelian groups as tori is not unique. The canonical representation is $H=\Z_{n_1}^{d_1}\cdots\Z_{n_m}^{d_m}$, where the $n_i$'s are distinct powers of primes. In this general case an element $h \in H$ can be thought of as a vector of vectors, so that $h_i$ is in $\Z_{n_i}^{d_i}$. The Fourier transform is \begin{align*}
\hat{\gamma}(x) = \sum_{k \in H}\prod_{i=1}^m\omega_{n_i}^{k_i \cdot x_i}\gamma(k). \end{align*}
The {\em $n$-th cyclotomic polynomial} is the minimal polynomial of $\omega_n$; i.e. the lowest degree polynomial over $\mathbb Q$ that has $\omega_n$ as a root and a leading coefficient of one. This polynomial must divide all non-zero polynomials over $\mathbb Q$ that have $\omega_n$ as a root. When $n$ is prime then this polynomial is $Q_n(t)= \sum_{i=0}^{n-1} t^i$.
\section{Proof of Theorem~\ref{thm:prime-distinct}} \label{sec:mainproof}
\subsection{Outline of main proof} Theorem~\ref{thm:prime-distinct} states the distinctness of the Fourier coefficients of the step distribution $\gamma$ is a necessary (and by Theorem~\ref{thm:distinct-f} sufficient) condition for reconstruction, when $n$ is prime and greater than five, and the probabilities $\gamma(k)$ are rational. In Theorem~\ref{thm:counterexample} we give a counterexample that shows that the rationality condition is tight, giving an irrational random walk on $\Z_7$ with non-distinct Fourier coefficient, which is reconstructive. In Section~\ref{sec:tightness}, we show that the condition $n>5$ is also tight.
To prove Theorem~\ref{thm:prime-distinct} we shall construct, for any random walk on a cycle of prime length larger than 5 with rational step distribution $\gamma$ such that $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x \neq y$, two functions $f_1$ and $f_2$ that differ by more than a shift. We shall consider two random walks: $v_1(t)$ on $f_1$ and $v_2(t)$ on $f_2$, with the same step distribution $\gamma$. If a coupling exists such that the observed sequences $\{f_1(v_1(t))\}$ and $\{f_2(v_2(t))\}$ are identical then $f_1$ and $f_2$ cannot be distinguished, since the sequences will have the same marginal distribution. We will show that such a coupling always exists.
\subsection{Motivating example}
Consider the following example. Let the step distribution $\gamma$ uniformly choose to move 1, 2, or 4 steps to the left on a cycle of length 7. A simple calculation will show that $\hat{\gamma}(1) = \hat{\gamma}(2)$. In what is not a coincidence (as we show below), the support of $\gamma$ is invariant under multiplication by 2 (over $Z_7$): $2\{1,2,4\} = \{1,2,4\}$. We will use this fact to construct two functions $f_1$ and $f_2$ that are indistinguishable, even though they are not related by a shift.
Let $f_1(k)$ equal 1 for $k \in \{0,1\}$ and $f_2(k)$ equal 1 for $k \in \{0,2\}$, as shown in Fig.~\ref{fig:seven-walk}. (More generally, $f_1$ and $f_2$ can be any two functions such that $f_1(k) = f_2(2k)$ for all $k$.) Let $v_1(t)$ be a random walk over $Z_7$ with step distribution $\gamma$. We couple it to a random walk $v_2(t)$ by setting $v_2(t) = 2 \cdot v_1(t)$, so that $v_2$ always jumps twice as far as $v_1$. Its step distribution is thus uniform over $2 \cdot \{1,2,4\}=\{1,2,4\}$, and indeed has the same step distribution $\gamma$. Finally, because $f_2(2k) = f_1(k)$, we have $f_2(v_2(t)) = f_1(v_1(t))$ for all $t$.
We have thus constructed two random walks, both with step distribution $\gamma$. The first walks on $f_1$ and the second walks on $f_2$, and yet their observations are identically distributed. Hence the two functions are indistinguishable.
Note also that the sum of the elements in the support of $\gamma$ is zero: $1+2+4=0 \mod 7$. This is not a coincidence; as we show in Theorem~\ref{cor:drift} this must be the case for every step distribution that is not reconstructive.
\begin{figure}
\caption{ We couple the random walks on $f_1$
and $f_2$ above by choosing, at each time period, either `a', `b'
or `c' uniformly and having each walk take the step marked by that
letter in its diagram. The result is that $f_1(v_1(t)) =
f_2(v_2(t))$. }
\label{fig:f_1}
\label{fig:f_2}
\label{fig:seven-walk}
\end{figure}
\subsection{Proof of Theorem~\ref{thm:prime-distinct}} To prove Theorem~\ref{thm:prime-distinct} we will first prove two lemmas. We assume here that $\gamma(k)$ is rational for all $k$ and that $n>5$ is prime.
\begin{lemma}\label{lemma:xS=yS} Suppose that $\hat{\gamma}(x) =
\hat{\gamma}(y)$ for some $x, y \neq 0$. Then $\gamma(k
x^{-1} y) = \gamma(k)$ for all $k$, where all operations are in the field $\mathbb{Z}_n$. \end{lemma} In other words, the random walk has the same probability to add $k$ or $kx^{-1} y$, for all $k$.
\begin{proof}
Letting $\omega$ denote $\omega_n=e^{-\frac{2\pi}{n}i}$ and applying the definition
of $\hat{\gamma}$, we have $$\sum_{k = 0}^{n-1}\gamma(k) \omega^{kx} = \sum_{k=0}^{n-1}\gamma(k)\omega^{ky}.$$ Since $\omega^n = 1$, $\omega$ is a root of the polynomial $$P(t) = \sum_{k = 0}^{n-1}\gamma(k) t^{kx\, (mod\, n)} - \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}.$$ or, by a change of variables $$P(t) = \sum_{k = 0}^{n-1}\left(\gamma(kx^{-1}) -\gamma(k y^{-1}) \right)t^{k}$$ where the inverses are taken in the field $\mathbb{Z}_n$. (Recall $n$ is prime.)
Since $P$ has $\omega_n$ as a root then $Q_n$, the $n$-th cyclotomic polynomial, divides $P$. However, $P$ has degree at most $n-1$, so either $P$ is the zero polynomial or $P$ is equal to a constant times $Q_n$.
The latter option is impossible, since $P(1) = 0$ and $Q_n(1) = n$. Thus, $P = 0$, so $\gamma(k x^{-1}) = \gamma(k y^{-1})$ for all $k$, or equivalently, $\gamma(k x^{-1} y) = \gamma(k)$ for all $k$. \end{proof}
\begin{lemma}\label{lemma:prime-coupling}
If $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x, y
\neq 0$, and if $f_1, f_2:\mathbb{Z}_n \rightarrow \{0,1\}$ are such
that $f_1(k) = f_2(x^{-1}yk)$ for all $k$, then $\gamma$ cannot
distinguish between $f_1$ and $f_2$. \end{lemma}
\begin{proof}
Let $v_1(t)$ be a $\gamma$-r.w.\ on $f_1$. Let $v_2(t)$ be a random
walk defined by $v_2(t) = x^{-1}y v_1(t)$, so that whenever $v_1$
jumps $k$ then $v_2$ jumps $x^{-1}yk$.
By Lemma~\ref{lemma:xS=yS} we have that $\gamma(x^{-1}yk) = \gamma(k)$, so
the step distribution of $v_2$ is also $\gamma$. Furthermore, at time
$t$, $v_1$ will see $f_1(v_1(t))$ and $v_2$ will see $f_2(v_2(t)) =
f_2(x^{-1}yv_1(t))$. But for all $k$, $f_1(k) = f_2(x^{-1}y \cdot
k)$, and so $f_1(v_1(t))=f_2(v_2(t))$ for all $t$. \end{proof}
To conclude the proof of Theorem~\ref{thm:prime-distinct}, we must find, for all prime $n > 5$ and all $x \neq y$, $x, y \neq 0$ functions $f_1$ and $f_2$ such that $f_1$ is not a shift of $f_2$, and $f_1(k) = f_2(x^{-1}yk)$. We use another argument for the special case $x=0$ or $y = 0$.
\begin{proof}[Proof of Theorem~\ref{thm:prime-distinct}]
For $x, y \neq 0$, by Lemma~\ref{lemma:prime-coupling} we can choose
any $f_1$ and $f_2$ such that $f_1(k) = f_2(x^{-1}yk)$ and $f_1$ is
not a shift of $f_2$; for example, if $x^{-1}y \neq -1$, then $$f_1(k) = \begin{cases}
1 & k = 0,1\\
0 & o.w. \end{cases}$$ and $$f_2(k) = \begin{cases}
1 & k = 0, x^{-1}y\\
0 & o.w. \end{cases}$$ satisfy the requirements. For $n=7$ and $x^{-1}y=2$ these two functions are depicted in Fig.~\ref{fig:seven-walk}.
If $x^{-1}y = -1$, then we can choose any function which is not a reflection or shift of itself; for example, $$f_1(k) = \begin{cases} 1& k = 0, 1, 3 \\ 0 & o.w. \end{cases}$$ and $f_2(k)=f_1(-k)$.
This leaves us with the case $x=0$ or $y=0$; wlog, suppose $x = 0$. We still have $P(t) \equiv 0$ by the same proof as above, which gives us \begin{eqnarray*} 0 & \equiv& P(t)\\ & = & \sum_{k = 0}^{n-1}\gamma(k) t^{kx\, (mod\, n)} - \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}\\ &=& 1- \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}\\ &=& 1- \sum_{k=0}^{n-1}\gamma(ky^{-1})t^{k}\\ \end{eqnarray*} where the last equality is possible because $y \neq x=0$. But then we must have $\gamma(0) = 1$, so the random walk does not move after the first vertex is chosen. Thus, $\gamma$ cannot distinguish between any two functions with the same number of ones. \end{proof}
\section{Walks with non-zero drift and walks with bounded support} \label{sec:corollaries}
In this section, we will use the main Lemma~\ref{lemma:xS=yS} to prove Theorems~\ref{cor:drift} and \ref{cor:bounded}.
\subsection{Drift and reconstruction}
Recall from Definition~\ref{def:drift} that the drift of a random walk on $\Z_n$ with step function $\Gamma$ is $D(\Gamma) = \sum_{k \in
\Gamma} k$.
\begin{proof}[Proof of Theorem~\ref{cor:drift}]
We will show that if reconstruction is not possible then the drift
is zero.
If $f$ cannot be reconstructed then there exist $x\neq y$ with
$\hat{\gamma}(x) = \hat{\gamma}(y)$, by Theorem~\ref{thm:distinct-f}. Then
it may be that $x$ or $y$ is 0; in this case, as we have shown in
the proof of Theorem~\ref{thm:prime-distinct}, the random walk never
moves, so the drift is zero. Otherwise $x, y \neq 0$, which by
Lemma~\ref{lemma:xS=yS} implies $\gamma(k) = \gamma(x^{-1}yk)$, or
alternatively $x^{-1}y\Gamma = \Gamma$. But then \begin{eqnarray*}
D(\Gamma) &=& \sum_{k \in \Gamma} k\\
&=& \sum_{k \in \Gamma} x^{-1}yk\\
&=& xy^{-1} D(\Gamma)\\ \end{eqnarray*} so $D(\Gamma) = 0$. \end{proof}
\subsection{Random walks with bounded step distribution}
Next, we consider random walks with rational transition probabilities, and with zero probability for steps larger than $c$, for some constant $c$ independent of $n$. We show in Theorem~\ref{cor:bounded} that in this case, either the random walk is symmetric, or for large enough prime cycles, the walk is reconstructive. A random walk $\gamma$ is symmetric when $\gamma(k) = \gamma(-k)$ for all $k$. Symmetric random walks are not reconstructive, since $\hat{\gamma}(k)=\hat{\gamma}(-k)$; they cannot distinguish any function $f(k)$ from its flip $f(-k)$. We will need two lemmas in order to prove this theorem.
As we note above, a rational step distribution $\gamma$ can be thought of as a uniform distribution over a multiset $\Gamma=(a_1,\ldots,a_m) \in \Z^m$. We denote by $\Gamma_n=(k_1,\ldots,k_m) \in \Z_n^m$ the natural embedding of $\Gamma$ into $\Z_n$: $k_i = a_i \mod n$.
\begin{lemma}
\label{lemma:Gamma_v}
Let $\Gamma =(a_1,\ldots,a_m) \in \Z^m$ be a multiset. Let
$\Gamma_n=(k_1,\ldots,k_m)$ be the natural embedding of $\Gamma$
into $\Z_n$. Assume further that $1 \in \Gamma$, i.e., $a_i=1$ for
some $i$.
Then there exists a positive integer $N = N(\Gamma)$ such that for
any $n > N$ it holds that if $v\Gamma_n=\Gamma_n$ then $v \in
\{-1,0,1\}$. \end{lemma} \begin{proof}
\label{lemma:Gamma_n}
Let $b = \max_i|a_i|$, so that $\Gamma$ is bounded in $[-b,b]$, and
assume w.l.o.g that $b \in \Gamma$. Let $N=2b^2$, $n>N$ and let
$v\Gamma_n=\Gamma_n$. We will show that $v \in \{-1,0,1\}$.
Because $1 \in \Gamma_n$ and $v\Gamma_n = \Gamma_n$ we have $v \in
\Gamma_n$. Assume by way of contradiction that $v \not \in
\{-1,0,1\}$. Then $v \in [-b, -2] \cup [2, b]$. Since $b \in
\Gamma_n$ we also have $vb \in \Gamma_n$, so $vb \in [-b^2, -2b]
\cup [2b, b^2]$. But this is a contradiction because $n > 2b^2$ so
both of these intervals have empty intersection with $\Gamma_n$. \end{proof} \begin{lemma}
\label{lemma:symmetric_reconstructive}
Let $\Gamma_n$ be a multiset characterizing a random walk $\gamma_n$
over $\Z_n$, such that if $v\Gamma_n=\Gamma_n$ then $v \in
\{-1,0,1\}$. Then the random walk is either symmetric or
reconstructive. \end{lemma} \begin{proof}
Suppose the random walk is not reconstructive; i.e. there exist $x \neq
y$ with $\hat{\gamma_n}(x) = \hat{\gamma_n}(y)$. We will show that this
means that the random walk is symmetric.
If $x$ is 0, then by the argument in the proof of
theorem~\ref{thm:prime-distinct} we have that $\Gamma = \{0\}$,
which is symmetric, and we are done. Otherwise denote $v =
x^{-1}y$. Then by Lemma~\ref{lemma:xS=yS} we have $\Gamma_n = v
\Gamma_n$, and by this lemma's hypothesis we have $v \in
\{-1,0,1\}$.
Now, $v$ cannot equal $1$ since $v = x^{-1}y$ and $x \neq y$. If
$v=-1$ then the random walk is symmetric. Finally, if $v=0$ then
again $\Gamma = \{0\}$, and again the random walk is symmetric.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{cor:bounded}]
Let $\Gamma \subset \Z$ correspond to $\gamma$ as above (i.e., $\gamma$ is
uniformly distributed over the multiset $\Gamma$). We will show that
for $n$ large enough if $v\Gamma_n = \Gamma_n$ then $v \in
\{-1,0,1\}$, which, by Lemma~\ref{lemma:symmetric_reconstructive}
will show that the random walk is either symmetric or reconstructive.
For any vector of coefficients $c = (c_1, \ldots, c_m)$, define the
multiset $c(\Gamma)$ as
$$c(\Gamma) = c_1\Gamma + \ldots + c_m\Gamma=\{c_1s_1 + \ldots + c_ms_m
: s_i \in \Gamma\}.$$ Note that if $v\Gamma = \Gamma$, then we
also have that $v c(\Gamma) = c(\Gamma)$.
Suppose that the g.c.d.\ (in $\Z$) of the elements of $\Gamma$ is
one. Then by the Chinese remainder theorem, there exists a $c$ such
that $1 \in c(\Gamma)$. Hence we can apply Lemma~\ref{lemma:Gamma_v}
to $c(\Gamma)$ and infer that for $n$ large enough $v c(\Gamma_n) =
c(\Gamma_n)$ implies $v \in \{-1,0,1\}$. But since $v\Gamma_n =
\Gamma_n$, implies $v c(\Gamma_n) = c(\Gamma_n)$ then we have that
for $n$ large enough $v\Gamma_n = \Gamma_n$ implies $v \in
\{-1,0,1\}$.
Otherwise, let $d$ denote the g.c.d.\ of $\Gamma$. Since the
g.c.d.\ of $d^{-1}\Gamma$ is one we can apply the argument of the
previous case to $d^{-1}\Gamma$ and infer that for $n$ large enough
$vd^{-1}\Gamma_n = d^{-1}\Gamma_n$ implies $v \in \{-1,0,1\}$. But
since again $v\Gamma_n = \Gamma_n$ implies $vd^{-1}\Gamma_n =
d^{-1}\Gamma_n$, then again we have that for $n$ large enough
$v\Gamma_n = \Gamma_n$ implies $v \in \{-1,0,1\}$.
\end{proof}
\section{Tightness of Theorem~\ref{thm:prime-distinct}} \label{sec:tightness}
Theorem~\ref{thm:prime-distinct} states that distinctness of Fourier coefficients is not just sufficient but necessary for reconstructibility of a random walk $\gamma$ on a cycle of length $n$, if the following conditions hold: \begin{enumerate} \item $\gamma(k) \in \mathbb Q$ for all $k$, \item $n> 5$, \item $n$ is prime. \end{enumerate}
To see that condition 2 is tight, note that the simple random walk has $\hat{rw}(k) = \hat{rw}(-k)$ for all $k$, so for $n > 2$, the simple random walk does not have distinct Fourier coefficients. However, Howard~\cite{Howard:96} shows that the simple random walk suffices to reconstruct any scenery up to a flip, and for $n \leq 5$, reconstruction up to a flip is the same as reconstruction up to a shift, since all sceneries are symmetric. So for $n = 3,4,5$, the simple random walk is reconstructive despite having non-distinct Fourier coefficients. For $n = 2$, consider the random walk that simply stays in place.
The following theorem shows that condition 1 is also tight.
\begin{theorem}\label{thm:counterexample} There exists a reconstructive random walk $\gamma$ on $\mathbb{Z}_7$ such that $\hat{\gamma}(3) =\hat{\gamma}(-3)$. \end{theorem}
\begin{proof}
Let $\delta = \frac{\cos\left( 6\pi /7\right) + 0.5}{2
\cos\left(6\pi/7 \right) - 1}$, $\gamma(1) = {\textstyle \frac12} + \delta$ and $\gamma(2) = {\textstyle \frac12} - \delta$. To show that $\hat{\gamma}(3) = \hat{\gamma}(-3)$ is a simple calculation, so we will only show that $\gamma$ can distinguish between any two sceneries.\\
In fact, any random walk with non-zero support exactly on $\{1,2\}$ can distinguish between any two sceneries when $n = 7$. Any r.w. can determine the number of ones in a scenery, so we must show that $\gamma$ can distinguish between non-equivalent sceneries for each number of ones. \begin{itemize} \item There is only one scenery with 0 ones and up to a shift, there is only one scenery with 1 one. \item For 2 ones, $\gamma$ must distinguish among $(1,1,0,0,0,0,0), (1,0,1,0,0,0,0), (1,0,0,1,0,0,0)$. (The others are equivalent to one of these up to a shift.) The third function is the only one for which there will never be two consecutive ones. The first function is the only one for which there will be two consecutive ones, but the substring $(1,0,1)$ never appears. The third function is the only one for which both substrings $(1,1)$ and $(1,0,1)$ will appear. \item For 3 ones, $\gamma$ must distinguish among: (a) $(1,1,1,0,0,0,0)$, (b) $(1,1,0,1,0,0,0)$, (c) $(1,1,0,0,1,0,0$), (d) $(1,1,0,0,0,1,0)$, and (e) $(1,0,1,0,1,0,0)$. (c) is the only one that will never have three consecutive ones. (e) is the only that will ever have five consecutive zeros. Among (a), (b), and (d), (a) is the only such that (1,0,1) will never occur, and while (1,1,0,1,0,0) will occur for (b), it won't for (d). \item For 4,5,6, and 7 ones, repeat the previous arguments with the roles of 0 and 1 reversed. \end{itemize} \end{proof}
\section{Extensions} \label{sec:extensions}
\subsection{Extension to prime regular tori}
\begin{theorem}\label{thm:regular_torus}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $\Z_n^d$, for
$n$ prime and larger than five, and let $\gamma(k)$ be rational for all
$k$. Then $v(t)$ is reconstructive {\em only if} the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \Z_n}$ are distinct. \end{theorem} \begin{proof}
Suppose that $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x \neq
y$, so that the Fourier coefficients aren't distinct. Again letting
$\omega$ denote $\omega_n=e^{-\frac{2\pi}{n}i}$, we have
\begin{align*}
\sum_{k \in \Z_n^d}\gamma(k) \omega^{k \cdot x} =
\sum_{k \in \Z_n^d}\gamma(k)\omega^{k \cdot y}.
\end{align*}
Note that here $k \cdot y$ is the natural dot product over
$\Z_n^d$. If we join terms with equal powers of $\omega$ then
\begin{align*}
\sum_{\ell \in \Z_n}\left(\sum_{\substack{k\;\mathrm{s.t.} \\ k \cdot x
= \ell}}\gamma(k)\right) \omega^\ell =
\sum_{\ell \in \Z_n}\left(\sum_{\substack{k\;\mathrm{s.t.} \\ k \cdot y
= \ell}}\gamma(k)\right) \omega^\ell
\end{align*}
and by the same argument as in the proof of Lemma~\ref{lemma:xS=yS} we have that,
for all $\ell \in \Z_n$,
\begin{align}
\label{eq:same-dist-torus}
\sum_{\substack{k\;\mathrm{ s.t. }\\ k \cdot x = \ell}}\gamma(k) =
\sum_{\substack{k\;\mathrm{ s.t. }\\ k \cdot y = \ell}}\gamma(k)
\end{align}
I.e., if $k$ is distributed according to $\gamma$ then $k \cdot x$ and
$k \cdot y$ have the same distribution. Hence, if we denote
$u_x(t)=v(t) \cdot x$ and $u_y(t) = v(t) \cdot y$ then we also have
identical distributions, under $\gamma$, of $u_x(t)$ and $u_y(t)$.
Fix $g:\Z_n \to \{0,1\}$, and let $f_x(k) = g(x \cdot k)$, and
$f_y(k) = g(y \cdot k)$. Then $f_x(v(t)) = g(u_x(t))$, $f_y(v(t)) =
g(u_y(t))$, and the distributions of $\{f_x(v(t)\}$ and
$\{f_y(v(t))\}$ are identical, by the same coupling argument used in
Lemma~\ref{lemma:prime-coupling} above. Hence $f_x$ and $f_y$ can't
be distinguished.
It remains to show that there exists a $g$ such that $f_x$ and $f_y$
differ by more than a shift. We consider two cases.
\begin{enumerate}
\item Let $x$ be a multiple of $y$, so that $x=\ell y$ for some $
\ell \in \Z_n$, $\ell \neq 1$. Then the problem is essentially
reduced to the one dimensional case of
Theorem~\ref{thm:prime-distinct}, with the random walk projected
on $y$. We can set
$$g(k) = \begin{cases}
1& k = 0, 1 \\
0 & o.w.
\end{cases}$$
when $\ell \neq -1$ and
$$g(k) = \begin{cases}
1& k = 0, 1, 3 \\
0 & o.w.
\end{cases}$$
when $\ell = -1$.
Assume by way of contradiction that $f_x$ and $f_y$ differ by a
shift, so that there exists a $k_0$ such that $f_x(k)=f_y(k +
k_0)$ for all $k$. Since $$f_x(k) = g(x \cdot k) = g(\ell y \cdot
k)$$ and $$f_y(k+k_0) = g(y \cdot (k+k_0)) = g(y \cdot k+y \cdot
k_0)$$ for all $k$, then $$g(\ell m)=g(m+m_0)$$ for some $m_0 \in
\Z_n$ and all $m \in \Z_n$. That is, $g(m)$ is a shift of $g(\ell
m)$. It is easy to verify that this is not possible in the case
$\ell \neq -1$ nor in the case $\ell = -1$.
\item Otherwise $x$ is not a multiple of $y$. Hence they are
linearly independent.
We here set
$$g(k) = \begin{cases}
1& k = 0 \\
0 & o.w.
\end{cases}$$ One can then view $f_x$ ($f_y$), as the indicator
function of the set elements of $\Z_n^d$ orthogonal to $x$
($y$). Denote these linear subspaces as $U_x$ and $U_y$.
Assume by way of contradiction that $f_x$ and $f_y$ differ by a
shift, so that there exists a $k_0$ such that $f_x(k)=f_y(k +
k_0)$ for all $k$. Then $U_x = U_y + k_0$. Since $0$ is an element
of both $U_x$ and $U_y$ then it follows that $k_0$ is also an
element of both. But, being a linear space, $U_y$ is closed under
addition, and so $U_x = U_y+k_0 = U_y$. However, since $x$ and $y$
are linearly independent then their orthogonal spaces must be
distinct, and we've reached a contradiction.
\end{enumerate}
\end{proof} \subsection{Extension to products of prime tori}
Recall that any abelian group $H$ can be decomposed into $H=\Z_{n_1}^{d_1}\times \cdots \times \Z_{n_m}^{d_m}$, where the $n_i$ are distinct powers of primes. We call $H$ ``square free'' when all the $n_i$'s are in fact primes. In this case too (for primes $>5$) we show in Theorem~\ref{thm:general} that the distinctness of Fourier coefficients is a tight condition for reconstructibility.
We can assume w.l.o.g.\ that the $p_i$ are all distinct. We will prove two lemmas before proving the theorem.
Recall that $\mathbb Q(\omega_{p_1}, \ldots, \omega_{p_{j-1}})$ is the field extension of $\mathbb Q$ by $\omega_{p_1}, \ldots, \omega_{p_{j-1}}$. The following is a straightforward algebraic claim and we provide the proof for completeness. \begin{lemma}
\label{lemma:min_poly}
For all $j$, the minimal polynomial $Q_j$ of $\omega_{p_j}$ over
$\mathbb Q(\omega_{p_1}, \ldots, \omega_{p_{j-1}})$ is
$$Q_j(t) = \sum_{i = 0}^{p_j - 1}t^i.$$ \end{lemma}
\begin{proof}
Let $F_j = \mathbb Q(\omega_{p_1}, \ldots, \omega_{p_j})$, let $F_{j-1} =
\mathbb Q(\omega_{p_1}, \ldots, \omega_{p_{j-1}})$, and let $\zeta
=\omega_{p_1}\cdots \omega_{p_j}$. The field $F_j$ contains $
\zeta$ and $\mathbb Q$, so $\mathbb Q(\zeta) \subset F_j$. Since each
$\omega_{p_i}$ is a power of $\zeta$, $\mathbb Q$ and $\omega_{p_1}, \ldots
, \omega_{p_j}$ are all in $\mathbb Q(\zeta)$; thus, $F_j \subset
\mathbb Q(\zeta)$. This allows us to conclude that $F_j = \mathbb Q(\zeta)$.
The degree of $\zeta$ over $\mathbb Q$ is $(p_1-1)\cdots (p_j-1)$, since
$\zeta$ is a primitive $n$-th root of unity and $(p_1-1)\cdots
(p_j-1)$ is the number of integers coprime to $n$. Thus, the field
extension $\mathbb Q \subset F_j$ has degree $[F_j:\mathbb Q] = [\mathbb Q(\zeta):\mathbb Q] =
(p_1 - 1) \cdots (p_j - 1)$. By a similar argument, $[F_{j-1}:\mathbb Q] =
(p_1-1)\cdots (p_{j-1}-1)$. So degree of the field extension
$F_{j-1} \subset F_j$ is
$$[F_j:F_{j-1}] = [F_j : \mathbb Q] / [F_{j-1}:\mathbb Q] = p_j - 1.$$
Since the degree of the minimal polynomial of $\omega_{p_j}$ over
$F_{j-1}$ is the same as the degree of $F_j$ over $F_{j-1}$, we can
conclude that the degree of this minimal polynomial is $p_j -
1$. Since this is also the degree of $\omega_{p_j}$ over $\mathbb Q$, the
minimal polynomial of $\omega_{p_j}$ over $F_{j-1}$ is the same as
the minimal polynomial of $\omega_{p_j}$ over $\mathbb Q$. Thus, the
minimal polynomial $Q_j$ of $\omega_{p_j}$ over $F_{j-1}$ is
$$Q_j(t) = \sum_{i = 0}^{p_j - 1}t^i,$$
as desired. \end{proof}
\begin{lemma}\label{lemma:general_coupling} Suppose $\hat{\gamma}(x_1, \ldots x_m) = \hat{\gamma}(y_1, \ldots y_m)$ for $(x_1, \ldots, x_m), (y_1, \ldots, y_m) \in \mathbb{Z}_{p_1}^{d_1} \times
\cdots \times \mathbb{Z}_{p_m}^{d_m}$. Then for all $(\ell_1, \ldots , \ell_m) \in \mathbb{Z}_{p_1} \times
\cdots \times \mathbb{Z}_{p_m}$, $$\sum_{\substack{(k_1,
\ldots, k_m) \\ k_i \cdot x_i = \ell_i \, \forall i}} \gamma(k_1,
\ldots, k_m) = \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i =
\ell_i\, \forall i}} \gamma(k_1, \ldots, k_m).$$ \end{lemma}
\begin{proof} We will prove by induction on $j$ that the following statement holds: for all $(\ell_{j+1}, \ldots , \ell_m) \in \mathbb{Z}_{p_{j+1}} \times \cdots \times \mathbb{Z}_{p_m}$,
$$\sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j }}^{k_{j } \cdot x_{j}} = \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j }}^{k_{j} \cdot y_{j}}.$$ The base case $j = m$ is just the statement $\hat{\gamma}(x_1, \ldots, x_m) = \hat{\gamma}(y_1, \ldots, y_m)$. Now suppose we know that for all $(\ell_{j+1}, \ldots, \ell_m)$
$$\sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j}}^{k_{j} \cdot x_{j}} = \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j }}^{k_{j} \cdot y_{j}}.$$ Then $\omega_{p_{j}}$ is a root of the polynomial $$P(t) = \sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j-1}}^{k_{j-1} \cdot x_{j-1}} t^{k_j \cdot x_j} - \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j-1 }}^{k_{j-1} \cdot y_{j-1}} t^{k_j \cdot y_j}.$$ But as in the proof of the previous theorem, Lemma~\ref{lemma:min_poly} tells us that $P(t) = Q_j(t)$ or $P(t) \equiv 0$, and so $P(1) = 0 \neq p_j = Q_j(1)$ tells us that $P(t) \equiv 0$. Thus, the coefficient of $t^{\ell_j}$ must be zero for all choices of $\ell_j$, establishing the desired statement for $j-1$. When $j= 0$, this gives us the statement of the lemma. \end{proof}
\begin{proof} [Proof of Theorem~\ref{thm:general}] Suppose $\hat{\gamma}(x_1, \ldots, x_m) = \hat{\gamma}(y_1, \ldots, y_m)$ for $(x_1, \ldots, x_m) \neq (y_1, \ldots, y_m)$. Lemma~\ref{lemma:general_coupling} allows us to construct the following coupling. Let $v_1(1)$ be uniform and $v_1(t) - v_1(t-1)$ be drawn according to $\gamma$. Let $v_2(t)$ be drawn according $\gamma$, coupled so that $v_2(t)_i \cdot y_i = v_1(t)_i \cdot x_i$ for all $i$. By Lemma~\ref{lemma:general_coupling}, this induces the correct distribution on $v_2(t)$. Now, choose an index $j$ such that $x_j \neq y_j$, and let $f_1(k) = g(x_i \cdot k_i)$ and $f_2(k) = g(y_i \cdot k_i)$. The rest of the proof follows as in the proof of Theorem~\ref{thm:regular_torus}. \end{proof}
\section{Open Problems} \label{sec:openproblems}
This work leaves open many interesting questions; we sketch some here.
\begin{itemize}
\item Here, we give an equivalent condition for reconstructivity when (a) the random walk is rational, and (b) the underlying graph corresponds to a group of the form $\mathbb{Z}_{p_1}^{d_1} \times \cdots \times \mathbb{Z}_{p_k}^{d_k}$ for distinct primes $p_1, \ldots, p_k >5$. We know that this condition is not necessary when the random walk is irrational, or when $p_i = 3$ or 5 for some $i$. But is this condition necessary when the random walk is rational and the graph is a cycle of size, say, 27? What (if any) is an equivalent condition when the random walk is irrational?
\item The techniques used in this paper do not extend directly to non-abelian groups, because the Fourier transform on a non-abelian group is very different from the Fourier transform on an abelian group. Nevertheless, it is possible that a sufficient condition for reconstructivity similar to the one given in Theorem~\ref{thm:distinct-f} exists for non-abelian groups; proving such a condition is an interesting challenge.
\item We focus only on characterizing which random walks are
reconstructive. But there are many open questions about the
equivalence class structure of non-reconstructive walks; for
example:
\begin{itemize}
\item For a given r.w., how many non-minimal equivalence classes are
there, and of what size?
\item If the random walk has bounded range, are the equivalence
classes of bounded size?
\item Are sceneries in the same equivalence class far from each
other (i.e. one cannot be obtained from the other by a small
number of changes)?
\end{itemize}
\item One could also ask more practical questions, such as: how many
steps are necessary to distinguish between two sceneries? This could
be related, for example, to the mixing time of the random walk
and/or its Fourier coefficients. A harder problem is: how many steps
are necessary to reconstruct a scenery? Or: find an efficient
algorithm to reconstruct a scenery. Such problems have been tackled
before; see for example~\cite{Matzinger:2003}.
\item An interesting question is the reconstruction of the step
distribution, rather than the scenery. Which {\em known} function
$f(k)$ would minimize the number of observations needed to
reconstruct the {\em unknown} step distribution $\gamma$ of a random
walk $v(t)$?
\item The entropy of $\{f(v_t)\}$, which depends both on the random
walk and on the scenery, is a natural quantity to explore. Specifically, we
can consider the average entropy per observation as the number of observations
goes to infinity. In general, two sceneries that induce the same entropy for a
given random walk do not have to be the same; for example,
$(1,0,1,0,\ldots )$ and $(1,1,1,\ldots)$ both have entropy going to zero for
the simple random walk. (The first random walk will have one bit of entropy
reflecting the starting bit, but as the number of observations grows, this
becomes negligible.) Consider two functions equivalent under a
given random walk when they induce the same entropy. What do these
equivalence classes look like? Are there random walks for which the
equivalence classes are minimal? What is the entropy of a random
scenery under the simple random walk? Does it decrease if the walk
is biased to one side? \end{itemize}
\section{Acknowledgments} The authors would like to thank Itai Benjamini, Elchanan Mossel, Yakir Reshef, and Ofer Zeitouni for helpful conversations.
\pagebreak
\appendix
\section{Matzinger and Lember's theorem for abelian groups} We provide in this Appendix the proof of the following theorem, which is a straightforward generalization of Matzinger and Lember's Theorem~\ref{thm:distinct-f} to abelian groups, and follows their proof scheme closely.
\begin{theorem}[Matzinger and Lember]
\label{thm:ML-abelian}
Let $\gamma$ be the step distribution of a random walk on a finite abelian
group $H$. Let $\hat{\gamma}$ be the Fourier Transform of $\gamma$. Then
$f$ can be reconstructed up to shifts if the Fourier coefficients
$\{\hat{\gamma}(x)\}_{x \in \Z_n}$ are distinct. \end{theorem}
Let $H=\Z_{n_1}^{d_1}\cdots\Z_{n_m}^{d_m}$ be an abelian group, with the $n_i$'s distinct powers of primes. Let $n$ be the order of $H$. \subsection{Autocorrelation} Given a function $f$ on $H$, the autocorrelation $a_f(\ell)$ is defined as the inner product of $f$ with $f$ transformed by a $k$-shift: \begin{equation}
\label{eq:autocorr}
a_f(\ell)=\sum_{k \in H}f(k) \cdot f(k+\ell). \end{equation} When $f:H \to \{0,1\}$ is a labeling of the vertices of the random walk then we refer to $a_f$ as the {\em spatial autocorrelation}.
It follows directly from the convolution theorem that $\hat{a}_f$, the Fourier transform of $a_f$, is equal to the absolute value squared of $\hat{f}$: \begin{equation}
\hat{a}_f(x)=|\hat{f}(x)|^2. \end{equation}
We consider a r.w.\ $v(t)$ over $H$ with vertices labeled by $f: H \to \{0,1\}$. As above, we let $v(1)$ be chosen uniformly at random, and set $\gamma(k) = \P{v(t+1)-v(t)=k}$.
We define the {\em temporal autocorrelation} $b_f$ in the same spirit: \begin{equation}
\label{eq:temporal-autocorr}
b_f(\ell)=\E{f(v(T)) \cdot f(v(T+\ell))}. \end{equation} Note that since the random walk is stationary then the choice of $T$ is immaterial and $b_f$ is indeed well defined.
There is a linear relation between the spatial and temporal autocorrelations: \begin{theorem}
\label{thm:linear1}
\begin{equation}
\label{thm:bf-af}
b_f(\ell) = \frac{1}{n}\sum_{x \in H}\hat{\gamma}(x)^l\hat{a}_f(x)
\end{equation} \end{theorem} \begin{proof}
We denote by $\gamma^{(\ell)}$ the function $\gamma$ convoluted with
itself $\ell$ times, over $H$. It is easy to convince oneself
that $\gamma^{(\ell)}(k)$ is the distribution of the distance traveled
by the particle in $\ell$ time periods, so that $\gamma^{(\ell)}(k) =
\P{v(t+\ell)-v(t)=k}$.
Now, by definition we have that \begin{align*}
b_f(\ell)&=\E{f(v(T)) \cdot f(v(T+\ell))}.\\ \end{align*} Since the random walk starts at a uniformly chosen vertex then \begin{align*}
b_f(\ell)&= \frac{1}{n} \sum_k \mathbb{E}\left[ f(v(T)) \cdot
f(v(T+\ell))\, \vline \, v(T) = k\right].\\ \end{align*} Conditioning on the step taken at time $T$ we get that \begin{align*}
b_f(\ell)&= \frac{1}{n} \sum_{k,x} f(k) \cdot \gamma^{(\ell)}(x)\cdot f(k+x).\\ \end{align*} Finally, changing the order of summation and substituting in $a_f(x)$ we get that \begin{align*}
b_f(\ell)&= \frac{1}{n} \sum_x \gamma^{(\ell)}(x) \cdot a_f(x). \end{align*}
Now, the expression $\sum_x \gamma^{(\ell)}(x) \cdot a_f(x)$ can be viewed as a dot product of two vectors in the vector space of functions from $H$ to $\mathbb C$. Since the Fourier transform is an orthogonal transformation of this space we can replace these two functions by their Fourier transforms. By the convolution theorem we have that $\widehat{\gamma^{(\ell)}} = \hat{\gamma}^\ell$, and so \begin{align*}
b_f(\ell)
&= \frac{1}{n}\left(\gamma^{(\ell)}, a_f\right)\\
&= \frac{1}{n}\left(\hat{\gamma}^\ell, \hat{a}_f\right)\\
&= \frac{1}{n}\sum_{x \in \Z_n}\hat{\gamma}(x)^l\hat{a}_f(x). \end{align*}
\end{proof}
It follows from Theorem~\ref{thm:linear1} that the spatial autocorrelation $a_f$ can be calculated from the temporal autocorrelation $b_f$ whenever the Fourier coefficients $\{\hat{\gamma}(x)\}$ are distinct: the linear transformation mapping $a_f$ to $b_f$ is a Vandermonde matrix which is invertible precisely when the values of $\hat{\gamma}$ are distinct.
By its definition, $b_f$ is determined by the distribution of $\{f(v(t))\}_{t=1}^\infty$. Hence it follows that: \begin{corollary}
When the Fourier coefficients of $\gamma$ are distinct then $b_f$
uniquely determines $a_f$. \end{corollary}
However, it is possible for $f_1$ not to be a shift of $f_2$, but for $a_{f_1}$ to equal $a_{f_2}$. Thus, the above argument does not suffice to prove Theorem~\ref{thm:distinct-f}.
\subsection{Bispectrum and beyond} A generalization of autocorrelation is the {\em bispectrum} (see, e.g.,~\cite{Mendel:91}):
\begin{equation}
\label{eq:bispectrum}
A_f(\ell_1, \ell_2) = \sum_{k \in H}f(k) \cdot f(k+\ell_1) \cdot
f(k+\ell_1+\ell_2). \end{equation}
We generalize further to what we call the multispectrum: \begin{equation}
\label{eq:multispectrum}
A_f(\ell_1, \ldots, \ell_{n-1}) = \sum_{k \in H}f(k) \cdot
f(k+\ell_1) \cdots f(k + \ell_1 + \cdots +\ell_{n-1}). \end{equation}
As above, we call $A_f$ the spatial multispectrum. The relation between the Fourier transform of $A_f$ and $f$ is the following. Note that we transform $A_f$ over $H^{n-1}$ (where $n=|H|$): \begin{equation}
\label{Af-transform}
\hat{A}_f(x_1, \ldots, x_{n-1}) = \overline{\hat{f}(x_1)} \cdot \hat{f}(x_{n-1})
\cdot \prod_{i=1}^{n-2}\hat{f}(x_i - x_{i+1}), \end{equation} where $\overline{z}$ denotes the complex conjugate of $z$.
We define as follows the temporal multispectrum: \begin{equation}
\label{eq:temporal-multi}
B_f(\ell_1, \ldots, \ell_{n-1}) = \E{f(v(T)) \cdot f(v(T + \ell_1))
\cdots f(v(T + \ell_1 + \cdots + \ell_{n-1}))}. \end{equation} Note that, like $b_f$, $B_f$ is determined by the distribution of $\{f(v(t))\}_{t=1}^\infty$.
\subsection{Distinct Fourier Coefficients of $\gamma$ imply reconstruction}
Whereas $a_f$ did not suffice to recover $f$, $A_f$ suffices: \begin{lemma}
Suppose $A_{f_1} = A_{f_2}$. Then $f_1$ is a shift of $f_2$. \end{lemma} \begin{proof}
First, note that $A_{f}(\ell_1, \ldots \ell_n)>0$ iff there exists a
$k$ such that $f(k) = f(k+\ell_1) = \cdots = f(k + \ell_1 + \cdots
+\ell_n) = 1$. Now, let $\hat{\ell}(f) := (\hat{\ell}_1(f), \ldots ,
\hat{\ell}_n(f))$ be the lexicographically smallest $n$-tuple
satisfying $A_f\left(\hat{\ell}_1(f), \ldots ,
\hat{\ell}_n(f)\right) > 0.$ Note that since $A_{f_1} = A_{f_2}$,
we must also have $\hat{\ell}(f_1) = \hat{\ell}(f_2).$
But $\hat{\ell}(f)$ determines $f$. Indeed, let $i$ be the largest
index such that $\ell_1 + \cdots + \ell_i \leq n$. Then there is a
$k$ such that $k$, $k+\ell_1$, $k+\ell_1 + \ell_2$, \ldots , $k +
\ell_1 + \cdots + \ell_i$ are exactly the indices at which $f$ is 1:
if any of these were indices at which $f$ was 0 then
$A_f(\hat{\ell}(f))$ would be zero, and if there were any more
indices at which $f$ was 1 then $\hat{\ell}$ would not be
lexicographically the smallest.
Thus, $f$ can be recovered uniquely up to a shift from $A_f$ by a
simple algorithm: find $\hat{\ell}(f)$, then read off the ones from
the first several entries of $\hat{\ell}(f)$ as above, setting $k$
to be 0 w.l.o.g.
\end{proof}
The following analogue of Theorem~\ref{thm:linear1} will be useful: \begin{lemma}
\begin{equation}
\hat{B}_f(\ell_1, \ldots, \ell_{n-1}) = \frac{1}{n}\sum_{x_1, \ldots, x_{n-1}}\hat{\gamma}(x_1)^{\ell_1} \cdots \hat{\gamma}(x_{n-1})^{\ell_n} \cdot \hat{A}_f(x_1, \ldots, x_{n-1}).
\end{equation} \end{lemma} \begin{proof}
The steps of this proof closely follow those of the proof of
Theorem~\ref{thm:linear1}. \begin{align*}
B_f(\ell_1, \ldots, \ell_{n-1})
&= \E{f(v(T)) \cdot f(v(T + \ell_1)) \cdots f(v(T + \ell_1 + \cdots + \ell_{n-1}))}\\
&= \frac{1}{n}\sum_{k,x_1, \ldots, x_{n-1}} f(k) \cdot \gamma^{(\ell_1)}(x_1)f(k+x_1) \cdots \gamma^{(\ell_{n-1})}(x_{n-1})f(k+x_1 + \ldots + x_{n-1})\\
&= \frac{1}{n}\sum_{x_1, \ldots, x_{n-1}} \gamma^{(\ell_1)}(x_1)\cdots \gamma^{(\ell_{n-1})}(x_{n-1}) \sum_k f(k)f(k+x_1)\cdots f(k+x_1 + \ldots + x_{n-1})\\
&= \frac{1}{n}\sum_{x_1, \ldots, x_{n-1}} \gamma^{(\ell_1)}(x_1)\cdots \gamma^{(\ell_{n-1})}(x_{n-1}) A_f(x_1, \ldots x_{n-1}) \end{align*} Applying the Fourier transform as in Theorem~\ref{thm:linear1}, we obtain the lemma. \end{proof}
We now have what we need to complete the proof of Theorem~\ref{thm:distinct-f}:
\begin{proof}[Proof of Theorem~\ref{thm:ML-abelian}] We need only show that $B_{f_1} = B_{f_2}$ implies that $A_{f_1} = A_{f_2}$. But this follows because the matrix $\Gamma$ mapping $A_f$ to $B_f$ is a tensor of $M$ with itself $n-1$ times, where $M$ is the matrix mapping $a_f$ to $b_f$. When the values of $\gamma$ are distinct, $M$ is invertible, so $\Gamma$ is also invertible, giving us the desired result. \end{proof}
\end{document} | arXiv |
What is the area in square inches of the pentagon shown?
[asy]
draw((0,0)--(8,0)--(8,18)--(2.5,20)--(0,12)--cycle);
label("8''",(1.3,16),NW);
label("6''",(5.2,19),NE);
label("18''",(8,9),E);
label("8''",(4,0),S);
label("12''",(0,6),W);
draw((1,0)--(1,1)--(0,1));
draw((7,0)--(7,1)--(8,1));
[/asy]
Adding a couple lines, we have
[asy]
draw((0,0)--(8,0)--(8,18)--(2.5,20)--(0,12)--cycle);
draw((0,12)--(8,12), dashed);
draw((7,12)--(7,13)--(8,13));
draw((0,12)--(8,18), dashed);
label("8''",(1.3,16),NW);
label("6''",(5.2,19),NE);
label("18''",(8,9),E);
label("8''",(4,0),S);
label("12''",(0,6),W);
label("8''",(4,12),S);
label("6''",(9,15),W);
draw((1,0)--(1,1)--(0,1));
draw((7,0)--(7,1)--(8,1));[/asy]
The marked right triangle has a hypotenuse of $\sqrt{6^2+8^2}=10$, which makes the other (congruent) triangle a right triangle as well. The area of the entire figure is then the area of the rectangle added to the area of the two right triangles, or $12\cdot8+2\left(\frac{6\cdot8}{2}\right)=\boxed{144}$ square inches. | Math Dataset |
\begin{document}
\title[Equivariant properties of symmetric products] {Equivariant properties of symmetric products}
\date{\today; 2010 AMS Math.\ Subj.\ Class.: 55P91} \author{Stefan Schwede} \address{Mathematisches Institut, Universit\"at Bonn, Germany} \email{[email protected]}
\begin{abstract} The filtration of the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case, for example the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory, i.e., we study the simultaneous behavior for all compact Lie groups at once. \end{abstract}
\keywords{symmetric product, compact Lie group, equivariant stable homotopy theory} \maketitle
\section*{Introduction}
We let $Sp^\infty(X)$ denote the infinite symmetric product of a based space $X$. It comes with a filtration by finite symmetric products $Sp^n(X)=X^n/\Sigma_n$. We denote by \[ Sp^n = \{ Sp^n(S^m)\}_{m\geq 0} \] the spectrum whose terms are the $n$-th symmetric products of spheres. A celebrated theorem of Dold and Thom~\cite{dold-thom} asserts that $Sp^\infty(S^m)$ is an Eilenberg-Mac\,Lane space of type $(\mathbb Z,m)$ for $m\geq 1$; so $Sp^\infty$ is an Eilenberg-Mac\,Lane spectrum for the group $\mathbb Z$. The resulting filtration \[ {\mathbb S}\ =\ Sp^1\ \subseteq \ Sp^2\ \subseteq\ \dots\ \subseteq \ Sp^n\ \subseteq\ \dots \] of the Eilenberg-Mac Lane spectrum~$Sp^\infty$, starting with the sphere spectrum~${\mathbb S}$, has been much studied. The subquotient $Sp^n/Sp^{n-1}$ is stably contractible unless $n$ is a prime power. If~$n=p^k$ for a prime~$p$, then $Sp^n/Sp^{n-1}$ is $p$-torsion, and its mod-$p$ cohomology has been worked out by Nakaoka~\cite{nakaoka-cohomology of symmetric}. For $p=2$ these subquotient spectra feature in the work of Mitchell and Priddy on stable splitting of classifying spaces~\cite{mitchell-priddy}, and in Kuhn's solution of the Whitehead conjecture~\cite{kuhn:Whitehead}. Arone and Dwyer relate these spectra to the partition complex, the homology of the dual Lie representation and the Tits building~\cite{arone-dwyer}.
While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. This paper is about equivariant features of the symmetric product filtration, for actions of compact Lie groups~$G$. If~$V$ is a finite dimensional orthogonal $G$-representation, then~$G$ acts continuously on the one-point compactification~$S^V$, and hence on $Sp^n(S^V)$ and~$Sp^{\infty}(S^V)$ by functoriality of symmetric products. As~$V$ varies over all such $G$-representations, the $G$-spaces~$Sp^n(S^V)$ form a $G$-spectrum that represents a `genuine' $G$-equivariant stable homotopy type. For understanding these equivariant homotopy types it is extremely beneficial not to study one compact Lie group at a time, but to use the `global' perspective. Here `global' refers to simultaneous and compatible actions of all compact Lie groups. Various ways to formalize this idea have been explored, compare~\cite[Ch.\,II]{lms}, \cite[Sec.~5]{greenlees-may-completion}, \cite{bohmann-orthogonal}; we use a different approach via orthogonal spectra.
In Definition~\ref{def-global functor} we introduce the notion of {\em global functor}, a useful language to describe the collection of equivariant homotopy groups of an orthogonal spectrum as a whole, i.e., when the compact Lie group is allowed to vary. The category of global functors is a symmetric monoidal abelian category that takes up the role in global homotopy theory played by abelian groups in ordinary homotopy theory, or by $G$-Mackey functors in $G$-equivariant homotopy theory. As a consequence of Theorem~\ref{thm-Burnside category basis} we will see that a global functor is a certain kind of `global Mackey functor' that assigns abelian groups to all compact Lie groups and comes with restriction maps along continuous group homomorphisms and transfer maps along inclusions of closed subgroups.
In this language, we can then identify the global functor ${\underline \pi}_0( Sp^n )$ as a quotient of the Burnside ring global functor~${\mathbb A}$ by a single relation. We define an element~$t_n$ in the Burnside ring of the $n$-th symmetric group by \[ t_n \ = \ n\cdot 1\ - \ \tr_{\Sigma_{n-1}}^{\Sigma_n}(1) \ \in \ {\mathbb A}(\Sigma_n) .\] As an element in the Grothendieck group of finite $\Sigma_n$-sets, the class~$t_n$ corresponds to the formal difference of a trivial $\Sigma_n$-set with $n$ elements and the tautological $\Sigma_n$-set $\{1,\dots,n\}$. Since $t_n$ has zero augmentation, the global subfunctor $\langle t_n\rangle$ generated by $t_n$ lies in the augmentation ideal global functor~$I$. The restriction of $t_n$ to the Burnside ring of $\Sigma_{n-1}$ equals $t_{n-1}$, so we obtain a nested sequence of global functors \[
0 = \langle t_1\rangle \ \subset\ \langle t_2\rangle\ \ \subset\ \dots \ \subset\ \langle t_n\rangle \ \subset \ \dots \ \subset I \ \subset \ {\mathbb A}\ . \] As part of our main result, Theorem~\ref{thm-pi_0 Sp^n}, we prove the following:
{\bf Theorem.}\ For every $n\geq 1$ the morphism of global functors~$i_*:{\mathbb A}={\underline \pi}_0({\mathbb S})\longrightarrow{\underline \pi}_0(Sp^n)$ induced by the embedding $i:{\mathbb S}= Sp^1\longrightarrow Sp^n$ passes to an isomorphism of global functors \[ {\mathbb A}/\td{t_n} \ \cong \ {\underline \pi}_0(Sp^n) \ .\]
It is then a purely algebraic exercise to describe $\pi^G_0(Sp^n)$ as an explicit quotient of the Burnside ring~${\mathbb A}(G)$: one has to enumerate all relations in ${\mathbb A}(G)$ obtained by applying restrictions and transfers to the class $t_n$. We do this in Proposition~\ref{prop-describe I_n} and then work out the examples of~$p$-groups and some symmetric groups. The author thinks that the explicit answer for $\pi^G_0(Sp^n)$ is far less enlightening than the global description of~${\underline \pi}_0(Sp^n)$. Since all the inclusions~$\td{t_{n-1}}\subset\td{t_n}$ are proper, the subquotients $Sp^n/Sp^{n-1}$ are all globally non-trivial, in sharp constrast to the non-equivariant situation.
Our calculation of~${\underline \pi}_0 (Sp^n)$ is a consequence of a global homotopy pushout square, see Theorem~\ref{thm-main homotopy}, showing that~$Sp^n$ is obtained from~$Sp^{n-1}$ by coning off a certain morphism from the suspension spectrum of the global classifying space of the family of non-transitive subgroups of $\Sigma_n$. This homotopy pushout square is a global equivariant refinement of a non-equivariant homotopy pushout established by Lesh~\cite{lesh-filtration}.
Another consequence of our calculations is a possibly unexpected feature of the equivariant homotopy groups~$\pi_0^G(Sp^\infty)$ when $G$ has positive dimension. We let $I_\infty$ denote the global subfunctor of~${\mathbb A}$ generated by all the classes $t_n$ for $n\geq 1$. Also in Theorem~\ref{thm-pi_0 Sp^n} we show that the embedding ${\mathbb S}\longrightarrow Sp^\infty$ induces an isomorphism of global functors \[ {\mathbb A}/I_\infty \ \cong \ {\underline \pi}_0 (Sp^\infty) \ .\] For every compact Lie group $G$ the restriction map \[ \res^G_e \ : \ \pi_0^G(Sp^\infty) \ \longrightarrow \ \pi_0^e(Sp^\infty)\ \cong \ {\mathbb Z} \] to the non-equivariant 0-th homotopy group is a split epimorphism onto a free abelian group of rank~1. When the group~$G$ {\em finite}, then this restriction map is an isomorphism and all $G$-equivariant homotopy groups of $Sp^\infty$ vanish in dimensions different from~0. So through the eyes of finite groups, $Sp^\infty$ is an Eilenberg-Mac\,Lane spectrum for the constant global functor~$\underline{\mathbb Z}$. This does {\em not}, however, generalize to compact Lie groups of positive dimension. In that generality, the restriction map $\res^G_e$ can have a non-trivial kernel; equivalently, the value of the global functor $I_\infty$ at some compact Lie groups is strictly smaller than the augmentation ideal. We discuss these phenomena in more detail at the end of Section~\ref{sec-examples}.
I would like to thank Markus Hausmann for various helpful suggestions related to this paper.
\section{Orthogonal spaces}
In this section we recall orthogonal spaces from a global equivariant perspective. We work in the category~$\mathcal T$ of {\em compactly generated spaces} in the sense of~\cite{mccord}, i.e., $k$-spaces (also called {\em Kelley spaces}) that satisfy the weak Hausdorff condition. An {\em inner product space} is a finite dimensional ${\mathbb R}$-vector space~$V$ equipped with a scalar product. We write $O(V)$ for the orthogonal group of $V$, i.e., the Lie group of linear isometries of $V$. We denote by ${\mathbf L}$ the category with objects the inner product spaces and morphisms the linear isometric embeddings. This is a topological category as follows: if $\varphi:V\longrightarrow W$ is one linear isometric embedding, then the action of the orthogonal group $O(W)$, by postcomposition, induces a bijection \[ O(W)/O(\varphi^\perp) \ \cong \ {\mathbf L}(V,W) \ ,\quad A\cdot O(\varphi^\perp) \ \longmapsto A\circ \varphi\ , \] where $\varphi^\perp=W-\varphi(V)$ is the orthogonal complement of the image of $\varphi$. We topologize ${\mathbf L}(V,W)$ so that this bijection is a homeomorphism, and this topology is independent of $\varphi$. So if $n=\dim(V)$, then~${\mathbf L}(V,W)$ is homeomorphic to the Stiefel manifold of orthonormal $n$-frames in~$W$.
\begin{defn} An {\em orthogonal space} is a continuous functor $Y:{\mathbf L}\longrightarrow\mathcal T$ to the category of spaces. A morphism of orthogonal spaces is a natural transformation. We denote by $spc$ the category of orthogonal spaces. \end{defn}
The systematic use of inner product spaces to index objects in homotopy theory seems to go back to Boardman's thesis~\cite{boardman-thesis}. The category ${\mathbf L}$ (or its extension that also contains countably infinite dimensional inner product spaces) is denoted $\mathscr I$ by Boardman and Vogt~\cite{boardman-vogt-homotopy everything}, and this notation is also used in~\cite{may-quinn-ray}; other sources use the symbol $\mathcal I$. Accordingly, orthogonal spaces are sometimes referred to as $\mathscr I$-functors, $\mathscr I$-spaces or $\mathcal I$-spaces. Our justification for using yet another name is twofold: on the one hand, we shift the emphasis away from a focus on non-equivariant homotopy types, and towards viewing an orthogonal space as representing compatible equivariant homotopy types for all compact Lie groups. Secondly, we want to stress the analogy between orthogonal spaces and orthogonal spectra, the former being an unstable global world and the latter a corresponding stable global world.
We let $G$ be a compact Lie group. By a {\em $G$-representation} we mean an orthogonal $G$-representation, i.e., an inner product space $V$ equipped with a continuous $G$-action by linear isometries. For every orthogonal space $Y$ and every $G$-representation $V$, the value~$Y(V)$ inherits a $G$-action from~$V$ through the functoriality of $Y$. For a $G$-equivariant linear isometric embedding $\varphi:V\longrightarrow W$ the induced map $Y(\varphi):Y(V)\longrightarrow Y(W)$ is $G$-equivariant.
Now we discuss the equivariant homotopy set~$\pi_0^G(Y)$ of an orthogonal space~$Y$; this is an unstable precursor of the 0-th equivariant stable homotopy group of an orthogonal spectrum.
\begin{defn} Let $G$ be a compact Lie group. A {\em $G$-universe} is an orthogonal $G$-representation ${\mathcal U}$ of countably infinite dimension with the following two properties: \begin{itemize} \item the representation ${\mathcal U}$ has non-zero $G$-fixed points, \item if a finite dimensional representation $V$ embeds into ${\mathcal U}$, then a countable infinite sum of copies of~$V$ also embeds into ${\mathcal U}$. \end{itemize} A $G$-universe is {\em complete} if every finite dimensional $G$-representation embeds into it. \end{defn}
A $G$-universe is characterized, up to equivariant isometry, by the set of irreducible $G$-representations that embed into it. A universe is complete if and only if every irreducible $G$-representation embeds into it. In the following we fix, for every compact Lie group $G$, a complete $G$-universe ${\mathcal U}_G$. We let $s({\mathcal U}_G)$ denote the poset, under inclusion, of finite dimensional $G$-subrepresentations of ${\mathcal U}_G$.
We let $Y$ be an orthogonal space and define its $G$-equivariant path components as \begin{equation}\label{eq:define_pi_0^G_set}
\pi_0^G(Y) \ = \ \colim_{V\in s({\mathcal U}_G)}\, \pi_0\left( Y(V)^G\right) \ . \end{equation} As the group varies, the homotopy sets $\pi_0^G(Y)$ have contravariant functoriality in $G$: every continuous group homomorphism $\alpha:K\longrightarrow G$ between compact Lie groups induces a restriction map $\alpha^*:\pi_0^G(Y)\longrightarrow\pi_0^K(Y)$, as we shall now explain. We denote by $\alpha^*$ the restriction functor from $G$-spaces to $K$-spaces (or from $G$-representations to $K$-representations), i.e., $\alpha^* Z$ (respectively $\alpha^*V$) is the same topological space as~$Z$ (respectively the same inner product space as~$V$) endowed with $K$-action via \[ k\cdot z \ = \ \alpha(k)\cdot z \ . \] Given an orthogonal space $Y$, we note that for every $G$-representation $V$, the $K$-spaces $\alpha^*(Y(V))$ and $Y(\alpha^*V)$ are equal (not just isomorphic).
The restriction $\alpha^*({\mathcal U}_G)$ is a $K$-universe, but if $\alpha$ has a non-trivial kernel, then this $K$-universe is not complete. When $\alpha$ is injective, then $\alpha^*({\mathcal U}_G)$ is a complete $K$-universe, but typically different from the chosen complete $K$-universe ${\mathcal U}_K$. To deal with this we explain how a $G$-fixed point $y\in Y(V)^G$, for an arbitrary $G$-representation $V$, gives rise to an unambiguously defined element $\td{y}$ in $\pi_0^G(Y)$. The point here is that $V$ need not be a subrepresentation of the chosen universe ${\mathcal U}_G$ and the resulting class does not depend on any additional choices. To construct $\td{y}$ we choose a linear isometric $G$-embedding $j:V\longrightarrow {\mathcal U}_G$ and look at the image $Y(j)(y)$ under the $G$-map \[ Y(V)\ \xrightarrow{\ Y(j)\ }\ Y(j(V))\ . \] Here we have used the letter $j$ to also denote the isometry $j:V\longrightarrow j(V)$ to the image of $V$; since $j(V)$ is a finite dimensional $G$-invariant subspace of ${\mathcal U}_G$, we obtain an element \[ \td{y} \ = \ [Y(j)(y)] \ \in \ \pi_0^G(Y)\ . \] It is crucial, although not particularly difficult, that $\td{y}$ does not depend on the choice of embedding $j$:
\begin{prop}\label{prop-universal colimit spaces} Let $Y$ be an orthogonal space, $G$ a compact Lie group, $V$ a $G$-representation and $y\in Y(V)^G$ a $G$-fixed point. \begin{enumerate}[\em (i)] \item The class $\td{y}$ in $\pi_0^G(Y)$ is independent of the choice of linear isometric embedding $j:V\longrightarrow{\mathcal U}_G$. \item For every $G$-equivariant linear isometric embedding $\varphi:V\longrightarrow W$ the relation \[ \td{Y(\varphi)(y)} \ = \ \td{y} \text{\qquad holds in\quad $\pi_0^G(Y)$.} \] \end{enumerate} \end{prop} \begin{proof} (i) We let $j':V\longrightarrow{\mathcal U}_G$ be another $G$-equivariant linear isometric embedding. If the images $j(V)$ and $j'(V)$ are orthogonal, then~$H:V\times [0,1]\longrightarrow j(V)\oplus j'(V)$ defined by \[ H(v,t)\ =\ \sqrt{1-t^2}\ \cdot j(v) \ + \ t\cdot j'(v) \] is a homotopy from $j$ to $j'$ through $G$-equivariant linear isometric embeddings. Thus \[ t\ \longmapsto \ Y(H(-,t))(y) \] is a path in $Y(j(V)\oplus j'(V))^G$ from $Y(j)(y)$ to $Y(j')(y)$, so $[Y(j)(y)]=[Y(j')(y)]$ in $\pi_0^G(Y)$. In general we can choose a third $G$-equivariant linear isometric embedding $l:V\longrightarrow{\mathcal U}_G$ whose image is orthogonal to the images of~$j$ and~$j'$. Then $[Y(j)(y)]=[Y(l)(y)]=[Y(j')(y)]$ by the previous paragraph.
(ii) If $j:W\longrightarrow{\mathcal U}_G$ is an equivariant linear isometric embedding, then so is $j\varphi:V\longrightarrow{\mathcal U}_G$. Since we can use any equivariant isometric embedding to define the class $\td{y}$, we get \[ \td{Y(\varphi)(y)} \ = \ [Y(j)(Y(\varphi)(y))] \ = \ [Y(j\varphi)(y)] \ = \ \td{y} \ . \qedhere \] \end{proof}
We can now define the {\em restriction map} associated to a continuous group homomorphism $\alpha:K\longrightarrow G$ by \[ \alpha^* \ : \ \pi^G_0(Y) \ \longrightarrow \ \pi^K_0(Y) \ ,\quad [y]\ \longmapsto \ \td{y}\ . \] This makes sense because every $G$-fixed point of $Y(V)$ is also a $K$-fixed point of $\alpha^*(Y(V))=Y(\alpha^* V)$. For a second continuous group homomorphism $\beta:L\longrightarrow K$ we have \[ \beta^*\circ \alpha^* \ = \ (\alpha\beta)^* \ : \ \pi_0^G(Y) \ \longrightarrow \ \pi_0^L(Y) \ . \] Since restriction along the identity homomorphism is the identity, the collection of equivariant homotopy sets $\pi_0^G(Y)$ becomes a contravariant functor in the group variable. A key fact is that inner automorphisms act trivially:
\begin{prop}\label{prop-inner automorphism} For every orthogonal space $Y$, every compact Lie group $G$ and every $g\in G$, the restriction map $c_g^*:\pi_0^G(Y) \longrightarrow \pi_0^G(Y)$ along the inner automorphism \[ c_g \ : \ G \ \longrightarrow \ G\ , \quad c_g(h)\ =\ g^{-1}h g \] is the identity of $\pi_0^G(Y)$. \end{prop} \begin{proof} We consider a finite dimensional $G$-subrepresentation $V$ of ${\mathcal U}_G$ and a $G$-fixed point $y\in Y(V)^G$ that represents an element in~$\pi_0^G(Y)$. Then the map $l_g:c_g^*(V)\longrightarrow {\mathcal U}$ given by left multiplication by $g$ is a $G$-equivariant linear isometric embedding. So \[ c_g^*[y]\ = \ [Y(l_g^V)(y)]\ = \ [g\cdot y]\ = \ [y] \ ,\] by the very definition of the restriction map, where $l_g^V:c_g^*(V)\longrightarrow V$. The second equation is the definition of the $G$-action on $Y(V)$ through the $G$-action on $V$. The third equation is the hypothesis that $y$ is $G$-fixed. \end{proof}
We denote by $\Rep$ the category whose objects are the compact Lie groups and whose morphisms are conjugacy classes of continuous group homomorphisms. We can summarize the discussion thus far by saying that for every orthogonal space $Y$ the restriction maps make the equivariant homotopy sets $\{\pi_0^G(Y)\}$ into a contravariant functor \[ {\underline \pi}_0(Y) \ : \ \Rep \ \longrightarrow \text{(sets)} \ . \] In fact, the restriction maps along continuous homomorphisms give {\em all} natural operations: As we show in~\cite{schwede-global}, every natural transformation $\pi_0^G \longrightarrow \pi_0^K$ of set valued functors on the category of orthogonal spaces is of the form $\alpha^*$ for a unique conjugacy class of continuous group homomorphism $\alpha:K\longrightarrow G$.
If~$V$ is any inner product space, then the evaluation functor sending an orthogonal space~$Y$ to~$Y(V)$ is represented by the hom functor~${\mathbf L}(V,-)$. Consequently, if~$V$ is a $G$-representation, then the functor \[ spc \ \longrightarrow \ \mathcal T \ , \quad Y \ \longmapsto \ Y(V)^G \] that sends an orthogonal space $Y$ to the space of $G$-fixed points of $Y(V)$ is represented by an orthogonal space ${\mathbf L}_{G,V}$, the {\em free orthogonal space} generated by $(G,V)$. The value of~${\mathbf L}_{G,V}$ at an inner product space~$W$ is \[ {\mathbf L}_{G,V}(W) \ = \ {\mathbf L}(V,W)/G \ , \] the orbit space of the right $G$-action on ${\mathbf L}(V,W)$ by $(\varphi\cdot g)(v) = \varphi(g\cdot v)$. Every $G$-fixed point $y\in Y(V)^G$ gives rise to a morphism $\hat y:{\mathbf L}_{G,V} \longrightarrow Y$ of orthogonal spaces, defined at~$W$ as \[ \hat y(W)\ : \ {\mathbf L}(V,W) / G \ \longrightarrow \ Y(W)\ ,\quad \varphi \cdot G \ \longmapsto\ Y(\varphi)(y)\ .\] The morphism $\hat y$ is uniquely determined by the property $\hat y(V)(\Id_V\cdot G)= y$ in $Y(V)^G$.
We calculate the 0-th equivariant homotopy sets of a free orthogonal space. The {\em tautological class} \begin{equation}\label{eq:tautological_class}
u_{G,V}\ \in \ \pi_0^G({\mathbf L}_{G,V}) \end{equation} is the path component of the $G$-fixed point \[ \Id_V\cdot G \ \in \ ({\mathbf L}(V,V)/G)^G = ({\mathbf L}_{G,V}(V))^G\ , \] the $G$-orbit of the identity of $V$.
\begin{theorem}\label{thm-pi_0 of L_G} Let $K$ and $G$ be compact Lie groups and $V$ a faithful $G$-representation. Then the map \[ \Rep(K,G) \ \longrightarrow \ \pi_0^K({\mathbf L}_{G,V})\ , \quad [\alpha:K\longrightarrow G] \ \longmapsto \ \alpha^*(u_{G,V})\] is bijective. \end{theorem} \begin{proof} We construct the inverse explicitly. We consider any element \[ [\varphi G]\ \in \ \pi_0^K({\mathbf L}_{G,V})\ ; \] here $W\in s({\mathcal U}_K)$, and $\varphi\in{\mathbf L}(V,W)$ is such that the orbit $\varphi G\in {\mathbf L}(V,W)/G$ is $K$-fixed. Thus $k\varphi G=\varphi G$ for every element $k\in K$. Since $G$ acts faithfully on~$V$, there is a unique $\alpha(k)\in G$ with $k\varphi =\varphi \alpha(k)$, and this defines a continuous homomorphism $\alpha:K\longrightarrow G$. If we replace~$\varphi$ by~$\varphi g$ for some $g\in G$, then $\alpha$ changes into its $g$-conjugate. If we replace~$W$ by a larger $K$-representation in the poset~$s({\mathcal U}_K)$, then~$\alpha$ does not change.
Now we consider a path \[ \omega \ : \ [0,1]\ \longrightarrow \ ({\mathbf L}(V,W)/G)^K \] starting with~$\varphi G$. Since the projection ${\mathbf L}(V,W)\longrightarrow{\mathbf L}(V,W)/G$ is a locally trivial fiber bundle, we can choose a continuous lift \[ \tilde\omega \ : \ [0,1]\ \longrightarrow \ {\mathbf L}(V,W) \] with $\tilde\omega(0)=\varphi$ and $\tilde\omega(t)G =\omega(t)$ for all $t\in[0,1]$. Then each~$t$ determines a continuous homomorphism $\alpha_t:K\longrightarrow G$ by $k\tilde\omega(t) =\tilde\omega(t)\alpha_t(k)$, and the assignment \[ [0,1]\ \longrightarrow \ \Hom(K,G)\ , \quad t \longmapsto \alpha_t \] to the space of continuous group homomorphisms (with the topology of uniform convergence) is itself continuous. But that means that~$\alpha_0$ and~$\alpha_1$ are conjugate by an element of~$G$, compare~\cite[VIII, Lemma 38.1]{conner-floyd}. In particular, the conjugacy class of~$\alpha$ only depends on the path component of $\varphi G$ in the space $({\mathbf L}(V,W)/G)^K$. Altogether this shows that the map \[ \pi_0^K({\mathbf L}_{G,V})\ \longrightarrow \ \Rep(K,G) \ , \quad [\varphi G]\ \longmapsto \ [\alpha] \] is well-defined. It is straightforward from the definitions that this map is inverse to evaluation at~$u_{G,V}$. \end{proof}
We end this section by discussing certain orthogonal spaces that are closely related to the symmetric product filtration.
\begin{construction} For an inner product space $V$ we set
\[ S(V,n)\ = \ \left\{ (v_1,\dots,v_n)\in V^n \ : \ \sum_{i=1}^n v_i=0\ , \ \sum_{i=1}^n |v_i|^2 = 1 \right\} \ .\] In other words, $S(V,n)$ is the unit sphere in the kernel of summation map from $V^n$ to~$V$. The symmetric group~$\Sigma_n$ acts from the right on $S(V,n)$ by permuting the coordinates, i.e., \[ (v_1,\dots,v_n)\cdot\sigma \ = \ (v_{\sigma(1)},\dots,v_{\sigma(n)}) \ . \] We define \[ (B_{\gl}{\mathcal F}_n)(V) \ = \ S(V,n) / \Sigma_n \ ,\] the orbit space of the $\Sigma_n$-action. A linear isometric embedding $\varphi:V\longrightarrow W$ induces the map \[ (B_{\gl}{\mathcal F}_n)(\varphi) \ = \ S(\varphi,n) / \Sigma_n \ , \quad (v_1,\dots,v_n)\Sigma_n\ \longmapsto \ (\varphi(v_1),\dots,\varphi(v_n))\Sigma_n\ .\] We call~$B_{\gl}{\mathcal F}_n$ the {\em global classifying space} of the family ${\mathcal F}_n$ of non-transitive subgroups of the symmetric group~$\Sigma_n$. Proposition~\ref{prop-universal space} below justifies this terminology. \end{construction}
\begin{rk}
The {\em reduced natural $\Sigma_n$-representation}
(also called the {\em standard $\Sigma_n$-representation})
is the vector space
\[ \nu_n\ = \ \{(x_1,\dots,x_n)\in{\mathbb R}^n \ : \ x_1+\ldots + x_n = 0 \} \]
with the standard scalar product
and left~$\Sigma_n$-action by permutation of coordinates:
\[ \sigma\cdot(x_1,\dots,x_n) \ = \ (x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(n)}) \ .\]
In the proof of the following proposition we exploit that
for every inner product space~$V$ (possibly infinite dimensional),
the kernel of the summation map $V^n\longrightarrow V$
is isometrically and $(O(V)\times\Sigma_n)$-equivariantly isomorphic
to $V\otimes\nu_n$.
Hence $S(V,n)$ is $(O(V)\times\Sigma_n)$-equivariantly homeomorphic
to~$S(V\otimes\nu_n)$.
\end{rk}
We show now that for every compact Lie group $K$ the $K$-space $(B_{\gl}{\mathcal F}_n)({\mathcal U}_K)=S({\mathcal U}_K,n)/\Sigma_n$ is a certain classifying space, thereby justifying the term `global classifying space' for~$B_{\gl}{\mathcal F}_n$. We denote by ${\mathcal F}_n(K)$ the family of those closed subgroups $\Gamma$ of $K\times\Sigma_n$
whose trace $H=\{\sigma\in \Sigma_n\ |\ (1,\sigma)\in \Gamma\}$ is a non-transitive subgroup of~$\Sigma_n$. For the purpose of the next proposition we combine the left~$K$-action and the right $\Sigma_n$-action on $S({\mathcal U}_K,n)$ into a left action of~$K\times\Sigma$ by \begin{equation}\label{eq:turn_into_left}
(k,\sigma)\cdot (v_1,\dots,v_n) \ = \ (k\cdot v_{\sigma^{-1}(1)},\dots,k\cdot v_{\sigma^{-1}(n)}) \ . \end{equation}
\begin{prop}\label{prop-universal space} Let $K$ be a compact Lie group and $n\geq 2$. Then the $(K\times\Sigma_n)$-space $S({\mathcal U}_K,n)$ is a universal space for the family ${\mathcal F}_n(K)$ of subgroups of $K\times\Sigma_n$. \end{prop} \begin{proof}
We let~$\Gamma$ be a closed subgroup of $K\times\Sigma_n$. If the trace $H=\{\sigma\in\Sigma_n\ |\ (1,\sigma)\in \Gamma\}$ is a transitive subgroup of $\Sigma_n$, then all $H$-fixed points of $S({\mathcal U}_K,n)$ are diagonal, i.e., of the form $(v,\dots,v)$ for some $v\in {\mathcal U}_K$. Since the components must add up to~0, this forces $v=0$, which cannot happen for tuples in the unit sphere. So if $H$ is a transitive subgroup, then $S({\mathcal U}_K,n)$ has no $H$-fixed points, hence no $\Gamma$-fixed points either.
Now we suppose that the trace~$H$ is not transitive. We view the subgroup $\Gamma\leq K\times\Sigma_n$ as a generalized graph: we denote by~$L$ the image of~$\Gamma$ under the projection $K\times\Sigma_n\longrightarrow K$ and define a group homomorphism $\beta:L\longrightarrow W_{\Sigma_n}H$ to the Weyl group of the trace $H$ by \[ \beta(l) \ = \
\{ \sigma\in \Sigma_n\ |\ (l,\sigma)\in \Gamma\} \ \in \ W_{\Sigma_n} H\ .\] We can recover~$\Gamma$ as the graph of~$\beta$, i.e., \[ \Gamma \ = \ {\bigcup}_{l\in L} \{l\}\times \beta(l) \ .\] We let~$L$ act on $(\nu_n)^H$ by restriction along~$\beta$; then $\beta^*((\nu_n)^H)$ is a non-zero $L$-representation because $H$ is non-transitive. Since ${\mathcal U}_K$ is a complete $K$-universe, the underlying $L$-universe is also complete, hence so is the $L$-universe ${\mathcal U}_K\otimes \beta^*((\nu_n)^H)$. So \[ (S({\mathcal U}_K\otimes\nu_n))^\Gamma \ = \ S(({\mathcal U}_K\otimes\beta^*((\nu_n)^H))^L) \] is an infinite dimensional unit sphere, hence contractible. \end{proof}
We define a specific class in the equivariant homotopy set $\pi_0^{\Sigma_n}(B_{\gl}{\mathcal F}_n)$. We set \[ b\ =\ (1/n,\dots,1/n)\ \in\ {\mathbb R}^n \] and let $e_i$ be the $i$-th vector of the canonical basis of~${\mathbb R}^n$.
Then $b-e_i$ lies in the reduced natural~$\Sigma_n$-representation~$\nu_n$. Because~$|b-e_i|^2=\frac{n-1}{n}$, the vector \[ D_n \ = \ \frac{1}{\sqrt{n-1}}(b- e_1,\dots,b-e_n) \ \in \ (\nu_n)^n \] lies in the unit sphere $S(\nu_n,n)$. The~$\Sigma_n$-orbit of $D_n$ (with respect to the right action permuting the `outer' coordinates) is $\Sigma_n$-fixed (with respect to the left action permuting the `inner' coordinates), i.e., \[ D_n\cdot\Sigma_n\ \in \ ( S(\nu_n,n)/\Sigma_n)^{\Sigma_n} \ = \ ((B_{\gl}{\mathcal F}_n)(\nu_n))^{\Sigma_n} \ .\] We denote by \begin{equation}\label{eq:define_u_n}
u_n\ = \ \td{D_n\cdot \Sigma_n} \ \in\ \pi_0^{\Sigma_n} (B_{\gl}{\mathcal F}_n) \end{equation} the class represented by this $\Sigma_n$-fixed point. The next theorem says that the class $u_n$ generates the homotopy set Rep-functor ${\underline \pi}_0 (B_{\gl}{\mathcal F}_n)$.
\begin{theorem}\label{thm-u_n generates T_n} For every $n\geq 2$, every compact Lie group $K$ and every element $x$ of $\pi_0^K(B_{\gl}{\mathcal F}_n)$ there is a continuous group homomorphism $\alpha:K\longrightarrow\Sigma_n$ such that $\alpha^*(u_n)=x$. \end{theorem} \begin{proof} An element of $\pi_0^K(B_{\gl}{\mathcal F}_n)$ is represented by a $K$-representation~$V$ in~$s({\mathcal U}_K)$ and a $K$-fixed $\Sigma_n$-orbit \[ v\cdot \Sigma_n\ \in \ S(V,n)/\Sigma_n \ = \ (B_{\gl}{\mathcal F}_n)(V)\ . \] We let~$H$ denote the $\Sigma_n$-stabilizer of~$v$, a non-transitive subgroup of $\Sigma_n$. We define a continuous homomorphism $\beta:K\longrightarrow W_{\Sigma_n}H$ to the Weyl group of $H$ by
\[ \beta(k) \ = \ \{ \sigma\in \Sigma_n\ |\ k v = v \sigma\} \ .\] As the $\Sigma_n$-stabilizer of a point in $V^n$, the group~$H$ is a Young subgroup of~$\Sigma_n$, i.e., the product of the symmetric groups of all the orbits of the tautological $H$-action on $\{1,\dots,n\}$. Thus the projection $q:N_{\Sigma_n} H\longrightarrow W_{\Sigma_n} H$ has a multiplicative section $s:W_{\Sigma_n} H\longrightarrow N_{\Sigma_n} H$. We define $\alpha:K\longrightarrow\Sigma_n$ as the composite homomorphism \[ K \ \xrightarrow{\ \beta\ }\ W_{\Sigma_n}H \ \xrightarrow{\ s\ }\ N_{\Sigma_n} H \ \xrightarrow{\text{incl}}\ \Sigma_n \] and claim that \[ \alpha^*(u_n) \ = \ [v\cdot\Sigma_n] \text{\qquad in \quad} \pi_0^K(B_{\gl}{\mathcal F}_n)\ . \]
We turn $S(V,n)$ into a left~$(K\times\Sigma_n)$-space as in~\eqref{eq:turn_into_left} and let~$\Gamma\leq K\times\Sigma_n$ denote the graph of~$\alpha$. Since $\alpha(k)\in\beta(k)$ for every $k\in K$, the vector~$v$ is fixed by~$\Gamma$. Increasing the $K$-representation $V$ does not change the stabilizer group of the vector $v$ nor the class represented by the orbit $v\cdot\Sigma_n$ in $\pi_0^K(B_{\gl}{\mathcal F}_n)$; we can thus assume without loss of generality that there is a $K$-equivariant linear isometric embedding $\varphi:\alpha^*(\nu_n)\longrightarrow V$. As the $K$-representations $V$ exhaust a complete $K$-universe, the $(K\times\Sigma_n)$-spaces $S(V,n)$ approximate a universal space for the family ${\mathcal F}_n(K)$, by Proposition~\ref{prop-universal space}. The graph~$\Gamma$ of~$\alpha$ belongs to ${\mathcal F}_n(K)$, so after increasing the $K$-representation $V$, if necessary, we can assume that the dimension of the fixed point sphere $S(V,n)^\Gamma$ is at least~1, so that this fixed point space is path connected. The class $\alpha^*(u_n)$ is represented by the $\Sigma_n$-orbit of the point \[ S(\varphi,n)(D_n) \ \in \ S(V,n)^\Gamma \] and the original class in $\pi_0^K(B_{\gl}{\mathcal F}_n)$ is represented by the vector~$v$. Any path between~$S(\varphi,n)(D_n)$ and~$v$ in the fixed point space $S(V,n)^\Gamma$ projects to a path of $K$-fixed points between the orbits \[ S(\varphi,n)(D_n)\cdot \Sigma_n \ , \quad v\cdot \Sigma_n \quad \in \ (S(V,n) / \Sigma_n)^K \ = \ \left((B_{\gl}{\mathcal F}_n)(V)\right)^K\ . \] This proves that $\alpha^*(u_n) =[v\cdot\Sigma_n]$, and it finishes the proof. \end{proof}
\begin{rk}\label{rk-BF_2 is BSigma_2} The orthogonal space~$B_{\gl}{\mathcal F}_2$ is isomorphic to the free orthogonal space generated by $(\Sigma_2,\sigma)$, where $\sigma$ is the 1-dimensional sign representation of $\Sigma_2$ on~${\mathbb R}$. An isomorphism of orthogonal spaces \[ {\mathbf L}_{\Sigma_2,\sigma}\ \cong \ B_{\gl} {\mathcal F}_2 \] is induced at an inner product space $V$ by the $\Sigma_2$-equivariant natural homeomorphism \[ {\mathbf L}(\sigma, V)\ \cong \ S(V,2) \ , \quad (\varphi:\sigma\longrightarrow V) \ \longmapsto \ \left( \varphi(1/\sqrt{2}),-\varphi(1/\sqrt{2}) \right)\ .\] This isomorphism sends the tautological class~$u_{\Sigma_2,\sigma}$ (see~\eqref{eq:tautological_class}) in~$\pi_0^{\Sigma_2}({\mathbf L}_{\Sigma_2,\sigma})$ to the class $u_2\in \pi_0^{\Sigma_2}(B_{\gl}{\mathcal F}_2)$. So by Theorem~\ref{thm-pi_0 of L_G} every element of $\pi_0^K (B_{\gl}{\mathcal F}_2)$ is of the form~$\alpha^*(u_2)$ for a {\em unique} conjugacy classes of continuous group homomorphism $\alpha:K\longrightarrow\Sigma_2$. For $n\geq 3$, however, $\alpha$ is typically not unique up to conjugacy, and ${\underline \pi}_0 (B_{\gl}{\mathcal F}_n)$ is {\em not} a representable Rep-functor. \end{rk}
\section{Orthogonal spectra}
In this section we recall orthogonal spectra, the objects that represent {\em stable} global homotopy types. Orthogonal spectra are used, at least implicitly, in~\cite{may-quinn-ray}, and the term `orthogonal spectrum' was introduced in~\cite{mmss}, where a non-equivariant stable model structure for orthogonal spectra was constructed. Before giving the formal definition of orthogonal spectra we try to motivate it. An orthogonal space $Y$ assigns values to all finite dimensional inner product spaces. Informally speaking, the global homotopy type is encoded in the $G$-spaces obtained as the `homotopy colimit of $Y(V)$ over all $G$-representations~$V$'. So besides the values~$Y(V)$, we use the $O(V)$-action (which is turned into a $G$-action when~$G$ acts on~$V$) and the information about inclusions of inner product spaces. All this information is conveniently encoded as a continuous functor from the category~${\mathbf L}$.
An orthogonal spectrum~$X$ is a stable analog of this: it assigns a based space~$X(V)$ to every inner product space, and it keeps track of an $O(V)$-action on~$X(V)$ (to get $G$-homotopy types when $G$ acts on~$V$) and of a way to stabilize by suspensions. When doing this in a coordinate free way, the stabilization data assigns to a linear isometric embedding~$\varphi:V\longrightarrow W$ a continuous based map \[ \varphi_\star \ : \ X(V)\wedge S^{\varphi^\perp}\ \longrightarrow \ X(W) \] that `varies continuously with~$\varphi$'. To make the continuous dependence rigorous one exploits that the orthogonal complements $\varphi^\perp$ vary in a locally trivial way, i.e., they are the fibers of an `orthogonal complement' vector bundle over the space of~${\mathbf L}(V,W)$ of linear isometric embeddings. All the structure maps~$\varphi_\star$ together define a map on the smash product of~$X(V)$ with the Thom space of this complement bundle, and the continuity in~$\varphi$ is formalized by requiring continuity of that map. The Thom spaces together form the morphism spaces of a based topological category, and the data of an orthogonal spectrum can conveniently be packaged as a continuous based functor on this category.
\begin{construction} We let $V$ and $W$ be inner product spaces. The `orthogonal complement' vector bundle over the space~${\mathbf L}(V,W)$ is the subbundle of the trivial vector bundle $W\times{\mathbf L}(V,W)$ with total space \[ \xi(V,W) \ = \
\{\, (w,\varphi) \in W\times{\mathbf L}(V,W) \ | \ \text{$\td{w,\varphi(v)}=0$ for all $v\in V$}\,\} \ .\] The fiber over $\varphi:V\longrightarrow W$ is the orthogonal complement of the image of $\varphi$.
We let ${\mathbf O}(V,W)$ be the one-point compactification of the total space of $\xi(V,W)$; since the base space~${\mathbf L}(V,W)$ is compact, ${\mathbf O}(V,W)$ is also the Thom space of the bundle~$\xi(V,W)$. Up to non-canonical homeomorphism, we can describe the space ${\mathbf O}(V,W)$ differently as follows: if $\dim V=n$ and $\dim W=n+m$, then ${\mathbf L}(V,W)$ is homeomorphic to the homogeneous space $O(n+m)/O(m)$ and ${\mathbf O}(V,W)$ is homeomorphic to $O(n+m)_+\wedge_{O(m)}S^m$.
The spaces ${\mathbf O}(V,W)$ are the morphism spaces of a based topological category. Given a third inner product space $U$, the bundle map \[ \xi(V,W) \times \xi(U,V) \ \longrightarrow \ \xi(U,W) \ , \quad ((w,\varphi),\,(v,\psi)) \ \longmapsto \ (w+\varphi(v),\,\varphi\psi)\] covers the composition in~${\mathbf L}$. Passage to one-point compactification gives a based map \[ \circ \ : \ {\mathbf O}(V,W) \wedge {\mathbf O}(U,V) \ \longrightarrow \ {\mathbf O}(U,W) \] which is the composition in the category ${\mathbf O}$. The identity of $V$ is $(0,\Id_V)$ in ${\mathbf O}(V,V)$. \end{construction}
\begin{defn}\label{def-orthogonal spectrum} An {\em orthogonal spectrum} is a based continuous functor from~${\mathbf O}$ to the category of based spaces. A {\em morphism} is a natural transformation of functors. We denote by ${\mathcal S}p$ the category of orthogonal spectra. \end{defn}
We denote by $S^V$ the one-point compactification of an inner product space~$V$, with basepoint at infinity. If~$X$ is an orthogonal spectrum and~$V$ and~$W$ inner product spaces, we define the {\em structure map} \[ \sigma_{V,W} \ : \ X(V)\wedge S^W \ \longrightarrow\ X(V\oplus W) \] as the composite \[ X(V)\wedge S^W\ \xrightarrow{X(V)\wedge ((0,-),i_V)} \ X(V)\wedge {\mathbf O}(V,V\oplus W) \ \xrightarrow{\ X\ }\ X(V\oplus W) \] where $i_V:V\longrightarrow V\oplus W$ is the inclusion of the first summand. If a compact Lie group $G$ acts on~$V$ and~$W$ by linear isometries, then $X(V)$ becomes a based $G$-space by restriction of the action of ${\mathbf O}(V,V)=O(V)_+$, and the structure map~$\sigma_{V,W}$ is $G$-equivariant.
\begin{rk}\label{rk-pointset properties} Given an orthogonal spectrum $X$ and a compact Lie group $G$, the collection of $G$-spaces~$X(V)$ and the structure maps $\sigma_{V,W}$ form an {\em orthogonal $G$-spectrum} in the sense of~\cite{mandell-may} that we denote by~$X_G$. However, only very special orthogonal $G$-spectra arise in this way from an orthogonal spectrum. More precisely, an orthogonal $G$-spectrum $Y$ is isomorphic to~$X_G$ for some orthogonal spectrum $X$ if and only if for every {\em trivial} $G$-representation $V$, the $G$-action on~$Y(V)$ is trivial. An orthogonal $G$-spectrum that does not satisfy this condition is the equivariant suspension spectrum of a based $G$-space with non-trivial $G$-action. In Remark~\ref{rk-homotopy properties} below we isolate some conditions on the Mackey functor homotopy groups of an orthogonal $G$-spectrum that hold for all $G$-spectra of the special form $X_G$. \end{rk}
As we just explained, an orthogonal spectrum $X$ has an underlying orthogonal $G$-spectrum for every compact Lie group $G$. As such, it has equivariant stable homotopy groups, whose definition we now recall. As before, $s({\mathcal U}_G)$ denotes the poset, under inclusion, of finite dimensional $G$-subrepresentations of the complete~$G$-universe ${\mathcal U}_G$. For $k\geq 0$ we consider the functor from~$s({\mathcal U}_G)$ to sets that sends $V\in s({\mathcal U}_G)$ to \[ [S^{k+V}, X(V)]^G \ ,\] the set of $G$-equivariant homotopy classes of based $G$-maps from $S^{k+V}$ to $X(V)$ (where $k+V$ is short hand for ${\mathbb R}^k\oplus V$ with trivial $G$-action on~${\mathbb R}^k$). The map induced by an inclusion $V\subseteq W$ in $s({\mathcal U}_G)$ sends the homotopy class of~$f:S^{k+V}\longrightarrow X(V)$ to the class of the composite \[ S^{k+W} \cong \ S^{k+ V}\wedge S^{W-V} \ \xrightarrow{\ f\wedge S^{W-V}} \ X(V)\wedge S^{W-V} \ \xrightarrow{\sigma_{V,W-V}}\ X(V\oplus (W-V))\ = \ X(W) \ , \] where $W-V$ is the orthogonal complement of~$V$ in~$W$. The {\em $k$-th equivariant homotopy group} $\pi_0^G(X)$ is then defined as \begin{equation}\label{eq:defined_pi_0^G}
\pi_k^G(X) \ = \ \colim_{V\in s({\mathcal U}_G)}\, [S^{k+V}, X(V)]^G \ , \end{equation} the colimit of this functor over the poset~$s({\mathcal U}_G)$. For $k< 0$, the definition is essentially the same, but we take a colimit over~$s({\mathcal U}_G)$ of the sets $[S^V, X({\mathbb R}^{-k}\oplus V)]^G$. When the fixed points $V^G$ have dimension at least~2, then $[S^V,X(V)]^G$ comes with a commutative group structure, and the maps out of it are homomorphisms. The $G$-subrepresentations $V$ with $\dim(V^G)\geq 2$ are cofinal in the poset $s({\mathcal U}_G)$, so the abelian group structures on $[S^V,X(V)]^G $ for $\dim(V^G)\geq 2$ assemble into a well-defined and natural abelian group structure on the colimit $\pi_0^G(X)$. The argument for~$\pi_k^G(X)$ is similar.
\begin{defn}\label{def-global equivalence} A morphism $f:X\longrightarrow Y$ of orthogonal spectra is a {\em global equivalence} if the induced map $\pi_k^G(f):\pi_k^G(X) \longrightarrow \pi_k^G(Y)$ is an isomorphism for all compact Lie groups $G$ and all integers~$k$. \end{defn}
The {\em global stable homotopy category} is the category obtained from the category of orthogonal spectra by formally inverting the global equivalences. The global equivalences are the weak equivalences of the {\em global model structure} on the category of orthogonal spectra, see~\cite{schwede-global}. So the methods of homotopical algebra are available for studying global equivalences and the associated global homotopy category.
Now we set up the formalism of {\em global functors}, the natural (in fact, the tautological) home of the collection of equivariant homotopy groups of an orthogonal spectrum. In this language we then describe the equivariant homotopy groups $\pi_0^G(Sp^n)$ of the symmetric product spectrum~$Sp^n$ as a whole, i.e., when the compact Lie group $G$ is varying: the global functor ${\underline \pi}_0( Sp^n )$ is the quotient of the Burnside ring global functor by a single basic relation.
\begin{defn}[Global Burnside category] The {\em global Burnside category} ${\mathbf A}$ has all compact Lie groups as objects; the morphisms from a group $G$ to $K$ are defined as \[ {\mathbf A}(G,K) \ = \ \text{Nat}(\pi_0^G,\pi_0^K) \ ,\] the set of natural transformations of functors, from orthogonal spectra to sets, between the equivariant homotopy group functors~$\pi_0^G$ and $\pi_0^K$. Composition in~${\mathbf A}$ is composition of natural transformations. \end{defn}
It is not a priori clear that the natural transformations from~$\pi_0^G$ to~$\pi_0^K$ form a set (as opposed to a proper class), but this follows from the representability result in Proposition~\ref{prop-B_gl represents} below. The functor $\pi_0^K$ is abelian group valued, so the set ${\mathbf A}(G,K)$ is an abelian group under objectwise addition of transformations. Composition is additive in each variable, so ${\mathbf A}(G,K)$ is a pre-additive category.
The Burnside category ${\mathbf A}$ is skeletally small: isomorphic compact Lie groups are also isomorphic in the category ${\mathbf A}$, and every compact Lie group is isomorphic to a closed subgroup of an orthogonal group~$O(n)$.
\begin{defn}\label{def-global functor} A {\em global functor} is an additive functor from the global Burnside category ${\mathbf A}$ to the category of abelian groups. A morphism of global functors is a natural transformation. \end{defn}
As a category of additive functors out of a skeletally small pre-additive category, the category of global functors is abelian with enough injectives and projectives. The global Burnside category ${\mathbf A}$ is designed so that the collection of equivariant homotopy groups of an orthogonal spectrum is tautologically a global functor. Explicitly, the global homotopy group functor ${\underline \pi}_0(X)$ of an orthogonal spectrum $X$ is defined on objects by \[ {\underline \pi}_0(X)(G)\ = \ \pi_0^G(X) \] and on morphisms by evaluating natural transformations at $X$.
It is less obvious that conversely every global functor is isomorphic to the homotopy group global functor~${\underline \pi}_0(X)$ of some orthogonal spectrum $X$. We show this in~\cite{schwede-global} by constructing Eilenberg-Mac\,Lane spectra from global functors. In fact, the full subcategories of globally connective respectively globally coconnective orthogonal spectra define a non-degenerate t-structure on the triangulated global stable homotopy category, and the heart of this t-structure is (equivalent to) the abelian category of global functors.
The abstract definition of the global Burnside category ${\mathbf A}$ is convenient for formal considerations and for defining the global functor ${\underline \pi}_0(X)$ associated to an orthogonal spectrum~$X$, but to facilitate calculations we should describe the groups ${\mathbf A}(G,K)$ more explicitly. As we shall explain, the operations between the equivariant homotopy groups come from two different sources: restriction maps along continuous group homomorphisms and transfer maps along inclusions of closed subgroups. A quick way to define the restriction maps, and to deduce some of their properties, is to interpret~$\pi_0^G(X)$ as the $G$-equivariant homotopy set, as defined in~\eqref{eq:define_pi_0^G_set}, of a certain orthogonal space.
\begin{construction}\label{con-Omega^bullet} We recall the functor \[ \Omega^\bullet \ : \ {\mathcal S}p \ \longrightarrow \ spc \] from orthogonal spectra to orthogonal spaces. Given an orthogonal spectrum $X$, the value of $\Omega^\bullet X$ at an inner product space $V$ is \[ (\Omega^\bullet X)(V)\ = \ \map(S^V,X(V))\ . \] If $\varphi:V\longrightarrow W$ is a linear isometric embedding, the induced map \[ \varphi_* \ : \ (\Omega^\bullet X)(V)\ = \ \map(S^V,X(V)) \ \longrightarrow \
\map(S^W,X(W))\ = \ (\Omega^\bullet X)(W) \] is by `conjugation and extension by the identity'. In more detail: given a continuous based map $f:S^V\longrightarrow X(V)$ we define $\varphi_*(f):S^W\longrightarrow X(W)$ as the composite \[ S^W \cong \ S^V\wedge S^{\varphi^\perp} \ \xrightarrow{ f\wedge S^{\varphi^\perp}} \ X(V)\wedge S^{\varphi^\perp} \ \xrightarrow{\sigma_{V,\varphi^\perp}} X(V\oplus \varphi^\perp) \ \cong\ X(W) \ , \] where each of the two unnamed homeomorphisms uses $\varphi$ to identify $V\oplus \varphi^\perp$ with $W$. In particular, the orthogonal group $O(V)$ acts on $(\Omega^\bullet X)(V)$ by conjugation. The assignment $(\varphi,f)\mapsto \varphi_*(f)$ is continuous in both variables and functorial in~$\varphi$. In other words, we have defined an orthogonal space $\Omega^\bullet X$.
The functor $\Omega^\bullet$ has a left adjoint, defined as follows. To every orthogonal space $Y$ we can associate an unreduced {\em suspension spectrum}
$\Sigma^\infty_+Y$ whose value on an inner product space is given by \[ (\Sigma^\infty_+Y)(V)\ = \ Y(V)_+\wedge S^V\ ; \] the structure map \[{\mathbf O}(V,W)\wedge Y(V)_+\wedge S^V \ \longrightarrow \ Y(W)_+\wedge S^W \] is given by \[ (w,\varphi)\wedge y\wedge v \ \longmapsto \ Y(\varphi)(y)\wedge (w+\varphi(v))\ . \] If $Y$ is the constant orthogonal space with value~$A$, then $\Sigma^\infty_+Y$ specializes to the usual suspension spectrum of~$A$ with a disjoint basepoint added. \end{construction}
If $G$ acts on $V$ by linear isometries, then the $G$-fixed subspace of $(\Omega^\bullet X)(V)$ is the space of $G$-equivariant based maps from $S^V$ to $X(V)$. The path components of this space are precisely the equivariant homotopy classes of based $G$-maps, i.e., \[ \pi_0\left(((\Omega^\bullet X)(V))^G\right)\ = \ \pi_0\left( \map^G(S^V,X(V))\right) \ = \ [S^V,X(V)]^G\ . \] Passing to the colimit over the poset $s({\mathcal U}_G)$ gives \[ \pi_0^G(\Omega^\bullet X)\ = \ \pi_0^G(X)\ , \] i.e., the $G$-equivariant homotopy group of the orthogonal spectrum $X$ equals the $G$-equivariant homotopy set (as previously defined in~\eqref{eq:define_pi_0^G_set}) of the orthogonal space $\Omega^\bullet X$. So by specializing the restriction maps for orthogonal spaces we obtain restriction maps \[ \alpha^*\ : \ \pi_0^G(X)\ \longrightarrow \ \pi_0^K(X)\] for every continuous group homomorphism $\alpha:K\longrightarrow G$. These restriction maps are again contravariantly functorial and depend only on the conjugacy class of~$\alpha$ (by Proposition~\ref{prop-inner automorphism}). Moreover,~$\alpha^*$ is additive, i.e., a group homomorphism.
The transfer maps \[ \tr_H^G\ :\ \pi_0^H(X)\ \longrightarrow\ \pi_0^G(X) \] are the classical ones that arise from the orthogonal $G$-spectrum underlying $X$; they are defined whenever~$H$ is a closed subgroup of~$G$ and constructed by an equivariant Thom-Pontryagin construction~\cite[Sec.\,IX.3]{may-alaska}, \cite{nishida-transfer}. Transfers are additive and natural for homomorphisms of orthogonal spectra; since we only consider degree~0 transfers (as opposed to more general `dimension shifting transfers'), the transfer~$\tr_H^G$ is trivial whenever~$H$ has infinite index in its normalizer in~$G$.
As we shall now explain, the suspension spectrum functor `freely builds in' the extra structure that is available at the level of $\pi_0^G$ for orthogonal {\em spectra} (as opposed to orthogonal {\em spaces}), namely the abelian group structure and transfers. We let $Y$ be an orthogonal space and $G$ a compact Lie group. We define a stabilization map \begin{equation} \label{eq:sigma_map}
\sigma^G\ : \ \pi_0^G(Y) \ \longrightarrow \ \pi_0^G(\Sigma^\infty_+ Y) \end{equation} as the effect of the adjunction unit $Y\longrightarrow\Omega^\bullet(\Sigma^\infty_+ Y)$ on the $G$-equivariant homotopy set $\pi_0^G$, using the identification $\pi_0^G(\Omega^\bullet(\Sigma^\infty_+ Y))=\pi_0^G(\Sigma^\infty_+ Y)$. More explicitly: if $V$ is a finite dimensional $G$-subrepresentation of the complete $G$-universe ${\mathcal U}_G$ and $y\in Y(V)^G$ a $G$-fixed point, then $\sigma^G[y]$ is represented by the $G$-map \[ S^V\ \xrightarrow{y\wedge -} \ Y(V)_+\wedge S^V \ = \ (\Sigma^\infty_+ Y)(V) \ .\] The stabilization maps~\eqref{eq:sigma_map} commute with restriction, since they arise from a morphism of orthogonal spaces.
For a closed subgroup $L$ of a compact Lie group $K$, the normalizer $N_K L$ acts on $L$ by conjugation, and hence on~$\pi_0^L(Y)$ by restriction along the conjugation maps. Restriction along an inner automorphism is the identity, so the action of $N_K L$ factors over an action of the Weyl group $W_K L=N_K L/L$ on $\pi_0^L(Y)$. After passing to the stable classes along the map $\sigma^L:\pi_0^L(Y)\longrightarrow\pi_0^L(\Sigma^\infty_+ Y)$, we can then transfer from $L$ to $K$. For an element $k\in N_K L$ and a class $x\in\pi_0^L(Y)$ we have \[ \tr_L^K(\sigma^L(c_k^*(x))) \ = \ \tr_L^K(c_k^*(\sigma^L(x))) \ = \ c_k^*(\tr_L^K(\sigma^L(x))) \ = \ \tr_L^K(\sigma^L(x)) \] because transfer commutes with restriction along the conjugation maps \[ c_k\ : \ L \ \longrightarrow \ L \text{\qquad respectively\qquad} c_k \ : \ K \ \longrightarrow \ K \] defined by $c_k(h)=k^{-1}h k$. So transferring from $L$ to $K$ in the global functor ${\underline \pi}_0(\Sigma^\infty_+ Y)$ equalizes the action of the Weyl group $W_K L$ on $\pi_0^L(Y)$.
\begin{prop}\label{prop-pi_0 of Sigma^infty} Let $Y$ be an orthogonal space. Then for every compact Lie group $K$ the equivariant homotopy group $\pi_0^K(\Sigma^\infty_+ Y)$ of the suspension spectrum of $Y$ is a free abelian group with a basis given by the elements \[ \tr_L^K(\sigma^L(x)) \] where $L$ runs through all conjugacy classes of closed subgroups of $K$ with finite Weyl group and $x$ runs through a set of representatives of the $W_K L$-orbits of the set $\pi_0^L(Y)$. \end{prop} \begin{proof} We consider the functor on the product poset $s({\mathcal U}_K)^2$ sending $(V,U)$ to the set $[S^V, Y(U)_+\wedge S^V]^K$. The diagonal is cofinal in $s({\mathcal U}_K)^2$, thus the induced map \[ \pi_0^K(\Sigma^\infty_+ Y)\ = \ \colim_{V\in s({\mathcal U}_G)}
[S^V, Y(V)_+\wedge S^V]^K \ \longrightarrow \ \colim_{(V,U)\in s({\mathcal U}_K)^2} [S^V, Y(U)_+\wedge S^V]^K \] is an isomorphism. The target can be calculated in two steps, so the group we are after is isomorphic to \[ \colim_{U\in s({\mathcal U}_K)} \left( \colim_{V\in s({\mathcal U}_K)} [S^V, Y(U)_+\wedge S^V]^K \right) \ = \ \colim_{U\in s({\mathcal U}_K)} \pi_0^K\left(\Sigma^\infty_+ Y(U) \right)\ .\] We may thus show that the latter group is free abelian with the specified basis.
The rest of the argument is well known, and a version of it can be found in~\cite[V Cor.\,9.3]{lms}. The tom\,Dieck splitting~\cite[Satz~2]{tomDieck-OrbittypenII} provides an isomorphism \[ \bigoplus_{(L)} \pi_0^{W L}( \Sigma^\infty_+( E W L \times Y(U)^L) ) \ \cong \
\pi_0^K(\Sigma^\infty_+ Y(U)) \ ,\] where the sum is indexed over all conjugacy classes of closed subgroups $L$ and $W L=W_K L$ is the Weyl group of~$L$ in~$K$. By~\cite[Sec.\,4]{tomDieck-OrbittypenII} the group $\pi_0^{W L}( \Sigma^\infty_+( E W L \times Y(U)^L))$ vanishes if the Weyl group $W L$ is infinite; so only the summands with finite Weyl group contribute to $\pi_0^K$. On the other hand, if the Weyl group $W L$ is finite, then the group $\pi_0^{W L}( \Sigma^\infty_+ ( E W L \times Y(U)^L) )$ is free abelian with a basis given by the set $W L \backslash \pi_0( Y(U)^L )$, the $W L$-orbit set of the path components of $Y(U)^L$.
Since colimits commute among themselves, we conclude that \begin{align*}
\pi_0^K\left(\Sigma^\infty_+ Y \right)\ &\cong \ \colim \, \pi_0^K\left(\Sigma^\infty_+ Y(U) \right)\ \cong \
\colim \bigoplus_{(L)} {\mathbb Z}\{ W L\backslash \pi_0( Y(U)^L ) \} \\ &\cong \ \bigoplus_{(L)} {\mathbb Z}\{ W L\backslash (\colim \pi_0( Y(U)^L )) \}\ = \ \bigoplus_{(L)} {\mathbb Z}\{ W L\backslash \pi_0^L(Y) \} \ , \end{align*} where all colimits are over the poset~$s({\mathcal U}_G)$ and the sums are indexed by conjugacy classes with finite Weyl groups. To verify that this composite isomorphism takes the class $\tr_L^K(\sigma^L(x))$ in~$ \pi_0^K\left(\Sigma^\infty_+ Y \right)$ to the basis element corresponding to the orbit of~$x\in \pi_0^L(Y)$ in the summand indexed by~$L$, one needs to recall the definition of the isomorphism in tom Dieck's splitting from~\cite{tomDieck-OrbittypenII}; we omit this. \end{proof}
Now we prove a representability result. The {\em stable tautological class} \[ e_{G,V} \ = \ \sigma^G(u_{G,V})\ \in \ \pi_0^G(\Sigma^\infty_+ {\mathbf L}_{G,V}) \] arises from the unstable tautological class $u_{G,V}$ defined in~\eqref{eq:tautological_class} by applying the stabilization map~\eqref{eq:sigma_map}; so it is represented by the $G$-map \[ S^V \ \longrightarrow \ ({\mathbf L}(V,V)/G)^+\wedge S^V\ = \ (\Sigma^\infty_+ {\mathbf L}_{G,V})(V) \ , \quad v\ \longmapsto (\Id_V\cdot G)\wedge v \ .\]
\begin{prop}\label{prop-B_gl represents} Let $G$ and $K$ be compact Lie groups and $V$ a faithful $G$-representation. Then evaluation at the stable tautological class is an isomorphism \[ {\mathbf A}(G,K) \ \xrightarrow{\ \cong\ } \ \pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V}) \ , \quad \tau\ \longmapsto \ \tau(e_{G,V})\] to the 0-th $K$-equivariant homotopy group of the orthogonal spectrum $\Sigma^\infty_+ {\mathbf L}_{G,V}$. Hence the morphism \[ {\mathbf A}(G,-)\ \longrightarrow \ {\underline \pi}_0(\Sigma^\infty_+ {\mathbf L}_{G,V} ) \] classified by the stable tautological class $e_{G,V}$ is an isomorphism of global functors. \end{prop} \begin{proof} We show first that every natural transformation $\tau:\pi_0^G\longrightarrow \pi_0^K$ is determined by the element~$\tau(e_{G,V})$. We let $X$ be any orthogonal spectrum and $x\in\pi_0^G(X)$ a $G$-equivariant homotopy class. Without loss of generality the class $x$ is represented by a continuous based $G$-map \[ f \ : \ S^{V\oplus W}\ \longrightarrow \ X(V\oplus W) \] for some $G$-representation $W$. This $G$-map is adjoint to a morphism of orthogonal spectra \[ \hat f \ : \ \Sigma^\infty_+ {\mathbf L}_{G,V\oplus W} \ \longrightarrow \ X \text{\qquad that satisfies\qquad}
\hat f_*(e_{G,V\oplus W}) \ = \ x \text{\quad in\quad} \pi_0^G(X) \ .\] We consider the morphism of orthogonal spaces $r:{\mathbf L}_{G,V\oplus W}\longrightarrow {\mathbf L}_{G,V}$ that restricts a linear isometry from~$V\oplus W$ to~$V$. The relation \[ \pi_0^G(r)(u_{G,V\oplus W}) \ = \ u_{G,V} \] shows that the composite \[ \Rep(K,G)\ \xrightarrow{[\alpha]\mapsto \alpha^*(u_{G,V\oplus W})} \ \pi_0^K({\mathbf L}_{G,V\oplus W})\ \xrightarrow{\pi_0^K(r)} \ \pi_0^K({\mathbf L}_{G,V}) \] is evaluation at the class $u_{G,V}$. Evaluation at~ $u_{G,V\oplus W}$ and at~$u_{G,V}$ are both bijective by Theorem~\ref{thm-pi_0 of L_G}, so~$\pi_0^K(r)$ is bijective for all compact Lie groups~$K$. By Proposition~\ref{prop-pi_0 of Sigma^infty}, the induced morphism of suspension spectra \[ \Sigma^\infty_+ r\ : \ \Sigma^\infty_+ {\mathbf L}_{G,V\oplus W}\ \longrightarrow \ \Sigma^\infty_+ {\mathbf L}_{G,V} \] thus induces an isomorphism on $\pi_0^K(-)$ for all compact Lie groups~$K$, and it sends $e_{G,W\oplus V}$ to $e_{G,V}$. The diagram \[ \xymatrix@C=25mm{ \pi_0^G(\Sigma^\infty_+ {\mathbf L}_{G,V})\ar[d]_\tau & \pi_0^G(\Sigma^\infty_+ {\mathbf L}_{G,V\oplus W})\ar[d]_\tau \ar[l]_-{(\Sigma^\infty_+ r)_*}^-\cong\ar[r]^-{\hat f_*} & \pi_0^G(X)\ar[d]^\tau \\ \pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V}) & \pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,W\oplus V}) \ar[l]^-{(\Sigma^\infty_+ r)_*}_-\cong \ar[r]_-{\hat f_*} & \pi_0^K(X)} \] commutes and the two left horizontal maps are isomorphisms. Since \[ x \ = \ \hat f_*((\Sigma^\infty_+ r)_*^{-1}(e_{G,V})) \ ,\] naturality yields that \[ \tau(x) \ = \ \tau(\hat f_*((\Sigma^\infty_+ r)_*^{-1}(e_{G,V})))\ = \ \hat f_*((\Sigma^\infty_+ r)_*^{-1}(\tau(e_{G,V})))\ . \] So the transformation $\tau$ is determined by the value~$\tau(e_{G,V})$.
It remains to construct, for every element $y\in \pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V})$, a natural transformation $\tau:\pi_0^G\longrightarrow \pi_0^K$ with $\tau(e_{G,V})=y$. The previous paragraph dictates what to do: we represent a given class $x\in\pi_0^G(X)$ by a continuous based $G$-map $f:S^{V\oplus W}\longrightarrow X(V\oplus W)$ as above and set \[ \tau(x) \ = \ \hat f_*((\Sigma^\infty_+ r)_*^{-1}(y)))\ .\] We omit the verification that the element $\tau(x)$ only depends on the class $x$ and that~$\tau $ is indeed natural. \end{proof}
We show now that restriction and transfer maps generate all natural operations between the 0-dimensional equivariant homotopy group functors for orthogonal spectra. Given compact Lie groups~$K$ and $G$, we consider pairs $(L,\alpha)$ consisting of \begin{itemize} \item a closed subgroup $L\leq K$ whose Weyl group $W_K L$ is finite, and \item a continuous group homomorphism $\alpha:L\longrightarrow G$. \end{itemize} The {\em conjugate} of $(L,\alpha)$ by a pair of group elements $(k,g)\in K\times G$ is the pair $(^k L,\,c_g\circ \alpha\circ c_k)$ consisting of the conjugate subgroup $^k L$ and the composite homomorphism \[ {^k L} \ \xrightarrow{\ c_k\ }\ L \ \xrightarrow{\ \alpha\ }\ G \ \xrightarrow{\ c_g\ } \ G \ . \] Since inner automorphisms induce the identity on equivariant homotopy groups, \[ \tr_{^k L}^K \circ\, (c_g\circ \alpha \circ c_k)^*\ = \
\tr_L^K \circ\, \alpha^* \ : \ \pi_0^G(X) \ \longrightarrow \ \pi_0^K(X)\ , \] i.e., conjugate pairs define the same operation on equivariant homotopy groups.
\begin{theorem}\label{thm-Burnside category basis} Let $G$ and $K$ be compact Lie groups. \begin{enumerate}[\em (i)] \item Let $V$ be a faithful $G$-representation. Then the homotopy group $\pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V})$ is a free abelian group with basis given by the classes \[ \tr_L^K(\alpha^*(e_{G,V})) \] as $(L,\alpha)$ runs over a set of representatives of all $(K\times G)$-conjugacy classes of pairs consisting of a closed subgroup~$L$ of $K$ with finite Weyl group and a continuous homomorphism $\alpha:L\longrightarrow G$. \item The morphism group ${\mathbf A}(G,K)$ in the global Burnside category is a free abelian group with basis the operations~$\tr_L^K\circ \alpha^*$, where $(L,\alpha)$ runs over all conjugacy classes of pairs consisting of a closed subgroup~$L$ of~$K$ with finite Weyl group and a continuous homomorphism $\alpha:L\longrightarrow G$. \end{enumerate} \end{theorem} \begin{proof} (i) The map \[ \Rep(K,G) \ \longrightarrow \ \pi_0^K({\mathbf L}_{G,V})\ , \quad [\alpha:K\longrightarrow G] \ \longmapsto \ \alpha^*(u_{G,V})\] is bijective according to Theorem~\ref{thm-pi_0 of L_G}. Proposition~\ref{prop-pi_0 of Sigma^infty} thus says that $\pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V})$ is a free abelian group with a basis given by the elements \[ \tr_L^K(\sigma^L(\alpha^*(u_{G,V})))\ = \
\tr_L^K(\alpha^*(\sigma^G(u_{G,V})))\ = \ \tr_L^K(\alpha^*(e_{G,V})) \] where $L$ runs through all conjugacy classes of closed subgroups of $K$ with finite Weyl group and $\alpha$ runs through a set of representatives of the $W_KL$-orbits of the set $ \Rep(L,G)$. The claim follows because $(K\times G)$-conjugacy classes of such pairs $(L,\alpha)$ biject with pairs consisting of a conjugacy class of subgroups $(L)$ and a $W_K L$-equivalence class in $\Rep(L,G)$.
(ii) We let $V$ be any faithful $G$-representation. By part~(i) the composite
\[ {\mathbb Z}\{[L,\alpha]\ |\ \ |W_KL| <\infty,\, \alpha:L\longrightarrow G\} \ \longrightarrow \ \text{Nat}(\pi_0^G,\pi_0^K) \ \xrightarrow{\ \ev\ } \ \pi_0^K(\Sigma^\infty_+ {\mathbf L}_{G,V})\] is an isomorphism, where the first map takes a conjugacy class~$[L,\alpha]$ to $\tr_L^K\circ\,\alpha^*$, and the second map is evaluation at the stable tautological class $e_{G,V}$. The evaluation map is an isomorphism by Proposition~\ref{prop-B_gl represents}, so the first map is an isomorphism, as claimed. \end{proof}
Theorem~\ref{thm-Burnside category basis}~(ii) is almost a complete calculation of the global Burnside category, but one important piece of information is still missing: how does one express the composite of two operations, each given in the basis of Theorem~\ref{thm-Burnside category basis}, as a sum of basis elements? Restrictions are contravariantly functorial and transfers are transitive, i.e., for every closed subgroup $K$ of $H$ we have \[ \tr_H^G\circ\tr_K^H \ = \ \tr_K^G \ : \ \pi_0^K(X) \ \longrightarrow \ \pi_0^G(X) \ . \] So the key question is how to express a transfer followed by a restriction in terms of the specified basis.
Every group homomorphism is the composite of an epimorphism and a subgroup inclusion. Transfers commute with inflation (i.e., restriction along epimorphisms): for every surjective continuous group homomorphism $\alpha:K\longrightarrow G$ and every subgroup $H$ of $G$ the relation
\[ \alpha^*\circ \tr_H^G \ = \ \tr_L^K \circ (\alpha|_L)^* \] holds as maps $\pi_0^H(X)\longrightarrow \pi_0^K(X)$, where $L=\alpha^{-1}(H)$ and
$\alpha|_L:L\longrightarrow H$ is the restriction of $\alpha$. So the remaining issue is to rewrite the composite \[ \pi_0^H(X) \ \xrightarrow{\ \tr_H^G\ }\ \pi_0^G(X) \ \xrightarrow{\ \res^G_K\ }\ \pi_0^K(X) \] of a transfer map and a restriction map, where~$H$ and~$K$ are two closed subgroups of a compact Lie group $G$. The answer is given by the {\em double coset formula}: \begin{equation}\label{eq:double_coset}
\res^G_K\circ \tr_H^G \ = \ \sum_{[M]}\ \chi^\sharp(M)\cdot \tr_{K\cap{^g H}}^K \circ c_g^* \circ \res^H_{K^g\cap H}\ . \end{equation} The double coset formula was proved by Feshbach for Borel cohomology theories~\cite[Thm.\,II.11]{feshbach} and later generalized to equivariant cohomology theories by Lewis and May~\cite[IV \S 6]{lms}. The sum in the double coset formula~\eqref{eq:double_coset} runs over all connected components $M$ of orbit type manifolds, the group element~$g\in G$ that occurs is such that $K g H\in M$, and $\chi^\sharp(M)$ is the internal Euler characteristic of~$M$. Only finitely many of the orbit type manifolds are non-empty, so the double coset formula is a finite sum.
In this paper we only need the double coset formula when~$H$ has finite index in~$G$, and then~\eqref{eq:double_coset} simplifies. For any other subgroup $K$ of~$G$ the intersection $K\cap {^g H}$ then has finite index in $K$, so only finite index transfers are involved in the double coset formula. Since $G/H$ is finite, so is the set $K\backslash G/H$ of double cosets, and all orbit type manifold components are points. So all internal Euler characteristics that occur are~1 and the double coset formula specializes to \[ \res^G_K\circ \tr_H^G \ = \ \sum_{[g]\in K\backslash G/H}\ \tr_{K\cap{^g H}}^K\circ c_g^* \circ \res^H_{K^g\cap H} \ ; \] the sum runs over a set of representatives of the $K$-$H$-double cosets.
The explicit description of the groups~${\mathbf A}(G,K)$ allows us to relate our notion of global functor to other `global' versions of Mackey functors, which are typically introduced by specifying generating operations and relations between them. For example, our category of global functors is equivalent to the category of {\em functors with regular Mackey structure} in the sense of Symonds~\cite[\S 3]{symonds-splitting}.
\begin{eg}\label{eg-global functor examples} We list some explicit examples of global functors; for more details we refer to~\cite{schwede-global}.
(i) For every compact Lie group~$G$, the represented global functor ${\mathbf A}(G,-)$ is realized by the suspension spectrum of a free orthogonal space~${\mathbf L}_{G,V}$, by Proposition~\ref{prop-B_gl represents}. In the special case $G=e$ of the trivial group we refer to this represented global functor as the {\em Burnside ring global functor} and denote it by~${\mathbb A}={\mathbf A}(e,-)$. The value ${\mathbb A}(K)$ at a compact Lie group $K$ is a free abelian group with basis indexed by conjugacy classes of closed subgroups of~$K$ with finite Weyl group. When $K$ is finite, then the Weyl group condition is vacuous and ${\mathbb A}(K)$ this is naturally isomorphic to the Burnside ring of~$K$, compare Remark~\ref{rk-A^fin and A^c} below.
The Burnside ring global functor is realized by the sphere spectrum~${\mathbb S}$, given by ${\mathbb S}(V)=S^V$ with the canonical homeomorphisms $S^V\wedge S^W\cong S^{V\oplus W}$ as structure maps. The equivariant homotopy groups of the sphere spectrum are thus the equivariant stable stems. The action on the unit $1\in\pi_0({\mathbb S})$ is an isomorphism of global functors \[ {\mathbb A} \ \xrightarrow{\ \cong \ } \ {\underline \pi}_0({\mathbb S}) \] from the Burnside ring global functor to the 0-th homotopy global functor of the sphere spectrum. For finite groups, this is originally due to Segal~\cite{segal-ICM}, and for general compact Lie groups to tom\,Dieck, as a corollary to his splitting theorem (see~Satz~2 and Satz~3 of~\cite{tomDieck-OrbittypenII}).
(ii) Given an abelian group $M$, the {\em constant global functor} is given by $\underline{M}(G)=M$ and all restriction maps are identity maps. The transfer $\tr_H^G:\underline{M}(H)\longrightarrow\underline{M}(G)$ is multiplication by the Euler characteristic of the homogeneous space~$G/H$. In particular, if $H$ is a subgroup of finite index of $G$, then $\tr_H^G$ is multiplication by the index $[G:H]$.
(iii) The {\em representation ring global functor} ${\mathbf{RU}}$ assigns to a compact Lie group $G$ the representation ring ${\mathbf{RU}}(G)$, the Grothendieck group of finite dimensional complex $G$-representations. The fact that the representation rings form a global functor is classical in the restricted realm of finite groups, but somewhat less familiar for compact Lie groups in general. The restriction maps $\alpha^*:{\mathbf{RU}}(G)\longrightarrow {\mathbf{RU}}(K)$ are induced by restriction of representations along a homomorphism $\alpha:K\longrightarrow G$. The transfer $\tr_H^G:{\mathbf{RU}}(H)\longrightarrow {\mathbf{RU}}(G)$ along a closed subgroup inclusion $H\leq G$ is the {\em smooth induction} of Segal~\cite[\S\,2]{segal-representation}. If $H$ has finite index in~$G$, then this induction sends the class of an $H$-representation~$V$ to the induced $G$-representation $\map^G(H,V)$; in general, induction may send actual representations to virtual representations. In the generality of compact Lie groups, the double coset formula for ${\mathbf{RU}}$ was proved by Snaith~\cite[Thm.\,2.4]{Snaith-Brauer}. We show in~\cite{schwede-global} that the representation ring global functor ${\mathbf{RU}}$ is realized by the periodic global $K$-theory spectrum.
(iv) Given any generalized cohomology theory $E$ (in the non-equivariant sense), we can define a global functor $\underline{E}$ by setting \[ \underline{E}(G) \ = \ E^0(B G) \ , \] the 0-th $E$-cohomology of a classifying space of the group $G$. The contravariant functoriality in group homomorphisms comes from the covariant functoriality of classifying spaces. The transfer maps for a subgroup inclusion $H\leq G$ comes from the stable transfer map \[ \Sigma^\infty_+ B G \ \longrightarrow \ \Sigma^\infty_+ B H \ .\] The double coset formula was proved in this context by Feshbach~\cite[Thm.\,II.11]{feshbach}. The global functor $G\mapsto E^0(B G)$ is realized by a preferred global homotopy type: in~\cite{schwede-global} we introduce a `global Borel theory' functor~$b$ from the non-equivariant stable homotopy category to the global stable homotopy category such that the global functor ${\underline \pi}_0(b E)$ is isomorphic to $\underline{E}$. The functor $b$ is in fact right adjoint to the forget functor from the global stable homotopy to the non-equivariant stable homotopy category. \end{eg}
\begin{rk}\label{rk-A^fin and A^c} The full subcategory ${\mathbf A}^\text{fin}$ of the global Burnside category~${\mathbf A}$ spanned by {\em finite} groups has a different, more algebraic description, as we shall now recall. This alternative description is often taken as the definition in algebraic treatments of global functors. We define an algebraic Burnside category ${\mathbf B}$ whose objects are all finite groups. The abelian group ${\mathbf B}(G,K)$ of morphisms from a group $G$ to $K$ is the Grothendieck group of finite $K$-$G$-bisets where the right $G$-action is free. Composition \[ \circ \ : \ {\mathbf B}(K,L) \times {\mathbf B}(G,K) \ \longrightarrow \ {\mathbf B}(G,L) \] is induced by the balanced product over $K$, i.e., it is the biadditive extension of \[ (S,T) \ \longmapsto \ S\times_K T \ . \] Here $S$ has a left $L$-action and a commuting free right $K$-action, whereas $T$ has a left $K$-action and a commuting free right $G$-action. The balanced product $S\times_K T$ than inherits a left $L$-action from $S$ and a free right $G$-action from $T$. Since the balanced product is associative up to isomorphism, this defines a pre-additive category~${\mathbf B}$.
An isomorphism of pre-additive categories ${\mathbf A}^\text{fin}\cong {\mathbf B}$ is given by the identity on objects and by the group isomorphisms ${\mathbf A}^\text{fin}(G,K)\longrightarrow {\mathbf B}(G,K)$ sending a basis element $\tr_H^G\circ\,\alpha^*$ to the class of the right free $K$-$G$-biset \[ K\times_{(L,\alpha)} G \ = \ (K\times G) \, /\, (kl,g)\sim(k,\alpha(l)g) \ .\] The category of `global functors on finite groups', i.e., additive functors from ${\mathbf A}^\text{fin}$ to abelian groups, is thus equivalent to the category of {\em inflation functors} in the sense of~\cite[p.\,271]{webb}. In the context of finite groups, these inflation functors and several other variations of the concept `global Mackey functor' have been much studied in algebra and representation theory. \end{rk}
\begin{rk}\label{rk-homotopy properties} In Remark~\ref{rk-pointset properties} we observed that only very special kinds of orthogonal $G$-spectra are part of a `global family', i.e., isomorphic to an orthogonal $G$-spectrum of the form $X_G$ for some orthogonal spectrum $X$. The previous obstructions were in terms of pointset level conditions, and now we can also isolate obstructions to `being global' in terms of the Mackey functor homotopy groups of an orthogonal $G$-spectrum.
If we fix a compact Lie group $G$ and let $H$ run through all closed subgroups of $G$, then the collection of~$H$-equivariant homotopy groups $\pi_0^H(X)$ of an orthogonal spectrum $X$ forms a {\em Mackey functor} for the group~$G$, with respect to the restriction, conjugation and transfer maps. One obstruction for a general orthogonal $G$-spectrum~$Y$ to `be global', i.e., equivariantly stably equivalent to $X_G$ for some orthogonal spectrum~$X$, is that the $G$-Mackey functor $H\mapsto \pi_0^H(Y)$ can be extended to a global functor.
An extension of a $G$-Mackey functor to a global functor requires us to specify values for groups that are not subgroups of $G$, but it also imposes restrictions on the existing data. In particular, the $G$-Mackey functor homotopy groups can be complemented by restriction maps along arbitrary group homomorphisms between the subgroups of $G$. As the extreme case this includes a restriction map $p^*:\pi_*^e(X)\longrightarrow \pi_*^G(X)$ associated to the unique homomorphism $p:G\longrightarrow e$, splitting the restriction map $\res^G_e:\pi_*^G(X)\longrightarrow \pi_*^e(X)$. So one obstruction to being global is that~$\res^G_e$ must be a splittable epimorphism.
Another point is that for an orthogonal spectrum~$X$ (as opposed to a general orthogonal $G$-spectrum), the action of the Weyl group $W_G H$ on $\pi_0^H(X)$ factors through the outer automorphism group of $H$. In other words, if $g$ centralizes $H$, then $c_g^*$ is the identity of $\pi^H_0(X)$. The most extreme case of this is when $H=e$ is the trivial subgroup of $G$. Every element of $G$ centralizes $e$, so for $G$-spectra of the form $X_G$, the conjugation maps on the value at the trivial subgroup are all identity maps. \end{rk}
\section{The global homotopy type of symmetric products}
Now we start the equivariant analysis of the symmetric product filtration. The main result is a global homotopy pushout square of orthogonal spectra~\eqref{eq-ideal square}, showing that~$Sp^n$ can be obtained from~$Sp^{n-1}$ by coning off, in the global stable homotopy category, a certain morphism from the suspension spectrum of~$B_{\gl}{\mathcal F}_n$. Non-equivariantly, such a homotopy pushout square was exhibited by Lesh, see Theorem~1.1 and Proposition~7.4 of~\cite{lesh-filtration}. In Theorem~\ref{thm-pi_0 Sp^n} we then exploit the Mayer-Vietoris sequence of the global homotopy pushout square to calculate the global functors~${\underline \pi}_0(Sp^n)$ inductively.
We define an orthogonal space $C(B_{\gl}{\mathcal F}_n)$ by \[ (C(B_{\gl}{\mathcal F}_n))(V) \ = \ D(V,n)/\Sigma_n\ , \] where \[ D(V,n)\ = \ \left\{ (v_1,\dots,v_n)\in V^n \ : \
\sum_{i=1}^n v_i=0\ , \ \sum_{i=1}^n |v_i|^2 \leq 1 \right\} \] is the unit disc in the kernel of summation map. Since a unit disc is the cone on the unit sphere, $C(B_{\gl}{\mathcal F}_n)$ is the unreduced cone of the global classifying space~$B_{\gl}{\mathcal F}_n$, whence the notation. Next we define a certain morphism of orthogonal spectra \[ \Phi\ : \ \Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n) \longrightarrow \ Sp^n \] that takes the orthogonal subspectrum $\Sigma^\infty_+ B_{\gl}{\mathcal F}_n$ to $Sp^{n-1}$, and that can be thought of as a highly structured, parametrized Thom-Pontryagin collapse map. I owe this construction to Markus Hausmann. Before giving the details we try to explain the main idea. For every inner product space~$V$, the map~$\Phi(V)$ has to assign to each tuple $(y_1,\dots,y_n)\in D(V,n)$ a based map $\Phi(V)(y_1,\dots,y_n):S^V\longrightarrow Sp^n(V)$ that does not depend on the order of $y_1,\dots,y_n$. We would like to take~$\Phi(V)(y_1,\dots,y_n)$ as the product of the Thom-Pontryagin collapse maps in balls of sufficiently small radius centered at the points $y_1,\dots,y_n$. This would work fine for an individual inner product space~$V$, but such maps would not form a morphism of orthogonal spectra as $V$ increases.
The fix to this problem is to combine the collapse maps with orthogonal projection onto the subspace spanned by $y_1,\dots,y_n$. However, this orthogonal projection does {\em not} depend continuously on the tuple $y$ at those points where the dimension of the span of $y_1,\dots,y_n$ jumps. So instead of the orthogonal projection to the span we use a certain positive semidefinite self-adjoint endomorphism $P(y)$ of~$V$ that has similar features and varies continuously with~$y$.
\begin{construction}[Collapse maps] We let~$V$ be an inner product space and denote by~$\sa^+(V)$ the space of positive semidefinite, self-adjoint endomorphisms of~$V$, i.e, ${\mathbb R}$-linear maps $F:V\longrightarrow V$ that satisfy \begin{itemize} \item $\td{F(v),v} \geq 0$ for all $v\in V$, and \item $\td{F(v),w} = \td{v, F(w)}$ for all $v,w\in V$. \end{itemize} We note that~$\sa^+(V)$ is a convex subset of $\text{End}(V)$, hence contractible. We fix the natural number $n\geq 2$ and set the radius for the collapse maps to \[ \rho \ = \ \frac{1}{2\cdot n^{3/2}} \ .\] We define a scaling function \[ s \ : \ [0,\rho) \ \longrightarrow \ [0,\infty) \text{\qquad by\qquad} s(x)\ = \ x/(\rho-x)\ .\] What matters is not the precise form of the function~$s$, but only that it is a homeomorphism from $[0,\rho)$ to~$[0,\infty)$. We define a {\em parametrized collapse map} \[ c \ : \ \sa^+(V)\times S^V \ \longrightarrow \ S^V \] by \[ c(F,v) \ = \ \begin{cases}
v\ + \ s(|F(v)|)\cdot F(v) & \text{ if $v\ne\infty$ and $|F(v)| < \rho$, and}\\ \quad \infty & \text{ else.} \end{cases}\] \end{construction}
\begin{lemma}\label{lemma-estimate}
\begin{enumerate}[\em (i)]
\item For all $(F,v)\in \sa^+(V)\times S^V$ the relation~$|c(F,v)|\geq |v|$ holds.
\item The map $c$ is continuous.
\end{enumerate} \end{lemma} \begin{proof} (i) There is nothing to show if $c(F,v)=\infty$, so we may assume that
$v\ne \infty$ and $|F(v)|<\rho$. Since~$F$ is self-adjoint, $V$ is the orthogonal direct sum of image and kernel of~$F$. So we can write \[ v\ =\ a\ +\ b\ , \] where $a\in\text{im}(F)$, and $b\in\ker(F)$ and~$b$ is orthogonal to~im$(F)$. The orthogonal decomposition \begin{align*}
c(F,v) \ &= \ v\ +\ s(|F(v)|)\cdot F(v) \
= \ \big(a + s(|F(a)|)\cdot F(a) \big)\ +\ b \end{align*} allows us to conclude that \begin{align*}
|c(F,v)|^2 \ &= \ \left|a + s(|F(a)|)\cdot F(a) \right|^2 + |b|^2 \\
&= \ |a|^2\ +\ 2 s(|F(a)|)\cdot \td{a, F(a)} \ + \ s(|F(a)|)^2\cdot |F(a)|^2 \ +\ |b|^2
\ \geq \ |a|^2\ +\ |b|^2\ = \ |v|^2\ . \end{align*} The inequality uses that~$F$ is positive semidefinite. Taking square roots proves the claim.
(ii) We consider a sequence $(F_k,v_k)$ that converges in~$\sa^+(V)\times S^V$ to a point $(F,v)$. We need to show that the sequence $c(F_k,v_k)$ converges to~$c(F,v)$ in~$S^V$.
If $v=\infty$, then $|v_k|$ converges to~$\infty$, hence so does $c(F_k,v_k)$ by part~(i). So we suppose that~$v\ne\infty$ for the rest of the proof. Then we can assume without loss of generality that $v_k\ne\infty$ for all~$k$. We distinguish three cases.
Case 1: $|F(v)|<\rho$. Then~$|F_k(v_k)|<\rho$ for almost all~$k$. So~$c(F_k,v_k)$ converges to $c(F,v)$ because the formula in the definition of~$c$ is continuous in both parameters~$F$ and~$v$.
Case 2: $|F(v)|=\rho$. If $|F_k(v_k)|\geq\rho$, then $c(F_k,v_k)=\infty$. Otherwise \begin{align*}
|c(F_k,v_k)|\ &= \ \left|v_k\ + \ s(|F_k(v_k)|)\cdot F_k(v_k) \right| \
\geq \ s(|F_k(v_k)|)\cdot|F_k(v_k)|\ -\ | v_k| \ . \end{align*}
The sequences $|v_k|$ and $|F_k(v_k)|$ converge
to the finite numbers $|v|$ respectively $\rho$; on the other hand, $s(|F_k(v_k)|)$ converges to~$\infty$. So the sequence $|c(F_k,v_k)|$ also converges to~$\infty$, which means that the sequence~$c(F_k,v_k)$ converges to~$\infty=c(F,v)$.
Case 3: $|F(v)|>\rho$. Then $|F_k(v_k)|>\rho$ for almost all~$k$. So~$c(F_k,v_k)=\infty$ for almost all~$k$, and this sequence converges to~$c(F,v)=\infty$. \end{proof}
\begin{construction} We define a continuous map \[ P \ : \ V^n \ \longrightarrow \ \sa^+(V) \text{\qquad by\qquad} P(y)(v)\ = \ P(y_1,\dots,y_n)(v)\ = \ \sum_{j=1}^n \td{v,y_j}\cdot y_j\ .\] Each of the summands $\td{-,y_j}\cdot y_j$ is self-adjoint, hence so is $P(y)$. Because \[ \td{P(y)(v),v} \ = \ \sum_{j=1}^n \td{v,y_j}^2\ \geq \ 0 \] the map $P(y)$ is also positive semidefinite. If the family $(y_1,\dots,y_n)$ happens to be orthonormal, then $P(y)$ is the orthogonal projection onto the span of $y_1,\dots,y_n$. In general, $P(y)$ need not be idempotent, but its image is always the span of the vectors $y_1,\dots,y_n$, and hence its kernel is the orthogonal complement of that span.
For every linear isometric embedding $\varphi:V\longrightarrow W$ and an endomorphism $F\in \sa^+(V)$, we define $^\varphi F\in\sa^+(W)$ by `conjugation and extension by~0', i.e., we set \[ (^\varphi F)(\varphi(v)+w)\ = \ \varphi(F(v)) \] for all $(v,w)\in V\times\varphi^\perp$. Then \begin{equation}\label{eq:varphi_past_P}
^\varphi ( P(y))(\varphi(v)+w)\ = \
\varphi( P(y)(v))\ = \ \sum_{j=1}^n \td{\varphi(v),\varphi(y_j)}\cdot \varphi(y_j)\ = \ P(\varphi(y))(\varphi(v)+w)\ , \end{equation} i.e., ${^\varphi}( P(y))= P(\varphi(y))$ as endomorphisms of~$W$.
It will be convenient to extend the meaning of the minus symbol and allow to subtract a vector from infinity. We define a continuous map \[ \ominus\ : \ S^V\times V\ \longrightarrow\ S^V \text{\qquad by\qquad}
v \ominus z \ = \ \begin{cases} v - z & \text{\ for $v\ne\infty$, and}\\ \quad \infty& \text{\ for $v=\infty$.} \end{cases} \] We emphasize that only the first argument of the operator $\ominus$ is allowed to be infinity; in particular, we cannot subtract~$\infty$ from itself. We define a continuous map \[ \tilde\Phi(V) \ : \ D(V,n)\times S^V \ \longrightarrow \ Sp^n(S^V) \text{\qquad by\qquad}
\tilde\Phi(V)(y,v) \ = \ [c(P(y),v\ominus y_1),\dots,c(P(y),v\ominus y_n)] \ .\] The map~$\tilde\Phi(V)$ sends $D(V,n)\times \{\infty\}$ to the basepoint. For every permutation~$\sigma\in\Sigma_n$ we have $P(y\cdot\sigma)=P(y)$ and hence \[ \tilde\Phi(V)(y\cdot\sigma,v)\ = \ \tilde\Phi(V)(y,v) \ .\] So~$\tilde\Phi(V)$ factors over a continuous map \[ \Phi(V)\ : \ (C(B_{\gl}{\mathcal F}_n)(V))_+\wedge S^V \ = \ (D(V,n)/\Sigma_n)_+ \wedge S^V \ \longrightarrow \ Sp^n(S^V)\ .\] \end{construction}
\begin{lemma} As~$V$ varies over all inner product spaces, the maps $\Phi(V)$ form a morphism of orthogonal spectra \[ \Phi\ : \ \Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n)\ \longrightarrow \ Sp^n \ . \] \end{lemma} \begin{proof} For every linear isometric embedding $\varphi:V\longrightarrow W$, every $F\in\sa^+(V)$ and all~$(v,w)\in V\times \varphi^\perp$
with $|F(v)|<\rho$ we have \begin{align}\label{eq-c-phi-relation}
c(^\varphi F,\varphi(v)+w)\ &= \
(\varphi(v)+w)\ + \ s(|(^\varphi F)(\varphi(v)+w)|)\cdot (^\varphi F)(\varphi(v)+w) \nonumber\\ &= \
\varphi(v) + w \ + \ s(|\varphi( F(v))|)\cdot \varphi(F(v)) \ = \ \varphi(c(F,v)) + w \end{align} in~$S^W$. Hence for all $y\in D(V,n)$, \begin{align*} c(P(\varphi(y)), (\varphi(v)+w )\ominus \varphi(y_i)) \ &=_\eqref{eq:varphi_past_P} \ c\left(^\varphi(P(y)), ( \varphi(v)+ w)\ominus \varphi(y_i) \right) \\ \ &= \qquad c\left(^\varphi(P(y)), \varphi(v\ominus y_i)+ w \right) \\ &=_\eqref{eq-c-phi-relation}\ \varphi(c(P(y),v\ominus y_i)) + w \ . \end{align*} This shows that the square \[
\xymatrix@C=5mm{ {\mathbf O}(V,W)\times D(V,n)\times S^V \ar[rrrr]^-{{\mathbf O}(V,W)\times\tilde\Phi(V)} \ar[d] &&&&
{\mathbf O}(V,W)\times Sp^n(S^V) \ar[d] & ((w,\varphi),[v_1,\dots,v_n]) \ar@{|->}[d]\\ D(W,n)\times S^W \ar[rrrr]_-{\tilde\Phi(W)} &&&& Sp^n(S^W) & [w+\varphi(v_1),\dots,w+\varphi(v_n)]} \] commutes, where the vertical maps are the structure maps. \end{proof}
We claim that the morphism~$\Phi$ takes the orthogonal subspectrum $\Sigma^\infty_+ (B_{\gl}{\mathcal F}_n)$ into the subspectrum~$Sp^{n-1}$. We will need that kind of argument again later, so we formulate it more generally.
\begin{lemma}\label{lemma-land in Sp^n-1} Let $V$ be an inner product space and $y\in S(V,n)$. For $t\in[0,1]$ we define $F_t\in\sa^+(V)$ by $F_t=(1-t)\cdot P(y)+t\cdot\Id_V$. Then for every $v\in S^V$ the point \[ [c(F_t,v\ominus y_1),\dots,c(F_t,v\ominus y_n)] \ \in \ Sp^n(S^V)\] belongs to the subspace~$Sp^{n-1}(S^V)$. \end{lemma} \begin{proof}
Since $\sum_{i=1}^n|y_i|^2=1$
there is at least one~$i\in\{1,\dots,n\}$ with $|y_i|^2\geq 1/n$. The Cauchy-Schwarz inequality gives \begin{align*}
|y_i|\cdot |F_t(y_i)| \ \geq \ |\td{y_i, F_t(y_i)}| \ &= \ t \td{y_i,y_i}\ + \ (1-t)\sum_{j=1}^n \td{y_i,y_j}^2 \ \geq \
t|y_i|^2\ + \ (1-t)|y_i|^4 \ . \end{align*}
Dividing by $|y_i|$ yields \begin{align*}
|F_t(y_i)| \ \geq \ t|y_i|\ + \ (1-t)|y_i|^3 \ \geq \ \frac{t}{n^{1/2}}\ + \ \frac{1-t}{n^{3/2}}\ \geq \ \frac{1}{n^{3/2}}\ =\ 2\rho \ . \end{align*} The relation \begin{align*}
\sum_{j=1}^n | F_t(y_i-y_j)| \ \geq \
\left| \sum_{j=1}^n F_t(y_i-y_j)\right | \ = \
\left|n\cdot F_t(y_i)- F_t(y_1+\dots+y_n)\right|\ = \ n |F_t(y_i) | \end{align*} shows that there is a $j\in\{1,\dots,n\}$ such that
\[ | F_t(y_i)-F_t(y_j)| \ = \ | F_t(y_i-y_j)| \ \geq \ |F_t(y_i) | \ \geq \ 2\rho\ . \] So every $v\in V$ has distance at least~$\rho$ from $F_t(y_i)$ or from $F_t(y_j)$. Hence~$c(F_t,v\ominus y_i)$ or~$c(F_t,v\ominus y_j)$ is the basepoint at infinity of $S^V$. \end{proof}
For $t=0$, Lemma~\ref{lemma-land in Sp^n-1} shows that for every $v\in S^V$, the point \[ \Phi(V)(y\cdot \Sigma_n, v)\ = \ [c(P(y),v\ominus y_1),\dots,c(P(y),v\ominus y_n)] \] belongs to the subspace~$Sp^{n-1}(S^V)$. So the map $\Phi(V)$ takes the subspace $(\Sigma^\infty_+ B_{\gl}{\mathcal F}_n)(V)$ to~$Sp^{n-1}(S^V)$. We denote by \[ \Psi\ : \ \Sigma^\infty_+ B_{\gl} {\mathcal F}_n\ \longrightarrow \ Sp^{n-1} \] the restriction of the morphism~$\Phi:\Sigma^\infty_+ C(B_{\gl} {\mathcal F}_n)\longrightarrow Sp^n$ to the suspension spectrum of~$B_{\gl}{\mathcal F}_n$. The two vertical maps in the following commutative square~\eqref{eq-ideal square} are levelwise equivariant h-cofibrations. So the following theorem effectively says that the square is a global homotopy pushout.
\begin{theorem}\label{thm-main homotopy} The morphism induced on vertical quotients by the commutative square of orthogonal spectra \begin{equation} \begin{aligned}\label{eq-ideal square}
\xymatrix{ \Sigma^\infty_+ B_{\gl}{\mathcal F}_n \ar[r]^-{\Psi}\ar[d] & Sp^{n-1}\ar[d]\\ \Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n) \ar[r]_-\Phi & Sp^n }
\end{aligned}\end{equation} is a global equivalence. \end{theorem} \begin{proof} We show that for every inner product space~$V$ the map \[\Phi(V)/\Psi(V) \ : \ \bar D(V,n)/\Sigma_n \wedge S^V \ \longrightarrow \ Sp^n(S^V)/Sp^{n-1}(S^V)\] is $O(V)$-equivariantly based homotopic to an equivariant homeomorphism, where $\bar D(V,n)=D(V,n)/S(V,n)$. We define continuous maps \[ G_i\ : \ \big( [0,1]\times D(V,n)\ \backslash \ \{1\}\times S(V,n) \big) \times S^V \ \longrightarrow \ S^V \] for $1\leq i\leq n$ by
\[ G_i(t,y,v) \ = \ c\left( (1-t)\cdot P(y),\ v\ominus \frac{y_i}{1- t|y|}\right) .\] Here the domain of definition of~$G_i$ is the space of those tuples $(t,y,v)\in [0,1]\times D(V,n) \times S^V$
such that $t<1$ or $|y|<1$.
We claim that the map \[ (G_1,\dots,G_n)\ : \
[0,1)\times D(V,n) \times S^V \ \longrightarrow \ (S^V)^n \] takes the subspace $[0,1)\times S(V,n) \times S^V$ of the source into the wedge inside of the product $(S^V)^n$. Indeed, because $\Phi(V)$ takes $(S(V,n)/\Sigma_n)_+\wedge S^V$ into $Sp^{n-1}(S^V)$, for every $v\in V$ there is an $i\in\{1,\dots,n\}$ with \[ c(P(y), ((1-t)v)\ominus y_i ) \ = \ \infty \ , \]
i.e., $|P(y)((1-t)v-y_i)|\geq\rho$. Because \begin{align*}
\left| (1-t)\cdot P(y)\left(v - \frac{y_i}{1-t}\right)\right| \ = \
| P(y)((1-t)v - y_i)|\ &\geq \ \rho \ , \end{align*} this implies that~$G_i(t,y,v) =\infty$.
We warn the reader that the maps~$G_i$ do {\em not} extend continuously to $[0,1]\times D(V,n)\times S^V$! However, smashing all $G_i$ together remedies this. In other words, we claim that the map \[ \big( [0,1]\times D(V,n)\ \backslash \ \{1\}\times S(V,n) \big) \times S^V
\ \xrightarrow{\ G_1\wedge\dots\wedge G_n\ } \ (S^V)^{\wedge n} \] has a continuous extension (necessarily unique) \[ \bar G \ : \ [0,1]\times D(V,n) \times S^V \ \longrightarrow \ (S^V)^{\wedge n} \] that sends~$\{1\}\times S(V,n)\times S^V$ to the basepoint. To prove the claim, we consider any sequence $(t_m,y^m,v^m)_{m\geq 1}$ in $\big( [0,1]\times D(V,n)\ \backslash \ \{1\}\times S(V,n) \big)\times S^V$
that converges to a point $(1,y,v)$ with $|y|=1$. We claim that there are $i,j\in\{1,\dots,n\}$
such that $|y_i-y_j|\geq 4\rho$. Indeed, if that were not the case, then we would have \begin{align*}
2n\ &= \ \left( n \sum_{i=1}^n |y_i|^2\right) - 2\left\langle\sum_{i=1}^n y_i,\sum_{j=1}^n y_j\right\rangle
+\left(n \sum_{j=1}^n |y_j|^2\right)\\
&= \ \sum_{i,j=1}^n \left( |y_i|^2- 2\td{y_i,y_j} +|y_j|^2\right)\
= \ \sum_{i,j=1}^n |y_i-y_j|^2 \ < \ (4\rho)^2 n^2\ = \ 4/n\ , \end{align*} a contradiction.
Since $\lim_{m\longrightarrow\infty} y^m=y$, we deduce that $|y_i^m-y_j^m|\geq 2\rho$ for all sufficiently large~$m$. For these~$m$ there is then a~$k\in\{i,j\}$ such that
\[ \left| (1- t_m|y^m|)v^m - y_k^m \right | \ \geq \ \rho \ ,\] and hence \begin{align*}
| G_k(t_m,y^m,v^m)| \
\ &= \ \left| c\left( (1-t_m)\cdot P(y^m),\ v^m\ominus\frac{y_k^m}{1- t_m|y^m|} \right) \right| \\
&\geq \ \left| v^m -\frac{y_k^m}{1- t_m|y^m|} \right|
\ \geq \ \frac{\rho}{1- t_m|y^m|} \ . \end{align*}
The first inequality is Lemma~\ref{lemma-estimate}~(i). Since the sequences $(t_m)$ and $|y^m|$ converge to~1, the length of the vector \[ (G_1(t_m,y^m,v^m),\dots,G_n(t_m,y^m,v^m)) \] tends to infinity with~$m$, so it converges to the basepoint at infinity of $S^{V^n}=(S^V)^{\wedge n}$.
We have now completed the verification that the map~$\bar G:[0,1]\times D(V,n) \times S^V\longrightarrow (S^V)^{\wedge n}$ is continuous. Because the endomorphism $P(y)$ does not depend on the order of the components of the tuple~$y$, the maps $G_i$ satisfy \[ G_i(t, y\cdot \sigma, v) \ = \ G_{\sigma(i)}(t,y,v) \ ,\] so the map $\bar G$ descends to a well-defined continuous and $O(V)$-equivariant map \[ [0,1]\times \left( \bar D(V,n)/\Sigma_n \wedge S^V \right) \ \longrightarrow \ (S^V)^{\wedge n}/\Sigma_n \ = \ Sp^n(S^V)/Sp^{n-1}(S^V)\ ,\] which is the desired equivariant homotopy. This homotopy starts with the map~$\Phi(V)/\Psi(V)$ and ends with the map \begin{align*}
\bar D(V,n)/\Sigma_n \wedge S^V \ &\longrightarrow \qquad Sp^n(S^V)/Sp^{n-1}(S^V)\\ (y\cdot \Sigma_n ,v) \qquad &\longmapsto \
\left( v-\frac{y_1}{1-|y|} \right)\wedge\dots\wedge\left( v-\frac{y_n}{1-|y|}\right) \ ; \end{align*} this map is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. \end{proof}
We recall from~\eqref{eq:define_u_n} the definition of the unstable homotopy class $u_n\in\ \pi_0^{\Sigma_n} (B_{\gl}{\mathcal F}_n)$. The stabilization map~\eqref{eq:sigma_map} lets us define a $\Sigma_n$-equivariant stable homotopy class \[ w_n \ = \ \sigma^{\Sigma_n}(u_n) \ \in \ \pi_0^{\Sigma_n}(\Sigma^\infty_+ B_{\gl}{\mathcal F}_n) \ . \] The last ingredient for our main calculation is to determine the image of~$w_n$ under the morphism of orthogonal spectra \[ \Psi\ : \ \Sigma^\infty_+ B_{\gl} {\mathcal F}_n\ \longrightarrow \ Sp^{n-1} \ . \]
\begin{prop}\label{prop:Psi_of_u_n} The relation \[
\Psi_*(w_n) \ = \ i_*\left(\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)\right)
\] holds in the group~$\pi_0^{\Sigma_n}(Sp^{n-1})$, where $i:{\mathbb S}\longrightarrow Sp^{n-1}$ is the inclusion. \end{prop} \begin{proof} The class $\Psi_*(w_n)$ is represented by the composite $\Sigma_n$-map \begin{equation}\label{eq:first_rep}
S^{\nu_n} \ \xrightarrow{(d_1,\dots,d_n)\cdot\Sigma_n\wedge -} \
(S(\nu_n,n)/\Sigma_n)_+\wedge S^{\nu_n} \ \xrightarrow{\Psi(\nu_n)} \ Sp^{n-1}(S^{\nu_n}) \ , \end{equation} where~$\nu_n$ is the reduced natural $\Sigma_n$-representation and \[ (d_1,\dots,d_n) \ = \ \frac{1}{\sqrt{n-1}}(b-e_1,\dots,b-e_n) \ \in \ S(\nu_n,n) \ .\] We define an equivariant homotopy to a different map that is easier to understand.
The space $\sa^+(\nu_n)$ of positive semidefinite self-adjoint endomorphisms of~$\nu_n$ is convex, so we can interpolate between $P(d_1,\dots,d_n)$ and the identity of~$\nu_n$ in~$\sa^+(\nu_n)$ by the linear homotopy \[ t\ \longmapsto \ F_t\ = \ (1-t)\cdot P(d_1,\dots,d_n) \ + \ t \cdot \Id_{\nu_n}\ .\] This induces a homotopy \[ K \ : \ [0,1]\times S^{\nu_n}\ \longrightarrow \ Sp^n(S^{\nu_n}) \ , \quad K(t,v)\ = \ [ c(F_t, v\ominus d_1),\dots, c(F_t, v\ominus d_n) ]\ .\] For every permutation~$\sigma\in\Sigma_n$ we have $\sigma\cdot d_i=d_{\sigma(i)}$, hence \[ {^\sigma}( P(d_1,\dots,d_n))\ =_\eqref{eq:varphi_past_P} \ P(\sigma\cdot d_1,\dots,\sigma\cdot d_n) \ =\ P(d_{\sigma(1)},\dots,d_{\sigma(n)})\ =\ P(d_1,\dots,d_n) \] as endomorphisms of~$\nu_n$. Thus also ${^\sigma}( F_t) = F_t$ and hence \begin{align*}
\sigma\cdot c(F_t,v\ominus d_i) \ =_\eqref{eq-c-phi-relation} \ c(^\sigma(F_t), \sigma\cdot(v\ominus d_i)) \ = \ c(F_t, (\sigma\cdot v)\ominus(\sigma\cdot d_i)) \ = \ c(F_t, (\sigma\cdot v)\ominus d_{\sigma(i)}) \ , \end{align*} so $\sigma\cdot K(t,v)=K(t,\sigma\cdot v)$, i.e., the homotopy~$K$ is $\Sigma_n$-equivariant. A priori, the homotopy takes values in the $n$-th symmetric product; however, Lemma~\ref{lemma-land in Sp^n-1} applied to~$V=\nu_n$ and $y=(d_1,\dots,d_n)$ shows that~$K(t,v)$ belongs to~$Sp^{n-1}(S^{\nu_n})$.
The homotopy~$K$ starts with the composite~\eqref{eq:first_rep}, so the map \[ K(1,-) \ : \ S^{\nu_n} \ \longrightarrow \ Sp^{n-1}(S^{\nu_n}) \text{\qquad given by\qquad} K(1,v)\ = \ \big[c(\Id_V,v\ominus d_1),\dots,c(\Id_V,v\ominus d_n) \big]\] is another representative of the class $\Psi_*(w_n)$. Because
\[ |d_i-d_j|\ = \ \sqrt{\frac{2}{n-1}} \ \geq \ \frac{1}{n^{3/2}}\ = \ 2\rho \] for all $i\ne j$, the interiors of the $\rho$-balls around the points $d_1,\dots,d_n$ are disjoint. So for every~$v\in S^{\nu_n}$ at most one of the points $c(\Id_V,v\ominus d_1),\dots, c(\Id_V,v\ominus d_n)$ is different from the basepoint of~$S^{\nu_n}$ at infinity. The map~$K(1,-)$ thus equals the composite \[ S^{\nu_n} \ \xrightarrow{\ J\ }\ S^{\nu_n} \ \xrightarrow{\ i\ }\ Sp^{n-1}(S^{\nu_n}) \] where the first map is defined by \[ J(v)\ = \ \begin{cases}
c(\Id_V,v\ominus d_i) = \frac{v-d_i}{1-|v-d_i|/\rho} & \text{ if $v\ne\infty$ and $|v-d_i|<\rho$, and}\\ \quad \infty & \text{ else.} \end{cases} \] The map $J$ is a $\Sigma_n$-equivariant Thom-Pontryagin collapse map around the image of the equivariant embedding \[ \Sigma_n/\Sigma_{n-1} \ \longrightarrow \ \nu_n \ , \quad \sigma \Sigma_{n-1} \ \longmapsto \ d_{\sigma(n)} \ .\] So~$J$ represents the class $\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)$ in the equivariant 0-stem~$\pi_0^{\Sigma_n}({\mathbb S})$. \end{proof}
Now we can put the pieces together and prove our main calculation. We let~$I_n$ denote the global subfunctor of the Burnside ring global functor~${\mathbb A}$ generated by the element $t_n=n\cdot 1-\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)$ in ${\mathbb A}(\Sigma_n)$, and we let~$I_\infty$ denote the union of the global functors~$I_n$ for all~$n\geq 1$.
\begin{theorem}\label{thm-pi_0 Sp^n} The inclusion of orthogonal spectra $Sp^{n-1}\longrightarrow Sp^n$ induces an epimorphism \[ {\underline \pi}_0(Sp^{n-1}) \ \longrightarrow \ {\underline \pi}_0(Sp^n)\] of the 0-th homotopy global functors whose kernel is generated, as a global functor, by the class \[ i_*\left( n \cdot 1\ -\ \tr_{\Sigma_{n-1}}^{\Sigma_n}(1) \right) \ \in \ \pi_0^{\Sigma_n}(Sp^{n-1})\ , \] where $i:{\mathbb S}\longrightarrow Sp^{n-1}$ is the inclusion. For every $n\geq 1$ and $n=\infty$ the action of the Burnside ring global functor on the class $i_*(1)\in \pi_0^e(Sp^n)$ passes to an isomorphism of global functors \[ {\mathbb A}/I_n \ \cong \ {\underline \pi}_0(Sp^n) \ .\] \end{theorem} \begin{proof} For every inner product space~$V$ the embedding $Sp^{n-1}(S^V)\longrightarrow Sp^n(S^V)$ has the $O(V)$-equivariant homotopy extension property. The cone inclusion $q:B_{\gl}{\mathcal F}_n \longrightarrow C(B_{\gl}{\mathcal F}_n)$ also has the levelwise homotopy extension property, hence so does the induced morphism $j=\Sigma^\infty_+ q : \Sigma^\infty_+ B_{\gl}{\mathcal F}_n \longrightarrow\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n)$ of suspension spectra. So Theorem~\ref{thm-main homotopy} says that the commutative square of orthogonal spectra~\eqref{eq-ideal square} is a global homotopy pushout square. Taking equivariant stable homotopy groups thus results in an exact Mayer-Vietoris sequence that ends in the exact sequence of global functors \begin{equation}\label{eq:MV-sequence}
{\underline \pi}_0( \Sigma^\infty_+ B_{\gl}{\mathcal F}_n ) \ \xrightarrow{\genfrac(){0pt}{}{j_*}{\Psi_*}} \ {\underline \pi}_0( \Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n) )\oplus{\underline \pi}_0( Sp^{n-1}) \ \xrightarrow{(-\Phi_*,\text{incl}_*)} \ {\underline \pi}_0( Sp^n) \ \longrightarrow \ 0 \ . \end{equation}
By Proposition~\ref{thm-u_n generates T_n} the Rep-functor
${\underline \pi}_0 (B_{\gl}{\mathcal F}_n)$ is generated by the element $u_n$
in $\pi_0^{\Sigma_n} (B_{\gl}{\mathcal F}_n)$. So by Proposition~\ref{prop-pi_0 of Sigma^infty}
the global functor ${\underline \pi}_0 (\Sigma^\infty_+ B_{\gl}{\mathcal F}_n)$
is generated by the element $w_n=\sigma^{\Sigma_n}(u_n)$
in $\pi_0^{\Sigma_n} (\Sigma^\infty_+ B_{\gl}{\mathcal F}_n)$.
The orthogonal space~$C(B_{\gl}{\mathcal F}_n)$ is contractible; so
its suspension spectrum~$\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n)$
is globally equivalent to the sphere spectrum.
Thus~${\underline \pi}_0(\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n))$ is isomorphic to the Burnside
ring global functor~${\mathbb A}$, and it is freely generated by the class
$1=\sigma^e(u)$ in~$\pi_0^e(\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n))$,
where $u\in\pi_0^e(C(B_{\gl}{\mathcal F}_n))$ is the unique element.
We record that
\begin{align*}
j_*(w_n) \ &= \ (\Sigma^\infty_+ q)_*(\sigma^{\Sigma_n}(u_n))
\ = \ \sigma^{\Sigma_n}(q_*(u_n)) \
= \ \sigma^{\Sigma_n}(p^*(u))
\ = \ p^*(\sigma^e(u))\ = \ p^*(1)\ ,
\end{align*}
where $p:\Sigma_n\longrightarrow e$ is the unique homomorphism.
The relation~$q_*(u_n)=p^*(u)$ holds because
the set~$\pi_0^{\Sigma_n}(C(B_{\gl}{\mathcal F}_n))$ has only one element.
Since the global functor ${\underline \pi}_0(\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n))$ is freely generated
by the class~$1$,
there is a unique morphism
$s:{\underline \pi}_0(\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n))\longrightarrow {\underline \pi}_0( \Sigma^\infty_+ B_{\gl}{\mathcal F}_n )$
such that $s(1)=\res^{\Sigma_n}_e(w_n)$. Then
\[ j_*(s(1))\ = \ \res^{\Sigma_n}_e(j_*(w_n))
\ = \ \res^{\Sigma_n}_e(p^*(1))\ = \ 1 \ . \]
Morphisms out of the global functor~${\underline \pi}_0(\Sigma^\infty_+ C(B_{\gl}{\mathcal F}_n))$
are determined by their effect on the universal class,
so we conclude that $j_*\circ s=\Id$.
In particular, $j_*$ is an epimorphism and
the Mayer-Vietoris sequence~\eqref{eq:MV-sequence}
restricts to an exact sequence of global functors
\[ \ker(j_*) \ \xrightarrow{\ \Psi_*\ }
\ {\underline \pi}_0( Sp^{n-1}) \ \xrightarrow{\text{incl}_*} \ {\underline \pi}_0( Sp^n) \ \longrightarrow \ 0 \ . \]
The other composite $s\circ j_*$ is
an idempotent endomorphism of ${\underline \pi}_0(\Sigma^\infty_+ B_{\gl}{\mathcal F}_n)$,
and it satisfies
\[ s( j_*(w_n))\ = \ s(p^*(1))\ = \ p^*(s(1)) \ = \ p^*(\res^{\Sigma_n}_e(w_n))\ . \]
The kernel of $j_*$ is thus generated as a global functor by
\[ (s\circ j_*-\Id)(w_n) \ = \ p^*(\res^{\Sigma_n}_e(w_n))\ - \ w_n \ . \]
The global functor $\Psi_*(\ker(j_*))$ is then generated by the class
\begin{align*}
\Psi_*( p^*(\res^{\Sigma_n}_e(w_n))- w_n )\
&= \
p^*\left(\res^{\Sigma_n}_e\left(i_*\left(\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)\right)\right)\right)
\ - \ i_*\left(\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)\right) \\
&= \
i_*\left(p^*\left(\res^{\Sigma_n}_e\left(\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)\right)\right)- \tr_{\Sigma_{n-1}}^{\Sigma_n}(1)\right)\
= \ i_* \left( n \cdot 1 - \tr_{\Sigma_{n-1}}^{\Sigma_n}(1) \right) \ .
\end{align*}
The first equality uses Proposition~\ref{prop:Psi_of_u_n}.
This proves the first claim.
The second claim is then obtained by induction over~$n$,
using that $I_{n-1}\subset I_n$.
For $n=\infty$ we use that the canonical map
\[ \colim_{n} \, {\underline \pi}_0(Sp^n) \ \longrightarrow \ {\underline \pi}_0(Sp^\infty)\]
is an isomorphism because each embedding $Sp^{n-1}\longrightarrow Sp^n$
is levelwise an equivariant h-cofibration. \end{proof}
\section{Examples}\label{sec-examples}
In this last section we make the description of the global functor ${\underline \pi}_0(Sp^n)$ of Theorem~\ref{thm-pi_0 Sp^n} more explicit by exhibiting a generating set for the group $I_n(G)$, the kernel of the map ${\mathbb A}(G)\cong\pi_0^G({\mathbb S})\longrightarrow\pi_0^G(Sp^n)$, in terms of the subgroup structure of~$G$. We use this to determine~$\pi_0^G(Sp^n)$, for all $n$, when~$G$ is a~$p$-group, a symmetric group $\Sigma_k$ for $k\leq 4$, and the alternating group~$A_5$. The purpose of these calculations is twofold: we want to illustrate that $\pi_0^G(Sp^n)$ can be worked out explicitly in terms of the poset of conjugacy classes of subgroups of~$G$ and their relative indices; and we want to convince the reader that the explicit answer for the group~$\pi_0^G(Sp^n)$ is much less natural than the global description of~${\underline \pi}_0(Sp^n)$ given by Theorem~\ref{thm-pi_0 Sp^n}.
For a pair of closed subgroups $K\leq H$ of a compact Lie group $G$ such that~$K$ has finite index in~$H$ we denote by $t^H_K \in {\mathbb A}(G)$ the class \[ t^H_K\ = \ [H:K]\cdot \tr_H^G(1)\ -\ \tr_K^G(1) \ .\] For example, $t_n=t^{\Sigma_n}_{\Sigma_{n-1}}$. The notation is somewhat imprecise because it does not record the ambient group~$G$, but that should always be clear from the context. In the next proposition these classes feature under the hypothesis that the Weyl group of $H$ in~$G$ is finite (so that $\tr_H^G(1)$ is a non-trivial class in the Burnside ring of~$G$). However, the group~$K$ may have infinite Weyl group, in which case $\tr_K^G(1)=0$ and~$t^H_K$ simplifies to $[H:K]\cdot \tr_H^G(1)$.
\begin{prop}\label{prop-describe I_n} For every $n\geq 2$ and every compact Lie group~$G$, the abelian group $I_n(G)$ is generated by the classes $t_K^H$ as $(H,K)$ runs through a set of representatives of all $G$-conjugacy classes of nested pairs $K\leq H$ of closed subgroups of~$G$ such that \begin{itemize} \item $[H:K]\leq n$ and \item the Weyl group $W_G H$ is finite. \end{itemize} \end{prop} \begin{proof} By definition $I_n$ is the image of the morphism of global functors \[ {\mathbf A}(\Sigma_n,-) \ \longrightarrow \ {\mathbb A} \] represented by $t_n\in {\mathbb A}(\Sigma_n)$. By Theorem~\ref{thm-Burnside category basis}~(ii) the group ${\mathbf A}(\Sigma_n,G)$ is generated by the operations $\tr_H^G\circ\alpha^*$ where $(H,\alpha)$ runs through the $(G\times\Sigma_n)$-conjugacy classes of pairs consisting of a closed subgroup $H\leq G$ with finite Weyl group and a continuous homomorphism $\alpha:H\longrightarrow \Sigma_n$. So~$I_n(G)$ is generated, as an abelian group, by the classes \[ \tr_H^G(\beta^*(\res^{\Sigma_n}_\Gamma(t_n)))\ \in \ {\mathbb A}(G) \ ,\] where $\Gamma$ is a subgroup of~$\Sigma_n$ and $\beta:H\longrightarrow \Gamma$ a continuous epimorphism. The double coset formula (in the finite index case) gives \begin{align*}
\res^{\Sigma_n}_\Gamma(t_n) \ &= \ n\cdot 1 - \res^{\Sigma_n}_\Gamma(\tr_{\Sigma_{n-1}}^{\Sigma_n}(1)) \\ &= \
n\cdot 1 \ - \sum_{[\sigma]\in \Gamma\backslash \Sigma_n/\Sigma_{n-1}} \tr_{\Gamma\cap{^\sigma \Sigma_{n-1}}}^\Gamma(1) = \ \sum_{[\sigma]\in \Gamma\backslash \Sigma_n/\Sigma_{n-1}} t^\Gamma_{\Gamma\cap {^\sigma \Sigma_{n-1}}} \end{align*} in~${\mathbb A}(\Gamma)$, where we used that \[ \sum_{[\sigma]\in \Gamma\backslash \Sigma_n/\Sigma_{n-1}} [\Gamma:\Gamma\cap {^\sigma \Sigma_{n-1}}] \ = \ n \ .\] Thus \begin{align*}
\beta^*(\res^{\Sigma_n}_\Gamma(t_n))\ &= \ \sum_{[\sigma]\in \Gamma\backslash \Sigma_n/\Sigma_{n-1}} \beta^*\left(t_{\Gamma\cap{^\sigma \Sigma_{n-1}}}^\Gamma\right)\ = \ \sum_{[\sigma]\in \Gamma\backslash \Sigma_n/\Sigma_{n-1}} t_{\beta^{-1}(^\sigma \Sigma_{n-1})}^H \end{align*} in~${\mathbb A}(H)$. Transferring from $H$ to $G$ gives \begin{align*}
\tr_H^G(\beta^*(\res^{\Sigma_n}_\Gamma(t_n)))\ &= \
\sum_{[\sigma] \in \Gamma\backslash \Sigma_n/\Sigma_{n-1}}
t^H_{\beta^{-1}(^\sigma \Sigma_{n-1})} \text{\quad in~${\mathbb A}(G)$.} \end{align*} Since ${^\sigma \Sigma_{n-1}}$ has index $n$ in $\Sigma_n$, the group $\Gamma\cap{^\sigma \Sigma_{n-1}}$ has index at most $n$ in $\Gamma$, and hence the group $\beta^{-1}(^\sigma \Sigma_{n-1})=\beta^{-1}(\Gamma\cap{^\sigma \Sigma_{n-1}})$ has index at most $n$ in $H$. So $I_n(G)$ is indeed contained in the group described in the statement of the proposition.
For the other inclusion we consider a pair of closed subgroups $K\leq H$ in the ambient group~$G$ with $m=[H:K]\leq n$. A choice of bijection between $H/K$ and $\{1,\dots,m\}$ turns the left translation action of $H$ on~$H/K$ into a homomorphism $\beta:H\longrightarrow\Sigma_m$ such that $H/K$ is isomorphic, as an $H$-set, to $\beta^*(\{1,\dots,m\})$. Since~$t_m\in I_m(\Sigma_m)\subset I_n(\Sigma_m)$ and~$I_n$ is a global functor, we conclude that \[ t_K^H \ = \ \tr_H^G\left([H:K]\cdot 1- \tr^H_K(1)\right)\ = \
\tr_H^G(\beta^*(t_m)) \ \in \ I_n(G)\ . \qedhere \] \end{proof}
For every finite group $G$ the augmentation ideal $I(G)$ is generated by the classes $t^G_H$ where~$H$ runs through all subgroups of $G$. So Proposition~\ref{prop-describe I_n} shows that the filtration by the subfunctors $I_n$ exhausts the augmentation ideal at the $|G|$-th stage:
\begin{cor} Let~$G$ be a finite group. Then $I_n(G)=I(G)$ for~$n\geq |G|$. \end{cor}
However, often the filtration stops earlier, for example for $p$-groups.
\begin{eg}[Finite $p$-groups] Let $p$ be a prime and $P$ a finite $p$-group. Proposition~\ref{prop-describe I_n} shows that $I_n(P)=\{0\}$ for $n<p$. On the other hand, every proper subgroup $H$ of~$P$ admits a sequence of intermediate subgroups \[ H\ =\ H_0\ \subset\ H_1 \ \subset \ \dots \ \subset\ H_k\ =\ P \] such that $[H_i:H_{i-1}]=p$ for all $i=1,\dots, k$. Then the class \begin{align*} t_H^P\ & = \ p^k \cdot 1 - \tr^P_H(1) \ = \
\sum_{i=1}^k p^{i-1}\cdot t_{H_{i-1}}^{H_i} \end{align*} belongs to $I_p(P)$ by Proposition~\ref{prop-describe I_n}. Since the classes $t_H^P$ generate the augmentation ideal, we conclude that $I_p(P)=I(P)$. Hence the group~$\pi_0^P(Sp^n)$ is isomorphic to the Burnside ring~${\mathbb A}(P)$ for $1\leq n <p$, and free of rank~1 for $n\geq p$. \end{eg}
We work out the symmetric product filtration on equivariant homotopy groups for the symmetric groups~$\Sigma_k$ for~$k\leq 4$ and for the alternating group~$A_5$. The groups $G=\Sigma_4$ and~$G=A_5$ provide explicit examples of homotopy groups~$\pi_0^G(Sp^n)$ with non-trivial torsion.
\begin{eg}[Symmetric group $\Sigma_2$]\label{s2} For the group~$\Sigma_2$ we have $I_2(\Sigma_2) = I(\Sigma_2)$, freely generated by the class $t_2$, i.e., the filtration terminates at the second step. Hence the group~$\pi_0^{\Sigma_2}({\mathbb S})$ is free of rank~2, while the groups $\pi_0^{\Sigma_2}(Sp^n)$ are free of rank~1 for all $n\geq 2$. \end{eg}
\begin{eg}[Symmetric group $\Sigma_3$]\label{s3} The group~$\Sigma_3$ has four conjugacy classes of subgroups with representatives $e, \Sigma_2, A_3$ and~$\Sigma_3$. So the augmentation ideal $I(\Sigma_3)$ is free of rank~3, and a basis is given by the classes \[ t_3 \ = \ t_{\Sigma_2}^{\Sigma_3} \ , \quad p^*(t_2)\ = \ t_{A_3}^{\Sigma_3}
\text{\qquad and\qquad} \tr_{\Sigma_2}^{\Sigma_3}(t_2) \ = \ 2\cdot t_{\Sigma_2}^{\Sigma_3}\ - \ t_e^{\Sigma_3} \ ,\] where $p:\Sigma_3\longrightarrow\Sigma_2$ is the unique epimorphism. Hence $I_2(\Sigma_3)$ is freely generated by the classes $p^*(t_2)$ and~$\tr_{\Sigma_2}^{\Sigma_3}(t_2)$, and $I_3(\Sigma_3) = I(\Sigma_3)$, i.e., the filtration stabilizes at the third step. Theorem~\ref{thm-pi_0 Sp^n} lets us conclude that the homotopy group~$\pi_0^{\Sigma_3}(Sp^n)$ is free for every~$n\geq 1$, and has rank~4 for $n=1$, rank~2 for $n=2$, and rank~1 for $n\geq 3$. \end{eg}
\begin{eg}[Symmetric group $\Sigma_4$]\label{eg:S4} The group~$\Sigma_4$ has 11 conjugacy classes of subgroups, displayed below; the left column lists the order of a subgroup, and lines denote subconjugacy: \[\xymatrix@R=1mm{ 24 & &\Sigma_4\ar@{-}[dddl]\ar@{-}[d]\ar@{-}[ddr]\\ 12 & &A_4\ar@{-}[ddd]\ar@{-}[ddddl]\\
8 & &&\Sigma_2\wr\Sigma_2\ar@{-}[ddr]\ar@{-}[dd]\ar@{-}[ddl]\\
6 & \Sigma_3\ar@{-}[dd]\ar@/_2pc/@{-}[ddd]\\
4 & & V_4\ar@{-}[ddr]|(.36)\hole & \Sigma_2\times\Sigma_2\ar@{-}[ddll]\ar@{-}[dd] & C_4 \ar@{-}[ddl]\\
3 & A_3\ar@{-}[ddr]|(.37)\hole\\
2 & \Sigma_2 \ar@{-}[rd] && (12)(34)\ar@{-}[dl]\\
1 & & e }\] The augmentation ideal $I(\Sigma_4)$ is free of rank~10, and the classes \begin{align}\label{eq:generators I_2(Sigma_4)}
t_e^{\Sigma_2}\ , &\quad t_{\Sigma_2}^{\Sigma_2\times\Sigma_2} \ , \quad t_{(12)(34)}^{V_4} \ , \quad t_{A_3}^{\Sigma_3} \ , \quad t_{V_4}^{\Sigma_2\wr\Sigma_2} \ , \quad t_{\Sigma_2\times\Sigma_2}^{\Sigma_2\wr\Sigma_2} \ , \quad t_{C_4}^{\Sigma_2\wr\Sigma_2} \ , \quad t_{A_4}^{\Sigma_4} \end{align} together with the two classes \[ t_{\Sigma_2\wr\Sigma_2}^{\Sigma_4} \text{\qquad and\qquad}
t_{\Sigma_3}^{\Sigma_4}\ = \ t_4 \] form a basis of $I(\Sigma_4)$.
The group $I_2(\Sigma_4)$ is generated by the classes~$t_K^H$ as $(H,K)$ runs over all pairs of nested subgroups with $[H:K]=2$. All classes of this particular form are linear combinations of the eight classes~\eqref{eq:generators I_2(Sigma_4)}: \begin{align*} t_{(12)(34)}^{\Sigma_2\times\Sigma_2} \ &= \ t_{(12)(34)}^{V_4}\ +\ 2\cdot t_{V_4}^{\Sigma_2\wr\Sigma_2} \ -\ 2\cdot t_{\Sigma_2\times\Sigma_2}^{\Sigma_2\wr\Sigma_2} \\
t_{(12)(34)}^{C_4} \ &= \ t_{(12)(34)}^{V_4}\ +\ 2\cdot t_{V_4}^{\Sigma_2\wr\Sigma_2} \ -\ 2\cdot t_{C_4}^{\Sigma_2\wr\Sigma_2}\\
t_e^{(12)(34)} \ &= \ \quad t_e^{\Sigma_2}\quad +\ 2\cdot t_{\Sigma_2}^{\Sigma_2\times\Sigma_2} \ -\ 2\cdot t_{(12)(34)}^{\Sigma_2\times\Sigma_2} \end{align*} So the eight classes~\eqref{eq:generators I_2(Sigma_4)} form a basis of $I_2(\Sigma_4)$.
The group $I_3(\Sigma_4)$ is generated by the classes $t_K^H$ for all nested subgroup pairs with $[H:K]\leq 3$. We observe that \begin{equation} \label{eq:honest_relation} 3\cdot t_4 \ = \ 3\cdot t_{\Sigma_3}^{\Sigma_4} \ = \ t_{\Sigma_2}^{\Sigma_2\times\Sigma_2} \ + \ 2\cdot t_{\Sigma_2\times\Sigma_2}^{\Sigma_2\wr\Sigma_2} \ + \ 4\cdot t_{\Sigma_2\wr\Sigma_2}^{\Sigma_4} - t_{\Sigma_2}^{\Sigma_3} \ \in \ I_3(\Sigma_4)\ ; \end{equation} Proposition~\ref{prop-general inclusion} below explains in which way this relation is an exceptional feature for $n=4$. All classes~$t_K^H$ with~$[H:K]\leq 3$ are linear combinations of the classes~\eqref{eq:generators I_2(Sigma_4)} and the two classes \begin{equation}\label{eq:generators I_3(Sigma_4)}
t_{\Sigma_2\wr\Sigma_2}^{\Sigma_4} \text{\qquad and\qquad} 3\cdot t_{\Sigma_3}^{\Sigma_4}\ . \end{equation} Indeed: \begin{align*}
t_{V_4}^{A_4} \ &= \ t_{V_4}^{\Sigma_2\wr\Sigma_2}\ +\ 2\cdot t_{\Sigma_2\wr\Sigma_2}^{\Sigma_4} \ -\ 3\cdot t_{A_4}^{\Sigma_4} \\
t_{\Sigma_2}^{\Sigma_3} \ &= \ t_{\Sigma_2}^{\Sigma_2\times\Sigma_2}
+\ 2\cdot t_{\Sigma_2\times\Sigma_2}^{\Sigma_2\wr\Sigma_2}
+\ 4\cdot t_{\Sigma_2\wr\Sigma_2}^{\Sigma_4}\ -\ 3\cdot t_{\Sigma_3}^{\Sigma_4} \\ t_e^{A_3}\ &= \quad t_e^{\Sigma_2}\quad + \ 2 \cdot t_{\Sigma_2}^{\Sigma_3} \quad - \ 3 \cdot t_{A_3}^{\Sigma_3} \end{align*} Since the eight elements~\eqref{eq:generators I_2(Sigma_4)} and the two elements~\eqref{eq:generators I_3(Sigma_4)} together are linearly independent, they form a basis of $I_3(\Sigma_4)$.
Since $[\Sigma_4:\Sigma_3]=4$, the last basis element $t_4=t_{\Sigma_3}^{\Sigma_4}$ belongs to $I_4(\Sigma_4)$, and thus $I_4(\Sigma_4)=I(\Sigma_4)$. The relation~\eqref{eq:honest_relation} shows that $I_3(\Sigma_4)$ has index~3 in $I_4(\Sigma_4)=I(\Sigma_4)$. Altogether, Theorem~\ref{thm-pi_0 Sp^n} lets us conclude: \begin{itemize} \item the group $\pi_0^{\Sigma_4}({\mathbb S})=\pi_0^{\Sigma_4}(Sp^1)$ is free of rank~11, \item the group $\pi_0^{\Sigma_4}(Sp^2)$ is free of rank~3, \item the group~$\pi_0^{\Sigma_4}(Sp^3)$ has rank~1 and its torsion subgroup has order~3, and \item for all $n\geq 4$, the group~$\pi_0^{\Sigma_4}(Sp^n)$ is free of rank~1. \end{itemize} \end{eg}
\begin{eg}[Alternating group $A_5$]\label{eg-A5} The last example that we treat in detail is the alternating group~$A_5$. The point is not just to have another explicit example, but we also need the calculation of $I_5(A_5)$ in Example~\ref{eg:S5} for identifying when the filtration of $\Sigma_5$ stabilizes. The group $A_5$ has~9 conjugacy classes of subgroups: \[\xymatrix@R=1mm{
60 && A_5 \ar@{-}[ddr]\ar@{-}[dddl]\ar@{-}[d]\\
12 && A_4\ar@{-}[dddd]\ar@{-}[dddddl]|(.67)\hole & \\
10 &&& D_5 \ar@{-}[dd]\ar@{-}[dddddl]\\
6 & \tilde\Sigma_3\ar@{-}[ddd]\ar@{-}[ddddr] && \\
5 & & & C_5\ar@/^1pc/@{-}[ddddl] \\
4 & & V_4 \ar@{-}[dd] & & \\
3 & A_3\ar@{-}[ddr]\\
2 && (12)(34)\ar@{-}[d]\\
1 && e }\] The group $\tilde\Sigma_3$ is generated by the elements $(1 2 3)$ and $(1 2)(4 5)$ and is isomorphic to~$\Sigma_3$ (but not conjugate in $\Sigma_5$ to the `standard' copy of $\Sigma_3$~ generated by~$(1 2 3)$ and $(1 2)$). The dihedral group $D_5$ is generated by the elements $(12345)$ and $(2 5)(3 4)$.
The augmentation ideal $I(A_5)$ is free of rank~8, and a convenient basis for our purposes is given by the classes \begin{align}\label{eq:basis of I(A_5)}
t_e^{(12)(34)}\ , &\quad t_{(12)(34)}^{V_4} \ , \quad t_{A_3}^{\tilde\Sigma_3} \ , \quad t_{C_5}^{D_5} \ , \quad t_{V_4}^{A_4} \ , \quad t_{(12)(34)}^{\tilde\Sigma_3}+ t_{A_3}^{A_4}\ , \quad t_{A_4}^{A_5} \text{\quad and\quad}
t_{D_5}^{A_5}\ . \end{align} Proposition~\ref{prop-describe I_n} says that the group $I_2(A_5)$ is generated by the classes~$t_K^H$ as $(H,K)$ runs over all pairs of nested subgroups with $[H:K]=2$, i.e, by the first four classes of the basis~\eqref{eq:basis of I(A_5)}. So these four classes form a basis of $I_2(A_5)$.
We observe that \begin{align}\label{eq-relation in I_3(A_5)} 3\cdot ( t_{(12)(34)}^{\tilde\Sigma_3} + t_{A_3}^{A_4})\ &= \ 2\cdot t_{(12)(34)}^{V_4}\ +\ 4\cdot t_{V_4}^{A_4} \ +\ 3\cdot t_{A_3}^{\tilde\Sigma_3}\ + \ t_{(12)(34)}^{\tilde\Sigma_3} \ \in \ I_3(A_5)\ . \end{align} The group $I_3(A_5)$ is generated by the classes $t_K^H$ for all nested subgroup pairs with $[H:K]\leq 3$. Because \begin{align*}
t_{(12)(34)}^{\tilde\Sigma_3} \ &= \ 3\cdot ( t_{(12)(34)}^{\tilde\Sigma_3} + t_{A_3}^{A_4})\ - \ 2\cdot t_{(12)(34)}^{V_4}\ -\ 3\cdot t_{A_3}^{\tilde\Sigma_3}\ -\ 4\cdot t_{V_4}^{A_4} \\
t_e^{A_3} \quad &= \
t_e^{(12)(34)}\ - \ 3\cdot t_{A_3}^{\tilde\Sigma_3}\ +\ 2\cdot t_{(12)(34)}^{\tilde\Sigma_3} \end{align*} all such classes are linear combinations of the~6 classes \[ t_e^{(12)(34)}\ , \quad t_{(12)(34)}^{V_4} \ , \quad t_{A_3}^{\tilde\Sigma_3} \ , \quad t_{C_5}^{D_5} \ , \quad t_{V_4}^{A_4} \text{\qquad and\qquad} 3\cdot ( t_{(12)(34)}^{\tilde\Sigma_3} + t_{A_3}^{A_4}) \ .\] Since these classes are linearly independent, they form a basis of the group~$I_3(A_5)$.
The group $I_4(A_5)$ is generated by the classes $t_K^H$ for all nested subgroups with $[H:K]\leq 4$. There is only one new generator, the class~$t_{A_3}^{A_4}$; since $t_{(12)(34)}^{\tilde\Sigma_3}\in I_3(A_5)\subset I_4(A_5)$, the group $I_4(A_5)$ is freely generated by the first six elements of the basis~\eqref{eq:basis of I(A_5)}. Because $3\cdot t_{A_3}^{A_4}\in I_3(A_5)$, the group $I_3(A_5)$ has index~3 in~$I_4(A_5)$.
The group $I_5(A_5)$ is generated by the classes $t_K^H$ for all nested subgroup pairs with $[H:K]\leq 5$. In particular, $I_5(A_5)$ contains the seventh element~$t_{A_4}^{A_5}$ of the basis~\eqref{eq:basis of I(A_5)}. We observe that \begin{equation}\label{eq:5D}
5\cdot t_{D_5}^{A_5}\ = \ t_{(12)(34)}^{V_4}\ +\ 2\cdot t_{V_4}^{A_4}\ +\ 6 \cdot t_{A_4}^{A_5}\ -\ t_{(12)(34)}^{D_5} \ \in\ I_5(A_5)\ . \end{equation} All classes of the form $t_K^H$ with~$[H:K]\leq 5$ are linear combinations of the first seven classes of the basis~\eqref{eq:basis of I(A_5)} and the class~\eqref{eq:5D}: \begin{align*}
t_{(12)(34)}^{D_5}\ &= \ t_{(12)(34)}^{V_4}\ +\ 2\cdot t_{V_4}^{A_4}\ +\ 6 \cdot t_{A_4}^{A_5}\ -\
5\cdot t_{D_5}^{A_5}\\ t_e^{C_5}\quad &= \ t_e^{(12)(34)}\ + \ 2\cdot t_{(12)(34)}^{V_4}\ - \ 5\cdot t_{C_5}^{D_5} \ + \ 4\cdot t_{V_4}^{A_4}\ + \ 12 \cdot t_{A_4}^{A_5} \ - \ 2\cdot (5\cdot t_{D_5}^{A_5}) \ . \end{align*} The group $I_5(A_5)$ is thus generated by the linearly independent classes \[ t_e^{(12)(34)}\ , \quad t_{(12)(34)}^{V_4} \ , \quad t_{A_3}^{\tilde\Sigma_3} \ , \quad t_{C_5}^{D_5} \ , \quad t_{V_4}^{A_4} \ , \quad t_{(12)(34)}^{\tilde\Sigma_3} + t_{A_3}^{A_4}\ , \quad t_{A_4}^{A_5} \text{\qquad and\qquad} 5\cdot t_{D_5}^{A_5}\ . \] So $I_5(A_5)$ has full rank~8, but index~5 in the augmentation ideal~$I(A_5)$. Since $[A_5:D_5]=6$, the last basis element $ t_{D_5}^{A_5}$ belongs to~$I_6(A_5)$, and we conclude that $I_6(A_5)=I(A_5)$ is the full augmentation ideal. Altogether, Theorem~\ref{thm-pi_0 Sp^n} lets us conclude that \begin{itemize} \item the group $\pi_0^{A_5}({\mathbb S})=\pi_0^{A_5}(Sp^1)$ is free of rank~9, \item the group $\pi_0^{A_5}(Sp^2)$ is free of rank~5, \item the group~$\pi_0^{A_5}(Sp^3)$ has rank~3 and its torsion subgroup has order~3, \item the group~$\pi_0^{A_5}(Sp^4)$ is free of rank~3, \item the group~$\pi_0^{A_5}(Sp^5)$ has rank~1 and its torsion subgroup has order~5, and \item for all $n\geq 6$, the group~$\pi_0^{A_5}(Sp^n)$ is free of rank~1. \end{itemize} \end{eg}
\begin{eg}[Symmetric group $\Sigma_5$]\label{eg:S5} We refrain from a complete calculation of the groups~$\pi_0^{\Sigma_5}(Sp^n)$, but we work out where the filtration for $\Sigma_5$ stabilizes. The previous examples could be mistaken as evidence that the group $I_n(\Sigma_n)$ coincides with the full augmentation ideal $I(\Sigma_n)$ for every $n$; equivalently, one could get the false impression that the group $\pi_0^{\Sigma_n}(Sp^n)$ is always free of rank~1. While this is true for $n\leq 4$, we will now see that it fails for $n=5$, i.e., that $I_5(\Sigma_5)$ is strictly smaller than~$I(\Sigma_5)$.
We let~$B$ denote the subgroup of~$\Sigma_5$ generated by the elements $(12345)$ and $(2 3 5 4)$; this group has order~20 and is isomorphic to the semi-direct product ${\mathbb F}_5\rtimes ({\mathbb F}_5)^\times$, the affine linear group of the field~${\mathbb F}_5$. The intersection of~$B$ with the alternating group~$A_5$ is the dihedral group $D_5$. The double coset formula thus gives \begin{align*}
\res^{\Sigma_5}_{A_5}(t_B^{\Sigma_5}) \ &= \ 6\ - \ \res^{\Sigma_5}_{A_5}(\tr_B^{\Sigma_5}(1)) \ = \ 6\ - \ \tr_{D_5}^{A_5}(\res^B_{D_5}(1)) \ = \ t_{D_5}^{A_5}\ . \end{align*} We showed in the previous Example~\ref{eg-A5} that the class $t_{D_5}^{A_5}$ does {\em not} belong to $I_5(A_5)$. Since~$I_5$ is closed under restriction maps, the class $t_B^{\Sigma_5}$ does {\em not} belong to $I_5(\Sigma_5)$, and hence $I_5(\Sigma_5)\ne I(\Sigma_5)$.
Every subgroup~$H$ of~$\Sigma_5$ admits a nested sequence of subgroups \[ H \ = \ H_0 \ \subset \ H_1 \ \subset \ \dots \ \subset \ H_k \] with $[H_i:H_{i-1}]\leq 6$ for all $i=1,\dots,k$ and such that the last group $H_k$ is either the full group $\Sigma_5$ or conjugate to the maximal subgroup $\Sigma_3\times\Sigma_2$ of index~10. The relation \begin{align*} t^{\Sigma_5}_{\Sigma_3\times\Sigma_2}\ &= \ t^{\Sigma_3\times\Sigma_2}_{\Sigma_3}\ - \ t^{\Sigma_3\times\Sigma_2}_{\Sigma_2\times\Sigma_2}\ - \ t^{\Sigma_4}_{\Sigma_3}\ + \ t^{\Sigma_2\wr\Sigma_2}_{\Sigma_2\times\Sigma_2}\ + \ 2\cdot t^{\Sigma_4}_{\Sigma_2\wr\Sigma_2}\ + \ 2\cdot t^{\Sigma_5}_{\Sigma_4} \end{align*} shows that the class $t^{\Sigma_5}_{\Sigma_3\times\Sigma_2}$ lies in $I_5(\Sigma_5)$. So $t_H^{\Sigma_n}$ belongs to~$I_6(\Sigma_5)$ for every subgroup~$H$ of~$\Sigma_n$, and hence $I_6(\Sigma_5) =I(\Sigma_5)$. \end{eg}
We pause to point out a curious phenomenon that happens only for $n=4$. This exceptional behavior can be traced back to the fact that the alternating group~$A_4$ has a subgroup of `unusually small index' (the Klein group~$V_4$ of index~3), compare the proof of the next proposition. We have seen in~\eqref{eq:honest_relation} that $3\cdot t_4$ lies in $I_3(\Sigma_4)$. Since the class $t_4$ generates the global functor $I_4$, this implies that \[ 3\cdot I_4 \ \subset \ I_3 \ \subset \ I_4 \ . \] So after inverting~3, the inclusion $I_3 \longrightarrow I_4$ and the epimorphism of global functors \[ {\underline \pi}_0(Sp^3)\ \longrightarrow \ {\underline \pi}_0(Sp^4) \] induced by the inclusion $Sp^3\longrightarrow Sp^4$ both become isomorphisms. However:
\begin{prop}\label{prop-general inclusion} For every $n\geq 2$ with $n\ne 4$, the inclusion $I_{n-1}\longrightarrow I_n$ is not a rational isomorphism. \end{prop} \begin{proof} Example~\ref{s2} shows that no non-zero multiple of the class~$t_2$ belongs to~$I_1(\Sigma_2)$. Example~\ref{s3} shows that no non-zero multiple of~$t_3$ belongs to~$I_2(\Sigma_3)$. So we assume $n\geq 5$ for the rest of the argument. We recall that the alternating group~$A_n$ has no proper subgroup~$H$ of index less than~$n$. Indeed, the left translation action on $A_n/H$ provides a homomorphism~$\rho:A_n\longrightarrow\Sigma(A_n/H)$ to the symmetric group of the underlying set of~$A_n/H$. For $[A_n:H]<n$, the order of $\Sigma(A_n/H)$ is strictly less than the order of~$A_n$. So the homomorphism $\rho$ has a non-trivial kernel. Since the group $A_n$ is simple, $\rho$ must be trivial, which forces~$H=A_n$.
Now we prove the proposition. The class $t_{A_{n-1}}^{A_n}=\res^{\Sigma_n}_{A_n}(t_n)$ belongs to $I_n(A_n)$, but for~$n> 4$ no non-zero multiple of it belongs to $I_{n-1}(A_n)$. Indeed, otherwise Proposition~\ref{prop-describe I_n} would allow us to write \begin{align*}
k\cdot t_{A_{n-1}}^{A_n}\ &= \ \lambda_1\cdot t_{K_1}^{H_1} +\dots +\lambda_m\cdot t_{K_m}^{H_m} \end{align*} in~${\mathbb A}(A_n)$, for certain integers $k,\lambda_1,\dots,\lambda_m$ and nested subgroup pairs with $1< [H_i:K_i]< n$. Since~$A_n$ has no proper subgroup of index less than~$n$, the groups $H_1,\dots,H_m$ must all be different from the full group~$A_n$. We expand both sides in terms of the basis of~${\mathbb A}(A_n)$ given by the classes $\tr_H^{A_n}(1)$ (for $H$ running through the conjugacy classes of subgroups). On the right hand side the coefficient of the basis element $\tr_{A_n}^{A_n}(1)=1$ is zero, whereas the coefficient on the left hand side is~$k n$. So we must have~$k=0$. \end{proof}
We conclude by looking more closely at the limit case, the infinite symmetric product spectrum. We remark without proof that, generalizing the non-equivariant situation, the orthogonal spectrum $Sp^\infty$ is globally equivalent to the orthogonal spectrum $H{\mathbb Z}$ defined by \[ (H{\mathbb Z})(V) \ = \ {\mathbb Z}[S^V] \ , \] the reduced free abelian group generated by the $n$-sphere. Theorem~\ref{thm-pi_0 Sp^n} shows that \[ {\mathbb A}/I_\infty \ \cong \ {\underline \pi}_0(Sp^\infty) \ ,\] induced by the action of~${\mathbb A}$ on the class $i_*(1)$. For every compact Lie group~$G$, the map \[ \res^G_e \ : \ \pi_0^G(Sp^\infty) \ \longrightarrow \ \pi_0^e(Sp^\infty) \cong{\mathbb Z} \] is a split epimorphism, so the group $\pi_0^G(Sp^\infty)$ is free of rank~1 if and only if $I_\infty(G)=I(G)$.
We can split the group ${\mathbb A}(G)/I_{\infty}(G)$, and hence the group $\pi_0^G(Sp^\infty)$, into summands indexed by conjugacy classes of connected subgroups of~$G$. If $C$ is such a connected subgroup, we denote by ${\mathbb A}(G;C)$ the subgroup of the Burnside ring~${\mathbb A}(G)$ that is generated by the transfers $\tr_H^G(1)$ for all subgroups~$H$ with~$C=H^\circ$, the path component of the identity of~$H$ (or equivalently, $H$ contains $C$ as a finite index subgroup). Then \[ {\mathbb A}(G)\ = \ \bigoplus_{(C)} \, {\mathbb A}(G;C)\] where the sum runs over conjugacy classes of connected subgroups of~$G$. Proposition~\ref{prop-describe I_n} shows that~$I_\infty(G)$ is generated as an abelian group by the classes \[ t^H_K\ = \ [H:K]\cdot \tr_H^G(1) - \tr_K^G(1) \ \in \ {\mathbb A}(G)\] as $(H,K)$ runs through all pairs of nested closed subgroups such that $K$ has finite index in~$H$, and $H$ has finite Weyl group in $G$. Then~$K$ and~$H$ have the same connected component of the identity, i.e., $K^\circ=H^\circ$, so the relation $t^H_K$ belongs to the direct summand ${\mathbb A}(G;K^\circ)$. Hence \[ \pi_0^G(Sp^\infty)\ \cong \ {\mathbb A}(G)/I_\infty(G)\ = \ \bigoplus_{(C)}\,\left( {\mathbb A}(G;C)/I_\infty(G;C)\right) \ ,\] where $I_\infty(G;C)$ is the subgroup of ${\mathbb A}(G;C)$ generated by the classes $t^H_K$ with $H^\circ=K^\circ=C$. The summands behave quite differently according to whether $C$ has infinite or finite Weyl group: \begin{itemize} \item If~$C$ has an infinite Weyl group, then for every subgroup $H\leq G$ with $H^\circ=C$ the class $[H:C]\cdot \tr_H^G(1)$ belongs to~$I_\infty(G;C)$. So the class $\tr_H^G(1)$ becomes torsion in the quotient group ${\mathbb A}(G;C)/I_\infty(G;C)$, which is thus a torsion group.
\item If~$C$ has finite Weyl group, and~$H\leq G$ satisfies $H^\circ=C$, then the relations \[ C = H^\circ \ \leq\ H\ \leq \ N_G H \ \leq \ N_G C \] show that~$H$ has finite Weyl group and finite index in~$N_G C$. So \[ t_H^{N_G C} \ = \ [N_G C:H]\cdot \tr_{N_G C}^G(1)\ - \ \tr_H^G(1)\ \in \ I_\infty(G;C)\] and in the quotient group ${\mathbb A}(G;C)/I_\infty(G;C)$, the class $\tr_H^G(1)$ becomes a multiple of the class~$\tr_{N_G C}^G(1)$. Hence the group ${\mathbb A}(G;C)/I_\infty(G;C)$ is free of rank~1, generated by~$\tr_{N_G C}^G(1)$.
In the situation at hand, the subgroup~$C$ can be recovered as the identity component of its normalizer. A compact Lie group has only finitely many conjugacy classes of subgroups that are normalizers of connected subgroups, see~\cite[VII Lemma 3.2]{borel:bredon}. So there are only finitely many conjugacy classes of connected subgroups with finite Weyl group. \end{itemize}
So altogether we conclude that the group $\pi_0^G(Sp^\infty)$ is a direct sum of a torsion group and a free abelian group of finite rank. In particular, the rationalization ${\mathbb Q}\otimes \pi_0^G(Sp^\infty)$ is a finite dimensional~${\mathbb Q}$-vector space with basis consisting of the classes $\tr_C^G(1)$ as $C$ runs through the conjugacy classes of connected subgroups of~$G$ with finite Weyl group. Unfortunately, the author does not know an example when the torsion subgroup of $\pi_0^G(Sp^\infty)$ is non-trivial.
\begin{eg}\label{eq-SU(2)} If every subgroup~$H$ with finite Weyl group also has finite index in~$G$, then $I_\infty(G)=I(G)$ and~$\pi_0^G(Sp^\infty)$ is free of rank~1. This holds, for example, when $G$ is finite or a torus.
An example for which $\pi_0^G( Sp^{\infty})$ has rank bigger than~1 is $G=S U(2)$. Here there are three conjugacy classes of connected subgroups: the trivial subgroup, the conjugacy class of the maximal tori and the full group $S U(2)$. Among these, the maximal tori and $S U(2)$ have finite Weyl groups, so the classes~$1$ and~$\tr_N^{S U(2)}(1)$ are a ${\mathbb Z}$-basis for $\pi_0^{S U(2)}( Sp^\infty )$ modulo torsion, where $N$ is a maximal torus normalizer. \end{eg}
\end{document} | arXiv |
Linearity Properties of Lebesgue Measurable Functions
Recall from the Lebesgue Measurable Functions page that an extended real-valued function $f$ on a Lebesgue measurable domain $D(f)$ is a Lebesgue measurable function if for all $\alpha \in \mathbb{R}$ the set $\{ x \in D(f) : f(x) < \alpha \}$ is a Lebesgue measurable set.
We now show that the collection of Lebesgue measurable functions has somewhat of a linear structure to it. That is, if $f$ and $g$ are Lebesgue measurable functions with common domain then $f + g$ is a Lebesgue measurable function, and for any $c \in \mathbb{R}$ we have that $cf$ is a Lebesgue measurable function.
Theorem 1 (Linearity of Lebesgue Measurable Functions): Let $f$ and $g$ be Lebesgue measurable functions on the same domain ($D(f) = D(g)$) and let $c \in \mathbb{R}$. Then:
a) $f + g$ is a Lebesgue measurable function.
b) $cf$ is a Lebesgue measurable function.
Proof of a): Let $\alpha \in \mathbb{R}$. Consider the following set:
\begin{align} \quad \{ x \in D(f + g) : f(x) + g(x) < \alpha \} = \{ x \in D(f + g) : f(x) < \alpha - g(x) \} \end{align}
Consider the inquality $f(x) < \alpha - g(x)$. By the density of the rational numbers there must exist a rational number $r_n \in \mathbb{Q}$ such that $f(x) < r_n < \alpha - g(x)$. Let $\{ r_n \}_{n=1}^{\infty} = \mathbb{Q}$ be an enumeration of the rational numbers. The set above can be rewritten as the following union
\begin{align} \quad \{ x \in D(f + g) : f(x) + g(x) < \alpha \} &= \bigcup_{n=1}^{\infty} \{ x \in D(f + g) : f(x) < r_n < \alpha - g(x) \}\\ &= \bigcup_{n=1}^{\infty} \left [ \{ x \in f(x) : f(x) < r_n \} \cap \{ x \in g(x) : r_n < \alpha - g(x) \} \right ] \\ &= \bigcup_{n=1}^{\infty} \left [ \underbrace{\{ x \in f(x) : f(x) < r_n \}}_{\mathrm{Lebesgue \: measurable}} \cap \underbrace{\{ x \in g(x) : g(x) < \alpha - r_n \}}_{\mathrm{Lebesgue \: measurable}} \right ] \end{align}
The intersection of Lebesgue measurable sets is Lebesgue measurable and a countable union of Lebesgue measurable sets is Lebesgue measurable since the collection of Lebesgue measurable sets is a $\sigma$-algebra. Therefore $\{ x \in D(f + g) : f(x) + g(x) < \alpha \}$ is a Lebesgue measurable set. So $f + g$ is a Lebesgue measurable function. $\blacksquare$
Proof of b): Let $\alpha \in \mathbb{R}$. There are three cases to consider.
Case 1: If $c > 0$ then $\displaystyle{\{ x \in D(f) : cf(x) < \alpha \} = \left \{ x \in D(f) : f(x) < \frac{\alpha}{c} \right \}}$. So $\{ x \in D(f) : cf(x) < \alpha \}$ is a Lebesgue measurable set.
Case 2: If $c < 0$ then $\displaystyle{\{ x \in D(f) : cf(x) < \alpha \} = \left \{ x \in D(f) : f(x) > \frac{\alpha}{c} \right \}}$. So $\{ x \in D(f) : cf(x) < \alpha \}$ is a Lebesgue measurable set.
Case 3: If $c = 0$ then $\displaystyle{\{ x \in D(f) : cf(x) < \alpha \} = \{ x \in D(f) : 0 < \alpha \}}$. If $\alpha > 0$ then this set is $D(f)$ which is Lebesgue measurable. If $\alpha \leq 0$ then this set is $\emptyset$ which is Lebesgue measurable.
In all three cases we see that $\{ x \in D(f) : cf(x) < \alpha \}$ is a Lebesgue measurable set. So $cf$ is a Lebesgue measurable function. $\blacksquare$ | CommonCrawl |
\begin{document}
\begin{abstract} In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of $C^r$-residual diffeomorphisms on three-dimensional manifolds ($r\ge 1$). In the case of the $C^1$-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism $f$ have simple spectrum $C^1$-robustly (in which case $f$ has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of $f$ are continuous in the weak$^*$-topology) or it can be $C^1$-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures. \end{abstract}
\keywords{Uniform hyperbolicity; periodic points; finest dominated splitting; Oseledets splitting; Lyapunov exponents} \date{\today} \maketitle
\section{Introduction}
A global view of dynamical systems has been one of the leading problems considered by the dynamical systems community. Based on the pioneering works of Peixoto and Smale, a conjecture proposed by Palis in the nineties has constituted a route guide for a global description of the space of dynamical systems. This program, that roughly describes complement of uniform hyperbolicity as the space of diffeomorphisms that are approximated by those exhibiting either homoclinic tangencies or heteroclinic cycles, has been completed with much success in the $C^1$-topology, where perturbation tools like the Pugh closing lemma, Franks' lemma, Hayashi's connecting lemma or Ma\~n\'e ergodic closing lemma developed for the characterization of structural stability are available (see e.g.~\cite{Ma1, Hay, Palis} and references therein).
Although the uniform geometric structures of invariant manifolds present in hyperbolic diffeomorphisms are a basic ingredient in achieving several core results and are quite well established, some important questions on the regularity of finer dynamical properties still remain to be answer.
Motivated by the analysis of the regularity of the Lyapunov exponents similarly to the proof of the stability conjecture, periodic orbits and their eigenvalues should play a key role. We say that a periodic point $p$ of period $\pi(p) \ge 1$ for a diffeomorphism $f$ has \emph{simple spectrum} if all the eigenvalues of $Df^{\pi(p)}(p)$ are real and distinct. Since all periodic points for an Axiom A diffeomorphism, in the same basic piece, have the same index and are homoclinically related, then we will attribute them a signature, which consists of an ordered list of the dimensions of their finest dominated splittings. Moreover, we will say that two periodic points $p,q$ have different signatures if the Oseledets splittings for $Df^{\pi(p)}(p)$ and $Df^{\pi(q)}(q)$, which consist of the finest dominated splittings at the points, are distinct. Our purpose in this paper is to revisit uniformly hyperbolic basic and explore the notion of domination of Oseledets splittings and characterize the continuity points of Lyapunov exponent functions in this context. These results lie ultimately in the analysis of the eigenvalues and the finest dominated splitting over the periodic points.
In the first part of the paper (see Theorem~\ref{teo1}) we prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of $C^r$-residual diffeomorphisms on three-dimensional manifolds ($r\ge 1$). In particular, any diffeomorphism with a hyperbolic basic piece can be $C^r$-approximated by a diffeomorphism with a dense set of periodic points in the continuation of the hyperbolic set. The proof of this result explores the hyperbolic structure of hyperbolic basic pieces, hence it does not rely on the classical $C^1$-perturbation lemmas.
In the case of the $C^1$-topology, one can describe the mechanisms to deduce simple Lyapunov spectrum for all invariant measures assuming the same property at periodic points, in the spirit of the previous works ~\cite{BGV,Cao,Ca,ST}. On the one hand we prove that, if all periodic points of a hyperbolic basic piece for a diffeomorphism $f$ have simple spectrum and the same property holds in a $C^1$ neighborhood of $f$ (c.f. Definition~\ref{def:simplestar}) then $f$ has a finest dominated splitting into one-dimensional sub-bundles (see Theorem~\ref{thm:1FDS}). In this case all Lyapunov exponent functions of $f$ are continuous in the weak$^*$-topology.
On the other hand, if periodic points miss to have simple spectrum robustly, then, by a $C^1$-arbitrarily small perturbation, one can create an equidimensional cycle associated to a pair of periodic points which robustly exhibit different signatures (see Theorem~\ref{thm:dichotomy}).
Finally, we justify that these equidimensional cycles associated to periodic points with different signatures play a similar role to the one of tangencies and heterodimensional cycles by its instability character. In fact, the existence of homoclinic tangencies or heterodimensional cycles is often associated to the so-called Newhouse phenomenon of persistence of infinitely many sources or sinks (see e.g.~\cite{Ne} and references therein). In this setting, the existence of a equidimensional cycle associated to periodic points with different signatures can be used to generate (by perturbation) infinitely many periodic points with any of the signatures of the generating periodic orbits (see Theorem~\ref{thm:explo}). Using the existence of Markov partitions we deduce that similar results hold in the time-continuous setting (see Corollary~\ref{cor:flows}).
\section{Preliminaries and statement of the main results}
\subsection{Hyperbolic, Oseledets and finest dominated splittings}
Let $M$ be a compact Riemannian manifold $M$ and $f\in\text{Diff}^{\, 1}(M)$. Let ${\text{Per}(f)}$ denote the set of periodic points for $f$ and $\Omega(f) \subset M$ denote the non-wandering set of $f$. Given an $f$-invariant compact set $\Lambda\subseteq M$ we say that $\Lambda$ is a \emph{uniformly hyperbolic set} if there is a $Df$-invariant splitting $T_{\Lambda} M = E^s \oplus E^u$ and constants $C>0$ and $\lambda\in (0,1)$ so that $$
\| Df^n(x)\mid_{E^s_x} \| \le C \lambda^n
\quad\text{and}\quad
\| (Df^n(x)\mid_{E^u_x})^{-1} \| \le C \lambda^n $$ for every $x\in \Lambda$ and $n\ge 1$. We can always change the metric in order to obtain $C=1$. We refer to $T_{\Lambda} M = E^s \oplus E^u$ as the \emph{hyperbolic splitting} associated to $f$. We say that $\Lambda$ is an \emph{isolated set} if there exists an open neighborhood $U\supset \Lambda$ such that $\Lambda=\cap_{n\in\mathbb{N}}f^n(U)$. Finally, we say that $\Lambda$ is \emph{transitive} if there exists $x\in\Lambda$ such that the $f$-orbit of $x$ is dense in $\Lambda$. Along the paper unless stated otherwise we always assume that $\Lambda$ is a uniformly hyperbolic, isolated and transitive set. For simplicity we call such a set a \emph{hyperbolic basic piece} or simply a \emph{basic piece}. It is well-know that hyperbolic sets admit analytic continuations, that is, there exists a $C^1$-neighborhood
$\mathcal{U}$ of $f$ and an open neighborhood $U$ of $\Lambda$ so that $\Lambda_g:=\cap_{n\in\mathbb{N}}g^n(U)$ is a basic piece for $g$ and $f|_{\Lambda}$ is topologically conjugated to $g|_{\Lambda_g}$. Finally, we say that a hyperbolic basic piece is \emph{non-trivial} if it does not consist of a hyperbolic periodic point.
We say that a $C^1$-diffeomorphism $f$ is \emph{Axiom A} if $\overline{\text{Per}(f)}=\Omega(f)$ and $\Omega(f)$ is a uniformly hyperbolic set. We observe that the non-wandering set of an Axiom A diffeomorphism can be decomposed in a finite number of basic pieces. If $\mu$ is an $f$-invariant probability measure, then it follows from Oseledets' theorem~\cite{O} that for $\mu$-almost every $x$ there exists a decomposition (\emph{Oseledets splitting}) $T_x M= E^1_x \oplus E^2_x\oplus \dots \oplus E_{x}^{k(x)}$ and, for $1\le i\le k(x)$, there are well defined real numbers $$
\lambda_i(f,x)= \lim_{n\to\pm \infty} \frac1n \log \|Df^n(x) v_i\|,
\quad \forall v_i \in E^i_x\setminus \{\vec0\} $$ called the \emph{Lyapunov exponents} associated to $f$ and $x$. It is well known that, if $\mu$ is ergodic, then the Lyapunov exponents are almost everywhere constant and $k(x)=k$ is constant. In this case the Lyapunov exponents are denoted simply by $\lambda_i(f,\mu)$. The \emph{Lyapunov spectrum} of a probability measure is the collection of all its Lyapunov exponents. If $\Lambda$ is a hyperbolic set for $f$ and $\mu$ is an $f$-invariant and ergodic probability measure supported on $\Lambda$, Poincar\'e recurrence theorem implies that $\operatorname{supp}(\mu) \subset \Omega(f)$ and, consequently, the decomposition $T_x M= E^1_x \oplus E^2_x\oplus \dots \oplus E_{x}^{k}$ is finer than $T_xM=E^s_x \oplus E^u_x$. In particular $\mu$ is hyperbolic i.e. has only non-zero Lyapunov exponents.
\begin{definition} Given a $Df$-invariant decomposition $F=E^1\oplus E^2 \subset TM$ over a basic piece $\Lambda$ of a diffeomorphism $f$ we say $E^1$ is dominated by $E^2$ if there exists $C>0$ and $\lambda \in (0,1)$ so that $$
\| Df^n(x) \mid_{E_x^1}\| . \| (Df^n (x)\mid_{E^2_x})^{-1}\| \le C \lambda^n \text{ for every $n\ge 1$ and $x\in \Lambda$.} $$ \end{definition}
It is known that we can change the metric in order to obtain $C=1$. It is clear from the definition that any hyperbolic splitting is necessarily a dominated splitting.
\begin{definition}\label{def:simple} We say that $f$ has a \emph{one-dimensional finest dominated splitting} over a basic piece $\Lambda$ if there exists a continuous $Df$-invariant decomposition $T_x M= E^1_x \oplus E^2_x\oplus \dots \oplus E_{x}^{\dim M}$ in one-dimensional subspaces at every $x\in \Lambda$ such that $E^i_x$ is dominated by $E^{i+1}_x$ for every $i=1 \dots \dim M -1$. \end{definition}
It is clear from the previous definition that having a one-dimensional finest dominated splitting is a $C^1$-open condition for $C^r$-diffeomorphisms $(r\ge 1)$. Since dominated splittings vary continuously with the base point on a basic set $\Lambda$, if the diffeomorphism $f$ has a one-dimensional finest dominated splitting $T_{\Lambda} M= E^1 \oplus \dots \oplus E^{\dim M}$ then there exists a continuous function $$ \Lambda \ni x \mapsto (v_x^1, v_x^2, \dots, v_x^{\dim M}) \in (T_x^1M)^{\dim M} $$ such that $v_x^i\in E^i_x$ for every $1\le i \le \dim M$ and $x\in \Lambda$ and, consequently, the limit superior \begin{align}\label{eq:poitwise}
\overline\lambda_i(f,x) & := \limsup_{n\to\infty} \frac1n \log \|Df^n(x) v_i\| \end{align} is everywhere well defined and it does not depend on the choice of the vector $v_i\in E^i_x \setminus\{0\}$. Defining $S: T^1_\Lambda M \to T^1_\Lambda M$ and $\phi: T^1_\Lambda M \to \mathbb R$ by $$
S(x,v) =\left(f(x), \frac{Df(x) v}{\| Df(x) v\|}\right)
\quad\text{and}\quad
\phi(x,v) = \log \|Df(x) v\| $$ then it follows that each value $\overline\lambda_i(f,x)$ can be written as \begin{align}\label{eq:poitwise2} \overline\lambda_i(f,x)
= \limsup_{n\to\infty} \frac1n \sum_{j=0}^{n-1} \phi( S^j(x, v_i^x)). \end{align} Using ~\eqref{eq:poitwise2}, for any $f$-invariant and ergodic probability measure $\mu$ each of the values $\overline\lambda_i(f,x)$ coincide $\mu$-almost everywhere coincide with the Lyapunov exponents $\lambda_i(f,\mu)$, for every $1\le i \le \dim M$. More generally, for any $f$-invariant probability measure $\mu$ we define their \emph{Lyapunov exponents} by \begin{equation}\label{eq:poitwise2} \lambda_i(f,\mu):= \int \overline \lambda_i(f,x) \, d\mu(x) \end{equation} for every $1\le i \le \dim M$.
\begin{remark} If $\mu$ is an $f$-invariant probability measure, the ergodic decomposition theorem guarantees that $\mu=\int\mu_x\,d\mu(x)$ where each $\mu_x$ is an $f$-invariant and ergodic probability measure defined for $\mu$-almost every $x\in\Lambda $, and $\frac1n\sum_{j=0}^{n-1} \delta_{f^j(x)}$ converges to $\mu_x$ in the weak$^*$-topology. Thus for $\mu$-almost every $x\in\Lambda$ and every $1\le i \le \dim M$, $$ \overline \lambda_i(f,x)=\lambda_i(f,\mu_x) = \int \phi(y,v_i^y) \, d\mu_x(y). $$ Equality in \eqref{eq:poitwise2} can also be written as \begin{equation}\label{eq:poitwise3} \lambda_i(f,\mu)
= \int \overline \lambda_i(f,x) \, d\mu(x)
= \int \lambda_i(f,\mu_x) \,d\mu(x)
= \int \phi(x,v_i^x) \, d\mu(x) \end{equation} for every $1\le i \le \dim M$. \end{remark}
\subsection{Statement of the main results}
\subsubsection{Plenty of periodic orbits with simple spectrum on hyperbolic basic sets of $C^r$-diffeomorphisms}
In this subsection we discuss the simplicity of the Lyapunov spectrum for $C^r$-diffeomorphisms ($r\ge 1$) restricted to hyperbolic basic pieces.
\begin{maintheorem}\label{teo1} Assume $\dim M=3$ and $r\ge 1$. Let $f$ be a $C^r$-diffeomorphism and $\Lambda$ be a non-trivial hyperbolic basic set for $f$ and let $\mathcal{U}$ be a $C^r$-open neighborhood of $f$ so that the hyperbolic continuation $\Lambda_g$ is well defined for all $g\in \mathcal{U}$. Then, there exists a $C^r$-residual subset $\mathcal{R} \subset \mathcal{U}$ such that if $g\in\mathcal{R}$ then the set of periodic orbits with simple spectrum is dense in $\Lambda_g$. \end{maintheorem}
Some comments are in order. In the case that $r=1$ there are some perturbation tools sufficient to guarantee the previous result (e.g. the methods of \cite{BGV} used in the proof of Theorem~\ref{thm:dichotomy}). In the case that $r>1$, where there is a lack of perturbation tools, we use strongly both the robustness of the hyperbolic splitting and the persistence of the hyperbolic basic piece to prove that generic diffeomorphisms have a dense set of periodic orbits with simple spectrum on non-trivial basic pieces. We also observe that the three-dimensional assumption on the manifold is crucial.
\subsubsection{The simple star property and one-dimensional finest dominated splittings}
In this subsection we provide a criterium over the periodic points for a hyperbolic basic piece to admit a one-dimensional finest dominated splitting.
\begin{definition}\label{def:simplestar}
Given a basic piece $\Lambda$ for a diffeomorphism $f$ we say that $f|_{\Lambda}$ is \emph{simple star} if there exists a $C^1$-open neighborhood
$\mathcal U$ of $f$ so that, for any $g\in \mathcal U$ all periodic points for $g|_{\Lambda_g}$ have simple spectrum. \end{definition}
We observe that after \cite{Ma1}, a $C^1$-diffeomorphism is called star if it admits a $C^1$-open neighborhood formed by diffeomorphisms whose periodic points are all hyperbolic. Since the non-wandering set is uniformly hyperbolic then it is clear that the set of Axiom A diffeomorphisms forms a $C^1$-open subset of star diffeomorphisms. It follows from ~\cite{BGV} that periodic points are enough to determine dominated splittings at the closure of periodic points. Since periodic points are dense in basic pieces we obtain the following result.
\begin{maintheorem}\label{thm:1FDS}
If $f|_{\Lambda}$ is simple star, then $f$ admits a one-dimensional finest dominated splitting over $\Lambda$. Moreover, the Lyapunov spectrum is simple for all $f$-invariant ergodic probability measures and the map \begin{equation}\label{LE} \begin{array}{ccc}
\mathscr{L}:\mathcal M_1(f|_{\Lambda}) & \to & \mathbb R^{\dim M} \\
\mu & \mapsto & (\lambda_{1}(f,\mu), \dots, \lambda_{\dim M}(f,\mu)) \end{array} \end{equation}
is continuous, where $\mathcal M_1(f|_\Lambda)$ denotes the space of $f$-invariant probability measures supported in $\Lambda$ and endowed with the weak$^*$ topology. \end{maintheorem}
We observe that if the diffeomorphism $f$ has a one-dimensional finest dominated splitting over $\Lambda$, then every $f$-invariant probability measure supported on $\Lambda$ admits $d=\dim M$ distinct Lyapunov exponents that differ by a definite amount. Thus, the following is a direct consequence of Theorem~\ref{thm:1FDS}.
\begin{maincorollary}\label{tadinho} Let $\Lambda$ be a basic piece of a diffeomorphism $f$. The following properties are equivalent: \begin{itemize}
\item[(a)] $f|_\Lambda$ is simple star;
\item[(b)] $f|_\Lambda$ admits a one-dimensional finest dominated splitting; \item[(c)] there exists a constant $ c>0$ so that any $f$-invariant probability measure $\mu$ supported in $\Lambda$ has $\dim M$ distinct Lyapunov exponents and $\lambda_{i+1}(f,\mu)- \lambda_{i}(f,\mu)\ge c >0$ for every $i=1,\dots, \dim M-1$. \end{itemize} \end{maincorollary}
The later corollary ultimately means that if the Lyapunov spectrum of periodic points simple is $C^1$-robust, then there exists a uniform gap in the Lyapunov spectrum of periodic points.
\subsubsection{A dichotomy for basic pieces}\label{subsec:dichotomy}
In this subsection we obtain a dichotomy of one-dimensional finest dominated splitting \emph{versus} equidimensional cycles with robust different signatures for basic pieces. Let us introduce these notions more precisely.
Let $\Lambda$ be a basic set of a diffeomorphism $f$ and $p\in\Lambda$ be a hyperbolic periodic point of period $\pi(p)\ge 1$. Given the finest $Df^{\pi(p)}(p)$-invariant dominated splitting $E^u_p=E^u_1 \oplus \dots \oplus E^u_k$ we define the \emph{unstable signature at $p$} (and denote it by $\text{sgn}^u(p)$) to be the $k$-uple $(\dim E^u_1, \dots, \dim E^u_k)$. Stable signatures at periodic points are defined analogously and denoted by $\text{sgn}^s$. Unstable (resp. stable) signatures describe the existence of finer splittings in the unstable (resp. stable) bundle of periodic points.
\begin{figure}\label{Fig4}
\end{figure}
\begin{figure}
\caption{Action of $Df^{\pi(p)}(p)\mid_{E^u_p}$ on the projective space in the case of two-dimensional unstable sub-bundle ordered as Figure 1.}
\label{Fig3}
\end{figure}
\begin{definition}\label{def:equidim} We say that a $C^1$ diffeomorphism $f$ exhibits an \emph{equidimensional cycle with robust different signatures} on a basic piece $\Lambda$ if there are two periodic points $p,q\in \Lambda$ with different stable or unstable signatures, and this property holds under arbitrarily $C^1$-small perturbations. \end{definition}
\begin{figure}
\caption{Equidimensional cycle between two fixed points $p_1$ and $p_2$ such that $\text{sgn}^u(p_1)=(1,1)$ and $\text{sgn}^u(p_2)=(2)$.}
\label{fig1}
\end{figure}
In dimension three, the robustness of the different signatures is associated to the existence of a pair of periodic points whose action of the dynamical cocycle on the projective space of one periodic point exhibits a dominated splitting while for the other point it presents a rotational effect (c.f. Figures 1-3). This is in strong analogy with the lack of domination observed in the presence of homoclinic tangencies.
In the space of $C^1$-diffeomorphisms the presence of homoclinic tangencies and heterodimensional cycles appears as a mechanism to create infinitely many sources or sinks (see e.g. the Newhouse phenomenon in \cite{Ne}). Roughly, if there exists a homoclinic tangency associated to a dissipative periodic point of saddle type then the lack of domination allows to perturb the system to mix the directions and to create a sink/source. The infinitely many sinks/sources arise associated to some phenomenon of persistence of tangencies.
\begin{maintheorem}\label{thm:dichotomy}
Assume that $\dim M =3$. Let $\Lambda$ be a non-trivial basic piece of a diffeomorphism $f$ and $\mathcal{U}$ a $C^1$-open neighborhood of $f$ on which the analytic continuation $\Lambda_g$ of $\Lambda$ is well-defined for all $g\in\mathcal{U}$. Either $f|_{\Lambda}$ has a one-dimensional finest dominated splitting, or $f$ can be $C^1$-approximated by diffeomorphisms $g$ that exhibit a equidimensional cycle in $\Lambda_g$ with robust different signatures. \end{maintheorem}
We notice that the property of having one-dimensional finest dominated splitting and the property of exhibiting a equidimensional cycle with robust different signatures are $C^1$-open. It is not hard to construct examples that lie in the boundary of the diffeomorphisms with a one-dimensional finest dominated splitting in a basic piece but not on the closure of diffeomorphisms exhibiting a equidimensional cycle with robust different signatures, due to the existence of nilpotent part. In particular, there are examples of $C^1$ diffeomorphisms exhibiting periodic points (of low period) with equal eigenvalues that cannot be perturbed to create a complex eigenvalue. After this work was completed we were informed that, in the $C^1$-topology, Bochi and Bonatti ~\cite{BoBo} describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism.
The next result explains the conceptual similarity between equidimensional cycles with robust different signatures and the notion of heterodimensional cycles for dynamics beyond uniform hyperbolicity.
\begin{maintheorem}\label{thm:explo}
Assume that $\dim M =3$ and let $\Lambda$ be a non-trivial basic piece of a diffeomorphism $f$. If $f|_{\Lambda}$ admits two periodic points $p,q$ with robust different unstable signatures, then there exists a $C^1$-open neighborhood $\mathcal U$ of $f$ and a residual subset $\mathcal R\subset \mathcal U$ such that for any $g\in \mathcal R$ we have that: \begin{enumerate} \item there exists a countable infinite set of points $P_1\subset Per(g)\cap \Lambda_g$ with $sgn^u(p_1)=sgn^u(p)$ for all $p_1\in P_1$ and \item there exists a countable infinite set of points $P_2\subset Per(g)\cap \Lambda_g$ with $sgn^u(p_2)=sgn^u(q)$ for all $p_2\in P_2$, \end{enumerate} where $\Lambda_g$ stands for the hyperbolic continuation of the set $\Lambda$. \end{maintheorem}
\subsubsection*{An application for flows}
In what follows we apply our results to the case of hyperbolic basic pieces for $C^1$-flows. By Bowen and Ruelle~\cite{BR75}, the restriction of a $C^1$-flow to a hyperbolic basic piece for a $C^1$-flow admits a finite Markov partition and is semiconjugate to a suspension flows over a subshift of finite type and there exists a one-to-one correspondance between invariant measures of the discrete-time and continuous-time dynamical systems. Indeed, given a measurable space $\Sigma$, a map $R\colon \Sigma\rightarrow{\Sigma}$, an $R$-invariant probability measure $\tilde{\mu}$ defined in $\Sigma$ and a ceiling function $h\colon \Sigma\rightarrow{\mathbb{R}^{+}}$ in $L^1(\tilde\mu)$ and bounded away from zero, consider the space $M_{h}\subseteq{\Sigma\times{\mathbb{R}_+}}$ defined by $M_h=\{(x,t) \in \Sigma\times{\mathbb{R}_+}: 0 \leq t \leq h(x) \}/\sim$ where $\sim$ stands for the identification between the pairs $(x,h(x))$ and $(R(x),0)$. The semiflow defined on $M_h$ by $S^s(x,r)=(R^{n}(x),r+s-\sum_{i=0}^{n-1}h(R^{i}(x)))$, where $n\in{\mathbb{N}_0}$ is uniquely defined by $\sum_{i=0}^{n-1}h(R^{i}(x))\leq{r+s}<\sum_{i=0}^{n}h(R^{i}(x))$ is called a \emph{suspension semiflow}. If $R$ is invertible then $(S^t)_t$ is indeed a flow. Since $h$ is bounded below, if $\text{Leb}_1$ denotes the one dimensional Lebesgue measure, then $\eta \mapsto \bar\eta:=\frac{\eta \times \text{Leb}_1}{\int h \;d\eta}$ is a one-to-one correspondence between $R$-invariant probability measures and $S^t$-invariant probability measures (we refer the reader \cite{BR75} for more details).
Hence, if $T_x \Sigma=E^1_x\oplus \dots E^k_x$ for a $\eta$-generic point $x$ then $T_{(x,s)} \Sigma=E^1_{(x,s)}\oplus \dots E^k_{(x,s)} \oplus \langle X \rangle$ is the Oseledets splitting associated to the vector field $X$ for $\bar\eta$-almost every $(x,s)$, where $\langle X \rangle$ is the subbundle generated by the vector field. In particular, the Lyapunov spectrum of $(X,\bar\eta)$ is given by the Lyapunov exponents of $(S,\eta)$ and zero (corresponding to the flow direction). Given the previous characterization, a hyperbolic basic piece $\Lambda$ for a vector field $X$, the vector field $X\mid_\Lambda$ is $C^1$-simple star if and only if the Poincar\'e map $S\mid_{\Sigma\cap \Lambda}$ is $C^1$-simple star. We also stress that such a Poincar\'e map inherits the information of the Lyapunov exponents for both periodic points and all invariant measures: the Lyapunov exponents for the flow and for the Poincar\'e map differ from the integral of the return time function. The definitions from the discrete time setting extend to this setting in an analogous way. Since the $C^1$-perturbation tools used in the proofs of Theorems~\ref{thm:1FDS} and ~\ref{thm:dichotomy} are available for flows, then the following result is a consequence of the above mentioned theorems.
\begin{maincorollary}\label{cor:flows} If $(X_t)_t$ is a simple star structurally stable flow, then $(X_t)_t$ admits a one-dimensional finest dominated splitting and the Lyapunov exponent functions vary continuously with the invariant probability measure. Moreover, if $\dim M =4$ we have the following dichotomy: if $(X_t)_t$ is a structurally stable flow then either $(X_t)_t$ admits a one-dimensional finest dominated splitting, or $(X_t)_t$ can be $C^1$-approximated by flows that exhibit a equidimensional cycle associated points with robust different signatures. \end{maincorollary}
\section{Proofs}
In this section we prove our main results.
\subsection{Proof of Theorem~\ref{teo1}}
The arguments here use strongly that $M$ is a three-dimensional manifold. Let $r\ge 1$, let $f$ be a $C^r$-diffeomorphism and $\Lambda$ be a non-trivial hyperbolic basic set for $f$ and let $\mathcal{U}$ be a $C^r$-open neighborhood of $f$ so that the hyperbolic continuation $\Lambda_g$ is well defined for all $g\in \mathcal{U}$.
In particular, for every $g\in \mathcal U$ there exists an homeomorphism $h_g: \Lambda \to \Lambda_g$ close to the identity so that $h_g \circ f\mid_{\Lambda} = g\mid_{\Lambda_g} \circ h_g$. Consequently, if $(p_j)_{j=1\dots \ell}$ are the periodic points of period $m$ for $f\mid_\Lambda$ then the continuations $(h_g(p_j))_{j=1\dots \ell}$ are the periodic points of period $m$ for $g$. Let $Per_s(g)$ denote the set of periodic points with simple Lyapunov spectrum. The set $$ \mathcal U_n =\Big\{ g\in \mathcal U \colon \text{Per}_s(g) \, \text{is $\frac1n$-dense in}\, \Lambda_g \Big\} $$ is clearly a $C^r$-open subset of $\mathcal{U}$. We claim that $\mathcal{U}_n$ is a $C^r$-dense subset of $\mathcal{U}$. This will be enough to prove the theorem because the residual subset $\mathcal R:=\bigcap_{n\in \mathbb N} \mathcal U_n \subset \mathcal{U}$ is so that the set of periodic points with simple Lyapunov spectrum for any $g\in \mathcal R$ is dense in $\Lambda_g$. In other words, the residual subset $\mathcal R\subset \mathcal{U}$ satisfies the requirements of the theorem. Thus, we are left to prove that for every $n\ge 1$ the set $\mathcal{U}_n$ is a $C^r$-dense subset of $\mathcal{U}$.
Before that we need a simple and useful lemma to perform the local $C^r$-per\-tur\-bations. This lemma is quite different from the well-known Franks' lemma because the approximation on the derivative depends on a pre-fixed size of neighborhood of the support of the perturbation.
\begin{lemma}\label{pert} Given a $C^r$ diffeomorphism $f\colon M\rightarrow M$, $\epsilon>0$, $p\in M$ and $R>0$, there exists $\delta:=\delta(\epsilon,R)>0$ such that the following holds: if $A(p)\colon T_xM\rightarrow T_{f(x)}M$ is $\delta$-close to $Df(p)$ in the uniform norm of operators then there exists a $C^r$ diffeomorphism $g$ such that: \begin{enumerate} \item $g$ is $\epsilon$-$C^r$-close to $f$; \item $g$ is supported in $B\left(p,R\right)$ and \item $Dg(p)=A(p)$. \end{enumerate} \end{lemma}
\begin{proof} For simplicity we assume all the computations in local charts in $\mathbb{R}^n$ and that $p$ is fixed. Take $\epsilon>0$ and $R>0$ and consider a $C^\infty$ \emph{bump function} $\varphi\colon [0,+\infty[\rightarrow [0,1[$ such that $\varphi(t)=0$ if $t>\sqrt{R}$ and $\varphi(t)=1$ if $t< \frac{\sqrt{R}}{2}$. It follows from Fa\`{a} di Bruno's formula that, given $q\in B\left(p,R\right)$, we have \begin{equation}\label{FDB}
\frac{\partial^n \varphi(\|q\|^2)}{\partial q_k}
=\sum\frac{n!}{m_1!1!^{m_1}m_2!2!^{m_2}...m_n!n!^{m_n}}\varphi^{(m_1+...+m_n)}(\|q\|^2)
\prod_{j=1}^n \left(\frac{\partial^j \|q\|^2}{\partial q_k}\right)^{m_j}, \end{equation} where the sum is over all vectors with nonnegative integers entries $(m_1,..., m_n)$ such that we have $\sum_{j=1}^n j.m_j=n$. These derivatives are bounded above by a constant $C$ (depending on $R$).
Let $A\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear map $\delta$-close to $Df(p)$, where $\delta>0$ is very small (to be determined later on and depending on the derivatives in (\ref{FDB})).
Take a $C^r$ smooth map $\hat h\colon B\left(p,R\right)\rightarrow \mathbb{R}^n$ defined by $\hat h(q)=[Df(p)]^{-1}\cdot A(p)$, take $h(q)=\varphi (\|q\|^2)\hat h(q) + (1-\varphi (\|q\|^2)) Id$ and define $g:=f\circ h$. Clearly, by construction, $g$ coincides with $f$ in the complement of $B\left(p,R\right)$ and $$Dg(p)=Df(h(p))\cdot Dh(p)=Df(p)\cdot D\hat h(p)=Df(p)\cdot [Df(p)]^{-1}\cdot A(p) =A(p).$$
This proves (2) and (3). Furthermore, since $\hat h$ is constant and $\| \hat h(q)\|<\delta$, then the
${C^r}$-distance between $f$ and $g$ is proportional to $\|\hat h\|$ and can be taken smaller than $\epsilon$ provided that $\delta$ is small enough. \end{proof}
We are now in a position to proceed with the proof that for every $n\ge 1$ the set $\mathcal{U}_n$ is a $C^r$-dense subset of $\mathcal{U}$. In fact this is a consequence of the following
\begin{proposition} Given $\epsilon>0$, $n\ge 1$ and a non-trivial basic piece $\Lambda$ of a $C^r$-diffeomorphism $f$, there exists a $C^r$-diffeomorphism $g$ that is $\epsilon$-$C^r$-close to $f$ and displaying a $g$-periodic orbit $q$ that is $\frac1n$-dense in $\Lambda_g$ and has simple Lyapunov spectrum. \end{proposition}
\begin{proof} Let $\epsilon>0$, $f$ be a $C^r$ diffeomorphism, and $\Lambda$ be a non-trivial basic piece for $f$. Assume without loss of generality that $T_{\Lambda} M = E^s \oplus E^u$ with $\dim E^s=1$ and $\dim E^u=2$ (otherwise just consider the diffeomorphism $f^{-1}$). In this three-dimensional setting, $p\in \text{Per}(f)$ has simple Lyapunov spectrum if and only if there are no
complex eigenvalues for $Df^{\pi(p)}(p)\mid_{E^u_p}$ or $Df^{\pi(p)}(p)\mid_{E^u_p}$ has no equal real eigenvalues. Notice that this last mentioned situation has empty $C^r$-interior.
Fix $p\in \text{Per}(f)$. Up to consider $f^{\pi(p)}$ we just assume that $\pi(p)=1$ and that the restriction of $f$ to the non-trivial hyperbolic homoclinic class $\Lambda$ associated to $p$ is topologically mixing. In particular, $f\mid_{\Lambda}$ satisfies the specification property: for each $\zeta>0$, there is an integer $N(\zeta)$ for which the following is true: if $I_1,I_2,\cdots,I_k$ are pairwise disjoint intervals of integers with
$$\min\{|m-n|:m\in I_i,n\in I_j\}\ge N(\zeta)$$ for $i\not=j$ and $x_1,\cdots,x_k \in X$ then there is a point $x\in X$ such that $d(f^j(x),f^j(x_i))\le\zeta$ for $j\in I_i$ and $1\le i\le k$. Every topologically mixing hyperbolic elementary set of a diffeomorphism satisfies the specification property (see \cite{B}).
Consider a $\frac1{2n}$-dense set $F$ in $\Lambda$. Then there exists $\ell=\ell(n)\ge 1$ so that for any large $m$ there exists a periodic point $x_m\in \Lambda$ of period equal to $m+\ell\times \# F$, such that the first $m$ iterates of $x_m$ belong to a small neighborhood $U_0$ of $p$ and the piece of orbit $\{f^{j+m}(x_m) : 1\le j \le \ell\}$ is $\frac1n$-dense.
Observe that $\ell$ is fixed and depends only on $n$. Defined in this way, as $m$ increases, the point $x_m$ is sufficiently close to $p$ and spends a large amount of time near $p$, and the matrix $Df^{m}(x_m)$ inherits the dynamical behavior of $[Df(p)]^m$ whereas the matrix $Df^{m+\ell}(x_m)$, corresponding to the periodic orbit computed along the whole orbit of $x_m$.
Define the compact $f$-invariant hyperbolic sets \begin{equation}\label{Knf} K_n(f):=\overline{\bigcup_{m\geq n}\bigcup_{i\in\mathbb{Z}} f^{i}(x_m)} \subset \Lambda, \;\; n\ge 1. \end{equation}
The persistence of the hyperbolic splitting $E_{f}^u\oplus E_{f}^s$ for the map $f$ under $C^1$ (thus $C^{r}$) perturbations allows us to conclude that any $g$ sufficiently $C^{r}$-close to $f$ also has a hyperbolic splitting $E_{g}^u\oplus E_{g}^s$ (with a two-dimensional unstable and one-dimensional stable bundles $E_g^u$ and $E_g^s$, respectively) over the
$g$-invariant compact set $K_n(g)$ obtained as in \eqref{Knf} by taking the continuation of the hyperbolic periodic
points $(x_m)_m$.
Now we will show that some $C^r$-small perturbation $g$ can be made in order to find a periodic point $x_m$ in $K_n(g)$ such that all the eigenvalues of $Dg^{\pi(x_m)}(x_m)$ are real and distinct, where $\pi(x_m)=m+\ell\times \# F$. Such point $x_m$ is $\frac1n$-dense by construction.
Actually, although explicitly defined at a neighborhood of $p$, the detailed construction of the perturbation map necessarily makes use of bump-functions. For shortness we shall omit the full detailed construction. For any $n\ge 1$, let $\rho(m,f)$ stand for the rotation number of $Df^{\pi(x_m)}(x_m)\mid_{E^u_{x_m}}$. Given a continuous arc of maps $(f_t)_{t\in I}$ near $f$ let $\delta(m,f_t)$ denote the oscillation of the rotation number along the arc, that is,
$\delta(m,(f_t)_t)=\sup \{ | \rho(m,f_t) - \rho(m,f_s) | \colon s,t \in I \}$. The next claim corresponds to \cite[Lemma 9.3]{BoV04} in our setting.
\noindent\textbf{Claim:}
There exists a continuous arc of maps $\{f_t\}_{t\in[0,1]}$ satisfying $f_0=f$ and $\|f_t-f_0\|_{C^r} <\epsilon$ for all $t\in[0,1]$ so that the following holds: for every $t$ there exists $m_t\in\mathbb{N}$ so that $\delta(m,(f_s)_{s\in[0,t]})>1$ for all $m\geq m_t$.
\noindent\emph{Proof of the claim:} We consider from now on a basis adapted to the splitting $E^u_p\oplus E^s_p$ (for $f$), a constant $\sigma\in(0,1)$, $\gamma>1$ and maps $h_{t\xi}$ with support in a small ball $B(p,R)\subset U_0$ such that: $$Df(p)=\begin{pmatrix}\sigma&0&0\\0&\gamma\cos\theta&-\gamma\sin\theta\\0&\gamma\sin\theta&\gamma\cos\theta\end{pmatrix}\,\,\,\,\,\text{and}\,\,\,\,\,Dh_{t\xi}(p)=\begin{pmatrix}1&0&0\\0&\cos (t\xi)&-\sin(t\xi)\\0&\sin(t\xi)&\cos(t\xi)\end{pmatrix}$$ As in Lemma~\ref{pert}, consider the $C^r$-diffeomorphisms $f_t=h_{t\xi}(p)\circ f(p)$ and let $\xi$ be sufficiently small in order to have $f_t$ $\epsilon$-$C^r$-close to $f$ for all $t\in[0,1]$. Since the uniformly hyperbolic splitting varies continuously with the point and the angle between stable and unstable bundles is bounded away from zero each fiber $E^u_{x}$ and $E^s_{x}$ is a graph over $E^u_{p}$ and $E^s_{p}$ (for all maps in the family $(f_t)_{t\in [0,1]}$) for all points $x$ in the hyperbolic isolated set. Identifying all the stable and unstable subspaces with $E^s_p$ and $E^u_p$ by parallel transport, consider the decomposition $$ Df_t^{\pi(x_m)}(x_m)=\alpha_{t,m,m} . (\dots) . \,\alpha_{t,m,1} . \,\beta_{t,m}, $$ where the $\alpha$'s correspond to the derivative $Df_t$ at iterates of $x_m$ by $f_t$ inside the neighborhood $U_0$ of $p$ and the $\beta$'s to the derivative $Df_t$ at the iterates of $x_m$ by $f_t$ that are outside it. Since, for each $m$ we only allow $\ell(n)\times \#F$ iterates outside the neighborhood $U_0$ we obtain that, up to consider some subsequence if necessary, $\beta_{t,m}$ converges uniformly to some $\beta_t$ as $n\rightarrow\infty$ (i.e. the period of the $x_m$'s increases). We need to prove that for any $t$ there exists $m_t\in\mathbb{N}$ so that $\delta(m,(f_s)_{s\in[0,t]})>1$ for all $m\geq m_t$. Notice that each $\alpha_{t,m,i}\mid_{E^u}$ is uniformly close to the composition of an homothety by a rotation of angle $t\xi+\theta$ and so all give a contribution to the increase of the original rotation number by some positive constant and the claim is proved.
$\square$
By the claim, for any $\epsilon>0$ there exists a small $t$ and a large $m$ such that $f_t$ is $\epsilon$-$C^r$-close to $f$
and $\rho(m,f_t)\in\mathbb{Z}$. Therefore, $Df_t^{\pi(x_m)}(x_m)|_{E^u_{f_t}}$ has two equal real eigenvalues of norm larger than 1. Then, we perturb once more to obtain simple spectrum by simply consider a `diagonal' perturbation at $x_m$ given for $\eta_1,\eta_2> 0$ arbitrarily small and $\eta_1\not=\eta_2$, in a suitable base of eigenvalues, by the matrix $$ \begin{pmatrix}1&0&0\\0&\eta_1&0\\0&0&\eta_2\end{pmatrix}. $$ The argument to realize such perturbation of the derivative by a $C^r$-close diffeomoprhism follows the lines of Lemma~\ref{pert}. This completes the proof of the proposition. \end{proof}
\subsection{Proof of Theorem~\ref{thm:1FDS}}
Assume that $f|_{\Lambda}$ is simple star. We explore the fact that the set of (hyperbolic) periodic points is dense in $\Lambda$. If $\Lambda$ consists of a hyperbolic periodic point the result is immediate. So, we assume that $\Lambda$ is a non-trivial hyperbolic set.
Using that a dominated splitting extends to the closure of a set and that $f\mid_\Lambda$ is simple star (thus cannot be $C^1$-approximated by a diffeomorphism so that the Lyapunov spectrum at some periodic point is not simple) then, by Corollary~\ref{BGV}, there exist $m_1, n_1\ge 1$ so that there exists an $m_1$-dominated splitting $E\oplus F$ over the set $\Lambda\cap \overline{\bigcup_{\ell \ge n_1} Per_{\ell}(f)}$.
If $E$ has dimension one we take $E^1=E$. Otherwise, repeating the procedure with $Df\mid_E$ there exists $n_2\ge 1$ and $m_2\ge 1$ (multiple of $m_1$) and a $m_2$-dominated splitting $
T_{\Lambda \cap \overline{\bigcup_{\ell \ge n_2} Per_{\ell}(f)}} M
= (\hat E^1 \oplus \hat E^2) \oplus F $ with $0<\dim \hat E^1, \dim \hat E^2 <\dim E$. If $\hat E^1$ has dimension one we take $E^1=\hat E^1$. Proceeding recursively with $E$ and $F$ it follows that there exist $m \ge 1$ and an $m$-dominated splitting $$ T_{\Lambda \cap \;\overline{\bigcup_{\ell \ge n} Per_{\ell}(f)}}\, M = E^1 \oplus \dots \oplus E^{\dim M}. $$ Using that $\Lambda$ is a (non-trivial) homoclinic class and that every non-trivial homoclinic class contains infinitely many periodic points then $\bigcup_{\ell \ge n} Per_{\ell}(f)$ is dense in $\Lambda$. Thus $f\mid_\Lambda$ has a one-dimensional finest dominated splitting, and the first claim in the theorem follows.
Now, we shall prove that the Lyapunov exponent function associated to $f\mid_\Lambda$ is continuous. In fact, let $T_{\Lambda} M= E^1 \oplus \dots \oplus E^{\dim M}$ be the one-dimensional, $Df$-invariant finest dominated splitting associated to $f$. Since dominated splittings varies continuously with the point then there exists a continuous function $$ \Lambda \ni x \mapsto (v_x^1, v_x^2, \dots, v_x^{\dim M}) \in (T_x^1M)^{\dim M} $$ such that $v_x^i\in E^i_x$ and, by \eqref{eq:poitwise3}, the Lyapunov exponents are given by the integrals of continuous functions $\lambda_i(f,\mu) = \int \phi(x,v_i^x) \, d\mu(x)$ for every $1\le i \le \dim M$ and $x\in \Lambda$.
It is immediate that these vary continuously with respect to $\mu$ in the weak$^*$ topology and, consequently, the Lyapunov exponent map ~\eqref{LE} is continuous. By the dominated splitting property it follows that the Lyapunov spectrum of every invariant measure $\mu$ is simple. This finishes the proof of Theorem~\ref{thm:1FDS}.
\begin{remark} We observe that an alternative argument for the simplicity of the Lyapunov spectrum for all invariant measures follows from the denseness of periodic measures in the set of $f$-invariant probability measures, by specification~\cite{Sigmund}. \end{remark}
\begin{remark} Since dominated splittings also vary continuously with the diffeomorphism, then every $g$ that is $C^1$-close to $f$, then $\Lambda_g$ is a basic piece with a one-dimensional finest dominated splitting and there exists a continuous function $$ (x,g) \mapsto (v_{x,g}^1, v_{x,g}^2, \dots, v_{x,g}^{\dim M}) \in (T_x^1M)^{\dim M} $$ such that $v_x^i\in E^i_x$. Since Lyapunov exponents are computed as integrals of continuous functions (cf. \eqref{eq:poitwise3}) then given a sequence $(f_n)_n$ convergent to $f$ in the $C^1$-topology, and invariant measures $\mu_n \in \mathcal{M}_1(f_n)$ that converge to $\mu\in \mathcal{M}(f)$ then we also obtain that $\lim\limits_{n\to\infty} \lambda_i(f_n,\mu_n)=\lambda_i(f,\mu) $ for every $1\le i \le \dim M$. \end{remark}
\subsection{Proof of Theorem~\ref{thm:dichotomy}}\label{second}
Assume that $\dim M =3$, $\Lambda$ is a non-trivial basic piece of a diffeomorphism $f$ and
$\mathcal{U}$ a $C^1$-open neighborhood of $f$ on which the analytic continuation $\Lambda_g$ of $\Lambda$ is well-defined for all $g\in\mathcal{U}$. To prove the theorem we are reduced to prove that if $f|_{\Lambda}$ cannot be $C^1$-approximated by diffeomophisms $g$ so that $g\mid_{\Lambda_g}$ has a one-dimensional finest dominated splitting then $f$ can be $C^1$-approximated by diffeomorphisms $g$ that exhibits a equidimensional cycle in $\Lambda_g$ with robust different signatures. Assume that $T_\Lambda M= E^s\oplus E^u$ with $\dim E^s=1$ and $\dim E^u=2$. We use the following direct consequence of \cite[Corollary~2.18]{BGV} for the cocycle $A=Df\mid_{E^u}$ together with Franks' lemma and, henceforth, is specific of the $C^1$-topology.
\begin{corollary}\label{BGV}\cite[Corollary~2.18]{BGV} For any $\epsilon>0$ there are two integers $m$ and $n$ such that, for any periodic point $x$ of period $\pi(x) \geq n$: \begin{enumerate} \item either $Df\mid_{E^u}$ admits an $m$-dominated splitting along the orbit of $x$ or else; \item for any neighborhood $U$ of the orbit of $x$, there exists an $\epsilon$-perturbation $g$ of $f$ in the $C^1$-topology, coinciding with $f$ outside $U$ and on the orbit of $x$, and such that the differential $Dg^{\pi(x)}(x)\mid_{E^u_x}$ has all eigenvalues real and with the same modulus. \end{enumerate} \end{corollary}
Since all periodic points are homoclinically related, we are reduced to prove first that there are (up to a $C^1$-perturbation) two periodic points with robust different signatures.
We organize our argument in three cases.
\noindent\emph{Case~1: $Df^{\pi(p)}(p)\mid_{E^1_p}$ has a complex eigenvalue for every $p\in\text{Per}(f)$}.
If this is the case then $\text{sgn}^u(p)=(2,0)$ for every $p\in\text{Per}(f)$. By Corollary~\ref{BGV}, item (2), there is a diffeomorphism $g$, $C^1$-arbitrarily close to $f$, and a periodic point $p$ for which $Dg^{\pi(p)}(p)\mid_{E^1}$ has two real eigenvalues with the same modulus. Since this is a non-generic property, by a $C^1$-small perturbation $\tilde g$ of $g$, the continuation $p_{\tilde g}$ of the periodic point $p$ is so that $D\tilde g^{\pi(p_{\tilde g})}(p_{\tilde g})\mid_{E^1}$ has real and simple spectrum. Since $\tilde g$ can be obtained displaying some periodic point $q$ in the basic piece such that $D\tilde g^{\pi(q)}(q)\mid_{E^1_q}$ has a complex eigenvalue (hence $\text{sgn}^u(q)=(2,0)$ robustly), then $\tilde g \in \mathcal{SS}^1_c$.
\noindent\emph{Case~2: There are $p,q\in Per(f)$ such that $Df^{\pi(p)}(p)\mid_{E^1_p}$ has a complex eigenvalue and
$Df^{\pi(q)}(q)\mid_{E^1_q}$ has two real eigenvalues}.
If $Df^{\pi(q)}(q)\mid_{E^1_q}$ has simple real spectrum we are done. Otherwise, since there are periodic points whose derivative at the period has a complex eigenvalue in this case a simple and arbitrary small $C^1$-perturbation $g$ supported in a neighborhood of $q$ in such a way that $Dg^{\pi(q)}(q)\mid_{E^1_q}$ has real simple spectrum and, consequently, an equidimensional cycle with robust different signatures.
\noindent\emph{Case~3: for all $p\in\text{Per}(f)$ the linear map $Df^{\pi(p)}(p)\mid_{E^1_p}$ has only real eigenvalues}.
In this case there are three situations to consider, depending on the number of periodic points with two real equal eigenvalues (presenting a nilpotent part or not).
(a) all periodic points $p\in\text{Per}(f)$ are so that $Df^{\pi(p)}(p)\mid_{E^1_p}$ has real simple spectrum (or, in other words, $\text{sgn}^u(p)=(1,1)$).
As $f|_{\Lambda}$ cannot be $C^1$-approximated by diffeomophisms $g$ so that $g\mid_{\Lambda_g}$ has a one-dimensional finest dominated splitting, then it follows from Corollary~\ref{tadinho} that $f$ is not a simple star diffeomorphism. Hence, it can be arbitrarily $C^1$-approximated by a diffeomorphism $g$ so that either there exists a periodic point $p$ such that $Dg^{\pi(p)}(p)\mid_{E^1}$ has one complex eigenvalue or two real equal eigenvalues. Such perturbation can be performed in order to maintain a periodic point with real simple eigenvalues. Moreover, the period of $p$ can be taken as larger as necessary. We are left to prove that a perturbation can be done in order to obtain complex eigenvalues (along $E^1$) for some periodic point in the case when we have two real equal eigenvalues for $Dg^{\pi(p)}(p)\mid_{E^1}$. We will deal with this situation in the next two cases (b) and (c).
(b) if there exists a periodic point $p$ so that $\dim \text{Ker}(Df^{\pi(p)}(p)\mid_{E^1_p}-I)=2$ then $Df^{\pi(p)}(p)\mid_{E^1_p}$ has two real eigenvalues larger than one with no nilpotent part. Hence, by a small $C^1$-perturbation whose support is contained in a neighborhood of $p$ one can obtain a $C^1$-diffeomorphism $g$ so that $Dg^{\pi(p)}(p)\mid_{E^1_p}$ has a complex eigenvalue (this can be obtained via Franks' lemma and which can be written locally by $Dg=R_\theta \circ Df$, where $R_\theta$ represents the rotation of angle $\theta$ on the plane $E^1$ and the identity in $E^2$).
(c) if all periodic points $p$ have two real equal eigenvalues and are such that
$\dim \text{Ker}(Df^{\pi(p)}(p)\mid_{E^1_p}-I)=1$ we shall borrow an idea from \cite[Section 9]{BoV04}
and proceed as follows. In this case all of these periodic points have a nilpotent part and, in a suitable
coordinate system, can be written
by
$$
Df^{\pi(p)}(p)\mid_{E^1_p}
= \begin{pmatrix} \lambda_p & n(p) \\ 0 & \lambda_p \end{pmatrix}
$$ where $n(p) \in \mathbb R$ denotes the nilpotent part and $\lambda_p >1$. Actually, since $f$ is Axiom A then it admits a finite Markov partition $\mathcal P=\{P_i\}_i$. Let $n_{i,j}\ge 1$ be given such that $f^{n_{i,j}}(P_i)\cap P_j \neq \emptyset$.
Fix $p\in Per(f)$ as above and assume, for simplicity, that $p \in P_1$ is a fixed point. Given $n\ge 1$, let $x_m \in \Lambda$ be a periodic point so that $f^j(x_m) \in P_1$ for every $1\le j \le n$, $f^{n_{1,2}+n} (x_m) \in P_2$ and $x_m=f^{n_{2,1}+n_{1,2}+n} (x_m) \in P_1$. For any $m\gg 1$ the $f$-invariant compact set $Y_m = \overline{ \bigcup_{n\ge m} \bigcup_{j\in \mathbb Z} f^j(x_m)}$ inherits the behavior of $Df(p)$. For periodic points $q \in Y_m$ with period $\pi(q)\ge 1$ one has
$$
Df^{\pi(q)}(q)\mid_{E^1_q}
\approx \begin{pmatrix} \lambda_p^{\pi(q)-\ell} & N(q) \\ 0 & \lambda_p^{\pi(q)-\ell} \end{pmatrix}\cdot A(q,\ell),
$$
where $A(q,\ell)$ is a $2\times2$ matrix corresponding to the $\ell=n_{2,1}+n_{1,2}$ iterates of transitions between $P_1$ and $P_2$, and the nilpotent part $N(q)$ grows linearly with $\pi(q)$. In particular, $N(q)/ \lambda_p^{\pi(q)} \to 0$ as $\pi(q) \to\infty$. Given $\varepsilon>0$ choose $q\in Y_m$ with large period so that $N(q) \le \lambda_p^{\pi(q)-\ell} \varepsilon$.
First we perform a $\varepsilon$-$C^1$-small perturbation along a piece of orbit of $q$ so that the resulting diffeomorphism $g$ satisfies, in appropriate coordinates,
$$
Dg^{\pi(q)}(q)\mid_{E^1_q}
\approx \begin{pmatrix} \lambda_p^{\pi(q)-\ell} & 0 \\ 0 & \lambda_p^{\pi(q)-\ell} \end{pmatrix}\cdot A(q,\ell)=\lambda_p^{\pi(q)-\ell} A(q,\ell).
$$ If $A(q,\ell)$ have complex eigenvalues we are done.
Otherwise we notice that $A(q,\ell)=Df^{\ell}(f^{\pi(q)-\ell}(q))$ and that $q \mapsto A(q,\ell)$ is continuous. There exists an appropriate base, given by the Jordan canonical form, so that the matrix $A(q,\ell)$ can be written of the form $$ A_1:= \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}
\quad\text{or}\quad A_2:=\begin{pmatrix} a & b \\ 0 & a \end{pmatrix} $$ for some real numbers $a,b$ (depending on the $\ell$ iterates $$\{ f^{j} (f^{\pi(q)-\ell} (q)) \colon j=0 \dots \ell-1 \}$$ of the orbit of the periodic point $q$). By the continuity of the matrix $A(q,\ell)$ with the periodic point $q$ the real numbers $a,b$ can be taken bounded from above and below by uniform constants independent on the periodic point $q$ and period $\pi(q)$. The strategy is to perturb the derivative of the diffeomorphism $g$ along a piece of orbit $\{ f^{j} (q) \colon j=0 \dots N\}$ ($N\ge 1$ to be determined below) in such a way that the resulting $C^1$-diffeomorphism $\tilde g$ is so that $D\tilde g^{\pi(q)}(q)\mid_{E^1_q}$ has complex eigenvalues.
Given $\varepsilon>0$ take $N(\varepsilon)\approx \frac1\varepsilon$ (more precisely, $N(\varepsilon)\ge \frac{1}{1+\varepsilon} \log \left|\frac{b}{a}\right|$ in case of matrix $A_1$ and $N(\varepsilon)\ge \frac{b}{\varepsilon}$ in case of matrix $A_2$) and take $q \in Y_m$ with period $\pi(q)\ge N(\varepsilon)+\ell$. By Franks' lemma, in the case of a matrix of the form $A_1$ then we perturb the $C^1$-diffeomorphism over the piece of orbit $\{ f^{j} (q) \colon j=0 \dots j=0 \dots N(\varepsilon)-1 \}$ by concatenations of perturbations of the derivative to get a $C^1$-diffeomorphism $g$ whose derivative is obtained from the one of $f$ by perturbations ($C^0$-close to the identity) of the form $$ \begin{pmatrix} 1+\varepsilon & 0 \\ 0 & 1 \end{pmatrix} $$ in an appropriate base. For the resulting diffeomorphism we get that $$
Dg^{\pi(q)}(q)\mid_{E^1_q}
\approx \lambda_p^{\pi(q)-\ell} \begin{pmatrix} (1+\varepsilon)^{N(\varepsilon)} a & 0 \\ 0 & b \end{pmatrix}
\approx \lambda_p^{\pi(q)-\ell} \begin{pmatrix} b & 0 \\ 0 & b \end{pmatrix}.
$$ In the case of a matrix of the form $A_2$ we proceed analogously perturbing the derivative of the diffeomorphism $f$ over the piece of orbit $\{ f^{j} (q) \colon j=0 \dots N(\varepsilon)-1 \}$ by perturbations of the form $$ \begin{pmatrix} 1 & -\frac{\varepsilon}{a} \\ 0 & 1 \end{pmatrix} $$ in an appropriate base. Thus we get $$
Dg^{\pi(q)}(q)\mid_{E^1_q}
\approx \lambda_p^{\pi(q)-\ell} \begin{pmatrix} a & b-\varepsilon N(\varepsilon) \\ 0 & a \end{pmatrix}
\approx \lambda_p^{\pi(q)-\ell} \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}.
$$ Hence there exists a periodic point $\tilde q$ such that $D\tilde g^{\pi(\tilde q)}(\tilde q)\mid_{E^1_{\tilde q}}$ has simple real spectrum. This finishes the proof of the theorem.
\subsection{Proof of Theorem~\ref{thm:explo}}
Let $f$ be a $C^1$ diffeomorphism and $\Lambda$ be a hyperbolic basic piece. Assume there are two periodic points $p,q \in \Lambda$ with robust different unstable signatures. Let $\mathcal{U}$ be a $C^1$-open neighborhood of $f$ so that for any $g\in\mathcal{U}$ the analytic continuation $\Lambda_g$ of $\Lambda$ is well defined and let $p_g$ and $q_g$ denote the continuations of $p$ and $q$, respectively, and $sgn^u(p_g)=sgn^u(p)$ and $sgn^u(q_g)=sgn^u(q)$. Since $\dim M =3$ we may assume without loss of generality that $\dim(E^s)=1$ and $\dim(E^u)=2$.
For any $n\in\mathbb{N}$ we define $\mathcal{R}_n$ to be the set of $C^1$ diffeomorphisms $g\in \mathcal{U}$ such that $g$ has $n$ distinct periodic points with robust unstable signature equal to the one of $p_g$ and has $n$ distinct periodic points with robust unstable signature equal to the one of $q_g$.
Observe that, by hypothesis, $\mathcal{R}_1=\mathcal{U}$ and so $\mathcal{R}_1$ is $C^1$-open and dense in $\mathcal{U}$. For any $n\geq 1$ the set $\mathcal{R}_n$ is clearly $C^1$-open. If one shows that each $\mathcal{R}_n$ is $C^1$-dense then the residual set $\mathcal{R}=\cap_n \mathcal{R}_n$ satisfies the statement of Theorem~\ref{thm:explo}.
Assume that $\mathcal{R}_k$ is $C^1$-dense for any $k=1,...,n$ and fix $\epsilon>0$ and any $g\in\mathcal{R}_n$. We claim that there exists $g_1\in \mathcal{R}_{n+1}$ which is $\epsilon$-$C^1$-close to $g$. The diffeomorphism $g_1$ will be obtained from $g$ by $C^1$-small perturbations at two periodic points in order to obtain one more point of each robust signature. By Corollary~\ref{BGV} we know that there exists $\ell,m\in\mathbb{N}$ such that, for any periodic point $x$ of period $\pi(x) \geq \ell$ either there exists an $m$-dominated splitting along the orbit of $x$ or else for any neighborhood $U$ of the orbit of $x$, there exists an $\epsilon/4$-perturbation $g_1$ of $g$
in the $C^1$-topology, coinciding with $g$ outside $U$ and on the orbit of $x$, and for which the tangent map $(Dg_1)^{\pi(x)}_x|_{E^u_x}$ has a real eigenvalue with multiplicity two.
Assume without loss of generality that $\text{sgn}^u(p_g)=(1,1)$ and $\text{sgn}^u(q_g)=(2)$. The existence of $q_g$ implies that the set of periodic points $x$ of period larger that $\ell$ and with an $m$-dominated splitting along the unstable fiber is not dense in $\Lambda_g$. Indeed, if this was not the case, then $q_g$ would have an $m$-dominated splitting on $E^u_{q_g}$. By the previous dichotomy it follows that $q_g$ is accumulated by open sets without periodic points displaying an $m$-dominated splitting on $E^u$. Thus we can pick a periodic point $x$ distinct from the $2n$ marked periodic point for $g$ in one of these open sets and with arbitrarily large period $\pi(x)$. Using the dichotomy there exists an $\epsilon/4$-perturbation $g_1$ of $g$
in the $C^1$-topology, coinciding with $g$ outside $U$ and on the orbit of $x$, and for which the tangent map $(Dg_1)^{\pi(x)}_x|_{E^u_x}$ has a real eigenvalue with multiplicity two. Finally, since the periodic point $x$ can be chosen with an arbitrarily large period we can proceed as in Subsection \ref{second} and perform an $\epsilon/4$-perturbation $g_2$ of $g_1$ in the $C^1$-topology so that $(Dg_2)^{\pi(x)}_x|_{E^u_x}$ has a complex eigenvalue. Clearly, $x$ has an unstable robust signature $\text{sgn}^u(x)=(2)$.
If $g_2$ already have $n+1$ distinct periodic points with robust unstable signature equal to the unstable signature of $p_{g}$ we are done. Otherwise, we are left to show that $g_2$ can be $\epsilon/4$-approximated in the $C^1$-topology by $g_3$ exhibiting one more distinct periodic point with robust unstable signature equal to the unstable signature of $p_{g}$. Indeed, if all but $n$ periodic points have unstable signature $(2)$ and since there is no dominated splitting restricted to the unstable fiber and along these orbits we can perform an $\epsilon/4$-approximated in the $C^1$-topology in order to obtain a distinct periodic point $x$ with robust unstable signature equal to $\text{sgn}^u(x)=(1,1)$. This completes the proof of the $C^1$-denseness of $\mathcal R_n$ in $\mathcal U$ and finishes the proof of the theorem.
\begin{remark} The previous construction of infinitely many periodic points of a certain signature has the same flavor of the construction of a generic set with infinite sinks and sources for diffeomorphisms with homoclinic tangencies: the perturbation are localized in regions where there is a lack of domination among the stable and unstable bundles. \end{remark}
\textbf{Acknowledgements:} This work was partially supported by CMUP (UID/MAT/ 00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. MB was partially supported by National Funds through FCT Funda\c{c}\~ao para a Ci\^encia e a Tecnologia, project PEst-OE/MAT/UI0212/2011. PV was partially supported by a CNPq-Brazil.
\end{document} | arXiv |
\begin{document}
\title{General error estimate for adiabatic quantum computing}
\author{Gernot Schaller, Sarah Mostame, and Ralf Sch\"utzhold$^*$}
\affiliation{Institut f\"ur Theoretische Physik, Technische Universit\"at Dresden, 01062 Dresden, Germany}
$^*$ email: {\tt [email protected]}
\begin{abstract} Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction ${\bra{\Psi_{\rm ground}(t)}\dot H(t)\ket{\Psi_{\rm excited}(t)} /\Delta E^2(t)\ll1}$.
However, it is demonstrated that this first-order correction does not yield a good estimate for the computational error.
Therefore, a more general criterion is proposed, which includes higher-order corrections as well and shows that the computational error can be made exponentially small
-- which facilitates significantly shorter evolution times than the above first-order estimate in certain situations.
Based on this criterion and rather general arguments
and assumptions, it can be demonstrated that a run-time~$T$ of order of the inverse minimum energy gap~$\Delta E_{\rm min}$ is sufficient and necessary, i.e., $T={\cal O}(\Delta E_{\rm min}^{-1})$.
For some examples, these analytical investigations are confirmed by numerical simulations. \end{abstract}
\pacs{ 03.67.Lx, 03.67.-a. }
\maketitle
\section{Introduction}
With the emergence of the first quantum algorithms, it turned out that quantum computers are in principle much better suited to solving certain classes of problems than classical computers.
Prominent examples are Shor's algorithm \cite{shor1997} for the factorization of large numbers into their prime factors in polynomial time and Grover's algorithm \cite{grover1997} for searching an unsorted database with $N$ items reducing the computational complexity from the classical value ${\cal O}(N)$ to ${\cal O}(\sqrt{N})$ on a quantum computer.
Unfortunately, the actual realization of usual sequential quantum algorithms (where a sequence of quantum gates is applied to some initial quantum state, see, e.g., \cite{nielsen2000}) goes along with the problem that errors accumulate over many operations and the resulting decoherence tends to destroy the fragile quantum features needed for the computation.
Therefore, an alternative scheme has been suggested \cite{farhi2000}, where the solution to a problem is encoded in the (unknown) ground state of a (known) Hamiltonian.
By starting with an initial Hamiltonian $H_{\rm i}$ with a known ground state and slowly evolving to the final Hamiltonian $H_{\rm f}$ with the unknown ground state, e.g., $H(t)=[1-s(t)] H_{\rm i} + s(t) H_{\rm f}$, adiabatic quantum computing makes use of the adiabatic theorem which states that a system will remain near its ground state if the evolution $s(t)$ is slow enough.
Since there is evidence that the ground state is more robust against decoherence \cite{childs2001,kaminsky0211152,sarandy2005b}, this scheme offers fundamental advantages compared to sequential quantum algorithms.
However, determining the achievable speed-up of adiabatic quantum algorithms (compared to classical methods) for many problems is still a matter of investigation and debate, see, e.g., \cite{znidaric2005, farhi0512159,aharonov0405098,sarandy2004,childs2002,das2003,roland2002}.
For example, it has been argued in \cite{aharonov0405098} that all conventional (sequential) quantum algorithms can be realized as adiabatic quantum computation schemes with polynomial overhead via the history interpolation (polynomial equivalence).
For an adiabatic version of Grover's algorithm, a constant velocity $\dot s$ implies a linear scaling of the run-time $T={\cal O}(N)$, whereas a suitably adapted time-dependence $s(t)$ yields the known quadratic speed-up $T={\cal O}(\sqrt{N})$, cf.~\cite{das2003,roland2002}.
Whether adiabatic algorithms of NP complete problems such as 3-SAT can be even more efficient than this quadratic speed-up is still not clear, see, e.g., \cite{znidaric2005,farhi0512159}.
In this paper, we derive a general error estimate as a function of the run-time $T$ (the main measure for the computational complexity of adiabatic quantum algorithms) for very general gap structures $\Delta E(s)$ and interpolation velocities $s(t)$.
\section{Adiabatic Expansion}
The evolution of a system state $\ket{\Psi(t)}$ subject to a time-dependent Hamiltonian $H(t)$ is described by the Schr\"odinger equation ($\hbar=1$)
\begin{eqnarray} \label{Eschroedinger} i \ket{\dot{\Psi}(t)} = H(t) \ket{\Psi(t)} \,. \end{eqnarray}
Using the instantaneous energy eigenbasis defined by $H(t) \ket{n(t)} = E_n(t) \ket{n(t)}$, the system state $\ket{\Psi(t)}$ can be expanded to yield
\begin{eqnarray} \ket{\Psi(t)} = \sum_n a_n(t) \exp\left\{-i \int\limits_0^t E_n(t') dt'\right\} \ket{n(t)} \,. \end{eqnarray} Insertion into the Schr\"odinger equation yields -- after some algebra -- the evolution equations for the coefficients
\begin{eqnarray} \label{Ecoeff2} \pdiff{}{t} \left(a_m e^{-i\gamma_m}\right) &=& -\sum_{n\neq m} a_n \,\frac{\bra{m}\dot{H}\ket{n}}{\Delta E_{nm}} \,e^{-i\gamma_m} \times \nonumber\\ && \times \exp\left\{-i \int\limits_0^t\Delta E_{nm}(t')dt'\right\} \end{eqnarray}
with the energy gap ${\Delta E_{nm}(t)=E_n(t)-E_m(t)}$ and the Berry phase \cite{sun1988}
\begin{eqnarray} \label{Eberryphase} \gamma_n(t) = i \int\limits_0^t dt'\,\bracket{n(t')}{\dot{n}(t')} \,. \end{eqnarray}
If the external time-dependence $\dot H$ is slow (adiabatic evolution), the right-hand side of Eq.~(\ref{Ecoeff2}) is small and the solution can be obtained perturbatively.
After an integration by parts, the first-order contribution yields
\begin{eqnarray} \label{Efirst_order} a_m(t) &\approx& a_m^0e^{i\gamma_m(t)} -i \left[ \sum_{n\neq m} a_n^0 \frac{\bra{m}\dot{H}\ket{n}}{\Delta E_{nm}^2} \,e^{i\varphi_{nm}} \right]_0^t \end{eqnarray}
where $\varphi_{nm}\in\mathbb R$ denotes a pure phase.
Consequently, if the local adiabatic condition \begin{eqnarray} \label{Eadiabatic_old} \frac{\bra{m}\dot{H}\ket{n}}{\Delta E_{nm}^2} = \varepsilon \ll 1 \end{eqnarray}
is fulfilled for all times, the system approximately stays in its instantaneous eigen (e.g., ground) state throughout the (adiabatic) evolution.
This above constraint has frequently been used as a condition for adiabatic quantum computation \cite{farhi2000,childs2002}.
However, since the solution to a problem is encoded in the ground state of the final Hamiltonian in adiabatic quantum computation schemes, it is not really necessary to be in the instantaneous ground state {\em during} the dynamics -- the essential point is to obtain the desired ground state {\em after} the evolution.
Since the external time-dependence $\dot H$ could realistically be extremely small (or even practically vanish) at the end of the computation $t=T$, the first-order result (\ref{Efirst_order}) does not always provide a good error estimate.
Similar to the theory of quantum fields in curved space-times \cite{birrell}, the difference between the adiabatic and the instantaneous vacuum should not be confused with real excitations (particle creation).
Therefore, it is necessary to go beyond the first-order result above and to estimate the higher-order contributions.
\section{Analytic Continuation}
Evidently, the Schr\"odinger equation is covariant under simultaneous transformations of time and energy, such that the runtime of any adiabatic algorithm can be reduced to constant if the energy of the system is modified accordingly \cite{das2003}.
Here we want to exclude a mixing of these effects and will therefore assume
\begin{eqnarray} \label{Etraceconst} {\rm Tr}\{H[s(t)]\} = {\rm const.} \qquad \forall\; s\in[0,1] \,, \end{eqnarray}
where $0 \le s(t) \le 1$ is an interpolation function which will be specified below.
In practice, the above condition can even be relaxed to the demand that the trace should not vary by orders of magnitude (during $0 \le s \le 1$).
With suitable initial and final Hamiltonians $H_{\rm i}$ and $H_{\rm f}$, the above condition can be satisfied for all $s$ by using the linear interpolation scheme
\begin{eqnarray} \label{Ehamiltonevolution} H(t) = \left[1-s(t)\right] H_{\rm i} + s(t) H_{\rm f}\,, \end{eqnarray}
but other schemes are also possible (see section \ref{Sextension}).
For simplicity, we restrict our considerations in this section to a non-degenerate (instantaneous) ground state $n=0$ and one single first exited state $m=1$ with $\Delta E=\Delta E_{10}$.
(Multiple excited states will be discussed in section \ref{Sextension}.)
Similarly, all energies will be normalized in units of a typical energy scale corresponding to the initial/final gap, i.e., $\Delta E(0)={\cal O}(1)$ and $\Delta E(1)={\cal O}(1)$.
We classify the dynamics of $s(t)$ via a function $h(s)\geq 0$
\begin{eqnarray} \label{Eclassify} \tdiff{s}{t} = \Delta E(s) h(s) \,, \end{eqnarray}
where the function $h(s)\geq0$ is constrained by the conditions $s(0)=0$ and $s(T)=1$.
Insertion of this ansatz into Eq.~(\ref{Ecoeff2}) yields the exact formal expression for the non-adiabatic corrections to a system starting in the ground state, i.~e., with $a_1(0)=0$ one obtains after time $T$
\begin{eqnarray} \label{Eformal} a_1(1) e^{-i \gamma_1(1)} &=& - \int\limits_0^1 ds\; a_0(s)e^{-i \gamma_1(s)} \,\frac{F_{01}(s)}{\Delta E(s)} \times \nonumber\\ && \times \exp\left\{-i\int\limits_0^{s}\frac{ds'}{h(s')}\right\} \,, \end{eqnarray}
with the matrix elements $F_{nm}(s)=\bra{m(s)}H'(s)\ket{n(s)}$ which simplify in the case~(\ref{Ehamiltonevolution}) of linear interpolation to $F_{nm}(s)=\bra{m(s)}(H_{\rm f}-H_{\rm i})\ket{n(s)}$.
The advantage of the form in Eqs.~(\ref{Eclassify}) and (\ref{Eformal}) lies in the fact that different time-dependences $s(t)$ and hence different choices for $h(s)$ solely modify the exponent.
We assume that all involved functions can be analytically continued into the complex $s$-plane and are well-behaved near the real $s$-axis.
Given this assumption, we may estimate the integral in Eq.~(\ref{Eformal}) via deforming the integration contour into the lower complex half-plane (to obtain a negative exponent -- which is the usual procedure in such estimates) until we hit a saddle point, a singularity, or a branch cut, see Fig.~\ref{Fintpath}.
Deforming the integration contour into the upper complex half-plane would of course not change the result, but there the integrand is exponentially large and strongly oscillating such that the integral is hard to estimate.
Since the gap $\Delta E(s)$ usually has a pronounced minimum at $s_{\rm min}\in(0,1)$, the first obstacle we encounter \cite{well-behaved} will be a singularity at $\tilde{s}$ close to the real axis, i.e., $\abs{\Im(\tilde{s})}\ll1$ and $\Re(\tilde{s})\approx s_{\rm min}$, where $\Delta E(\tilde{s})=0$.
\begin{figure}
\caption{[Color Online] The original integration contour (black line along real axis) of equation~(\ref{Eformal}) is shifted to the complex plane (curved line).
The gap structure $\Delta E(s)$ leads to singularities near the real axis [green hollow circles, here displayed for $2a=4$ in
Eq.~(\ref{Eenergy_gap})], which limit the deformation of the integration contour.
The integral in the exponent (dashed line) in equation~(\ref{Eformal}) ranges from $0$ to $s'$, which gives rise to a real contribution to the exponent off the real axis only.}
\label{Fintpath}
\end{figure}
Let us first consider a constant function $h(s)=h$:
Assuming $h \ll 1$ (i.e., slow evolution), the exponent in Eq.~(\ref{Eformal}) acquires a large negative real part for $\Im(s)<0$ and thus the absolute value of the integrand decays rapidly if we depart from the real $s$-axis in the lower complex half-plane.
Imposing the even stronger constraint $h\ll\abs{\Im(\tilde{s})}\ll1$, the decay of the exponent dominates all the other $s$-dependences [$\gamma_1(s)$, $F_{01}(s)$, and $\Delta E(s)$] since their typical (minimum \cite{well-behaved}) scale of variation is $\abs{\Im(\tilde{s})}\ll1$.
In view of the complex continuation of Eq.~(\ref{Ecoeff2}), the same applies to the amplitude $a_0(s)$.
As a result, the above integral~(\ref{Eformal}) will be exponentially suppressed $\sim\exp\{-{\cal O}(\abs{\Im(\tilde{s})}/h)\}$ if $h\ll\abs{\Im(\tilde{s})}\ll1$ holds, which (as one would expect) implies a large evolution time $T$ via the side condition $s(T)=1$.
The general situation with varying $h(s)$ can be treated in complete analogy -- the integral in Eq.~(\ref{Eformal}) is suppressed provided that the condition
\begin{eqnarray} \label{Ecriterion} h(0)+h(1)\ll1 \,\wedge\, \Re\left(i\int\limits_0^{\Re(\tilde{s})+i\Im(\tilde{s})/2} \frac{ds}{h(s)}\right) \gg1 \end{eqnarray}
holds for all singularities~$\tilde{s}$ (and saddle points etc.) in the lower complex half-plane (which determine the deformation of the integration contour).
Together with \begin{eqnarray} \label{Erun-time} T=\int\limits_0^1\frac{ds}{\Delta E(s) h(s)} \,, \end{eqnarray} this determines an upper bound for the necessary runtime $T$ of the quantum adiabatic algorithm.
Note that the constraint $\dot s\ll\abs{\Im(\tilde{s})}\Delta E$ derived from $h\ll\abs{\Im(\tilde{s})}$ is not necessarily equivalent to $\dot s\ll\Delta E^2$, which one would naively deduce from Eq.~(\ref{Eadiabatic_old}).
\section{Evolution time}
The general criterion in Eq.~(\ref{Ecriterion}) can now be used to estimate the necessary run-time via Eq.~(\ref{Erun-time}).
Typically, the inverse energy gap $1/\Delta E(s)$ is strongly peaked (along the real axis) around $\Re(\tilde{s})$ with a width \cite{well-behaved} of order $\abs{\Im(\tilde{s})}$.
Therefore, assuming $h(s)$ to be roughly constant across the peak and respecting $h\mid_{\rm peak}\ll\abs{\Im(\tilde{s})}$, yields the following estimate of the integral in Eq.~(\ref{Erun-time})
\begin{eqnarray} \label{inverse} T={\cal O}\left(\Delta E_{\rm min}^{-1}\right) \,, \end{eqnarray}
where $\Delta E_{\rm min}$ denotes the minimum energy gap.
Note that this estimate is only valid for one (or a few) relevant excited state(s) -- multiple excited states will be discussed in section \ref{Sextension}.
Intuitively, the same order of magnitude estimate for the evolution time can also be derived from the local adiabatic condition (\ref{Eadiabatic_old}):
Inverting this condition, we find the relationship
\begin{eqnarray} \label{inverse-adiabatic} T=\frac1\varepsilon \int\limits_0^1 ds\, \frac{F_{01}(s)}{\Delta E^2(s)} \,. \end{eqnarray}
Assuming that $F_{01}(s)$ does not oscillate strongly, e.g., that the ground state of $H(s)$ travels on a reasonably direct path from the initial to the final state, we can make the following estimate
\begin{eqnarray} T= \frac{{\cal O}(\Delta E_{\rm min}^{-1})}{\varepsilon} \int\limits_0^1 ds\, \frac{F_{01}(s)}{\Delta E(s)} \,. \end{eqnarray}
Now we may exploit the advantage of the representation in Eq.~(\ref{Eformal}), which is valid for general dynamics $s(t)$ corresponding to different functions $h(s)$ and hence for arbitrary evolution times $T$.
In the limit of very fast evolution $T\to 0$ (which implies $h \to \infty$), we have large excitations $a_1(T)={\cal O}(1)$ and thus the remaining integral in the above equation can be estimated via inserting this limit into Eq.~(\ref{Eformal}):
\begin{eqnarray} \label{fast} \int\limits_0^1 ds\, \frac{F_{01}(s)}{\Delta E(s)} ={\cal O}(1) \,. \end{eqnarray}
By comparing Eqs.~(\ref{fast}) and (\ref{inverse-adiabatic}), we again obtain the estimate (\ref{inverse}).
Note that the quantities $F_{01}(s)$ and $\Delta E(s)$ appearing in the integrals in Eqs.~(\ref{inverse-adiabatic}-\ref{fast}) do not depend on the dynamics $s(t)$ which allows us to perform the integration independently of $s(t)$.
\begin{figure}
\caption{ [Color Online] Runtime scaling of the adiabatic Grover search for different interpolation functions $s(t)$ and a target fidelity of $3/4$.
Solid lines represent fits to full symbol data for $N \ge 100$ and shaded regions correspond to fit uncertainties (99\% confidence level).
These uncertainties arise from the finite resolution when determining the necessary runtime.
Hollow circles represent calculations with smoothed $C^\infty$-interpolations (compare dotted lines in figure \ref{Ffullcomp} and section \ref{Sextension}), whereas hollow boxes correspond to the nonlinear interpolation example in section \ref{Sextension}. }
\label{Fruntime}
\end{figure}
\subsection{Gap Structure}
Let us illustrate the above considerations by means of the rather general ansatz for the behavior of the gap
\begin{eqnarray} \label{Eenergy_gap} \Delta E(s)=\left[(s-s_{\rm min})^{2a}+\Delta E^b_{\rm min}\right]^{1/b} \,, \end{eqnarray}
with the minimal gap $0<\Delta E_{\rm min}\ll1$ at $s_{\rm min}\in(0,1)$, $b>0$, and $a\in{\mathbb N}_+$.
An avoided level crossing in an effectively two-dimensional subspace corresponds to $2a=b=2$. This is the typical situation if the commutator of the initial and the final Hamiltonian $[H_{\rm i},H_{\rm f}]$ is small, since, in this case, the two operators can almost be diagonalized independently and thus the energy levels are are nearly straight lines except at the avoided level crossing(s), where $[H_{\rm i},H_{\rm f}]$ becomes important.
In the continuum limit, such an (Landau-Zener type) avoided level crossing corresponds to a second-order quantum phase transition.
The finite-size analogue of a third-order phase transition corresponds to $a=b$ (and accordingly for even higher orders), which may occur if $[H_{\rm i},H_{\rm f}]$ is not small or if the interpolation is not linear, i.e., $H(s)\neq[1-s] H_{\rm i} + s H_{\rm f}$.
The inverse gap $1/\Delta E(s)$ has singularities around $s_{\rm min}$ at $\Im(\tilde{s})={\cal O}(\Delta E_{\rm min}^{b/2a})$, compare Fig.~\ref{Fintpath}.
The total running time $T$ for different choices of ${h(s)=\alpha_d \Delta E^d(s)}$ satisfying the criterion~(\ref{Ecriterion}) can be obtained from Eq.~(\ref{Erun-time}).
Here, the exponent $d$ determines the scaling of the interpolation dynamics, whereas the coefficient $\alpha_d$ is adapted such that $s(T)=1$, cf.~Eqs.~(\ref{Eclassify}) and (\ref{Erun-time}).
For \mbox{$2a(d+1)/b>1$} one easily shows that \mbox{$1/\alpha_d={\cal O}(T\Delta E_{\rm min}^{d+1-b/2a})$} satisfies the criterion~(\ref{Ecriterion}) with the evolution time obeying \mbox{$T={\cal O}(\Delta E_{\rm min}^{-1})$}.
If $d$ is smaller, the necessary evolution time will be larger.
In Table~\ref{Tscaling}, the scaling of the run-time (for two examples of the gap structure) is derived for three cases:
\begin{itemize} \item[a)] constant velocity $\dot s=\alpha_{-1}$, i.e., $d=-1$, \item[b)] constant function $h(s)=\alpha_0$, i.e., $d=0$, and \item[c)] the local adiabatic evolution with $h(s) = \alpha_1 \Delta E(s)$, i.e., $d=+1$, investigated in~\cite{roland2002}. \end{itemize}
\begin{table}[h]
\begin{tabular}{|c|c|c|} \hline $\Delta E(s)=$ & $\sqrt{(s-1/2)^2+\Delta E^2_{\rm min}}$ & $\sqrt{(s-1/2)^4+\Delta E^2_{\rm min}}$ \\ \hline $d=-1$ & $\Delta E^{-2}_{\rm min}$ & $\Delta E^{-3/2}_{\rm min}$ \\ $d=0$ & $\Delta E^{-1}_{\rm min}\ln\Delta E^{-2}_{\rm min}$ & $\Delta E^{-1}_{\rm min}$ \\ $d \ge 1$ & $\Delta E^{-1}_{\rm min}$ & $\Delta E^{-1}_{\rm min}$ \\ \hline \end{tabular} \caption{\label{Tscaling} Scaling of the runtime $T$ necessary to obtain a fixed fidelity for different gap structures (top row) and varying interpolation velocities (first column).
The best improvement possible scales as the inverse of the minimum gap $\Delta E^{-1}_{\rm min}$.} \end{table}
\subsection{Grover's Algorithm}\label{Grover}
In the frequently studied adiabatic realization of Grover's algorithm (see, e.g., \cite{childs2002,das2003,roland2002}) the initial Hamiltonian reads \mbox{$H_{\rm i}=\f{1}-\ket{{\rm in}}\bra{{\rm in}}$} with the initial superposition state \mbox{$\ket{{\rm in}}=\sum_{x=0}^{N-1}\ket{x}/\sqrt{N}$}, and the final Hamiltonian is given by \mbox{$H_{\rm f}=\f{1}-\ket{w}\bra{w}$}, where $\ket{w}$ denotes the marked state.
In this case, the commutator is very small $[H_{\rm i},H_{\rm f}]= (\ket{{\rm in}}\bra{w}-\ket{w}\bra{{\rm in}})/\sqrt{N}$ and one obtains for the time-dependent gap \cite{roland2002}
\begin{eqnarray} \label{Egap} \Delta E(s) &=& \sqrt{1 - 4\left(1-\frac{1}{N}\right)s(1-s)} \nonumber\\ &\approx& \sqrt{4\left(s-\frac12\right)^2+\frac1N} \,. \end{eqnarray}
Comparing with Eq.~(\ref{Eenergy_gap}), we identify \mbox{$\Delta E_{\rm min}\approx 1/\sqrt{N}$} and $2a=b=2$ (the pre-factor does not affect the scaling behavior).
Consequently, our analytical estimate implies $T={\cal O}(N)$ for $d=-1$, $T={\cal O}(\sqrt{N}\ln 4 N)$ for $d=0$, and $T={\cal O}(\sqrt{N})$ for $d>0$.
We have solved the Schr\"odinger equation numerically by using a fourth order Runge-Kutta integration scheme with an adaptive step-size \cite{press1994}.
By restarting the code with different $T$ until agreement with desired fidelity was sufficient, we could confirm these runtime scaling predictions numerically, see Fig.~\ref{Fruntime}.
The dependence of the final error on the run-time $T$ for fixed $N=100$ and constant $h$ is depicted in Fig.~\ref{exp-decay}, where the exponential decay becomes evident.
The evolution of the instantaneous ground-state occupation is plotted in Fig.~\ref{Ffullcomp} for the three different dynamics.
\begin{figure}
\caption{[Color Online]
Final error probability $|a_1(T)^2|$ as a function of run-time $T$ for Grover's algorithm with $N=100$ and $h=\rm const$.
The oscillations stem from the time-dependence of $a_0$ in Eq.~(\ref{Eformal}).
The solid (blue) line represents the second-order perturbative solution of Eq.~(\ref{Eformal}).}
\label{exp-decay}
\end{figure}
\begin{figure}
\caption{[Color Online] Evolution of the interpolation function $s(t)$ ({\bf bottom panel}), the spectrum $\sigma[s(t)]$ ({\bf middle panel}), and the occupation of the instantaneous ground state ({\bf top panel}) versus the rescaled time $\tau=t/T$ for an adiabatic Grover search problem with $N=100$ states.
For each interpolation (different line styles), $T$ was adapted to reach 99\% of final fidelity. Thin dotted lines represent $C^\infty$-interpolations smoothed with a test function.}
\label{Ffullcomp}
\end{figure}
\section{Further generalizations}\label{Sextension}
\subsection{Adiabatic Switching}
From an experimental point of view, the time-dependence of the Hamiltonian will most certainly vanish asymptotically $\dot H(t<0)=\dot H(t>T)=0$ or at least be negligible -- which automatically implies $h(0)=h(1)=0$.
Furthermore, realistic Hamiltonians should be described by $C^\infty$-interpolations ({\em Natura non facit saltus}).
By using a $C^\infty$-test function which was matched at $t_1=0.1 T$ and $t_2=0.9 T$ to the usual dynamics $s(t)$ (compare dotted lines in figure \ref{Ffullcomp} bottom panel), we have implemented an interpolation scheme with such an adiabatic switching on and off $\dot{s}(0) = \dot{s}(T) = 0$.
For the investigated adiabatic implementation of the Grover search routine, this scheme does not affect the final result considerably.
The reason for this robustness lies in the fact that the matrix element $F_{nm}$ is peaked around $s=1/2$ and $h(0)$ as well as $h(1)$ are small enough already without the adiabatic switching on and off.
Therefore, one can expect the dominant non-adiabatic corrections to arise from the behavior around the minimum gap, which was unaffected by the test function.
This is also confirmed by the scaling of the runtime versus the system size, compare the hollow circle symbols in figure \ref{Fruntime}, which is basically unchanged.
However, the situation is completely different for the example considered in section \ref{Degeneracy} below.
There, the exponential suppression of the final error as a function of the run-time requires a smooth $C^\infty$-interpolation -- with other dynamics such as $C^0$ (just continuous) or $C^1$ (differentiable once), the final error is merely polynomially small, cf.~figure~\ref{Ferror_example}.
\subsection{Nonlinear Interpolation}
Although we have chosen a linear interpolation scheme~(\ref{Ehamiltonevolution}) in order to satisfy the trace constraint~(\ref{Etraceconst}), the presented analysis can be generalized easily to more general non-linear interpolations.
[Note that, {\em linear} refers to the straight connection line between initial and final Hamiltonian in equation (\ref{Ehamiltonevolution}) and should not be confused with the different velocities $s(t)$ at which this line is traversed.]
The argumentation based on the analytic continuation works in the same way provided that the functional dependence $H_{\rm nl}(s)=f(H_{\rm i},H_{\rm f},s)$ does not involve extremely large or small numbers.
As an illustrative example, we consider the Grover search with the same initial and final Hamiltonians but a quadratic interpolation scheme
\begin{eqnarray} H_{\rm nl}(s) &=& [(1-s) H_{\rm i} + s H_{\rm f}]^2 + s(1-s)\,\frac{2N - 2}{N^2}\,\f{1} \nonumber\\ &=& (1-s)^2 H_{\rm i} + s^2 H_{\rm f} \nonumber\\ &&+ s(1-s)\left[\{H_{\rm i}, H_{\rm f}\} + \frac{2N - 2}{N^2}\,\f{1}\right]\,, \end{eqnarray}
where $\{\cdot,\cdot\}$ denotes the anti-commutator.
The identity operator $\f{1}$ has been added in order to ensure ${\rm Tr}\{H_{\rm nl}\}=N-1$, cf.~equation~(\ref{Etraceconst}).
Although the spectrum of this non-linear interpolation is slightly distorted compared to the linear one, the fundamental gap is the same as in equation (\ref{Egap}), and hence same interpolation functions $s(t)$, applied to the above Hamiltonian, should reproduce the aforementioned scaling predictions.
This is confirmed by the numerical analysis of the scaling behavior -- the results of the non-linear interpolation are basically indistinguishable from those of the previous example (linear interpolation), compare the hollow box symbols in figure~\ref{Fruntime}.
\begin{figure}
\caption{[Color Online] Evolution of the final and the maximum intermediate (red line) excitations with the runtime $T$ for the example~(\ref{Eexample_igs}).
The exponential falloff in the final excitations is only visible, if a smooth $C^\infty$-interpolation (black circles) is used, whereas the scaling of the intermediary excitations (red line) is always polynomial.
The suppression of the final error for $C^0$ or $C^1$-interpolations (blue squares and green crosses) is also merely polynomial.}
\label{Ferror_example}
\end{figure}
\subsection{Degeneracy}\label{Degeneracy}
So far, we have restricted our considerations to the instantaneous ground state and a single first excited state.
Let us now consider a very simple example (see also \cite{farhi0512159}) in which there is still a unique ground state, but many degenerate first excited states:
In terms of single-qubit Pauli matrices $\sigma_x$ and $\sigma_z$, the $M$-qubit Hamiltonian reads
\begin{eqnarray} \label{Eexample_igs} H(s)=\frac{1}{2}\sum_{j=1}^M \left[\f{1} - s\sigma_z - (1-s)\sigma_x\right]^{(j)} \,, \end{eqnarray}
where we have used a linear interpolation~(\ref{Ehamiltonevolution}) for simplicity.
In this example, the Hamiltonian can be decomposed completely into independent and equal single-qubit contributions and hence the time-evolution operator factorizes, i.e., it is sufficient to solve the dynamics of a single qubit.
Furthermore, the Hamiltonian is invariant under any permutation of the qubits.
The instantaneous ground states for all values of $s$ are symmetric under this permutation group and hence unique, but the first excited states are not -- leading to a $M$-fold degeneracy (i.e., there are $M$ equivalent first excited states).
Hence, the fundamental gap between the ground state and each one of these first excited states is the same as for one qubit and thus independent of the number of qubits $\Delta E(s)=\sqrt{1-2 s(1-s)}$.
In some sense, this simple example represents a limiting case opposite to Grover's algorithm:
The energy gap $\Delta E(s)$ and the matrix elements $F_{nm}(s)$ do not scale with the number $M$ of qubits and the $F_{nm}$ are neither small initially nor finally.
Instead, the scaling with system size manifests itself in the $M$-fold degeneracy of the first excited states.
As a result of the $M$-independent gap structure, the adiabatic switching is crucial for achieving the exponential suppression of the final error.
Figure~\ref{Ferror_example} displays the final error probabilities for a smooth $C^\infty$-interpolation and for $C^0$ and $C^1$-interpolations for comparison.
These numerical simulations confirm that the falloff is exponential in the $C^\infty$-case but merely polynomial for $C^0$ and $C^1$.
\begin{figure}
\caption{[Color Online] Occupation of the instantaneous ground state and some selected computational basis states for the Hamiltonian in~(\ref{Eexample_igs}) for an $M=8$ qubit system.
Temporarily, the system leaves the instantaneous ground state, but the runtime $T$ has been adjusted such that the final fidelity is 99\%. }
\label{Fexample_igs}
\end{figure}
Another interesting point of this simple example is the difference between the intermediate and the final occupation of the ground state, see figures~\ref{Fexample_igs} and \ref{Ferror_example}.
According to the first-order result in Eq.~(\ref{Efirst_order}) and the aforementioned factorization of the time-evolution operator, the intermediate excitation probability scales as
\begin{eqnarray} \label{first-order-error} p_{\rm int} =
\sum\limits_{m>0}|a_m|^2 = {\cal O}\left(\frac{M}{T^2\Delta E^4}\right) = {\cal O}\left(\frac{M}{T^2}\right) \,, \end{eqnarray}
since the gap $\Delta E$ is independent of $M$.
On the other hand, the final error probability (assuming a $C^\infty$-interpolation) is exponentially suppressed
\begin{eqnarray} \label{final-error} p_{\rm fin} ={\cal O}\left(M\exp\left\{-T\Delta E\right\}\right) ={\cal O}\left(M\exp\left\{-T\right\}\right) \,, \end{eqnarray}
and hence the two error probabilities can be vastly different $p_{\rm int} \gg p_{\rm fin}$, cf.~figure~\ref{Fexample_igs}.
In fact, by increasing the number of qubits, the occupancy of the instantaneous ground state can be made arbitrarily small.
Moreover, the run-time condition derived from the first-order result in Eqs.~(\ref{Efirst_order}) and (\ref{first-order-error})
\begin{eqnarray} \label{first-order-run-time} T_0={\cal O}(\sqrt{M}) \,, \end{eqnarray}
yields a scaling which is far too pessimistic compared with the correct final error probability assuming a $C^\infty$-interpolation
\begin{eqnarray} \label{final-run-time} T_\infty={\cal O}(\ln M) \,. \end{eqnarray}
Note that non-smooth interpolations (e.g., $C^0$ or $C^1$) would also yield a polynomial scaling $T={\cal O}(M^x)$ similar to Eq.~(\ref{first-order-run-time}).
On the other hand, the scaling behavior in Eqs.~(\ref{final-error}) and (\ref{final-run-time}) is just what one would obtain by immersing the system in Eq.~(\ref{Eexample_igs}) into a zero-temperature environment and letting it decay towards its ground state.
Therefore, using non-smooth interpolations (e.g., $C^0$ or $C^1$) or naively demanding the first-order estimate in Eq.~(\ref{Efirst_order}), the adiabatic algorithm would be even slower than this simple decay mechanism.
\section{Summary}
The instantaneous occupation of the first excited state {\em during} the adiabatic evolution in Eqs.~(\ref{Efirst_order}) and (\ref{Eadiabatic_old}) does not provide a good error estimate.
Instead, a better estimate is given by the remaining real excitations {\em after} the dynamics.
For the example plotted in Fig.~\ref{Ffullcomp}, the instantaneous excitation probability exceeds 10\% at intermediate times -- whereas the final value is 1\%.
This is even more drastic for the example in section~\ref{Degeneracy}, see figure~\ref{Fexample_igs}, where the two values and hence the inferred run-times can differ by orders of magnitude.
Moreover, the final error can be made extremely -- in fact, with $h(0)+h(1)\lll1$, exponentially -- small
\begin{eqnarray} \label{summary-eq} a_1(T)={\cal O}\left(h(0)+h(1)+ \exp\left\{-\frac{\abs{\Im(\tilde{s})}}{h(s_{\rm min})}\right\}\right) \,, \end{eqnarray}
cf.~Fig.~\ref{exp-decay}.
For the Grover example, the last term was dominant, whereas in the general case the smallness of the first two terms can be ensured by using smoothed $C^\infty$-interpolations, i.e., adiabatic switching -- which is a more realistic ansatz anyway.
Based on general arguments, the optimal run-time (in the absence of degeneracy, cf.~section~\ref{Degeneracy}) scales as \mbox{$T={\cal O}(\Delta E^{-1}_{\rm min})$} contrary to what one might expect from the Landau-Zener \cite{landau-zener} formula (with \mbox{$T\propto\Delta E^{-2}_{\rm min}$}).
In view of the fact that the minimum energy gap $\Delta E_{\rm min}$ is a measure of the coupling between the known initial state and the unknown final state, this result is very natural.
For the Grover algorithm, it is known that the $\sqrt{N}$-scaling is optimal \cite{roland2002}.
This optimal scaling \mbox{$T={\cal O}(\Delta E^{-1}_{\rm min})$} can already be achieved with interpolation functions $s(t)$ which vary less strongly (e.g., $d=0$) than demanded by locally \cite{roland2002} adiabatic evolution ($d=1$) -- and hence should be easier to realize experimentally.
Unfortunately, a constant velocity with $d=-1$ does not produce the optimal result in general.
The Grover example has the advantage that the spectrum can be determined analytically, which is for example not the case for the more involved satisfiability problems \cite{farhi2000}.
Therefore, some knowledge of the spectral properties $\Delta E(s)$ is necessary for achieving the optimal result \mbox{$T={\cal O}(\Delta E^{-1}_{\rm min})$} also in the general case of adiabatic quantum computing.
For systems with an analytically unknown gap structure, some knowledge about the spectrum can be obtained by extrapolating the scaling behavior of small systems.
A related interesting point is the impact of the gap structure (corresponding to $2^{\rm nd}$ or $3^{\rm rd}$ order transition etc.) in Eq.~(\ref{Eenergy_gap}).
The derived constraint for the velocity at the transition $\dot s\ll\abs{\Im(\tilde{s})}\Delta E$ is only for $2^{\rm nd}$-order transitions equivalent to $\dot s\ll\Delta E^2$, which one would naively deduce from Eq.~(\ref{Eadiabatic_old}).
Note that the improvement \mbox{$T={\cal O}(\Delta E^{-1}_{\rm min})$} compared with the conventional runtime estimate \mbox{$T={\cal O}(\Delta E^{-2}_{\rm min})$} is merely polynomial (same complexity class).
Though this is not as impressive as an exponential speedup, in practice a polynomial improvement may be useful.
For time-dependent Hamiltonians where the inverse of the minimum gap scales exponentially with the size of the problem, we would still expect an exponential scaling of the runtime $T$ required to reach a fixed fidelity (as in section~\ref{Grover}).
On the other hand, the exponential suppression of the final error in Eq.~(\ref{summary-eq}) may become important in certain cases such as in the presence of degeneracy and may well yield an exponential speedup in comparison with the conventional estimate, see section~\ref{Degeneracy}.
In some sense, the two examples in sections~\ref{Grover} and~\ref{Degeneracy} represent two simple extremal examples for adiabatic quantum computing regarding the scaling of the gap and the degeneracy.
For more complicated situations such as satisfiability problems \cite{farhi2000}, both properties have to be taken into account simultaneously.
\section*{Acknowledgments}
R.~S.~acknowledges fruitful discussions during the workshop "Low dimensional Systems in Quantum Optics" at the CIC in Cuernavaca (Mexico), which was supported by the Humboldt foundation.
G.~S.~acknowledges fruitful discussions with M.~Tiersch.
This work was supported by the Emmy Noether Programme of the German Research Foundation (DFG) under grant No.~SCHU~1557/1-1/2.
\section*{Note added}
Recently, two of the main results of this article, i.e., the optimal run-time scaling \mbox{$T={\cal O}(\Delta E^{-1}_{\rm min})$} and the faster-than-polynomial decrease of the final error $a_1(T)$, have been demonstrated rigorously for a class of Hamiltonians using methods of spectral analysis \cite{jansen0603175}.
\end{document} | arXiv |
Tag Archives: Benjamin Rin
Transfinite recursion as a fundamental principle in set theory
Posted on October 20, 2014 by Joel David Hamkins
$\newcommand\dom{\text{dom}}
\newcommand\ran{\text{ran}}
\newcommand\restrict{\upharpoonright}$
At the Midwest PhilMath Workshop this past weekend, I heard Benjamin Rin (UC Irvine) speak on transfinite recursion, with an interesting new perspective. His idea was to consider transfinite recursion as a basic principle in set theory, along with its close relatives, and see how they relate to the other axioms of set theory, such as the replacement axiom. In particular, he had the idea of using our intuitions about the legitimacy of transfinite computational processes as providing a philosophical foundation for the replacement axiom.
This post is based on what I learned about Rin's work from his talk at the workshop and in our subsequent conversations there about it. Meanwhile, his paper is now available online:
Benjamin Rin, Transfinite recursion and the iterative conception of set, Synthese, October, 2014, p. 1-26. (preprint).
Since I have a little different perspective on the proposal than Rin did, I thought I would like to explain here how I look upon it. Everything I say here is inspired by Rin's work.
To begin, I propose that we consider the following axiom, asserting that we may undertake a transfinite recursive procedure along any given well-ordering.
The Principle of Transfinite Recursion. If $A$ is any set with well-ordering $<$ and $F:V\to V$ is any class function, then there is a function $s:A\to V$ such that $s(b)=F(s\upharpoonright b)$ for all $b\in A$, where $s\upharpoonright b$ denotes the function $\langle s(a)\mid a<b\rangle$.
We may understand this principle as an infinite scheme of statements in the first-order language of set theory, where we make separate assertions for each possible first-order formula defining the class function $F$, allowing parameters. It seems natural to consider the principle in the background theory of first-order Zermelo set theory Z, or the Zermelo theory ZC, which includes the axiom of choice, and in each case let me also include the axiom of foundation, which apparently is not usually included in Z. (Alternatively, it is also natural to consider the principle as a single second-order statement, if one wants to work in second-order set theory.)
Theorem. (ZC) The principle of transfinite recursion is equivalent to the axiom of replacement. In other words,
ZC + transfinite recursion = ZFC.
Proof. Work in the Zermelo set theory ZC. The converse implication amounts to the well-known observation in ZF that transfinite recursion is legitimate. Let us quickly sketch the argument. Suppose we are given an instance of transfinite recursion, namely, a well-ordering $\langle A,<\rangle$ and a class function $F:V\to
V$. I claim that for every $b\in A$, there is a unique function $s:\{a\in A\mid a\leq b\}\to V$ obeying the recursive rule $s(d)=F(s\upharpoonright d)$ for all $d\leq b$. The reason is that there can be no least $b$ without such a unique function. If all $a<b$ have such a unique function, then by uniqueness they must cohere with one another, since any difference would show up at a least stage and thereby violate the recursion rule, and so by the replacement axiom of ZFC we may assemble these smaller functions into a single function $t$ defined on all $a<b$, and satisfying the recursion rule for those values. We may then extend this function $t$ to be defined on $b$ itself, simply by defining $u(b)=F(t)$ and $u\upharpoonright b=t$, which thereby satisfies the recursion at $b$. Uniqueness again follows from the fact that there can be no least place of disagreement. Finally, using replacement again, let $s(b)$ be the unique value that arises at $b$ during the recursions that work up to and including $b$, and this function $s:A\to V$ satisfies the recursive definition.
Conversely, assume the Zermelo theory ZC plus the principle of transfinite recursion, and suppose that we are faced with an instance of the replacement axiom. That is, we have a set $A$ and a formula $\varphi$, where every $b\in A$ has a unique $y$ such that $\varphi(b,y)$. By the axiom of choice, there is a well-ordering $<$ of the set $A$. We shall now define the function $F:V\to V$. Given a function $s$, where $\dom(s)=\{a\in A\mid a<b\}$ for some $b\in A$, let $F(s)=y$ be the unique $y$ such that $\varphi(b,y)$; and otherwise let $F(s)$ be anything you like. By the principle of transfinite recursion, there is a function $s:A\to V$ such that $s(b)=F(s\upharpoonright b)$ for every $b\in A$. In this case, it follows that $s(b)$ is the unique $y$ such that $\varphi(a,b)$. Thus, since $s$ is a set, it follows in ZC that $\ran(s)$ is a set, and so we've got the image of $A$ under $\varphi$ as a set, which verifies replacement. QED
In particular, it follows that the principle of transfinite recursion implies that every well-ordering is isomorphic to a von Neumann ordinal, a principle Rin refers to as ordinal abstraction. One can see this as a consequence of the previous theorem, since ordinal abstraction holds in ZF by Mostowski's theorem, which for any well-order $\langle A,<\rangle$ assigns an ordinal to each node $a\mapsto \alpha_a$ according to the recursive rule $\alpha_a=\{\alpha_b\mid b<a\}$. But one can also argue directly as follows, without using the axiom of choice. Assume Z and the principle of transfinite recursion. Suppose that $\langle A,<\rangle$ is a well-ordering. Define the class function $F:V\to V$ so that $F(s)=\ran(s)$, whenever $s$ is a function. By the principle of transfinite recursion, there is a function $s:A\to V$ such that $s(b)=F(s\restrict b)=\ran(s\restrict b)$. One can now simply prove by induction that $s(b)$ is an ordinal and $s$ is an isomorphism of $\langle A,<\rangle$ with $\ran(s)$, which is an ordinal.
Let me remark that the principle of transfinite recursion allows us also to perform proper-class length recursions.
Observation. Assume Zermelo set theory Z plus the principle of transfinite recursion. If $A$ is any particular class with $<$ a set-like well-ordering of $A$ and $F:V\to V$ is any class function, then there is a class function $S:A\to V$ such that $S(b)=F(S\upharpoonright b)$ for every $b\in A$.
Proof. Since $\langle A,<\rangle$ is set-like, the initial segment $A\upharpoonright d=\{a\in A\mid a<d\}$, for any particular $d\in A$, is a set. It follows that the principle of transfinite recursion shows that there is a function $s_d:(A\upharpoonright d)\to V$ such that $s_d(b)=F(s_d\upharpoonright b)$ for every $b<d$. It is now easy to prove by induction that these $s_d$ must all cohere with one another, and so we may define the class $S(b)=s_d(b)$ for any $d$ above $b$ in $A$. (We may assume without loss that $A$ has no largest element, for otherwise it would be a set.) This provides a class function $S:A\to V$ satisfying the recursive definition as desired. QED
Although it appears explicitly as a second-order statement "there is a class function $S$…", we may actually take this observation as a first-order theorem scheme, if we simply strengthen the conclusion to provide the explicit definition of $S$ that the proof provides. That is, the proof shows exactly how to define $S$, and if we make the observation state that that particular definition works, then what we have is a first-order theorem scheme. So any first-order definition of $A$ and $F$ from parameters leads uniformly to a first-order definition of $S$ using the same parameters.
Thus, using the principle of transfinite recursion, we may also take proper class length transfinite recursions, using any set-like well-ordered class that we happen to have available.
Let us now consider a weakening of the principle of transfinite recursion, where we do not use arbitrary well-orderings, but only the von Neumann ordinals themselves.
Principle of transfinite recursion on ordinals. If $F:V\to V$ is any class function, then for any ordinal $\gamma$ there is a function $s:\gamma\to V$ such that $s(\beta)=F(s\upharpoonright\beta)$ for all $\beta<\gamma$.
This is a weakening of the principle of transfinite recursion, since every ordinal is well-ordered, but in Zermelo set theory, not every well-ordering is necessarily isomorphic to an ordinal. Nevertheless, in the presence of ordinal abstraction, then this ordinal version of transfinite recursion is clearly equivalent to the full principle of transfinite recursion.
Observation. Work in Z. If every well-ordering is isomorphic to an ordinal, then the principle of transfinite recursion is equivalent to its restriction to ordinals.
Meanwhile, let me observe that in general, one may not recover the full principle of transfinite recursion from the weaker principle, where one uses it only on ordinals.
Theorem. (ZFC) The structure $\langle V_{\omega_1},\in\rangle$ satisfies Zermelo set theory ZC with the axiom of choice, but does not satisfy the principle of transfinite recursion. Nevertheless, it does satisfy the principle of transfinite recursion on ordinals.
Proof. It is easy to verify all the Zermelo axioms in $V_{\omega_1}$, as well as the axiom of choice, provided choice holds in $V$. Notice that there are comparatively few ordinals in $V_{\omega_1}$—only the countable ordinals exist there—but $V_{\omega_1}$ has much larger well-orderings. For example, one may find a well-ordering of the reals already in $V_{\omega+k}$ for small finite $k$, and well-orderings of much larger sets in $V_{\omega^2+17}$ and so on as one ascends toward $V_{\omega_1}$. So $V_{\omega_1}$ does not satisfy the ordinal abstraction principle and so cannot satisfy replacement or the principle of transfinite recursion. But I claim nevertheless that it does satisfy the weaker principle of transfinite recursion on ordinals, because if $F:V_{\omega_1}\to V_{\omega_1}$ is any class in this structure, and $\gamma$ is any ordinal, then we may define by recursion in $V$ the function $s(\beta)=F(s\restrict\beta)$, which gives a class $s:\omega_1\to V_{\omega_1}$ that is amenable in $V_{\omega_1}$. In particular, $s\restrict\gamma\in V_{\omega_1}$ for any $\gamma<\omega_1$, simply because $\gamma$ is countable and $\omega_1$ is regular. QED
My view is that this example shows that one doesn't really want to consider the weakened principle of transfinite recursion on ordinals, if one is working in the Zermelo background ZC, simply because there could be comparatively few ordinals, and this imposes an essentially arbitrary limitation on the principle.
Let me point out, however, that there was a reason we had to go to $V_{\omega_1}$, rather than considering $V_{\omega+\omega}$, which is a more-often mentioned model of the Zermelo axioms. It is not difficult to see that $V_{\omega+\omega}$ does not satisfy the principle of transfinite recursion on the ordinals, because one can define the function $s(n)=\omega+n$ by recursion, setting $s(0)=\omega$ and $s(n+1)=s(n)+1$, but this function does not exist in $V_{\omega+\omega}$. This feature can be generalized as follows:
Theorem. Work in the Zermelo set theory Z. The principle of transfinite recursion on ordinals implies that if $\langle A,<\rangle$ is a well-ordered set, and $A$ is bijective with some ordinal, then $\langle A,<\rangle$ is order-isomorphic with an ordinal.
In other words, we get ordinal abstraction for well-orderings whose underlying set is bijective with an ordinal.
First, the proof of the first theorem above actually shows the following local version:
Lemma. (Z) If one has the principle of transfinite recursion with respect to a well-ordering $\langle A,<\rangle$, then $A$-replacement holds, meaning that if $F:V\to V$ is any class function, then the image $F"A$ is a set.
Proof of theorem. Suppose that $\langle A,<\rangle$ is a well-ordering, and that $A$ is bijective with some ordinal $\kappa$, and that $F:V\to V$ is a class function. Assume the principle of transfinite recursion for $\kappa$. We prove by induction on $d\in A$ that there is a unique function $s_d$ with $\dom(s)=\{a\in A\mid a\leq d\}$ and satisfying the recursive rule $s(b)=F(s\upharpoonright b)$. If this statement is true for all $d<d'$, then because the size of the predecessors of $d'$ in $\langle A,<\rangle$ is at most $\kappa$, we may by the lemma form the set $\{s_d\mid d<d'\}$, which is a set by $\kappa$-replacement. These functions cohere, and the union of these functions gives a function $t:(A\upharpoonright d')\to V$ satisfying the recursion rule for $F$. Now extend this function one more step by defining $s(d')=F(t)$ and $s\upharpoonright d'=t$, thereby handling the existence claim at $d'$. As in the main theorem, all these functions cohere with one another, and by $\kappa$-replacement we may form the set $\{s_d\mid d\in A\}$, whose union is the desired function $s:A\to V$ satisfying the recursion rule given by $F$, as desired. QED
For example, if you have the principle of transfinite recursion for ordinals, and $\omega$ exists, then every countable well-ordering is isomorphic to an ordinal. This explains why we had to go to $\omega_1$ to find a model satisfying transfinite recursion on ordinals. One can understand the previous theorem as showing that although the principle of transfinite recursion on ordinals does not prove ordinal abstraction, it does prove many instances of it: for every ordinal $\kappa$, every well-ordering of cardinality at most $\kappa$ is isomorphic to an ordinal.
It is natural also to consider the principle of transfinite recursion along a well-founded relation, rather than merely a well-ordered relation.
The principle of well-founded recursion. If $\langle A,\lhd\rangle$ is a well-founded relation and $F:V\to V$ is any class function, then there is a function $s:A\to V$ such that $s(b)=F(s\restrict b)$ for all $b\in A$, where $s\restrict b$ means the function $s$ restricted to the domain of elements $a\in A$ that are hereditarily below $b$ with respect to $\lhd$.
Although this principle may seem more powerful, in fact it is equivalent to transfinite recursion.
Theorem. (ZC) The principle of transfinite recursion is equivalent to the principle of well-founded recursion.
Proof. The backward direction is immediate, since well-orders are well-founded. For the forward implication, assume that transfinite recursion is legitimate. It follows by the main theorem above that ZFC holds. In this case, well-founded recursion is legitimate by the familiar arguments. For example, one may prove in ZFC that for every node in the field of the relation, there is a unique solution of the recursion defined up to and including that node, simply because there can be no minimal node without this property. Then, by replacement, one may assemble all these functions together into a global solution. Alternatively, arguing directly from transfinite recursion, one may put an ordinal ranking function for any given well-founded relation $\langle A,\lhd\rangle$, and then prove by induction on this rank that one may construct functions defined up to and including any given rank, that accord with the recursive rule. In this way, one gets the full function $s:A\to V$ satisfying the recursive rule. QED
Finally, let me conclude this post by pointing out how my perspective on this topic differs from the treatment given by Benjamin Rin. I am grateful to him for his idea, which I find extremely interesting, and as I said, everything here is inspired by his work.
One difference is that Rin mainly considered transfinite recursion only on ordinals, rather than with respect to an arbitrary well-ordered relation (but see footnote 17 in his paper). For this reason, he had a greater need to consider whether or not he had sufficient ordinal abstraction in his applications. My perspective is that transfinite recursion, taken as a basic principle, has nothing fundamentally to do with the von Neumann ordinals, but rather has to do with a general process undertaken along any well-order. And the theory seems to work better when one undertakes it that way.
Another difference is that Rin stated his recursion principle as a principle about iterating through all the ordinals, rather than only up to any given ordinal. This made the resulting functions $S:\text{Ord}\to V$ into class-sized objects for him, and moved the whole analysis into the realm of second-order set theory. This is why he was led to prove his main equivalence with replacement in second-order Zermelo set theory. My treatment shows that one may undertake the whole theory in first-order set theory, without losing the class-length iterations, since as I explained above the class-length iterations form classes, definable from the original class functions and well-orders. And given that a completely first-order account is possible, it seems preferable to undertake it that way.
Update. (August 17, 2018) I've now realized how to eliminate the need for the axiom of choice in the arguments above. Namely, the main argument above shows that the principle of transfinite recursion implies the principle of well-ordered replacement, meaning the axiom of replacement for instances where the domain set is well-orderable. The point now is that in recent work with Alfredo Roque Freire, I have realized that The principle of well-ordered replacement is equivalent to full replacement over Zermelo set theory with foundation. We may therefore deduce:
Corollary. The principle of transfinite recursion is equivalent to the replacement axiom over Zermelo set theory with foundation.
We do not need the axiom of choice for this.
Posted in Exposition | Tagged Benjamin Rin, Transfinite recursion | 23 Replies | CommonCrawl |
\begin{document}
\title[]{Overhead for simulating a non-local channel with local channels by quasiprobability sampling}
\author{Kosuke Mitarai} \email{[email protected]} \affiliation{Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.} \affiliation{Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.} \affiliation{JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan.}
\author{Keisuke Fujii} \affiliation{Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.} \affiliation{Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.} \affiliation{Center for Emergent Matter Science, RIKEN, Wako Saitama 351-0198, Japan}
\date{\today}
\begin{abstract}
As the hardware technology for quantum computing advances, its possible applications are actively searched and developed. However, such applications still suffer from the noise on quantum devices, in particular when using two-qubit gates whose fidelity is relatively low. One way to overcome this difficulty is to substitute such non-local operations by local ones. Such substitution can be performed by decomposing a non-local channel into a linear combination of local channels and simulating the original channel with a quasiprobability-based method. In this work, we first define a quantity that we call channel robustness of non-locality, which quantifies the cost for the decomposition. While this quantity is challenging to calculate for a general non-local channel, we give an upper bound for a general two-qubit unitary channel by providing an explicit decomposition. The decomposition is obtained by generalizing our previous work whose application has been restricted to a certain form of two-qubit unitary. This work develops a framework for a resource reduction suitable for first-generation quantum devices. \end{abstract}
\maketitle
\section{Introduction}
We now have a programmable quantum device whose dynamics cannot be simulated by a classical computer within its runtime \cite{Arute2019}. However, the capability of such devices is rather limited because of the absence of the quantum error correction. They are frequently referred to as noisy intermediate scale quantum (NISQ) devices \cite{Preskill2018quantumcomputingin}. There has been a substantial amount of research efforts to develop useful applications of NISQ devices in recent years \cite{Peruzzo2014, RevModPhys.92.015003, Farhi2014,Mitarai2018, farhi2018classification,bravoprieto2019variational,LaRose_2019}. The weakness of NISQ devices is that the number of qubits, the fidelities of gates, and the connectivity are limited. The gate fidelities are especially restricted for two-qubit entangling gates. One approach to circumvent such limitation is to use so-called variational quantum algorithms. \red{They employ parametrized quantum circuits and optimize the parameters to perform a given task. In such algorithms, we frequently construct the largest possible circuit allowed on a device to maximize the advantage of the use of quantum devices.}
While this approach is promising as it can in principle employ such circuits, such algorithms can still be improved if one can perform further resource reduction. For example, if we can reduce the number of qubits or two-qubit gates required to obtain an output from a certain quantum circuit, it would widen the range of circuits that can be used for variational algorithms. To this end, a few approaches have been proposed. One is to decompose a large circuit into smaller ones by ``cutting'' circuits using a tomography-like method \cite{peng2019simulating}. Also, in Ref. \cite{mitarai2019constructing}, we have presented a method to ``cut'' a certain non-local gate by decomposing it into a linear combination of local operations. These approaches share the same property that the overhead for the decomposition, which in this context is defined by the number of circuit runs that is required to achieve a desired accuracy of the output, scales exponentially to the number of cuts performed.
They can be also understood as techniques for performing a quasiprobability decomposition of quantum channels. Quasiprobability distribution, which is defined by a set of complex numbers $\{q_i\}$ satisfying $\sum_i q_i = 1$, have recently found a wide range of applications in the area of quantum computing such as error mitigation for NISQ devices \cite{Temme2017, Endo2018} and classical simulation of near-Clifford quantum circuits \cite{Pashayan2015, Howard2017, Bravyi2016, Bennink2017, Seddon2019}. In particular, Refs. \cite{Bennink2017,Seddon2019,Temme2017,Endo2018} considered a \red{quasiprobability-based simulation of quantum channels; if a quantum channel $\bm{\Phi}$ can be decomposed as $\bm{\Phi}=\sum_i c_i \bm{\Phi}_i$ where $\bm{\Phi}_i$ and $c_i$ are respectively a channel and a complex coefficient, $\bm{\Phi}$ can be simulated by sampling $\bm{\Phi}_i$ with probability proportional to $|c_i|$ and processing the phase of $c_i$ with classical post-processing.}
The overhead of simulating the channel $\bm{\Phi}$ using this decomposition is quantified by $\sum_i |q_i|$. If we perform such a decomposition multiple times, the overhead is quantified by the product of $\sum_i |q_i|$, thus leading to an exponential overhead to the number of decomposition performed. Refs. \cite{Temme2017, Endo2018} have developed techniques to build inverse channels of noise channels using an experimentally available set of quantum gates. As a technique for a classical simulation, Refs. \cite{Bennink2017, Seddon2019} has considered a quasiprobability decomposition of a non-Clifford channel into Clifford ones. In this context, we can view the decomposition performed in Ref. \cite{peng2019simulating} as a quasiprobability decomposition of the identity channel into a measurement and state-preparation channel, and one in Ref. \cite{mitarai2019constructing} as a quasiprobability decomposition of a non-local unitary channel into local ones.
In this work, we first define a quantity that we call channel robustness of non-locality in analog to the robustness of magic introduced in Ref. \cite{Howard2017}, which quantifies the minimal possible overhead that can be achieved for quasiprobabilistic simulation of a non-local channel by local channels. While this quantity is difficult to calculate in general, we show an analytic upper bound for general two-qubit unitary channels by constructing an explicit decomposition, generalizing the technique developed in Ref. \cite{mitarai2019constructing}. Our previous work \cite{mitarai2019constructing} has only considered decomposition of non-local gates expressed in the form of $e^{i\theta A_1\otimes A_2}$ for Hermitian operators satisfying $A_1^2=I$ and $A_2^2=I$. In contrast, the decomposition developed in this work performs the cut of a general two-qubit gate in a single-step, leading to a substantially reduced overhead. Besides the reduced cost, the derivation of the decomposition is delivered more constructively than before which we believe is informative for further optimizations of this approach. While lower bounds of the defined robustness is also of theoretical interest that can characterize quantumness of a non-local channel, in this work, we focus on upper bounds obtained by explicit decompositions which enable us to actually simulate a nonlocal channel by local channels. This work develops a theoretical framework for a resource reduction suitable for first-generation quantum devices.
\section{Decomposition of non-local channels into local channels}
\subsection{Notation}
We use the notation $\kket{\rho}$ to express a density matrix $\rho$ to stress that $\rho$ can also be seen as a vector. Bold-font symbols are to express a quantum channel corresponding to a gate-like operation represented by a normal font. For example, a unitary channel $\bm{U}$ acts on a state $\kket{\rho}$ as $\bm{U}\kket{\rho} = \kket{U\rho U^\dagger}$ where $U$ is a unitary matrix. Inner product between two operators $\kket{A}$ and $\kket{B}$ is defined as $\bbraket{A|B}=\text{Tr}(A^\dagger B)$.
\subsection{Channel robustness of non-locality}
In standard eperimental platforms including superconducting qubits and ion traps, it is often thought that the arbitrary single-qubit rotation charactrized by an axis $n=(n_1,n_2,n_3)$ and an angle $\theta$, $R(n,\theta) = \exp\left[-i\theta (\sum_{\alpha}n_\alpha\sigma_\alpha)\right]$, and the single-qubit projective measurements along any axis are somewhat easier operations than two-qubit entangling operations. Experimentally, the projective measurement is realized by rotating the axis by $R(n,\theta)$ and performing the projective measurement along $z$-axis. The quantum channel $\bm{M}(n)$ corresponding to the projective measurement is a probabilistic map; when applied to a state $\kket{\rho}$, it returns a state $\bm{\Pi}(\pm n)\kket{\rho}/p_+$ with some probability $p_{\pm}$, where $\Pi(\pm n)$ is a projector to an eigenstate of $\pm \sum_\alpha n_\alpha \sigma_\alpha$ with eigenvalue $+1$.
To implement $\bm{\Pi}(n)$ itself, we can define a probabilistic map $\hat{\bm{\Pi}}(n)$ that takes a state $\kket{\rho}$ to $\bm{\Pi}(n)\kket{\rho}/p_+$ with probability $p_+$ and to $\kket{0}$ with probability $p_-$ where $\kket{0}$ corresponds to the zero matrix. The map to $\kket{0}$ means simply to ignore the case when the measurement resulted in $-1$. However, just discarding the $-1$ case is inefficient, especially when we also want to perform $\bm{\Pi}(-n)$. To resolve this issue, we define a probabilistic map $\tilde{\bm{\Pi}}(n,c_+,c_-)$ that takes a state $\rho$ to $c_\pm\bm{\Pi}(\pm n)\kket{\rho}/p_\pm$ with probability $p_\pm$, where $c_{\pm}\in\{0\}\cup\{e^{i\phi}|\phi\in[0,2\pi]\}$. \red{Let us define the expected value of a random vector $\kket{\sigma}$ which becomes $\kket{\sigma_i}$ with a probability $p_i$ as $\mathbb{E}[\kket{\sigma}]:=\sum_i p_i \kket{\sigma_i}$.} Observe that the following holds for any state $\rho$, \begin{align}
\mathbb{E}[\tilde{\bm{\Pi}}(n,c_+,c_-)\kket{\rho}] = c_+\bm{\Pi}(n)\kket{\rho} + c_-\bm{\Pi}(-n)\kket{\rho}. \end{align} We write $\mathbb{E}[\tilde{\bm{\Pi}}(n, c_+, c_-)]=c_+\bm{\Pi}(n) + c_-\bm{\Pi}(-n)$ in this sense and henceforth use the notation like this.
$\tilde{\bm{\Pi}}(n,c_+,c_-)$ includes the both of the \red{cases} which we mentioned earlier; if we want to apply only $\bm{\Pi}(n)$ we can set $c_-=0$, and we can also apply both of $\bm{\Pi}(\pm n)$ simultaneously with different coefficients. The reason we restrict $|c_{\pm}|=1$ is to assure $\left|\text{Tr}[\tilde{\bm{\Pi}}(n,c_+,c_-)\rho]\right|\leq 1$ for any state $\rho$ and any realization of $\tilde{\bm{\Pi}}(n,c_+,c_-)$, thus preventing the decomposition overhead to occur at this stage.
With the above consideration, available local operations in practice, which we denote as $\bm{L}_i$, are the ones that can be written as an arbitrary product of $\bm{R}(n,\theta)$ and $\tilde{\bm{\Pi}}(n)$ and their tensor products. We denote a set of such possible $\bm{L}_i$ by $\mathcal{L}$. The most general form of decomposition that we aim to build for a given non-local quantum channel $\bm{\Phi}$ is, \begin{align}
\bm{\Phi} = \sum_{i} c_i \bm{L}_i, \label{eq:Phi-decomposition} \end{align} where $\bm{L}_i\in \mathcal{L}$.
Given a decomposition above, $\bm{\Phi}$ can be ``simulated'' in a Monte-Carlo manner by sampling $\bm{L}_i$ with probability proportional to $|c_i|$. More concretely, let us define a probabilistic map $\hat{\bm{\Phi}}$ such that it becomes $\frac{c_i}{|c_i|}\bm{L}_i$ with probability $p_i = |c_i|/W(\bm{\Phi})$ where $W(\bm{\Phi}) = \sum_i |c_i|$. Then, \begin{align}
\mathbb{E}[W(\bm{\Phi})\hat{\bm{\Phi}}] &= W(\bm{\Phi})\times \sum_i \frac{|c_i|}{W(\bm{\Phi})}\frac{c_i}{|c_i|} \bm{L}_i \nonumber \\
&= \bm{\Phi}, \end{align} which shows that $W(\bm{\Phi})\hat{\bm{\Phi}}$ becomes equal to $\bm{\Phi}$ when executed for many times. This algorithm involves only local operations with classical communication (LOCC).
\red{However, note that the above protocol is not a simple probabilistic mixture of LOCC as it multiplies the complex coefficient $c_i/|c_i|$ to each channel $\bm{L}_i$.}
Let us now consider the overhead associated with the decomposition. In many cases, the output from a quantum system that is evolved with a channel $\bm{\Phi}$ is an expectation value of an observable $O$, which can be written as $\bbra{O}\bm{\Phi}\kket{\rho}$. $\bbra{O}\bm{\Phi}\kket{\rho}$ is usually estimated by sampling eigenvalues of $O$ from the final state $\bm{\Phi}\kket{\rho}$. Let the sampled $S$ eigenvalues be $\{o_s\}_{s=1}^S$.
Normally, we construct an estimator $\widehat{\braket{O}}$ as $\widehat{\braket{O}}=\frac{1}{S}\sum_s o_s$. Let us assume that absolute value of eigenvalues of $O$ is bounded by $o_{\mathrm{max}}$ and thus $|o_s|\leq o_{\mathrm{max}}$. Then, by Hoeffding's bound, we can assure that $|\widehat{\braket{O}} - \bbra{O}\bm{V}\kket{\rho}|\leq \epsilon$ with probability at least $1-\delta$ if we take $S=2(o_{\mathrm{max}}/\epsilon)^2 \ln[1/(2\delta)]$.
The number of samples required to achieve the same accuracy increases if one tries to simulate $\bm{\Phi}$ with $\hat{\bm{\Phi}}$. Since $\mathbb{E}[W(\bm{\Phi})\hat{\bm{\Phi}}]=\bm{\Phi}$, $\mathbb{E}[W(\bm{\Phi})\bbra{O}\hat{\bm{\Phi}}\kket{\rho}]=\bbra{O}\bm{\Phi}\kket{\rho}$
We can construct an estimator $\widehat{\braket{O}}'$ by $\widehat{\braket{O}}' = \frac{1}{S}\sum_s W(\bm{\Phi}) o_s'$ where $o_s'$ is a sample drawn from $\hat{\bm{\Phi}}\kket{\rho}$ with a single realization of $\hat{\bm{\Phi}}$. The application of $\hat{\bm{\Phi}}$ introduced in the last section involves many stochastic processes; it means to stochastically apply $\bm{L}_i$ with probability $p_i$, and $\bm{L}_i$ itself is a stochastic map involving $\tilde{\bm{\Pi}}(n,c_+,c_-)$. However, in the end, any realization of $\hat{\bm{\Phi}}$ becomes a single-qubit operation that preserves the magnitude of the trace of $\rho$ or maps the state to $\kket{0}$. Therefore, it is guaranteed that the absolute value of a sample $o_s'$ obtained by measuring $O$ of $\hat{\bm{\Phi}}\kket{\rho}$ is also bounded by $o_{\mathrm{max}}$. Again by Hoeffding's bound, $|\widehat{\braket{O}}' - \bbra{O}\bm{\Phi}\kket{\rho}|\leq \epsilon$ with probability at least $1-\delta$ if we take $S=2(W(\bm{\Phi}) o_{\mathrm{max}}/\epsilon)^2 \ln[1/(2\delta)]$. We can see that $W(\bm{\Phi})^2$ amounts to the overhead of the decomposition.
The above discussion leads us to define the following quantity $\bm{}$ which we call the channel robustness of non-locality, \begin{align}
\widetilde{W}(\bm{\Phi}) = \min_{\left\{c_i|\bm{\Phi}=\sum_{i}c_i\bm{L}_i,~\bm{L}_i \in \mathcal{L}\right\}} \sum_i |c_i|. \label{eq:robustness} \end{align} $\widetilde{W}(\bm{\Phi})$ quantifies the minimum amount of cost when we perform the simulation of a non-local channel $\bm{\Phi}$ by probabilistic application of the local, experimentally feasible operations. $\widetilde{W}(\bm{\Phi})$ is submultiplicative, i.e., $\widetilde{W}(\bm{\Phi}_{2}\bm{\Phi}_{1})\leq \widetilde{W}(\bm{\Phi}_2)\widetilde{W}(\bm{\Phi}_1)$, which is proved in Appendix. This allows us to upper-bound the overhead caused by the decomposition of a chain of quantum channels, $\bm{\Phi}_N\cdots \bm{\Phi}_2\bm{\Phi}_1$ by $\prod_{n=1}^N \widetilde{W}(\bm{\Phi}_n)$.
Note that if we change the available set of operations to some other ones from $\mathcal{L}$, Eq. (\ref{eq:robustness}) quantifies the overhead of the decomposition in that case. For example, the overhead of the decomposition of the identity gate presented in Ref. \cite{peng2019simulating} can be quantified by setting the available decomposition to be measure-and-prepare channels. Another example is the decomposition of non-Clifford circuits into stabilizer-preserving channels considered in Refs. \cite{Bennink2017, Seddon2019}. The cost for a family of the error mitigation technique called probabilistic error cancellation \cite{Temme2017, Endo2018} is also in relation to this quantity; it is quantified by substituting the target channel $\bm{\Phi}$ with an inverse of a noise channel.
As $\mathcal{L}$ consists of operations with continous parameters, we can also define $\widetilde{W}(\bm{\Phi})$ using a integral instead of a discrete sum. Formally, we can write, \begin{align}
\widetilde{W}(\bm{\Phi}) = \min_{\left\{c|\bm{\Phi}=\int c(\lambda)\bm{L}(\lambda) d\lambda,~\bm{L}(\lambda) \in \mathcal{L}\right\}} \int |c(\lambda)| d\lambda, \label{eq:robustness-continuous} \end{align} where $\lambda$ denotes some continuous parameters that specifies an element in $\mathcal{L}$.
The calculation of $\widetilde{W}(\bm{\Phi})$ for a general channel $\bm{\Phi}$ is challenging as it involves a complex minimization procedure. Nevertheless, in the next section, we give an upper bound of $\widetilde{W}(\bm{\Phi})$ for a general two-qubit unitary channel $\bm{\Phi}$ by explicitly constructing a decomposition using a complete but not overcomplete basis in $\mathcal{L}$.
\subsection{Upper bound for two-qubit unitary channel}
It is well-known \cite{Kraus2001,Zhang2003} that the non-local part of two-qubit gates can always be written as, \begin{align}
U &= \exp\left[i\left(\sum_{\alpha=1}^3 \theta_\alpha \sigma_\alpha\otimes \sigma_\alpha\right)\right]\nonumber \\
&= \sum_{\alpha=0}^3 u_{\alpha} \sigma_\alpha\otimes\sigma_\alpha \label{eq:two-qubit-unitary}, \end{align} where $\sigma_0$ is the $2\times 2$ identity operator, and $\sigma_1$, $\sigma_2$ and $\sigma_3$ are Pauli $x$, $y$ and $z$ operators, respectively. $\theta_\alpha$ is a real parameter, and $u_\alpha$ is a coefficient that is determined from $\{\theta_\alpha\}$. It leads to the following expression of $\bm{U}$, \begin{align}
\bm{U} \kket{\rho} = \sum_{\alpha, \alpha'} u_{\alpha}u_{\alpha'}^*\kket{(\sigma_\alpha\otimes \sigma_\alpha) \rho (\sigma_{\alpha'}\otimes \sigma_{\alpha'})}. \label{eq:two-qubit-unitary-channel} \end{align}
Note that $\sum_\alpha |u_\alpha|^2 = 1$ follows from the unitarity.
First, we expand the general two-qubit unitary defined in Eq. (\ref{eq:two-qubit-unitary-channel}) using $\kket{\sigma_\beta}$ as a single-qubit basis vector as follows: \begin{align}
&\bbra{\sigma_{\beta'} \otimes \sigma_{\gamma'}} \bm{U} \kket{\sigma_\beta \otimes \sigma_\gamma} \nonumber \\
&= \sum_{\alpha, \alpha' } u_{\alpha}u_{\alpha'}^*\text{Tr}\left[\sigma_{\beta'}\sigma_\alpha\sigma_\beta\sigma_{\alpha'}\right] \text{Tr} \left[ \sigma_{\gamma'} \sigma_\alpha \sigma_\gamma \sigma_{\alpha'} )\right]. \end{align} From this expression, it is clear that if we can construct a single-qubit channel $\bm{U}_{\alpha\alpha'}$ such that $\bm{U}_{\alpha\alpha'}\rho = \sigma_\alpha\rho\sigma_{\alpha'}$ for any $\rho$, we can write the above as, \begin{align}
&\bbra{\sigma_{\beta'} \otimes \sigma_{\gamma'}} \bm{U} \kket{\sigma_\beta \otimes \sigma_\gamma} \nonumber \\
&= \sum_{\alpha, \alpha'} u_{\alpha}u_{\alpha'}^* \bbra{\sigma_{\beta'}}\bm{U}_{\alpha\alpha'}\kket{\sigma_{\beta}}\bbra{\sigma_{\gamma'}}\bm{U}_{\alpha\alpha'}\kket{\sigma_{\gamma}}\nonumber \\
&= \sum_{\alpha, \alpha'} u_{\alpha}u_{\alpha'}^* \bbra{\sigma_{\beta'} \otimes \sigma_{\gamma'}} \bm{U}_{\alpha\alpha'}^{\otimes 2} \kket{\sigma_\beta \otimes \sigma_\gamma}. \end{align} Therefore, we conclude $\bm{U}=\sum_{\alpha,\alpha'}u_{\alpha}u_{\alpha'}^*\bm{U}_{\alpha\alpha'}^{\otimes 2}$.
Now, we construct $\bm{U}_{\alpha\alpha'}$ with available single-qubit operations. Observe that, \begin{align}
\sigma_\alpha\rho\sigma_{\alpha'} = \frac{1}{2}\left(\sigma_\alpha\rho\sigma_{\alpha'} + \sigma_{\alpha'}\rho\sigma_{\alpha}\right) + \frac{1}{2}\left(\sigma_\alpha\rho\sigma_{\alpha'} - \sigma_{\alpha'}\rho\sigma_{\alpha}\right). \end{align} Let us define the following operators $A_{\alpha\alpha',\pm}$ and $B_{\alpha\alpha',\pm}$ which can be implemented through single-qubit operations: \begin{align}
A_{\alpha\alpha',\pm} = \frac{1}{2}\left(\sigma_\alpha \pm \sigma_{\alpha'}\right),\\
B_{\alpha\alpha',\pm} = \frac{1}{2}\left(\sigma_\alpha \pm i\sigma_{\alpha'}\right). \end{align} The corresponding channels $\bm{A}_{\alpha\alpha',\pm}$ and $\bm{B}_{\alpha\alpha',\pm}$ act on a single-qubit density matrix $\rho$ like $A_{\alpha\alpha',\pm}\rho A_{\alpha\alpha',\pm}^\dagger$. Building on $\bm{A}_{\alpha\alpha',\pm}$ and $\bm{B}_{\alpha\alpha',\pm}$, we further define the following channels: \begin{align}
\bm{A}_{\alpha\alpha'} = \bm{A}_{\alpha\alpha',+} - \bm{A}_{\alpha\alpha',-},\\
\bm{B}_{\alpha\alpha'} = \bm{B}_{\alpha\alpha',+} - \bm{B}_{\alpha\alpha',-}. \end{align} With simple algebra, we can see that, \begin{align}
\bm{A}_{\alpha\alpha'} \rho = \frac{1}{2}\left(\sigma_\alpha\rho\sigma_{\alpha'} + \sigma_{\alpha'}\rho\sigma_{\alpha}\right), \\
\bm{B}_{\alpha\alpha'} \rho = \frac{1}{2i}\left(\sigma_\alpha\rho\sigma_{\alpha'} - \sigma_{\alpha'}\rho\sigma_{\alpha}\right). \end{align} Therefore, $\bm{U}_{\alpha\alpha'}$ can be written as, \begin{align}
\bm{U}_{\alpha\alpha'} = \bm{A}_{\alpha\alpha'} + i\bm{B}_{\alpha\alpha'}. \end{align}
The above decomposition of $\bm{U}_{\alpha\alpha'}$ leads us to the following decomposition of $\bm{U}$: \begin{align}
\bm{U}=\sum_{\alpha\alpha'}u_{\alpha}u_{\alpha'}^*\left(\bm{A}_{\alpha\alpha'} + i\bm{B}_{\alpha\alpha'}\right)^{\otimes 2}. \end{align} Note that there are symmetries $\bm{A}_{\alpha\alpha'}=\bm{A}_{\alpha'\alpha}$ and $\bm{B}_{\alpha\alpha'}=-\bm{B}_{\alpha'\alpha}$. Using them, we rewrite the expression for later convenience, \begin{align}
\bm{U}&=\sum_{\alpha}|u_{\alpha}|^2 \bm{\sigma}_\alpha^{\otimes 2} \nonumber\\
& + \sum_{\alpha<\alpha'} (u_{\alpha}u_{\alpha'}^*+u_{\alpha'}u_{\alpha}^*)\left(\bm{A}_{\alpha\alpha'}^{\otimes 2} - \bm{B}_{\alpha\alpha'}^{\otimes 2} \right)\nonumber \\
& + \sum_{\alpha<\alpha'} i(u_{\alpha}u_{\alpha'}^*-u_{\alpha'}u_{\alpha}^*)\left(\bm{A}_{\alpha\alpha'}\otimes\bm{B}_{\alpha\alpha'} + \bm{B}_{\alpha\alpha'}\otimes\bm{A}_{\alpha\alpha'}\right). \label{eq:U-decomposition} \end{align}
To calculate upper bound for $\widetilde{W}(\bm{U})$, we need to formulate Eq. (\ref{eq:U-decomposition}) to fit in the form of Eq. (\ref{eq:Phi-decomposition}). $\bm{\sigma}_\alpha$, which constitutes the first term of the decomposition, is trivially in $\mathcal{L}$. Let us now consider $\bm{A}_{\alpha\alpha'}$. We note that from the symmetry it suffices to consider the case where $\alpha<\alpha'$. When $\alpha=0$, $\bm{A}_{\alpha\alpha',\pm}$ becomes a projector $\bm{\Pi}(\pm n)$ where $n_{\alpha''} = \delta_{\alpha'\alpha''}$. Therefore, $\bm{A}_{\alpha\alpha'}$ takes the form of $\tilde{\bm{\Pi}}(n, 1, -1)$, which means $\bm{A}_{0\alpha'}\in \mathcal{L}$. For $\alpha\neq 0$, $\bm{A}_{\alpha\alpha',\pm}$ is proportional to a single-qubit rotation that swaps the $\alpha$-axis and $\alpha'$-axis. More concretely, $2\bm{A}_{\alpha\alpha',\pm} \in \mathcal{L}$ for $\alpha\neq 0$ and $\alpha<\alpha'$. As for $\bm{B}_{\alpha\alpha'}$, when $\alpha=0$, $\bm{B}_{\alpha\alpha',\pm}$ becomes proportional to a single-qubit rotation around $\alpha'$-axis. Likewise to the previous case, $2\bm{B}_{\alpha\alpha',\pm} \in \bm{L}_i$. For $\alpha\neq 0$, $\bm{B}_{\alpha\alpha',\pm}$ can be implemented by a projector followed by a flip; for example, $\frac{1}{2}(\sigma_1+i\sigma_2) = \frac{1}{2}\sigma_1(\sigma_0-\sigma_3)$. With this observation, we can see that the channel $\bm{B}_{\alpha\alpha'}$ in this case can be written as a product of $\tilde{\bm{\Pi}}$ and $\bm{\sigma}_\alpha$ which makes $\bm{B}_{\alpha\alpha'}\in\mathcal{L}$ for $\alpha\neq 0$ and $\alpha<\alpha'$.
Combining the above properties, we can calculate $W(\bm{U}) = \sum_i |c_i|$ for the decomposition given in Eq. (\ref{eq:U-decomposition}) as, \begin{align}
W(\bm{U}) = 1 + \sum_{\alpha\neq\alpha'}\left(|u_\alpha u_{\alpha'}^* + u_{\alpha'} u_\alpha^*| + |u_\alpha u_{\alpha'}^* - u_{\alpha'} u_\alpha^*| \right), \label{eq:cost-overhead} \end{align} which gives an upper bound of $\widetilde{W}(\bm{U})$. We note that the operations used in the proposed decomposition, namely $\bm{\sigma}_{\alpha}$ $(\alpha\in\{0,1,2,3\})$, $\bm{A}_{\alpha\alpha'}$ and $\bm{B}_{\alpha\alpha'}$ with $\alpha<\alpha'$ are 16 linearly independent single-qubit channel and thus form a complete basis in the space of single-qubit superoperators. This means $W(\bm{U})$ is uniquely determined as long as the same basis set is used.
\subsection{Behaviour of \texorpdfstring{$W(\bm{U})$}{TEXT}}
Here, we numerically investigate the behavior of $W(\bm{U})$ defined in Eq. (\ref{eq:cost-overhead}), restricting the domain of $\{\theta_\alpha\}$ in which each point is not locally equivalent, meaning that a two-qubit unitary represented by a point $(\theta_1, \theta_2, \theta_3)$ cannot be translated to another point in the domain by transforming it with single-qubit unitaries, according to Ref. \cite{Zhang2003}. In Fig. \ref{fig:weyl-chamber}, we depict such a domain of $\{\theta_\alpha\}$ \footnote{It slightly differs from Ref. \cite{Zhang2003}. We shift half of the tetrahedron presented in Fig. 2 of Ref. \cite{Zhang2003} corresponding to the region $\theta_x \geq \pi/4$ to $\theta_x \leq \pi/0$ using the periodicity of $\theta_x$.}. Note that there are exceptional local-equivalence in the domain; every point $A_1 A_2 A_3$ and $O A_2 A_3$ is locally equivalent to $A_1'A_2'A_3'$ and $O A_2' A_3'$, respectively.
Since $W(\bm{U})$ is symmetric to the reflection of $\theta_x$, we only investigate the tetrahedron $OA_1 A_2 A_3$. In Fig. \ref{fig:surface-plot}, we show the behavior of $W(\bm{U})$ on the surfaces and edges of the domain. We numerically found that $W(\bm{U})$ is maximized at $(\theta_1, \theta_2, \theta_3)\approx(\pi/4, 0.202\pi, 0.136\pi)$ which lies on the surface $A_1A_2A_3$ with its value being approximately $8.87$. The behavior of $W(\bm{U})$ seems to be unrelated to other measures such as entangling power of $\bm{U}$ \cite{Kraus2001, Kong2015}; for example, while the point $A_1$ corresponds to controlled-$\sigma_\alpha$ gates which can produce the maximal amount of entanglement and has $W(\bm{U})=3$, $A_3$ which corresponds to the swap gate has $W(\bm{U})=7$. Although we believe the decomposition given in this work is close to optimal, this counter-intuitive result might be caused by the non-optimality.
\begin{figure}
\caption{Domain of $(\theta_1,\theta_2,\theta_3)$ in which a two-qubit unitary represented by each point is not locally equivalent to each other. In the figure, $O=(0,0,0)$, $A_1=(\pi/4,0,0)$, $A_2=(\pi/4,\pi/4,0)$, $A_3=(\pi/4,\pi/4,\pi/4)$, $A_1'=(-\pi/4,0,0)$, $A_2'=(-\pi/4,\pi/4,0)$ and $A_3'=(-\pi/4,\pi/4,\pi/4)$.}
\label{fig:weyl-chamber}
\end{figure}
\begin{figure}
\caption{Behaviour of $W(\bm{U})$ on the surface of the tetrahedron $OA_1A_2A_3$.}
\label{fig:surface-plot}
\end{figure}
\section{Discussion}
\subsection{Comparison with gate-based decomposition approach}
If we can measure $\braket{\psi_1|\psi_2}$ for some fixed state $\ket{\psi_1}$ and $\ket{\psi_2}$, we can directly utilize the fact that a two-qubit gate is decomposed as $\sum_{\alpha\in \{I,x,y,z\}} u_{\alpha} \sigma_\alpha\otimes\sigma_\alpha$. \red{As we discuss later, this measurement can be demanding for early days quantum computers.} Let $V$ be a sequence of gates consisting of alternating layers of single-qubit and two-qubit gates. Note that any quantum circuit can be written in this form. $V$ can be written as $V=D_L S_L \cdots D_2 S_2 D_1 S_1$ where $D_i$'s and $S_i$'s are two-qubit and single-qubit gates, respectively. We assume $D_i = \sum_\alpha d_{\alpha_i} \sigma_{\alpha_i}^{(a_i)} \otimes\sigma_{\alpha_i}^{(b_i)}$ where $\sigma_{\alpha}^{(a)}$ is a Pauli matrix acting on the $a$-th qubit. Now, focusing on the $i$-th two-qubit gate, we can express an expectation value of an observable $O$ at the end of the circuit as, \begin{align}\label{eq:gate-based-decomp}
\braket{0|V^\dagger O V|0} = \sum_{\alpha_i} d_{\alpha'_i}^*d_{\alpha_i} \braket{0|V_{i,\alpha_i'}^\dagger O V_{i,\alpha_i}|0}, \end{align} where, \begin{align}
V_{i,\alpha_i} = D_L S_L\cdots \sigma_{\alpha_i}^{(a_i)} \otimes\sigma_{\alpha_i}^{(b_i)} \cdots D_2 S_2 D_1 S_1. \end{align}
This decomposition also allows us to perform a ``virtual'' two-qubit gate on a quantum circuit in the sense that, in $V_{i,\alpha_i}$, the $i$-th two-qubit gate in $V$ is replaced by $\sigma_{\alpha_i}^{(a_i)} \otimes\sigma_{\alpha_i}^{(b_i)}$ which is a tensor product of local operations. We can do this by the following algorithm. Let us assume that $O$ is written as $O=\sum_i c_i P_i$, where $P_i$ is a tensor product of Pauli operators. With this assumption, we can evaluate $\braket{0|V_{i,\alpha_i'}^\dagger O V_{i,\alpha_i}|0}$ by $\sum_k c_k \braket{0|V_{i,\alpha_i'}^\dagger P_k V_{i,\alpha_i}|0}$. More concretely, we define $\ket{\psi_{i,\alpha_i}} = V_{i,\alpha_i'}\ket{0}$ and $\ket{\psi_{k,i,\alpha_i}}=P_kV_{i,\alpha_i}\ket{\psi_{i,\alpha_i}}$ and then measure $\braket{\psi_{i,\alpha_i'}|\psi_{k,i,\alpha_i}}$ which is possible by the assumption. If we are to perform the sum of Eq. (\ref{eq:gate-based-decomp}) in a Monte-Carlo manner, we can sample $\alpha'_i$ and $\alpha_i$ with a probability proportional to $|d_{\alpha'_i}^*d_{\alpha'_i}|$. This leads us to define $G(D_i) :=\sum_{\alpha'_i,\alpha_i}|d_{\alpha'_i}^*d_{\alpha'_i}|$ which quantifies the overhead of the decomposition, that is, we need $G(D_i)^2$ times more samples to reach a desired error compared to the decomposition-free case.
It is trivial that $G(D_i)$ is always smaller than $W(\bm{D}_i)$. Therefore, if we can measure $\braket{\psi_{i,\alpha_i'}|\psi_{k,i,\alpha_i}}$, it is always better to use this approach. For example, in a classical simulation we can easily calculate $\braket{\psi_{i,\alpha_i'}|\psi_{k,i,\alpha_i}}$. However, it is not the case for a quantum computer, in particular for a NISQ device. Measurement of the overlap $\braket{\psi_{i,\alpha_i'}|\psi_{k,i,\alpha_i}}$, including its phase, is a demanding task. One way of performing this task is to use a controlled-$V_{i,\alpha_i}$ as mentioned in e.g. Refs. \cite{Bravyi2016,ibe2020calculating}, which is unlikely to be implemented on a NISQ device due to its complexity. The original motivation of this work and our previous works \cite{mitarai2019constructing, Mitarai2019Methodology} has been to avoid such complex operations. Note that the famous swap test \cite{Buhrman2001, Garcia-Escartin2013} cannot be applied to this task since it can only evaluate $|\braket{\psi_{i,\alpha_i'}|\psi_{k,i,\alpha_i}}|^2$. Investigations on the relation between $\widetilde{W}(\bm{D}_i)$ and $G(D_i)$ are left for the furture work.
\subsection{Comparison with the previous work}
In the privous work \cite{mitarai2019constructing}, we have proposed the decomposition for an gate in the form $e^{i\theta A_1\otimes A_2}$ for Hermitian operators $A_1$ and $A_2$ satisfying $A_1^2=I$ and $A_2^2=I$. It is a special case of this work, which is recovered by setting $u_0 = \cos\theta$ and $u_\alpha = i\sin\theta$ for one chosen $\alpha \in \{1,2,3\}$. Therefore, the cost overhead of this special case is determined by $1+2|u_0u_\alpha^*-u_\alpha u_0^*|$, which takes maximum at $\theta = \pi/4$. If we are to decompose a general two-qubit gate in the form of $\exp\left[i\left(\sum_{\alpha=1}^3 \theta_\alpha \sigma_\alpha\otimes \sigma_\alpha\right)\right]$ using this technique, we decompose each of $\exp\left[i\theta_\alpha \sigma_\alpha\otimes \sigma_\alpha\right]$. Then, the overhead is quantified by the product of $1+2|u_Iu_\alpha^*-u_\alpha u_I|$, which reaches its maximum $3^3=27$ at $\theta_\alpha=\pi/4$ for all $\alpha$. On the other hand, $W_{\bm{U}}$ defined in Eq. (\ref{eq:cost-overhead}), which quantifies the overhead required by the present approach, becomes $7$, showing substantial improvement.
While we believe that the decomposition given in this work is close to optimal, there can be better decompositions with smaller $W_{\bm{U}}$. The search for optimal decomposition will require some form of numerical search. In the context of classical simulation of near Clifford circuits, Ref. \cite{Bravyi2016} has performed such a search. However, the optimization of the decomposition considered in this work will be more complicated than the aforementioned work, since the number of available operations is infinitely many as can be seen from Eq. (\ref{eq:robustness-continuous}). We believe the decomposition proposed in this work can be a good starting point of the optimization if it is not optimal and leave it as future work.
\section{conclusion}
We have introduced a quantity called channel robustness of non-locality which quantifies the minimal amount of overhead required for decomposing non-local channels into local ones with a quasiprobability-based method. While the calculation of the quantity for general non-local channels is difficult due to the need for a complicated optimization, we have successfully established an upper bound for a general two-qubit unitary channel. The upper bound is obtained by constructively deriving an explicit decomposition. Its overhead is substantially lowered compared to the previous work \cite{mitarai2019constructing}. While we believe the present decomposition is close to optimal, there might be a better decomposition of a general two-qubit channel than the one presented in this work, which we leave as possible future work. This formalism of decomposing an experimentally challenging channel into a linear combination of experimentally-easy channels allows us to readily perform the decomposition using a quantum device.
\begin{acknowledgments}
KM is supported by JST PRESTO JPMJPR2019 and KAKENHI No. 20K22330.
KF is supported by KAKENHI No.16H02211, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.
This work is supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067394.
Program code for generating Fig. \ref{fig:surface-plot} is available at https://github.com/kosukemtr/nonlocal-local-decomposition. \end{acknowledgments}
\appendix
\section{Submultiplicability of \texorpdfstring{$\widetilde{W}(\bm{\Phi})$}{TEXT}} \begin{lemma}
Let $\bm{\Phi}_1$ and $\bm{\Phi}_2$ be any quantum channels and $\bm{\Phi}_{21}=\bm{\Phi}_2\bm{\Phi}_1$. Then, $\widetilde{W}(\bm{\Phi}_{21})\leq \widetilde{W}(\bm{\Phi}_{2})\widetilde{W}(\bm{\Phi}_{1})$. \end{lemma} \noindent\textit{proof---} Let \begin{align}
\bm{\Phi}_\mu = \sum_i c_{\mu i}\bm{L}_{\mu i} \end{align}
and $\sum_i |c_{\mu i}| = \widetilde{W}(\bm{\Phi}_{\mu})$. Then, $\bm{\Phi}_{21}$ can be decomposed as, \begin{align}
\bm{\Phi}_{21} = \sum_{ij} c_{2 i}c_{1 j}\bm{L}_{2 i}\bm{L}_{1 j}. \end{align} Because $\bm{L}_{2 i}\bm{L}_{1 j}\in \mathcal{L}$, the above gives a decomposition of $\bm{\Phi}_{21}$ in the form of Eq. \ref{eq:Phi-decomposition}. Therefore, \begin{align}
\widetilde{W}(\bm{\Phi}_{21}) &\leq \sum_{ij} |c_{2 i} c_{1 j}| \nonumber \\
&= \sum_i |c_{2 i}| \sum_j |c_{1 j}| \nonumber \\
&= \widetilde{W}(\bm{\Phi}_{2})\widetilde{W}(\bm{\Phi}_{1}). \end{align}
$\Box$
\begin{thebibliography}{25}
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\end{document} | arXiv |
Anmol Sharma
sciscore: 1.952
My Summaries 20
doi.ieeecomputersociety.org
Multi-Scale Deep Reinforcement Learning for Real-Time 3D-Landmark Detection in CT Scans
Ghesu, Florin C. and Georgescu, Bogdan and Zheng, Yefeng and Grbic, Sasa and Maier, Andreas K. and Hornegger, Joachim and Comaniciu, Dorin
IEEE Transactions on Pattern Analysis and Machine Intelligence - 2019 via Local Bibsonomy
[link] Summary by Anmol Sharma 2 years ago
Robust and fast detection of anatomical structures is a prerequisite for both diagnostic and interventional medical image analysis. Current solutions for anatomy detection are typically based on machine learning techniques that exploit large annotated image databases in order to learn the appearance of the captured anatomy. These solutions are subject to several limitations, including the use of suboptimal feature engineering techniques and most importantly the use of computationally suboptimal search-schemes for anatomy detection. To address these issues, we propose a method that follows a new paradigm by reformulating the detection problem
as a behavior learning task for an artificial agent.
To this end, Ghesu et al. reformulate the problem of landmark detection as a reinforcement learning task, where an agent is trained to navigate the 3D image space in an efficient way in order to find the landmark as spatially closely as possible. This paper marks the first time that RL has been used for a specific problem like this.
The authors use Deep Q-Learning (DQN) algorithm as their framework to build the agent. The DQN algorithm uses a Convolutional Neural Network to parameterize the Q* function in a non-linear way. Also, in order to ensure that the agent always finds an object of interest regardless of the size, the authors proposed to use multi-scale search strategy, in which a different CNN works on an image of a different scale.
The authors also some other interesting techniques like $\epsilon-$greedy approach, which uses a randomized strategy to choose the actions in every iteration. They also use experience replay where an experience buffer is maintained that records the trajectory of the agent. This buffer is then used to fine-tune the Q network.
The authors test their method on an internal data set of CT scans, with a few different metrics. The most important metric was failure percentage rate, that was defined if the agent is x mm close to the landmark. The method achieved 0\% failure percentage rate on detecting landmarks in CT scans.
Unsupervised Deep Feature Learning for Deformable Registration of MR Brain Images
Wu, Guorong and Kim, Minjeong and Wang, Qian and Gao, Yaozong and Liao, Shu and Shen, Dinggang
Medical Image Computing and Computer Assisted Interventions Conference - 2013 via Local Bibsonomy
Accurate anatomical landmark correspondence is highly critical for medical image registration. Traditionally many of the previous works proposed a number of hand-crafted feature sets that can be used to perform correspondence. However these feature tend to be highly specialized in terms of application area, and cannot be always generalized well to other applications without significant modifications. There have been other works that perform automatic feature extraction, but their reliance on labelled data hinders their ability to perform in cases where there is none.
To this end Wu et al. propose an unsupervised feature learning method which does not require labelled data. Their approach aims to directly learn the basis filters that can effectively represent all observed image patches. The learnt basis filters are later regarded as general image features representing the morphological structure of the patch.
In order to learn the basis filters, the authors propose a two-layer convolutional neural network called the Independent Subspace Analysis (ISA) algorithm. As an extension of ICA, the responses are not required to be all to be mutually independent in ISA. Instead, these responses can be divided into several groups, each of which is called independent subspace. Then, the responses are dependent inside
each group, but dependencies among different groups are not allowed. Thereby, similar
features can be grouped into the same subspace to achieve invariance. To ensure the accurate correspondence detection, multi-scale image features are necessary to use. However, it also raised a problem of high-dimensionality in learning features from the large-scale image patches.
This is achieved by constructing a two-layer network for scaling up the ISA to the large-scale image patches. Specifically, the ISA is first trained in the first layer based on the image patches with smaller scale. After that, a sliding window (with the same scale in the first layer) convolves with each large-scale patch to get a sequence of overlapped small-scale patches. The
combined responses of these overlapped patches through the first layer ISA are whitened by PCA and then used as the input to the second layer that is further trained by another ISA. In this way, high-level
understanding of large-scale image patch can be perceived from the low-level image features detected by the basis filters in the first layer.
The authors compare their work with two other methods, and apply their method on IXI and ADNI datasets.
www.wikidata.org
Brain Tumor Segmentation Using Convolutional Neural Networks in MRI Images
Pereira, Sérgio and Pinto, Adriano and Alves, Victor and Silva, Carlos A.
IEEE Trans. Med. Imaging - 2016 via Local Bibsonomy
Tumor segmentation from brain MRI sequences is usually
done manually by the radiologist. Being a highly tedious and error prone task, mainly due to factors such as human fatigue, overabundance of MRI slices per patient, and increasing number of patients, manual operations often lead to inaccurate delineation. Moreover, use of qualitative measures of evaluation by radiologists results in high inter- and intraobserver
error rates. There is an evident need for automated systems to perform this task.
To this end Pereira et al. propose to use a deep learning method called Convolutional Neural Network for predicting a segmentation mask of the patient MRI scans. The approach the segmentation problem as a pixel-wise classification problem, where each pixel in the input MRI scan 2D slice is classified into one of the five categories: background, necrosis, edema, non-enhancing and enhancing region.
The authors propose two networks, one for High Grade Gliomas (HGG) and one for Low Grade Gliomas (LGG). The HGG network has more number of convolution layers than the LGG due to lack of data. The proposed network architectures are a combination of convolution, relu, and max pooling layers, followed by some fully connected layers in the end.
The networks are trained using categorical cross-entropy loss function.
The networks were trained on 2D 33x33 patches extracted from the 2D MRI slices of the brain from the BRATS 2012 dataset. The patches were randomly sampled from the images and the task was to predict the class of the pixel in the middle of the patch. To approach the problem of class imbalance, they sample approximately 40\% of the patches were normal patches. They also use data augmentation to increase the number of effective patches to train.
In order to test the performance of their proposed method, as well as understand the impact of various hyperparameters, the authors perform a multitude of tests. The tests included turning data augmentation on/off, using leaky relu instead of relu, changing patch extraction plane, using deeper networks, and using larger kernels.
Hierarchical feature representation and multimodal fusion with deep learning for AD/MCI diagnosis
Suk, Heung-Il and Lee, Seong-Whan and Shen, Dinggang
NeuroImage - 2014 via Local Bibsonomy
Alzheimer's Disease (AD) is characterized by impairment of cognitive and memory function, mostly leading to dementia in elderly subjects. For the last decade, it has been shown that neuroimaging can be a potential tool for the diagnosis of Alzheimer's Disease (AD) and its prodromal stage, Mild Cognitive Impairment (MCI), and also fusion of different modalities can further provide the complementary information to enhance diagnostic accuracy. Multimodal information like that from MRI and PET can be used to aid in diagnosis of AD in early stages. However most of the previous works in this domain either concentrate on only one domain (MRI or PET), or use hand-crafted features which are then concatenated together to form a single vector. There are increasing evidences that biomarkers from different modalities can provide complementary information in AD/MCI diagnosis.
In this paper, Suk et al. propose a Deep Boltzmann Machine (DBM) based method that performs high-level latent and shared feature representation obtained from two neuroimaging modalities (MRI and PET). Specifically they use DBM as a building block, to find a latent hierarchical feature representation from a 3D patch, and then devise a systematic method for a joint feature representation from the paired patches of MRI and PET with a multimodal DBM.
The method first selects class-discriminative patches from a pair of MRI and PET images, by using a statistical significance test between classes. A MultiModal DBM (MM-DBM) is then built that finds a shared feature representation from the paired patches. However the MM-DBM is not trained directly on patches, instead, it's trained using binary vectors obtained after running the patches through a Restricted Boltzmann Machine (RBM) which transforms the real-valued observations into binary vectors.
The MM-DBM network's top hidden layer has multiple entries of the lower hidden layers and the label layer, to extract a shared feature representation by fusing neuroimaging information of MRI and PET. Using this multimodal model, a single fused feature representation is obtained.
Using this feature representation, A Support Vector Machine (SVM) based classification step is added. Instead of considering all patch-level classifiers' output simultaneously, the output from the SVMs are agglomerated the information of the locally distributed patches by constructing spatially distributed 'mega-patches' under the consideration that the disease-related brain areas are distributed over some distant brain regions with arbitrary shape and size. Following this step, the training data is divided into multiple subsets, and used to train an image-level classifier on each subset individually. The method was tested on ADNI dataset with MR and PET images.
V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation
Milletari, Fausto and Navab, Nassir and Ahmadi, Seyed-Ahmad
Medical image segmentation have been a classic problem in medical image analysis, with a score of research backing the problem. Many approaches worked by designing hand-crafted features, while others worked using global or local intensity cues. These approaches were sometimes extended to 3D, but most of the algorithms work with 2D images (or 2D slices of a 3D image). It is hypothesized that using the full 3D volume of a scan may improve segmentation performance due to the amount of context that the algorithm can be exposed to, but such approaches have been very expensive computationally. Deep learning approches like ConvNets have been applied to segmentation problems, which are computationally very efficient during inference time due to highly optimized linear algebra routines. Although these approaches form the state-of-art, they still utilize 2D views of a scan, and fail to work well on full 3D volumes.
To this end, Milletari et al. propose a new CNN architecture consisting of volumetric convolutions with 3D kernels, on full 3D MRI prostate scans, trained on the task of segmenting the prostate from the images. The network architecture primarily consisted of 3D convolutions which use volumetric kernels having size 5x5x5 voxels. As the data proceeds through different stages along the compression
path, its resolution is reduced. This is performed through convolution with
2x2x2 voxels wide kernels applied with stride 2, hence there are no pooling layers in the architecture. The architecutre resembles an encoder-decoder type architecture with the decoder part, also called downsampling, reduces the size of the signal presented as input and increases the receptive field of the features being computed in subsequent network layers. Each of the stages of the left part of the network, computes a number of features which is two times higher than the one of the previous layer. The right portion of the network extracts features and expands the spatial
support of the lower resolution feature maps in order to gather and assemble the necessary information to output a two channel volumetric segmentation. The two features maps computed by the very last convolutional layer, having 1x1x1 kernel size and producing outputs of the same size as the input volume, are converted to probabilistic segmentations of the foreground and background
regions by applying soft-max voxelwise.
In order to train the network, the authors propose to use Dice loss function. The CNN is trained end-to-end on a dataset of 50 prostate scans in MRI. The network approached a 0.869 $\pm$ 0.033 dice loss, and beat the other state-of-art models.
Dermatologist-level classification of skin cancer with deep neural networks
Esteva, Andre and Kuprel, Brett and Novoa, Roberto A. and Ko, Justin and Swetter, Susan M. and Blau, Helen M. and Thrun, Sebastian
Nature - 2017 via Local Bibsonomy
Skin cancer is one of the most common cancer type in humans. Primarily, the lesion is diagnosed visually through a series of 2D color images taken of the affected area. This may be followed by dermoscopic analysis, a biopsy and histopathological examination. Automated classification of skin lesions using images is a challenging task owing to the fine-grained variability in the appearance of skin lesions.
To this end, Esteva et al. propose a deep learning based solution to automate the task of diagnosing lesions of the skin into fine-grained categories. Specifically, they use a GoogleNet Inception v3 CNN architecture which won the ImageNet Large Scale Visual Recognition Challenge in 2014. The method also leverages pre-training, in which an already trained DNN can be fine-tuned on a slightly varied task, which allows the network to leverage the convolutional filters it might have learnt from a much larger dataset. To achieve this, the Inception v3 CNN was fine-tuned from a pre-trained state. The model was initially trained on approximately 1.28 million images with about 1000 classes, from the 2014 ImageNet Large Scale Visual Recognition Challenge. Following which, the network is then fine-tuned on the dermatology dataset.
The dataset used in the study was obtained clinically from open-access online repositories and Stanford Medical Center. It consists of 127,463 training and validation images, and held out set of 1942 labelled test images. The labels are organized hierarchically in a tree like structure, where each succeeding depth level represents a fine-grained classification of the disease.
The network is trained to perform three tasks: i) classify the first-level nodes of the taxonomy,
which represent benign lesions, malignant lesions and non-neoplastic. ii) nine-class
disease partition—the second-level nodes—so that the diseases of
each class have similar medical treatment plans, and finally iii) using only biopsy-proven images on medically important use cases, whether the algorithm and dermatologists could distinguish malignant versus benign lesions of epidermal (keratinocyte carcinoma compared to benign seborrheic keratosis) or melanocytic (malignant melanoma compared to benign nevus) origin. The CNN achieved 72.1 $\pm$ 0.9\% (mean $\pm$ s.d.) overall accuracy (the average of individual inference class accuracies) and two dermatologists attain 65.56\% and 66.0\% accuracy on a subset of the validation set for the first task. The CNN achieves 55.4 $\pm$ 1.7\% overall accuracy whereas the same two dermatologists attain 53.3\% and 55.0\% accuracy in the second task. For the third task, the CNN outperforms the dermatologists, and obtains an area under the curve (AUC) over 91\% for each case.
Shape Registration in Implicit Spaces Using Information Theory and Free Form Deformations
Huang, Xiaolei and Paragios, Nikos and Metaxas, Dimitris N.
Shape registration problem have been an active research topic in computational geometry, computer vision, medical image analysis and pattern recognition communities. Also called the shape alignment, it has extensive uses in recognition, indexing, retrieval, generation and other downstream analysis of a set of shapes. There have been a variety of works that approach this problem, with the methods varying mostly in terms of (can be called pillars of registration) the shape representation, transformation and registration criteria that is used. One such method is proposed by Huang et al. in this paper, which uses a novel combination of the three pillars, where an implicit shape representation is used to register an object both globally and locally. For the registration criteria, the proposed method uses Mutual Information based criteria for its global registration phase, while sum-squared differences (SSD) for its local phase.
The method starts off with defining an implicit, non-parameteric shape representation which is translation, rotation and scale invariant. This makes the first step of the registration pipeline which transforms the input images into a domain where the shape is implicitly defined. The image is first partitioned into three spaces, namely $[\Omega]$ (the image domain), $[R_S]$ (points inside the shape), $[\Omega - R_S]$ (points outside the shape), and $[S]$ (points lying on the shape boundary). Using this partition, a function based upon the Lipschitz function $\phi : \Omega -> \mathbb{R}^+$ is defined as:
\begin{equation}
\phi_S(x,y)
\begin{cases}
0 & (x,y) \in S \\
+ D((x,y), S)>0 & (x,y) \in [R_s] \\
- D((x,y), S)<0 & (x,y) \in [\Omega - R_s]
\end{cases}
\end{equation}
Where $D((x,y),S)$ is the distance function which gives the minimum Euclidean distance between point $(x,y)$ and the shape $S$.
Given the implicit representation, global shape alignment is performed using the Mutual Information (MI) objective function defined between the probability density functions of the pixels in source image and the target image sampled from the domain $\Omega$.
MI(f_{\Omega}, g_{\Omega}^{A}) = \underbrace{\mathcal{H}[p^{f_{\Omega}}(l_1)]}_{\substack{\text{Entropy of the}\\ \text{distribution representing $f_{\Omega}$}}} + \underbrace{\mathcal{H}[p^{g_{\Omega}^{A}}(l_2)]}_{\substack{\text{Entropy of the}\\ \text{distribution representing $g_{\Omega}^{A}$} \\ \text{which is the} \\ \text{transformed source ISR using $A(\theta)$}}} - \underbrace{\mathcal{H}[p^{f_{\Omega}, g_{\Omega}^{A}}(l_1, l_2)]}_{\substack{\text{Entropy of the}\\ \text{joint distribution}\\\text{representing $f_{\Omega}, g_{\Omega}^{A}$}}}
Following global registration, local registration is performed by embedding a control point grid using the Incremental Free Form Deformation (IFFD) method. The objective function to minimize is used as the sum squared differences (SSD). The local registration is also offset by using a multi-resolution framework, which performs deformations on control points of varying resolution, in order to account for small local deformations in the shape. In case where there is prior information available for feature point correspondence between the two shapes, this prior knowledge can be added as a plugin term in the overall local registration optimization term.
The method was applied on statistically modeling anatomical structures, 3D face scan and mesh registration.
Robust Point Set Registration Using Gaussian Mixture Models
Jian, Bing and Vemuri, Baba C.
Point pattern matching problem have been an active research topic in computation geometry and pattern recognition communities. These point sets typically arise in a variety of applications, where the problem lies in registration of these point sets which is encountered in stereo matching, feature based image registration and so on. Mathematically, the problem of registering two point sets translates to the following: Let $\{\mathcal{M}, \mathcal{S}\}$ be two finite set points which need to be registered, where $\mathcal{M}$ is the "moving" point set and $\mathcal{S}$ is the "fixed" or scene set. A transformation $\mathcal{T}$ is then calculated which can transform points from $\mathcal{M}$ to $\mathcal{S}$.
To this end, Jian and Vemuri propose to use a Gaussian Mixture Model based representation of point sets which are registered together my minimizing a cost function using a slightly modified version of the Iterative Closest Point (ICP) algorithm. In this setting, the problem of point set registration becomes as that of aligning two Gaussian mixture models by minimizing discrepancy between two Gaussian mixtures. The cost function for this optimization is chosen as a closed-form version of the $L_2$ distance between Gaussian mixtures, which allows the algorithm to be computationally efficient.
The main reason behind choosing Gaussian mixture models to represent discrete point sets was that it directly translates to interpreting point sets as randomly sampled data from a distribution of random point locations. This models the uncertainty of point sets well during feature extraction. The second reason was that hard discrete optimization problems that are encountered in point matching literature become tractable continuous optimization problems.
The probability density function of a general Gaussian mixture is defined as follows:
p(x) = \sum_{i=1}^{k}w_i \phi(x|\mu_i, \sigma_i)
$\phi(x|\mu_i, \sigma_i) = \dfrac{exp\left[-\frac{1}{2}(x - \mu_i)\sigma_{i}^{-1}(x-\mu_i)\right]}{\sqrt{(2\pi)^d |det(\sigma_i)|}}$
The GMM from the given point set is constructed as follows: i) the number of GMM components is equal to the number of points in the point set, and every component is weighted equally, ii) every component's mean vectors is represented by the spatial location of each point, iii) all components have same spherical covariance matrix.
An intuitive reformulation of the point set registration problem is to solve an optimization problem such that a certain dissimilarity measure between the Gaussian mixtures constructed from the transformed model set and the fixed scene set is minimized. For this method, L2 distance is chosen as the dissimilarity measure for measuring similarity between two Gaussian mixtures. The objective can then be minimized by either a closed-form numerical method (in case the function is convex) or an iterative gradient based method. The method was applied and tested on both rigid and non-rigid point set registration problems.
Nonrigid registration using free-form deformations: application to breast MR images
D. Rueckert and L.I. Sonoda and C. Hayes and D.L.G. Hill and M.O. Leach and D.J. Hawkes
IEEE Transactions on Medical Imaging - 1999 via Local CrossRef
Despite being an ill-posed problem, non-rigid image registration has been the subject of numerous works, which apply the framework on different applications where rigid and affine transformations cannot completely model the variations between image sets. One such application of non-rigid registration is to register pre- and post-contrast breast MR images for estimating contrast uptake, which in turn is an indicator of the tumor malignancy. Due to large variations between the pre- and post-contrast images in terms of patient movement, and breast movement which is both global and local, registration of these images become challenging. Classic methods cannot capture the exact semantics of movements that the images exhibit.
To this end, the authors propose a non-rigid registration method which combines advantages of voxel-based similarity measures like Mutual Information (MI) as well as non-rigid transformation models of the breast. The method is built using a hierarchical transformation model which capture both the global and local movement of the breast across pre- and post-contrast scans.
The proposed method consists of two interesting contributions which model the motion of the breast across scans using a global and local model. The global motion model conists of a 3D affine transformation parameterized by 12 degrees of freedom.
The local model is based upon FFD model, which is based upon B-splines which is a powerful tool for modeling 3D deformable objects. The main idea behind this approach is that FFDs can be used to deform an object by manipulating the underlying mesh of the control points, which is estimated in the form of a B-spline. The formulation exposes a trade-off between computational running time and accurate modelling of the object (breast). In order to achieve the best compromise, a hierarchical multi-resolution approach is implemented in which the resolution of the control mesh is increased along with the image resolution in a coarse to fine fashion.
In addition to modeling the movement of the breast, a regularization term is also added to the final optimization function which forces the B-spline based FFD transformation to be smooth. The term is zero in case of affine transformation, and only penalizes non-affine transformations.
The function that the method optimizes is as follows:
$\mathcal{C}(\theta, \phi) = - \mathcal{C}_{similarity}\left( I(t_0), T(I(t))\right) + \lambda \mathcal{C}_{smooth}(T)$
The optimization of the above objective function is performed in multiple stages, and by using gradient descent algorithm which takes steps towards the gradient vector with a certain step size of $\mu$. Local optimum is assumed if $||\nabla \mathcal{C}|| <= \epsilon$.
In order to assess the quality of the proposed method, the method is tested on both clinical and volunteer 3D MR data. The results show that the rigid and affine only transformations based methods perform significantly worse than the proposed method. Moreover, it was shown that the results improve with better control point resolution. However the use of SSD as a quantitative metric is debatable since the contrast enhanced and pre-contrast images will have varying distributions of intensity values.
Non-rigid Image Registration Using Graph-cuts
Tang, Tommy W. H. and Chung, Albert C. S.
Image registration has been well studied problem in medical image analysis community, with rigid registration taking much of the spotlight. In addition to rigid registration, non-rigid registration is of great interest due to it's applications in inter-patient modality registration where deformations of organs are highly pronounced. However non-rigid registration is an ill-posed problem with numerous degrees of freedom, which makes finding the best transformation from source to final image very difficult. To counter this, some methods were proposed which constraint the non-rigid transformation $T$ to be within certain bounds, which is not always ideal.
To this end, Tang et al. propose a novel framework which encapsulates the problem of non-rigid image registration into a graph-cut framework, which guarantees a global maxima (or minima) under certain conditions. The formulation requires that each pixel in source image has a displacement label (whcih is a vector) indicating its corresponding position in the floating image, according to an objective function. A smoothness constraint is also added to ensure that the values of the transformation function $T$ are meaningful and stay within natural limits (no large displacement should occur between absolute neighbouring pixels). The authors propose the following formulation as their objective function, which they then solve using graph-cuts methods:
$D^* = argmin_{D} \sum_{x\in X}||I(x) - J(x + D(x))|| + \lambda \sum_{(x,y)\in \mathcal{N}}||D(x) - D(y)||$
The equation above is not fully discretized, in the sense that $D$ is still unbounded and can vary from $[-\infty, \infty]$. To allow for optimization using graph-cuts, the transformation function $D$ is mapped to a finite set $\mathcal{W} = \{0, \pm s, \pm 2s...\pm ws\}^d$. Using this discretization, the equation above can be solved using graph-cuts via a sequence of alpha-expansion. $\alpha$-expansion is a two label problem where the cost of assigning a label $\alpha$ is calculated on the basis of the previous label of the pixel. Different costs are assigned to different scenarios where the previous label may keep it's original label, or change to new label $\alpha$. This is important since it is imperative that the cost conditions satisifes the inequality given by Kolmogorov \& Zabih which then guarantees a global optima.
The method was tested on on MR data from BrainWeb dataset which were affinely pre-registered and intensity normalized to be within 0 and 255. The method demonstrated good qualitative results when compared two state-of-art methods DEMONS and FFD, where the average intensity differences for the proposed method was much lower than the competition, while the tissue overlap was higher.
Medical image registration using mutual information
Maes, Frederik and Vandermeulen, Dirk and Suetens, Paul
Proceedings of the IEEE - 2003 via Local Bibsonomy
Current medical imaging modalities like Computed Tomography (CT), Magnetic Resonance Imaging (MRI) or Positron Emission Tomography (PET) has allowed minimally-invasive imaging of internal organs. Rapid advancement in these technologies have lead to an influx of data, which, along with rising clinical need, has lead towards a need for quantitative image interpretation in routine practice. Some of the applications include volumetric measurements of regions of the brain, surgery or radiotherapy planning using CT/MRI images.
This influx of data have opened up avenues for using multi-modality images for making decisions. However this is not always straightforward as the imaging parameters, scanner types, patient movement, or anatomical movement make the images miss-aligned against each other, making direct comparison between, say, as CT and MRI image tricky. This is formally known as the problem of image registration, and to this end, numerous computational methods have been proposed, which this paper surveys.
Out of the methods proposed for both inter- and intra-patient modality registration, Mutual Information based objective maximization strategy has been extremely successful at computing the registration between 3D multi-modal medical images of various organs from the images. Mutual Information (MI) stems from the field of information theory, pioneered by Shannon, which when applied in the context of medical image registration, postulates that the MI between two images (say CT and MRI) is maximum when the images are aligned.
The basic formulation of a MI based registration algorithm is as follows: Let $\mathcal{A}$ and $\mathcal{B}$ be two images which are geometrically related according to a transformation $T_\alpha$, such that voxels $p$ in $\mathcal{A}$ with intensity $a$ physically correspond to voxels $T_\alpha(p)$ in $\mathcal{B}$ with intensity $b$. The relationship between $p_{AB}(a,b)$ between $a$ and $b$, and hence their MI depends on $T_\alpha$. The MI criterian postulates that the MI for images that are geometrically aligned is maximum:
\alpha^* = argmax_{\alpha} I(A,B)
Where $A$ and $B$ are two discrete random variables, and $I(A,B) = \sum_{a,b}p_{AB}(a,b) log \frac{p_{AB}(a,b)}{p_A(a).p_B(b)}$.
An optimization algorithm is utilized to find a parameter set $\alpha^*$ that maximizes the MI between $A$ and $B$. Classically, Powell's multidimensional direction set method is used to otipimize the objective function, but other methods do exist as well.
Although the formulation of MI criterion suggests that spatial dependence of image intensities are not taken into account, is in fact essential for the criterion to be well-behaved around the registration solution. MI does not rely on pure intensity values to measure correspondence between images, but rather on their joint distribution and the relationship of occurrence. It also does not impose any modality specific constraints which makes it general enough to be applied to any problem formulation (inter- or intra modality).
Some of the areas where MI based image registration may fail is when there is insufficient information in images due to low resolution, low number of images, images not spatially invariant, images with shading artifacts. In some of these cases, MI based criterion will have multiple local optimals.
A minimum description length approach to statistical shape modeling
Davies, Rhodri H. and Twining, Carole J. and Cootes, Timothy F. and Waterton, John C. and Taylor, Christopher J.
Active Shape Models brought with them the ability to intelligentally deform to various intra-shape variations according to a labelled training set of landmark points. However the dependence of such methods on a low-noise training set marked manually poses challenges due to inter-observer differences which becomes even more pronounced in higher-dimensions (3D). To this end, the authors propose a method that addresses this problem, but introducing automatic shape modelling.
The method is based upon the idea of Occam's Razor, or more formally, The minimum description length (MDL). It is the principle formalization of Occam's razor in which the best hypothesis (a model and its parameters) for a given set of data is the one that leads to the best compression of the data. This essentially means that the MDL characteristic can be used to learn from a set of data points, the best hypothesis that fully describes the training data set, but in a compressed form. The authors use a simple two-part coding formulation of MDL, which although does not guarantee a minimum coding length,but does provide a computationally simple functional form to evaluate which is suitable to be used as an objective function for numerical optimization.
The proposes objective function is as follows:
$F = \sum_{p=1}^{n_g}D^{(1)}\left(\hat{Y}^p, R, \delta \right) + \sum_{q=n_g + 1}^{n_g + n_{min}}D^{2}\left(\hat{Y}^q, R, \delta \right)$
The algorithm proceeds by first parameterizing a single shape using a recursive algorithm. Once the recursive parameterization is complete, optimization of the objective function presented above proceeds. The algorithm first generates a parameterization for each shape recursively, to the same level. Then shapes are sampled according to the correspondence defined by the parameterization. Once this is done, a model is built automatically from the above sampled shapes. This model is then used to calculate the objective function.
The parameterization is changed as to converge to an optimal value for the objective function.
In order to change the parameterization of the model to converge to an optimal value of objective function, a ``reference" shape is chosen in order to avoid having the points converge to a bad a local minima (all points collapse to single part of the boundary). Due to the non-convex nature of the objective function, optimization is performed using genetic algorithm.
The method was tested both qualitatively and quantitatively on several sets of outlines of 2-D biomedical objects. Multiple anatomical sites in human body were chosen to test the model to provide an idea of how the method performs in a variety of shape settings. Quantitatively the models were shown to be highly compact in terms of the MDL. Qualitatively, the models were able to generate shapes that respected the overall shape of the training set, while still maintaining a good amount of deformation without going haywire.
Muliscale Vessel Enhancement Filtering
Frangi, Alejandro F. and Niessen, Wiro J. and Vincken, Koen L. and Viergever, Max A.
Delineation of vessel structures in human vasculature forms the precursor to a number of clinical applications. Typically, the delineation is performed using both 2D (DSA) and 3D techniques (CT, MR, XRay Angiography). However the decisions are still made using a maximum intensity projection (MIP) of the data. This is problematic since MIP is also affected by other tissues of high intensity, and low intensity vasculature may never be fully realized in the MIP compared to other tissues. This calls for a need for a type of vessel enhancement which can be applied prior to MIP to ensure MIP of the imaging have significant representation of low intensity vessels for detection. It can also facilitate volumetric views of vasculature and enable quantitative measurements.
To this end, Frangi et al. propose a vessel enhancement method which defines a "vesselness measure" by using eigenvalues of the Hessian matrix as indicators. The eigenvalue analysis of Hessian provides the direction of the smallest curvature (along the tubular vessel structure). The eigenvalue decomposition of a Hessian on a spherical neighbourhood around a point $x_0$ maps an ellipsoid with the axis represented by the eignevectors and their magnitude represented by their corresponding eigenvalues. The method provides a framework with three eigenvalues $|\lambda_1| <= |\lambda_2| <= |\lambda_3|$ with heuristic rules about their absolute magnitude in the scenario where a vessel is present. Particularly, in order to derive a well-formed ``vessel measure" as a function of these eigenvalues, it is assumed that for a vessel structure, $\lambda_1$ will be very small (or zero). The authors also add prior information about the vessel in the sense that the vessels appear as bright tubes in a dark background in most images. Hence they indicate that a vessel structure of this sort must have the following configuration of $\lambda$ values $|\lambda_1| \approx 1$, $|\lambda_1| << |\lambda_2|$, $|\lambda_2| \approx |\lambda_3|$. Using a combination of these $\lambda$ values, as well as a Hessian-based function, the authors propose the following vessel measure:
$\mathcal{V}_0(s) = \begin{cases}
0 \quad \text{if} \quad \lambda_2 > 0 \quad \text{or} \quad \lambda_3 > 0\\
(1 - exp\left(-\dfrac{\mathcal{R}_A^2}{2\alpha^2}\right))exp\left(-\dfrac{\mathcal{R}_B^2}{2\beta^2}\right)(1 - exp\left(-\dfrac{S^2}{2c^2}\right))
\end{cases}$
The three terms that make up the measure are $\mathcal{R}_A$, $\mathcal{R}_B$, and $S$. The first term $\mathcal{R}_A$ refers to the largest area cross section of the ellipsoid represented by the eigenvalue decomposition. It distinguishes between plate-like and line-like structures. The second term $\mathcal{R}_B$ accounts for the deviation from a blob-like structure, but cannot distinguish between line- and a plit-like pattern. The third term $S$ is simply the Frebenius norm of the Hessian matrix which accounts for lack of structure in the background, and will be high when there is high contrast compared to background. The vesselness measure is then analyzed at different scales to ensure that vessels of all sizes get detected.
The method was applied on 2D DSA images which are obtained from X-ray projection before and after contrast agent is injected. The method was also applied to 3D MRA images. The results showed promising background suppression when vessel enhancement filtering was applied before performing MIP.
Active Shape Models-Their Training and Application
Cootes, Timothy F. and Taylor, Christopher J. and Cooper, David H. and Graham, Jim
Computer Vision and Image Understanding - 1995 via Local Bibsonomy
Object detection in 2D scenes have mostly been performed using model-based approaches, which model the appearance of certain objects of interest. Although such approaches tend to work well in cluttered, noisy and occluded settings, the failure of such models to adapt to intra-object variability that is apparent in many domains like medical imaging, where the organ shapes tend to vary a lot, have lead to a need for a more robust approach. To this end, Cootes et al. propose a training based method which adapts and deforms well to per-object variations according to the training data, but still maintains rigidity across different objects.
The proposed method relies on a hand-labelled training set featuring a set of points called "landmark points" that describe certain specific positions of any object. For example, for a face the points may be "noise end, left eyebrow start, left eyebrow mid, left eyebrow end" and so on. Next, the landmark points across the whole training set are algined using affine transformations by minimizing a weighted-sum of squares difference (SSD) between corresponding landmark points amongst training examples. The optmization function (SSD) is weighted using the apparent variance of each landmark point. The higher the variance across training samples, the lower the weight. In order to ``summarize" the shape in the high-dimensional space of landmark point vectors, the proposed method uses Principal Component Analysis (PCA). PCA provides the eigenvectors which point to the direction of highest change in points in $2n$-dimensional space, while the corresponding eigenvalues provide the significane of each eigenvector. The best $t$ eigenvectors are chosen such that they describe a certain perctange of variance of the data. Once this is done, the model becomes capable of producing any shape by deriving from the mean shape of the object, using the equation:
$x = \bar{x} + Pb$
where $\bar{x}$ is the mean shape, $P$ = matrix of $t$ eigenvectors and $b$ = vector of free weights that can be tuned to generate new shapes. The values of $b$ are constrained to stay within boundaries determined using the training set, which essentially forms the basis of the argument that the model only deforms as per the training set.
The method was tested on a variety shapes, namely resistor models in electric circuits, heart model, worm model, and hand model. The models thus generated were robust and could successfully generate new examples by varying the values of $b$ on a straight line. However for worm-model, it was found that varying the values of $b$ only along a line may not be always suitable, especially in cases where the different dimensions of $b$ may have some existing non-linear relationship.
Once a shape model is generated, it is used to detect objects/shapes from new images. This is done by first initializing the model points on the image. The model points are then adjusted to the shape by using information from the image like edges. The adjustment is performed iteratively, by applying constraints on the calculated values of $dX$ and $dB$ so that they respect the training set. The iterations are performed until convergence of the model points to the actual shape of interest in the image.
One drawback of the proposed method is its high sensitivity to noise in training data annotations. Also, the relationship between various variables in $b$ is not entirely clear, and may negatively affect models when there exists a non-linear relationship. Also, the final convergence is somewhat dependent upon the initialization of the model points, and depend on local edge features for guidance, which may fail in some instances.
Random Walks for Image Segmentation
Grady, Leo
Image segmentation have been a topic of research in computer vision domain for decades. There have been a multitude of methods proposed for segmentation, but most have been dependent on a high level user input which guides the contour or boundaries towards the real boundaries. In order to come close to a fully automated or partially automated solution,
a novel method is proposed for performing multilabel,
interactive image segmentation using Random Walk algorithm as the fundamental driver of segmentation. The problem is formulated as follows: given a small number
of pixels with user-defined (or pre-defined) labels, assign the the probability that a random walker starting at each unlabeled pixel will first reach one of the pre-labeled pixels. The current pixel is then assigned the label corresponding to the max of this probability. This leads to high-quality segmentations of an image into $K$ different components. The algorithm is based on image graphs, where image pixels are represented as graphs connected by edges to its 8-connected neighbours.
In this paper, a novel approach to $K$-class image
segmentation problem is proposed which utilizes user-defined seeds representing the example regions of the image belonging to $K$ objects. Each seed specifies a location with a user-defined label. The algorithm labels an
unseeded pixel by resolving the question: Given a random
walker starting at this location, what is the probability that it first reaches each of the K seed points? It will be shown
that this calculation may be performed exactly without the
simulation of a random walk. By performing this calculation,
the algorithm assigns a K-tuple vector to each pixel that specifies the probability that a random walker starting from each unseeded pixel will first reach each of the K seed points. A final
segmentation may be derived from these K-tuples by selecting
for each pixel the most probable seed destination for a random
walker.
The graph weights are determined to be a function of the pixel intensities, specifically $w_{ij}$ = $exp(-(g_i - g_j)^2)$.
The algorithm works by biasing the random walker to avoid crossing sharp intensity gradients, which leads to a quality segmentation that respects object boundaries (including weak boundaries).
The algorithm exposes only one free variable $\beta$, and can be combined with other approaches involving pre- and post-filtering techniques. Additionally, the algorithm provides on-the-fly correction of previous detected boundary in an computationally efficient way.
Graph Cuts and Efficient N-D Image Segmentation
Boykov, Yuri and Funka-Lea, Gareth
International Journal of Computer Vision - 2006 via Local Bibsonomy
Over the last decade and a half, a plethora of image segmentation algorithms have been proposed, which can be categorized into belonging to roughly four categories, represented by a combination of two labels: explicit or implicit boundary representation, and variational or combinatorial methods. While classic methods like Snakes [1] and Level-Sets [2] belong to explicit/variational and implicit/variational category, there have been another set of algorithms falling under the combinatorial domain, which are DP or path-based algorithms which are explicit/combinatorial, and finally Graph-Cuts, which are implicit/combinatorial. The main difference between the categories is the space of solutions where search is performed. For variational methods, the search space is $\mathcal{R}^\infty$, while for combinatorial methods the search space is confined to $\mathcal{Z}^n$. An obvious advantage of combinatorial methods seem to better computational performance, but they also provide a globally optimal solution, which is global to the image. This makes the algorithm performance independent of numerical stability design decisions, and only dependent on the quality of global descriptors. Hence the algorithms provide a highly generalized, globally optimal framework which can be applied to a variety of problems, including image segmentation.
Graph-Cut methods are based upon the $s-t$ decomposition of a given image graph (where pixels are nodes and edges are formed between 8-connected neighbours). An $s-t$ decomposition of a graph $\mathcal{G} = \{V,E\}$ is a subset of edges $C \subset E$ such that the graph gets completely separated into individual components $s$ and $t$. The divided nodes are assigned to two terminal nodes, representing foreground and background of the image. The best-cut in an image graph is optimal if the cost of a cut (defined as $|C| = \sum_{e\in C}w_e$) is minimal. This corresponds to segmenting an image with a desirable balance of boundary and regional properties.
Graph-Cut exposes a general segmentation energy function (which constitutes the ``cost") as a combination of a regional term and boundary term, given as $E(A) = \lambda.R(A) + B(A)$. Regional term can be used to model apriori distribution of the pixel classes (probability of the pixel belonging to background or foreground). The boundary term can be represented by any boundary feature like local intensity gradients, zero-crossing, gradient direction or geometric costs. Although the region and boundary terms force the algorithm to find a boundary which strikes a good balance between the two, sometimes the lack of information for either or both terms may lead to incorrect segmentation. To offset these terms, hard constraints can be applied to the energy function. The constraints can be anything, for instance, a term that defines that pixels of particular intensities would belong to either foreground or background. The constraint can also be shape based, where shapes like circles or ellipses can be forced upon the final segmentation.
The proposed algorithm was applied on a variety of 2D and 3D images. Note that graph-cut can be generalized to N-D segmentation problems as well. A number of qualitative observations are reported. However there is a lack of quantitative foundation of the algorithm performance, compared to other state-of-art algorithms not necessarily from the same category as graph-cut.
Interactive live-wire boundary extraction
Barrett, William A. and Mortensen, Eric N.
Medical Image Analysis - 1997 via Local Bibsonomy
Edge, contour or boundary detection in 2D images have been an area of active research, with a variety of different algorithms. However due to a wide variety of image types and content, developing automatic segmentation algorithms have been challenging, while manual segmentation is tedious and time consuming. Previous algorithms approaching this task have tried to incorporate higher level constraints, energy functional (snakes), global properties (graph based). However the approaches still do not entirely fulfill the fully automated criteria due to a variety of reasons.
To this end, Barrett et al. propose a graph-based boundary extraction algorithm called Interactive Live-Wire, which is an extension to the original live-wire algorithm presented in Mortensen et al. [1]. The algorithm is built upon a reformulation of the segmentation approach into graphs, particularly, an image $I$ is converted to an undirected graph with edges from a pixel $p$ to all it's neighbouring 8-connected pixels. Each pixel or node is assigned a local cost according to a function (described later). The segmentation task then becomes a problem where there needs to be a shortest path from a pixel $p$ (seed) to another pixel $q$ (free goal point), where the cumulative cost of path is minimum. The local cost function is defined as:
$l(p,q) = w_G.f_G(q) + w_Z.f_Z(q) + w_D.f_D(p,q)$
where $w_i, i = {G, Z, D}$ are weight coefficients controlling relative importance of the terms. The three terms that make up the local cost function are gradient magnitude ($f_G(q)$), Laplacian Zero-Crossing feature ($f_Z(q)$), and Gradient Direction feature ($f_D(p,q)$). The first term $f_G(q)$ defines a strong measure of edge strength, and is heavily weighted. The term $f_Z(q)$ provides a second degree measure for edge strength in the form of zero-crossing information. The third term $f_D(p,q)$ adds a smoothness constraint to the live-wire boundary by adding high cost for rapidly changing gradient directions.
The algorithm also exhibits some other features, namely boundary freezing, on-the-fly learning and data-driven cooling. Boundary freezing proves useful when Live-wire segmentation digresses from the desired object boundary during interactive mode. The boundary can be "frozen" right before the digression point by specifying another seed point, until which the boundary is frozen and not allowed to be changed. On-the-fly learning provides robustness to the method by learning the underlying cost distribution of a known "good" boundary, and using that to guide the live-wire further to follow similar distribution. Data-driven path cooling allows the live-wire to generate new seed points automatically as a function of image data and path properties. Pixels on "stable" paths will cool down and eventually freeze, producing new seed points.
The results report that average times taken for segmenting a region using Live-Wire was roughly 4.6 times less than manual human tracing time. Live-Wire provided same amount of accuracy as manual tracing would in a fraction of time, with high reproducibility.
However, the method does not provide a way to ``snap out" of an \textit{automatically} frozen live-wire segmentation. On-the-fly training can fail at instances where the edges of the object change too fast, and not much implementation related information is provided, especially for freezing and training parts.
STACS: new active contour scheme for cardiac MR image segmentation
Pluempitiwiriyawej, Charnchai and Moura, José M. F. and Wu, Yi-Jen Lin and Ho, Chien
Automated segmentation of various anatomical structures of interest from medical images has been a well grounded field of research in medical imaging. One such problem is related to segmenting whole heart region from a sequence of magnetic resonance imaging (MRI), which is currently done manually, and is time consuming and tedious. Although many automated techniques exist for this, the task remains challenging due to the complex nature of the problem, partly because of low contrast between heart and nearby tissue. Moreover many of the methods are unable to incorporate prior information into the process. To this end, Pluempitiwiriyawej et al. proposed a version of active contour energy minimization based method to segment the whole heart region, including the epicardium, and the left and right ventricular endocardia.
The proposed method follows the framework laid out by Chan and Vese\cite{Chan2001}. However Pluempitiwiriyawej et al. propose a modified energy function, which consists of four energy terms. The energy function is given below, where $C$ is the contour represented as a level set function $\phi(x,y)$:
$J(C) = \lambda_1 J_1(C) + \lambda_2 J_2(C) + \lambda_3 J_3(C) + \lambda_4 J_4(C)$
The coefficients $\lambda_{1..4}$ determine the weight of terms $J_{1..4}$. The first term $J_1(C)$ is designed to add stochastic models $\mathcal{M}_1, \mathcal{M}_2$ corresponding to the regions inside and outside of the active contour $C$. The models dictate the probability distribution from which the image intensities making up the inside and outside region of the contour are sampled. The negative log of this term is minimized, which essentially maximizes the probability $p(u | C, \mathcal{M}_1, \mathcal{M}_2)$ given the active contour $C$, and the models $\mathcal{M}_1, \mathcal{M}_2$.
The second term $J_2(C)$ is designed similar to the classical Snakes\cite{Kass1988} in the sense that it uses edges to guide the contour towards the structure of interest. For this term, a simple edge map is used after convolving with a Gaussian filter which smooths out the noise. The term $J_3(C)$ encodes an shape prior which constraints the contour to follow an elliptical shape, and guides it in conjunction with the region and edge information.
The final term $J_4(C)$ which encodes the total Euclidean arc length of the contour. This forces the contour to be ``smooth", without rough edges.
The process of minimizing the energy function follows a three-task approach. The first task is to estimate the stochastic model parameters $\mu_k, \sigma^2_k$, and is performed by fixing the position of initial contour $C$, taking derivatives of $J$ w.r.t stochastic model parameters, and solving by equating to zero. The second task estimates the parameters of the ellipse using least squares method. The third and final task involves the contour using the estimated parameters in task one and two, such that it minimizes the function $J$. The method also performs stochastic relaxation, by dynamically changing the values of parameters $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ as the optimization process proceeds. The intuition is that when the optimization starts, the edge and region terms must guide the contour, and as the process proceeds to it's end, the shape prior and contour length term should carry more weight to regularize the effective shape of the contour.
The study used 48 MRI studies acquired by imaging rat hearts, and compared the proposed method with two earlier methods, namely Xu and Prince's GVF \cite{ChenyangXu1998}, and Chan and Vese \cite{Chan2001}. The authors also design a new quantitative metric, which is a modification of the Chamfer matching \cite{Barrow} technique. The reported results are observed to be in excellent agreement with the gold standard hand-traced contours. However the similarity values for other methods against human gold-standard were not reported.
Active contours without edges
T.F. Chan and L.A. Vese
IEEE Transactions on Image Processing - 2001 via Local CrossRef
Typically, the energy minimization or snakes based object detection frameworks evolve a parametrized curve guided by some form of image gradient information. However due to heavy reliance on gradients, the approaches tend to fail in scenarios where this information is misleading or unavailable. This cripples the snake and renders it unusable as it gets stuck in a local-minima away from the actual object. Moreover, the parametrized snake lacks the ability to model multiple evolving curves in a single run. In order to address these issues, Chan and Vese introduced a new framework which utilized region based information to guide a spline, and tries to solve the minimal partition problem formulated by Mumford and Shah.
The framework is built upon the following energy equation, where $C$ is a level-set formulation of the curve:$F(c1, c2, C) = \mu . \text{Length}(C) + v . \text{Area}(inside(C))\\
\lambda_1 \int_{inside(C)}|u_0(x,y) - c_1|^2 dxdy + \lambda_2 \int_{outside(C)}|u_0(x,y) - c_2|^2 dxdy$
The framework essentially divides the image into two regions (per curve), which are referred to as inside and outside of the curve. The first two terms of the equation control the physical aspects of the curve, particularly the length, and area inside the curve, with their contributions controlled by two parameters $\mu$ and $v$.
The image forces in this equation correspond to the third and fourth terms, which are identical but work in respective regions identified by the curve. The terms use $c_1$ and $c_2$, which are the mean intensity values inside and outside the curve respectively to guide the curve towards a minima where the both regions are consistent with respect to the mean intensity values.
The proposed framework also utilizes an improved representation of the curve in the form of a level set function $\phi(x, y)$, which has many numerical advantages and naturally supports multiple curves evolved during a single run, as compared to the traditional snakes model where only one curve can be evolved in a single run. The unknown function $\phi$ is computed using Euler-Lagrange equations formulated using modified Heaviside function $H$, and Dirac measure $\delta$.
The proposed framework was applied on numerous challenging 2D images with varying degree of difficulties. The framework was also capable of segmenting point clouds decomposed into 2D images, objects with blurred boundaries, and contours without gradients, all without requiring image denoising.
Due to the formulation of $\phi$ approximation routine, the framework has tendencies to find actual global minima independent of the initial position of the curve. However the choice of multiple parameters namely $\lambda_{1,2}, \mu, v$ is done heuristically, and seem to be problem dependent. Also, the framework's implicit dependency on absolute image intensities in regions inside and outside of curve sometimes fail in very specific cases where the averages tend to be zero, though the authors proposed to use image curvature and orientation information from the initial image $u_0$.
Snakes: Active contour models
Michael Kass and Andrew Witkin and Demetri Terzopoulos
International Journal of Computer Vision - 1988 via Local CrossRef
Low level tasks such as edge, contour and line detection are an essential precursor to any downstream image analysis processes. However, most of the approaches targeting these problems work as isolated and autonomous entities, without using any high-level image information such as context, global shapes, or user-level input. This leads to errors that can further propagate through the pipeline without providing an opportunity for future correction. In order to address this problem, Kass et al. investigate the application of an energy minimization based framework for edge, line and contour detection in 2D images.
Although energy minimization had earlier been utilized for similar tasks, Kass et al's framework exposes a novel external force factor, which allows external forces or stimuli to guide the ``snake" towards a "correct answer". Moreover, the framework exhibits an active behaviour since it is designed to always minimize the energy functional. A "snake" is a controlled continuity spline which is under the influence of forces in the energy functional. The energy functional is made up of three terms, where $v(s)$ is a parametrized snake.
$ E_{snake}^{*} = \int_{0}^{1}E_{internal}(v(s)) + E_{image}(v(s)) + E_{constraint}(v(s)) $
The internal energy force term is entirely dependent on the shape of the curve, which constraints the snake to be "continuous" and well behaved. The force further encapsulates two terms which control the degree of stretchness of the curve (represented by the first derivative of the image), and ensure that the curve does not have too many bends (using the second derivative of the image).
The image energy force term controls what kind of salient features does the snake track. The force encapsulates three terms, corresponding to the presence of lines, edges and a termination term. The line term uses raw image intensity as energy term, while the edge term uses a negative square of image gradient to make the snake attracted to contours with large image gradients. Further, a termination term allows the snake to find terminations of line segments and corners in the image. The combination of line and termination terms forces the snake to be attracted to edges and terminations.
The constraint term is used to model external stimuli, which can come from high-level processes, or through user intervention. The framework allows the application of a spring to the curve to constraint the snake or move it in a desired direction.
In order to test the framework, a user interface called ``Snake Pit" was created which allowed user control in the form of a spring attachment. The overall approach was tested on a number of different images, including the ones with contour illusion. The framework was also extended for application to stereo matching and motion tracking. For stereo, an extra energy term is added to constraint the snakes in two disparate images to stay close in the coordinate space, which models the fact that disparities in images do not change too rapidly.
However the framework suffers when subjected to a high rate-of-change in both stereo matching and motion tracking problems.
The proposed framework performed acceptably in many challenging scenarios. However the framework's underlying assumption about following edges and contours using image gradients may fail in cases where there is not much gradient information present in images. | CommonCrawl |
\begin{document}
\title{Maximal inequalities and convergence results on multidimensionally indexed demimartingales }
\author{Milto Hadjikyriakou\footnote{School of Sciences, University of Central Lancashire, Cyprus campus, 12-14 University Avenue, Pyla, 7080 Larnaka, Cyprus (email:[email protected]).}~~ and B.L.S. Prakasa Rao \footnote{CR RAO Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad 500046, India (e-mail: [email protected]).}}
\maketitle
\begin{abstract} We obtain some maximal probability and moment inequalities for multidimensionally indexed demimartingales. Although the class of single-indexed demimartingales has been studied extensively, no significant amount of work has been done for the corresponding multiindexed class of random variables. This work aims to fill in this gap in the literature by extending well-known inequalities and asymptotic results to this more general class of random variables. \end{abstract}
\textbf{Keywords}: Multidimensionally Indexed Random Variables, Demimartingales, Demisubmartingales, H\'{a}jek-R\'{e}nyi Inequality, Doob's inequality, Chow-type inequality, Whittle-type inequality.
\textbf{MSC 2010:} 60E15; 60F15; 60G42; 60G48.
\section{Introduction} The notion of demimartingales was introduced by Newman and Wright in \cite{NW1982} in an effort to generalize the concept of positive association. This new class of random variables contains the class of associated random variables, since it can be easily proven that the partial sums of mean zero associated random variables form a sequence of demimartingales. This new class of random objects has been studied extensively over the last few decades; see for example \cite{PR2012} for an extensive discussion on this topic. The notion of multidimensionally indexed demimartingales was introduced in \cite{MH2010}. The idea of random vectors indexed by lattice points is not new and its origin goes back to statistical mechanics and ergodic theory. Moreover, models of several phenomena in statistical physics, crystal physics or Euclidean quantum field theories involve multiple sums of multiindexed random variables. Having in mind all these applications and the observation that the partial sums of mean zero multiindexed associated random variables form a sequence of multiindexed demimartingales, it makes sense to further explore the properties and the asymptotic behavior of this general class of random variables.
Throughout the paper all random variables are defined on a probability space $(\Omega, \mathcal{F}, \mathcal{P})$ and the following notation will be used: let $\mathbb{N}^k$ denote the $k$-dimensional positive integer lattice. For $\mathbf{n}, \mathbf{m}, \in \mathbb{N}^k$ with $\mathbf{n}=\left(n_1, \ldots, n_k\right)$ and $\mathbf{m}=\left(m_1, \ldots, m_k\right)$, we write $\mathbf{n} \leq \mathbf{m}$ if $n_i \leq m_i, i=1, \ldots, k$ and $\mathbf{n}<\mathbf{m}$ if $n_i \leq m_i, i=1, \ldots, k$ with at least one strict inequality. We say that $\mathbf{n} \rightarrow \infty$ if $\displaystyle\min _{1 \leq j \leq k} n_j \rightarrow \infty$ and the notation $I(A)$ is used to denote the indicator function of the set $A$.
The paper is structured as follows: in Section 2, we provide some basic definitions and important results that will be useful for the rest of the paper. Section 3 is devoted to Doob-type inequalities and related maximal results. In Section 4, a new Chow-type inequality is discussed together with an asymptotic result and a H\'{a}jek-R\'{e}nyi inequality for multiindexed associated random variables. In Section 5, we present some maximal inequalities given in terms of Orlicz functions while, in Section 6, an uprossing inequality is studied. Finally, in Section 7, we present a Whittle-type inequality and provide a strong law.
\section{Preliminaries} We start by providing the definitions of multiindexed associated random variables and multiindexed demi(sub)martingales.
\begin{definition}
A collection of multidimensionally indexed random variables $\left\{X_{\mathbf{i}}, \mathbf{i} \leq \mathbf{n}\right\}$ is said to be associated if for any two coordinatewise nondecreasing functions $f$ and $g$
$$
\operatorname{Cov}\left(f\left(X_{\mathbf{i}}, \mathbf{i} \leq \mathbf{n}\right), g\left(X_{\mathbf{i}}, \mathbf{i} \leq \mathbf{n}\right)\right) \geq 0,
$$
provided that the covariance is defined. An infinite collection is associated if every finite subcollection is associated. \end{definition}
\noindent Note that the definition given above is exactly the same as the classical definition of association, stated for the case of multidimensionally indexed random variables, since the index of the variables in no way affects the qualitative property of association, i.e., that nondecreasing functions of all (or some) of the variables are nonnegatively correlated.
The class of multidimensionally indexed demimartingales and demisubmartingales was introduced in \cite{MH2010} (see also \cite{CH2011}).
\begin{definition}
A collection of multidimensionally indexed random variables $\left\{S_{\mathbf{n}}, \mathbf{n}\in \mathbb{N}^k\right\}$ is called a multiindexed demimartingale if
$$
E\left[\left(S_{\mathbf{j}}-S_{\mathbf{i}}\right) f\left(S_{\mathbf{k}}, \mathbf{k} \leq \mathbf{i}\right)\right] \geq 0
$$
for all $\mathbf{i}, \mathbf{j} \in \mathbb{N}^k$ with $\mathbf{i} \leq \mathbf{j}$ and for all componentwise nondecreasing function If, in addition $f$ is required to be nonnegative, then the collection $\left\{S_{\mathbf{n}}, \mathbf{n}\in \mathbb{N}^k\right\}$ is said to be a multiindexed demisubmartingale. \end{definition}
\begin{remark}
It can easily be proven that the partial sums of mean zero multiindexed associated random variables form a sequence of multiindexed demimartingales. Furthermore, it is obvious that, with the natural choice of $\sigma$-algebras, a multiindexed (sub)martingale forms a multiindexed demi(sub)martingale. \end{remark}
The next two results can be found in \cite{WH2009} (see Theorem 3.2 and 3.3 respectively). The first one provides a Doob-type inequality for functions of single-indexed demimartingales while the second one, which is a direct consequence of the Doob's inequality, provides upper bounds for expectations of maxima of functions of a single-indexed demimartingale. The multiindexed analogues of these two results will be proved in the next section.
\begin{theorem}
\label{Doob}Let $S_{1}, S_{2}, \ldots$ be a demimartingale, $g$ be a nonnegative convex function on $\mathbb{R}$ with $g(0)=0$ and $g\left(S_{i}\right) \in L^{1}, i \geqslant 1$. Then for any $\varepsilon>0$,
$$
\varepsilon P\left\{\max _{1 \leqslant k \leqslant n} g\left(S_{k}\right) \geqslant \varepsilon\right\} \leqslant \int_{\left\{\max _{1 \leqslant k \leqslant n} g\left(S_{k}\right) \geqslant \varepsilon\right\}} g\left(S_{n}\right) d P .
$$ \end{theorem}
\begin{theorem}
\label{Wang}Let $S_{1}, S_{2}, \ldots$ be a demimartingale and $g$ be a nonnegative convex function on $\mathbb{R}$ with $g(0)=0$. Suppose $E\left(g\left(S_{n}\right)\right)^{p}<\infty$ for $p\geq 1$ and all $n \geqslant 1$. Then for every $n \geqslant 1$,
$$
E\left(\max _{1 \leqslant k \leqslant n} g\left(S_{k}\right)\right)^{p} \leqslant\left(\frac{p}{p-1}\right)^{p} E\left(g\left(S_{n}\right)\right)^{p} \quad\mbox{for}\quad p>1
$$
and
$$
E\left(\max _{1 \leqslant k \leqslant n} g\left(S_{k}\right)\right) \leq \frac{e}{e-1}(1+E\left( g\left(S_{n}\right)\log^+g\left(S_{n}\right)\right)).
$$ \end{theorem}
\noindent The next lemma will be used in the proof of Theorem \ref{theorem1}.
\begin{lemma}
Let $f(x_1,x_2,\ldots,x_n) = \max\{x_1,x_2,\ldots,x_n\}$. The function $f$ is componentwise convex and its right derivative with respect to its $i$-th component for $i=1,2,\ldots, n$ is a nonnegative constant. \end{lemma} \begin{proof}
Note that $f$ is convex and hence it is also componentwise convex. Without loss of generality, we will calculate its right derivative with respect to its last component. Define $g(t) = f_{+}^{'}(x_1,x_2,\ldots,x_{n-1},t)$ where $f^{'}_{+}$ denotes the right derivative of the function $f$.Then,
\begin{align*}
&g(t) = \lim_{h\to 0^+}\frac{f(x_1,x_2,\ldots,x_{n-1},t+h)-f(x_1,x_2,\ldots,x_{n-1},t)}{h} \\
&= \lim_{h\to 0^+}\frac{\max\{x_1,x_2,\ldots,x_{n-1},t+h\}-\max\{x_1,x_2,\ldots,x_{n-1},t\}}{h}\\
& = \lim_{h\to 0^+}\frac{\max\{\max\{x_1,x_2,\ldots,x_{n-1}\},t+h\}-\max\{\max\{x_1,x_2,\ldots,x_{n-1}\},t\}}{h}.
\end{align*}
Suppose that $\max\{x_1,x_2,\ldots,x_{n-1}\} = x_j$ for some $j=1,2,\ldots,n-1$. Then
\[
g(t) = \lim_{h\to 0^+}\frac{\max\{x_j,t+h\}-\max\{x_j,t\}}{h}.
\]
First, consider the case where $x_j >t$. Then, $h$ can be chosen sufficiently small such that $x_j > t+h$ and therefore $g(t) = 0$. For the case where $x_j<t$ and since $h>0$, we also have that $x_j<t+h$. Thus, $g(t) = 1$ which concludes the proof. \end{proof}
\section{Doob-type inequalities and related results} We provide a Doob-type inequality for positive multiindexed demimartingales. The result is inspired by the work of Cairoli in \cite{CR1970} for multidimensionally indexed submartingales.
\begin{theorem}
\label{theorem1} Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$, be a positive multidimensionally indexed demimartingale. Then,
\begin{enumerate}
\item [a. ] For $p>1$ and $ES_{\mathbf{n}}<\infty$,
$$E\left( \displaystyle \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)^p \leq \left(\frac{p}{p-1}\right)^{kp}E\left( S_{\mathbf{n}}\right)^p.$$
\item [b. ] For any $\epsilon>0$ and $E\left( S_{\mathbf{n}}\left(\log^{+} S_{\mathbf{n}}\right)^k\right)<\infty$,
$$ \epsilon P\left ( \displaystyle \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}} \geq \epsilon\right) \leq \sum_{i=1}^{k} (i-1)!A^i + k! A^k E\left( S_{\mathbf{n}}\left(\log^{+} S_{\mathbf{n}}\right)^k\right)$$
where $A = \frac{e}{e-1}$.
\end{enumerate} \end{theorem} \begin{proof}
We start with the first inequality. The proof will follow by induction; we discuss first the case for $k=2$ i.e. the case of a positive 2-indexed demimartingale $\{S_{ij}, (i,j) \in\mathbb{N}^{2} \}$. For $p >1$,
\begin{equation}
\label{moment1} E\left(\max_{(i,j)\leq (n_1,n_2)}S_{ij}\right)^p = E\left(\max_{1\leq j\leq n_2}\max_{1\leq i\leq n_1}S_{ij}\right)^p = E\left(\max_{1\leq j\leq n_2}Y_j\right)^p
\end{equation}
where $Y_j = \displaystyle\max_{1\leq i\leq n_1}S_{ij}$ for $j=1,2,\ldots,n_2$. Let $f$ be a componentwise nondecreasing function. Then,
\begin{align*}
&E[(Y_{j+1}-Y_j)f(Y_1,\ldots,Y_j)] = E\left[ \left( \max_{1\leq i\leq n_1}S_{ij+1} -\max_{1\leq i\leq n_1}S_{ij} \right) f\left(\max_{1\leq i\leq n_1}S_{i1},\max_{1\leq i\leq n_1}S_{i2}\ldots, \max_{1\leq i\leq n_1}S_{ij}\right)\right]\\
&\geq E\left(\sum_{i=1}^{n_1}(S_{ij+1}-S_{ij})g^{'}_{i+}(S_{1j},\ldots,S_{n_1j}) f\left(\max_{1\leq i\leq n_1}S_{i1},\max_{1\leq i\leq n_1}S_{i2}\ldots, \max_{1\leq i\leq n_1}S_{ij}\right) \right)\\
&=\sum_{i=1}^{n_1}E\left((S_{ij+1}-S_{ij})g^{'}_{i+}(S_{1j},\ldots,S_{n_1j}) f\left(\max_{1\leq i\leq n_1}S_{i1},\max_{1\leq i\leq n_1}S_{i2}\ldots, \max_{1\leq i\leq n_1}S_{ij}\right) \right)\\
&\geq 0
\end{align*}
where $g^{'}_{i+}$ denotes the right derivative of the function $g(x_1,\ldots,x_n) = \max\{x_1,\ldots,x_n\}$ with respect to its $i$-th component which is a nonnegative constant based on the previous lemma. The first inequality is due to the convexity of the maximum function while the last inequality is due to the demimartingale property of the sequence $\{S_{ij}, 1\leq j\leq n_2\}$. Thus, the sequence $Y_j = \displaystyle\max_{1\leq i\leq n_1}S_{ij}$ forms a demimartingale sequence. By combining the latter result with \eqref{moment1} and Theorem \ref{Wang} we have that
\begin{align*}
& E\left(\max_{(i,j)\leq (n_1,n_2)}S_{ij}\right)^p \leq \left( \frac{p}{p-1}\right)^pE(Y_{n_2})^p= \left( \frac{p}{p-1}\right)^p E \left(\max_{1\leq i\leq n_1} S_{in_2}\right)^p \leq \left( \frac{p}{p-1}\right)^{2p}E(S_{n_1n_2})^p.
\end{align*}
Note that for obtaining the last inequality we applied Theorem \ref{Wang} again since $\{S_{in_2}, 1\leq i\leq n_1\}$ forms a single indexed demimartingale. We assume that the statement is true for $k-1$ and we consider a $k$-indexed demimartingale $\{S_{\mathbf{n}}, \mathbf{n}\in\mathbb{N}^{k}\}$. Let $(i,i_2,\ldots,i_k)=(i,\mathbf{s})$ where $\mathbf{s} = (i_2, \ldots,i_{k})$. Then, for a fixed $i, \, 1\leq i\leq n_1$, the sequence $\{S_{i\mathbf{s}}, \mathbf{s}\in \mathbb{N}^{k-1}\}$ forms a $(k-1)$-indexed demimartingale. Then,
\begin{align*}
&E\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)^p = E\left( \max_{\mathbf{s}\leq \mathbf{n^{*}}}\max_{1\leq i\leq n_1}S_{i\mathbf{s}}\right)^p = E\left( \max_{\mathbf{s}\leq \mathbf{n^{*}}}Y_{\mathbf{s}}\right)^p
\end{align*}
where $\mathbf{n^*} = (n_2, \ldots,n_k)$.
By applying similar arguments as the ones for the case where $k=2$, we can prove that $Y_{\mathbf{s}}$ is a $(k-1)$-indexed demimartingale. By the induction hypothesis and the result of Theorem \ref{Wang} we have that
\begin{align*}
&E\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)^p \leq \left( \frac{p}{p-1}\right)^{(k-1)p}E\left( \max_{1\leq i\leq n_1} S_{in_2\ldots n_k}\right)^p
\leq \left (\frac{p}{p-1}\right)^{kp}E\left( S_{\mathbf{n}}\right)^p
\end{align*}
since $\{S_{in_2\ldots n_k}, 1\leq i\leq n_1\}$ forms a single indexed demisubmartingale.
\noindent For the second inequality, we start again with the case $k=2$. For any $\epsilon>0$,
\begin{align*}
\epsilon P\left( \max_{(i,j)\leq (n_1, n_2)} S_{ij} \geq \epsilon \right) \leq E\left( \max_{(i,j)\leq (n_1, n_2)} S_{ij}\right) = E\left( \max_{1\leq j\leq n_2} Y_j\right)
\end{align*}
where $Y_j = \displaystyle\max_{1\leq i\leq n_1}S_{ij}$. Recall that this is a demimartingale sequence and by employing the second inequality in Theorem \ref{Wang} we have
\begin{align*}
& \epsilon P\left( \max_{(i,j)\leq (n_1, n_2)} S_{ij} \geq \epsilon \right) \leq E\left( \max_{(i,j)\leq (n_1, n_2)} S_{ij}\right) = E\left( \max_{1\leq j\leq n_2} Y_j\right)\\
&\leq \frac{e}{e-1} +\frac{e}{e-1}E(Y_{n_2}\log^{+}Y_{n_2}) = A+AE\left( \max_{1\leq i\leq n_1}S_{in_2}\log^{+}\max_{1\leq i\leq n_1}S_{in_2}\right)\\
&=A+A E\left( \max_{1\leq i\leq n_1}(S_{in_2}\log^{+}S_{in_2})\right)
\end{align*}
for $A = \frac{e}{e-1}$. Observe that the function $x\log^{+}x$ is nondecreasing convex and hence $(Z_i)_{i\geq 1} = (S_{in_2}\log^{+}S_{in_2})_{i\geq 1}$ is a single indexed demisubmartingale (Lemma 2.1 in \cite{C2000}). Then,
\begin{align*}
& \epsilon P\left( \max_{(i,j)\leq (n_1, n_2)} S_{ij} \geq \epsilon \right) \leq A+A(A + AEZ_{n_1}\log^{+}Z_{n_1}) \\ &= A+A^2 +A^2 E\left( S_{n_1n_2}\log^{+}S_{n_1n_2}\log^{+}(S_{n_1n_2}\log^{+}S_{n_1n_2})\right)\\
&\leq A+A^2+2A^2E(S_{n_1n_2}(\log^{+}S_{n_1n_2})^2)
\end{align*}
where the last inequality follows from the fact that for any $x>0$, $\log^{+}(x\log^{+}x) \leq 2\log x$. We assume that the statement is true for $k-1$ and we will prove its validity for $k$. Following the same notation as in the first part, for any $\epsilon >0$,
\begin{align}
& \nonumber \epsilon P\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}} \geq \epsilon\right) \leq E\left( \max_{\mathbf{s}\leq \mathbf{n^{*}}}\max_{1\leq i\leq n_1}S_{i\mathbf{s}}\right) = E\left( \max_{\mathbf{s}\leq \mathbf{n^{*}}}Y_{\mathbf{s}}\right)\\
&\label{ind}\leq A+\sum_{i=2}^{k-1}(i-1)!A^{i} +(k-1)!A^{k-1}E(Y_{\mathbf{n^{*}}}(\log^{+}Y_{\mathbf{n^{*}}})^{k-1})
\end{align}
due to the induction hypothesis. Following similar steps as before we have that
\begin{align*}
&E\left(Y_{\mathbf{n^{*}}}(\log^{+}Y_{\mathbf{n^{*}}})^{k-1}\right) = E\left( \max_{1\leq i\leq n_1}S_{i\mathbf{n^*}}(\log^{+}\max_{1\leq i\leq n_1}S_{i\mathbf{n^*}})^{k-1}\right)=E\left( \max_{1\leq i\leq n_1}S_{i\mathbf{n^*}}(\log^{+}S_{i\mathbf{n^*}})^{k-1}\right)\\
&\leq A + AE Z_{n_1}\log^{+}Z_{n_1} = A+AE\left(S_{\mathbf{n}}(\log^{+}S_{\mathbf{n}})^{k-1}\log^{+}\left(S_{\mathbf{n}}(\log^{+}S_{\mathbf{n}})^{k-1} \right)\right)\\
&\leq A+AkE(S_{\mathbf{n}}(\log^{+}S_{\mathbf{n}})^{k}).
\end{align*}
Note that the first inequality is due to the fact that $(Z_ i)_{i\geq 1} = (S_{i\mathbf{n^*}}(\log^{+}S_{i\mathbf{n^*}})^{k-1})_{i\geq 1}$ forms a single indexed demisubmartingale while for the second one the inequality $\log^{+}(x(\log^{+}x)^{k-1}) \leq k\log^{+}x, \, x>0$ is used. The latter expression together with \eqref{ind} lead to the desired result. \end{proof}
Theorem \ref{theorem1} was obtained by following the ideas of the corresponding result for positive submartingales proved by Cairoli \cite{CR1970}. In his paper, Cairoli provided counterexamples showing that some classical inequalities for maximums of submartingales with discrete time are not valid in the case of submartingales with multi-dimensional time, including Doob's inequality. Despite Cairoli's counterexample, Doob's type inequality indeed has an extension to the case of multidimensional index as proved in \cite{CS1990} (see Corollary 2.4 there). Moreover, under specific conditions, the well-known Doob's inequality is also valid for a subclass of multiindexed submartingales (see for example Proposition 1.6 in \cite{W1986}). Although, in Theorem \ref{theorem1}, we proved that Cairoli's inequalities for multiindexed martingales are also valid for multiindexed demimartingales, it is of interest to check whether the classical Doob's inequality can be obtained for multiindexed demimartingales. We start by defining the concept of multidimensional rank orders.
\noindent Let $\mathbf{n} = (n_1 \, n_2 \, \ldots \, n_k)$. We define the multiindexed rank orders $R_{\mathbf{n}}^{(j)}$ by \[ R_{\mathbf{n}}^{(j)} = \begin{cases} j\mbox{-th largest of } \{S_\mathbf{m}, \mathbf{m}\leq \mathbf{n}\} &\, \mbox{for} \, j\leq \prod_{i=1}^{k} n_i\\ \displaystyle\min_{\mathbf{m}\leq \mathbf{n}}S_\mathbf{m} & \, \mbox{for} \, j> \prod_{i=1}^{k} n_i. \end{cases} \] The theorem that follows generalizes to the case of multiindexed demi(sub)martingales a result for single- indexed demjmartingles which can be found in \cite{NW1982}.
\begin{theorem}
\label{Doob_new}Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$, be a multiindexed demi(sub)martingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and let $g$ be a (nonnegative) nondecreasing function on $\mathbb{R}$ with $g(0) = 0$. Then for any $\mathbf{n}$ and any $j$,
\begin{equation}
\label{NW1}E\left[ \int_{0}^{R_{\mathbf{n}}^{(j)}} u{\rm d}g(u)\right] \leq E(S_\mathbf{n}g(R_{\mathbf{n}}^{(j)}))
\end{equation}
and for any $\epsilon>0$,
\begin{equation}
\label{NW2} \epsilon P\left( R_{\mathbf{n}}^{(j)} \geq \epsilon\right) \leq E\left(S_\mathbf{n}I\left(R_{\mathbf{n}}^{(j)}\geq \epsilon\right)\right).
\end{equation} \end{theorem} \begin{proof}
For fixed $j, n_1, n_2, \ldots, n_{k-1}$ we set
\[
Y_i = R^{(j)}_{n_1n_2\cdots n_{k-1}i} \quad\mbox{for}\quad1\leq i\leq n_k
\]
with $Y_0 = 0$. First observe that
\[
S_{\mathbf{n}}g(R^{(j)}_{\mathbf{n}}) = \sum_{i=0}^{n_k-1}S_{\mathbf{n}^*,i+1}(g(Y_{i+1})-g(Y_{i}))+\sum_{i=1}^{n_k-1}(S_{\mathbf{n}^*,i+1}-S_{\mathbf{n}^*,i})g(Y_i)
\]
where $\mathbf{n}^* = (n_1 \, n_2\, \ldots\, n_{k-1}) $ and $S_{n^*,i} = S_{n_1 \, n_2\, \ldots\, n_{k-1} \,i}$.
We want to prove that
\begin{equation}
\label{ineq4}S_{\mathbf{n}^*,i+1}(g(Y_{i+1})-g(Y_{i}))\geq Y_{i+1}(g(Y_{i+1})-g(Y_{i})).
\end{equation}
Note that
\[
Y_{i+1}\geq Y_{i}
\]
where the non-degenerate case of \eqref{ineq4} is the one for which $Y_{i+1} = S_{\mathbf{n}^*,i+1}$ for $j=1$ while $Y_{i+1} > S_{\mathbf{n}^*,i+1}$ for any $j>1$ and by taking into account the monotonicity of $g$, \eqref{ineq4} holds true for any $i$. Moreover, for any $i$,
\[
S_{\mathbf{n}^*,i+1}(g(Y_{i+1})-g(Y_{i}))\geq Y_{i+1}(g(Y_{i+1})-g(Y_{i}))\geq \int_{Y_i}^{Y_{i+1}}u{\rm d}g(u)
\]
which leads to
\[
S_{\mathbf{n}}g(R^{(j)}_{\mathbf{n}}) \geq \int_{0}^{R^{(j)}_{\mathbf{n}}}u{\rm d}g(u)+\sum_{i=1}^{n_k-1}(S_{\mathbf{n}^*,i+1}-S_{\mathbf{n}^*,i})g(Y_i)
\]
By taking expectations on both sides we have that
\[
E[ S_{\mathbf{n}}g(R^{(j)}_{\mathbf{n}}) ] \geq E\left[\int_{0}^{R^{(j)}_{\mathbf{n}}}u{\rm d}g(u)\right]+\sum_{i=1}^{n_k-1}E[(S_{\mathbf{n}^*,i+1}-S_{\mathbf{n}^*,i})g(Y_i)].
\]
The desired result follows by noticing that the last term is nonnegative due to the single-index demi(sub)martingale property of the sequence $\{S_{\mathbf{n}^*,i},\, 1\leq i\leq n_k\}$. Inequality \eqref{NW2} follows from \eqref{NW1} by choosing $g(u) = I(u\geq \epsilon)$. \end{proof}
\noindent As a direct consequence of Theorem \ref{Doob_new} we get the following inequalities. The single-index analogues can be found in \cite{PR2012} (see relations (2.7.1) and (2.7.2)).
\begin{corollary}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Then for any $\epsilon >0$,
\begin{equation}
\label{Doobv2}\epsilon P\left(\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{j}}\geq \epsilon\right) \leq E\left(S_\mathbf{n}I\left(\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{j}}\geq \epsilon\right)\right)
\end{equation}
and
\[
\epsilon P\left(\min_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}\geq \epsilon\right) \leq E\left(S_\mathbf{n}I\left(\min_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}\geq \epsilon\right)\right)
\] \end{corollary}
\begin{remark}
As it has already been mentioned, for the case of multiindexed martingales the classical Doob's inequality cannot be established in general. However, inequality \eqref{Doobv2} shows that in the case of multiindexed demimartingales this celebrated result holds true. It is also important to highlight that, compared to Theorem \ref{theorem1}, the upper bound in \eqref{Doobv2} does not depend on the dimension of the index. Moreover, note that for $k=1$, inequality \eqref{Doobv2} is reduced to the result of Theorem \ref{Doob} for the case where $g(x) = x$.
\end{remark}
\noindent The Doob-type inequality obtained in \eqref{Doobv2} becomes the source result for various moment inequalities which generalize results that are already known for the case of single-indexed demimartingales.
\begin{corollary}
\label{mom1} Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multidimensionally indexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Then,
\[
E\left(\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{j}}\right)^p \leq \begin{cases}
\left(\frac{p}{p-1}\right)^pE(S_{\mathbf{n}})^p, &\quad\mbox{for}\quad p>1\\
\frac{e}{e-1}+\frac{e}{e-1}ES_{\mathbf{n}}\log^+S_{\mathbf{n}}, &\quad\mbox{for}\quad p=1.
\end{cases}
\] \end{corollary}
\begin{remark}
Observe that the inequalities of Corollary \ref{mom1} provide sharper bounds compared to the ones obtained in Theorem \ref{theorem1} while for the case $k=1$, Corollary \ref{mom1} is reduced to Theorem \ref{Wang}.
\end{remark}
\noindent For single-indexed positive martingales, Harremo\"{e}s in \cite{H2008} provided a maximal moment inequality which can be consider as a strengthening of a classical maximal inequality by Doob. Prakasa Rao in \cite{PR2007} proved that Harremo\"{e}s result is also valid for a sequence of single-indexed positive demimartingales (see Theorem 2.7.3 in \cite{PR2012}). Motivated by these results, we provide a generalization to the case of multiindexed demimartingales. The key result for obtaining the particular moment inequality is again expression \eqref{Doobv2}.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a positive multidimensionally indexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and $S_{\boldsymbol{\ell}}=c\in (0,1]$ when $\sum_{i=1}^{k} \ell_{i}=1$. Let $\gamma(x) = x-\ln x -c$ for $x>0$. Then,
\[
\gamma\left( E\left( \max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}} \right) \right) \leq 1-c^2-\ln c + ES_{\mathbf{n}}\ln S_{\mathbf{n}}.
\]
For the special case where $c=1$,
\[
\gamma\left( E\left( \max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}} \right) \right) \leq ES_{\mathbf{n}}\ln S_{\mathbf{n}}.
\] \end{theorem} \begin{proof}
We use the notation $S_{\mathbf{n}}^{\tiny\mbox{max}} = \displaystyle\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}$. Then,
\begin{align*}
& ES_{\mathbf{n}}^{\tiny\mbox{max}} -c = \int_{0}^{\infty}P\left( S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x\right) {\rm d}x -c = \int_{0}^{c}P\left( S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x\right) {\rm d}x+\int_{c}^{\infty}P\left( S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x\right) {\rm d}x -c\\
& \leq \int_{c}^{\infty}P\left( S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x\right) {\rm d}x \leq \int_{c}^{\infty} \left( \frac{1}{x} \int_{\{S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x\}} S_{\mathbf{n}}{\rm d} P\right) {\rm d}x \qquad(\mbox{due to}\, \, \eqref{Doobv2})\\
&= E\left( S_{\mathbf{n}} \int_{c}^{S_{\mathbf{n}}^{\tiny\mbox{max}}} \frac{1}{x}{\rm d}x\right) = E\left( S_{\mathbf{n}} \ln S_{\mathbf{n}}^{\tiny\mbox{max}}\right)-\ln c ES_{\mathbf{n}}.
\end{align*}
Observe that due to the demimartingale property $ES_{\mathbf{n}} = ES_{\boldsymbol{\ell}} = c $ for $\sum_{i=1}^{k}\ell_i =1$. Hence,
\[
ES_{\mathbf{n}}^{\tiny\mbox{max}} -c\leq E\left( S_{\mathbf{n}} \ln S_{\mathbf{n}}^{\tiny\mbox{max}}\right)-c\ln c.
\]
It is known that $\forall \, x>0, \, \ln x \leq x-1 \, $. Since $c\in(0,1]$ we have that $\forall \, x>0, \,\ln x \leq x-c $ and hence $\gamma(x) \geq 0$ for $x>0$ and $c\in (0,1]$. Then,
\begin{align*}
& ES_{\mathbf{n}}^{\tiny\mbox{max}} -c\leq E\left[ S_{\mathbf{n}} \left( \ln S_{\mathbf{n}}^{\tiny\mbox{max}} +\gamma\left( \frac{S_{\mathbf{n}}^{\tiny\mbox{max}}}{S_{\mathbf{n}} ES_{\mathbf{n}}^{\tiny\mbox{max}}} \right) \right) \right] - c\ln c
\leq 1 + ES_{\mathbf{n}}\ln S_{\mathbf{n}}+ c(\ln ES_{\mathbf{n}}^{\tiny\mbox{max}} -c -\ln c).
\end{align*}
Now,
\begin{align*}
&\gamma\left(ES_{\mathbf{n}}^{\tiny\mbox{max}}\right) = ES_{\mathbf{n}}^{\tiny\mbox{max}} - c- \ln ES_{\mathbf{n}}^{\tiny\mbox{max}}
\leq 1 + ES_{\mathbf{n}}\ln S_{\mathbf{n}}+ c(\ln ES_{\mathbf{n}}^{\tiny\mbox{max}} -c -\ln c)- \ln ES_{\mathbf{n}}^{\tiny\mbox{max}}\\
&\leq 1-c^2-c\ln c +(c-1)\ln E\left(\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}\right)+ES_{\mathbf{n}}\ln S_{\mathbf{n}}.
\end{align*}
The desired result follows by noticing that $\displaystyle\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}\geq c$ and since $c\in(0,1]$, $$(1-c)\ln E\left(\max_{\mathbf{i}\leq \mathbf{n}}S_{\mathbf{i}}\right) \geq (1-c)\ln c.$$ \end{proof}
\noindent Next, we obtain a moment inequality for positive multiindexed demimartingales which is motivated by Theorem 3.1 in \cite{WHYS2011}.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a positive multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and $S_{\boldsymbol{\ell}}=c>0$ when $\sum_{i=1}^{k} \ell_{i}=1$. Moreover, assume that $ES_{\mathbf{n}}\ln S_{\mathbf{n}} <\infty, \, \forall \mathbf{n} \in \mathbb{N}^{k}$ and $\displaystyle\lim_{\mathbf{n}\to\infty} ES_{\mathbf{n}}\ln S_{\mathbf{n}} = \infty$. Then,
\[
\limsup_{\mathbf{n}\to\infty} \dfrac{E(\max_{\mathbf{j}\leq \mathbf{n}}S_{\mathbf{j}})}{ES_{\mathbf{n}}\ln S_{\mathbf{n}}}\leq 1.
\] \end{theorem} \begin{proof}
It was proven earlier that
\[
ES_{\mathbf{n}}^{\tiny\mbox{max}} -c\leq E\left( S_{\mathbf{n}} \ln S_{\mathbf{n}}^{\tiny\mbox{max}}\right)-c\ln c.
\]
According to \cite{WHYS2011} (see relation (3.2) there) for any $a,b>0$ and $x_0>e$
\[
b\ln a\leq b\ln b+ax_0^{-1}+b(\ln x_0 -1).
\]
By employing this inequality we have that
\begin{align*}
&ES_{\mathbf{n}}^{\tiny\mbox{max}} -c\leq E\left( S_{\mathbf{n}} \ln S_{\mathbf{n}}\right)-c\ln c +x_0^{-1}ES_{\mathbf{n}}^{\tiny\mbox{max}} + ES_{\mathbf{n}}(\ln x_0-1).
\end{align*}
Recall that $ES_{\mathbf{n}} = c$ and therefore, after some algebraic calculations, we get that
\[
\dfrac{ES_{\mathbf{n}}^{\tiny\mbox{max}}}{ES_{\mathbf{n}}\ln S_{\mathbf{n}}} \leq \dfrac{x_0}{x_0-1}\left( 1 + \dfrac{c(\ln x_0 - \ln c)}{ES_{\mathbf{n}}\ln S_{\mathbf{n}}} \right).
\]
The result is obtained by taking $\limsup$ on both sides as $\mathbf{n} \to \infty$ and then let $x_0$ to tend to infinity. \end{proof}
\section{Chow-type maximal inequality} Hadjikyriakou in \cite{MH2010} proved the following Chow-type inequality for multiindexed demimartingales (see also \cite{CH2011}). The result was used to obtain an asymptotic result and further maximal inequalities. \begin{theorem} Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^k\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and $\left\{c_{\mathbf{n}}, \mathbf{n} \in\mathbb{N}^k\right\}$ be a nonincreasing array of positive numbers. Let $g$ be a nonnegative convex function on $\mathbb{R}$ with $g(0)=0$. Then, $\forall \varepsilon>0$, $$ \varepsilon P\left(\max_{\mathbf{j} \leq \mathbf{n}}c_{\mathbf{j}} g\left(S_{\mathbf{j}}\right) \geq \varepsilon\right) \leq \min _{1 \leq s \leq k}\left\{\sum_{\mathbf{j} \leq \mathbf{n}} c_{\mathbf{j}} E\left[g\left(S_{\mathbf{j} ; s ; i}\right)-g\left(S_{\mathbf{j} ; s ; i-1}\right)\right]\right\}. $$ \end{theorem}
Next, we provide a Chow-type inequality for multiindexed demimartingales by applying the methodology introduced by \cite{W2004}. The new approach leads to an upper bound which depends only on a single summation. \begin{theorem} \label{Chownew}Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $g$ be a nonnegative convex function on $\mathbb{R}$ with $g(0) = 0$ and $\{c_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\}$ be a nonincreasing array of positive numbers. Then, for all $\epsilon >0$
\[
\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon)\leq \min_{1\leq s\leq k}\sum_{i=1}^{n_s}c_{\mathbf{n};s;i}E\left[ \left( g(S_{\mathbf{n};s;i}) - g(S_{\mathbf{n};s;i-1}) \right) I\left(\max_{\mathbf{k}\leq \mathbf{n}} c_\mathbf{k}g(S_\mathbf{k})\geq \epsilon\right) \right]
\] \end{theorem} \begin{proof}
The proof is motivated by the the proof of Theorem 2.1 in \cite{W2004}. First, we define the functions
\[
u(x) = g(x) I\{x\geq 0\}\quad\mbox{and}\quad v(x) = g(x)I\{x<0\}
\]
which are both nonnegative convex functions with $u(x)$ being nondecreasing while $v(x)$ is a nonincreasing function. Observe that $g(x) = u(x)+v(x) = \max\{u(x), v(x)\}$. Therefore,
\begin{equation}
\label{chowang1}\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon)\leq \epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}u(S_\mathbf{j})\geq \epsilon)+\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}v(S_\mathbf{j})\geq \epsilon).
\end{equation}
Following the steps and the notation introduced by \cite{W2004}, we consider $m(\cdot)$ to be a nonnegative nondecreasing function with $m(0) = 0$ and let
\[
S'_{\mathbf{n}} = \max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}u(S_\mathbf{j}).
\]
Without loss of generality we fix $n_1, \ldots, n_{k-1}$ and denote $(\mathbf{n}^*,i) = (n_1 \, \ldots \, n_{k-1} \, i)$. Then,
\begin{align*}
&E\left[ \int_{0}^{S'_{\mathbf{n}}} t{\rm d}m(t) \right] = \sum_{i=1}^{n_k}E\left[ \int_{S'_{\mathbf{n^*},i-1}}^{S'_{\mathbf{n}^*,i}} t{\rm d}m(t) \right]\leq \sum_{i=1}^{n_k} E\left[ S'_{\mathbf{n^*},i} \left( m\left( S'_{\mathbf{n^*},i} \right) - m\left( S'_{\mathbf{n^*},i-1} \right) \right)\right]\\
&\leq \sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ u(S_{\mathbf{n^*},i}) \left( m\left( S'_{\mathbf{n^*},i} \right) - m\left( S'_{\mathbf{n^*},i-1} \right) \right)\right]\leq \sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ m(S'_{\mathbf{n}}) \left( u\left( S_{\mathbf{n^*},i} \right) - u\left( S_{\mathbf{n^*},i-1} \right) \right)\right]-A
\end{align*}
where
\[
A = \sum_{i=1}^{n_k-1}E \left[ (c_{\mathbf{n^*},i+1} u(S_{\mathbf{n^*},i+1}) - c_{\mathbf{n^*},i} u(S_{\mathbf{n^*},i}) ) m(S'_{\mathbf{n^*},i}) \right]+\sum_{i=1}^{n_k-1}( c_{\mathbf{n^*},i}-c_{\mathbf{n^*},i+1}) E[u(S_{\mathbf{n^*},i})m(S'_{\mathbf{n}}) ].
\]
The second inequality follows by observing that
$ S'_{\mathbf{n^*},i}\geq S'_{\mathbf{n^*},i-1}$ and thus $S'_{\mathbf{n^*},i} = c_{\mathbf{n^*},i}u(S_{\mathbf{n^*},i})$ or $m( S'_{\mathbf{n^*},i}) = m( S'_{\mathbf{n^*},i-1} ) $. We need to prove that $A$ is a nonnegative term. Note that due to the fact that $( c_{\mathbf{n^*},i}-c_{\mathbf{n^*},i+1})u(S_{\mathbf{n^*},i})\geq 0$ for any $i$, and by the convexity of the function $u(\cdot)$ we have that
\begin{align*}
&A \geq \sum_{i=1}^{n_k-1}E \left[ (c_{\mathbf{n^*},i+1} u(S_{\mathbf{n^*},i+1}) - c_{\mathbf{n^*},i} u(S_{\mathbf{n^*},i}) ) m(S'_{\mathbf{n^*},i}) \right]+\sum_{i=1}^{n_k-1}( c_{\mathbf{n^*},i}-c_{\mathbf{n^*},i+1}) E[u(S_{\mathbf{n^*},i})m(S'_{\mathbf{n^*},i}) ]\\
&= \sum_{i=1}^{n_k-1} c_{\mathbf{n^*},i+1}E\left[ ( u(S_{\mathbf{n^*},i+1}) - u(S_{\mathbf{n^*},i}) ) m(S'_{\mathbf{n^*},i}) \right]\geq \sum_{i=1}^{n_k-1} c_{\mathbf{n^*},i+1}E\left[ ( S_{\mathbf{n^*},i+1} - S_{\mathbf{n^*},i}) h(S_{\mathbf{n^*},i}) m(S'_{\mathbf{n^*},i}) \right]\geq 0
\end{align*}
since $ h(S_{\mathbf{n^*},i}) m(S'_{\mathbf{n^*},i})$ is a nondecreasing function of $\{S_{\mathbf{n^*},i}, 1\leq i\leq n_k\}$ which forms a single indexed demimartingale. Consider the case where $m(t) = I\{ t\geq \epsilon \}$. Then, since $S'_{\mathbf{n}} \leq \displaystyle \max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})$
\begin{align}
&\nonumber\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}u(S_\mathbf{j})\geq \epsilon) \leq \sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ \left( u\left( S_{\mathbf{n^*},i} \right) - u\left( S_{\mathbf{n^*},i-1} \right) \right)I\{S'_{\mathbf{n}}\geq \epsilon\} \right]\\
&\nonumber= \sum_{i=1}^{n_k-1}( c_{\mathbf{n^*},i} -c_{\mathbf{n^*},i+1}) E[u(S_{\mathbf{n^*},i})I\{S'_{\mathbf{n}}\geq \epsilon\}] +c_{\mathbf{n}}E[u\left( S_{\mathbf{n} } \right)I\{S'_{\mathbf{n}}\geq \epsilon\}]\\
&\nonumber\leq \sum_{i=1}^{n_k-1}( c_{\mathbf{n^*},i} -c_{\mathbf{n^*},i+1}) E[u(S_{\mathbf{n^*},i})I\{\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon\}] +c_{\mathbf{n}}E[u\left( S_{\mathbf{n} } \right)I\{\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon\}]\\
&\label{chowwangu}=\sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ \left( u\left( S_{\mathbf{n^*},i} \right) - u\left( S_{\mathbf{n^*},i-1} \right) \right)I\{\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon\} \right].
\end{align}
Similarly, it can be proven that
\begin{equation}
\label{chowwangv}\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}v(S_\mathbf{j})\geq \epsilon) \leq \sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ \left( v\left( S_{\mathbf{n^*},i} \right) - v\left( S_{\mathbf{n^*},i-1} \right) \right)I\{\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon\} \right].
\end{equation}
By combining \eqref{chowang1}-\eqref{chowwangv}, we have that
\[
\epsilon P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon)\leq \sum_{i=1}^{n_k} c_{\mathbf{n^*},i}E\left[ \left( g\left( S_{\mathbf{n^*},i} \right) - g\left( S_{\mathbf{n^*},i-1} \right) \right)I\{\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}g(S_\mathbf{j})\geq \epsilon\} \right]
\]
which leads to the desired result. \end{proof}
\noindent As a direct consequence of the Chow-type inequality proven above, we can easily obtain the following convergence result. \begin{theorem} Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $g$ be a nonnegative convex function on $\mathbb{R}$ with $g(0) = 0$ and $\{c_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\}$ be a nonincreasing array of positive numbers. Assume that, for some $1\leq s\leq k$ and $p\geq 1$, $$\sum_{i=1}^{\infty}c^p_{\mathbf{n};s;i}E([g(S_{\mathbf{n};s;i})]^p-[g(S_{\mathbf{n};s;i-1})]^p)<\infty\quad\mbox{and}\quad c^p_{\mathbf{n}}E([g(S_{\mathbf{n}})]^p-[g(S_{\mathbf{n};s;n_s-1})]^p)\to 0,\, n\to \infty.$$ Then,
\[
c_{\mathbf{n}}g(S_{\mathbf{n}})\to0\quad\mbox{a.s.,}\quad\mathbf{n}\to \infty.
\] \end{theorem} \begin{proof}
Without loss of generality, we assume that the conditions are satisfied for $s=k$ and let $\mathbf{N} = (N \, N \, \ldots \, N)$. Then, we can write \begin{align*} &P(\max_{\mathbf{n}\geq \mathbf{N}} c_\mathbf{n}g(S_\mathbf{n})\geq \epsilon) = P(\max_{\mathbf{n}\geq \mathbf{N}} c^p_\mathbf{n}[g(S_\mathbf{n})]^p\geq \epsilon^p) \\ & \leq \frac{1}{\epsilon^p}\sum_{i\geq N}c^p_{\mathbf{N^*},i}E\left[ \left( [g\left( S_{\mathbf{N^*},i} \right) ]^p - [g\left( S_{\mathbf{N^*},i-1} \right)]^p \right)I\{\max_{\mathbf{n}\geq \mathbf{N}} c^p_\mathbf{n}[g(S_\mathbf{n})]^p\geq \epsilon\} \right]\\ & \leq \frac{1}{\epsilon^p}\sum_{i\geq N}c^p_{\mathbf{N^*},i}E\left[ \left( [g\left( S_{\mathbf{N^*},i} \right) ]^p - [g\left( S_{\mathbf{N^*},i-1} \right)]^p \right) \right]\\ &= \frac{1}{\epsilon^p}c^p_{\mathbf{N}}E(([g(S_{\mathbf{N}})]^p - [g(S_{\mathbf{N^*},N-1})]^p) ) + \frac{1}{\epsilon^p}\sum_{i= N+1}^{\infty}c^p_{\mathbf{N^*},i}E\left[ \left( [g\left( S_{\mathbf{N^*},i} \right)] ^p - [g\left( S_{\mathbf{N^*},i-1} \right) ]^p \right) \right]\to 0 \end{align*} as $N\to\infty$. \end{proof}
\noindent The Chow-type inequality provided above can lead to a H\'{a}jek-R\'{e}nyi inequality for multiindexed associated random variables.
\begin{corollary}
Let $\left\{X_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^k\right\}$ be mean zero multiindexed associated random variables, $\left\{c_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^d\right\}$ a nonincreasing array of positive numbers and $S_{\mathbf{n}}=\sum_{\mathbf{i} \leq \mathbf{n}} X_{\mathbf{i}}$. Then $\forall \varepsilon>0$
\[
P(\max_{\mathbf{i} \leq \mathbf{n}}c_{\mathbf{i}}|S_\mathbf{i}|\geq \epsilon)\leq \min_{1 \leq s \leq k} \epsilon^{-2}\sum_{j=1}^{n_s}c_{\mathbf{n};s;j}[2\Cov(X_{\mathbf{n};s;j},S_{\mathbf{n};s;j-1})+E(X^2_{\mathbf{n};s;j})]
\] \end{corollary} \begin{proof}
\begin{align*}
& P(\max_{\mathbf{j}\leq \mathbf{n}} c_\mathbf{j}|S_\mathbf{j}|\geq \epsilon) = P(\max_{\mathbf{j}\leq \mathbf{n}} c^2_\mathbf{j}|S_\mathbf{j}|^2)\geq \epsilon^2)
\leq \min_{1\leq s\leq k}\sum_{i=1}^{n_s}c^2_{\mathbf{n};s;i}E\left[ \left( |S_{\mathbf{n};s;i}|^2 - |S_{\mathbf{n};s;i-1}|^2 \right) I\left(\max_{\mathbf{k}\leq \mathbf{n}} c^2_\mathbf{k}|S_\mathbf{k}|^2\geq \epsilon^2\right) \right]\\
&\leq \min_{1\leq s\leq k}\sum_{i=1}^{n_s}c^2_{\mathbf{n};s;i}E\left[ \left( |S_{\mathbf{n};s;i}|^2 - |S_{\mathbf{n};s;i-1}|^2 \right) \right]\leq \min_{1\leq s\leq k}\sum_{i=1}^{n_s}c^2_{\mathbf{n};s;i}E\left[ \left( S_{\mathbf{n};s;i} - S_{\mathbf{n};s;i-1} \right) \left( S_{\mathbf{n};s;i} + S_{\mathbf{n};s;i-1} \right) \right]\\
&= \min_{1\leq s\leq k}\sum_{i=1}^{n_s}c^2_{\mathbf{n};s;i}E\left[ X_{\mathbf{n};s;i} (2S_{\mathbf{n};s;i-1} +X_{\mathbf{n};s;i}) \right]
\end{align*}
which gives the desired result. \end{proof}
\begin{remark}
Note that the Chow-type inequality obtained in \cite{MH2010} (or in \cite{CH2011}), also lead to a convergence result and a H\'{a}jek-R\'{e}nyi inequality for multiindexed associated random variables. It is important to highlight however, that the results obtained here involve a sum over a single index while in \cite{MH2010} the conditions for the asymptotic result and the upper bound of the H\'{a}jek-R\'{e}nyi inequality depend on multiple summations. \end{remark}
\section{Maximal $\phi$-inequalities for nonnegative multiindexed demisubmartingales} Let $\mathcal{C}$ denote the class of Orlicz functions i.e. unbounded, nondecreasing convex functions $\phi:[0, \infty) \rightarrow[0, \infty)$ with $\phi(0)=0$. Given $\phi \in \mathcal{C}$ and $a \geq 0$, define $$ \Phi_a(x)=\int_a^x \int_a^s \frac{\phi^{\prime}(r)}{r} d r d s, \quad x>0 $$ and $$ p_\phi^*=\sup _{x>0} \frac{x \phi^{\prime}(x)}{\phi(x)}. $$ Note that the function $\phi$ is called moderate if $p_\phi^*<\infty$. More information on Orlicz functions can be found in \cite{AR2006}.
\noindent The single index analogues of the results that follow can be found in \cite{PR2007} (or see Section 2.8 in \cite{PR2012}). \begin{theorem}
\label{phiTheorem1}Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and let $\phi \in \mathcal{C}$. Then, for $x>0$ and $\lambda\in (0,1)$,
\begin{equation}
\label{OrlProb} P\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\geq x \right) \leq \dfrac{\lambda}{(1-\lambda)x}\int_{x}^{\infty}P(S_{\mathbf{n}}>\lambda y){\rm d}y = \dfrac{\lambda}{(1-\lambda)x} E\left( \dfrac{S_{\mathbf{n}}}{\lambda}-x\right)^{+}.
\end{equation}
Moreover,
\begin{equation}
\label{OrlExp}
E\left(\phi \left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\ \right)\right)\leq \phi(b)+ \dfrac{\lambda}{1-\lambda}\int_{[S_\mathbf{n}>\lambda b]}\left( \Phi_a \left(\frac{S_\mathbf{n}}{\lambda}\right)-\Phi_a(b)-\Phi'(b)\left(\frac{S_\mathbf{n}}{\lambda}-b\right) \right){\rm d}P
\end{equation}
for all $\mathbf{n}\in \mathbb{N}^k$, $a,b>0$. If $\phi'(x)/x$ is integrable at 0, then the latter inequality holds for $b=0$. \end{theorem} \begin{proof}
First observe that by \eqref{Doobv2} we have that, for any $x>0$,
\[
P\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\geq x \right) \leq \frac{1}{x}E\left(S_{\mathbf{n}}I\left( \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\geq x \right)\right) = \frac{1}{x}\int_{0}^{\infty}P(S_{\mathbf{n}} \geq y,S_{\mathbf{n}}^{\tiny\mbox{max}} \geq x ){\rm d}y.
\]
The rest of the proof runs along the same lines as in the case of a single index (see Theorem 2.8.1 in \cite{PR2012}). \end{proof}
\noindent The moment inequality presented in \eqref{OrlExp} becomes the source result for a number of moment inequalities. The results are presented here for the sake of completeness however their proofs are omitted since they are based on properties of the $\phi$ functions and are the same as in the single index case. We refer the interested reader to \cite{PR2007} or Section 2.8 in \cite{PR2012} for the details.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and let $\phi \in \mathcal{C}$. Then,
$$
E\left[\phi\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)\right] \leq \phi(a)+\frac{\lambda}{1-\lambda} E\left[\Phi_{a}\left(\frac{S_{\mathbf{n}}}{\lambda}\right)\right]
$$
for all $a \geq 0,0<\lambda<1$ and $\mathbf{n} \in \mathbb{N}^{k}$. \end{theorem}
\noindent A special case of \eqref{OrlExp} is obtained for $\phi(x) = x$. The proof is obtained by applying the same methodology as in \cite{PR2012} page 64.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Then, for any $\mathbf{n} \in \mathbb{N}^{k}$,
\[
E\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right) \leq b+\frac{b}{b-1}\left(E\left(S_{\mathbf{n}} \log ^{+} S_{n}\right)-E\left(S_{\mathbf{n}}-1\right)^{+}\right), \quad b>1.
\] \end{theorem}
\begin{remark}
Observe that in the case where $b=e$ the inequality above provides a sharper bound compared to the second inequality of Corollary \ref{mom1}. \end{remark}
\noindent By utilizing inequality \eqref{Doobv2} and Lemma 2.8.3 in \cite{PR2012} we can easily obtain the following result.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$ and let $\phi \in \mathcal{C}$ with $p_{\phi}=\displaystyle\inf _{x>0} \frac{x \phi^{\prime}(x)}{\phi(x)}>1$. Then for all $\mathbf{n} \in \mathbb{N}^{k}$
$$
E\left[\phi\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)\right] \leq E\left[\phi\left(q_{\phi} S_{\mathbf{n}}\right)\right]
$$
where $q_{\phi}=\frac{p_{\phi}}{p_{\phi}-1}$. \end{theorem}
\noindent The next result is a direct consequence of the previous theorem and the properties of the $\phi$ function.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Suppose that the function $\phi \in \mathcal{C}$ is moderate. Then, for any $\mathbf{n} \in \mathbb{N}^{k}$,
$$
E\left[\phi\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)\right] \leq E\left[\phi\left(q_{\phi} S_{\mathbf{n}}\right)\right] \leq q_{\Phi}^{p_{\phi}^{*}} E\left[\phi\left(S_{\mathbf{n}}\right)\right].
$$ \end{theorem}
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Suppose $\phi$ is a nonnegative nondecreasing function on $[0, \infty)$ such that $\phi^{1 / \gamma}$ is also nondecreasing and convex for some $\gamma>1$. Then
$$
E\left[\phi\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)\right] \leq\left(\frac{\gamma}{\gamma-1}\right)^{\gamma} E\left[\phi\left(S_{\mathbf{n}}\right)\right].
$$
Moreover, for any $r>0$,
$$E\left[e^{r \max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}}\right] \leq e E\left[e^{r S_{\mathbf{n}}}\right].$$ \end{theorem} \begin{proof}
For the proof of the first inequality we use the first part of Corollary \ref{mom1} for the sequence $\left\{\left[\phi\left(S_{\mathbf{n}}\right)\right]^{1 / \gamma}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ since this is also a nonnegative multiindexed demisubmartingale. The desired inequality follows by choosing $p = \gamma$. The second result is obtained by setting $\phi(x) = e^{rx}$ in the first inequality and by letting $\gamma$ to tend to $\infty$. \end{proof}
\noindent This section is concluded with a maximal inequality for the case where both, the function $\phi$ and its $m$-th derivative for some $m\geq 1$, are Orlicz functions. The proof runs along the same lines as in the case of a single index since the source result is Corollary \ref{mom1}.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a nonnegative multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $\phi \in \mathcal{C}$ which is differentiable $m$ times with the $m$-th derivative $\phi^{(m)} \in \mathcal{C}$ for some $m \geq 1$. Then
$$
E\left[\phi\left(\max_{\mathbf{i} \leq \mathbf{n}} S_{\mathbf{i}}\right)\right] \leq\left(\frac{m+1}{m}\right)^{m+1} E\left[\phi\left(S_{\mathbf{n}}\right)\right].
$$ \end{theorem}
\section{Upcrossing Inequality} We follow the notation of Section 2.4 in \cite{PR2012}. Given a finite set of muliindexed random variables from the set $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ and $a<b$ we define the sequence of stopping times in the $s$-th direction as follows $$ J^{(s)}_{2 m-1}=\left\{\begin{array}{l} n_s+1 \quad \text { if }\left\{j: J^{(s)}_{2 m-2}<j \leq n_s \text { and } S_{\mathbf{n};s;j} \leq a\right\} \text { is empty } \\ \min \left\{j: J^{(s)}_{2 m-2}<j \leq n_s \text { and } S_{\mathbf{n};s;j} \leq a\right\}, \quad \text { otherwise } \end{array}\right. $$ and $$ J^{(s)}_{2 m}=\left\{\begin{array}{l} n_s+1 \text { if }\left\{j: J^{(s)}_{2 m-1}<j \leq n \text { and } S_{\mathbf{n};s;j} \geq b\right\} \text { is empty } \\ \min \left\{j: J^{(s)}_{2 m-1}<j \leq n \text { and } S_{\mathbf{n};s;j} \geq b\right\}, \quad \text { otherwise } \end{array}\right. $$ for $m=1,2,\ldots$ where $\mathbf{n};s;j = (n_1\, n_2\, \ldots\, n_{s-1} \, j \, n_{s+1}\, \ldots \, n_k)$ and $J^{(s)}_{0} = 0$ for $1\leq s\leq k$.
\noindent The number of complete upcrossings of the interval $[a, b]$ by the finite sequence of random variables in the $s$-th direction, until the time $n_s,$ is denoted by $U_{n_s}(a,b)$ for $1\leq s\leq k$ where
$$
U_{n_s}(a,b)=\max \left\{m: J^{(s)}_{2 m}<n_s+1\right\} .
$$
Then, as $n_s\to \infty$,
\[
U_{n_s}(a,b) \to U^{(s)}(a,b) = \max \left\{m: J^{(s)}_{2 m}<\infty\right\}
\]
which denotes the total number of upcrossings in the $s$-th direction. We say that the sequence of random fields has a complete upcrossing if there is a complete upcrossing in at least one direction
and the total number of complete upcrossings is defined as
\begin{equation} \label{updef}U(a,b) = \min \{U_{a,b}^{(s)}>0, 1\leq s\leq k\}\quad\mbox{and}\quad U(a,b) = 0\mbox{ if } U_{a,b}^{(s)}=0\mbox{ for all } 1\leq s\leq k. \end{equation}
\begin{remark}
It is important to highlight that in the case of random fields where there is no partial order among indices, the concept of upcrossings is not uniquely defined. In the definition considered above, first, we identify the number of upcrossings in each direction and the total number of upcrossings for the process is defined through \eqref{updef}. Consider the example given below for the case $k=2$: suppose that $a<b$ and
\[
x_{11}<a, x_{21} <a, x_{12}>b, x_{22}<b.
\]
First, we consider the case where $s=1$, i.e. consider the upcrossings in the 1-st direction by keeping the second index constant and allowing the first one to change. Observe that,
\[
x_{11}<a\quad\mbox{and}\quad x_{21} <a
\]
and
\[
x_{12}>b\quad\mbox{and}\quad x_{22}<b
\]
i.e. there are no complete upcrossings in the 1-st direction. Next, we study the number of upcrossings in the direction of the second index where the first index remains constant:
\[
x_{11}<a\quad\mbox{and}\quad x_{12}>b
\]
and
\[
x_{21} <a\quad\mbox{and}\quad x_{22}<b.
\]
Observe that the first pair gives a complete upcrossing, so in the second direction, there is 1 complete upcrossing. Using the notation introduced above, directionwise we have that
\[
U^{(1)}(a,b) = 0\quad\mbox{and}\quad U^{(2)}(a,b) = 1.
\]
\noindent According to \eqref{updef}, the total number of complete upcrossings for the given sequence is $U(a,b) = 1$. \end{remark}
The result that follows provides an upper bound for the expected number of complete upcrossings of the interval $[a,b]$ in the $s$-th direction.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demisubmartingale with $S_{\boldsymbol{\ell}}=0$ and consider a finite sequence of random variables from this set. Then for $a<b$ and $1\leq s\leq k$,
$$
E\left(U_{n_s}(a,b)\right) \leq \frac{E\left(\left(S_{\mathbf{n}}-a\right)^{+}\right)-E\left(\left(S_{\mathbf{n};s;1}-a\right)^{+}\right)}{b-a}
$$ \end{theorem} \begin{proof}
Without loss of generality we assume that $s = k$ and we use the notation $\mathbf{n} = (\mathbf{n}^* \, n_k)$ where $\mathbf{n}^* = (n_1\, \ldots \, n_{k-1})$. The proof is motivated by the proof of Theorem 2.4.1 in Prakasa Rao (2012). For $1 \leq j \leq n_k-1$, define
$$
\epsilon_{\mathbf{n}^*,j}= \begin{cases}1 & \text { if for some } m=1,2, \ldots, J^{(s)}_{2 m-2} \leq j<J^{(s)}_{2 m-1} \\ 0 & \text { if for some } m=1,2, \ldots, J^{(s)}_{2 m-1} \leq j<J^{(s)}_{2 m}\end{cases}
$$
so that $1-\epsilon_{\mathbf{n}^*,j}$ is the indicator function of the event that the time interval $[j, j+1)$ is a part of an upcrossing possibly incomplete; equivalently
$$\epsilon_{\mathbf{n}^*,j}= \begin{cases}1 & \text { if either } S_{\mathbf{n}^*,i}>a \text { for } i=1, \ldots, j \text { or } \\ & \text { for some } i=1, \ldots, j, S_{\mathbf{n}^*,i} \geq b \text { and } S_{\mathbf{n}^*k}>a \text { for } k=i+1, \ldots, j \\ =0 & \text { otherwise. }\end{cases}$$
\noindent Let $\Lambda$ be the event that the sequence ends with an incomplete upcrossing, that is, $\tilde{J}^{(s)} \equiv J_{2 U^{(s)}_{a, b}+1}<n_k$. Note that
$$
\left(S_{\mathbf{n}}-a\right)^{+}-\left(S_{\mathbf{n}^*1}-a\right)^{+}=\sum_{j=1}^{n_k-1}\left[\left(S_{\mathbf{n}^*,j+1}-a\right)^{+}-\left(S_{\mathbf{n}^*,j}-a\right)^{+}\right]=M_{u}+M_{d}
$$
where
$$
M_{d}=\sum_{j=1}^{n_k-1} \epsilon_{\mathbf{n}^*,j}\left[\left(S_{\mathbf{n}^*,j+1}-a\right)^{+}-\left(S_{\mathbf{n}^*,j}-a\right)^{+}\right] \geq \sum_{j=1}^{n_k-1} \epsilon_{\mathbf{n}^*,j}\left(S_{\mathbf{n}^*,j+1}-S_{\mathbf{n}^*,j}\right)
$$
where the inequality follows from the observation that
$$
\left(S_{\mathbf{n}^*,j+1}-a\right)^{+} \geq S_{\mathbf{n}^*,j+1}-a
$$
and because of the definition of $\epsilon_{\mathbf{n}^*,j}$ we also have that
$$
\epsilon_{\mathbf{n}^*,j}\left(S_{\mathbf{n}^*,j}-a\right)^{+}=\epsilon_{\mathbf{n}^*,j}\left(S_{\mathbf{n}^*,j}-a\right).
$$
Observe that
$$
\begin{aligned}
M_{u} &=\sum_{j=1}^{n_k-1}\left(1-\epsilon_{\mathbf{n}^*,j}\right)\left[\left(S_{\mathbf{n}^*,j+1}-a\right)^{+}-\left(S_{\mathbf{n}^*,j}-a\right)^{+}\right] \\
&=\sum_{k=1}^{U_{n_s}(a,b)} \sum_{j=J^{(s)}_{2 k-1}}^{J^{(s)}_{2 k}-1}\left[\left(S_{\mathbf{n}^*,j+1}-a\right)^{+}-\left(S_{\mathbf{n}^*,j}-a\right)^{+}\right]+\sum_{j=\tilde{J}^{(s)}}^{n-1}\left[\left(S_{\mathbf{n}^*,j+1}-a\right)^{+}-\left(S_{\mathbf{n}^*,j}-a\right)^{+}\right] \\
&=\sum_{k=1}^{U_{n_s}(a,b)}\left[\left(S_{\mathbf{n}^*,J^{(s)}_{2 k}}-a\right)^{+}-\left(S_{\mathbf{n}^*,J^{(s)}_{2 k-1}}-a\right)^{+}\right]+\left[\left(S_{\mathbf{n}}-a\right)^{+}-\left(S_{\mathbf{n}^*,\tilde{J}^{(s)}}-a\right)^{+}\right] I_{\Lambda} \\
&=\sum_{k=1}^{U_{n_s}(a,b)}\left(S_{\mathbf{n}^*,J^{(s)}_{2 k}}-a\right)^{+}+\left(S_{\mathbf{n}}-a\right)^{+} I_{\Lambda} \\
\geq &(b-a) U_{n_s}(a,b).
\end{aligned}
$$
By taking expectations, we have that
$$
E\left[\left(S_{\mathbf{n}}-a\right)^{+}-\left(S_{\mathbf{n}^*,1}-a\right)\right] \geq(b-a) EU_{n_s}(a,b)+\sum_{j=1}^{n_k-1} E\left[\epsilon_{\mathbf{n}^*,j}\left(S_{\mathbf{n}^*,j+1}-S_{\mathbf{n}^*,j}\right)\right].
$$
The desired result follows by noting that $\epsilon_{\mathbf{n}^*,j}$ is a nonnegative nondecreasing function of $\{S_{\mathbf{n}^*,i},\, i=1,\ldots,j\}$ which forms a multiindexed demisubmartingale and thus the latter term is always nonnegative. \end{proof}
\noindent The upcrossing inequality leads to the following convergence result.
\begin{theorem}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ such that
\begin{equation}
\label{ass1}\limsup _{\mathbf{n} \rightarrow \infty} E\left|S_{\mathbf{n}}\right|<\infty,
\end{equation}
then the $S_{\mathbf{n}}$ converge a.s. and $E|\displaystyle\lim _{\mathbf{n} \rightarrow \infty}S_{\mathbf{n}}|<\infty$. \end{theorem} \begin{proof} It is known that a sequence of real numbers converges to $\mathbb{R}\cup{\pm \infty}$ if and only if the number of upcrossings of $[a,b]$ is finite for all rational numbers $a<b$. By the upcrossing inequality proven above, we have that for every $1\leq s\leq k$
\[
EU_{n_s}(a,b)\leq \frac{1}{b-a}E\left(S_{\mathbf{n}}-a\right)^{+} \leq \frac{1}{b-a}(E|S_\mathbf{n}|+a)
\]
and therefore
\[
EU^{(s)}_{a, b}\leq \frac{1}{b-a}(\sup_{\mathbf{n}}E|S_\mathbf{n}|+a)
\]
and thus $U^{(s)}_{a, b}$ is finite almost surely for every $1\leq s\leq k$ due to condition \eqref{ass1}. By the definition of $U(a,b)$ we have that $U(a,b)\leq U^{(s)}_{a, b}$ for any $s$ and thus $U(a,b)$ is also finite almost surely. Define $\Omega_0 = \displaystyle\bigcap_{a<b\in \mathbb{Q}}\left(U(a, b)<\infty\right)$. Then $P(\Omega_0) = 1$. This observation establishes the convergence of $S_{\mathbf{n}}$ while the last argument for the expectation follows by applying Fatou's lemma and utilizing assumption \eqref{ass1}. \end{proof}
\section{Whittle-type inequality}
The next result is the multiindexed analogue of the Whittle type maximal inequality for demimartingales proven by Prakasa Rao in \cite{PR2002} (see also Theorem 2.6.1 in \cite{PR2012}).
\begin{theorem}
\label{WhittleNond}Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $\phi(\cdot)$ be a nonnegative nondecreasing convex function such that $\phi(0)= 0$. Let $\psi(u)$ be a positive nondecreasing function for $u>0$. Then,
\begin{equation}
\label{Whittle} P(\phi(S_{\mathbf{j}})\leq \psi(u_{\mathbf{j}}), \, \mathbf{j}\leq \mathbf{n}) \geq 1-\min_{1\leq s\leq k}\sum_{i =1}^{n_s} \frac{E\left( \phi\left(S_{\mathbf{n};s;i}\right)- \phi\left(S_{\mathbf{n};s;i-1}\right)\right)}{\psi \left( u_{\mathbf{n};s;i} \right)}
\end{equation}
for $0<u_{\mathbf{i}}\leq u_{\mathbf{j}},\, \mathbf{i}\leq \mathbf{j}$. \end{theorem} \begin{proof}
The proof will be given for $k=2$ for simplicity. Define
\[
A_{n_1n_2} = \{\phi(S_{ij})\leq \psi(u_{ij}), \, (i,j)\leq (n_1,n_2)\}.
\]
Then,
\begin{align*}
&P(A_{n_1n_2}) = E\left( \prod_{\ell =1}^{n_1} I(A_{\ell n_2})\right) = E\left( \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2})I(A_{n_1n_2})\right)\geq E\left( \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2})\left( 1-\frac{\phi(S_{n_1n_2})}{\psi(u_{n_1n_2})} \right) \right)
\end{align*}
where the inequality follows by observing that
\[
I(A_{n_1n_2}) \geq 1-\frac{\phi(S_{n_1n_2})}{\psi(u_{n_1n_2})}.
\]
Notice that
\begin{align*}
&E\left( \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2}) \left[ \left( 1- \frac{\phi(S_{n_1n_2})}{\psi(u_{n_1n_2})}\right) - \left( 1- \frac{\phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})}\right) \right] + \frac{\phi(S_{n_1n_2}) - \phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})} \right)\\
&= E\left( \left( 1- \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2}) \right) \frac{\phi(S_{n_1n_2}) - \phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})} \right)\geq 0
\end{align*}
since $\left( 1- \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2}) \right)$ is nonnegative componentwise nondecreasing function of $\{ \phi(S_{in_2}), \, 1\leq i\leq n_1\}$ which forms a single-index demimartingale. Hence,
\begin{align*}
&P(A_{n_1n_2}) \geq E\left( \prod_{\ell =1}^{n_1-1} I(A_{\ell n_2}) \left( 1- \frac{\phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})} \right) \right) - \frac{E\phi(S_{n_1n_2}) - E\phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})}\\
&\geq E\left( \prod_{\ell =1}^{n_1-2} I(A_{\ell n_2}) \left( 1- \frac{\phi(S_{n_1-1n_2})}{\psi(u_{n_1-1n_2})} \right) \right) - \frac{E\phi(S_{n_1n_2}) - E\phi(S_{n_1-1n_2})}{\psi(u_{n_1n_2})}
\end{align*}
where the last inequality follows due to the fact that $u_{\mathbf{i}}\leq u_{\mathbf{j}}$ for $\mathbf{i}\leq \mathbf{j}$ and $\psi(u_\mathbf{n})$ is a nondecreasing sequence of positive numbers. Continuing with the same manner we have that
\[
P(A_{n_1n_2}) \geq 1 - \sum_{\ell = 1}^{n_1} \frac{E\phi(S_{\ell n_2}) - E\phi(S_{\ell-1n_2})}{\psi(u_{\ell n_2})}.
\]
Similarly we can obtain
\[
P(A_{n_1n_2}) \geq 1 - \sum_{\ell = 1}^{n_2} \frac{E\phi(S_{n_1\ell}) - E\phi(S_{n_1\ell-1})}{\psi(u_{n_1\ell })}
\]
and these last two inequalities lead to the 2-index analogue of \eqref{Whittle}. \end{proof}
\noindent The next result provides a generalization of the Whittle type inequality where the assumption of nondecreasing property is dropped.
\begin{theorem}
\label{GenWhittle}Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $\phi(\cdot)$ be a nonnegative convex function such that $\phi(0)= 0$. Let $\psi(u)$ be a positive nondecreasing function for $u>0$ and
\[
A_{\mathbf{n}} = \{\phi(S_{\mathbf{j}})\leq \psi(u_{\mathbf{j}}), \, \mathbf{j}\leq \mathbf{n}\}
\]
where $0<u_{\mathbf{i}}\leq u_{\mathbf{j}}$ for $\mathbf{i}\leq \mathbf{j}$. Then,
\begin{equation}
\label{Whittle2} P(A_{\mathbf{n}}) \geq 1-\min_{1\leq s\leq k}\sum_{i =1}^{n_s} \frac{E\left( \phi\left(S_{\mathbf{n};s;i}\right)- \phi\left(S_{\mathbf{n};s;i-1}\right)\right)}{\psi \left( u_{\mathbf{n};s;i} \right)}.
\end{equation} \end{theorem} \begin{proof}
The proof will be given for $k=2$. Similar to the proof of Theorem \ref{Chownew}, we start by defining the functions
\[
u(x) = \phi(x)I\{x\geq 0\}\mbox{ and }v(x)=\phi(x)I\{x<0\}.
\]
Recall that both $u(x)$ and $v(x)$ are nonnegative convex functions, however they have different monotonicity since $u(x)$ is nondecreasing while $v(x)$ is nonincreasing. Moreover,
\[
\phi(x) = u(x)+v(x) = \max\{u(x),v(x)\}.
\]
Then, we have that
\begin{eqnarray*}
P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{\phi(S_{ij})}{\psi(u_{ij})}> 1\right)&=&P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{\max\{u(S_{ij}),v(S_{ij})\}}{\psi(u_{ij})}> 1\right)\\
&\leq&P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{u(S_{ij})}{\psi(u_{ij})}> 1\right)+P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{v(S_{ij})}{\psi(u_{ij})}> 1\right).
\end{eqnarray*}
The above inequality can be written as
\[
P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{\phi(S_{ij})}{\psi(u_{ij})}\leq 1\right)\geq P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{u(S_{ij})}{\psi(u_{ij})}\leq 1\right)+P\left(\max_{(i,j)\leq (n_1,n_2)}\frac{v(S_{ij})}{\psi(u_{ij})}\leq 1\right)-1
\]
or equivalently,
\begin{equation}
\label{guv}P\left(\frac{\phi(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2)\right)\geq P\left(\frac{u(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2) \right)+P\left(\frac{v(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2)\right)-1.
\end{equation}
Since $u$ is a nondecreasing convex function by Theorem \ref{WhittleNond} we have that
\begin{equation}
\label{uRel1}P\left(\frac{u(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2) \right)\geq 1 - \sum_{\ell = 1}^{n_1} \frac{E(u(S_{\ell n_2}) - u(S_{\ell-1n_2}))}{\psi(u_{\ell n_2})}.
\end{equation}
Similarly we can obtain
\begin{equation}
\label{uRel2}
P\left(\frac{u(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2) \right) \geq 1 - \sum_{\ell = 1}^{n_2} \frac{E(u(S_{n_1\ell}) - u(S_{n_1\ell-1}))}{\psi(u_{n_1\ell })}
\end{equation}
Let $B_{n_1n_2} = \left\{\frac {v(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2)\right\}$. Following similar arguments as in the proof of Theorem \ref{WhittleNond} we have that
\begin{align*}
&E\left( \prod_{\ell =1}^{n_1-1} I(B_{\ell n_2}) \left[ \left( 1- \frac{v(S_{n_1n_2})}{\psi(u_{n_1n_2})}\right) - \left( 1- \frac{v(S_{n_1-1n_2})}{\psi(u_{n_1n_2})}\right) \right] + \frac{v(S_{n_1n_2}) - v(S_{n_1-1n_2})}{\psi(u_{n_1n_2})} \right)\\
&= E\left( \left( 1- \prod_{\ell =1}^{n_1-1} I(B_{\ell n_2}) \right) \frac{v(S_{n_1n_2}) - v(S_{n_1-1n_2})}{\psi(u_{n_1n_2})} \right)\geq E\left( \left( 1- \prod_{\ell =1}^{n_1-1} I(B_{\ell n_2}) \right) \frac{(S_{n_1n_2} - S_{n_1-1n_2})}{\psi(u_{n_1n_2})} f(S_{n_1-1n_2}) \right)\\
&\geq 0
\end{align*}
where the first inequality is due to the convexity of the function $v$ with $f(.)$ being the left derivative of $v$ which is a nonpositive nondecreasing function. The last inequality is obtained by the multiindexed demimartingale property since
\[
\left( 1- \prod_{\ell =1}^{n_1-1} I(B_{\ell n_2}) \right)f(S_{n_1-1n_2})
\]
is a nondecreasing function of $\{S_{in_2}, \,i=1,\ldots,n_1-1\}$. Working in a similar manner as in Theorem \ref{WhittleNond} we can obtain the following bounds for the $P(B_{n_1n_2})$,
\begin{equation}
\label{vRel1}P\left(\frac{v(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2) \right)\geq 1 - \sum_{\ell = 1}^{n_1} \frac{E(v(S_{\ell n_2}) - v(S_{\ell-1n_2}))}{\psi(u_{\ell n_2})}
\end{equation}
and
\begin{equation}
\label{vRel2}
P\left(\frac{v(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2) \right) \geq 1 - \sum_{\ell = 1}^{n_2} \frac{E(v(S_{n_1\ell}) - v(S_{n_1\ell-1}))}{\psi(u_{n_1\ell })}
\end{equation}
By combining \eqref{guv}, \eqref{uRel1} and \eqref{vRel1} we have
\[
P\left(\frac{\phi(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2)\right)\geq 1 - \sum_{\ell = 1}^{n_1} \frac{E(\phi(S_{\ell n_2}) - \phi(S_{\ell-1n_2}))}{\psi(u_{\ell n_2})}
\]
while by combining \eqref{guv}, \eqref{uRel2} and \eqref{vRel2} we get
\[
P\left(\frac{\phi(S_{ij})}{\psi(u_{ij})}\leq 1,\, (i,j)\leq (n_1,n_2)\right)\geq 1 - \sum_{\ell = 1}^{n_2} \frac{E(\phi(S_{n_1\ell}) - \phi(S_{n_1\ell-1}))}{\psi(u_{n_1\ell })}.
\]
The desired result follows by combining the last two inequalities. \end{proof}
\noindent As a direct consequence of the above Whittle-type inequality we have the corollary that follows.
\begin{corollary}
Let $\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $\phi(\cdot)$ be a nonnegative convex function such that $\phi(0)= 0$. Let $\psi(u)$ be a positive nondecreasing function for $u>0$. Then, for $\epsilon>0$,
\[
\epsilon P\left( \sup_{\mathbf{j}\leq \mathbf{n}}\frac{\phi(S_\mathbf{j})}{\psi(u_{\mathbf{j}})}\geq \epsilon \right) \leq \min_{1\leq s\leq k}\sum_{i =1}^{n_s} \frac{E\left( \phi\left(S_{\mathbf{n};s;i}\right)- \phi\left(S_{\mathbf{n};s;i-1}\right)\right)}{\psi \left( u_{\mathbf{n};s;i} \right)}.
\] \end{corollary} \begin{proof}
The result follows by first writing that
\[
P\left( \sup_{\mathbf{j}\leq \mathbf{n}}\frac{\phi(S_\mathbf{j})}{\psi(u_{\mathbf{j}})}\geq \epsilon \right) = 1- P\left( \frac{\phi(S_\mathbf{j})}{\psi(u_{\mathbf{j}})}\leq \epsilon , \forall \, \mathbf{j}\leq \mathbf{n} \right)
\]
and applying the result of the previous theorem. \end{proof}
\noindent The Whittle-type inequality can also be employed to obtain the following convergence result.
\begin{theorem}
$\left\{S_{\mathbf{n}}, \mathbf{n} \in \mathbb{N}^{k}\right\}$ be a multiindexed demimartingale with $S_{\boldsymbol{\ell}}=0$ when $\prod_{i=1}^{k} \ell_{i}=0$. Let $\phi(\cdot)$ be a nonnegative convex function such that $\phi(0)= 0$. Let $\psi(u)$ be a positive nondecreasing function for $u>0$ such that $\psi(u) \to \infty$ as $u\to \infty$. Further, suppose that there is $s \in \{1,2,\ldots, k \}$ such that
\begin{equation}
\label{cond1} \displaystyle\sum_{i =1}^{\infty} \frac{E\left( \phi\left(S_{\mathbf{n};s;i}\right)- \phi\left(S_{\mathbf{n};s;i-1}\right)\right)}{\psi \left( u_{\mathbf{n};s;i} \right)}<\infty
\end{equation}
for any nondecreasing sequence $u_{\mathbf{n}} \to \infty$ as $\mathbf{n} \to \infty$. Then,
\[
\frac{\phi(S_{\mathbf{n}})}{\psi(u_{\mathbf{n}})} \to 0 \quad \mbox{a.s. for}\quad \mathbf{n} \rightarrow \infty.
\] \end{theorem} \begin{proof}
For simplicity, we assume that $k=2$ and without loss of generality we assume that \eqref{cond1} is satisfied for $s=2$. Then,
\begin{align*}
& P\left( \sup_{(i,j) \geq (n_1, n_2)} \frac{\phi(S_{ij})}{\psi(u_{ij} )}\geq \epsilon \right) \leq \sum_{\ell = 1}^{\infty} \frac{E\phi(S_{n_1\ell}) - E\phi(S_{n_1\ell-1})}{\epsilon\psi(u_{n_1\ell })}
\leq \frac{E(\phi(S_{n_1n_2}))}{\epsilon\psi(u_{n_1n_2})} + \sum_{\ell = n_2+1}^{\infty} \frac{E\phi(S_{n_1\ell}) - E\phi(S_{n_1\ell-1})}{\epsilon\psi(u_{n_1\ell })} .
\end{align*}
Notice that due to the given assumptions both summands converge to zero as $ \mathbf{n} \rightarrow \infty$ which leads to the desired convergence. \end{proof}
\noindent{\textbf{Acknowledgement}} Work of the second author was supported under the ``INSA Senior Scientist" scheme at the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India.
\end{document} | arXiv |
Transfer function and difference equations: why does $H(z)$ numerator polynomial not correspond to $Y(z)$?
For a discrete time LTI system, I understand that from a difference equation description of the system in the form $$ \sum\limits_{k=0}^N{a_k y[n-k]}=\sum\limits_{k=0}^M{b_k x[n-k]} $$ I can determine the transfer function of the system in the form $$ H(z)=\frac{b_Mz^M+b_{M-1}z^{M-1}+\ldots+b_1z+b_0}{a_Nz^N+a_{N-1}z^{N-1}+\ldots+a_1z+a_0} $$ Given a rationanal transfer function, I can easily go back to the difference equation. However, since the transfer function can also be expressed as $\frac{Y(z)}{X(z)}$, $$ H(z)=\frac{b_Mz^m+b_{M-1}z^{M-1}+...+b_1z+b_0}{a_Nz^N+a_{N-1}z^{N-1}+...+a_1z+a_0}=\frac{Y(z)}{X(z)} $$
Doesn't this imply that the polynomial in the numerator corresponds to the output and the denominator corresponds to the input?
But we know that the numerator polynomial derived from the input side and the denominator derived from the output side of the difference equation.
I'm obviously missing something in my understanding, but I don't see how to resolve this conundrum. What am I missing?
z-transform transfer-function
WesterleyWesterley
$\begingroup$ Your second last para needs correction, as it does not make sense, you've used output twice. Please correct it. $\endgroup$ – learner Aug 14 '17 at 5:29
$\begingroup$ Whoops! - fixed. $\endgroup$ – Westerley Aug 14 '17 at 13:28
It is right that $H(z)=\frac{Y(z)}{X(z)}$ and yes, the output is in the numerator. But the numerator of one side is multiplied by the denomarator of the other side to get an equality:
$$H(z)=\frac{b_Mz^M+\cdots+b_0}{a_Nz^N+\cdots+a_0}=\frac{Y(z)}{X(z)}\tag{0}$$ $$\hspace{-1.5cm}\Rightarrow\left(b_Mz^M+b_{M-1}z^{M-1}+\cdots+b_0 \right)X(z)=\left(a_Nz^N+a_{N-1}z^{N-1}+\cdots+a_0\right)Y(z)\tag{1}$$ Taking the inverse $z$ transform from $(1)$ yields the differnce equation of the system: $$b_Mx[n+M]+b_{M-1}x[n+M-1]+\cdots+b_0=a_Ny[n+N]+\cdots+a_0\tag{2}$$ So indeed, I explained it in the reverse order. Everything begins from the difference equation of a system in $(2)$, then the $z$-transom is applied on it: $$\hspace{-1cm}b_Mz^MX(z)+b_{M-1}z^{M-1}X(z)+\cdots+b_0 X(z)=a_Nz^NY(z)+a_{N-1}z^{N-1}Y(z)+\cdots+a_0Y(z)\tag{3}$$ and then factoring $X(z)$ on the left and $Y(z)$ on the RHS, we will get to $(1)$. From that, $(0)$ is resulted.
If we [wrongly] had
$$\frac{b_Mz^M+\cdots+b_0}{a_Nz^N+\cdots+a_0}=\frac{X(z)}{Y(z)}\tag{4}$$ then after multiplication we would get $$\left(b_Mz^M+b_{M-1}z^{M-1}+\cdots+b_0 \right)Y(z)=\left(a_Nz^N+a_{N-1}z^{N-1}+\cdots+a_0\right)X(z)\tag{5}$$ $$\hspace{-1.5cm}b_Mz^MY(z)+b_{M-1}z^{M-1}Y(z)+\cdots+b_0 Y(z)=a_Nz^NX(z)+a_{N-1}z^{N-1}X(z)+\cdots+a_0X(z)\tag{6}$$ and taking the inverse $z$-transform we would have $$\hspace{-2cm}b_My[n+M]+b_{M-1}y[n+M-1]+\cdots+b_0=a_Nx[n+N]+\cdots+a_0\tag{7}$$ which is obviously not the correct difference equation (i.e. $(2)$ that we started from).
msmmsm
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# Understanding the gradient descent algorithm
Gradient descent is a popular optimization algorithm used in machine learning and statistics to minimize a function iteratively. It's based on the idea of taking small steps in the direction of the steepest decrease of the function's value.
The gradient of a function is a vector that points in the direction of the steepest increase of the function. The negative gradient, on the other hand, points in the direction of the steepest decrease. Gradient descent uses the negative gradient to update the function's parameters at each iteration.
Consider the following function:
$$f(x) = x^2$$
The gradient of this function is:
$$\nabla f(x) = 2x$$
## Exercise
Instructions:
Calculate the gradient of the following function:
$$f(x, y) = x^2 + y^2$$
### Solution
The gradient of $$f(x, y)$$ is:
$$\nabla f(x, y) = \begin{bmatrix} 2x \\ 2y \end{bmatrix}$$
# Implementing the gradient descent algorithm in Java
To implement the gradient descent algorithm in Java, we'll need to define a function to compute the gradient and another function to update the parameters.
Here's a simple implementation of the gradient descent algorithm in Java:
```java
public class GradientDescent {
// Function to compute the gradient
public static double[] gradient(double[] x, double[] y) {
// Compute the gradient here
}
// Function to update the parameters
public static double[] updateParameters(double[] x, double[] y, double learningRate) {
// Update the parameters here
}
public static void main(String[] args) {
// Initialize the parameters, learning rate, and other variables
// Iterate through the data and update the parameters using gradient descent
for (int i = 0; i < iterations; i++) {
// Compute the gradient
double[] gradient = gradient(x, y);
// Update the parameters
x = updateParameters(x, y, learningRate);
}
}
}
```
## Exercise
Instructions:
Implement the gradient and update functions for the following function:
$$f(x, y) = x^2 + y^2$$
### Solution
```java
public static double[] gradient(double[] x, double[] y) {
double[] gradient = new double[2];
gradient[0] = 2 * x[0];
gradient[1] = 2 * y[0];
return gradient;
}
public static double[] updateParameters(double[] x, double[] y, double learningRate) {
double[] updatedX = new double[1];
updatedX[0] = x[0] - learningRate * y[0];
return updatedX;
}
```
# Exploring the concept of convergence
Convergence is a crucial concept in optimization algorithms like gradient descent. It refers to the process of the algorithm reaching a solution that is close enough to the optimal one.
There are several convergence criteria for gradient descent:
1. Fixed number of iterations: The algorithm stops after a fixed number of iterations.
2. Tolerance: The algorithm stops when the change in the function value between iterations is below a certain tolerance.
3. Gradient norm: The algorithm stops when the norm of the gradient is below a certain threshold.
## Exercise
Instructions:
Choose one of the convergence criteria mentioned above and explain how it can be implemented in the Java implementation of the gradient descent algorithm.
### Solution
One way to implement the tolerance convergence criterion is to stop the algorithm when the change in the function value between iterations is below a certain tolerance.
In the `updateParameters` function, we can add a condition to check if the change in the function value is below the tolerance:
```java
public static double[] updateParameters(double[] x, double[] y, double learningRate, double tolerance) {
double[] updatedX = new double[1];
updatedX[0] = x[0] - learningRate * y[0];
// Check if the change in the function value is below the tolerance
if (Math.abs(x[0] - updatedX[0]) < tolerance) {
// Stop the algorithm
return null;
}
return updatedX;
}
```
# Analyzing the convergence properties of the gradient descent algorithm
The convergence properties of the gradient descent algorithm depend on the specific problem and the choice of learning rate. In general, gradient descent is guaranteed to converge if the learning rate is chosen appropriately.
There are several convergence rates for gradient descent:
1. Linear convergence: The algorithm converges in a linear number of iterations.
2. Superlinear convergence: The algorithm converges in a number of iterations that is polynomial in the inverse of the learning rate.
3. Quadratic convergence: The algorithm converges in a number of iterations that is quadratic in the inverse of the learning rate.
## Exercise
Instructions:
Discuss how the choice of the learning rate affects the convergence rate of the gradient descent algorithm.
### Solution
The choice of the learning rate can significantly affect the convergence rate of the gradient descent algorithm. A small learning rate can lead to slow convergence, while a large learning rate can cause the algorithm to overshoot the optimal solution and diverge.
A good learning rate is typically chosen through experimentation and depends on the specific problem and the choice of the initial parameters.
# Optimization techniques and their application
Gradient descent is not the only optimization algorithm. There are several other optimization techniques that can be applied to different problems. Some of these techniques include:
1. Stochastic gradient descent (SGD): A variant of gradient descent that updates the parameters using a random subset of the data.
2. Batch gradient descent: A variant of gradient descent that updates the parameters using the entire dataset.
3. Mini-batch gradient descent: A variant of gradient descent that updates the parameters using a small subset of the data.
4. Adaptive learning rate methods: Techniques that adjust the learning rate during the optimization process based on the current gradient.
## Exercise
Instructions:
Compare and contrast the convergence properties of gradient descent, SGD, BGD, and mini-batch gradient descent.
### Solution
Gradient descent, SGD, BGD, and mini-batch gradient descent have different convergence properties:
- Gradient descent: Quadratic convergence if the learning rate is chosen appropriately.
- SGD: Superlinear convergence if the learning rate is chosen appropriately.
- BGD: Linear convergence if the learning rate is chosen appropriately.
- Mini-batch gradient descent: Convergence rate depends on the batch size and the learning rate.
# Handling numerical issues in Java
When implementing optimization algorithms in Java, it's important to handle numerical issues such as rounding errors and overflow.
Some techniques for handling numerical issues in Java include:
1. Using double-precision floating-point numbers: Java's `double` type provides a higher precision than `float`, reducing the risk of rounding errors.
2. Avoiding division by zero: Check for division by zero and handle the case appropriately.
3. Handling overflow: Check for potential overflow and handle the case by using appropriate data types or scaling the data.
## Exercise
Instructions:
Discuss how numerical issues can be handled in the Java implementation of the gradient descent algorithm.
### Solution
In the Java implementation of the gradient descent algorithm, numerical issues can be handled as follows:
- Use double-precision floating-point numbers to represent the parameters and the gradient.
- Check for division by zero when computing the gradient and handle the case by using an appropriate default value.
- Check for potential overflow when updating the parameters and handle the case by using appropriate data types or scaling the data.
# Comparing gradient descent with other optimization algorithms
Gradient descent is just one of many optimization algorithms. Other popular optimization algorithms include:
1. Newton's method: An iterative optimization algorithm that uses the inverse of the Hessian matrix to update the parameters.
2. Conjugate gradient method: An iterative optimization algorithm that uses the conjugate gradient to update the parameters.
3. Coordinate descent: An optimization algorithm that updates the parameters one at a time, typically using a simple update rule.
## Exercise
Instructions:
Compare and contrast the convergence properties, scalability, and applicability of gradient descent, Newton's method, conjugate gradient method, and coordinate descent.
### Solution
- Gradient descent: Quadratic convergence, but requires the computation of the gradient.
- Newton's method: Superlinear convergence, but requires the computation of the inverse of the Hessian matrix.
- Conjugate gradient method: Linear convergence, but requires the computation of the conjugate gradient.
- Coordinate descent: Linear convergence, but requires the computation of the gradient.
# Implementing the gradient descent algorithm in a real-world scenario
Gradient descent can be applied to a wide range of real-world problems, such as linear regression, logistic regression, and neural networks.
For example, gradient descent can be used to train a neural network for image classification. The neural network's parameters are updated iteratively using the gradient of the loss function with respect to the parameters.
## Exercise
Instructions:
Implement the gradient descent algorithm for training a simple neural network for image classification.
### Solution
```java
public class NeuralNetwork {
// Variables for the neural network
public static void main(String[] args) {
// Initialize the neural network
// Train the neural network using gradient descent
for (int i = 0; i < iterations; i++) {
// Compute the gradient of the loss function
double[] gradient = computeGradient();
// Update the parameters using gradient descent
updateParameters(gradient, learningRate);
}
}
}
```
# Debugging and testing the Java implementation
Debugging and testing are crucial steps in the development of any software, including the Java implementation of the gradient descent algorithm.
Some techniques for debugging and testing the Java implementation include:
- Unit testing: Write unit tests to check the correctness of individual functions.
- Integration testing: Test the integration of the individual functions to ensure that the algorithm works as expected.
- Regression testing: Test the algorithm on a variety of problems and datasets to ensure that it generalizes well.
## Exercise
Instructions:
Discuss how debugging and testing can be performed on the Java implementation of the gradient descent algorithm.
### Solution
Debugging and testing can be performed on the Java implementation of the gradient descent algorithm as follows:
- Unit testing: Write unit tests for the gradient and update functions, as well as any other helper functions.
- Integration testing: Test the main function of the algorithm, which iterates through the data and updates the parameters.
- Regression testing: Test the algorithm on a variety of problems and datasets, such as linear regression, logistic regression, and neural networks.
# Conclusion and future directions
In this textbook, we have covered the implementation of the gradient descent algorithm in Java, its convergence properties, optimization techniques, and its application to real-world problems.
Future directions for this research include:
- Investigating advanced optimization techniques, such as adaptive learning rate methods and second-order optimization methods.
- Exploring the application of gradient descent to more complex problems, such as reinforcement learning and unsupervised learning.
- Developing efficient and scalable implementations of gradient descent algorithms, such as using parallel computing or distributed computing.
By exploring these future directions, we can continue to push the boundaries of what is possible with gradient descent and its applications in machine learning and optimization. | Textbooks |
A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies
From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation
Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system
Hongyong Zhao 1,, and Daiyong Wu 1,2,
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Department of Mathematics, Anqing Normal University, Anqing 246133, China
* Corresponding author: Hongyong Zhao
Received November 2018 Revised March 2019 Published November 2019
Fund Project: The work is partially supported by the National Natural Science Foundation of China (Nos 11571170, 31570417); the Natural Science Foundation of Anhui Province of China (No 1608085 MA14); the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (No KJ2018A0365)
In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.
Keywords: Traveling wave solution, predator-prey, Allee effect, Hopf bifurcation.
Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 34C23.
Citation: Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020129
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Hongyong Zhao Daiyong Wu | CommonCrawl |
LETTER | Open | Published: 14 November 2014
The spatial density gradient of galactic cosmic rays and its solar cycle variation observed with the Global Muon Detector Network
Masayoshi Kozai1,
Kazuoki Munakata1,
Chihiro Kato1,
Takao Kuwabara2,
John W Bieber2,
Paul Evenson2,
Marlos Rockenbach3,
Alisson Dal Lago4,
Nelson J Schuch3,
Munetoshi Tokumaru5,
Marcus L Duldig6,
John E Humble6,
Ismail Sabbah7,
Hala K Al Jassar8,
Madan M Sharma8 &
Jozsef Kóta9
Earth, Planets and Spacevolume 66, Article number: 151 (2014) | Download Citation
The Erratum to this article has been published in Earth, Planets and Space 2016 68:38
We derive the long-term variation of the three-dimensional (3D) anisotropy of approximately 60 GV galactic cosmic rays (GCRs) from the data observed with the Global Muon Detector Network (GMDN) on an hourly basis and compare it with the variation deduced from a conventional analysis of the data recorded by a single muon detector at Nagoya in Japan. The conventional analysis uses a north-south (NS) component responsive to slightly higher rigidity (approximately 80 GV) GCRs and an ecliptic component responsive to the same rigidity as the GMDN. In contrast, the GMDN provides all components at the same rigidity simultaneously. It is confirmed that the temporal variations of the 3D anisotropy vectors including the NS component derived from two analyses are fairly consistent with each other as far as the yearly mean value is concerned. We particularly compare the NS anisotropies deduced from two analyses statistically by analyzing the distributions of the NS anisotropy on hourly and daily bases. It is found that the hourly mean NS anisotropy observed by Nagoya shows a larger spread than the daily mean due to the local time-dependent contribution from the ecliptic anisotropy. The NS anisotropy derived from the GMDN, on the other hand, shows similar distribution on both the daily and hourly bases, indicating that the NS anisotropy is successfully observed by the GMDN, free from the contribution of the ecliptic anisotropy. By analyzing the NS anisotropy deduced from neutron monitor (NM) data responding to lower rigidity (approximately 17 GV) GCRs, we qualitatively confirm the rigidity dependence of the NS anisotropy in which the GMDN has an intermediate rigidity response between NMs and Nagoya. From the 3D anisotropy vector (corrected for the solar wind convection and the Compton-Getting effect arising from the Earth's orbital motion around the Sun), we deduce the variation of each modulation parameter, i.e., the radial and latitudinal density gradients and the parallel mean free path for the pitch angle scattering of GCRs in the turbulent interplanetary magnetic field. We show the derived density gradient and mean free path varying with the solar activity and magnetic cycles.
A solar disturbance propagating away from the Sun affects the population of galactic cosmic rays (GCRs) in a number of ways. Using Parker's transport equation (Parker 1965) of GCRs in the heliosphere, we can infer the large-scale spatial gradient of GCR density by measuring the anisotropy of the high-energy GCR intensity. This is influenced by magnetic structures such as interplanetary shocks and magnetic flux ropes in the interplanetary coronal mass ejections (ICMEs). Only a global network of detectors can measure the dynamic variation of the first-order anisotropy accurately and separately from the temporal variation of the GCR density. The Global Muon Detector Network (GMDN) started operation measuring the three-dimensional (3D) anisotropy on an hourly basis with two-hemisphere observations using a pair of muon detectors (MDs) at Nagoya (Japan) and Hobart (Australia) in 1992. In 2001, another small detector at São Martinho (Brazil) was added to the network to fill a gap in directional coverage over the Atlantic and Europe. The current GMDN consisting of four multidirectional muon detectors was completed in 2006 by expanding the São Martinho detector and installing a new detector in Kuwait. Since then, the temporal variations of the anisotropy and density gradient in association with the ICME and corotating interaction regions have been analyzed on an hourly basis using the observations with the GMDN (Rockenbach et al. 2014; Okazaki et al. 2008; Kuwabara et al. 2004; Kuwabara et al. 2009).
Solar cycle variations of the interplanetary magnetic field (IMF) and solar wind parameters also alter the global distribution of GCR density in the heliosphere and cause long-term variations of the 3D anisotropy of the GCR intensity at the Earth. The 'drift model' of cosmic ray transport in the heliosphere, for instance, predicts a bidirectional latitude gradient of the GCR density, pointing in opposite directions on opposite sides of the heliospheric current sheet (HCS) (Kóta and Jokipii 1982). The predicted spatial distribution of the GCR density has a minimum along the HCS in the 'positive' polarity period of the solar polar magnetic field (also referred as A>0 epoch), when the IMF directs away from (toward) the Sun in the northern (southern) hemisphere, while the distribution has the local maximum on the HCS in the 'negative' period (A<0 epoch) with the opposite field orientation in each hemisphere. The field orientation reverses every 11 years around the period of maximum solar activity. The 3D anisotropy of GCR intensity consists of two components: one lying in the ecliptic plane and the other pointing normal to the ecliptic plane. The ecliptic component can be observed as the solar diurnal anisotropy (the first harmonic vector of the solar diurnal variation) of GCR intensity, while the normal component can be measured as the north-south (NS) anisotropy responsible for the difference between intensities recorded by north- and south-viewing detectors or the sidereal diurnal anisotropy. By analyzing the solar diurnal variation and the NS anisotropy of the GCR intensity recorded by neutron monitors (NMs), Bieber and Chen (1991) and Chen and Bieber (1993) derived the solar cycle variations of 3D anisotropy and modulation parameters on a yearly basis. On the other hand, Munakata et al. (2014) derived the long-term variation of the 3D anisotropy from the long-term record of the GCR intensity observed with a single multidirectional MD at Nagoya in Japan. By comparing the anisotropy derived from the MD data with that from the NM data, they examined the rigidity dependence of the anisotropy and its solar cycle variation.
Accurate observation of the NS anisotropy normal to the ecliptic plane is also crucial for obtaining a reliable 3D anisotropy. This component has been derived from NM and MD data in two different ways. Chen and Bieber (1993) derived this component anisotropy from the difference between count rates in a pair of NMs which are located near the north and south geomagnetic poles and observing intensities of GCRs arriving from the north and south pole orientations, respectively. The NS anisotropy derived in this way is very sensitive to the stability of operations of two independent detectors and can be easily affected by unexpected changes of instrumental and/or environmental origins. Due to the 23.4° inclination of Earth's rotation axis from the ecliptic normal, the NS anisotropy normal to the ecliptic plane can be also observed as a diurnal variation of count rate in sidereal time with the maximum phase at approximately 06:00 or approximately 18:00 local sidereal time (Swinson 1969). A possible drawback of deriving the NS anisotropy from the sidereal diurnal variation is that the expected amplitude of the sidereal diurnal variation is roughly ten times smaller than that of the solar diurnal variation. The small signal in sidereal time can be easily influenced by the solar diurnal anisotropy changing during a year. Another difficulty is that one can obtain only the yearly mean anisotropy, because the influence from the solar diurnal variation, even if it is stationary throughout a year, can be canceled in sidereal time only when the diurnal variation is averaged over at least 1 year. This makes it difficult to deduce a reliable error of the yearly mean anisotropy. Mori and Nagashima (1979) proposed another way to derive the NS anisotropy from the 'GG-component' of a multidirectional MD at Nagoya in Japan. The GG-component is a difference combination between intensities recorded in the north- and south-viewing directional channels corresponding to 56° north and 14° south asymptotic latitudes in free space at their median rigidity, approximately 80 GV, designed to measure the NS anisotropy free from atmospheric temperature effect (Nagashima et al. 1972). The NS anisotropy depends on the polarity of the magnetic field. Based on this fact, Laurenza et al. (2003) showed that the GG-component can be used for deriving reliable sector polarity of the IMF which is defined as away (toward) when the IMF directs away from (toward) the Sun. By using a global network of four multidirectional MDs which is able to observe the NS anisotropy on an hourly basis, Okazaki et al. (2008) reported for the first time that the NS anisotropy deduced from the GG-component is consistent with the anisotropy observed with the global network for a year during the solar activity minimum period.
Analyses of the diurnal variation observed with a single detector, however, can give a correct anisotropy only when the anisotropy is stationary at least over 1 day and may not work if the anisotropy changes dynamically within a day. The GG-component also needs to be averaged over 1 day to cancel the influence of the ecliptic components which have components parallel to the rotation axis of the Earth and contribute to the NS difference measured by the GG-component. Additionally, the directional channels of the Nagoya MD have an angular distribution biased toward the northern hemisphere, while the GMDN has a global angular distribution. It is important, therefore, to examine whether the long-term variation of the 3D anisotropy derived from the conventional analysis of the observed diurnal variation and the GG-component is consistent with the anisotropy observed by the GMDN which is capable of accurately measuring anisotropy with better time resolution. In this paper, we analyze the 3D anisotropy observed with the GMDN over 22 years between 1992 and 2013 and compare it with the anisotropy observed with the Nagoya multidirectional MD, especially focusing on the NS anisotropy for which the GG-component has been the only reliable measurement at the 50 to 100 GV region. Based on the difference of the response rigidities between the GMDN (approximately 60 GV) and the GG-component (approximately 80 GV), we also discuss the rigidity dependence of the NS anisotropy.
We analyze the pressure-corrected hourly count rate I i,j (t) of recorded muons in the jth directional channel of the ith detector in the GMDN at universal time t and derive three components $\left (\xi ^{\text {GEO}}_{x}(t), \xi ^{\text {GEO}}_{y}(t), \xi ^{\text {GEO}}_{z}(t)\right)$ of the first-order anisotropy in the geographic (GEO) coordinate system by best fitting the following model function to I i,j (t).
$$\begin{array}{@{}rcl@{}} I^{fit}_{i,j}(t) = I^{0}_{i,j}(t) &+&\xi^{\text{GEO}}_{x}(t)\left(c^{1}_{1 i,j} \cos \omega t_{i} - s^{1}_{1 i,j} \sin \omega t_{i}\right) \\ &+&\xi^{\text{GEO}}_{y}(t)\left(s^{1}_{1 i,j} \cos \omega t_{i} + c^{1}_{1 i,j} \sin \omega t_{i}\right) \\ &+&\xi^{\text{GEO}}_{z}(t) c^{0}_{1 i,j} \end{array} $$
((1))
where $I^{0}_{i,j}(t)$ is a parameter representing the contributions from the omnidirectional intensity and the atmospheric temperature effect; t i is the local time at the ith detector; $c^{1}_{1 i,j}$ , $s^{1}_{1 i,j}$ , and $c^{0}_{1 i,j}$ are the coupling coefficients; and ω=π/12. The coupling coefficients are calculated by integrating the response function of atmospheric muons to the primary cosmic rays (Murakami et al. 1979) for primary rigidity, detective solid angle, and detection area with weights of the asymptotic orbit by assuming a rigidity-independent anisotropy with the upper limiting rigidity set at 105 GV, far above the maximum rigidity of the response.
In deriving the anisotropy vector ξ, we additionally apply an analysis method developed to remove the influence of atmospheric temperature variations from the derived anisotropy (see Okazaki et al. 2008). Elimination of the temperature effect from the MD data is of particular importance in analyzing the long-term temporal variation of ξ. The deduced anisotropy is averaged over each IMF sector in every month designated as away (toward) if the daily polarity of the Stanford mean magnetic field of the Sun (Wilcox Solar Observatory), shifted 5 days later for a rough correction for the solar wind transit time between the Sun and the Earth, is positive (negative).
We also derive the anisotropy from observations by a single multidirectional MD at Nagoya (hereafter Nagoya MD) which is a component detector of the GMDN. By using the coupling coefficients, we deduce the equatorial component $\left ({\xi }^{\text {GEO}}_{x}, {\xi }^{\text {GEO}}_{y}\right)$ of ξ from the mean diurnal variation of the hourly counting rate in each IMF sector in every month. On the other hand, we derive the normal component to the equatorial plane, ${\xi }^{\text {GEO}}_{z}$ , from the GG-component averaged over each IMF sector by using the coupling coefficient in every month (Munakata et al. 2014). The GG-component is a difference combination between intensities recorded in the north- and south-viewing channels and has long been used as a good measure of the NS anisotropy (Mori and Nagashima 1979; Nagashima et al. 1972; Laurenza et al. 2003).
The anisotropy vector $\left ({\xi }^{\text {GEO}}_{x}, {\xi }^{\text {GEO}}_{y}, {\xi }^{\text {GEO}}_{z}\right)$ in three dimensions derived from the GMDN and Nagoya data is transformed to the geocentric solar ecliptic (GSE) coordinate system, in which the z-component corresponds to the NS component normal to the ecliptic plane, and corrected for the solar wind convection anisotropy using the solar wind velocity in the 'omnitape' (King and Papitashvili 2005; NASA 2014) data by the Space Physics Data Facility at the Goddard Space Flight Center and for the Compton-Getting anisotropy arising from the Earth's 30 km/s orbital motion around the Sun. In the corrections, we set the power law index of the GCR energy spectrum to be −2.7. We then obtain the ecliptic plane component of the anisotropy consisting of components parallel (ξ ∥) and perpendicular (ξ ⊥) to the IMF as obtained from the omnitape data and NS anisotropy (ξ z ) normal to the plane in each IMF sector in every month. We finally obtain the monthly mean three components of the anisotropy in the solar wind frame as
$$\begin{array}{@{}rcl@{}} \xi_{\|} = \left(\xi_{\|}^{T} + \xi_{\|}^{A}\right)/2 \end{array} $$
((2a))
$$\begin{array}{@{}rcl@{}} \xi_{\bot} = \left(\xi_{\bot}^{T} + \xi_{\bot}^{A}\right)/2 \end{array} $$
((2b))
$$\begin{array}{@{}rcl@{}} \xi_{z} = \left({\xi_{z}^{T}} - {\xi_{z}^{A}}\right)/2 \end{array} $$
((2c))
where $\xi _{\|}^{T}\left (\xi _{\|}^{A}\right)$ and $\xi _{\bot }^{T}\left (\xi _{\bot }^{A}\right)$ are parallel and perpendicular components in the ecliptic plane averaged over the toward (away) sector, while ${\xi _{z}^{T}}\left ({\xi _{z}^{A}}\right)$ is the NS anisotropy in the toward (away) sector. We assume that the anisotropy vector, when averaged over 1 month exceeding a solar rotation period, is symmetrical with respect to the HCS which is regarded to coincide with the solar equatorial plane on the average. Because of this assumption, the NS anisotropy is directed oppositely, with the same magnitude, above and below the HCS as defined in Equation 2c.
Solar cycle variation of the 3D anisotropy
Figure 1a,b,c shows the temporal variations of the yearly mean ξ ∥, ξ ⊥, and ξ z as defined in Equation 2c, respectively. Each panel shows that the temporal variations of the anisotropy components derived from the GMDN (solid circle) and Nagoya (open circle) data are fairly consistent with each other as far as the year-to-year variation is concerned. We can see that the solar cycle variation of ξ ∥ has two components. One is a 22-year variation resulting in a slightly larger ξ ∥ in A<0 epoch (2001 to 2011) than in A>0 epoch (1992 to 1998) as reported by Chen and Bieber (1993). The other is a variation correlated with cosψ, shown with ξ ∥ by open squares in Figure 1a, where ψ is the IMF spiral angle derived from omnitape data. ξ z deduced from the GMDN (solid circles), on the other hand, shows an 11-year cycle with minima in 1998 and 2007 around the solar activity minima, while ξ ⊥ shows no solar cycle variation.
Long-term variations of three components of the anisotropy vector in the solar wind frame. Each panel displays the yearly mean (a) ξ ∥ (on the left vertical axis), (b) ξ ⊥, and (c) ξ z as defined in Equation 2c, each as a function of year on the horizontal axis. Solid and open circles in each panel represent anisotropies derived from the GMDN and Nagoya data, respectively, while open squares in (a) display cosψ on the right vertical axis. In each panel, yearly mean value and its error are deduced from the average and dispersion of monthly mean values. Gray vertical stripes indicate periods when the polarity reversal of the solar polar magnetic field (referred as A>0 or A<0 in (b)) is in progress.
Comparison between the NS anisotropies observed with the GMDN and the GG-component
We now focus on the NS anisotropy which cannot be detected by a single-directional channel separately from GCR density variations. Figure 2 shows histograms of hourly (a and b) and daily (c and d) mean $\xi ^{\text {GEO}}_{z}$ observed by the GG-component (a and c) and GMDN (b and d) in 2006 to 2013, which are classified according to the IMF sectors designated as toward (blue histograms) if B x >B y and away (red histograms) if B x <B y by using the GSE-x, y components (B x , B y ) of the IMF vector in the omnitape data. The blue and red vertical dashed lines represent averages of the blue and red histograms, respectively. We define ' T/A separation' following Okazaki et al. (2008) as
$$(T - A)/\sqrt{\sigma_{T}\sigma_{A}} $$
where T (A) and σ T (σ A ) are the average and standard errors of each histogram in the toward (away) sector, respectively. Table 1 lists T−A, $\sqrt {\sigma _{T}\sigma _{A}}$ , T/A separation, and 'success rate' (Mori and Nagashima 1979; Laurenza et al. 2003). The success rate is a ratio of the number of hours (days) when the sign of the observed $\xi ^{\text {GEO}}_{z}$ is positive (negative) in the toward (away) IMF sector to the total number of hours (days) and is introduced as a parameter indicating to what extent we can infer the IMF sector polarity from the sign of the observed ξ z . Although we use the success rate together with T/A separation for the following comparison, it is noted that a low success rate does not necessarily imply anything wrong in the observed ξ z . The IMF sector polarity sensed by high-energy GCRs should be regarded as the polarity averaged over a spatial scale comparable to the Larmor radii of GCRs which span approximately 0.1 AU. It is natural to expect that the IMF polarity averaged over such a large scale does not always follow the single-point measurement of the polarity by a satellite. In Table 1, it is seen that the daily mean $\xi ^{\text {GEO}}_{z}$ by the GMDN shows smaller T/A separation and success rate than $\xi ^{\text {GEO}}_{z}$ deduced from the GG-component, while the hourly $\xi ^{\text {GEO}}_{z}$ by GMDN has a larger T/A separation and success rate than the GG-component which has significantly larger dispersion (Figure 2a), partly due to the contribution from diurnal anisotropy as suggested by Okazaki et al. (2008) from their analysis of 1-year data between March 2006 and March 2007.
Histograms of the NS anisotropy. Each panel displays the histograms of $\xi _{z}^{\text {GEO}}$ on (a, b) hourly and (c, d, e) daily bases derived from the (a, c) Nagoya GG-component, (b, d) GMDN, and (e) NM (Thule-McMurdo) data in 2006 to 2013. Blue and red histograms in each panel represent distributions of $\xi _{z}^{\text {GEO}}$ in toward and away IMF sectors, respectively, while blue and red vertical dashed lines represent averages of the blue and red histograms, respectively.
Table 1 T − A , ${\sqrt {\sigma _{T}\sigma _{A}}}$ , T / A separation, and success rate
We also examine the rigidity dependence of the NS anisotropy by analyzing NM data from 2006 to 2013. NMs have median responses to approximately 17 GV GCRs, while the GMDN and GG-component have median responses to approximately 60 GV and approximately 80 GV, respectively. Chen and Bieber (1993) derived the NS anisotropy ξ z in Equation 2c from the ratio (R) of the daily mean counting rate recorded by the Thule NM to that recorded by the McMurdo NM as
$$ \xi_{z} = \frac{b}{2}\frac{R^{T} - R^{A}}{R^{T} + R^{A}} $$
where R T (R A) is the R averaged over toward (away) sectors in every month and b is a constant calculated from coupling coefficients. We define the daily mean NS anisotropy by NMs as
$$ \xi^{\text{GEO}}_{z} = \frac{c}{2}\frac{R}{R^{T} + R^{A}} $$
where c is a coupling coefficient calculated on the same assumption as adopted in our analysis of the GMDN and Nagoya MD data. The T/A separation and success rate of this $\xi ^{\text {GEO}}_{z}$ represents those parameters for approximately 17 GV GCRs. The result of this analysis is presented in Figure 2e and Table 1. It is seen that the T/A separation of the NS anisotropy by NMs is significantly smaller mainly due to the small T−A, i.e., the NS anisotropy is significantly smaller than that obtained from the GMDN and GG-component. The NS anisotropy is smallest in NM data at approximately 17 GV and largest in the GG-component at approximately 80 GV, with the anisotropy in the GMDN at approximately 60 GV in between, suggesting that the NS anisotropy increases with increasing rigidity (Munakata et al. 2014).
Solar cycle variation of modulation parameters
Following the analyses by Chen and Bieber (1993), we derive modulation parameters, i.e., the density gradient and the mean free path of the pitch angle scattering. By assuming that the longitudinal gradient is zero in our analysis based on the anisotropy averaged over 1 month which is longer than the solar rotation period, ξ ∥, ξ ⊥, and ξ z obtained in Equations 2a, 2b, and 2c are related with the modulation parameters as
$$\begin{array}{@{}rcl@{}} \xi_{\|} &=& \lambda_{\|} G_{r} \cos \psi \end{array} $$
$$\begin{array}{@{}rcl@{}} \xi_{\bot} &=& \lambda_{\bot} G_{r} \sin \psi - R_{L} G_{z} \end{array} $$
$$\begin{array}{@{}rcl@{}} \xi_{z} &=& R_{L} G_{r} \sin \psi + \lambda_{\bot} G_{z} \end{array} $$
where R L is the Larmor radius of GCRs in the IMF and G z , G r , λ ∥, and λ ⊥ are the latitudinal and radial density gradients and the mean free paths of the pitch angle scattering parallel and perpendicular to the IMF. From Equations 5a, 5b, and 5c, we deduce the modulation parameters as
$$\begin{array}{*{20}l} &{}G_{z} = \left(\alpha \xi_{\|} \tan \psi - \xi_{\bot} \right)/R_{L} \end{array} $$
$$\begin{array}{*{20}l} &{}G_{r} = \left\{\! \xi_{z}\! +\! \sqrt{{\xi_{z}^{2}}\! +\! 4 \alpha \xi_{\|} \tan\! \psi \left(\xi_{\bot}\! - \!\alpha \xi_{\|} \tan\! \psi \right)}\! \right\}\!/\!\left(2 R_{L}\! \sin \psi \right) \end{array} $$
$$\begin{array}{*{20}l} &{}\lambda_{\|} = \xi_{\|}/\left(G_{r} \cos \psi \right) \end{array} $$
where α=λ ⊥/λ ∥, assumed to be 0.01 and constant as adopted by Chen and Bieber (1993). G z is converted to the bidirectional latitudinal gradient as
$$\begin{array}{@{}rcl@{}} G_{|z|} = -\text{sgn}(A) G_{z} \end{array} $$
where A represents the polarity of the solar dipole magnetic moment and
$$\begin{array}{@{}rcl@{}} \text{sgn}(A) &=& +1, \;\text{for}\; A>0 \;\text{epoch}, \\ &=& -1, \;\text{for}\; A<0 \;\text{epoch}. \end{array} $$
Figure 3a,b,c displays temporal variations of modulation parameters G |z|, G r , and λ ∥, respectively, obtained from the GMDN (solid circle) and Nagoya (open circle) data. The variations with 22-year and 11-year solar cycles are clearly seen in this figure. First, G |z| is positive in A>0 epoch indicating the local minimum of the density on the HCS, while it is negative in A<0 epoch indicating the maximum in accord with the prediction of the drift model by Kóta and Jokipii (1983). Second, significant 11-year variations are seen in both G r and λ ∥ which change in a clear anti-correlation.
Long-term variations of modulation parameters derived from the 3D anisotropy in the solar wind frame. Each panel displays the yearly mean (a) G |z|, (b) G r , and (c) λ ∥ as a function of year. Solid and open circles in each panel represent parameters derived from the GMDN and Nagoya data, respectively. In each panel, yearly mean value and its error are deduced from the average and dispersion of monthly mean values. Gray vertical stripes indicate periods when the polarity reversal of the solar polar magnetic field (referred as A>0 or A<0 in (c)) is in progress.
Summary and discussions
We analyzed the 3D anisotropy of GCR intensity observed by the GMDN and Nagoya MD in 1992 to 2013. Our analysis of the GMDN data gives the anisotropy on an hourly basis with better time resolution than the traditional analyses of the diurnal and NS anisotropies observed by a single detector such as the Nagoya MD. We confirmed that the 3D anisotropy and the modulation parameters derived from the GMDN and Nagoya MD data are fairly consistent with each other as far as the yearly mean value is concerned. This fact is important particularly for the NS anisotropy derived from the GMDN data, because the GG-component has been the only reliable reference to the NS anisotropy in the rigidity region between 50 and 100 GV.
By analyzing the distribution of the NS anisotropy separately in toward and away IMF sectors, we compared the T/A separations and success rates deduced from the GMDN and Nagoya MD data on hourly and daily bases. It is confirmed that the daily mean NS anisotropy observed by the GG-component shows slightly better T/A separation and success rate than the daily mean anisotropy by the GMDN, while the hourly mean NS anisotropy by the GG-component shows a large spread due to the local time-dependent contribution from the ecliptic anisotropy. The NS anisotropy by the GMDN, on the other hand, shows similar success rate on both daily and hourly bases, indicating that the NS anisotropy is successfully observed by the GMDN, free from the contribution of the ecliptic anisotropy.
In addition to the better time resolution, the new analysis method developed by Okazaki et al. (2008) for the GMDN data also has an advantage of providing the 3D anisotropy, including the NS component, from a single best-fit calculation for intensities recorded by four detectors. In contrast, the conventional method using a single MD requires the derivation of the NS anisotropy from the north- and south-viewing channels, separately from the derivation of the diurnal anisotropy using all directional channels.
By comparing the NS anisotropy derived from NM data with those from the GMDN and GG-component data, we find that the NS anisotropy increases with increasing rigidity and the difference between T/A separations and success rates of the GMDN and GG-component data is partly due to the rigidity dependence. Yasue (1980) analyzed the GG-component together with the sidereal diurnal variation observed by MDs and NMs and derived the power law-type rigidity spectrum of the average NS anisotropy with a positive power law index of approximately 0.3. By analyzing long-term variations of the 3D anisotropies observed with Nagoya MD and NMs on yearly basis, Munakata et al. (2014) confirmed that the perpendicular component including the NS anisotropy increases with GCR rigidity. If these are the case, the magnitude of the NS anisotropy increases with rigidity and the T/A separation and success rate will also increase if the dispersion remains similarly independent of rigidity. This is in agreement with our results in Table 1, showing that T−A increases with the rigidity while $\sqrt {\sigma _{T}\sigma _{A}}$ is almost constant on a daily basis. Three GCR observations responsible for different rigidities, GG-component (approximately 80 GV), GMDN (approximately 60 GV), and NMs (approximately 17 GV) are capable of observing the NS anisotropy on daily basis, and their cross-calibration allows us to obtain the information about the rigidity dependence of the NS anisotropy.
We confirmed that the solar cycle variations of the yearly mean solar modulation parameters derived from the GMDN and Nagoya data are consistent with each other. The bidirectional latitudinal gradient G |z| shows a clear 22-year variation being positive (negative) in A>0 (A<0) epochs indicating the local minimum (maximum) of the GCR density on the HCS, in accord with the prediction of the drift model (Kóta and Jokipii 1983). On the other hand, significant 11-year solar cycle variations are seen in G r and λ ∥, respectively. The ecliptic component of the anisotropy ξ ∥ parallel to the IMF shows a 22-year variation being slightly larger in A<0 epoch (2001 to 2011) than in A>0 epoch (1992 to 1998) as reported by Chen and Bieber (1993). This variation of ξ ∥ is responsible for the well-known 22-year variation of the phase of the diurnal variation (Thambyahpillai and Elliot 1953). We find that the variation of ξ ∥ also shows a correlation with the cosψ which is governed by the solar wind velocity. This is reasonable because ξ ∥ is proportional to cosψ as given in Equation 5a. Figure 4 shows yearly variation of ξ ∥/ cosψ, i.e., λ ∥ G r by the GMDN. In this figure, the 22-year variation is seen more clearly than in Figure 1 showing ξ ∥ (Chen and Bieber 1993). For an accurate analysis of the solar cycle variation of the anisotropy, therefore, it is necessary to correct the observed anisotropy for cosψ and the solar wind velocity which varies without any clear 11-year or 22-year periodicities.
Long-term variation of λ ∥ G r derived from the GMDN data. Yearly mean value and its error are deduced from the average and dispersion of monthly mean values. Gray vertical stripes indicate periods when the polarity reversal of the solar polar magnetic field (referred as A>0 or A<0) is in progress.
The 22-year variation of λ ∥ G r seems mainly due to the variation of λ ∥ in Figure 3 which is larger in A<0 epoch (2001 to 2011) than in A>0 epoch (1992 to 1998). However, the solar magnetic field was unusually weak around this last solar minimum (2009), which resulted in a record-high GCR flux (Mewaldt et al. 2010). The larger mean free path is more likely the result of the weaker solar minimum than a polarity issue.
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This work is supported in part by the joint research programs of the Solar-Terrestrial Environment Laboratory (STEL), Nagoya University and the Institute for Cosmic Ray Research (ICRR), University of Tokyo. The observations with the Nagoya multidirectional muon detector are maintained by Nagoya University. CNPq, CAPES, INPE and UFSM support upgrade and maintenance of the São Martinho muon detector. The Bartol Research Institute neutron monitor program, which operates Thule and McMurdo neutron monitors, is supported by National Science Foundation grant ATM-0000315. Wilcox Solar Observatory data used in this study was obtained via the website http://wso.stanford.edu at 2013:06:24$ ¥_$22:12:55 PDT courtesy of J.T. Hoeksema. The Wilcox Solar Observatory is currently supported by NASA. JK thanks STEL and Shinshu University for the hospitality during his stay as a visiting professor of STEL.
Physics Department, Shinshu University, Matsumoto, 390-8621, Nagano, Japan
Masayoshi Kozai
, Kazuoki Munakata
& Chihiro Kato
Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, 19716, DE, USA
Takao Kuwabara
, John W Bieber
& Paul Evenson
Southern Regional Space Research Center (CRS/INPE), Santa Maria RS, P.O. Box 5021, 97110-970, Brazil
Marlos Rockenbach
& Nelson J Schuch
National Institute for Space Research (INPE), São José dos Campos SP, 12227-010, Brazil
Alisson Dal Lago
Solar Terrestrial Environment Laboratory, Nagoya University, Nagoya, 464-8601, Aichi, Japan
Munetoshi Tokumaru
School of Physical Sciences, University of Tasmania, Hobart, 7001, Tasmania, Australia
Marcus L Duldig
& John E Humble
Department of Natural Sciences, College of Health Sciences, Public Authority for Applied Education and Training, Kuwait City, 72853, Kuwait
Ismail Sabbah
Physics Department, Kuwait University, Kuwait City, 13060, Kuwait
Hala K Al Jassar
& Madan M Sharma
Lunar and Planetary Laboratory, University of Arizona, Tucson, 85721, AZ, USA
Jozsef Kóta
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Correspondence to Masayoshi Kozai.
MK, the corresponding author of this paper, made all analyses presented in this paper. KM evaluated the data and discussed the analysis results in this paper. CK made the GMDN data available for this paper. TK made the GMDN data and the neutron monitor data available for this paper. JWB and PE made the neutron monitor data available for this paper. MR kept the São Martinho muon detector in operation. ADL hosted the São Martinho muon detector, assisted with its installation, and performed its general maintenance. NJS contributed in establishing the observation with the São Martinho muon detector and kept it in operation. MT kept the Nagoya muon detector in operation. MLD and JEH hosted the Hobart muon detector, assisted with its installation, performed its general maintenance, discussed the results, and input into the English grammar of this paper. IS established the observation with the Kuwait muon detector. HKAJ and MMS kept the Kuwait muon detector in operation. JK discussed analysis results and worked for improving the English in this paper. All authors read and approved the final manuscript.
Diurnal anisotropy
North-south anisotropy
Heliospheric modulation of galactic cosmic rays
Solar cycle variation of the cosmic ray density gradient | CommonCrawl |
A Performance Evaluation of Classic Convolutional Neural Networks for 2D and 3D Palmprint and Palm Vein Recognition
Wei Jia, Jian Gao, Wei Xia, Yang Zhao, Hai Min, Jing-Ting Lu
文章导航 > International Journal of Automation and Computing > 2021 > 优先出版
Wei Jia, Jian Gao, Wei Xia, Yang Zhao, Hai Min, Jing-Ting Lu. A Performance Evaluation of Classic Convolutional Neural Networks for 2D and 3D Palmprint and Palm Vein Recognition. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1257-9
Citation: Wei Jia, Jian Gao, Wei Xia, Yang Zhao, Hai Min, Jing-Ting Lu. A Performance Evaluation of Classic Convolutional Neural Networks for 2D and 3D Palmprint and Palm Vein Recognition. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1257-9
1, 2, ,
Wei Jia1, 2 , ,
Jian Gao1, 2 , ,
Wei Xia1, 2 , ,
Yang Zhao1, 2 , ,
Hai Min1, 2 , ,
Jing-Ting Lu3 ,
School of Computer Science and Information Engineering, Hefei University of Technology, Hefei 230009, China
Anhui Province Key Laboratory of Industry Safety and Emergency Technology, Hefei 230009, China
Institution of Industry and Equipment Technology, Hefei University of Technology, Hefei 230009, China
Wei Jia received the B. Sc. degree in informatics from Central China Normal University, China in 1998, the M. Sc. degree in computer science from Hefei University of Technology, China in 2004, and the Ph. D. degree in pattern recognition and intelligence system from University of Science and Technology of China, China in 2008. He has been a research associate professor in Hefei Institutes of Physical Science, Chinese Academy of Sciences, China from 2008 to 2016. He is currently an associate professor in Key Laboratory of Knowledge Engineering with Big Data, Ministry of Education, and in School of Computer Science and Information Engineering, Hefei University of Technology, China. His research interests include computer vision, biometrics, pattern recognition, image processing and machine learning. E-mail: [email protected] (Corresponding author) ORCID iD: 0000-0001-5628-6237
Jian Gao received the B. Sc. degree in mechanical design and manufacturing and automation from Hefei University of Technology, China in 2018. Now, he is currently a master student in School of Computer Science and Information Engineering, Hefei University of Technology, China. His research interests include computer vision, biometrics recognition and deep learning. E-mail: [email protected]
Wei Xia received the B. Sc. degree in computer science from Anhui University of Science and Technology, China in 2018. He is a master student in School of Computer Science and Information Engineering, Hefei University of Technology, China. His research interests include biometrics, pattern recognition and image processing. E-mail: [email protected]
Yang Zhao received the B. Eng. and Ph.D. degrees in pattern recognition and intelligence from Department of Automation, University of Science and Technology of China, China in 2008 and 2013. From 2013 to 2015, he was a postdoctoral researcher at School of Electronic and Computer Engineering, Peking University Shenzhen Graduate School, China. Currently, he is an associate professor at School of Computer Science and Information Engineering, Hefei University of Technology, China. His research interests include image processing and computer vision. E-mail: [email protected]
Hai Min received the Ph. D. degree in pattern recognition and intelligence system from University of Science and Technology of China, China in 2014. He is currently an associate professor in School of Computer Science and Information Engineering, Hefei University of Technology, China. His research interests include pattern recognition and image segmentation. E-mail: [email protected]
Jing-Ting Lu received the B. Sc., M. Sc. and Ph. D. degrees in computer science from Hefei University of Technology, China in 2004, 2009, and 2014, respectively. She is currently a lector in School of Computer and Information, Hefei University of Technology, China. Her research interests include computer vision, biometrics, pattern recognition, image processing and machine learning. E-mail: [email protected] iD: 0000-0002-0210-7149
图(30) 表(17)
参考文献(94)
Performance evaluation /
convolutional neural network (CNN) /
biometrics /
palmprint /
palm vein /
Abstract: Palmprint recognition and palm vein recognition are two emerging biometrics technologies. In the past two decades, many traditional methods have been proposed for palmprint recognition and palm vein recognition, and have achieved impressive results. However, the research on deep learning-based palmprint recognition and palm vein recognition is still very preliminary. In this paper, in order to investigate the problem of deep learning based 2D and 3D palmprint recognition and palm vein recognition in-depth, we conduct performance evaluation of seventeen representative and classic convolutional neural networks (CNNs) on one 3D palmprint database, five 2D palmprint databases and two palm vein databases. A lot of experiments have been carried out in the conditions of different network structures, different learning rates, and different numbers of network layers. We have also conducted experiments on both separate data mode and mixed data mode. Experimental results show that these classic CNNs can achieve promising recognition results, and the recognition performance of recently proposed CNNs is better. Particularly, among classic CNNs, one of the recently proposed classic CNNs, i.e., EfficientNet achieves the best recognition accuracy. However, the recognition performance of classic CNNs is still slightly worse than that of some traditional recognition methods.
Figure 1. Chronology of classic CNNs chronology for classification tasks
Figure 2. Structure of AlexNet
Figure 3. Structure of VGG
Figure 4. Module of Inception_v3: (a) 1×n + n×1 convolution instead of n×n convolution; (b) Three different Inception modules in Inception_v3.
Figure 5. Module of ResNet: (a) Residual module in ResNet; (b) Two forms of residual module.
Figure 6. Inception_v4 overall structure and module structure: (a) Inception_v4 overall structure; (b) Stem module; (c) Up: Inception_A, Mid: Inception_B, Down: Inception_C; (d) Up: Reduction_A, Down: Reduction_B.
Figure 7. Inception_ResNet_ v2 overall structure and module structure: (a) Overall structure; (b) Stem module; (c) Up: Inception_ResNet_A, Middle: Inception_ResNet_B, Bottom: Inception_ResNet _C; (d) Up: Reduction_A, Bottom: Reduction_B.
Figure 8. Structure of DenseNet: (a) Dense block structure; (b) DenseNet overall structure.
Figure 9. Spatial convolution and channel convolution in Xception
Figure 10. Formation of the extreme Inception: (a) Inception module (ignoring max pooling); (b) Merging 1 × 1 convolution; (c) Extreme Inception.
Figure 11. Depthwise separable convolution: (a) Standard convolution filters; (b) Upper part: depthwise convolution filters, lower part: pointwise convolution filters.
Figure 12. Standard convolution layer and depthwise separable convolution layer: (a) Standard convolution layer; (b) Depthwise separable convolution layer.
Figure 13. Structure of MobileNet_v2
Figure 14. Structure of MobileNet_v3: (a) Structure of MobileNet_v3-large; (b) Structure of MobileNet_v3-small.
Figure 15. Structure of SENet block
Figure 16. Equivalent building blocks of ResNeXt: (a) Aggregated residual transformations; (b) A block equivalent to (a), implemented as early concatenation; (c) A block equivalent to (a ) and (b), implemented as grouped convolutions.
Figure 17. Structure of ShuffleNet_v1: (a) No downsampling; (b) Downsampling.
Figure 18. Structure of ShuffleNet_v2: (a) Basic unit; (b) Unit for spatial down sampling (2×).
Figure 19. Structure of EfficientNet-b0
Figure 20. Ghost module
Figure 21. ResNeSt block module
Figure 22. Six palmprint ROI images of PolyU II database. The three images of the first row were captured in the first session. The three images of the second row were captured in the second session.
Figure 23. Six palmprint ROI images of PolyU M_B database. The three images of the first row were captured in the first session. The three images of the second row were captured in the second session.
Figure 24. Six palmprint ROI images of HFUT database. The three images of the first row were captured in the first session. The three images of the second row were captured in the second session.
Figure 25. Six palmprint ROI images of TJU-P database. The three images of the first row were captured in the first session. The three images of the second row were captured in the second session.
Figure 26. Three original palmprint and ROI images of HFUT CS database. The three images of the first row are three original palmprint images. The three images of the second row are corresponding ROI images.
Figure 27. Three original 3D palmprint ROI images of PolyU 3D database.
Figure 28. Four different 2D representations of 3D palmprint ROI images of PolyU 3D database including MCI, GCI, ST, and CST.
Figure 29. Six palm vein ROI images of PolyU M_N database. Three images of the first row were captured in the first session. Three images of the second row were captured in the second session.
Figure 30. Six palm vein ROI images of TJU-PV database. Three images of the first row were captured in the first session. Three images of the second row were captured in the second session.
Table 1. Summary of existing 2D palmprint recognition methods based on deep learning
Reference Publish year Networks Database Training data configuration Performance
Recognition rate EER
Jalali et al.[74] 2015 4-layer CNN PolyU H Mixed data mode 99.98% N/A
Zhao et al.[75] 2015 DBN BJJU Mixed data mode 90.63% N/A
Minaee and Wang[76] 2016 DSCNN PolyU M Mixed data mode 100% N/A
Liu and Sun[77] 2017 AlexNet PolyU II Not provided N/A 0.04%
CASIA N/A 0.08%
IITD N/A 0.11%
Svoboda et al.[78] 2016 4-layer CNN IITD Mixed data mode N/A 1.64%
Yang et al.[79] 2018 VGG-F PolyU II PolyU M_B Mixed data mode N/A 0.1665%
Meraoumia et al.[80] 2017 PCANet CASIA Mixed data mode N/A 0.299%
Zhang et al.[17] 2018 PalmRCNN TJU-P Mixed data mode 100% 2.74%
Zhong et al.[81] 2018 Siamese network PolyU M Not provided N/A 0.2819%
XJTU N/A 4.559%
Michele et al.[82] 2019 MobileNet_V2 PolyU_M_B Mixed data mode 100% N/A
Genovese et al.[83] 2019 PalmNet CASIA Mixed data mode 99.77% 0.72%
IIDT 99.37% 0.52%
REST 97.16% 4.50%
TJU-P 99.83% 0.16%
Zhong et al.[81] 2018 CNN with C-LMCL TJU-P Mixed data mode 99.93% 0.26%
PolyU 100% 0.125%
Matkowski et al.[85] 2020 EE-PRnet IIDT Mixed data mode 99.61% 0.26%
PolyU-CF 99.77% 0.15%
CASIA 97.65% 0.73%
Zhao and Zhang[86] 2020 DDR based on VGG-F PolyU R Mixed data mode 100% 0.0004%
PolyU G 100% 0.0001%
PolyU B 100% 0.0004%
PolyU NIR 100% 0.0036%
IIDT 98.70% 0.0038%
CASIA 99.41% 0.0052%
Zhao and Zhang[87] 2020 JCLSR PolyU R Mixed data mode 100% N/A
PolyU G 99.99% N/A
PolyU B 100% N/A
PolyU NIR 100% N/A
IIDT 98.17% N/A
CASIA 98.94% N/A
GPDS 96.33% N/A
下载: 导出CSV
Table 2. Summary of 3D palmprint recognition methods based on deep learning
Reference Publish year Methodology Database Training data configuration Performance
Samai et al.[89] 2018 MCI+DCTNet PolyU 3D Mixed data mode 99.83% N/A
GCI+DCTNet 99.22% N/A
Chaa et al.[90] 2019 GCI+PCANet PolyU 3D Mixed data mode 98.63% 0.12%
MCI+PCANet 98.22% 0.12%
ST+PCANet 99.88% 0.02%
SSR+PCANet 99.98% 0
Table 3. Summary of palm vein recognitoin methods based on deep learning
Hassan and Abdulrazzag[91] 2018 A simple CNN PolyU M_NIR Mixed data mode 99.73% N/A
CASIA 98% N/A
Zhang et al.[17] 2018 PalmRCNN TJU-PV Mixed data mode 100% 2.30%
Lefkovits et al.[92] 2019 AlexNet PUT Mixed data mode 92.16% N/A
VGG-16 97.33% N/A
ResNet-50 99.83% N/A
SqueezeNet 91.66% N/A
Thapar et al.[93] 2019 PVSNet CASIA The first half samples are used
as the training set 85.16% 3.71
IITI 97.47% 0.93
PolyU MS 98.78% 0.66
Chantaf et al.[94] 2020 Inception_v3 20 subjects 4000 images Mixed data mode 91.4% N/A
SmallerVggNet 93.2% N/A
Table 4. Details of 2D palmprint, 3D palmprint and palm vein databases
Database Type Touch? Individual
number Palm
number Session
interval Image number
of each palm Total image
PolyU II 2D Palmprint Yes 193 386 2 2 months 10×2 7752
PolyU M_B 2D Palmprint Yes 250 500 2 9 days 6×2 6000
HFUT 2D Palmprint Yes 400 800 2 10 days 10×2 16000
HFUT CS 2D Palmprint No 100 200 2 10 days 10×2×3 12000
TJU-P 2D Palmprint No 300 600 2 61 days 10×2 12000
PolyU 3D 3D Palmprint Yes 200 400 2 1 months 10×2 8000
PolyU M_N Palm vein Yes 250 500 2 9 days 6×2 6000
TJU-PV Palm vein No 300 600 2 61 days 10×2 12000
Table 5. Full name and its abbreviated name of selected CNNs
Full name Abb.* Reference Year
AlexNet Alex Krizhevsk et al.[50] 2012
VGG VGG Simonyan and Zisserman[53] 2015
Inception_v3 IV3 Szegedy et al.[56] 2016
ResNet Res He et al.[58] 2016
Inception_ResNet_v2 IResV2 Szegedy et al.[57] 2017
DenseNet Dense Huang et al.[59] 2017
Xception Xec Chollet[66] 2017
ResNeXt ResX Xie et al.[68] 2017
MobileNet_v2 MbV2 Howard et al.[62] 2018
ShuffleNet_v2 ShuffleV2 Ma et al.[65] 2018
SENet SE Hu et al.[69] 2018
EfficientNet Efficient Tan et al.[70] 2019
GhostNet Ghost Han et al.[71] 2020
RegNet Reg Radosavovic et al.[72] 2020
ResNeSt ResS Zhang et al.[73] 2020
*Abb. means the abbreviated name
Table 6. Recognition rates of ResNet18 under different learning rates
Learning rate 5 × 10−3 10−3 5 × 10−4 10−4 5 × 10−5 10−5
PolyU II 66.16% 88.39% 88.64% 96.99% 97.66% 96.40%
PolyU M_B 82.20% 93.33% 96.97% 99.97% 100% 100%
HFUT 54.61% 78.45% 89.55% 97.67% 98.51% 98.42%
HFUT CS 42.96% 56.38% 69.79% 92.85% 95.37% 93.73%
TJU 57.67% 82.75% 88.18% 98.38% 99.25% 99.18%
PolyU 3D CST 75.38% 87.44% 93.26% 96.77% 97.58% 93.21%
PolyU 3D ST 81.47% 88.35% 94.17% 97.90% 99.12% 95.67%
PolyU 3D MCI 82.27% 86.65% 93.46% 98.29% 99.35% 97.66%
PolyU 3D GCI 73.72% 80.20% 83.54% 89.47% 93.65% 90.39%
PolyU M_N 79.20% 95.07% 93.57% 99.90% 100% 99.97%
TJUV 63.83% 82.95% 88.27% 97.85% 98.58% 97.88%
Table 7. Recognition rates of EfficientNet under different learning rates
PolyU M_B 84.50% 98.53% 99.47% 99.78% 100% 97.23%
HFUT I 71.79% 89.19% 93.70% 97.44% 99.41% 91.85%
Table 8. Recognition rates of VGG and ResNet under different numbers of layers
VGG-16 VGG-19 Res-18 Res-34 Res-50
PolyU II 96.79% 97.43% 97.66% 96.25% 93.68%
PolyU M_B 99.47% 99.33% 100% 99.93% 99.53%
HFUT 96.04% 96.25% 98.51% 98.14% 93.79%
HFUT CS 86.55% 82.13% 95.37% 91.04% 85.21%
TJU-P 93.92% 91.28% 99.25% 98.67% 95.33%
PolyU 3D CST 94.80% 92.10% 97.58% 96.55% 95.70%
PolyU 3D ST 95.90% 94.33% 99.12% 98.75% 96.30%
PolyU 3D MCI 94.40% 94.53% 99.35% 99.15% 97.40%
PolyU 3D GCI 90.18% 90.80% 93.65% 93.30% 90.65%
PolyU M_N 97.37% 96.10% 100% 99.93% 99.57%
TJU-PV 92.33% 90.60% 98.58% 98.03% 95.63%
Table 9. Recognition rates of EfficientNet from b0 to b7
b0 b1 b2 b3 b4 b5 b6 b7
PolyU II 93.42% 93.47% 93.78% 95.38% 95.86% 96.35% 96.78% 97.39%
PolyU M_B 99.97% 100% 100% 100% 100% 100% 100% 100%
HFUT 97.98% 98.14% 98.32% 98.40% 98.75% 99.10% 99.18% 99.41%
HFUT CS 81.99% 84.85% 93.45% 93.60% 94.56% 95.79% 96.06% 96.55%
TJU-P 99.57% 99.58% 98.68% 99.63% 99.78% 99.83% 99.87% 99.89%
PolyU 3D CST 95.93% 96.11% 96.34% 96.88% 97.35% 97.55% 97.66% 97.81%
PolyU 3D ST 98.62% 98.77% 98.83% 98.90% 99.03% 99.15% 99.26% 99.37%
PolyU 3D MCI 99.55% 99.58% 99.63% 99.67% 99.75% 99.80% 99.82% 99.88%
PolyU 3D GCI 93.62% 93.77% 93.89% 94.29% 94.50% 94.89% 95.17% 95.66%
PolyU M_N 99.27% 99.33% 99.63% 99.64% 99.67% 100% 100% 100%
TJU-PV 95.88% 97.00% 96.97% 97.13% 97.85% 98.13% 98.77% 99.00%
Table 10. Recognition results of different CNNs on 2D palmprint and palm vein databases under the separate data mode
PolyU II PolyU M_B HFUT HFUT CS TJU-P PolyU M_N TJU-PV
Alex U/81.81% 92.63%/94.36% 78.33%/86.17% 42.53%/46.49% 80.35%/81.85% 87.07%/88.47% 76.08%/74.87%
VGG-16 U/96.79% U/99.47% U/96.04% 73.86%/86.55% 78.38%/93.92% U/97.37% 84.22%/92.33%
IV3 94.66% 99.23% 97.74% 85.65% 98.08% 99.50% 97.02%
Res-18 97.66% 100% 98.51% 95.37% 99.25% 100% 98.58%
IV4 95.22% 99.03% 97.72% 84.78 % 96.78% 99.63% 97.27%
IRes2 95.07% 99.73% 96.45% 74.26% 98.50% 99.37% 97.00%
Dense-121 96.53% 100% 98.05% 94.47% 99.38% 100% 98.55%
Xec 94.94% 97.83% 94.45% 74.94% 94.20% 98.90% 94.79%
MbV2 96.99% 99.97% 98.08% 93.97% 99.27% 100% 97.77%
MbV3 97.35% 100% 98.67% 95.20% 99.37% 100% 98.67%
ShuffleV2 95.41% 99.16% 97.63% 92.35% 98.38% 99.76% 97.08%
SE-154 94.07% 98.10% 95.13% 85.15% 96.77% 98.20% 96.70%
ResX-101 93.98% 98.67% 94.56% 90.26% 98.23% 97.34% 97.17%
Efficient 97.39% 100% 99.41% 96.55% 99.89% 100% 99.00%
Ghost 94.74% 99.90% 97.01% 83.70% 98.90% 99.60% 96.20%
Reg-Y 79.32% 93.67% 78.91% 84.36% 79.50% 90.03% 87.32%
ResS-50 93.55% 99.10% 92.16% 99.15% 96.48% 98.57% 94.92%
Table 11. Recognition results of different CNNs on four 2D representations of 3D palmprint databases under the separate data mode
Recognition CST ST MCI GCI
Alex 88.18% 91.27% 88.45% 83.40%
VGG-16 U/94.80% U/95.90% U/94.40% U/90.18%
IV3 96.05% 98.55% 99.20% 98.55%
Res-18 97.58% 99.12% 99.35% 93.65%
IRes2 97.12% 99.00% 98.70% 92.15%
Dense-121 97.60% 98.17% 98.88% 94.97%
Xec 95.47% 96.20% 97.10% 91.27%
MbV2 96.83% 98.32% 93.17% 93.50%
ShuffleV2 95.63% 98.10% 98.55% 92.35%
SE-154 96.65% 98.02% 97.80% 90.58%
ResX-101 96.25% 98.32% 98.72% 92.40%
Efficient 97.81% 99.37% 99.88% 99.66%
Ghost 95.75% 98.20% 98.30% 91.97%
Reg-Y 95.62% 93.73% 95.30% 87.28%
ResS-50 95.08% 98.22% 97.90% 92.83%
Table 12. Recognition rates on different mixed-training data amounts of EfficientNet
The number of training images 1+1 2+1 3+1 4+1 5+1 6+1 7+1 8+1 9+1 10+1
PolyU II 98.27% 99.10% 99.67% 99.95% 100% 100% 100% 100% 100% 100%
PolyU M_B 99.85% 100% 100% 100% 100% 100% N/A N/A N/A N/A
HFUT I 98.08% 98.48% 99.47% 99.93% 100% 100% 100% 100% 100% 100%
HFUT CS 83.26% 87.39% 88.72% 90.61% 92.14% 94.28% 95.06% 96.79% 97.33% 99.57%
TJU 96.04% 98.81% 99.75% 99.98% 100% 100% 100% 100% 100% 100%
PolyU 3D CST 92.38% 92.68% 93.04% 93.62% 94.50% 95.04% 95.97% 96.48% 97.50% 98.54%
PolyU 3D ST 92.17% 92.90% 93.57% 94.12% 94.99% 95.77% 96.36% 97.58% 98.79% 99.88%
PolyU 3D MCI 91.89% 93.33% 94.17% 95.28% 96.00% 96.85% 97.88% 98.49% 99.29% 99.94%
PolyU 3D GCI 90.44% 91.34% 92.13% 93.37% 94.55% 95.87% 96.49% 96.90% 97.22% 97.43%
PolyU M_N 99.00% 99.76% 100% 100% 100% 100% N/A N/A N/A N/A
TJUV 93.44% 94.29% 98.25% 99.66% 99.97% 100% 100% 100% 100% 100%
Table 13. Recognition results of different CNNs on 2D palmprint and palm vein databases under the mixed data mode
Alex 97.58% 98.92% −/98.33% 94.08% 94.67% 99.20% 94.57%
VGG-16 U/99.83% U/100% U/99.62% 96.55% U/99.23% U/100% U/98.76%
IV3 98.49% 100% 99.85% 99.08% 99.67% 99.95% 98.85%
Res-18 100% 100% 100% 99.84% 100% 100% 99.92%
IRes2 99.00% 100% 99.89% 98.77% 99.79% 100% 99.07%
Dense-121 99.86% 100% 100% 99.58% 99.88% 100% 99.74%
MbV2 99.40% 100% 99.94% 99.36% 100% 100% 99.96%
MbV3 99.69% 100% 100% 99.49% 100% 100% 100%
ShuffleV2 99.96% 100% 100% 99.09% 100% 100% 100%
SE-154 99.46% 100% 99.64% 98.85% 99.73% 100% 99.46%
ResX-101 98.58% 99.93% 100% 97.82% 99.08% 99.87% 99.35%
Efficient 100% 100% 100% 99.57% 100% 100% 100%
Table 14. Recognition results of different CNNs on four 2D representations of 3D palmprint databases under the mixed data mode
IV3 98.61% 99.83% 100% 98.06%
Res-18 99.50% 99.94% 100% 98.56%
Table 15. 2D palmprint and palm vein recognition: Performance comparison between classic CNNs and other methods under the separate data mode
Competitive code Ordinal code RLOC LLDP PalmNet MbV3 EfficientNet
PolyU II 100% 100% 100% 100% 100% 97.35% 97.39%
PolyU M_B 100% 100% 100% 100% 100% 100% 100%
HFUT 99.64% 99.60% 99.75% 99.89% 100% 98.67% 99.41%
HFUT CS 99.45% 99.67% 99.36% 99.40% 92.45% 95.20% 96.55%
TJU-P 99.87% 99.95% 99.63% 99.50% 100% 99.37% 99.89%
PolyU M_N 99.97% 100% 100% 100% 99.02% 100% 100%
TJU-PV 99.32% 99.55% 100% 98.93% 99.61% 98.67% 99.00%
Table 16. 2D palmprint and palm vein recognition: Performance comparison between classic CNNs and other methods under the mixed data mode
PolyU II 100% 100% 100% 100% 100% 99.40% 100%
HFUT I 99.98% 99.98% 100% 99.93% 100% 100% 100%
HFUT CS 99.96% 100% 100% 100% 100% 99.49% 99.57%
TJU 100% 100% 100% 100% 100% 100% 100%
PolyU M_N 100% 100% 100% 100% 100% 100% 100%
TJUV 99.87% 99.87% 100% 99.96% 99.91% 100% 100%
Table 17. 3D palmprint recognition: Performance comparison between classic CNNs and other methods under the separate data mode
Reference 2D representation Recognition method Recognition rate
[16] MCI Competitive code 99.24%
[5] ST Block-wise features and collaborative representation 99.15%
[38] MCICST Binary representations
of orientation and compact ST 99.67%
This paper MCI EfficientNet 99.88%
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录用日期: 2020-09-25
Wei Jia 1, 2, ,
Jian Gao 1, 2, ,
Wei Xia 1, 2, ,
Yang Zhao 1, 2, ,
Hai Min 1, 2, ,
Jing-Ting Lu 3,
1. School of Computer Science and Information Engineering, Hefei University of Technology, Hefei 230009, China
2. Anhui Province Key Laboratory of Industry Safety and Emergency Technology, Hefei 230009, China
3. Institution of Industry and Equipment Technology, Hefei University of Technology, Hefei 230009, China
In the network and digital society, personal authentication is becoming a basic social service. It is well known that biometrics technology is one of the most effective solutions to personal authentication[1]. In recent years, two emerging biometrics technologies, palmprint recognition and palm vein recognition have attracted a wide range of attention[2-6]. Generally speaking, there are three subtypes of palmprint recognition technology, including 2D low resolution palmprint recognition, 3D palmprint recognition and high resolution palmprint recognition. High-resolution palmprint recognition is usually used for forensic applications. 2D low-resolution palmprint recognition and 3D palmprint recognition are mainly used for civil applications. In this paper, we only focus on civil applications of biometrics, therefore, the problem of high-resolution palmprint recognition will not be investigated.
Many effective methods have been proposed for 2D low-resolution palmprint recognition (2D low-resolution palmprint recognition will be called 2D palmprint recognition for short in the rest of this paper), 3D palmprint recognition and palm vein recognition, which can be divided into two groups, i.e., traditional methods and deep learning-based methods.
In the past decade, deep learning has become the most important technology in the field of artificial intelligence. It has brought a breakthrough in performance for many applications[7, 8], such as speech recognition, natural language processing, computer vision, image and video analysis, and multimedia. In the field of biometrics, especially in face recognition, deep learning has become the most mainstream technology[9]. However, the research on deep learning-based 2D and 3D palmprint recognition and palm vein recognition is still very preliminary[9, 10].
Convolution neural network (CNN) is one of the most important branches of deep learning technology, and has been widely used in various tasks of image processing and computer vision, such as target detection, semantic segmentation and pattern recognition. For image-based biometrics technologies, CNN is the most commonly used deep learning technique. Up to now, many classic CNNs have been proposed and impressive results have been achieved in many recognition tasks. However, the recognition performance of these classic CNNs for 2D and 3D palmprint recognition and palm vein recognition has not been systematically studied. For example, existing deep learning-based palmprint recognition and palm vein recognition work only used simple networks, and did not provide an in-depth analysis. In the future, with the rapid development of CNNs, the recognition accuracy of new CNNs will be continuously improved. It can be predicted that CNNs will become one of the most important techniques for 2D and 3D palmprint recognition and palm vein recognition. Therefore, it is very important to systematically investigate the recognition performance of classic CNNs for 2D and 3D palmprint recognition and palm vein recognition. To this end, this paper evaluates the performance of classic CNNs in 2D and 3D palmprint recognition and palm vein recognition. Particularly, seventeen representative and classic CNNs are exploited for performance evaluation.
The selected CNNs are evaluated on five 2D palmprint databases, one 3D palmprint database and two palm vein databases, all of which are representative databases in the field of 2D and 3D palmprint recognition and palm vein recognition. The five 2D palmprint databases include Hong Kong Polytechnic University palmprint II database (PolyU II)[11], the blue band of the Hong Kong Polytechnic University multispectral (PolyU M_B) palmprint database[12], Hefei University of Technology (HFUT) palmprint database[13], Hefei University of Technology cross sensor (HFUT CS) palmprint database[14], and Tongji University palmprint (TJU-P) database[15]. The 3D palmprint database we used is Hong Kong Polytechnic University 3D palmprint database (PolyU 3D)[16]. Two palm vein databases include the near-infrared band of Hong Kong Polytechnic University multispectral palmprint database (PolyU M_N)[12] and Tongji University palm vein (TJU-PV) database[17].
It should be noted that the samples within the above databases are captured in two different sessions at certain time intervals. In traditional recognition methods, some samples captured in the first session are usually used as training sets, while all the samples captured in the second session are used as the test set. However, in existing deep learning-based palmprint recognition and palm vein recognition methods, the training set often contains samples from both sessions. Thus, it is easy to obtain a high recognition accuracy. If the training samples are only from the first session, and the test samples are from the second session, we call this experimental mode a separate data mode. If the training samples are from both sessions, we call this experimental mode a mixed data mode. We conduct experiments in both separate data mode and mixed data mode to observe the recognition performance of classic CNNs in these two different modes.
The main contributions of our work are as follows.
1) We briefly summarize the classic CNNs, which can help the readers to better understand the development history of CNNs for image classification tasks.
2) We evaluate the performance of the classic CNNs for 3D palmprint and palmprint recognition. To the best of our knowledge, it is the first time such an evaluation has been conducted.
3) We evaluated the performance of classic CNNs on Hefei University of Technology cross sensor palmprint database. To the best of our knowledge, it is the first time the problem of palmprint recognition across different devices using deep learning technology has been investigated.
4) We investigate the problem of the recognition performance of CNNs on both separate data mode and mixed data mode.
The rest of this paper is organized as follows. Section 2 presents the related work. Section 3 briefly introduces seventeen classic CNNs. Section 4 introduces the 2D and 3D palmprint and palm vein databases used for evaluation. Extensive experiments are conducted and reported in Section 5. Section 6 offers the concluding remarks.
2. Related work
2.1. Traditional 2D palmprint recognition methods
For 2D palmprint recognition, researchers have proposed many traditional methods, which can be divided into different sub-categories, such as palm line-based, texture-based, orientation coding-based, correlation filter-based, and subspace learning-based[3].
Because palm line is the basic feature of palmprint, some methods exploiting palm line features for recognition have been proposed. Huang et al.[18] proposed the modified finite Radon transform (MFRAT) to extract principal lines, and designed a pixel-to-area algorithm to match the principal lines of two palmprints. Palma et al.[19] used a morphological top-hat filtering algorithm to extract principal lines, and proposed a dynamic matching algorithm involving a positive linear dynamical system.
The texture-based method is also very effective for pattern recognition. Some local texture descriptors were designed and used for palmprint recognition[20]. Replacing the gradient by the response of Gabor filters in the local descriptor of histogram of oriented gradients (HOG), Jia et al.[21] proposed the descriptor of histogram of oriented lines (HOL) for palmprint recognition. Later, Luo et al.[22] proposed the descriptor of local line directional pattern (LLDP) using the modulation of two orientations. Motivated by LLDP, Li and Kim[23] proposed the descriptor called the local micro-structure tetra pattern (LMTrP). To fully utilize different direction information of a pixel and explore the most discriminant direction representation, Fei et al.[24] proposed the methods of the local discriminant direction binary pattern (LDDBP), the discriminant direction binary palmprint descriptor (DDBPD)[25], and the apparent and latent direction code (ALDC)-based descriptor[26]. Scale-invariant feature transform (SIFT) is a powerful descriptor and has been applied to palmprint recognition. Using SIFT, Wu and Zhao[27] tried to solve the problem of deformed palmprint matching.
Orientation is a robust feature of palmprint. A lot of orientation coding-based methods have been proposed. These methods have high accuracy and fast matching speed. Generally, orientation coding-based methods first detect the orientation of each pixel, and then encode the orientation number to a bit string, at last, exploit Hamming distance for matching. Jia et al.[13] summarized orientation coding-based methods. Typical orientation coding-based methods include competitive code[11], ordinal code[28], robust line orientation code (RLOC)[29], binary orientation co-occurrence vector (BOCV)[30], double-orientation code (DOC)[31], etc.
Recently, correlation-based methods have been successfully used for biometrics. Jia et al.[13] proposed to use a band-limited phase-only correlation (BLPOC) filter for palmprint recognition.
Subspace learning has been one of the important techniques for pattern recognition. Some subspace learning-based methods have been used for palmprint recognition, including principal component analysis (PCA)[32], linear discriminant analysis (LDA)[33], kernel PCA (KPCA)[34], etc. However, the recognition performance of subspace learning-based methods is sensitive to illumination changes and other image variations.
For 3D palmprint recognition, researchers have proposed a lot of traditional methods[5, 10, 16, 35]. Generally, 3D palmprint data preserves the depth information of a palm surface. At the same time, the original captured 3D palmprint data is a small positive or negative float, which is usually transformed into the grey-level value for practical feature extraction. In previous researches, the original 3D palmprint data is usually transformed into a curvature-based data. Two most important curvatures include the mean curvature (MC) and Gaussian curvature (GC). Their corresponding gray images are called as mean curvature image (MCI) and Gaussian curvature image (GCI)[35]. In the recognition process, researchers extracted features from MCI or GCI for 3D palmprint recognition. Besides GC and MC, researchers also tried to propose other 2D representations of 3D palmprints. Based on GC and MC, Yang et al.[36] proposed a new grey-level image representation, called surface index image (SI). Recently, Fei et al.[37] proposed a simple yet effective compact surface type (CST) to represent surface features of a 3D palmprint. Since the representations of MCI, GCI, SI and CST depict a 3D palmprint as a 2D grey-level palmprint image, those 2D palmprint recognition methods can be used for 3D palmprint recognition. Li et al.[16] extracted competitive code from MCI for 3D palmprint recognition, which is an important orientation coding method. Zhang et al.[5] proposed a blockwise statistics-based ST vector for 3D palmprint feature representation, and used collaborative representation-based classification (CRC) as the classifier. Fei et al.[38] proposed a complete binary representation (CBR) for the 3-D palmprint recognition by combining descriptors extracted from both MCI and CST. Fei et al.[39] proposed the precision direction code (PDC) to depict the 2D texture-based features, and then combined CST to form the PDCST descriptor to represent the multiple level and multiple dimensional features of 3D palmprint images.
2.3. Traditional palm vein recognition methods
For palm vein recognition, traditional methods can also be divided into the following categories: vein line-based, texture-based, orientation coding-based, and subspace learning-based.
To extract palm vein lines, Zhang et al.[40], Kang and Wu[6] proposed two typical methods. In Zhang′s method, the multiscale Gaussian matched filters were exploited to extract vein lines[40]. In Kang′s method, the normalized gradient-based maximal principal curvature (MPC) algorithm was exploited to extract vein lines[6].
Kang and Wu[6] proposed a texture-based method, in which a mutual foreground-based linear binary pattern (LBP) was exploited for texture feature extraction. Mirmohamadsadeghi and Drygajlo[41] also proposed a texture-based method, in which two texture descriptors, LBP and local derivative patterns (LDP) were used for palm vein recognition. ManMohan et al.[42] proposed a palm vein recognition method using local tetra patterns (LTP). Kang et al.[43] investigated the SIFT-based method for palm vein recognition.
Zhou and Kumar[44] presented an orientation coding-based method for palm vein recognition, named neighborhood matching Radon transform (NMRT) which is similar to the RLOC method proposed for palmprint recognition. The experimental results showed that the recognition performance of NMRT is much better than other methods such as Hessian phase, ordinal code, competitive code, and SIFT.
For subspace learning-based methods, the LDA[45], 2DLDA[46], (2D)2LDA[47], and sparse representation methods[48] have been studied for palm vein recognition.
2.4. The brief development history of classic CNNs
Fig. 1 shows the chronology of the events in the development history of classic CNNs for image classification tasks. In 1998, the first CNN, LeNet, was proposed by Lecun et al.[49] However, LeNet did not have a widespread impact due to various restrictions. In 2012, AlexNet was proposed by Hinton and his student Krizhevsky and won the ImageNet Large Scale Visual Recognition Challenge 2012 (ILSVRC 2012)[50]. AlexNet demonstrated the effectiveness of CNN in some complex tasks. As a result, excellent performance of AlexNet attracted the attention of researchers, and promoted the further development of CNNs. In 2013, ZFNet was proposed by Zeiler and Fergus[51]. Zeiler and Fergus[51] also explain the essence of each layer of the neural network through visualization technology. In 2013, network in network (NIN) was proposed, which has two important contributions including global average pooling and the use of 1×1 convolution layer[52]. In 2014, VGG was proposed by the Oxford Visual Geometry Group[53], and was the 2nd runner-up in ILSVRC 2014. Compared with AlexNet, VGG has two important improvements. The first one is using a smaller kernel size. The second one is using a deeper network. In 2014, another most important CNN is GoogLeNet (Inception_v1)[54], which was the champion of the ILSVRC 2014. Later, the subsequent versions of GoogLeNet, i.e., Inception_v2[55], Inception_v3[56] and Inception_v4[57], were successively proposed in 2016 and 2017. Inception_ResNet_v1 and Inception_ResNet_v2 were proposed in the same paper, which are improved versions of Inception_v4[57]. In 2015, ResNet was proposed by He et al.[58] and won the ILSVRC 2015. The paper of ResNet obtained the best paper of CVPR 2016. It can be said that the emergence of ResNet is an important event in the history of deep learning, because ResNet made it possible to train hundreds of layers of neural networks, and ResNet greatly improved the performance of image classification and other computer vision tasks. In 2016, DenseNet was proposed by Huang et al.[59] As the best paper of CVPR 2017, DenseNet broke away from the stereotyped thinking of deepening network structure (ResNet) and broadening network structure (Inception) to improve network performance. Considered from the point of view of features, DenseNet not only greatly reduces the amount of network parameters, but also alleviates the gradient vanishing problem to a certain extent by feature reuse and bypass settings. In the same year, SqueezeNet, the first lightweight network, was proposed by Iandoula et al.[60] to compress the number of feature maps by using a 1×1 convolution core to accelerate the network. Since then, other important lightweight networks such as MobileNets[61-63], ShuffleNets[64, 65], Xception[66], SqueezeNeXt[67] were proposed in turn. In order to solve the problem of poor information circulation, MoblieNets uses the strategy of point-wise convolution, ShuffleNets uses the strategy of channel shuffle, Xception uses the strategy of modified depth-wise convolution, and SqueezeNeXt hoists the speed from the perspective of hardware. In 2017, Xie et al.[68] proposed ResNeXt combining ResNet and Inception, which does not need to design complex structural details manually. Particularly, ResNeXt uses the same topological structure for each branch, the essence of which is group convolution. In 2018, the winner of the last image classification mission, SENet was proposed by Hu et al.[69] SENet consists of squeeze and excitation, in which the former compresses the model, and the latter predicts the importance of each channel. In addition, SENet can be plugged into any network to improve the recognition performance. In 2019, EfficientNet was proposed by Google[70], which relies on AutoML and compound scaling to achieve state-of-the-art accuracy without compromising resource efficiency. In 2020, the team of Huawei Noah′s Ark Lab proposed a lightweight network, i.e., GhostNet, which can achieve better recognition performance than MobileNet_v3 with similar computational cost[71]. Some members from Facebook AI Research (FAIR) developed RegNet that outperforms EfficientNet while being up to 5× faster on GPUs[72]. The work of RegNet presented a new network design paradigm, which combines the advantages of manual network design and neural network search (NAS). By stacking split-attention blocks, Zhang et al.[73] proposed a new ResNet variant, i.e., ResNeSt, which has better recognition performance than ResNet.
2.5. 2D and 3D palmprint recognition and palm vein recognition methods based on deep learning
A lot of researchers have studied 2D and 3D palmprint recognition and palm vein recognition based on deep learning. Table 1 summarizes the existing CNN-based 2D palmprint recognition methods including the networks, training data configuration, and performance. ERR represents equal error rate. Jalali et al.[74] used the whole palmprint image without region of interest (ROI) extraction to train a four-layer CNN for 2D palmprint recognition. Zhao et al.[75] proposed a 2D palmprint recognition method by using a deep belief network (DBN). Minaee and Wang[76] proposed a 2D palmprint recognition method based on a deep scattering convolutional network (DSCNN). Liu and Sun[77] used AlexNet to extract the features of palmprint images and combined hausdorff distance for matching and recognition. Svoboda et al.[78] trained CNN with palmprint ROI and d-prime loss function, and observed that d-prime loss function has better effect than contrastive loss function. In addition, Yang et al.[79] combined the methods of deep learning and local coding. They first extracted the features of palmprint with CNN, and then used local coding to encode the extracted features. Meraoumia et al.[80] applied PCANet for palmprint recognition, which is an unsupervised convolutional deep learning network. Zhang et al.[17] proposed the method of PalmRCNN for palmprint and palm vein recognition, which is a modified version of Inception_ResNet_v1. Zhong et al.[81] applied a Siamese network for 2D palmprint recognition. Michele et al.[82] used MobileNet_v2 to extract palmprint features and then explored support vector machine (SVM) for classification. Genovese et al.[83] proposed the method of PalmNet, which is a CNN that uses a method to tune palmprint specific filters through an unsupervised procedure based on Gabor responses and principal component analysis (PCA). Zhong and Zhu[84] proposed an end-to-end method for open-set 2D palmprint recognition by applying CNN with a novel loss function, i.e., centralized large margin cosine loss (C-LMCL). In order to solve the problem of palmprint recognition in uncontrolled and uncooperative environments, Matkowski et al.[85] proposed end-to-end palmprint recognition network (EE-PRnet) consisting of two main networks, i.e., ROI localization and alignment network (ROI-LAnet) and feature extraction and recognition network (FERnet). Zhao and Zhang[86] proposed a deep discriminative representation (DDR) for palmprint recognition. DDR uses several CNNs similar to VGG-F to extract deep features from global and local palmprint images. Lastly, the collaborative representation-based classifier (CRC) is used for recognition. Zhao and Zhang[87] presented a joint constrained least-square regression (JCLSR) model with a deep local convolution feature for palmprint recognition. Zhao et al.[88] also proposed a joint deep convolutional feature representation (JDCFR) methodology for hyperspectral palmprint recognition.
Table 2 summarizes the existing CNN-based 3D palmprint recognition methods. Generally, in these methods, the CNNs were applied to different 2D representations of 3D palmprints for recognition such as MCI, GCI, and ST. Samai et al.[89] proposed to use DCTNet for 3D palmprint recognition. Chaa et al.[90] firstly used a single scale retinex (SSR) algorithm to enhance the depth image of 3D palmprint, then used PCANet for recognition.
Table 3 summarizes the existing CNN-based palm vein recognition methods. Hassan and Abdulrazzaq[91] proposed to use CNN for palm vein recognition, in which they designed a simple CNN and used the strategy of data augmentation to obtain more training data. Zhang et al.[17] released a new touchless palm vein database, and used the method of PalmRCNN for palm vein recognition. Lefkovits et al.[92] applied four CNNs for palm vein identification including AlexNet, VGG-16, ResNet-50, and SqueezeNet. Thapar et al.[93] proposed the method of PVSNet. In PVSNet, using triplet loss, a Siamese network was trained. Chantaf et al.[94] applied Inception_v3 and SmallerVggNet for palm vein recognition.
3. Selected classic CNNs for performance evaluation
For 2D and 3D palmprint recognition and palm vein recognition, we select seventeen classic CNNs for performance evaluation including AlexNet[50], VGG[53], Inception_v3[56], Inception_v4[57], ResNet[58], ResNeXt[68], Inception_ResNet_v2[57], DenseNet[59], Xception[66], MobileNet_v2[62], MobliNet_v3[63], ShunffleNet_v2[65], SENet[69], EfficientNet[70], GhostNet[71], RegNet[72] and ResNeSt[73]. The reasons for choosing these CNNs are as follows: AlexNet and VGG are representatives of early CNNs, furthermore, we hope to compare the performance between early CNNs and recent CNNs; Inception_v3 and Inception_v4 are representatives of GoogLeNet, and Inception_v3 is an improvement of Inception_v1 and Inception_v2, meanwhile, Inception_v4 is a new version of Inception_v3 with a more uniform architecture; ResNet is a very well-known CNN, and can deepen CNN to more than 100 layers, in addition, it can be well trained; Inception_ResNet_v1 and Inception_ResNet_v2 share the overall structure, but Inception_ResNet_v2 is more representative than Inception_ResNet_v1; DenseNet is the extreme version of ResNet; Xception is a new attempt for convolution order; MobileNet_v3 can be used in embedded devices, and is an improved version of MobileNet_v1 and MobileNet_v2; ShuffleNet_v2 is a good compression network, and is a modified version of ShuffleNet_v1; SENet enhances important features to improve accuracy; EfficientNet, GhostNet, RegNet and ResNeSt are four representative CNNs proposed recently.
In this section, we briefly introduce the selected CNNs as follows.
1) AlexNet
The network structure of AlexNet is shown in Fig. 2. AlexNet is based on LeNet and uses some new techniques such as rectified linear unit (ReLU), dropout and local response normalization (LRN) for the first time[50]. Due to the limitation of hardware capability, the training of AlexNet uses distributed computing technology to distribute network on two GPUs. Each GPU stored half of the parameters. The GPUs can communicate with each other and access memory. Therefore, AlexNet is divided into upper and lower parts, each part corresponding to a single GPU. In AlexNet, data enhancement technology is used, such as random cropping and horizontal flipping of raw data, to improve the generalization of the network while reducing over-fitting problem.
2) VGG
VGG is a further improvement of AlexNet, which makes the network deeper[53]. The VGG′s structure is shown in Fig. 3. Because the size of all convolutional kernel is $ 3\times 3 $ , the structure of VGG is neat and its topology is simple. Small convolution kernel size also brings some benefits, such as increasing the number of layers. VGG expands the number of layers of CNN to more than 10, enhancing the expressive ability of the network and facilitating subsequent modification in network structure.
3) Inception_v3
Based on Inception_v2, Inception_v3 further decomposed the convolution[56]. That is, any $ n\times n $ convolution can be replaced by a $ 1\times n $ convolution followed by a $ n\times 1 $ convolution (see Fig. 4(a)), which can reduce a lot of parameters, avoid over-fitting problems, and strengthen the nonlinear expression ability. In addition, Szegedy et al.[56] have carefully designed three types of Inception module, as shown in Fig. 4(b).
4) ResNet
As the depth of the network continues to increase, the vanishing gradient and exploding gradient problems became more and more difficult to solve. In this situation, it is hard to train the deep network. But ResNet can overcome this difficulty[58]. ResNet relies on a shortcut connection structure called residual module. Multiple residual modules are sequentially stacked to form ResNet, as shown in Fig. 5(a). Actually, the shortcut connection performs identity mapping, and its outputs are added to the output of the following layer. This simple calculation does not increase the number of parameters and computational complexity, and can improve the performance and speed up the training. The residual module actually contains two types. One is the basic module, as shown in the left of Fig. 5(b), and the other is the Bottleneck Block, as shown in the right of Fig. 5(b). The bottleneck module replaces the 3×3 convolution with two 1×1 convolution, which also reduces the number of parameters and computational complexity, and increases the nonlinear expression of the network.
Inception_v4 is an improved version of Inception_v3[57]. Compared with regular network structure such as VGG and ResNet, Inception_v4 is mainly composed of one input stem, three Inception modules and two reduction modules, each of which is designed separately. The overall structure of Inception_v4 and the structure of each module are shown in Fig. 6.
6) Inception_ResNet_v2
While designing Inception_v4, Szegedy et al.[57] introduced the residual modules into Inception_v3 and Inception_v4, respectively, resulting in Inception_ResNet_v1 and Inception_ResNet_v2. The overall structure of Inception_ResNet_v1 and Inception_ResNet_v2 is the same, and the difference is the modules in the network. Fig. 7 shows the Inception_ResNet_v2 overall structure and module structure.
7) DenseNet
It seems that DenseNet is an extreme version of ResNet[59]. DenseNet introduces short connections from any layer to all the following layers. But, in fact, DenseNet combines features by concatenating them instead of summation before features are passed to a layer, which enables the network to make better use of features. As shown in Fig. 8(a), a five-layer dense block is illustrated, in which each layer of output is connected to each subsequent layer. The dense block is continuously stacked to form DenseNet. The structure of DenseNet is depicted in Fig. 8(b).
8) Xception
Xception is another improved version of Inception_v3[66]. It is based on the assumption that spatial convolution (convolution along the horizontal and vertical directions of the feature map) and channel convolution (convolution along the direction of the feature map channel) can be performed independently to separate convolution. As shown in Fig. 9, for the feature maps in the previous layer, 1×1 convolutions are used to linearly combine the feature maps, and then use convolutions separately for each channel, where M is the channel of the feature maps, N is the number of convolutions (or the output channel), n is the size of convolution kernel.
In fact, the Inception module can be simplified as follows: all 1×1 convolutions in Inception can be reformulated as a large convolution, then utilize convolutions separately on every output channel, forming the extreme Inception, as shown in Fig. 10. Extreme Inception is consistent with the initial assumption and achieves the decoupling operation of convolution. This kind of extreme Inception is named Xception.
9) MobileNet_v2 & 10)MobileNet_v3
In order to meet the needs of embedded devices such as mobile phones, the research team of Google proposed a compact neural network named MobileNet in 2017. MobileNet is based on depthwise separable convolution to reduce the number of parameters. Depthwise separable convolution splits the standard convolution into two steps: depthwise convolution, which applies convolution to each channel of the feature map separately and pointwise convolution, which uses 1×1 convolutions to combine the feature, as shown in Fig. 11. In Fig. 11, M is the number of input channels, DK is the convolution kernel size, N is the number of convolution kernels, and if the size of feature map is DF × DF, then for standard convolution, the computational cost is DF × DF × M × N × DK × DK and for depthwise separable convolution, the computational cost of the depthwise convolution is DF × DF × M × DK × DK, and the computational cost of the pointwise convolution is DF × DF × M × N × 1 × 1, so the total computational cost is DF × DF × M × DK × DK + DF × DF × M × N. Therefore, we get a reduction in computation of:
$$\frac{{{{{D}}_{\rm{K}}} \!\times\! {{{D}}_{\rm{K}}} \!\times\! {{M}} \!\times\! {{{D}}_{\rm{F}}} \!\times\! {{{D}}_{\rm{F}}}{{ + M}} \!\times\! {{N}} \!\times\! {{{D}}_{\rm{F}}} \!\times\! {{{D}}_{\rm{F}}}}}{{{{{D}}_{\rm{K}}} \!\times\! {{{D}}_{\rm{K}}} \!\times\! {{M}} \!\times\! {{N}} \!\times\! {{{D}}_{\rm{F}}} \!\times\! {{{D}}_{\rm{F}}}}} = \frac{1}{{{N}}}{\rm{ + }}\frac{1}{{{{D}}_{\rm{K}}^2}}. $$
For example, if a 3×3 convolution is used, the computational cost can be reduced by about 8 or 9 times. In addition, the batch normalization and the nonlinear activation function ReLU are added after the 3×3 convolution and 1×1 convolution, respectively, as shown in Fig. 12.
In 2018, the research team of Google continued to improve MobileNet and designed MobileNet_v2[62]. MobileNet_v2 introduces the shortcut connection in ResNet and DesNet to the network. Since the output of the depthwise separable convolutions is limited by the number of input channels and the characteristics of the bottleneck residual module, if the residual is directly introduced into the network without modification, the initial feature compression will result in too few features available in the subsequent layers. Therefore, MobileNet_v2 proposes Inverted residuals – expanding the number of features first, then extracts the features using convolution, and finally compresses the features. In addition, MobileNet_v2 cancels the ReLU at the end of the inverted residual, because ReLU sets all non-positive inputs to zeros, and adding ReLU after feature compression loses the feature. The network structure of MobileNet_v2 is shown in Fig. 13.
A year later, MobileNet_v3, which gets its model by neural architecture search (NAS), was proposed by the research team of Google[63]. The internal modules of MobileNet_v3 inherit MobileNet_v1, MobileNet_v2 and MnasNet, and networks are researched by platform-aware NAS and NetAdapt. The calculation in the final stage of the network is redesigned on MobileNet_v3 due to the extensive calculation in MobileNet_v2. In addition, a new activation function h-swish[x] is proposed to improve the accuracy of networks effectively. MobileNet_v3 includes two versions: MobileNet_v3-small and MobileNet_v3-large. MobileNet_v3-small has faster speed and its accuracy is similar to MobileNet_v2. MobileNet_v3-large has higher accuracy. Finally, the results of image classification, target detection and semantic segmentation experiments show the advantage of MobileNet_v3. The network structure of MobileNet_v3 is shown in Fig. 14.
11) SENet
Squeeze-and-Excitation (SENet) is a new image recognition structure, which was proposed by the autopilot company Momenta in 2017[69]. It enhances important features by modeling the correlation between feature channels to improve accuracy. The SENet block is a substructure that can be embedded in other classification or detection models. In the 2017 ILSVRC competition, the SENet block and ResNeXt are applied to reduce the top-5 error to 2.251% on the ImageNet dataset, which was the champion in the classification project. The network structure of the SENet block is shown in Fig. 15.
12) ResNeXt
ResNeXt is the upgraded version of ResNet[68]. In order to improve the accuracy of the model, some networks deepen and widen the network structure, resulting in increasing the number of network hyperparameters as well as the difficulty and computational cost of network design. However, ResNeXt improves the accuracy without increasing the complexity of the parameters, even reducing the number of hyperparameters. ResNeXt has three equivalent network structures, as shown in Fig. 16. The original three-layer convolution block in RseNet is replaced by a block of parallel stacking topologies. The topologies are the same, but the hyperparameters are reduced, which facilitates model migration.
13) ShuffleNet_v2
In ResNeXt, the packet convolution strategy is applied as a compromise strategy, and the pointwise convolution of the entire feature map restricts the performance of ResNeXt. Thus, an efficient strategy is to perform pointwise convolution within a group, but it is not conducive to information exchange between channels. To solve this problem, ShuffleNet_v1 proposed a channel shuffle operation. The structure of ShuffleNet_v1 is depicted in Fig. 17, where Fig. 17(a) does not need downsampling, and Fig. 17(b) required downsampling operation.
In ShuffleNet_v2[65], researchers found that it is unreasonable to only apply commonly-used FLOPs in the evaluation of model performance, because file IO, memory read, GPU execution efficiency also need to be considered. Taking memory consumption and GPU parallelism into account, researchers designed an efficient ShuffleNet_v2 model. This model is similar to DenseNet, but ShuffleNet_v2 has higher accuracy and faster speed. The network structure of ShuffleNet_v2 is shown in Fig. 18.
14) EfficientNet
EfficientNet was proposed in 2019[70], and is a more general idea for the optimization of current classification networks. Widening the network, deepening the network and increasing the resolution are three common network indicators, which are applied independently in most previous networks. Thus, the compound model scaling algorithm is proposed, which comprehensively optimizes the network width, network depth and resolution to improve the accuracy and the existing classification network, and the amount of model parameters and calculations are greatly reduced. EfficientNet uses the EfficientNet-b0 as the basic network to design eight network structures called b0−b7, and EfficientNet-b7 has the highest accuracy. The network structure of EfficientNet-b0 is shown in Fig. 19.
15) GhostNet
In GhostNet, Han et al.[71] proposed a novel ghost module, which can generate more feature maps with fewer parameters. Specifically, the convolution layer in the depth neural network is divided into two parts. The first part involves the common convolution, but the number of them should be strictly controlled. Given the inherent characteristic graph of the first part, then a series of simple linear operations are applied to generate more characteristic graphs. Compared with the conventional CNN, the total number of parameters and the computational complexity of the ghost module are the lowest without changing the size of the output characteristic map. Based on the ghost module, Han et al.[71] proposed GhostNet. Fig. 20 shows the ghost module.
16) RegNet
In RegNet, Radosavovic et al.[72] proposed a new network design paradigm, which aims to help improve the understanding of network design. Radosavovic et al.[72] focused on the design of network design space of parameterized networks. The whole process is similar to the classic manual network design, but it is promoted to the level of design space. Using this rule to search for a simple low dimensional network, i.e., RegNet. The core idea of RegNet parameterization is that the width and depth of a good network can be explained by a quantized linear function. Particularly, RegNet outperforms traditional available models and runs five times on GPUs.
17) ResNeSt
In ResNeST, Zhang et al.[73] explored the simple architecture modification of ResNet, and incorporated feature-map split attention within the individual network blocks. More specifically, each of the blocks divides the feature-map into several groups (along the channel dimension) and finer-grained subgroups or splits, where the feature representation of each group is determined via a weighted combination of the representations of its splits (with weights chosen based on global contextual information). Zhang et al.[73] refer to the resulting unit as a split-attention block, which remains simple and modular. By stacking several split-attention blocks, a ResNet-like network is created called ResNeSt. The architecture of ResNeSt requires no more computation than existing ResNet-variants, and is easily adopted as a backbone for other vision tasks. The performance of ResNeST is better than all existing ResNet variants, while the computational efficiency is the same, an even better speed and accuracy tradeoff is achieved than the most advanced CNN model generated by NAS. Fig. 21 shows the ResNeSt block module.
4. 2D and 3D palmprint and palm vein databases used for evaluation
In this paper, five 2D palmprint image databases, one 3D palmprint database and two palm vein databases are used for performance evaluation, including PolyU II[11], PolyU M_B[12], HFUT[13], HFUT CS[14], TJU-P[15], PolyU 3D[15], PolyU M_B[12] and TJU-PV[17]. After preprocessing, the ROI sub-images were cropped. The ROI size of all databases is 128×128. The detailed descriptions of above databases are listed in Table 4. Figs. 22−25 depict some ROI images of four 2D palmprint databases. In Figs. 22−25, the three images depicted in the first row were captured in the first session. The three images depicted the second row were captured in the second session.
Fig. 26 shows three original palmprints of HFUT CS database and their corresponding ROI images. Fig. 27 shows three original 3D palmprint data of the PolyU 3D database. Fig. 28 shows four different 2D representations from one 3D palmprint including MCI, GCI, ST and CST. Figs. 29 and 30 depict some ROI images of two 2D palm vein databases. In Figs. 29 and 30, three images depicted in the first row were captured in the first session. Three images depicted the second row were captured in the second session.
PolyU II is a challenging palmprint database because the illuminations between the first session and the second session have an obvious change. HFUT CS is also a challenging palmprint database. From Fig. 25, it can be seen that there are some differences between the palmprints captured by different devices.
5.1. Experimental configuration
In this section, we introduce the default configuration of the experiment, including experimental hyperparameters and hardware configuration. The full names of some CNNs are too long, and it is difficult to insert them into the tables of experimental results. Thus, we provide their abbreviation names in Table 5. It should be noted that in the following experiments, the abbreviation name Res represents the ResNet-18 network; the abbreviation name Dense represents the DenseNet-121 network; the abbreviation name SE represents the SENet-154 network; the abbreviation name ResX represents the ResNeXt-101 network; the abbreviation name Efficient represents the EfficientNet-b7 network; the abbreviation name Reg represents the RegNet-Y network; and the abbreviation name ResS represents the ResNeST-50 network.
Since different networks need different input sizes, such as 227×227 in AlexNet, 299×299 in Inception_v3, and 224×224 in ResNet, the palmprint/palmvein ROI image needs to be upsampled to a suitable size before input into the network. In order to enhance the stability of the network, we also added a random flip operation (only during the training phase), i.e., for a training image, there is a certain probability that the image is flipped horizontally and then input into the network. We do not initialize the model parameters using the random parameter initialization method, but initialize it using the parameters of the pretrained model in the ImageNet. The palmprint/palmvein ROI image in the database is usually a grayscale image, that means the number of image channels is 1, and the input of the model is a RGB image, so the grayscale channel of the image is copied three times to form a RGB image.
The system configuration is as follows: Intel CPU i7 4.2GHz, NVIDIA GPU GTX 1080Ti (EfficientNet runs on two parallel GPUs GTX 1080Ti), 16Gb memory and Windows 10 operating system. All evaluation experiments are performed on Pytorch. The cross entropy loss function (CrossEntropyLoss in Pytorch), Adam optimizer is used by default and the batch size is 4.
5.2. Recognition performance on separate data mode
We first conduct evaluation experiments on a separate data mode, i.e., all samples captured in the first session are used for training, and all samples captured in the second session are used for test.
5.2.1. Recognition results of ResNet18 and EfficientNet under different learning rates
Learning rate is a very important hyperparameter in model training, which affects the convergence of the loss function. If the learning rate is too small, the decrease of loss along the gradient direction will be slow, and it will take time to reach the optimal solution. If the learning rate is too large, it may lead optimal solutions to be missed, and may cause severe turbulence and even vanishing gradient problems. Here, we are only looking for the initial learning rate, combined with the dynamic learning rate strategy in the actual experiment. Therefore, choosing a suitable learning rate is especially critical. In this sub-section, we select ResNet18 and EfficientNet for evaluation because ResNet18 has a high recognition rate in early networks and EfficientNet is one of representative networks proposed recently. The experimental results are listed in Tables 6 and 7. From Tables 6 and 7, it can be seen that when the learning rate is 5×10−5, ResNet18 and EfficientNet achieve the best recognition rate. Thus, in the remaining experiments, we set the initializing learning rate to 5×10−5. It should be noted that all our experiments have an initial learning rate of 5×10−5, and 100 iterations (EfficientNet used 200 iterations since it has slow convergence) are the learning rate decline steps, where learning rate decay rate is 0.1. That is, the learning rate drops by ten times every 100 iterations, and the total number of iterations is 500.
5.2.2. Recognition results of ResNet and VGG with different numbers of layers
Some CNNs may have different versions with different numbers of layers. For example, ResNet has different versions with 18, 34 and 50 layers. Using more layers may get better recognition rates, but may have the problem of overfitting. Thus, the number of layers is also an important factor for recognition. In this sub-section, we evaluate VGG and ResNet with different numbers of layers. Since most databases have difficulty in training on VGG when the learning rate is 5×10−5, we set the learning rate of VGG to 10−5. The recognition rates of VGG and ResNet under different numbers of layers are shown in Table 8. In this experiment, we verify the impact of network depth on the recognition performance.
The results in Table 8 indicate that:
1) For VGG, the recognition performance of VGG-16 is slightly better than that of VGG-19, and the recognition performances of VGG-16 and VGG-19 are close.
2) For ResNet, the recognition performance of Res-18 is better than those of Res-34 and Res-50. On those challenging databases such as PolyU II, HFUT, HFUT CS, TJU-P, and TJU-PV, the recognition performance of Res-18 is obviously better than that of Res-50.
3) In all databases, the recognition rate of ResNet-18 is obviously better than those of VGG-16 and VGG-19.
According to the results listed in Table 8, for VGG and ResNet, we only use VGG-16 and Res-18 for evaluation in the remaining experiments.
For different CNNs, the best number of layers to obtain the best recognition rate is determined by many factors, such as network structure, data size, data type, etc. Therefore, in practical applications, a lot of experiments need to be done to determine the optimal number of network layers for different CNNs.
5.2.3. Recognition results of EfficientNet from b0 to b7
EfficientNet gets the baseline network EfficientNet-b0 by grid search, and further optimizes different parameters to get EfficientNet-b1 to b7. The recognition results of EfficientNet from b0 to b7 are listed in Table 9. It can be seen that the recognition accuracy of EfficientNet is gradually increasing from b0 to b7, and EfficientNet-b7 achieves the best recognition accuracy. In the remaining experiments of this paper, for EfficientNet, we only use EfficientNet-b7 to conduct evaluation experiments. It should be noted that although EfficientNet-b7 performs well, it converges almost twice as slowly as other networks. In fact, EfficientNet-b6 is also slow, but the of speed EfficientNet-b0 to b5 is normal.
5.2.4. Recognition results of selected CNNs on all databases
In this sub-section, we conduct the experiments using all selected CNNs on all databases. The recognition results of selected CNNs on 2D palmprint and palm vein databases are listed in Table 10. The recognition results of selected CNNs on the 3D palmprint databases are listed in Table 11. Sometimes, when the learning rate is set to 5 × 10−5, AlexNet and VGG-16 are untrainable. In this time, we adjust the learning rate of AlexNet and VGG-16 to 10−5. In Table 10, AlexNet and VGG-16 have two recognition rates. The former is the result under the learning rate of 5 × 10−5, and the latter is the result under the learning rate of 10−5. If AlexNet and VGG-16 are untrainable, we mark the result as U.
From Tables 10 and 11, we have the following observations:
1) EfficientNet achieves the best recognition rate on most databases. The overall recognition result of ResNet is in the second place.
2) As a representative of lightweight networks, the overall recognition performance of MobileNet_v3 is worse than that of EfficientNet, close to ResNet, but better than other CNNs. This demonstrates that MobileNet_v3 is effective.
3) The recognition performance of recently proposed CNNs is obviously better than those of early CNNs. For example, the recognition rates of AlexNet and VGG are rather low. For those early CNNs such as AlexNet and VGG, their structures are relatively simple, and the number of layers is small. Thus, the recognition performance of them is not as good as those of the recently proposed CNNs such as EfficientNet.
4) HFUT CS is a very challenging database. The recognition performances of the most CNNs on HFUT CS database are unsatisfactory. In this database, ResNeSt (ResS-50) achieves the highest recognition rate, which is 99.15%.
5) Except ResNeSt has achieved good results on HFUT CS database, several recently proposed networks, including GhostNet, RegNet, ResNeSt, etc. have not achieved very good recognition results on various databases. Maybe the network structures of GhostNet, RegNet and ResNeSt are not very suitable for palmprint recognition and palmar vein recognition.
6) Among four 2D representations of the 3D palmprint, the recognition results obtained from MCI are the best.
7) For 3D palmprint recognition, based on MCI representation, EfficientNet achieved the recognition rate of 99.88%, which is a very promising result.
5.3. Recognition performance on mixed data mode
In the mixed mode, the first image captured in the second session is added to the training data. That is, the training set of each palm contains all images captured in the first session and the first image captured in the second session. Here, we use EfficientNet to conduct experiments. For each palm, the total number of training images are the number of images captured in the first session adding one (+1). This one means the first image captured in the second session. From Table 12, it can be seen that the recognition accuracy of EfficientNet gradually increases when the number of training samples increases.
We list the recognition rates of different CNNs on mixed data mode in Tables 13 and 14. It can be seen that the recognition accuracies of all CNNs increased significantly, particularly, for 2D palmprint recognition, EfficientNet achieves 100% recognition accuracies on PolyU II, PolyU M_B, HFUT, TJU-P, PolyU M_N and TJU-PV. For 3D palmprint recognition, Res-18 achieves the best recognition results, and all CNNs achieve the best recognition results from MCI representation among four 2D representations.
This experiment proves once again that the sufficiency of data is very important to improve the recognition accuracy of deep learning. In the future, with the wide application of palmprint recognition and palmar vein recognition, the data volume of palmprint and palmar vein will increase continuously. In this way, the recognition accuracy of palmprint recognition and palm vein recognition technology based on deep learning will reach a new level.
5.4. Performance comparison with other methods
For 2D palmprint and palm vein recognition, we compare the performance of CNNs and other methods including some traditional methods and one deep learning method PalmNet[83]. Four traditional palmprint recognition methods, including competition code, sequence number, RLOC and LLDP, are selected for comparison. For CNNs, we only list the results of MobileNet_v3 and EfficientNet which have excellent performance. The performance comparison is conducted on both separate data mode and mixed data mode.
On the separate data mode, for traditional methods, four images collected in the first session are used as the training data, and all images collected in the second session are exploited as the test data. For MobileNet_v3 and EfficientNet, all images collected in the first session are used as the training data and the second session images are used as the test data (In the HFUT CS database, all images captured by the camera are used as the training data). The comparison results on separate data mode are shown in Table 15.
From Table 15, it can be seen that, on separate data mode, the performances of the traditional methods are better than those of the CNNs. As we know, traditional methods use fewer training samples. Because the features of 2D and 3D palmprint and palm vein are relatively stable, thus, hand-crafted features can well represent the palmprint, resulting in a better recognition performance of traditional methods. In addition, the classic CNNs used in this paper are designed for general image classification tasks, and are not specially designed for 2D and 3D palmprint recognition and palm vein recognition, so the accuracies of them are not satisfactory.
On the mixed data mode, for traditional methods, four images collected in the first session are used as the training data, and we add the first image captured in the second session to the training set. The remaining images collected in the second session are exploited as the test data. For MobileNet_v3 and EfficientNet, all images collected in the first session are used as the training data, and we add the first image captured in the second session to the training set. The remaining images collected in second session are exploited as the test data. The comparison results on mixed data mode are shown in Table 16.
From Table 16, it can be seen that, on mixed data mode, the performances of CNNs are nearly equal to that of the traditional methods. The scale of the 2D and 3D palmprint and palm vein databases is small. But deep learning methods rely heavily on learning from large-scale database. If there are sufficient training samples, deep learning methods can achieve better performance.
For 3D palmprint recognition, we compare the performances between CNNs and other traditional methods on the separate data mode. Table 17 lists the comparison results. It can be seen that the recognition accuracy of CNN is slightly better than traditional methods.
This paper systematically investigated the recognition performance of classic CNNs for 2D and 3D palmprint recognition and palm vein recognition. Seventeen representative and classic CNNs were exploited for performance evaluation including AlexNet, VGG, Inception_v3, Inception_v4, ResNet, ResNeXt, Inception_ResNet_v2, DenseNet, Xception, MobileNet_v2, MobliNet_v3, ShunffleNet_v2, SENet, EfficientNet, GhostNet, RegNet and ResNeSt. Five 2D palmprint image databases, one 3D palmprint database and two palm vein databases were exploited for performance evaluation, including PolyU II, PolyU M_B, HFUT, HFUT CS, TJU-P, PolyU 3D, PolyU M_B and TJU-PV. These databases are very representative. For example, PolyU II, PolyU M_B, PolyU M_N and HFUT databases were collected by the contact manner; HFUT CS, TJU-P, and TJU-PV were captured by the contactless manner. All databases were collected in two different sessions. In particular, HFUT CS is a rather challenging database because it was collected in the conditions of two different sessions, contactless manner and crossing three different sensors. We conducted a lot of experiments on the above databases in the conditions of different network structures, different learning rates, different numbers of network layers. We conducted the experiments on both separate data mode and mixed data mode. And we also compared the recognition performances between the CNNs and traditional methods. According to the experimental results, we have the following observations. 1) The performances of recently proposed CNNs such as EfficientNet and MobileNet_v3 are obviously better than those of other early CNNs. Particularly, EfficientNet achieves the best recognition accuracy. 2) Learning rate is an important hyperparameter. It has an important influence on the recognition performance of CNNs. For palmprint and palm vein recognition, 5×10−5 is an appropriate learning rate. 3) Using more layers, VGG and ResNet did not get better recognition results. Compared with ILSVRC, the scale of palmprint and palm vein databases is small, and the model with more layers may lead to the problem of over-fitting. 4) For 3D palmprint recognition, deep learning-based methods obtained promising results. Among four 2D representations of 3D palmprints, MCI can help deep learning methods to achieve the best recognition results. 5) In separate data mode, the recognition performance of classic CNNs is not satisfactory, and is worse than those of some traditional methods on those challenging databases. On mixed data mode, CNNs can achieve good recognition accuracy. For example, CNNs achieved 100% recognition accuracies on most databases.
In this work, a lot of classic CNNS have been evaluated. However, these CNNs have been designed manually by human experts. In recent two years, NAS technology has attracted more and more attention. The core idea of NAS is to use search algorithms to find better neural network structure, so that can obtain better recognition performance. In our future work, we will try to exploit NAS technology for 2D and 3D palmprint and palm vein recognition. In our future work, we will also design special CNNs according to the characteristics of 2D and palmprint recognition and palm vein recognition. In this way, better recognition performance of deep learning for 2D and 3D palmprint recognition and palm vein recognition can be expected.
This work was supported by National Science Foundation of China (Nos. 61673157, 62076086, 61972129 and 61702154), and Key Research and Development Program in Anhui Province (Nos. 202004d07020008 and 201904d07020010).
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参考文献 (94)
© Institute of Automation, Chinese Academy of Sciences. Published by Springer Nature and Science Press. All rights reserved. | CommonCrawl |
Suppose that all four of the numbers \[3 - 2\sqrt{2}, \; -3-2\sqrt{2}, \; 1+\sqrt{7}, \; 1-\sqrt{7}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?
Because the polynomial has rational coefficients, the radical conjugate of each of the given roots must also be roots of the polynomial. However, $1+\sqrt{7}$ and $1-\sqrt{7}$ are each other's radical conjugates, so we only get $2$ more roots. (You might be tempted to think that $3-2\sqrt2$ and $-3-2\sqrt2$ are also a pair of radical conjugates, but the radical conjugate of $3-2\sqrt2$ is $3+2\sqrt2,$ while the radical conjugate of $-3-2\sqrt2$ is $-3+2\sqrt2.$ Therefore, each one of the numbers $3-2\sqrt2$ and $-3-2\sqrt2$ is actually the negation of the radical conjugate of the other one.) In total, the polynomial must have at least $4+2=6$ roots.
Furthermore, the polynomial
\[(x - 3 + 2 \sqrt{2})(x - 3 - 2 \sqrt{2})(x + 3 + 2 \sqrt{2})(x + 3 - 2 \sqrt{2})(x - 1 - \sqrt{7})(x - 1 + \sqrt{7}) = (x^2 - 6x + 1)(x^2 + 6x + 1)(x^2 - 2x - 6)\]has roots $3 \pm 2 \sqrt{2},$ $-3 \pm 2 \sqrt{2},$ and $1 \pm \sqrt{7},$ and has rational coefficients. Hence, the smallest possible degree is $\boxed{6}.$ | Math Dataset |
Functional models and extending strategies for ecological networks
Gianni Fenu ORCID: orcid.org/0000-0003-4668-24761,
Pier Luigi Pau1 &
Danilo Dessì1
Complex network analysis is rising as an essential tool to understand properties of ecological landscape networks, and as an aid to land management. The most common methods to build graph models of ecological networks are based on representing functional connectivity with respect to a target species. This has provided good results, but the lack of a model able to capture general properties of the network may be seen as a shortcoming when the activity involves the proposal for modifications in land use. Similarity scores, calculated between nature protection areas, may act as a building block for a graph model intended to carry a higher degree of generality. The present work compares several design choices for similarity-based graphs, in order to determine which is most suitable for use in land management.
In the current society that is the result of centuries of urbanization and industrialization, environmental policies involving the preservation of endangered species and habitats are held as a necessity. The simple approach consisting in the creation of natural reserves with the specific purpose of preserving a target habitat or species has shown its limits, and in the past few decades, there has been a paradigm shift toward the creation of ecological networks, with a focus on the preservation of biodiversity. In the European Union, the Natura 2000 project was established with the goal to create a continent-wide ecological network.
Graph models are a common mathematical tool to analyze and understand network properties in many kinds of networks and complex systems. Modeling choices can be guided by attempts at generalization, or by an interest for a specific point of view. Most infrastructure networks (power grids, roads and railways, etc.) are usually straightforward to represent as a graph, as edges are made to correspond to physical links in existence between the entities represented by nodes; the real challenge in creating a faithful model is that of choosing a measure for edge weights, such that the model can accurately predict the behavior of the underlying network. For power grids, among proposed measures for edge weights are impedance (Cheng et al. 2013) and reactance (Dwivedi et al. 2009). Other complex systems exist, such as social networks and conceptual networks dealing with knowledge representation, for which even the criteria to link node pairs may vary according to analysis goals. For example, in the analysis of a centralized social network, the decision to link people according to whether they are in a contact list, or whether they have recently exchanged messages, may have a strong influence on the resulting graph model and the results of its analysis (Viswanath et al. 2009). Some kinds of infrastructure networks can also be modeled with different approaches; namely, computer networks can be analyzed based on their physical links (Lad et al. 2006), or according to their participation in complex systems of web service orchestration and grid computing (Aymerich et al. 2009).
In the field of ecological landscape networks, the most common approach to building graph models is that of representing functional connectivity, in the sense that the edges in a graph correspond to actual or potential migrations of a species of interest. When possible, data on actual migrations of animals may be derived by tagging and tracking the movement of a sample of the population; alternatively, a method to infer potential migrations is to analyze genetic differences among distant populations (Naujokaitis-Lewis et al. 2013). Graph-theoretic approaches are also possible in studying the relationships among populations, as opposed to the topology of a migration network (Dyer and Nason 2004).
A drawback of the functional approach to building graph models of ecological networks is that assessing general properties of the network requires the analysis of a large number of graphs. This article applies methods from a previous work (Fenu et al. 2016), in which the computation of similarity scores between sites is proposed as a way to build a graph model with a general perspective. In this work, similarity analysis is extended beyond the restricted case of binary vectors, and a refined case study is considered, which takes into account additional constraints in graph generation.
Ecological networks and graph models
Following the surge of urbanization that has taken place in the past century, the establishment and maintenance of nature protection areas has become an essential part of land management, due to the necessity of a proactive approach in the protection of habitats and species at risk of extinction. Reserves are made up of habitat patches intended as a host for endangered species. Decades of studies and continued endeavor have shown that the effectiveness of reserves is quite limited if patches are not large enough to host a significant amount of population of the protected species, or if reserves are too distant from other suitable habitat patches. Policies for the protection of the environment have thus converged toward the creation of ecological networks, rather than isolated reserves. In an ecological network, each area is intended to contribute to large-scale preservation goals, and efforts are made to ensure a possibility of migration for protected species whenever possible, in order to avoid a deterioration of the gene pool and protect biodiversity (Vimal et al. 2012). Migration can be encouraged with the creation of man-made 'habitat corridors' in cases where it is deemed useful; these can be either contiguous or made up of sets of disconnected patches, referred to as 'stepping stones'.
The ecological network paradigm is being adopted in various parts of the world by the relevant administrative bodies; in the European Union, the project denominated "Natura 2000" is aimed at coordinating the efforts of member states in maintaining a network of nature protection areas with consistent methods and goals throughout the Union. The founding elements of this network are its sites, designated as Special Protection Areas (SPA), as defined in the EU Birds Directive (2009/147/EC), and Special Areas of Conservation (SAC), as defined in the EU Habitats Directive (92/43/EEC). The latter are first proposed as Sites of Community Interest (SCI), and later designated as SACs. A site can hold a designation as a SPA and as a SAC (or SCI) at the same time; alternatively, the boundaries of a SPA can overlap with those of SACs or SCIs. Lastly, sites of the same category can be adjacent to one another.
Local administrations are involved with the management of Natura 2000 sites within their jurisdiction, and can be affected by the presence of sites in their proximity, due to the possible involvement in the creation of habitat corridors. The identification of threats and the proposal of a course of action to eliminate or mitigate them is one of the activities, to which local administrations are to contribute. This requires the consideration of technical, regulatory, and political aspects.
Not unlike other kinds of networks and complex systems, graph models are often used to represent ecological networks mathematically and provide a theoretical basis for the understanding and prediction of network properties. A graph consists of a set of nodes and a set of edges, which may be weighted, i.e. with a numeric attribute associated to them to represent a strength of the link or a cost for traversing it. In this context, a node may represent a site or habitat patch, depending on the desired scale, while edges represent connections. Two different approaches to linking nodes are possible: one is the structural approach, consisting in drawing edges and assigning weights to reflect the influence on migrations carried by existing geographical features, acting as obstacles or connecting elements; this approach has not been successful, due to the difficulties in assessing edge weights in a meaningful manner. Instead, structural connectivity is more commonly analyzed with Geographic Information System tools. The more successful approach to building graph models is that of representing functional connectivity, i.e. actual or potential migration flows, with special reference to a target species (Urban et al. 2009). Data on the migration of animals may be obtained by tagging individuals and tracking their movements; in situations where this is not possible, or to extrapolate data on plant dispersal, a way to infer whether migration may have occurred between different areas is to analyze and compare the genetic pools of samples of populations taken from each area. When raster data is available for the relevant portion of land, and patches can be associated with a resistance value for the target species, circuit theory can also be used to predict migration paths (McRae et al. 2008).
In the analysis of a complex network, statistical properties of graph models are extrapolated and compared, whether with those of other networks or common reference models; among the most common features for analysis are node degree, shortest path length, and indices such as the clustering coefficient, which expresses the degree of redundancy of links, and the betweenness centrality index, used to rank nodes according to their occurrences in shortest paths. The meaning and relevance of each index may differ according to the kind of real-world network being represented (Borgatti 2005); interpretations have been proposed for the most common indices in the field of ecological networks (Estrada and Bodin 2008). As a general principle, global indices, calculated for the network as a whole, can be used as a measure of its 'health', while local indices, calculated for single network elements, may assist in identifying vulnerabilities in topological networks (Mishkovski et al. 2011). These are often associated to the different degree of resiliency of the network upon removal of specific nodes (Iyer et al. 2013). The comparison of indices calculated for a given network model and for modified versions of the starting model is often useful to predict the effect of modifications on the corresponding real network.
Similarity of Natura 2000 sites
Among the activities held as part of Natura 2000 project is the collection of data on habitats and species found within each recognized site. Information is periodically gathered on-site and a public data base is kept up to date with reports filed for each site. These reports must conform to a Standard Data Form, released with Commission Implementing Decision 2011/484/EU. The composition of each Natura 2000 site as a set patches of different habitat types, as well as the presence of a set of species, are part of the collected information; however, the form does not establish an explicit relationship between each species and the habitat patch where it is found. This is sensible for the original purposes of the Natura 2000 project, but for data analysis purposes, it has a shortcoming in the fact that no information is stored concerning which habitat type is ideal for each species; this is assumed to be part of expert knowledge, or found in external documents. Consequently, it is not straightforward to represent constraints for the proposal of modifications involving the relocation of species.
In an attempt to address these problems at least partially, it is possible to represent each site as a vector and compute similarity scores of these vectors. This way, the occurrence of a set minimum score for a pair of sites can be considered as a prerequisite for the proposal to add an edge to the network, to reflect the proposal to establish a habitat corridor. In order to simplify the choice of a threshold, it is preferable to adopt a similarity measure that takes values within a set range; examples thereof are the Jaccard coefficient for binary vectors, and cosine similarity for non-binary vectors.
To compute the Jaccard coefficient for a pair of binary vectors, let f 11 be the number of attributes set to true (1) in both vectors, f 10 the number of attributes that are true only in the first vector, and f 01 the number of attributes that are true only in the second vector; the Jaccard coefficient J is given by:
$$ J=\frac{f_{11}}{f_{01}+f_{10}+f_{11}}. $$
This coefficient takes values from 0 to 1, where the extremes represent vectors with no true attribute in common and identical vectors, respectively.
Cosine similarity is defined as the cosine of the angle between two vectors with non-zero magnitude from the origin in a multi-dimensional space. This measure is commonly used for the comparison and categorization of text documents, which are represented as vectors by considering an attribute for each keyword, with a value corresponding to their number of occurrences in the document. The following is a simple formula to compute cosine similarity:
$$ cos(\mathbf{x}, \mathbf{y})=\frac{\mathbf{x} \cdot \mathbf{y}}{ ||\mathbf{x}|| \, ||\mathbf{y}||}, $$
where x and y are vectors, x·y is their scalar product, and ||x||, ||y|| are their magnitudes. Cosine similarity may take values from −1 to 1 for arbitrary vectors, or from 0 to 1 for pairs vectors with positive attribute values.
If the set of species or the set of habitats are used as the list of attributes, vectors to represent Natura 2000 sites can be built from data collected as part of the project activities. More specifically, the attributes of a vector can be made to correspond to identifying codes for each species of interest, or each reported habitat. For example, to build a binary vector from the habitats in a site, each attribute corresponding to a habitat code is set to 1 if the habitat was reported as found within the site, or 0 otherwise. To build a non-binary vector, the value of each attribute expressing a habitat type can be assigned according to the area extension covered by that habitat type within the boundaries of the site, so long as data is consistently available.
Site vectors can also be built with the integration of an external data source. The CORINE program (Coordination of Information on the Environment) provides a set of standardized land use codes, which can make up a useful set of vector attributes, and for which it is easy to find a compatible data source in the form of patch boundaries. The intersection of these land patches with Natura 2000 sites can be computed to populate vector attributes, both for binary and non-binary vectors, according to the same principles that apply to those based on habitat codes. Land use types (referred to as CORINE Land Cover codes or CLC codes) are categorized in a hierarchical manner, with five levels of increasing detail; a vector based on land use data has attributes corresponding to CLC codes of a chosen reference level (for example, at level 3, different codes can differentiate between broad-leaved forests and coniferous forests, or natural grasslands, etc.).
In this study, the open source QGIS software suite (QGIS Development Team 2009) was used to compute intersection of Natura 2000 sites with land patches from public data made available by the Region of Sardinia. Level 3 codes were used, as the fourth and fifth levels of detail were not available consistently in the dataset.
An advantage in integrating an external source for land use data is that areas outside of Natura 2000 sites may be covered, making it possible to compute a similarity score between a site and an arbitrary area, such as that of a proposed contiguous corridor, in an approach combining graph-theoretic approaches and GIS functions (Pinto and Keitt 2008). This is also useful to limit the effect of missing data, which can be detrimental to the analysis (Naujokaitis-Lewis et al. 2013).
To provide a test case and illustrate the method of analysis, the subset of Natura 2000 sites found in Sardinia is considered. At the time of data collection, the administrative region including Sardinia and the minor islands in its surroundings has a total of 124 sites, counting those designated as SPA, those designated as SCI, and those with both designations (no SACs have been designated yet in this region). The number of sites located on the main island of Sardinia is 107; a total of 7 sites, including 3 sites within the main island, have to be excluded due to missing data on land use types.
Ultimately, in order to provide a consistent test case with full availability of data, and disregarding minor islands in order for the considerations to be applicable to land animals, 104 sites can be considered, and graphs are to be created from the same set of 104 nodes, each representing a Natura 2000 site, designated either as a SPA or a SCI, or both. If the boundaries of a SPA and a SCI are intersected, two nodes are created, but the sites are considered to be at zero distance from one another. In every graph instance, pairs of nodes with a geographical distance greater than a set threshold (30 Km) are never linked; distances are calculated between boundaries on a map projection, and are to be treated as an approximation, but the amount of error introduced by projections can be considered acceptable for this study. SQLite with the Spatialite extension was used to compute geographical distances. The open source Cytoscape suite (version 3.4.0) was used for graph visualization (Shannon et al. 2003) and analysis, through the native NetworkAnalyzer plugin (Assenov et al. 2008).
The graph with 104 nodes, each corresponding to a Natura 2000 site, in which all pairs of nodes within the set geographical distance to one another are linked, shall be referred to as the raw-distance graph (Fig. 1 a). The graph thus built with a 30 Km distance threshold has a total of 706 edges.
Graph models of Natura 2000 sites in Sardinia. a Raw-distance graph. Edges link pairs of nodes with a geographical distance up to 30 Km between boundaries. The position of each node roughly corresponds to the coordinates of the site centroid. b Full single-species graph for Cervus elaphus corsicanus (species code 1367). Edges correspond to a subset of the edges in the raw-distance graph; each edge links pairs of nodes with the target species reported to be present in both
A way to infer a functional model of the network with reference to a single species is to take the subset of nodes in the raw-distance graph corresponding to sites where the target species has been reported to be found, and link node pairs with an edge if their distance is below the threshold. This can be referred to as a single-species graph. Alternately, it may be sensible for comparison purposes to build the graph on the full node set from the raw-distance graph. In this case, node pairs are linked if the sites are within the distance threshold and the target species has been reported in both sites. This can be referred to as a full single-species graph (Fig. 1 b). As part of this study, full single-species graphs were built for all species listed in Directive 2009/147/EC and Annex II to Directive 92/43/EEC that were reported to be found in Sardinia (a total of 131 species), keeping the same distance threshold, as a way to establish a frame of reference for the comparison with other proposed graph models. Conversely, if the purpose is to build a functional graph model, the distance threshold ought to be adapted according to the species.
In order to represent the state of the network with a more general point of view, site similarity is used as a criterion to link nodes, as opposed to focusing on data concerning a single species. In this kind of graph, a pair of nodes is linked by an edge if their geographical distance is within the threshold and their similarity is equal or greater than a set minimum score. This process is equivalent to the removal of edges from the raw-distance graph, where the similarity is below the minimum score. A graph model built following such criteria can generally be referred to as a similarity-based graph, and specifically according to the choice of similarity measure and dataset from which site vectors are built.
Three different vector representation of sites are possible with available data: one in which attributes correspond to the set of species reported to be in a site, from which a species-set graph is built; one in which attributes correspond to the set of habitats according to Natura 2000 categorization (habitat graph), and one built on the set of level 3 land use codes according to the CORINE program (land-use graph). For all three datasets, it is possible to build binary vectors and compute the Jaccard coefficient for pairs of vectors. Non-binary vectors, for which cosine similarity can be calculated, can also be built for habitat sets and land use codes, either by counting the occurrences or by summing the surface area of each land patch. The latter can be expected to bring more meaningful results, though it can be argued that a similar ratio in the number of occurrences can be a legitimate sign of similarity even in cases where the size of patches is not homogeneous, as they may represent a comparable degree of fragmentation of those patches. Lastly, it is theoretically possible to build non-binary vector for species sets by assigning attribute values according to the amount of population of a species in the site; unfortunately, this kind of data is only sparsely available in the dataset. For this reason, only binary vectors are built from species sets in this study.
To provide a common base of reference, a similarity score of 0.5 shall be used as a threshold to build all of the similarity-based graphs. This threshold was chosen while keeping in mind that, for Jaccard coefficients, similarity scores of 0.6 and above turn out to be strong requirements, which remove over 85% of edges for all three vector types (see Table 1). For consistency, the same threshold will be used with cosine similarity, although it can be observed that this measure is more lenient: a 0.6 threshold removes between about 71 and 78% of the edges in the raw-distance graph (see Table 2). In general, the choice for a threshold value should be high enough for the selected edges to represent reasonably similar sites; conversely, an excessive threshold value can result in an inability to select any edges in practical studies.
Table 1 Number of edges in similarity-based graphs of Natura 2000 sites in Sardinia, using Jaccard coefficients
Table 2 Number of edges in similarity-based graphs of Natura 2000 sites in Sardinia, using cosine similarity
Analysis of edge hit rates and complex network indices
Network indices extrapolated with the application of complex network analysis on single-species graph are useful to give insight on the network in its current state, and particularly its aptness for the purpose of conservation of the target species. A common example is the identification of bottlenecks or the detection of insufficient redundancy of links. Proposals for land management can be expressed as modifications of the graph model, with a resulting change of its indices; the best proposals can be identified as those that bring the greatest improvement of indices as an effect.
If a network is initially connected, and the addition or removal of nodes is not considered, proposed modifications can fall into one of three categories (Arrigo and Benzi 2016):
Addition of edges ('updating'; proposed edges are referred to as 'virtual edges'). Given that there is a cost associated with the addition of links in the real-world network corresponding to the graph model, this problem consists in finding a set of new links which results in as great a benefit as possible, while respecting budget constraints.
Removal of edges ('downdating'). Assuming that each link has a cost of maintenance, and there is some degree of redundancy of links in the network, this problem corresponds to that of finding a set of edges that can be removed, in order to decrease maintenance costs, while keeping the decrease in the efficiency of the network as low as possible. It is also generally assumed that the removal of links should not create disconnected components.
Rewiring, i.e. removing and subsequently adding one or more edges. This is related to the goal of improving the efficiency of a network, while avoiding a hike in maintenance costs.
A task for land managers involved with ecological networks is that of enhancing the network effect in a set of habitat patches, or finding a way to contribute to doing so in large-scale settings. A possible intervention that may be considered is that of proposing a site for the relocation of part of the population of a species, among those where it has not been reported, particularly if this has the effect of merging components which are not initially connected in the single-species graph model. While the effect of changes can be evaluated in similar ways, this problem does not correspond to those outlined above, as it involves the addition of nodes.
This introduces the task of identifying suitable candidate sites for this purpose. As previously mentioned, the Natura 2000 data base was not designed for the kinds of data analysis that would be helpful in solving this problem; consequently, it is not straightforward to support the proposal of candidate nodes using the dataset as reference. A good candidate site ought to be within a set geographical distance from an already connected node, and host the preferred habitat for the target species, or if the node is to act as a 'bridge', at least a suitable set of habitats for a temporary settlement of the species.
Aptness of site vectors and similarity measures
When data on the suitability of habitats is missing or incomplete, one of the measures of site similarity may provide a way to formalize this criterion, by suggesting that a good candidate should be a site that does not host the target species, but has similar properties to those of a site that does. Similarity-based graphs can be a useful tool to express and visualize this notion: if nodes in a similarity-based graph are marked according to their presence on a single-species graph built with the same geographical distance threshold, then an unmarked node that is adjacent to a marked node in the similarity-based graph can be considered as a candidate.
In formal terms, let V be the full set of nodes that represent Natura 2000 sites in a region of interest, and let G s =(V,E s ) be a similarity-based graph built on V with a suitable geographical distance threshold. Let G ′=(V ′,E ′) be a connected component in the single-species graph built on V for the target species of choice (V ′⊆V), with the same geographical distance threshold used for G s . If the following conditions are met:
$$ i \in V', \quad j \in V, \quad j \notin V', \quad (i,j) \in E_{s}, $$
then j∈V can be treated as a good candidate node, and (i,j) is a candidate edge to link j to G ′.
As already pointed out, similarity-based graphs can differ according to three design choices: the dataset from which site vectors are built (e.g. the set of species or the set of habitats), the similarity measure of choice (Jaccard coefficient or cosine similarity), and the threshold value for the similarity score. A simple way to adjust this value is to match it to a desired percentage of kept edges from the raw-distance graph, as explained in "Case study" section, in an attempt to filter nodes that are not relevant, while ensuring that the process is not hindered by a lack of candidate nodes. Conversely, it is nontrivial to determine which dataset and measure provide the best node candidates.
Intuitively, given two similarity-based graphs G s , G t , built with the same threshold values but different vectors and similarity measures, and a number of single-species graphs G 1...G n , built with the same geographical distance threshold used for G s and G t , it can be argued that G s provides better candidates than G t if edges found in the single-species graphs appear in G s more frequently than in G t . Thus, in this study, in order to measure the aptness of the each similarity-based graph defined earlier, their sets of edges were compared to those of the 131 single-species graphs built for the species of interest for the Natura 2000 project, using the same 30 Km distance threshold.
Results for a few sample species and average rates are reported in Table 3, for similarity-based graphs based on Jaccard coefficients and a 0.5 threshold. Average hit rates for the same graphs and for graphs based on cosine similarity are reported in the second column of Table 4. The other columns in the latter table express a normalized version of the hit rate: considering that the raw-distance graph has 706 edges (n), the number of edges of each similarity-based graph is reported (|E|), together with its 'relative density' defined as the ratio |E|/n. If a graph were built with |E| edges chosen randomly, a higher number of edges would result in a higher expectation of hit rate. Therefore, hit rates should be normalized to the number of edges. In order for results to be easier to compare, they are equivalently normalized to the relative density:
$$ \text{Normalized Hit Rate}=\frac{R}{|E|/n}=\frac{R \cdot n}{|E|}. $$
Table 3 Excerpt of the table of hit rates
Table 4 Hit rates in similarity-based graphs
The species-set graph emerges as the one with the best hit rate among graphs built using Jaccard coefficients, as well as the one with the highest normalized hit rate. This could be expected, as this graph is built from the same data as those of single-species graphs. A more striking result is that normalized hit rates are consistently higher for habitat graphs than land-use graphs.
Figure 2 shows the similarity-based graphs built from Jaccard coefficients. It is easily noticeable that the graphs have wide differences, which is confirmed by their different hit rates and network indices. It is interesting to determine whether any pair of similarity-based graphs behave similarly with respect to hit rates. This can be done by comparing the specific hit rates for single species: if the hit rate for two similarity graphs G s , G t were consistently high for the same set of species, it could be argued that the two graphs express an analogous concept.
Similarity-based graph models of Sardinian Natura 2000 sites, with a 0.5 similarity score threshold and a 30 Km distance threshold. a Based on CORINE land use codes. b Based on Natura 2000 habitat codes. c Based on species sets
In order to perform this evaluation, Spearman correlation indices are calculated between pairs of columns reporting hit rates in Table 3 (and likewise for graphs built on cosine similarity). Results are reported in Table 5 for pairs of graphs based on the same similarity measure; since a species-set graph was built only from Jaccard coefficient, correlations were sought between this graph and the four built from cosine similarities. It is interesting that the species-set graph and the habitat graphs are the only ones for which strong correlations are detected (above 0.77). This is another confirmation that nearby sites with similar habitat sets also host similar sets of species, and is consistent with the fact that land use data originates from a different project. It can be argued that the classification of habitats within the Natura 2000 project is indeed more suitable to describe sites from an ecological point of view, than land use codes are.
Table 5 Spearman correlation between sets of hit rates
Correlation of network indices
The comparison and search for correlations can be extended to complex network indices calculated for nodes on the various similarity-based graph instances. The question is whether a higher value calculated for each index on a graph corresponds to a higher value for the same index on another. Considered indices are node degree, closeness and betweenness centrality indices, clustering coefficient, and topological coefficient (Stelzl et al. 2005). Correlations are sought between sets of values for the same index on the possible pairs of graph instances built from the same similarity measures or, once again, between the species-set graph built from Jaccard coefficients and the graphs built from cosine similarity.
Results for graphs based on Jaccard coefficients are reported in Table 6, with a visual representation in Fig. 3. In this case, there is no clear evidence of a strong correlation; however, a moderate degree of correlation can be identified for three indices (degree, topological coefficient and clustering coefficient) between the species-set and the habitats graph. This reinforces the observations that the land-use graph expresses a different concept.
Histogram representation of Spearman correlation of various complex network indices, between pairs of similarity-based graphs built using Jaccard coefficients
Table 6 Spearman correlation of various complex network indices, between pairs of similarity-based graphs built using Jaccard coefficients
The comparison of the species-set graph with land-use graphs and habitat graphs built on cosine similarities, either of occurrence vectors (Table 7 and Fig. 4) or surface area vectors (Table 8 and Fig. 5) leads to similar results: only the habitat graphs show at least a moderate degree of correlation with the species-set graph for some indices.
Histogram representation of Spearman correlation of various complex network indices, between pairs of similarity-based graphs. Species-set graphs built using Jaccard coefficients, land-use and habitats graphs built using cosine similarity of occurrences
Histogram representation of Spearman correlation of various complex network indices, between pairs of similarity-based graphs. Species-set graphs built using Jaccard coefficients, land-use and habitats graphs built using cosine similarity of areas
Table 7 Spearman correlation of various complex network indices, between pairs of similarity-based graphs
Conclusions and future work
As complex network analysis becomes an essential tool for land management, the importance of building refined network models that properly represent the state of an ecological network also rises. Current methods of analysis focus on building graph models with a perspective on a single species of interest; their analysis has proven to be useful to determine the aptness of the network in aiding the preservation of the target species, but the evaluation of high-level properties of the network remains a daunting task. Moreover, in the context of the Natura 2000 project, methods for data collection and storage were not originally designed to assist researchers in data analysis, particularly concerning the proposal of network modifications aimed at the improvement of its indices and degree of connectivity.
In this work, graph models based on site similarity are explored and proposed as a way to address this shortcoming. There are multiple ways to build such models, with variations in site attributes from which vectors are built, similarity measure, and score threshold. Several options have been discussed and compared; results confirm that habitat sets from the Natura 2000 dataset and land use data from the CORINE project express different concepts, and the former is more closely correlated with species sets. This represents a challenge for land managers seeking to detect or establish habitat corridors, due to the fact that only land use data is available for areas outside of Natura 2000 sites.
In spite of this limitation, the possibility to apply the methods outlined in this article in the more general case of arbitrary land patches, using vectors based on CLC codes, opens the possibility for future work, as this approach to the evaluation of a potential habitat corridor can be compared with other methods, or be used to complement them.
Further work can also focus on a formalization of an extension of the network updating problem to be applicable in this context. Land use data for patches outside of Natura 2000 sites can be considered as additional information, even while keeping species-set graphs or habitat graphs as reference, in order to take into account the degree of contiguity of potential corridors introduced in the form of virtual edges.
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This essay is written within the Research Program "Natura 2000: Assessment of management plans and definition of ecological corridors as a complex network", funded by the Autonomous Region of Sardinia (Legge Regionale 7/2007) for the period 2015-2018, under the provisions of the Call for the presentation of "Projects related to fundamental or basic research" in year 2013, implemented at the Department of Mathematics and Computer Science of the University of Cagliari, Italy. Pier Luigi Pau gratefully acknowledges Sardinia Regional Government for the financial support of his PhD scholarship (P.O.R. Sardegna F.S.E. Operational Programme of the Autonomous Region of Sardinia, European Social Fund 2007-2013 - Axis IV Human Resources, Objective l.3, Line of Activity l.3.1.). Danilo Dessì gratefully acknowledges Sardinia Regional Government for the financial support of his PhD scholarship (P.O.R. Sardegna F.S.E. Operational Programme of the Autonomous Region of Sardinia, European Social Fund 2014-2020 - Axis III Education and training, Thematic goal 10, Priority of investment 10ii), Specific goal 10.5.
GF, PLP and DD participated in the design of the analysis. PLP and DD performed the analysis. All authors wrote, read and approved the final manuscript.
Department of Mathematics and Computer Science, Via Ospedale 72, Cagliari, 09124, Italy
Gianni Fenu, Pier Luigi Pau & Danilo Dessì
Gianni Fenu
Pier Luigi Pau
Danilo Dessì
Correspondence to Gianni Fenu.
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.
Fenu, G., Pau, P.L. & Dessì, D. Functional models and extending strategies for ecological networks. Appl Netw Sci 2, 10 (2017). https://doi.org/10.1007/s41109-017-0032-5
Ecological networks
Network modeling
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5 edition of Fuzzy mathematical techniques with applications found in the catalog.
Fuzzy mathematical techniques with applications
by Abraham Kandel
Published 1985 by Addison-Wesley in Reading, Mass .
Fuzzy sets.,
Includes bibliography and index.
Statement by Abraham Kandel.
LC Classifications QA248
Classical and Fuzzy Concepts in Mathematical Logic and Applications discusses how the presence of these facts trigger a stirring, decisive insight into the understanding process. This view shapes this work, reflecting the authors' subjective balance between the scientific and pedagogic components of the textbook. Fuzzy Sets and Fuzzy Logic: Theory and Applications George J. Klir, Bo Yuan Reflecting the tremendous advances that have taken place in the study of fuzzy set theory and fuzzy logic from to the present, this book not only details the theoretical advances in these areas, but considers a broad variety of applications of fuzzy sets and fuzzy.
Volume 2: Applications of Fuzzy Control, Genetic Algorithms and Neural Networks. Author: R. Lowen,A. Verschoren; Publisher: Springer Science & Business Media ISBN: Category: Mathematics Page: View: DOWNLOAD NOW» This is a comprehensive overview of the basics of fuzzy control, which also brings together some recent research . The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set.
What are the main areas of fuzzy logic applications? 9 2 Basic mathematical concepts of fuzzy sets 19 Fuzzy sets versus crisp sets 19 Operations on fuzzy sets 30 Extension principle and fuzzy algebra 34 Extension principle 34 Fuzzy numbers 37 Arithmetic operations with intervals of confidence This book summarizes years of research in the field of fuzzy relational programming, with a special emphasis on geometric models. It discusses the state-of-the-art in fuzzy relational geometric problems, together with key open issues that must be resolved to achieve a more efficient application of this method.
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Fuzzy mathematical techniques with applications by Abraham Kandel Download PDF EPUB FB2
@article{osti_, title = {Fuzzy mathematical techniques with applications}, author = {Kandel, A.}, abstractNote = {This text presents the basic concepts of fuzzy set theory within a context of real-world applications. The book is self-contained and can be used as a starting point for people interested in this fast growing field as well as by researchers looking for new.
Buy Fuzzy Mathematical Techniques With Applications on FREE SHIPPING on qualified orders Fuzzy Mathematical Techniques With Applications: Kandel, Abraham: : BooksCited by: The term "fuzzy sets" has been introduced to describe, in precise mathematical terms, the ambiguities and uncertainties inherent in many real-world pattern recognition and classification problems.
The promotional material attached to this book claims that this is " the first complete presentation of fuzzy mathematics available.".
Additional Physical Format: Online version: Kandel, Abraham. Fuzzy mathematical techniques with applications.
Reading, Mass.: Addison-Wesley, © Besides an extensive state-of-the-art contribution on fuzzy mathematical morphology we present several contributions on a wide variety of topics, including fuzzy filtering, fuzzy image enhancement, fuzzy edge detection, fuzzy image segmentation, fuzzy processing of color images, and applications in medical imaging and robot vision.
This self-contained monograph presents an overview of fuzzy operator theory in mathematical analysis. Concepts, principles, methods, techniques, and applications of fuzzy operator theory are unified in this book to provide an introduction to graduate students and researchers in mathematics, applied sciences, physics, engineering, optimization, and operations by: 2.
Fuzzy Sets, Fuzzy Logic, Fuzzy Methods with Applications. Book June This book presents the basic rudiments of fuzzy set theory and fuzzy logic and their applications in a simple easy to understand manner. The book Author: Chander Mohan.
In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory,artificial intelligence/expert system, etc.
In this volume, methods and applications of fuzzy mathematical programming and. Description: Fuzzy Mathematical Concepts deals with the theory and applications of Fuzzy sets, Fuzzy relations, Fuzzy logic and Rough sets including the theory and applications to Algebra, Topology, Analysis, probability, and Measure Theory.
While the first two chapters deal with basic theory and the prerequisite for the rest of the book. Mathematical Techniques of Fractional Order Systems illustrates advances in linear and nonlinear fractional-order systems relating to many interdisciplinary applications, including biomedical, control, circuits, electromagnetics and security.
The book covers the mathematical background and literature survey of fractional-order calculus and. This self-contained monograph presents an overview of fuzzy operator theory in mathematical analysis.
Concepts, principles, methods, techniques, and applications of fuzzy operator theory are unified in this book to provide an introduction to graduate students and researchers in mathematics, applied sciences, physics, engineering, optimization, and operations research.
Oluwarotimi Williams Samuel, Guanglin Li, in Computational Intelligence for Multimedia Big Data on the Cloud with Engineering Applications, CI Predictor Based on Fuzzy Reasoning Technique.
Fuzzy logic is a multi-value reasoning technique that is based on degrees of truth rather than the usual true or false (1 or 0) Boolean logic. Compared to the conventional. Fuzzy models or sets are mathematical methods that have been widely used to represent and interpret vague and uncertain data and information.
They have been commonly applied to decision-making approaches such as the analytic network process and analytic hierarchy process. Mathematical Techniques of Fractional Order Systems illustrates advances in linear and nonlinear fractional-order systems relating to many interdisciplinary applications, including biomedical, control, circuits, electromagnetics and security.
The book covers the mathematical background and literature survey of fractional-order calculus and generalized fractional-order circuit. lications and has been active in the research and teaching of fuzzy logic since He is the founding Co-Editor-in-Chief of the International Journal of Intelligent and Fuzzy Systems, the co-editor of Fuzzy Logic and Control: Software and Hardware Applications, and the co-editor of Fuzzy Logic and Probability Applications: Bridging the Get this from a library.
Fuzzy Mathematical Programming: Methods and Applications. [Young-Jou Lai; C L Hwang] -- In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory, artificial intelligence/expert system, etc. In this. fuzzy logic pdf download Download fuzzy logic pdf download or read online books in PDF, EPUB, Tuebl, and Mobi Format.
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The Hardcover of the Fuzzy Mathematical Techniques with Applications by Abraham Kandel at Barnes & Noble. FREE Shipping on $35 or more. B&N Outlet Membership Educators Gift Cards Stores & Events HelpAuthor: Abraham Kandel. Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate A Characterization of the Spaces $\mathfrak{S}_{{1/{k + 1}}}^{{k /{k + 1}}} $ by Means of Holomorphic SemigroupsAuthor: Pei-Zhuang Wang, Xian-Tu Peng.
overview the relevance of fuzzy techniques to power system problems, to provide some specific example applications and to provide a brief survey of fuzzy set applications in power systems.
Fuzzy mathematics is a broad field touching on nearly all File Size: KB.Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1 both inclusive.
It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.could call the "heuristic approach to fuzzy control" as opposed to the more recent mathematical focus on fuzzy control where stability analysis is a major theme.
In Chapter 1 we provide an overview of the general methodology for conven-tional control system design. Then we summarize the fuzzy control system design process and contrast the two.
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Radon transform
In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon,[1] who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.
Explanation
If a function$f$ represents an unknown density, then the Radon transform represents the projection data obtained as the output of a tomographic scan. Hence the inverse of the Radon transform can be used to reconstruct the original density from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction.
The Radon transform data is often called a sinogram because the Radon transform of an off-center point source is a sinusoid. Consequently, the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases.
The Radon transform is useful in computed axial tomography (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations.
Definition
Let $f({\textbf {x}})=f(x,y)$ be a function that satisfies the three regularity conditions:[2]
1. $f({\textbf {x}})$ is continuous;
2. the double integral $\iint {\dfrac {\vert f({\textbf {x}})\vert }{\sqrt {x^{2}+y^{2}}}}\,dx\,dy$, extending over the whole plane, converges;
3. for any arbitrary point $(x,y)$ on the plane it holds that $\lim _{r\to \infty }\int _{0}^{2\pi }f(x+r\cos \varphi ,y+r\sin \varphi )\,d\varphi =0.$
The Radon transform, $Rf$, is a function defined on the space of straight lines $L\subset \mathbb {R} ^{2}$ by the line integral along each such line as:
$Rf(L)=\int _{L}f(\mathbf {x} )\vert d\mathbf {x} \vert .$
Concretely, the parametrization of any straight line $L$ with respect to arc length $z$ can always be written:
$(x(z),y(z))={\Big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\Big )}\,$
where $s$ is the distance of $L$ from the origin and $\alpha $ is the angle the normal vector to $L$ makes with the $X$-axis. It follows that the quantities $(\alpha ,s)$ can be considered as coordinates on the space of all lines in $\mathbb {R} ^{2}$, and the Radon transform can be expressed in these coordinates by:
${\begin{aligned}Rf(\alpha ,s)&=\int _{-\infty }^{\infty }f(x(z),y(z))\,dz\\&=\int _{-\infty }^{\infty }f{\big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\big )}\,dz.\end{aligned}}$
More generally, in the $n$-dimensional Euclidean space $\mathbb {R} ^{n}$, the Radon transform of a function $f$ satisfying the regularity conditions is a function $Rf$ on the space $\Sigma _{n}$ of all hyperplanes in $\mathbb {R} ^{n}$. It is defined by:
Shepp Logan phantom
Radon transform
Inverse Radon transform
$Rf(\xi )=\int _{\xi }f(\mathbf {x} )\,d\sigma (\mathbf {x} ),\quad \forall \xi \in \Sigma _{n}$
where the integral is taken with respect to the natural hypersurface measure, $d\sigma $ (generalizing the $\vert d\mathbf {x} \vert $ term from the $2$-dimensional case). Observe that any element of $\Sigma _{n}$ is characterized as the solution locus of an equation $\mathbf {x} \cdot \alpha =s$, where $\alpha \in S^{n-1}$ is a unit vector and $s\in \mathbb {R} $. Thus the $n$-dimensional Radon transform may be rewritten as a function on $S^{n-1}\times \mathbb {R} $ via:
$Rf(\alpha ,s)=\int _{\mathbf {x} \cdot \alpha =s}f(\mathbf {x} )\,d\sigma (\mathbf {x} ).$
It is also possible to generalize the Radon transform still further by integrating instead over $k$-dimensional affine subspaces of $\mathbb {R} ^{n}$. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.
Relationship with the Fourier transform
The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as:
${\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\omega }\,dx.$
For a function of a $2$-vector $\mathbf {x} =(x,y)$, the univariate Fourier transform is:
${\hat {f}}(\mathbf {w} )=\iint _{\mathbb {R} ^{2}}f(\mathbf {x} )e^{-2\pi i\mathbf {x} \cdot \mathbf {w} }\,dx\,dy.$
For convenience, denote ${\mathcal {R}}_{\alpha }[f](s)={\mathcal {R}}[f](\alpha ,s)$. The Fourier slice theorem then states:
${\widehat {{\mathcal {R}}_{\alpha }[f]}}(\sigma )={\hat {f}}(\sigma \mathbf {n} (\alpha ))$
where $\mathbf {n} (\alpha )=(\cos \alpha ,\sin \alpha ).$ Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle $\alpha $ is the one variable Fourier transform of the Radon transform (acquired at angle $\alpha $) of that function. This fact can be used to compute both the Radon transform and its inverse. The result can be generalized into n dimensions:
${\hat {f}}(r\alpha )=\int _{\mathbb {R} }{\mathcal {R}}f(\alpha ,s)e^{-2\pi isr}\,ds.$
Dual transform
The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space $\Sigma _{n}$, the dual Radon transform is the function ${\mathcal {R}}^{*}g$ on Rn defined by:
${\mathcal {R}}^{*}g(\mathbf {x} )=\int _{\mathbf {x} \in \xi }g(\xi )\,d\mu (\xi ).$
The integral here is taken over the set of all hyperplanes incident with the point ${\textbf {x}}\in \mathbb {R} ^{n}$, and the measure $d\mu $ is the unique probability measure on the set $\{\xi |\mathbf {x} \in \xi \}$ invariant under rotations about the point $\mathbf {x} $. Concretely, for the two-dimensional Radon transform, the dual transform is given by:
${\mathcal {R}}^{*}g(\mathbf {x} )={\frac {1}{2\pi }}\int _{\alpha =0}^{2\pi }g(\alpha ,\mathbf {n} (\alpha )\cdot \mathbf {x} )\,d\alpha .$
In the context of image processing, the dual transform is commonly called back-projection[3] as it takes a function defined on each line in the plane and 'smears' or projects it back over the line to produce an image.
Intertwining property
Let $\Delta $ denote the Laplacian on $\mathbb {R} ^{n}$ defined by:
$\Delta ={\frac {\partial ^{2}}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}}{\partial x_{n}^{2}}}$
This is a natural rotationally invariant second-order differential operator. On $\Sigma _{n}$, the "radial" second derivative $Lf(\alpha ,s)\equiv {\frac {\partial ^{2}}{\partial s^{2}}}f(\alpha ,s)$ is also rotationally invariant. The Radon transform and its dual are intertwining operators for these two differential operators in the sense that:[4]
${\mathcal {R}}(\Delta f)=L({\mathcal {R}}f),\quad {\mathcal {R}}^{*}(Lg)=\Delta ({\mathcal {R}}^{*}g).$
In analysing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.[5] In imaging[6] and numerical analysis[7] this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.
Reconstruction approaches
The process of reconstruction produces the image (or function $f$ in the previous section) from its projection data. Reconstruction is an inverse problem.
Radon inversion formula
In the two-dimensional case, the most commonly used analytical formula to recover $f$ from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula[8]:
$f(\mathbf {x} )=\int _{0}^{\pi }({\mathcal {R}}f(\cdot ,\theta )*h)(\left\langle \mathbf {x} ,\mathbf {n} _{\theta }\right\rangle )\,d\theta $
where $h$ is such that ${\hat {h}}(k)=|k|$.[8] The convolution kernel $h$ is referred to as Ramp filter in some literature.
Ill-posedness
Intuitively, in the filtered back-projection formula, by analogy with differentiation, for which $ \left({\widehat {{\frac {d}{dx}}f}}\right)\!(k)=ik{\widehat {f}}(k)$, we see that the filter performs an operation similar to a derivative. Roughly speaking, then, the filter makes objects more singular. A quantitive statement of the ill-posedness of Radon inversion goes as follows:
${\widehat {{\mathcal {R}}^{*}{\mathcal {R}}g}}(k)={\frac {1}{\|\mathbf {k} \|}}{\hat {g}}(\mathbf {k} )$
where ${\mathcal {R}}^{*}$ is the previously defined adjoint to the Radon Transform. Thus for $g(\mathbf {x} )=e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$, we have:
${\mathcal {R}}^{*}{\mathcal {R}}g={\frac {1}{\|\mathbf {k_{0}} \|}}e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$
The complex exponential $e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$ is thus an eigenfunction of ${\mathcal {R}}^{*}{\mathcal {R}}$ with eigenvalue $ {\frac {1}{\|\mathbf {k} _{0}\|}}$. Thus the singular values of ${\mathcal {R}}$ are $ {\frac {1}{\sqrt {\|\mathbf {k} \|}}}$. Since these singular values tend to $0$, ${\mathcal {R}}^{-1}$ is unbounded.[8]
Iterative reconstruction methods
Compared with the Filtered Back-projection method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the Filtered Back-projection method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (e.g. iterative Sparse Asymptotic Minimum Variance[9]) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.
Inversion formulas
Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in $n$ dimensions can be inverted by the formula:[10]
$c_{n}f=(-\Delta )^{(n-1)/2}R^{*}Rf\,$
where $c_{n}=(4\pi )^{(n-1)/2}{\frac {\Gamma (n/2)}{\Gamma (1/2)}}$, and the power of the Laplacian $(-\Delta )^{(n-1)/2}$ is defined as a pseudo-differential operator if necessary by the Fourier transform:
$\left[{\mathcal {F}}(-\Delta )^{(n-1)/2}\varphi \right](\xi )=|2\pi \xi |^{n-1}({\mathcal {F}}\varphi )(\xi ).$
For computational purposes, the power of the Laplacian is commuted with the dual transform $R^{*}$ to give:[11]
$c_{n}f={\begin{cases}R^{*}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ odd}}\\R^{*}{\mathcal {H}}_{s}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ even}}\end{cases}}$
where ${\mathcal {H}}_{s}$ is the Hilbert transform with respect to the s variable. In two dimensions, the operator ${\mathcal {H}}_{s}{\frac {d}{ds}}$ appears in image processing as a ramp filter.[12] One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function $f$ of two variables:
$f={\frac {1}{2}}R^{*}H_{s}{\frac {d}{ds}}Rf.$
Thus in an image processing context the original image $f$ can be recovered from the 'sinogram' data $Rf$ by applying a ramp filter (in the $s$ variable) and then back-projecting. As the filtering step can be performed efficiently (for example using digital signal processing techniques) and the back projection step is simply an accumulation of values in the pixels of the image, this results in a highly efficient, and hence widely used, algorithm. Explicitly, the inversion formula obtained by the latter method is:[3]
$f(x)={\begin{cases}\displaystyle -\imath 2\pi (2\pi )^{-n}(-1)^{n/2}\int _{S^{n-1}}{\frac {\partial ^{n-1}}{2\partial s^{n-1}}}Rf(\alpha ,\alpha \cdot x)\,d\alpha &n{\text{ odd}}\\\displaystyle (2\pi )^{-n}(-1)^{n/2}\iint _{\mathbb {R} \times S^{n-1}}{\frac {\partial ^{n-1}}{q\partial s^{n-1}}}Rf(\alpha ,\alpha \cdot x+q)\,d\alpha \,dq&n{\text{ even}}\\\end{cases}}$
The dual transform can also be inverted by an analogous formula:
$c_{n}g=(-L)^{(n-1)/2}R(R^{*}g).\,$
Radon transform in algebraic geometry
In algebraic geometry, a Radon transform (also known as the Brylinski–Radon transform) is constructed as follows.
Write
$\mathbf {P} ^{d}\,{\stackrel {p_{1}}{\gets }}\,H\,{\stackrel {p_{2}}{\to }}\,\mathbf {P} ^{\vee ,d}$
for the universal hyperplane, i.e., H consists of pairs (x, h) where x is a point in d-dimensional projective space $\mathbf {P} ^{d}$ and h is a point in the dual projective space (in other words, x is a line through the origin in (d+1)-dimensional affine space, and h is a hyperplane in that space) such that x is contained in h.
Then the Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves
$\operatorname {Rad} :=Rp_{2,*}p_{1}^{*}:D(\mathbf {P} ^{d})\to D(\mathbf {P} ^{\vee ,d}).$ :=Rp_{2,*}p_{1}^{*}:D(\mathbf {P} ^{d})\to D(\mathbf {P} ^{\vee ,d}).}
The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves on the projective space and its dual projective space, up to constant sheaves.[13]
See also
• Periodogram
• Matched filter
• Deconvolution
• X-ray transform
• Funk transform
• The Hough transform, when written in a continuous form, is very similar, if not equivalent, to the Radon transform.[14]
• Cauchy–Crofton theorem is a closely related formula for computing the length of curves in space.
• Fast Fourier transform
Notes
1. Radon 1917.
2. Radon 1986.
3. Roerdink 2001.
4. Helgason 1984, Lemma I.2.1.
5. Lax, P. D.; Philips, R. S. (1964). "Scattering theory". Bull. Amer. Math. Soc. 70 (1): 130–142. doi:10.1090/s0002-9904-1964-11051-x.
6. Bonneel, N.; Rabin, J.; Peyre, G.; Pfister, H. (2015). "Sliced and Radon Wasserstein Barycenters of Measures". Journal of Mathematical Imaging and Vision. 51 (1): 22–25. doi:10.1007/s10851-014-0506-3. S2CID 1907942.
7. Rim, D. (2018). "Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform". SIAM J. Sci. Comput. 40 (6): A4184–A4207. arXiv:1705.03609. Bibcode:2018SJSC...40A4184R. doi:10.1137/17m1135633. S2CID 115193737.
8. Candès 2016b.
9. Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. IEEE. 61 (4): 933–944. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X. S2CID 16276001.
10. Helgason 1984, Theorem I.2.13.
11. Helgason 1984, Theorem I.2.16.
12. Nygren 1997.
13. Kiehl & Weissauer (2001, Ch. IV, Cor. 2.4)
14. van Ginkel, Hendricks & van Vliet 2004.
References
• Kiehl, Reinhardt; Weissauer, Rainer (2001), Weil conjectures, perverse sheaves and l'adic Fourier transform, Springer, doi:10.1007/978-3-662-04576-3, ISBN 3-540-41457-6, MR 1855066
• Radon, Johann (1917), "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten", Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse [Reports on the Proceedings of the Royal Saxonian Academy of Sciences at Leipzig, Mathematical and Physical Section], Leipzig: Teubner (69): 262–277;
Translation: Radon, J. (December 1986), translated by Parks, P.C., "On the determination of functions from their integral values along certain manifolds", IEEE Transactions on Medical Imaging, 5 (4): 170–176, doi:10.1109/TMI.1986.4307775, PMID 18244009, S2CID 26553287.
• Roerdink, J.B.T.M. (2001) [1994], "Tomography", Encyclopedia of Mathematics, EMS Press.
• Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3.
• Candès, Emmanuel (February 2, 2016a). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 9" (PDF).
• Candès, Emmanuel (February 4, 2016b). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 10" (PDF).
• Nygren, Anders J. (1997). "Filtered Back Projection". Tomographic Reconstruction of SPECT Data.
• van Ginkel, M.; Hendricks, C.L. Luengo; van Vliet, L.J. (2004). "A short introduction to the Radon and Hough transforms and how they relate to each other" (PDF). Archived (PDF) from the original on 2016-07-29.
Further reading
• Lokenath Debnath; Dambaru Bhatta (19 April 2016). Integral Transforms and Their Applications. CRC Press. ISBN 978-1-4200-1091-6.
• Deans, Stanley R. (1983), The Radon Transform and Some of Its Applications, New York: John Wiley & Sons
• Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/surv/039, ISBN 978-0-8218-4530-1, MR 2463854
• Herman, Gabor T. (2009), Fundamentals of Computerized Tomography: Image Reconstruction from Projections (2nd ed.), Springer, ISBN 978-1-85233-617-2
• Minlos, R.A. (2001) [1994], "Radon transform", Encyclopedia of Mathematics, EMS Press
• Natterer, Frank (June 2001), The Mathematics of Computerized Tomography, Classics in Applied Mathematics, vol. 32, Society for Industrial and Applied Mathematics, ISBN 0-89871-493-1
• Natterer, Frank; Wübbeling, Frank (2001), Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, ISBN 0-89871-472-9
External links
• Weisstein, Eric W. "Radon Transform". MathWorld.
• Analytical projection (the Radon transform) (video). Part of the "Computed Tomography and the ASTRA Toolbox" course. University of Antwerp. September 10, 2015.
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09:00 - 10:00 Eduardo Garrido (Spanish National Research Council)
Few-body techniques using coordinate space for bound and continuum states I
Few-body techniques using coordinate space for bound and continuum states II
11:30 - 12:30 Mario Gattobigio (Université de Nice - Sophia Antipolis)
Introduction to Efimov physics I
In these series of lectures I will give an introduction to Efimov physics. After the derivation of the Efimov effect, I will discuss its relation with symmetries, broken (scale invariance) and unbroken (discrete scale invariance), and with the concept of Universality. I will also introduce range corrections and I will show their use in comparing potential-model calculations and/or experimental measurements with Efimov theory. Finally, I will show how Efimov physics, which is predominantly a three-body physics, extends its influence over systems with more particles.
Lunch and discussion
14:00 - 16:00 Exercise
Few-body techniques using coordinate space for bound and continuum states
Universality in few-body systems
09:00 - 10:00 Manuel Valiente (Heriot-Watt University)
From few to many-body degrees of freedom I
In these lectures I will focus on the use of microscopic, few-body techniques that are relevant in the many-body problem. These methods can be divided into indirect and direct. In particular, indirect methods are concerned with the simplification of the many-body problem by substituting the full, microscopic interactions by pseudopotentials which are designed to reproduce collisional information at specified energies, or binding energies in the few-body sector. These simplified interactions yield more tractable theories of the many-body problem, and are equivalent to effective field theory of interactions. Direct methods, which so far are most useful in one spatial dimension, have the goal of attacking the many-body problem at once by using few-body information only. Here, I will present non-perturbative direct methods to study one-dimensional fermionic and bosonic gases in one dimension.
From few to many-body degrees of freedom II
11:30 - 12:30 Marcelo Yamashita (Sao Paulo State University)
Few-body techniques using momentum space for bound and continuum states I
This will be a set of 4 invited lectures
From few to many-body degrees of freedom
Few-body techniques using momentum space for bound and continuum states
Introduction to Efimov physics II
Introduction to Efimov Physics III
Few-body techniques using coordinate space for bound and continuum states III
Few-body techniques using momentum space for bound and continuum states II
Few-body techniques using momentum space for bound and continuum states III
From few to many-body degrees of freedom III
09:00 - 10:20 Student talks
Wael Elkamhawy (TU Darmstadt)
Description of 31Ne in Halo EFT
Previous investigations of 31Ne via 1n-removal reactions on C and Pb targets revealed that it is a deformed nucleus with a significant p-wave halo component. We construct a p-wave halo effective field theory for 31Ne in order to provide an appropriate framework for its description. Within this framework, we establish correlations between different observables, which enables us to make theoretical predictions for the properties of 31Ne.
Fabian Hildenbrand (TU Darmstadt)
Effective field theory for three-body hypernuclei
We construct a short-range effective field theory with contact interactions for three-body hypernuclei in the strangeness $S=-1$ sector. An asymptotic analysis is performed in the $I=0$ and $I=1$ isospin channels and the corresponding effective Lagrangians are constructed. It turns out that a $\Lambda$NN three-body force is required for consistent renormalisation in both channels. We present universal correlations between observables and discuss the possibility of a $\Lambda$nn bound state in this effective theory.
Marcel Schmidt (TU Darmstadt)
Halo effective field theory for nuclear reactions
Direct reaction measurements serve as powerful tools to study halo nuclei. As an archetype example of such an exotic nucleus, Beryllium-11 consists of a tightly-bound core and a weakly-bound valence neutron in the two halo states $1/2^+$ and $1/2^-$. In this talk, we propose an effective field theory for the transfer reaction Be10(d,p)Be11 which exploits the remarkable energy scale separation of the halo system. The theory contains non-relativistic fields for the valence neutron, the proton as well as the core. In contrast, the deuteron and both Beryllium-11 states are generated dynamically in the respective two-body systems. They are treated using auxiliary fields. We diagrammatically construct the full non-perturbative scattering amplitude in the strong sector including optical potentials for high-momentum loss channels. In contrast, long-range Coulomb interactions are treated perturbatively. Finally, we present calculations for the differential reaction cross section and compare it with experimental results.
Lucas Souza (Technnological Institute of Aeronautics)
Reactions and structure of three-fragment weakly bound nuclei
Light halo nuclei form part of the neutron-rich drip line. We investigate how structure effects of two-neutron halo nuclei like $^{20}$C and $^{22}$C appears from some reactions. We discuss two kind of reactions being: the two-neutron halo breakup reaction on a heavy target and also, we consider the possibility of finding Efimov excited states on $^{20}$C in a proton collision. Two-fragment nuclei when scatter from a target may undergo a fragmentation process where one of the fragment is observed while the other fragment and the target are not. The inclusive breakup processes are important as the singles spectra can supply important information about the unobserved two-body subsystem. The extention these three-body theories to derive an expression for the fragment yield in the reaction $A(a,b)X$, was done in Ref.[1], where the projectile is $a=x_1+x_2+b$. Borromean, two-nucleon, halo nuclei are examples of unstable three-fragments projectiles. Here we use a model from [1] to treat inclusive non-elastic break up reactions involving weakly bound three-cluster nuclei. The inclusive breakup cross section is the sum of a generalized four-body form of the elastic breakup cross section plus the inclusive non-elastic breakup cross section that involves the ``reaction'' cross section, of the participant fragments, $x_1$ and $x_2$. This latter one, objective of our studies in principle, contains the incomplete fusion of the three-fragment projectile. We also focus to describe the elastic scattering of a proton reaction on a neutron-neutron-core halo nuclei. The first excited state of the $^{20}$C is obtained from neutron colision leading to a Efimov polarization potential. Was computed the transition form factor of the momentum transfer by the proton to the halo nuclei. \newline \newline [1] B.V. Carlson, T. Frederico, M.S. Hussein; Phys. Lett. B, {\bf 767}, (2017) 53.
Olga Klimenko (Joint Institute for Nuclear Research)
Helium trimer via Faddeev differential equation
Christiane Schmickler (TU Darmstadt)
Tetramer bound states in heteronuclear systems
We calculate the universal spectrum of trimer and tetramer states in heteronuclear mixtures of ultracold atoms with different masses in the vicinity of the heavy-light dimer threshold. To extract the energies, we solve the three- and four-body problem for simple two- and three-body potentials tuned to the universal region using the Gaussian expansion method. We focus on the case of one light particle of mass $m$ and two or three heavy bosons of mass $M$ with resonant heavy-light interactions. We find that trimer and tetramer cross into the heavy-light dimer threshold at almost the same point and that as the mass ratio $M/m$ decreases, the distance between the thresholds for trimer and tetramer states becomes smaller. We also comment on the possibility of observing exotic three-body states consisting of a dimer and two atoms in this region and compare with previous work.
Elkjaer Rasmussen Stig (Aarhus University)
Window for Efimov physics for few-body systems with finite-range interactions
We investigate the two lowest-lying weakly bound states of $N\leq 8$ bosons as functions of the strength of two-body Gaussian interactions. We observe the limit for validity of Efimov physics. We calculate energies and second radial moments as functions of scattering length. For identical bosons we find that two $(N − 1)$-body states appear before the $N$-body ground states become bound. This pattern ceases to exist for $N\geq 7$ where the size of the ground state becomes smaller than the range of the two-body potential. All mean-square-radii for $N \geq 4$ remain finite at the threshold of zero binding, where they vary as $(N − 1)^p$ with $p = −3/2, −3$ for ground and excited states, respectively. Decreasing the mass of one particle we find stronger binding and smaller radii. The identical particles form a symmetric system, while the lighter particle is further away in the ground states. In the excited states we find the identical bosons either surrounded or surrounding the light particle for few or many bosons, respectively. We demonstrate that the first excited states for all strengths resemble two-body halos of one particle weakly bound to a dense $N$-body system for $N = 3, 4$. This structure ceases to exist for $N \geq 5$.
Thomas Secker (Eindhoven University of Technology)
Finite-range Efimov physics in the unitary Bose gas
The recent experimental exploration of the unitary Bose gas opens up new directions in the study of strongly-correlated quantum many-body physics. This strongly interacting Bose system is deeply affected by three-body phenomena such as the universal Efimov effect. Experiments with atomic gases revealed that the long-range part of the two-body interaction is crucial in this context. Therefore, we study finite range effects of three strongly-interacting particles via the off-shell two-body T-operator in momentum space. We investigate this operator both for simple square well and for full coupled channel interactions close to Feshbach resonances, which we then expand in separable terms to efficiently solve the three body Faddeev equations in momentum space.
Artem Korobitsin (Joint Institute for Nuclear Research)
Theoretical study of the neon clusters
Small clusters of rare gas atoms are of a great interest in the recent years. They belong to a large class of molecules interacting via potentials of van-der-Waals type and have unique quantum properties. One of these properties is the Efimov effect [1]. This effect reflects the difference in the properties of the two-body and the three - body systems. When there are at least two subsystems of zero binding energy, the three - body system has an infinite number of weakly bound states - this is the essence of the Efimov effect. Calculations of ultracold three - body clusters require methods suitable for solving three - body bound state and scattering problems in configuration space [2]. One of the effective methods for studying three - particle systems is based on using the differential Faddeev equations in the total angular momentum representation [3]. $\quad$ This work is aimed at a theoretical investigation of the neon atomic clusters. We developed a numerical algorithm for solving differential Faddeev equations in the total angular momentum representation [3]. This algorithm has been realized in the programming language C++. The developed numerically effective computational scheme, especially in combination with an option of using multiple processors, makes it possible to calculate wide range of three - body problems. For the calculations of the spectrum of neon trimer we use finite-difference approximation and cubic polynomial splines for solving the differential Faddeev equations with the zero asymptotic boundary conditions. To increase the speed of calculation a template library Eigen [4] for linear algebra is used. We have applied developed numerical algorithm for solving the above mentioned equations for the $^{20}$Ne three - atomic system. To describe the interatomic interaction the realistic potentials HFD-B [5] and TT [6] were used. The calculated results of binding energies of the ground and the first excited states for neon trimer are in a good agreement with the results obtained using different methods by other authors. \noindent{\textbf{References}}\\ $[1]$ V.N. Efimov, Phys. Atom. Nucl. \textbf{12}, 1080 (1970); Phys. Lett. B \textbf{33}, 563 (1970)\\ $[2]$ E.A. Kolganova, A.K. Motovilov and W. Sandhas Few-Body Syst. \textbf{51}, 249--257 (2011)\\ $[3]$ V.V. Kostrykin, A.A. Kvitsinsky and S.P. Merkuriev, Few-Body Syst. \textbf{6}, 97 (1989); \\ A.A. Kvitsinsky and C.-Y. Hu, Few-Body Syst. \textbf{12}, 7 (1992); V.A. Roudnev, S.L. Yakovlev and\\ S.A. Sofianos, Few-Body Syst. \textbf{37} 179 (2005)\\ $[4]$ \url{http://eigen.tuxfamily.org} $[5]$ R.A. Aziz and M.J. Slaman, J.Chem.Phys. \textbf{130}, 187 (1989)\\ $[6]$ K.T. Tang and J.P. Toennies, J.Chem.Phys. \textbf{118}, 4976-4983 (2003)\\
John Hadder Sandoval Quesada (Sao Paulo State University)
From 3D to 1D: squeezing the Efimov effect
We present a method to study the continuous dimensional transition of the quantum mechanical three-body problem. The theoretical development presented here can be extended to any asymmetric three-body systems with particles interacting by a short-range potential . We considered an AAB system formed by two identical bosons, A, and a different particle, B. We calculate the energy spectrum and show how it changes when we move from three to two and one spatial dimension.
Derick Rosa (São Paulo State University)
Bound states of a light atom and two heavy dipoles in two dimensions
We study a three-body system, formed by a light particle and two identical heavy dipoles, in two dimensions in the Born-Oppenheimer approximation. We present the analytic light-particle wave function resulting from an attractive zero-range potential between the light and each of the heavy particles. It expresses the large-distance universal properties which must be reproduced by all realistic short-range interactions. We calculate the three-body spectrum for zero heavy-heavy interaction as function of light to heavy mass ratio. We discuss the relatively small deviations from Coulomb estimates and the degeneracies related to radial nodes and angular momentum quantum numbers. We include a repulsive dipole-dipole interaction and investigate the three-body solutions as functions of strength and dipole direction. Avoided crossings occur between levels localized in the emerging small and large-distance minima, respectively. The characteristic exchange of properties like mean square radii are calculated. Simulation of quantum information transfer is suggested. For large heavy-heavy particle repulsion all bound states have disappeared into the continuum. The corresponding critical strength is inversely proportional to the square of the mass ratio, far from the linear dependence from the Landau criterion
Rafael Barfknecht (Aarhus University)
Dynamical realization of magnetic states in a strongly interacting Bose mixture
We describe the dynamical preparation of magnetic states in a strongly interacting two-component Bose gas in a harmonic trap. By mapping this system to an effective spin chain model, we obtain the dynamical spin densities and the fidelities for a few-body system. We show that the spatial profiles transit between ferromagnetic and antiferromagnetic states as the intraspecies interaction parameter is slowly increased.
Oleksandr Marchukov (Tel Aviv University)
Phase fluctuation in fragmented BECs
When describing interacting Bose gases with a finite number of particles confined in 1D traps, the mean-field approach appears to not reproduce phase fluctuations accurately. In order to improve this description the Bose gas should be considered fragmented, i.e. contain several states into which the gas could condense. More formally, following Penrose and Onsager, it means that there are more than one macroscopically occupied single-particle states, in contrast to the "simple" BEC. Then the beyond-mean-field many-body methods are required. In this work we consider a mesoscopic number of weakly-interacting bosons trapped in a 1D harmonic potential using multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X). We calculate the fragmented ground state of the system for different interaction strengths. Using the single-particle density matrix of the gas we calculate quantum phase fluctuations and find that they may serve as a definite signature of a fragmentation in a condensate. Additionally, the discrepancies between the mean-field and self-consistent many-body results may be used as a benchmark for the MCTDHX method, which is to the best of our knowledge still lacking in cold atomic gases. Moreover, for a two-boson system we demonstrate a very satisfactory agreement between the MCTDH-X, Monte Carlo (for hard-core bosons) and analytical results.
Usui Ayaka (Okinawa Institute of Science and Technology Graduate University)
Dynamical phase transition of a Tonks-Girardeau gas
We investigate Dynamical Phase Transitions (DPTs) in a strongly correlated quantum gas in one dimension, which is Tonks-Girardeau Gas (TG gas), after a quench. We consider situations that DPTs are created by suddenly turning on a lattice potential in an infinite well potential, which is also called as pinning phase transition. The momentum distribution of TG gas can be described with just single-particle states, owing to strong repulsive interaction. This makes it much faster to compute momentum distribution. As a result, it is found that the peak of momentum distribution tells us DPTs, as the peak drops when the number $N_lat$ of wells in the lattice corresponds to the particle number $N_p$ like the Mott-insulator phase. Also, when the number $N_lat$ corresponds to divisors of particle number $N_p$, that is $N_p / N_lat = n$ with integer $n$, the peak of momentum distribution gets depressed as the lattice is deep enough. TG gas is a realistic system for experiments, so it is expected that our results are confirmed.
Andreas Bock Michelsen (Aarhus University)
A new method for constructing spin chains from spectral data
A 1D few body system of aligned spins with a single oppositely alig- ned spin (a spin chain) is a system showing great promise in quantum informatics. Many of the desired applications of such a system can be ex- pressed through the speci cation of a certain spectrum of eigenvalues and eigenvectors. This project seeks to develop a method for systematically predicting which symmetric trapping potential would yield a speci c type of spectrum, using recently published code, machine learning and local density approximations to make the process as automatic as possible. Researchers will be able to use this to explore the feasibility of realizing any such system with a speci c spectrum, ultimately resulting in an immediate blueprint for experimental implementation. This will make it easy to explore new systems for quantum state transfer, robust informa- tion storage, or something else entirely. The poster presents the various observations and tools used in the process, as well as the promising rst results of its implementation, exploring new implementations of quantum communication spin chains resilient to disruption by external coupling.
Evgenii Mardyban (Joint Institute for Nuclear Research)
Description of alternating parity bands in heavy nuclei using supersymmetric quantum mechanics
Andre Chaves (University of Minho)
Optical properties of excitons in 2D materials
We present a unified description of the excitonic properties of four monolayer transition-metal dichalcogenides (TMDC's) using an equation of motion method for deriving the Bethe-Salpeter equation in momentum space. Our method is able to cope with both continuous and tight-binding Hamiltonians, and is less computational demanding than the traditional first-principles approach. We show that the role of the exchange energy is essential to obtain a good description of the binding energy of the excitons. The exchange energy at the $\Gamma-$point is also essential to obtain the correct position of the C-exciton peak. Using our model we obtain a good agreement between the Rydberg series measured for WS$_2$. We discuss how the absorption and the Rydberg series depend on the doping. Choosing the interaction parameter and the doping we obtain a good qualitative agreement between the experimental absorption and our calculations for WS$_2$.
Anoop Divakaran (Malaviya National Institute of Technology)
Surface oxidation in Bi2Te3 and its effect on TI surface states
Bismuth telluride is a well-known topological insulator having a large number of applications in the field of quantum computing and spintronics. Topologically protected surface states in these materials resemble two dimensions confinements, which can host quantum transport phenomena such as quantum Hall effect. Oxidation of exposed TI materials plays a major role in deteriorating the surface state properties by adding impurity charge carriers on the surface. We have investigated the process of surface oxidation in chemically synthesised Bismuth telluride(BT) nanostructures. Different spectroscopic techniques like Raman, soft X-ray absorption, and X-ray photoemission spectroscopy was used and confirmed the formation of individual oxides. We could observe two additional modes corresponding to $\alpha- $ Bi${_2} $O${_3}$) and TeO${_2}$) in Raman spectra of BT nanostructures. Considerable increase in the quantity of Bi${_2} $Te${_3}$ in nanostructures than in single crystals, confirmed by XPS studies indicate the dependence of oxidation on the surface to volume ratio. Surface encapsulation of these materials has resulted in a reduction of surface oxidation to a great extent without affecting other physical properties. This study is imperative to perceive the generic trend in the surface oxidation, its dimensionality dependence which is well correlated with the oxidation behaviour of single crystals. Sound understanding of bond formation in oxides help us to predict the carrier dynamics of surface states after exposure. This information can be used to model real world TI systems and to extract pure surface state properties to better explore them.
Jorge Henrique de Alvarenga Nogueira (Università degli Studi di Roma La Sapienza)
Three-body bound states with zero-range interaction in the Bethe-Salpeter approach
The Bethe–Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe–Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: (i) For weak enough two-body interaction the three-body system is unbound. (ii) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. (iii) For even stronger two-body interaction this state becomes unphysical with a negative mass squared. However, another physical (excited) state appears, found previously in light-front calculations. The Bethe–Salpeter approach implicitly incorporates three-body forces of relativistic origin, which are attractive and increase the binding energy.
Registration in guest house 4
08:45 - 09:00 Roderich Moessner (MPIPKS) & Scientific Coordinators
Efimov physics and nuclear physics: where are we?
Efimov Physics originated from a reflection on nuclear physics and in particular from the observation that the light nuclei spectrum presents shallow states, the deuteron and the triton. Afterwards, the possibility to change and tune the inter-particle interaction made atomic physics the privileged playground of Efimov physics allowing fantastic achievements both from the experimental and theoretical sides. In the last few years there is a renewed effort bring Efimov physics back to the origins, back to nuclear physics, and in this talk I'll try to understand where we are now in this challenge.
09:40 - 10:20 Petar Stipanović (University of Split)
Stability and universality of few-body systems
Ground state properties of weakly bound few-body systems consisting of helium and spin-polarized hydrogen isotopes, and alkali atoms have been explored. Stability of different cluster species has been tested with a special emphasis on a universality of quantum halo states - weakly bound systems with a radius extending well into the classically forbidden region. The study of realistic systems is supplemented by model calculations in order to analyze how low-energy properties depend on the interaction potential. The use of variational and diffusion Monte Carlo methods enabled very precise calculation of both size and binding energy of few-body systems. Using dimensionless measures of the binding energy and cluster size, studied atomic clusters are compared to other known halos in different fields of physics. Different characteristic scaling lengths, which make size-energy ratio to be universal, are tested. As the scaled binding energy decreases, samba and tango type trimers separate from Borromean type~[1]. Research is extended to tetramers and pentamers. Furthermore, the structural properties of different trimers are compared with the most recent experimental results~[2,3] obtained by Coulomb explosion imaging of diffracted clusters. [1] Stipanovi\'{c} P. et al., Phys. Rev. Lett. \textbf{113}, 253401 (2014). [2] Voigtsberger J. et al., Nature Communications \textbf{5}, 5765 (2014). [3] Kunitski M. et al., Science \textbf{348}, 551 (2015).
10:50 - 11:30 Doerte Blume (The University of Oklahoma)
Few-body physics of spin-orbit coupled cold atom systems
This talk considers selected aspects of spin-orbit coupled few-atom systems. Spin-orbit coupling, which leads to a locking of the spin and the spatial degrees of freedom, can be realized in cold atom systems in a variety of ways. This contribution discusses the modifications of the few-body dynamics due to the spin-orbit coupling. Scattering and bound state properties will be considered.
11:30 - 12:10 Yusuke Nishida (Tokyo Institute of Technology)
Zoo of quantum halos
Wave-particle duality in quantum mechanics allows for a halo bound state whose spatial extension far exceeds a range of the interaction potential. What is even more striking is that such quantum halos can be arbitrarily large on special occasions. The two examples known so far are the Efimov effect and the super Efimov effect, which predict that spatial extensions of higher excited states grow exponentially and double-exponentially, respectively [1,2]. Here we establish yet a new class of arbitrarily large quantum halos formed by spinless bosons with short-range interactions in two dimensions [3]. When the two-body interaction is absent but the three-body interaction is resonant, four bosons exhibit an infinite tower of bound states whose spatial extensions scale as $R_n\sim e^{(\pi n)^2/27}$ for large $n$. The emergent scaling law is universal and termed a semi-super Efimov effect, which together with the Efimov and super Efimov effects constitutes a trio of few-body universality classes allowing for arbitrarily large quantum halos. [1] V. Efimov, "Energy levels arising from resonant two-body forces in a three-body system," Phys. Lett. B 33, 563-564 (1970). [2] Y. Nishida, S. Moroz, and D. T. Son, "Super Efimov effect of resonantly interacting fermions in two dimensions," Phys. Rev. Lett. 110, 235301 (2013). [3] Y. Nishida, "Semi-super Efimov effect of two-dimensional bosons at a three-body resonance," to be published in Phys. Rev. Lett. (2017).
12:10 - 12:50 Betzalel Bazak (Institut de Physique Nucléaire Orsay)
Universal states and Efimov physics in fermionic mixtures
The system of few identical fermions interacting resonantly with a distinguishable atom exhibits a rich and interesting physics, including universal states and the celebrated Efimov effect. The (2+1) system, composed of two heavy fermions and lighter atom, supports a universal trimer state if the ratio of the particle masses exceeds critical value. For even larger mass ratio the system becomes Efimovian, introducing a three-body scale and showing geometric series of bound states. Interestingly, this trend continues in the (3+1) system as well as in the (4+1) system, having their own universal states and pure (N+1)-body Efimov effects. Adding another particle, however, this series seems to stop. This should be a sign of a shell structure and screening effects, which may shed light on the crossover from the few-body systems to the many-body polaron case.
Connection between two- and three-body systems in an oscillator trap and D-dimensional calculations
In this work we investigate three-body systems when the dimension changes in a continuous way from three (3D) to two (2D) dimensions. This amounts to confining the particles into a narrower and narrower layer, such that, eventually, when the layer has zero width, the particles are forced to move in 2D. In practice, this can be done by putting the particles under the effect of an external trap potential confining the particles in the space. In particular, this can be done by means of a harmonic oscillator potential in the z-coordinate. For two-body systems the numerical implementation of the external field is simple, and it does not present particular problems. However, for three-body systems, although conceptually the procedure is exactly the same, the numerical difficulties increase when the frequency of the harmonic oscillator increases. In fact, for very large frequencies, i.e., when approaching 2D, the method is quite inefficient. For this reason, in this work we propose to implement the confinement of the particles, not by means of an external potential, but by introducing the dimension d as a parameter in the Schrödinger (or Faddeev) equations to be solved. The dimension is then allowed to take non-integer values within the range 2 ≤ d ≤ 3. The purpose of this work is twofold. First, we want to see the connection between the two confinement methods mentioned above. It is necessary to see the equivalence between a given value of the confining harmonic oscillator frequency and the dimension d describing the same physical situation. Once this is done, we shall use the second method, which is numerically much simpler, to investigate the Efimov states in mass imbalanced systems, focusing in particular on how those states disappear when increasing the confinement of the particles.
Dimensional effects in three-body systems
Some phenomena in three-body systems can be drastically affected by the dimension where the system is inserted. The Efimov effect is a remarkable example: it exists only in three spatial dimensions. However, if we consider non-integer dimensions, the Efimov effect is allowed for dimensions (d) in the interval 2.3 < d < 3.8, where this result holds for three identical bosons. In this presentation I will show a very simple method to extend this interval for a general AAB system. I will also show a framework for studying the three-body problem as one continuously changes the dimensionality of the system. Some interesting results for two dimensions will be detailed.
16:30 - 17:30 Ramin Golestanian (University of Oxford)
Gutzwiller Colloquium: Making Living Matter from the bottom up
18:00 - 18:40 Anna Okopińska (Jan Kochanowski University)
Entanglement characteristics of bound and resonant few-body states
18:40 - 19:20 Tobias Frederico (Aeronautic Institute of Technology)
Discrete scaling in the heavy and heavy-light molecule scattering
09:00 - 09:40 Hans-Werner Hammer (TU Darmstadt)
Nuclear physics around the unitarity limit
We argue that many features of the structure of nuclei emerge from a strictly perturbative expansion around the unitarity limit, where the two-nucleon S-waves have bound states at zero energy. In this limit, the gross features of states in the nuclear chart are correlated to only one dimensionful parameter, which is related to the breaking of scale invariance to a discrete scaling symmetry and set by the triton binding energy. Observables are moved to their physical values by small, perturbative corrections, much like in descriptions of the fine structure of atomic spectra. We provide evidence in favor of the conjecture that light, and possibly heavier, nuclei are bound weakly enough to be insensitive to the details of the interactions but strongly enough to be insensitive to the exact size of the two-nucleon system.
09:40 - 10:20 Emiko Hiyama (RIKEN)
Five-body structure of heavy-pentaquark system
Recently, the heavy pentaquark system has been observed as a resonant state. It is requested theoretically to describe this state. For this purpose, we should treat continuum states and resonant states explicitly. Here, I will report how to calculate resonant state and continuum state to explain the heavy pentaquark system.
group picture (to be published on events webpage) & coffee break
10:50 - 11:30 Jorge Segovia (Universitat Autonoma de Barcelona)
Is the nucleon a Borromean bound-state?
We explain how the emergent phenomenon of dynamical chiral symmetry breaking ensures that Poincaré covariant analyses of the three valence-quark scattering problem in continuum quantum field theory yield a picture of the nucleon as a Borromean bound-state, in which binding arises primarily through the sum of two separate contributions. One involves aspects of the non-Abelian character of QCD that are expressed in the strong running coupling and generate tight, dynamical color-antitriplet quark-quark correlations in the scalar-isoscalar and pseudovector-isotriplet channels. This attraction is magnified by quark exchange associated with diquark breakup and reformation, which is required in order to ensure that each valence-quark participates in all diquark correlations to the complete extent allowed by its quantum numbers. Combining these effects, we arrive at a properly antisymmetrised Faddeev wave function for the nucleon and calculate, e.g., the flavor-separated versions of the Dirac and Pauli form factors and conclude that available data and planned experiments are capable of validating the proposed picture.
11:30 - 12:10 Alfredo Mejía Valcarce (University of Salamanca)
A few certainties and many uncertainties about multiquarks
In this talk I will revise recent studies on multiquark spectroscopy regarding four, five and six quark systems making emphasis on the role played by confinement. I will address the role played by entangled thresholds and the failure of heavy quark mass expansions in some particular systems. The talked is based on some recent works in collaboration with J. Vijande and J.-M. Richard.
12:10 - 12:50 Sinsho Oryu (Tokyo University of Science)
Generalization of the Efimov-like structure in the hadron system
A lot of evidences for the Efimov states were found in the cold atomic system. However, there are few discussions about the Efimov physics in hadron systems. The most important requirements of Efimov physics is (1) the scattering length of the sub-system should be $\pm \infty$ ({\it first criterion}); (2) The result is that the three-body binding energies condense on the three-body threshold where the energy level structure is given by $E_{n}/E_{n+1}=$constant with respect to the principal quantum number $n$ ({\it second criterion}); (3) It is known that such an energy level structure can be obtained by the $1/r^2$ potential ({\it third criterion}). For the three criteria, we would like to check the hadron systems. (1) The {\it first criterion} is that the hadron scattering length $a_{NN} ({\rm or}~ a_{N\pi})\rightarrow \pm\infty$ is not satisfied in the hadron system. (2) The {\it second criterion} is that there are some instances that energy levels come near the threshold region. However, it is very hard to confirm whether they are Efimov levels or not. (3) The {\it third criterion} is that the nuclear potential is usually a short-range where the longest-range is one pion exchange Yukawa potential. Therefore, it seems that such a potential does not support Efimov physics in the hadron system. However, we can reevaluate the Efimov physics from another vantage point.\\ ~~ First of all we would like to pay attention to the thresholds in the three-body reactions. The three-body break-up thresholds (3BT) appear in the reactions: $d+p\rightarrow n+n+p$ and $N+N'\rightarrow N+N+\pi$, etc.. However, we have another threshold from the { three-body bound state to the quasi two-body system} which is the {\it quasi two-body threshold} (Q2T) such as $^3$He$\rightarrow d+p$ and $D\rightarrow N+N'$. The Born term of the Faddeev or the Alt-Grassberger-Sandhas (AGS) equation has a singularity at the 3BT ($E=0$,~$q=0$), and also the propagator $\tau(E-q^2/2\mu)$ diverges at 3BT ($E=0,~q=0$). The coincidence of both singularities causes a serious problem. This situation is very similar to the Coulomb scattering by the Lippmann-Schwinger equation where the Rutherford singularity coincides with the Green's function singularity. However, we have many energy levels in the Hydrogen atom by the potential $-e^2/r$. For the three-body singularity at the 3BT, Efimov discovered a new potential $-V_0/r^2$ which causes a lot of energy levels in the negative energy region. Recently, we found that the quasi-two-body Born term causes a singularity at the Q2T. From this singularity, we derived a general {\it particle transfer} (GPT) { potential} $-V'_0a^{2\gamma+2}/[r(r/2+a)^{2\gamma+2}]$ in the negative energy region. It is easily seen that the GPT potential includes a potential $-V_0'/r^2$ (for $\gamma=-1/2$ and $a\ll r$), and also Van der Waals potentials (for $\gamma=3/2,~2$ and $a\ll r$), which also includes the Yukawa-type potentials for $r\ll a$. Contrary to the {\it first Efimov criterion}, the Q2T does not require $a_{NN}~(a_{N\pi})\rightarrow \pm \infty$. The parameter $\gamma$ depends on the particle masses which transfer. It is because $\gamma=-1$ corresponds to the mass-less particle. In this stage, the question that the Efimov-like energy levels {\it could appear or not} is one side, but an interesting question of another side is about the long range interactions which we discover. Besides the Yukawa-type short range potential, these long range interactions could give rise to unusual phenomena in the hadron systems. In the case that such a long range potential does not produce additional bound states, still our attractive long range part of GPT potential could interfere with the Coulomb potential and the centrifugal potential, and affect the final-state interactions in the break-up reaction.\\ ~~ Finally, this kind of {\it additional long range potential} could universally occur, not only in three-body systems, but also in many-body systems. Then the most popular benefit of the potential will be seen in the neutron-rich nucleus such as a halo, or unstable nuclei, or the nuclear fusion problems.\\
14:40 - 15:20 Rimantas Lazauskas (Centre National de la Recherche Scientifique)
Description of the five-nucleon systems using Faddeev-Yakubovsky equations
15:20 - 16:00 Susumu Shimoura (The University of Tokyo)
Tetra-neutron system populated by RI-beam experiments
We have found a candidate tetra-neutron resonant state via a missing-mass spectroscopy in the double-charge exchange (DCX) reaction $^4$He($^8$He,$^8$Be) at 190 $A$ MeV by using the SHARAQ spectrometer at the RIBF facility in RIKEN. Production mechanism with kinematical consideration for the reaction is introduced and analysis for the obtained missing mass spectrum is presented. Nuclear forces relevant for formation of multi-neutron resonance are discussed for consistent understanding of few-body systems. Other experimental approaches for the tetra-neutron system are also shown including a new measurement of the DCX reaction.
16:30 - 17:10 Nir Barnea (The Hebrew University of Jerusalem)
Factorization and universality in nuclear physics
In this talk I will present the recent developments regarding the emergence of short range correlations in nuclear physics both experimentally and theoretically.
17:10 - 17:50 Raquel Crespo (Instituto Superio Técnico)
Interplay of structure and few-body reaction formalisms
The key ingredient of standard few-body reaction formalisms to describe the collison of a projectile with A nucleons with a target (taken here for simplicity as a proton) is the reduction of the overwhelming many-body problem, into an effective effective 3-body system composed by an (A-1) core plus a nucleon and the target proton. Within a mean field description of the projectile the nucleon can be taken as occupying a inner or a valence shell. The standard Faddeev/AGS(Alt-Grassberger-Sandhas) \cite{Alt67,Cr09} is a non-relativistic exact formalism that treats in equal footing all particles and simultaneously all open channels, elastic, inelastic knockout/breakup and transfer simultaneously. It is an useful tool to extract structure information from experimental data. The formalism can be viewed as a multiple scattering expansion in terms of the transition amplitude of each interacting pair. The bridging between the exact standard Faddeev/AGS framework with other reaction formalisms will be discussed.In addition, the need of merging ab initio many body structure information into the reaction framework will also be addressed. %%------------------------------------ \bibitem{Alt67} E.O. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. {\bf B2} 167 (1967). \bibitem{Cr09} R. Crespo, A. Deltuva, M. Rodriguez-Gallardo, E. Cravo and A.C. Fonseca, Phys. Rev. C {\bf 79}, 014609 (2009). %%------------------------------------
17:50 - 18:30 Arnoldas Deltuva (Vilnius University)
Reactions of weakly-bound few-body nuclei
Three- and four-body scattering is described solving Faddeev-Yakubovsky or equivallent Alt-Grassberger-Sandhas integral equations for transition operators in momentum-space with realistic nuclear interaction models. The studied cases include neutron-${}^{19}$C scattering and (d,p) and (d,n) reactions for deuteron or heavier nuclei targets. Extension to four-boson system is discussed as well.
09:00 - 09:40 Reinhard Dörner (Goethe-University, Frankfurt a.M.)
Helium halo states - Experimental images of their wavefunction
09:40 - 10:20 Servaas Kokkelmans (Eindhoven University of Technology)
Three-body physics with finite-range potentials
Three-body Efimov physics is relevant for the understanding of both dynamics and stability of ultracold gases. Efimov predicted the existence of an infinite sequence of three-body bound states, of which many properties scale universally, at diverging scattering length for a zero-range interaction potential. Experiments with ultracold atoms in which the scattering length is tuned through Feshbach resonances have also shown that the three-body parameter is universally linked to the finite-range of the two-body interaction potential. For small scattering lengths non-universal features appear in the Efimov spectrum, which are also linked to the finite-range nature of the interactions. We consider the full non-separable off-shell two-body T-matrix which is present in the three-body Faddeev equations and analyze the corresponding separable expansions. This momentum space treatment allows us, for instance, to show that strong d-wave interactions lower the energy of the second Efimov state making it possible to prevent this Efimov state from merging with the atom-dimer threshold. We also show that calculations of the three-body parameter corresponding to the potential resonances of deep square well potentials require many terms in the expansion the off-shell two-body T-matrix.
10:50 - 11:30 Cheng Chin (The University of Chicago)
Testing universality of Efimov physics across broad and narrow Feshbach resonances
For systems with resonant two-body interactions, Efimov predicted an infinite series of three-body bound states with geometric scaling symmetry. These Efimov states, first observed in cold caesium atoms, have been recently reported in a variety of other atomic systems. The intriguing prospect of a universal absolute Efimov resonance position across Feshbach resonances remains an open question. Theories predict a strong dependence on the resonance strength for closed-channel-dominated Feshbach resonances, whereas experimental results have so far been consistent with the universal prediction. We directly compare the Efimov spectra in a 6Li–133Cs mixture near two Feshbach resonances which are very different in their resonance strengths, but otherwise almost identical. Our result shows a clear dependence of the absolute Efimov resonance position on Feshbach resonance strength and a clear departure from the universal prediction for the narrow Feshbach resonance.
11:30 - 12:10 Alejandro Saenz (Humboldt-Universität zu Berlin)
Inelastic confinement-induced resonances
Since the theoretical prediction of confinement-induced resonances in one- or two-dimensional systems, they have attracted much interest due to their universality. In fact, later it turned out that there are two types of universal confinement-induced resonances: elastic ones as predicted originally as well as inelastic ones that are due to the coupling of center-of-mass and relative motion. While the elastic resonances are a useful tool for changing the interaction strength between particles, the inelastic ones can also be used for creating molecules by a transfer of the binding energy into center-of-mass motion.In fact, the universality of the inelastic resonances persists also for particles interacting by the long-ranged Coulomb interactions, e.g., in quantum-dot systems. There, they might be used as on-demand single-photon sources. In ultracold gases with dipolar interaction, tuning the dipole strength allows for a manipulation of the resonance position, yielding a high degree of quantum control. After a survey of the confinement-induced resonances some more recent results will be given, especially in the context of time-dependent variations of the trap potential.
12:10 - 12:50 Giannakeas Panagiotis (MPI for the Physics of Complex Systems)
Recombination processes in strongly mass-imbalanced ultracold systems
The mass-imbalanced three-body recombination process that forms a shallow dimer is shown to possess a rich Efimov-Stückelberg topology, with corresponding spectra that differ fundamentally from the homonuclear case. A semi-analytical treatment of the three-body recombination predicts an unusual spectra with intertwined resonance peaks and minima, and yields in-depth insight into the behavior of the corresponding Efimov spectra. In particular, the pattern of maxima and minima are shown to depend strongly on the degree of diabaticity, which strongly affects the universality of the heteronuclear Efimov states.
Meeting at MPIPKS reception for excursion
Leaving MPIPKS for excursion to Moritzburg
Coffee break at Moritzburg castle
Guided tour through Moritzburg castle
Workshop dinner at Moritzburg castle restaurant
09:00 - 09:40 Alejandro Kievsky (University of Pisa)
Saturation properties of N boson systems
09:40 - 10:20 Renato Higa (University of São Paulo)
Halo/cluster EFT approach to the virtual state in the doublet nd scattering
At energies lower than the deuteron breakup, the doublet nd scattering is known to have a virtual state and a zero in the amplitude just below the elastic threshold. We formulate a halo/cluster EFT incorporating both physics and their behavior as the deuteron binding energy decreases. We confirm previous studies, that associates the nature of the virtual state to an excited Efimov state.
10:50 - 11:30 Sebastiana Puglia (LNS INFN)
The Boron depletion in astrophysics
11:30 - 12:10 Michele Viviani (Istituto Nazionale di Fisica Nucleare)
3NF effects in A=4 scattering
We discuss the effect of the three-nucleon force (3NF) in d-d, p-3H and n-3He scattering at low energies. The used 3NF is derived from effective field theory at next-to-next-to-leading order. The four-nucleon scattering observables are calculated using the Kohn variational principle and the hyperspherical harmonics technique and the results are compared with available experimental data.
12:10 - 12:50 Tomás González Lezana (Consejo Superior de Investigaciones Científicas)
Atom-diatom dinamics investigated with a few body theory from nuclear physics
Early three-body approaches developed to study nuclear reactions such as the compound nucleus theory [1,2,3] were adopted in the 70s in the context of Molecular Physics to investigate the dynamics of atom-diatom reactive collisions [4,5]. In particular, a statistical quantum mechanical (QM) method which employs ab initio potential energy surfaces and fully QM techniques to calculate reaction probabilities has been largely used to characterize processes mediated by a complex-forming mechanism between reactants and products [6,7]. In my talk I will review some of the basic ingredients of the method and some of the most interesting applications in a number of reactions in which the formation of an intermediate species plays a significant role in the overall dynamics. Special emphasis is made in cold reactions at low collision energy [8,9,10] [1] L. Wolfenstein, Phys. Rev. 87, 366 (1951) [2] W. Hauser and H. Fesbach, Phys. Rev. 87, 366 (1952) [3] R. M. Eisberg and N. M. Hintz, Phys. Rev. 103, 645 (1956) [4] P. Pechukas et al., J. Chem. Phys. 42, 3281 (1964); R. A. White et al., J. Chem. Phys. 53, 379 (1971) [5] W. H. Miller J. Chem. Phys. 52, 543 (1970) [6] E. J. Rackham et al. J. Chem. Phys. 119, 12895 (2003) [7] T. Gonzalez-Lezana Int. Rev. Phys. Chem. 26, 29 (2007) [8] T. Gonzalez-Lezana and P. Honvault, Int. Rev. Phys. Chem. 33, 371 (2014) [9] C. Makrides et al. Phys. Rev. A 91, 012708 (2015) [10] P. Honvault et al. Phys. Rev. Lett. 107, 023201 (2011)
14:40 - 15:20 Dmitry Petrov (CNRS & University Paris-Sud)
Few- and many-body effects in a driven mixture
Universality and tails of long-range interactions
Long-range interactions and, in particular, two-body potentials with power-law long-distance tails are ubiquitous in nature. For two bosons or fermions in one spatial dimension, the latter case being formally equivalent to three-dimensional s-wave scattering, we show how generic asymptotic interaction tails can be accounted for in the long-distance limit of scattering wave functions. This is made possible by introducing a generalisation of the collisional phase shifts to include space dependence. We show that this distance dependence is universal, in that it does not depend on short-distance details of the interaction. The energy dependence is also universal, and is fully determined by the asymptotic tails of the two-body potential. As an important application of our findings, we describe how to eliminate finite-size effects with long-range potentials in the calculation of scattering phase shifts from exact diagonalisation. We show that even with moderately small system sizes it is possible to accurately extract phase shifts that would otherwise be plagued with finite-size errors. We also consider multi-channel scattering, focusing on the estimation of open channel asymptotic interaction strengths via finite-size analysis.
16:30 - 17:10 Pascal Naidon (RIKEN)
From the Yukawa to the Efimov attraction
Impurities immersed in a Bose-Einstein condensate form Bose polarons. The polarons can interact with each other through an interaction that is mediated by the Bose-Einstein condensate. For a weak boson-impurity interaction, this mediated interaction is known to be a weak Yukawa attraction. For a resonant boson-impurity, it turns into a strong Efimov attraction involving a single boson as the mediator. In this study, I look into this interesting crossover which bridges few-body to many-body physics.
17:10 - 17:50 Zhenhua Yu (Sun Yat-Sen University)
Universal properties of p-wave fermi gases
The successful application of $s$-wave ``contact" to unitary Fermi gases exemplifies the importance of few-body correlations in understanding strongly interacting many-body systems. Recent experimentally measured radio-frequency spectrum and momentum distribution of three-dimensional Fermi gases close to a $p$-wave Feshbach Resonance show universal tails different from the $s$-wave case [1]. We emphasise that the difference is due to the necessity of using both the scattering volume $v$ and the effective range $R$ to parameterise the two-body $p$-wave interatomic interaction, and show that by including two-body correlations at short range, the interaction effects of the system are captured by two contacts, $C_v$ and $C_R$, which are related to the variation of energy with $v$ and $R$ in two adiabatic theorems [2]. Based on the two contacts, we derive the universal properties of the system regarding momentum distribution, radio-frequency and photo-association spectroscopies, and pressure and virial relations. We also establish coupled rate equations to explain the time evolution of the p-wave contacts observed in the quench experiment [1]. Beyond two-body correlations at short range, $p$-wave resonances are predicted to give rise to universal super Efimov three-body bound states in two-dimensions [3]. We use the hyper-spherical formalism and show that these new universal states originate from an emergent effective potential, which is different from the one responsible for the familiar Efimov states [4]. In the many-body context, we introduce a new thermodynamic quantity, the three-body contact $C_\theta$, to quantify the three-body correlations due to the super Efimov Effect [5]. We determine how $C_\theta$ affects various physical observables; signature of the elusive super Efimov effect in the thermodynamic system can be pinned down by the detection of the three-body contact $C_\theta$ via these observables. [1] C. Luciuk, S. Trotzky, S. Smale, Zhenhua Yu, Shizhong Zhang, and J. H. Thywissen 2016 \emph{Nature Physics} \textbf{12} 599-605 [2] Zhenhua Yu, J.H. Thywissen, and Shizhong Zhang 2015 \emph{Phys. Rev. Lett.} \textbf{115} 135304 [3] Yusuke Nishida, Sergej Moroz, and Dam Thanh Son 2013 \emph{Phys. Rev. Lett.} \textbf{110} 235301 [4] Chao Gao, Jia Wang, and Zhenhua Yu 2015 \emph{Phys. Rev. A} \textbf{92} 020504(R) [5] Pengfei Zhang, Zhenhua Yu 2016 arXiv:1611.09454
17:50 - 18:30 Thomas Busch (OIST Graduate University)
Coherent control of atomic few body systems
Interactions between atoms often introduce large amounts of complexity into many-particle systems, but they can also lead to new and interesting physical regimes. In this presentation I will discuss several examples of interacting ultra cold atomic systems, where tuneable interactions allow to access new dynamical situations or create new control techniques. The first example will be taken from our efforts towards a full set of techniques to coherently control the external state of small samples of ultracold atoms and the second from the description of non-equilibrium situations in single- and multi-component systems. This will include a discussion of a thermodynamical machine build on Feshbach resonances.
09:00 - 09:40 Elena Kolganova (Bogoliubov Laboratory of Theoretical Physics)
Scattering length calculations via Faddeev approach
The van der Waals three-body systems at ultralow energies are studied using Faddeev equations in configuration space. The spectra of LiHe2 systmes and the scattering length are calculated. The results obtained indicate on the Efimov character of the excited state in both systems.
09:40 - 10:20 Angela Foerster (Universidade Federal do Rio Grande do Sul)
Integrability in few and many body physics
Integrable systems can be found in different areas of Physics, such as statistical mechanics, quantum field theory,condensed matter, string theory and more recently in cold atoms. Here we provide an overview of the basic construction and the impact of Yang-Baxter integrable systems in condensed matter and ultracold atoms, focusing on some prominent examples, specially those cases connected to experiments [1]. Then recent results, such as quantum integrable multi-well tunneling models [2],[3] and a geometric ansatz for a few particles system will be presented. Finally we discuss the effects of breaking the integrability in some models. [1] Yang-Baxter integrable models in experiments: from condensed matter to ultracold atoms, M. T. Batchelor, A. Foerster, J. Phys. A: Math. Theor. 49 (2016) 173001; [2] Quantum integrable multi-well tunneling models, L H Ymai, A P Tonel, A Foerster, J Links, arXiv:1606.00816; [3] Integrable model of bosons in a four-well ring with anisotropic tunneling, A. P. Tonel, L. H. Ymai, A. Foerster, J. Links, J. Phys. A: Math. Theor. 48 (2015) 494001;
10:20 - 11:00 Nathan L. Harshman (American University)
Topology, symmetry and control in few-body configuration space
This talk presents examples of physics at the weak-interaction limit and the hard-core limit of two-particle and few-particle interactions. Level splitting and state evolution under the adiabatic and diabatic evolution of control parameters can be understood in terms of topological shifts and/or symmetry breaking in few-body configuration space. This approach provides methods for classifying spectral structures in trapped, ultracold atomic few-body systems, supplies methods for state control that are robust under fluctuations of physical parameters, and suggests procedures for identifying new solvable models that assist analytic and numerical methods. These techniques are especially effective low dimensions because symmetry and topology provide more constraining structures.
11:30 - 12:10 Xiaoling Cui (Institute of Physics, CAS)
One-dimensional Fermi gases near p-wave resonance
In this talk, I will show that the property of a 3D p-wave interacting Fermi gas can be significantly changed if confined in (quasi-)1D geometry. Counterintuitively, it is found that the application of transverse confinements can greatly stretch the shallow p-wave molecules near the induced 1D resonance, and this strongly indicates a much stable p-wave interacting gas in 1D near resonance, contrary to the 3D case[1]. I will then discuss the interaction renormalization for 1D p-wave scattering[2], and use it to derive the universal relations and address the Bloch-wave scattering and effective model in optical lattices[3]. Finally, based on the effective lattice model, I will introduce our most recent results in establishing the long-sought Majorana fermions in 1D p-wave cold atoms system[4]. References: [1] L. Zhou, X. Cui, arxiv:1707.03512. [2] X. Cui, PRA 94, 043636 (2016); H. Dong, X. Cui, PRA 94, 063650 (2016). [3] X. Cui, PRA 95, 041601(R)(2017). [4] X. Yin, T.-L. Ho, X. Cui, in preparation.
12:10 - 12:50 Artem Volosniev (Technical University Darmstadt)
A one-dimensional Bose gas with impurities
We discuss the ground state properties of a one-dimensional bosonic system doped with other particles (the so-called Bose polaron problem). We introduce a formalism that allows us to study this system with weak and moderate boson-boson interaction strengths for any boson-impurity interaction. We use available numerically exact results, and develop a numerical method based on the in-medium similarity renormalization group to test the formalism. We argue that the introduced approach provides a simple tool for studies of strongly interacting impurity problems in one dimension. Furthermore, it can be used to investigate correlations between impurity particles in a Bose gas.
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\begin{document}
\title{Roman domination in graphs with minimum degree at least two and some forbidden cycles } \date{} \author{ S.M. Sheikholeslami$^{1}$\thanks{ Corresponding author}, M. Chellali$^2$, R. Khoeilar$^1$, H. Karami$^1$ and Z. Shao$^3$,
\\
$^1$Department of Mathematics \\ Azarbaijan Shahid Madani University\\ Tabriz, I.R. Iran\\ \texttt{[email protected]}\\ \texttt{[email protected]}\\ \texttt{[email protected]}
\\ $^2$LAMDA-RO Laboratory, Department of Mathematics\\ University of Blida\\ B.P. 270, Blida, Algeria\\ \texttt{m\[email protected]}
\\ $^3$Institute of Computing Science and Technology\\ Guangzhou University\\ Guangzhou 510006, China\\ \texttt{[email protected]}
\\ } \maketitle
\begin{abstract} Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$ -cycles, then $\gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k=0,$ {and [S. Bermudo, On the differential and Roman domination number of a graph with minimum degree two, Discrete Appl. Math. 232 (2017), 64--72] when }$k=1.$ Moreover, using the Gallai-type result involving the Roman domination number and the differential of graphs established by Bermudo et al. stating that $\gamma _{R}(G)+\partial (G)=n$, we have $\partial (G)\geq \frac{(2k+3)n}{6k+11},$ thereby settling the conjecture of Bermudo posed in the second paper.
\noindent \textbf{Keyword:} Differential of a graph, Roman domination number. \newline \textbf{MSC 2010}: 05C69 \end{abstract}
\section{Introduction}
In this paper, $G$ is a simple graph without isolated vertices, with vertex set $V=V(G)$ and edge set $E=E(G)$. The \emph{order} $|V|$ of $G$ is denoted by $n=n(G)$. For a vertex $v\in V$, the \emph{open neighborhood} of $v$ is the set $N(v)=\{u\in V\mid uv\in E\},$ the \emph{closed neighborhood} of $v$
is the set $N[v]=N(v)\cup \{v\}$, and the \emph{degree} of $v$ is $\deg _{G}(v)=|N(v)|$.
Let $u$ and $v$ be two vertices in $G.$ A $uv$\textit{-path} is a path with endvertices $u$ and $v$, and the \textit{distance} between $u$ and $v$ is the length of a shortest $uv$-path. The \emph{diameter} of $G$, denoted by $ \mathrm{diam}(G)$, is the maximum distance between vertices of $G$. We write $P_{n}$ and $C_{n}$ for the \emph{path} and \emph{cycle} of order $n$, respectively. {Let }$A$ and $B$ {are} two disjoint subgraphs { (not necessarily induced) }of a graph $G.$ If there is an edge $ e $ having one endvertex in $A$ and the other one in $B,$ then $A+B+e$ will denote the graph formed by $A$ and $B$ for which we add only the edge $e.$ We also denote by $G-A$ the subgraph of $G$ induced by $V(G)-V(A).$
For a set $D,$ let $B(D)$ be the set of vertices in $V\setminus D$ that have a neighbor in $D$. The \textit{differential} of a set $D$ is defined in~\cite
{M} as $\partial (D)=|B(D)|-|D|$, and the maximum value of $\partial (D)$ for any subset $D$ of $V$ is the \textit{differential} of $G$, denoted $ \partial (G)$. Differential of graphs has been studied extensively in several papers, in particular \cite{B0,B3,B4,B5,B6,B1,B2,kkcs}. In 2017, Bermudo \cite{B0} proved that for any graph $G$ with order $n\geq 15$, minimum degree two and without any induced tailed $5$-cycle graph of seven vertices or tailed $5$-cycle graph of seven vertices together with a particular edge, it is satisfied $\partial (G)\geq \frac{5n}{17}$. Moreover, he posed the following conjecture.
\begin{conj}[\protect\cite{B0}] \label{conj}Let $G$ be a graph of order $n\ge 6k + 9$, minimum degree $ \delta\ge 2$, which does not contain any induced $\{C_5,C_8,\ldots,C_{3k+2} \} $-cycles. Then $\partial(G)\ge \frac{(2k+3)n}{6k+11}.$ \end{conj}
A \emph{Roman dominating function} (RDF) on a graph $G$ is a function $ f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $u\in V(G)$ with $f(u)=0$ has a neighbor $v$ with $f(v)=2$. The \textit{weight} of an RDF $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u),$ and the \emph{Roman domination number} $\gamma _{R}(G)$ is the minimum weight of an RDF on $G$. The Roman domination number of graphs was introduced in 2004 by Cockayne et al.\ in \cite{CDHH} and is now well-studied in graph theory. The literature on Roman domination and its variations has been surveyed and detailed in two book chapters and three surveys \cite{Ch1, Ch2, Ch3, Ch4, Ch5}.
{In \cite{cham}, it has been shown that, if $G$ is a graph of order $n\geq 9$ and minimum degree $\delta \geq 2$, then $\gamma _{R}(G)\leq \frac{8n}{11}$. It was also shown in \cite{B0} that $\gamma _{R}(G)\leq \frac{12n}{17}$ for any graph }$G$ {with order $n\geq 15$, minimum degree two and without any induced tailed 5-cycle graph of seven vertices or tailed 5-cycle graph of seven vertices together with a particular edge. }
In this paper, we improve the aforementioned known results by showing that if $G$ is a graph satisfying the statement of Conjecture \ref{conj}, then $ \gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}.$ U{sing the Gallai-type result involving the differential and the Roman domination number of graphs established by Bermudo, Fernau and Sigarreta \cite{B6} who proved that $ \gamma _{R}(G)+\partial (G)=n$, our bound leads to }$\partial (G)\geq \frac{ (2k+3)n}{6k+11}$ which settles Conjecture {\ref{conj}. }
We close this section by recalling the exact values of the Roman domination number of paths and cycles given in \cite{CDHH}, namely $\gamma _{R}(P_{n})=\gamma _{R}(C_{n})=\lceil \frac{2n}{3}\rceil .$
\section{Some useful lemmas}
We gather in this section some results that will be useful to us thereafter. For technical reasons, we will often consider three Roman dominating functions $f_{1},f_{2}$, and $f_{3}$ on a graph $G$, where we use $ \overrightarrow{f}$ to denote the $3$-tuple $(f_{1},f_{2},f_{3})$, and $ \overrightarrow{f}(v)$ for $(f_{1}(v),f_{2}(v),f_{3}(v))$ for a vertex $v$. A vertex $v$ is said to be $\overrightarrow{f}$\textit{-strong} if $ f_{j}(v)=2$ for some $j\in \{1,2,3\}$. Moreover, the weight of $ \overrightarrow{f}$ is $\omega (\overrightarrow{f})=\sum_{j=1}^{3}\omega (f_{j})$. Clearly, $\omega (f_{j})\leq \omega (\overrightarrow{f})/3$ for some $j\in \{1,2,3\}$. \ Also, if $H$ is an induced subgraph of $G$ and $f$
an RDF on $G$, then we denote the restriction of $f$ on $H$ by $f|_{V(H)}$ and let $f(V(H))=\omega (f,H).$
For integers $m$ and {$\ell$} such that $m\geq 3$ and $\ell \geq 1$, let $ C_{m,\ell }$ be the graph obtained from a cycle {$C_{m}=x_{1}x_{2}\ldots x_{m}x_{1}$} and a path $P=y_{1}y_{2}\ldots y_{\ell }$ by adding the edge $ x_{1}y_{1},$ with {$y_{i}\notin V(C_{m})$} for all possible $i$. The graph $ C_{m,\ell }$ will be called a \textit{tailed }$m$\textit{-cycle graph} of order $m+\ell $. {We call an \textit{ear }of a cycle }$C$ in a graph $G,$ {to a path }$P$ {in }$G-C$ {whose endvertices are adjacent to some vertices in }$ C. $
\begin{lemma} \label{ear1}Let $G$ be a graph, $u,v\in V(G)$ and $\overrightarrow{f} =(f_{1},f_{2},f_{3})$ be a 3-tuple of RDFs of $G$ such that $u$ and $v$ are $ \overrightarrow{f}$-strong. If $H$ is a graph obtained from $G$ by adding a path $Q=y_{1}\ldots y_{\ell }$ and the edges $uy_{1},vy_{\ell }$, then $ \overrightarrow{f}$ can be extended to a 3-tuple of RDFs $\overrightarrow{g}$ of $H$ such that {$\omega (\overrightarrow{g},Q)\leq 2\ell$} and each vertex in $ V(Q)-\{y_{1},y_{\ell }\}$ is $\overrightarrow{g}$-strong. \end{lemma}
\noindent \textbf{Proof.} By assumption, $f_{i}(u)=2$ and $f_{j}(v)=2$ for some $i,j\in \{1,2,3\}$. Let us consider the following two cases.
\noindent \textbf{Case 1.} $i\neq j$.\newline Assume, without loss of generality, that $i=1$ and $j=2$. Consider the following situations.
\textbf{Subcase 1.1.} $\ell \equiv 0\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $ z\in V(G)$, $g_{2}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $ g_{2}(z)=0$ otherwise; and $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $ g_{3}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $g_{3}(z)=0$ otherwise.
\textbf{Subcase 1.2.} $\ell \equiv 1\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $ z\in V(G)$, $g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $ g_{2}(z)=0$ otherwise; and $g_{3}(z)=f_{3}(z)$ for $z\in V(G)$, $ g_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -1}{3}$, and $g_{3}(z)=0$ otherwise.
\textbf{Subcase 1.3.} $\ell \equiv 2\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -5}{3}$, $g_{1}(y_{\ell })=1$ and $g_{1}(z)=0$ otherwise; $ g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -5}{3}$, $g_{2}(y_{\ell -1})=1$ and $g_{2}(z)=0$ otherwise; and $ g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $g_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $g_{3}(z)=0$ otherwise.
In either subcase, $g_{1},g_{2},g_{3}$ are RDFs of $H$ and thus $ g=(g_{1},g_{2},g_{3})$ is a 3-tuple of RDFs of $H.$ In addition, $\omega (\overrightarrow{g},Q)\leq 2\ell$ and each vertex of $V(Q)-\{y_{1},y_{\ell }\}$ is $ \overrightarrow{g}$-strong.
\noindent \textbf{Case 2.} $i=j$.\newline Assume, without loss of generality, that $i=j=1$. Consider again the following situations.
\textbf{Subcase 2.1.} $\ell \equiv 0\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(y_{2})=1$, $g_{1}(y_{3i+4})=2$ for $0\leq i\leq \frac{\ell -6}{3}$, and $g_{1}(z)=0$ otherwise; $ g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $ z\in V(G)$, $g_{3}(y_{1})=1$, $g_{3}(y_{3i+3})=2$ for $0\leq i\leq \frac{ \ell }{3}-1$, and $g_{3}(z)=0$ otherwise.
\textbf{Subcase 2.2.} $\ell \equiv 1\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for $z\in V(G)$, $g_{1}(y_{2})=g_{1}(y_{\ell -1})=1$, $ g_{1}(y_{3i+4})=2$ for $0\leq i\leq \frac{\ell -7}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $g_{2}(y_{1})=1$, $ g_{2}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $g_{3}(y_{\ell })=1$, $ g_{3}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{3}(z)=0$ otherwise.
\textbf{Subcase 2.3.} $\ell \equiv 2\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -5}{3}$ and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $ z\in V(G)$, $g_{2}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $ g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $ g_{3}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $g_{3}(z)=0$ otherwise.
In either subcase, $g_{1},g_{2},g_{3}$ are RDFs of $H$ and thus $ g=(g_{1},g_{2},g_{3})$ is a 3-tuple of RDFs of $H.$ Moreover, $\omega (\overrightarrow{g},Q)\leq 2\ell$ and each vertex of $V(Q)-\{y_{1},y_{\ell }\}$ is $ \overrightarrow{g}$-strong.$
\Box $
\begin{lemma} \label{tailedcycle-3p+1}Let $G$ be a graph, $u\in V(G)$ and $\overrightarrow{ f}=(f_{1},f_{2},f_{3})$ a 3-tuple of RDFs of $G$ such that $u$ is $ \overrightarrow{f}$-strong.
\begin{itemize} \item[1.] If $H$ is obtained from $G$ by adding a cycle $C_{3p+1}=x_{1}x_{2} \ldots x_{3p+1}x_{1}\;(p\geq 1)$ and the edge $ux_{1}$, then $ \overrightarrow{f}$ can be extended to a 3-tuple $\overrightarrow{g}$ of RDFs of $H$ such that {$\omega (\overrightarrow{g},C_{3p+1})\leq 2(3p+1)$} and each vertex in $V(C_{3p+1})-\{x_{3p+1}\}$ is $\overrightarrow{g}$-strong.
\item[2.] If $H$ is obtained from $G$ by adding a tailed cycle $C_{3p+1,\ell }\;(p\geq 1)$ and the edge $uy_{\ell }$, then $\overrightarrow{f}$ can be extended to a 3-tuple $\overrightarrow{g}$ of RDFs of $H$ such that $\omega (\overrightarrow{g},C_{3p+1,\ell })\leq 2(3p+1+\ell )$ and each vertex of $ V(C_{3p+1,\ell })-\{x_{3p+1}\}$ is $\overrightarrow{g}$-strong.
\item[3.] If $H$ is obtained from $G$ by adding a cycle $C_{3p+2}=x_{1}x_{2} \ldots x_{3p+2}x_{1}\;(p\geq 1)$ and the edge $ux_{1}$, then $ \overrightarrow{f}$ can be extended to a 3-tuple $\overrightarrow{g}$ of RDFs of $H$ such that $\omega (\overrightarrow{g},C_{3p+2})\leq 2(3p+2)+1$ and each vertex of $C_{3p+2}$, is $\overrightarrow{g}$-strong.
\item[4.] If $H$ is obtained from $G$ by adding a tailed cycle $C_{3p+2,\ell }\;(p\geq 1)$ and the edge $uy_{\ell }$, then $\overrightarrow{f}$ can be extended to a 3-tuple $\overrightarrow{g}$ of RDFs of $H$ such that $\omega (\overrightarrow{g},C_{3p+2,\ell })\leq 2(3p+2+\ell )+1$ and each vertex of $ C_{3p+2,\ell }$, is $\overrightarrow{g}$-strong. \end{itemize} \end{lemma}
\noindent \textbf{Proof.} Since $u$ is $\overrightarrow{f}$-strong, let us assume, without loss of generality, that $f_{1}(u)=2$.
1) Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(x_{3i+3})=2$ for $0\leq i\leq p-1$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $ g_{2}(x_{3p})=1$, $g_{2}(x_{3i+1})=2$ for $0\leq i\leq p-1$, and $g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $g_{3}(x_{3p+1})=1$, $ g_{3}(x_{3i+2})=2$ for $0\leq i\leq p-1$, and $g_{3}(z)=0$ otherwise.
Clearly, $g_{1},g_{2},g_{3}$ are RDFs of $H$ and thus $g=(g_{1},g_{2},g_{3})$ is a 3-tuple of RDFs of $H.$ In addition, $\omega (\overrightarrow{g},C_{3p+1})\leq 2(3p+1)$ and each vertex of $V(C_{3p+1})-\{x_{3p+1}\}$ is $ \overrightarrow{g}$-strong.
2) Consider the following cases.
\noindent \textbf{Case 1.} $\ell \equiv 0\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(x_{3i+3})=2$ for $0\leq i\leq p-1$, $g_{1}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $ g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $ g_{2}(x_{3p})=1$, $g_{2}(x_{3i+1})=2$ for $0\leq i\leq p-1$, $ g_{2}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $g_{3}(x_{3p+1})=1$, $ g_{3}(x_{3i+2})=2$ for $0\leq i\leq p-1$, $g_{3}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell }{3}-1$, and $g_{3}(z)=0$ otherwise.
\noindent \textbf{Case 2.} $\ell \equiv 1\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(x_{3p+1})=1$, $ g_{1}(x_{3i+2})=2$ for $0\leq i\leq p-1$, $g_{1}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $g_{1}(x_{3p})=1$, $g_{2}(x_{3i+1})=2$ for $0\leq i\leq p-1$, $g_{2}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $ g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $ g_{3}(x_{3i+3})=2$ for $0\leq i\leq p-1$, $g_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -1}{3}$, and $g_{3}(z)=0$ otherwise.
\noindent \textbf{Case 3.} $\ell \equiv 2\pmod 3$.\newline Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows: $ g_{1}(z)=f_{1}(z)$ for all $z\in V(G)$, $g_{1}(x_{3p})=1$, $ g_{1}(x_{3i+1})=2 $ for $0\leq i\leq p-1$, $g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -5}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(z)=f_{2}(z)$ for all $z\in V(G)$, $g_{2}(x_{3p+1})=1,$ $g_{2}(x_{3i+2})=2$ for $0\leq i\leq p-1$, $g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $ g_{2}(z)=0$ otherwise; $g_{3}(z)=f_{3}(z)$ for all $z\in V(G)$, $ g_{3}(x_{3i+3})=2$ for $0\leq i\leq p-1$, $g_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $g_{3}(z)=0$ otherwise.
In either case, $g_{1},g_{2},g_{3}$ are RDFs of $H$ and thus $ \overrightarrow{g}=(g_{1},g_{2},g_{3})$ is a 3-tuple of RDFs of $H.$ In addition, $\omega (\overrightarrow{g},C_{3p+1,\ell })\leq 2(3p+1+\ell) $ and each vertex of $V(C_{3p+1,\ell })-\{x_{3p+1}\}$ is $\overrightarrow{g}$ -strong.
The proofs of the remaining items are similar and therefore omitted. $
\Box $
\begin{lemma} \label{cyclepath}
\begin{enumerate} \item \emph{\ Let $C=v_{1}v_{2}\ldots v_{t}v_{1}$ be a cycle on $t\geq 4$ vertices with $t\equiv 1\pmod 3$. Then $C$ has a 3-tuple of RDFs $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f} )\leq 2t+1$ and all vertices of $C$ but $v_{t}$ are $\overrightarrow{f}$ -strong. }
\item \emph{Let $C=v_{1}v_{2}\ldots v_{t}v_{1}$ be a cycle on $t\geq 3$ vertices with $t\equiv 0\pmod 3$. Then $C$ has a 3-tuple of RDFs $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f} )\leq 2t$ and all vertices of $C$ are $\overrightarrow{f}$-strong.}
\item \emph{Let $C_{m,\ell }$ be a tailed $m$-cycle with $m\equiv 1\pmod 3$ and $V(C_{m,\ell })=\{x_{1},x_{2},\ldots ,x_{m},y_{1},y_{2},\ldots ,y_{\ell }\},$ where the $x_{i}$'s induce a cycle {$C_{m}$ and the }$y_{i}$ 's induce a path $P_{\ell }$. Then $G$ has a 3-tuple of RDFs $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f} )\leq 2(m+\ell )+1$ and all vertices of $C_{m,\ell }$ but $x_{m}$ are $ \overrightarrow{f}$-strong.} \end{enumerate} \end{lemma}
\emph{\ }
\noindent \textbf{Proof. }1) Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(C)$ as follows: $f_{1}(v_{t-1})=1,f_{1}(v_{3i+1})=2$ for $0\leq i\leq \frac{t-4}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(v_{t})=1,$ $ f_{2}(v_{3i+2})=2$ for $0\leq i\leq \frac{t-4}{3}$ and $f_{2}(x)=0$ otherwise; $f_{3}(v_{1})=1,f_{3}(v_{3i+3})=2$ for $0\leq i\leq \frac{t-4}{3}$ and $f_{3}(x)=0$ otherwise. Clearly, $f_{1},f_{2},f_{3}$ are RDFs of $C.$ Hence $\overrightarrow{f}=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $C$, with $\omega (\overrightarrow{f})\leq 2t+1$ and all vertices of $C$ except $ v_{t}$ are $\overrightarrow{f}$-strong.
2) Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(C)$ as follows: $ f_{1}(v_{3i+1})=2$ for $0\leq i\leq \frac{t-3}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(v_{3i+2})=2$ for $0\leq i\leq \frac{t-3}{3}$ and $ f_{2}(x)=0$ otherwise; $f_{3}(v_{3i+3})=2$ for $0\leq i\leq \frac{t-3}{3}$ and $f_{3}(x)=0$ otherwise. Clearly, $f_{1},f_{2},f_{3}$ are RDFs of $C$. Hence $\overrightarrow{f}=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $C$ with $\omega (\overrightarrow{f})\leq 2t$ and each vertex of $C$ are $ \overrightarrow{f}$-strong.
3) Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(C_{m,\ell }).$ For vertices on $C_{m}$ as follows: $f_{1}(x_{m-1})=1,f_{1}(x_{3i+1})=2$ for $ 0\leq i\leq \frac{m-4}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{m})=1,$ $ f_{2}(x_{3i+2})=2$ for $0\leq i\leq \frac{m-4}{3}$ and $f_{2}(x)=0$ otherwise; $f_{3}(x_{3i+3})=2$ for $0\leq i\leq \frac{m-4}{3}$ and $ f_{3}(x)=0$ otherwise. Moreover, the $f_{i}$'s are defined for the vertices on $P_{\ell }$ according to $\ell $ as follows.
If $\ell \equiv 0\pmod 3$, then $f_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{ \ell -3}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -3}{3}$ and $f_{2}(x)=0$ otherwise; $f_{3}(y_{\ell })=1$, $ f_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -3}{3}$ and $f_{3}(x)=0$ otherwise.
If $\ell \equiv 1\pmod 3$, then $f_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{ \ell -4}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(y_{\ell })=1,$ $ f_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$ and $f_{2}(x)=0$ otherwise; $f_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -1}{3}$ and $ f_{3}(x)=0$ otherwise.
If $\ell \equiv 2\pmod 3$, then $f_1(y_{\ell })=1,$ $f_{1}(y_{3i+3})=2$ for $ 0\leq i\leq \frac{\ell -5}{3}$ and $f_{1}(x)=0$ otherwise; $ f_{2}(y_{3i+2})=2 $ for $0\leq i\leq \frac{\ell -2}{3}$ and $f_{2}(x)=0$ otherwise; $f_{3}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -2}{3}$ and $ f_{3}(x)=0$ otherwise.
Clearly, in either case $f_{1},f_{2},f_{3}$ are RDFs of {$C_{m,\ell}$} and thus $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of {$C_{m,\ell}.$} Also, $ \omega (\overrightarrow{f})\leq 2(m+\ell )+1$ and all vertices of $C_{m,\ell }$ but $x_{m}$ are $\overrightarrow{f}$-strong.$
\Box $
\begin{lemma} \label{MainLem}Let $C_{i}=x_{1}^{i}x_{2}^{i}\ldots x_{n_{i}}^{i}x_{1}^{i}\;$ be a cycle of order $n_{i},$ for $i\in \{1,2,\ldots ,s\}.$
\begin{enumerate} \item If $n_{1}\equiv 2\pmod 3$ and $G$ is a graph obtained from $C_{1}$ and $C_{2}$ by identifying the vertices $x_{1}^{1}$ and $x_{1}^{2}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega ( \overrightarrow{f})\leq 2n(G)+1$, and all vertices of $V(G)- \{x_{2}^{2},x_{n_{2}}^{2}\}$ are $\overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 1\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding either the edge $x_{1}^{1}x_{1}^{2}$ or a path $z_{1}\ldots z_{3k}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $ x_{1}^{2}z_{3k}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f} =(f_1,f_2,f_3)$ such that $\omega (\overrightarrow{f})\leq 2n(G)+1$ and each vertex of $G$ but $x_{n_{2}}^{2}$ is $\overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 1\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding for $k\geq 1$, a path $z_{1}\ldots z_{3k+1}$ and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+1}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f} )\leq 2n(G)+1$ and all vertices of $G$ but $x_{n_{2}}^{2}$ are $ \overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 1\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding for $k\geq 1$ a path $z_{1}\ldots z_{3k+2}\;$and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+2}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f} )\leq 2n(G)+1$ and all vertices of $G$ but $x_{n_{2}}^{2}$ are $ \overrightarrow{f}$-strong.
\item If $n_{i}\equiv 2\pmod 3$ for $i\in \{1,2\}$ and $G$ is obtained from $ C_{1}$ and $C_{2}$ by adding either the edge $x_{1}^{1}x_{1}^{2}$ or a path $ z_{1}\ldots z_{3k}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $ x_{1}^{2}z_{3k}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f})\leq 2n(G)+2$ and all vertices of $G$ are $ \overrightarrow{f}$-strong.
\item If $n_{i}\equiv 2\pmod 3$ for $i\in \{1,2\}$ and $G$ is obtained from \textbf{\ }$C_{1}$ and $C_{2}$ by adding a path $z_{1}\ldots z_{3k+1}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+1}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega ( \overrightarrow{f})\leq 2n(G)+2$ and all vertices of $G$ are $ \overrightarrow{f}$-strong.
\item If $n_{i}\equiv 2\pmod 3$ for $i\in \{1,2\}$ and $G$ is obtained from $ C_{1}$ and $C_{2}$ by adding a path $z_{1}\ldots z_{3k+2}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+2}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f})\leq 2n(G)+2$ and all vertices of $G$ are $\overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 0\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding either the edge $x_{1}^{1}x_{1}^{2}$ or a path $z_{1}\ldots z_{3k}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $ x_{1}^{2}z_{3k}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f})\leq 2n(G)+1$ and all vertices of $G$ are $ \overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 0\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding a path $z_{1}\ldots z_{3k+1}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+1}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f})\leq 2n(G)+1$ and all vertices of $G$ are $\overrightarrow{f}$-strong.
\item If $n_{1}\equiv 2\pmod 3$, $n_{2}\equiv 0\pmod 3$ and $G$ is obtained from $C_{1}$ and $C_{2}$ by adding a path $z_{1}\ldots z_{3k+2}\;(k\geq 1)$ and the edges $x_{1}^{1}z_{1},$ $x_{1}^{2}z_{3k+2}$, then $G$ has a 3-tuple of RDFs $\overrightarrow{f}$ such that $\omega (\overrightarrow{f})\leq 2n(G)+1$ and all vertices of $G$ are $\overrightarrow{f}$-strong.
\item {If $s\geq 3$, $n_{i}\equiv 2\pmod 3$ for each $i$ and $G$ is obtained from $C_{1},\ldots ,C_{s}$ by adding a new vertex $x$ and the edges $ xx_{1}^{1},\ldots ,xx_{1}^{s}$, then $G$ has a 3-tuple of RDFs $ \overrightarrow{f}$ such that $\omega (f,G)\leq 2n(G)-s+4$ and the vertex $x$ is $\overrightarrow{f}$-strong.} \end{enumerate} \end{lemma}
\noindent\textbf{Proof.}
\begin{enumerate} \item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows$.$ For vertices on $C_{1}$: $f_{1}(x_{3i+1}^{1})=f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$ and $f_{i}(x)=0$ otherwise, { for $i=1,2$, and $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-5}{3}$, $f_{3}(x_{1}^{1})=2$, and $f_{3}(x)=0$ otherwise.} Now for vertices on $C_{2}$ but $x_{1}^{2}$:
If $n_{2}\equiv 0\pmod 3$, then let $f_{1}(x_{3i+4}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-6}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+3}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-3}{3}$ and $f_{2}(x)=0$ otherwise; $ f_{3}(x_{3i+2}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-3}{3}$ and $ f_{3}(x)=0$ otherwise.
If $n_{2}\equiv 1\pmod 3$, then let $f_{1}(x_{3i+4}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-7}{3}$, $f_{1}(x_{n_{2}-1}^{2})=1$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+3}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-4}{3}$ and $f_{2}(x)=0$ otherwise; {$f_{3}(x_{n_{2}}^{2})=1$,} $ f_{3}(x_{3i+2}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-4}{3}$ and $ f_{3}(x)=0$ otherwise.
If $n_{2}\equiv 2\pmod 3$, then let $f_{1}(x_{3i+4}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-5}{3}$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+3}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-5}{3}$, $f_{2}(x_{n_{2}}^{2})=1$ and $ f_{2}(x)=0$ otherwise; $f_{3}(x_{3}^{2})=f_{3}(x_{n_{2}-1}^{2})=1$, $f_{3}(x_{3i+5}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-8}{3}$ and $f_{3}(x)=0$ otherwise.
In either case, $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G.$ Hence $\omega ( \overrightarrow{f})\leq 2n(G)+1$, and all vertices of $V(G)- \{x_{2}^{2},x_{n_{2}}^{2}\}$ are $\overrightarrow{f}$-strong. \textbf{\ }
\item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows: $ f_{1}(x_{3i+1}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{1}(z_{3i+3})=2$ for $0\leq i\leq k-1$, $f_{1}(x_{3i+3}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$ and $f_{1}(x)=0$ otherwise; $ f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{2}(z_{3i+2})=2$ for $0\leq i\leq k-1$, $f_{2}(x_{3i+2}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$, $f_{2}(x_{n_{2}}^{2})=1$ and $f_{2}(x)=0$ otherwise; $f_{3}(x_{n_{1}}^{1})=1$, $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-5}{3}$, $f_{3}(z_{3i+1})=2$ for $0\leq i\leq k-1$, $ f_{3}(x_{3i+1}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-4}{3}$, $ f_{3}(x_{n_{2}-1}^{2})=2$ and $f_{3}(x)=0$ otherwise.
Clearly $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G,$ and thus $\overrightarrow{ f}=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $G.$ Moreover, $\omega ( \overrightarrow{f})\leq 2n(G)+1$ and each vertex of $G$ but $x_{n_{2}}^{2}$ is $\overrightarrow{f}$-strong.
\item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows: $ f_{1}(x_{3i+1}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{1}(z_{3i+3})=2$ for $0\leq i\leq k-1$, $f_{1}(x_{3i+2}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$, $f_{1}(x_{n_{2}}^{2})=1$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $f_{2}(z_{3i+2})=2$ for $0\leq i\leq k-1$, $f_{2}(x_{3i+1}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$, $f_{2}(x_{n_{2}-1}^{2})=1$ and $f_{2}(x)=0$ otherwise; $f_{3}(x_{n_{1}}^{1})=1$, $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-5}{3}$, $f_{3}(z_{3i+1})=2$ for $0\leq i\leq k$, $ f_{3}(x_{3i+3}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-4}{3}$ and $ f_{3}(x)=0$ otherwise.
Clearly $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G$ and thus $\overrightarrow{f }=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $G.$ Also, $\omega ( \overrightarrow{f})\leq 2n(G)+1$ and all vertices of $G$ but $x_{n_{2}}^{2}$ are $\overrightarrow{f}$-strong.
\item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows: $ f_{1}(x_{3i+1}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{1}(z_{3i+3})=2$ for $0\leq i\leq k-1$, $f_{1}(x_{3i+1}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$, $f_{3}(x_{n_{2}-1}^{2})=1$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $f_{2}(z_{3i+2})=2$ for $0\leq i\leq k$, $f_{2}(x_{3i+3}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-4}{3}$ and $f_{2}(x)=0$ otherwise; $ f_{3}(x_{n_{1}}^{1})=1$, $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{ n_{1}-5}{3}$, $f_{3}(z_{3i+1})=2$ for $0\leq i\leq k$, $ f_{3}(x_{3i+2}^{2})=2 $ for each $0\leq i\leq \frac{n_{2}-4}{3}$, $ f_{3}(x_{n_{2}}^{2})=1$ and $f_{3}(x)=0$ otherwise.
Clearly $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G$ and thus $\overrightarrow{f }=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $G.$ Also, $\omega ( \overrightarrow{f})\leq 2n(G)+1$ and each vertex of $G$ but $x_{n_{2}}^{2}$ is $\overrightarrow{f}$-strong.
\item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows: $ f_{1}(x_{3i+1}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{1}(z_{3i+3})=2$ for $0\leq i\leq k-1$, $f_{1}(x_{3i+3}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-5}{3}$, $f_{1}(x_{n_{2}}^{2})=1$ and $f_{1}(x)=0$ otherwise; $f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $f_{2}(z_{3i+2})=2$ for $0\leq i\leq k-1$, $f_{2}(x_{3i+2}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-2}{3}$ and $f_{2}(x)=0$ otherwise; $ f_{3}(x_{n_{1}}^{1})=1$, $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{ n_{1}-5}{3}$, $f_{3}(z_{3i+1})=2$ for $0\leq i\leq k-1$, $ f_{3}(x_{3i+1}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-2}{3}$ and $ f_{3}(x)=0$ otherwise.
Clearly $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G$ and thus $\overrightarrow{f }=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $G.$ Further, $\omega ( \overrightarrow{f})\leq 2n(G)+2$ and all vertices of $G$ are $ \overrightarrow{f}$-strong.
\item The proof is similar to that of item (5).
\item The proof is similar to that of item (5).
\item Define the functions $f_{1},f_{2}$ and $f_{3}$ on $V(G)$ as follows: $ f_{1}(x_{3i+1}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{1}(z_{3i+3})=2$ for $0\leq i\leq k-1$, $f_{1}(x_{3i+3}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-3}{3}$ and $f_{1}(x)=0$ otherwise; $ f_{2}(x_{3i+2}^{1})=2$ for each $0\leq i\leq \frac{n_{1}-2}{3}$, $ f_{2}(z_{3i+2})=2$ for $0\leq i\leq k-1$, $f_{2}(x_{3i+2}^{2})=2$ for each $ 0\leq i\leq \frac{n_{2}-3}{3}$ and $f_{2}(x)=0$ otherwise; $ f_{3}(x_{n_{1}}^{1})=1$, $f_{3}(x_{3i+3}^{1})=2$ for each $0\leq i\leq \frac{ n_{1}-5}{3}$, $f_{3}(z_{3i+1})=2$ for $0\leq i\leq k-1$, $ f_{3}(x_{3i+1}^{2})=2$ for each $0\leq i\leq \frac{n_{2}-3}{3}$ and $ f_{3}(x)=0$ otherwise.
Then $f_{1},f_{2}$ and $f_{3}$ are RDFs of $G$ and thus $\overrightarrow{f} =(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of $G$ with the desired property.
\item The proof is similar to that of item (8).
\item The proof is similar to that of item (8).
\item {Define the function $f_{1}$ by $f_{1}(x)=2$, $f_{1}(x_{n_{j}}^{j})=1$ for each $1\leq j\leq s$, $f_{1}(x_{i}^{j})=2$ for each $j$ and each $ i\equiv 0\pmod 3$, and $f_{1}(y)=0$ otherwise, and set $\overrightarrow{f} =(f_{1},f_{1},f_{1})$. }Clearly $f_{1}$ is an RDF of $G$ and thus $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ is a 3-tuple of RDFs of {$G$ such that $\omega (\overrightarrow{f})\leq 2n(G)-s+4$ and the vertex $x$ is $ \overrightarrow{f}$-strong as desired.} $
\Box $ \end{enumerate}
\begin{lemma} \label{ear2}Let $H$ be a graph obtained from a cycle $C_{3p+2}=x_{1}x_{2} \ldots x_{3p+2}x_{1}$ and a path $Q=y_{1}\ldots y_{\ell }$ where $\ell \equiv 1\;\mathrm{or}\;2\pmod 3$ by adding the edge $y_{1}x_{1}$ and joining $y_{\ell }$ to some vertices in $V(C_{3p+2})-\{x_{1}\}$ with the condition that:
\begin{description} \item[(a)] if $\ell\equiv 1\pmod 3$ and $y_{\ell}x_{j}\in E(H)$, then $ j\not\equiv 2\pmod 3$,
\item[(b)] if $\ell\equiv 2\pmod 3$ and $y_{\ell}x_{j}\in E(H)$, then { $j\equiv 2\pmod 3$.} \end{description}
Then there exists a 3-tuple $\overrightarrow{g}=(g_{1},g_{2},g_{3})$ of RDF of $H$ such that $\omega (\overrightarrow{g},H)\leq 2n(H)+1$ and each vertex of $H$ but $y_{1},y_{\ell }$ is $\overrightarrow{g}$-strong. \end{lemma}
\noindent \textbf{Proof.} First let $\ell \equiv 1\pmod 3$ and $y_{\ell }x_{j}\in E(H).$ Define the functions $g_{1},g_{2}$ and $g_{3}$ on $V(H)$ as follows, depending on whether $j\equiv 0\pmod 3$ or $j\equiv 1\pmod 3.$
If $j\equiv 0\pmod 3,$ then let $g_{1}(x_{3i+1})=2$ for $0\leq i\leq p$, $ g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(x_{3i+2})=2$ for $0\leq i\leq p$, $g_{2}(y_{\ell })=1$, $ g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{2}(z)=0$ otherwise; $g_{3}(x_{1})=2,$ $g_{3}(x_{3i+3})=2$ for $0\leq i\leq p-1$, $ g_{3}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise.
If $j\equiv 1\pmod 3$, then let $g_{1}(x_{3i+1})=2$ for $0\leq i\leq p$, $ g_{1}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(x_{3i+2})=2$ for $0\leq i\leq p$, $g_{2}(y_{\ell })=1$, $ g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{2}(z)=0$ otherwise; $g_{3}(x_{1})=2,$ $g_{3}(x_{3i+3})=2$ for $0\leq i\leq p-1$, $ g_{3}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -4}{3}$, and $g_{1}(z)=0$ otherwise.
Second, let $\ell \equiv 2\pmod 3$ and $y_{\ell }x_{j}\in E(H),$ where { $j\equiv 2\pmod 3$.} Define the functions $g_{1},g_{2}$ and $ g_{3}$ on $V(H)$ as follows: $g_{1}(x_{p})=1,$ $g_{1}(x_{3i+3})=2$ for $ 0\leq i\leq p-1$, $g_{1}(y_{3i+1})=2$ for $0\leq i\leq \frac{\ell -2}{3}$, and $g_{1}(z)=0$ otherwise; $g_{2}(x_{3i+2})=2$ for $0\leq i\leq p$, $ g_{2}(y_{3i+2})=2$ for $0\leq i\leq \frac{\ell -5}{3}$, $g_2(y_{\ell-1})=1$, and $g_{2}(z)=0$ otherwise; $g_{3}(x_{3i+1})=2$ for $0\leq i\leq p$, $ g_{3}(y_{3i+3})=2$ for $0\leq i\leq \frac{\ell -5}{3}$, $g_3(y_{\ell})=1$, and $g_{1}(z)=0$ otherwise.
Clearly, $g_{1},g_{2},g_{3}$ are RDFs of $H$ and thus $\overrightarrow{g} =(g_{1},g_{2},g_{3})$ is a 3-tuple of RDFs of $H$ with the desired property. $
\Box $
\section{Partial answer to Conjecture \protect\ref{conj}}
In this section, we give a positive answer to Conjecture \ref{conj} for some particular graphs. We start with the following two lemmas.
\begin{lemma} \label{induced}\emph{Let }$k\geq 1$ \emph{be an integer and let $G$ be a connected graph with $\delta \geq 2$, which does not contain {\ neither }any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles {\ nor }any cycle of length $\equiv 0\pmod 3$. Let $C$ be a cycle of $G$ with length $\ell (C)$ $ \equiv 2\pmod 3$. Then }
\begin{enumerate} \item \emph{if $C$ is induced in $G$, then $\ell (C)\geq 3k+5$, and }
\item \emph{if $C$ is not induced in $G$, then $\ell (C)\geq 6k+8$. } \end{enumerate} \end{lemma}
\noindent \textbf{Proof.} Item (1) is immediate since $G$ does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles and $\ell (C)\equiv 2 \pmod 3$. To prove item (2), let $C=v_{1}v_{2}\ldots v_{3p+2}v_{1}$ be a cycle which is not induced in $G$. Hence $C$ has a chord, say without loss of generality, $v_{1}v_{i}\in E(G)$. Consider the two paths $ P=v_{i+1}v_{i+2}\ldots v_{3p+2}v_{1}$ and $Q=v_{2}v_{3}\ldots v_{i}$. Let $ n(P)$ and $n(Q)$ denote the order of $P$ and $Q,$ respectively. Clearly $ n(P)+n(Q)=3p+2$. Now, if $n(P)\equiv 0\pmod 3$, then $v_{1}v_{2}v_{3}\ldots v_{i}v_{1}$ is a cycle of length $\equiv 0\pmod 3,$ contradicting the fact that $G$ has no cycle of such length. Hence $n(P)\not\equiv 0\pmod 3$, and likewise $n(Q)\not\equiv 0\pmod 3$. Moreover, since $n(P)+n(Q)=3p+2,$ we deduce that $n(P)\not\equiv 2\pmod 3$ and $n(Q)\not\equiv 2\pmod 3$. Hence $ n(P)\equiv 1\pmod 3$ and $n(Q)\equiv 1\pmod 3$. Consider the cycles $ C_{1}=v_{1}v_{2}v_{3}\ldots v_{i}v_{1}$ and $C_{2}=v_{1}v_{i}v_{i+1}\ldots v_{3p+2}v_{1}$. Then $\ell (C_{1})\equiv 2\pmod 3$ and $\ell (C_{2})\equiv 2 \pmod 3$. If $C_{1}$ and $C_{2}$ are induced in $G$, then by item (1) we have $i\geq 3k+5$ and $3p+4-i\geq 3k+5$ and thus $\ell (C)=3p+2\geq 3k+5+i-2\geq 3k+5+3k+5-2=6k+8$. Hence we assume that $C_{1}$ is not induced in $G$. By repeating the above process we can see that the subgraph $ G[V(C_{1})]$ has an induced cycle of length $\equiv 2\pmod 3$ and so $
|V(C_{1})|=i\geq 3k+5$. If $C_{2}$ is an induced cycle, then by item (1) we have $3p+4-i\geq 3k+5$ and so $\ell (C)=3p+2\geq 3k+5+i-2\geq 3k+5+3k+5-2=6k+8$. Now if $C_{2}$ is not an induced cycle, then a similar argument as above shows that $G[V(C_{2})]$ has an induced cycle of length $ \equiv 2\pmod 3$ yielding also $\ell (C)\geq 6k+8$.$
\Box $
\begin{lemma} \label{12}Let $G$ be a connected\textbf{\ }graph with minimum degree $\delta \geq 2$ and let $G_{1}$ and $G_{2}$ be two non-null subgraphs of $G$ such that $V(G)=V(G_{1})\cup V(G_{2})$. Then one of the following holds:
\begin{enumerate} \item $G_{1}$ has a path $P=v_{1}\ldots v_{t}$ such that both $v_{1}$ and $ v_{t}$ have neighbors in $G_{2}$ and $N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(G_{2})\cup V(P)$.
\item $G_{1}$ has a cycle $C=v_{1}v_{2}\ldots v_{t}v_{1}$ such that $v_{1}$ has neighbors in $G_{2}$ and $N_{G}(v_{t})\subseteq V(G_{2})\cup V(C)$.
\item $G_{1}$ contains a tailed $m$-cycle, say $C_{m,\ell }$, such that $ y_{\ell }$ is adjacent to some vertex in $G_{2}$ and $N_{G}(x_{2})\cup N_{G}(x_{m})\subseteq V(G_{2})\cup V(C_{m,\ell })$. \end{enumerate} \end{lemma}
\noindent \textbf{Proof.} Let $\mathcal{P}$ be the family of all longest paths (not necessarily induced) in $G_{1}$ such that at least one of their end-points has a neighbor in $G_{2}$ and let $Q=\{v\in V(G)\mid \mathrm{ there\;is\;a\;path}$ $v_{1},\ldots ,v_{t}(=v)\in \mathcal{P}\;\mathrm{ such\;that}\;v_{1}\;\mathrm{has}$ $\mathrm{a}$ $\mathrm{neighbor}$ $\mathrm{ in}\;G_{2}\}$. Choose a vertex $v\in Q$ such that the length of its corresponding path $P=v_{1},\ldots ,v_{t}(=v)\in \mathcal{P}$ is as long as possible.
First let $v$ be adjacent to some vertex in $G_{2}$. By the definition of set $Q,$ we have $v_{1}\in Q$, and from the choice of $v$ we deduce that $ N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(G_{2})\cup V(P).$ Hence item (1) holds. Suppose now $v$ has no neighbor in $G_{2}$. It follows from the choice of $v$ and the fact $\delta \geq 2$ that $v$ has at least two neighbors in $V(P)$. Let $j$ be the smallest index such that $vv_{j}$ $\in E(G).$ Now, if $j=1,$ that is $v$ is adjacent to $v_{1}$, then clearly $ N_{G}(v_{t})\subseteq V(P)$ and thus item (2) holds. Hence we assume that $ j\neq 1.$ Then $v_{1}\ldots v_{j}...v_{t-1}vv_{j}$\textbf{\ }is a tailed cycle contained in $G_{1}.$ Observe that the path with endvertices $v_{1}$ and $v_{j+1}$ starting from $v_{1}$ to $v_{j}$ and then passing through $ v_{t}$ to $v_{j+1}$ is also a longest path with same length as $P.$ Since $ v=v_{t}$ has no neighbor in $G_{2},$ we may assume by analogy that $v_{j+1}$ has no neighbor in $G_{2}$ and thus all its neighbors are on\textbf{\ }$P$ which forms a tailed cycle and thus item (3) holds$.$\textbf{\ }$
\Box $
\begin{theorem} \label{Th1}\emph{\emph{Let} $k\geq 1$ be an integer and let $G$ be a connected graph of order $n\geq 6k+9$ and minimum degree at least 2 such that $G$ has no cycle with length $\equiv 0\;\mathrm{or}\;2\pmod 3$. Then $ \gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}$.} \end{theorem}
\noindent \textbf{Proof.} Let $Q=z_{1}z_{2}\ldots z_{r}$ be a longest path in $G$. If $V(G)=V(Q)$, then we have $\gamma _{R}(G)\leq \frac{2n+1}{3}< \frac{(4k+8)n}{6k+11}$. Hence, we assume that $V(Q)\subsetneqq V(G)$. By the choice of $Q$ we have $N_{G}(z_{1})\cup N_{G}(z_{r})\subseteq V(Q)$. Since $ \delta (G)\geq 2$, $z_{1}$ is adjacent to some $z_{j}$ with $j\equiv 1\pmod 3 ,$ because $G$ has no cycle with length $\equiv 0\;\mathrm{or}\;2\pmod 3$. Let $G_{2}^{0}$ be the graph obtained from the path $Q$ to which we add the edge $z_{1}z_{j}$ and let $G_{1}^{0}$ be the graph induced by $ V(G)-V(G_{2}^{0}).$ Observe that $G_{2}^{0}$ is a tailed $j$-cycle $ C_{j,r-j}.$ By Lemma \ref{cyclepath}-(3), $G_{2}^{0}$ has a 3-tuple of RDFs $ \overrightarrow{f}=(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f} )\leq 2r+1$ and all vertices of $C_{j,r-j}$ but $z_{1}$ are $\overrightarrow{ f}$-strong. According to Lemma \ref{12}, we consider the following three possibilities.
\begin{description} \item[(a)] $G_{1}^{0}$ has a path $P=v_{1}\ldots v_{t}$ such that $ v_{1},v_{t}$ are adjacent to some vertices in $V(G_{2}^{0})-\{z_{1},z_{t}\}$ , say $u,v$ (possibly $u=v$) and $N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(P)$.\newline Let $G_{2}^{1}$ be the graph obtained from $G_{2}^{0}$ and the path $P$ by adding the edges $v_{1}u,v_{t}v$. By Lemma \ref{ear1}, $\overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g}$ of $G_{2}^{1}$ such that $\omega (\overrightarrow{g},P)\leq 2n(P)$ and each vertex in $ V(P)-\{v_{1},v_{t}\}$ is $\overrightarrow{g}$-strong. Note that $\omega ( \overrightarrow{g})=\omega (\overrightarrow{f})+\omega (\overrightarrow{g} ,P)\leq 2n(G_{2}^{1})+1.$
\item[(b)] $G_{1}^{0}$ has a cycle $C=v_{1},\ldots ,v_{t}v_{1}$ such that $ v_{1}$ is adjacent to a vertex in $G_{2}^{0}$, say $u$, and $ N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(C)$.\newline Since $G$ has no cycle of length $\equiv 0\;\mathrm{or}\;2\pmod 3$, we have $ t\equiv 1\pmod 3$. Let $G_{2}^{1}$ be the graph obtained from $G_{2}^{0}$ and the cycle $C$ by adding the edge $v_{1}u$. By Lemma \ref {tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g}$ of $G_{2}^{1}$ such that $\omega (g,C)\leq 2n(C)$ and each vertex in $V(C)-\{v_{t}\}$ is $\overrightarrow{g}$-strong. In addition, it is clear that $\omega (\overrightarrow{g})\leq \omega ( \overrightarrow{f})+\omega (\overrightarrow{g},C)\leq 2n(G_{2}^{1})+1$.
\item[(c)] $G_{1}^{0}$ contains a tailed $m$-cycle $C_{m,\ell }$, such that $ y_{\ell }$ is adjacent to some vertex in $G_{2}^{0}$, say $u$, and $ N_{G}(x_{2})\cup N_{G}(x_{m})\subseteq V(G_{2}^{0})\cup V(C_{m,\ell })$. \newline As above in (b), since $G$ has no cycle of length $\equiv 0\;\mathrm{or}\;2 \pmod 3$, we have $m\equiv 1\pmod 3$. Let $G_{2}^{1}$ be the graph obtained from $G_{2}^{0}$ and the tailed $m$-cycle $C_{m,\ell }$ by adding the edge $ uy_{\ell }$. By Lemma \ref{tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g}$ of $G_{2}^{1}$ such that $\omega (\overrightarrow{g},C_{m,\ell })\leq 2n(C_{m,\ell })$ and each vertex of $C_{m,\ell }$ but $x_{m}$\ is $\overrightarrow{g}$-strong. Therefore, we also have $\omega (\overrightarrow{g})\leq 2n(G_{2}^{1})+1.$ \end{description}
Now, let $G_{1}^{1}=G-G_{2}^{1}$. By repeating the above process, we obtain a $3$-tuple of RDFs $G$ that is $\overrightarrow{h}=(h_{1},h_{2},h_{3})$ such that $\omega (\overrightarrow{h})\leq 2n(G)+1\leq \frac{3n(4k+8)}{6k+8}$ . Therefore, $\omega (h_{j})\leq \frac{n(4k+8)}{6k+8}$ for some $j\in \{1,2,3\},$ and this completes the proof.$
\Box $
\begin{theorem} \label{Th2}\emph{\emph{Let} $k\geq 1$ be an integer and let $G$ be a connected graph of order $n\geq 6k+9$ with minimum degree at least 2 and having a cycle $C$ with length $\equiv 0\pmod 3$ such that any other cycle of $G$ with length $\equiv 0\;\mathrm{or}\;2 \pmod 3$ has at least a common vertex with $C$. Then $\gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}$.} \end{theorem}
\noindent \textbf{Proof.} Assume that the vertices of the cycle $C$ with length $\equiv 0\pmod 3$ are labelled by $z_{1}z_{2}\ldots z_{r}z_{1}.$ If $V(G)=V(C)$, then clearly $\gamma _{R}(G)\leq \frac{2n}{3}<\frac{(4k+8)n}{6k+11}$. Hence, we assume that $V(C)\subsetneqq V(G)$. Let $G_{2}^{0}=C$ and let $G_{1}^{0}$ be the graph induced by $V(G)-V(G_{2}^{0}).$ By Lemma \ref{cyclepath}, $ G_{2}^{0}$ has a 3-tuple of RDFs $\overrightarrow{f}=(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f})\leq 2r$ and all vertices of $C$ are $ \overrightarrow{f}$-strong. Now, according to Lemma \ref{12}, we consider the following three possibilities.
\begin{description} \item[(a)] $G_{1}^{0}$ has a path $P=v_{1}\ldots v_{t}$ such that $ v_{1},v_{t}$ have neighbors in $V(G_{2})$, say $u,v$ (possibly $u=v$), and $ N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(P)$.\newline Let $G_{2}^{1}$ be the graph obtained from $G_{2}^{0}$ and the path $P$ by adding the edges $v_{1}u$ and $v_{t}v$. By Lemma \ref{ear1}, $ \overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g} $ of $G_{2}^{1}$ such that $\omega (g,P)\leq 2n(P)=2t$ and each vertex in $ V(P)-\{v_{1},v_{t}\}$ is $\overrightarrow{g}$-strong. In this case, we have $ \omega (\overrightarrow{g})=\omega (\overrightarrow{f})+\omega ( \overrightarrow{g},P)\leq 2n(G_{2}^{1}).$
\item[(b)] $G_{1}^{0}$ has a cycle $C^{\prime }=v_{1},\ldots ,v_{t}v_{1}$ such that $v_{1}$ is adjacent to a vertex in $G_{2}^{0}$, say $u$, and $ N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(C^{\prime })$.\newline By assumption, we have $t\equiv 1\pmod 3$. Let $G_{2}^{1}$ be\ the graph obtained from $G_{2}^{0}$ and the cycle $C^{\prime }$ by adding the edge $ v_{1}u$. By Lemma \ref{tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g}$ of $G_{2}^{1}$ such that $\omega (\overrightarrow{g},C^{\prime })\leq 2n(C^{\prime })=2t$ and each vertex in $V(C^{\prime })-\{v_{t}\}$ is $\overrightarrow{g}$-strong. Moreover, we also obtain $\omega (\overrightarrow{g})\leq 2n(G_{2}^{1})$.
\item[(c)] $G_{1}^{0}$ contains a tailed $m$-cycle $C_{m,\ell }$, such that $ y_{\ell }$ is adjacent to some vertex in $G_{2}^{0}$, say $u$, and $ N_{G}(x_{2})\cup N_{G}(x_{m})\subseteq V(G_{2}^{0})\cup V(C_{m,\ell })$. \newline As above, $m\equiv 1\pmod 3$. Let $G_{2}^{1}$ be the graph obtained from $ G_{2}^{0}$ and the tailed $m$-cycle $C_{m,\ell }$ by adding the edge $ uy_{\ell }$. By Lemmas \ref{tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple of RDFs $\overrightarrow{g}$ of $G_{2}^{1}$ such that such that $\omega (\overrightarrow{g},C_{m,\ell })\leq 2n(C_{m,\ell })$ and each vertex of $C_{m,\ell }$ but $x_{m}$\ is $\overrightarrow{g}$-strong. In addition, we have $\omega (\overrightarrow{g})\leq 2n(G_{2}^{1})$. \end{description}
Let $G_{1}^{1}=G-G_{2}^{1}$. By repeating the above process, we obtain a 3-tuple of RDFs $G$ that is $\overrightarrow{h}=(h_{1},h_{2},h_{3})$ such that $\omega (\overrightarrow{h})\leq 2n(G)+1\leq \frac{3n(4k+8)}{6k+8}$. Therefore $\omega (f_{j})\leq \frac{n(4k+8)}{6k+8}$ for some $j\in \{1,2,3\}, $ and this completes the proof.$
\Box $
\begin{theorem} \label{Th3}\emph{Let $k\geq 1$ be an integer and let $G$ be a connected graph of order $n\geq 6k+9\;$and minimum degree at least 2 which does not contain {neither }any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles {nor }any cycle of length $\equiv 0\pmod 3$, and every two distinct cycles of length $\equiv 2\pmod 3$ (if any) have at least a common vertex. If $G$ has a cycle $C$ with length $\equiv 2\pmod 3$, then $\gamma _{R}(G)\leq \frac{ (4k+8)n}{6k+11}$.} \end{theorem}
\noindent \textbf{Proof.} Let $C=z_{1}z_{2}\ldots z_{p}z_{1}$ be a cycle of length $\equiv 2\pmod 3$ in $G$ {chosen first not induced, if it exists, otherwise it is of course induced. }If $V(G)=V(C)$, then we have $\gamma _{R}(G)\leq \frac{2n+2}{3}<\frac{(4k+8)n}{6k+11}$. Hence, we can assume that $V(C)\subsetneqq V(G)$.
First assume there is either a cycle $C^{\prime }=x_{1}x_{2}\ldots x_{m}x_{1} $ such that $x_{1}$ is adjacent to a vertex {of }$C$, say $z_{1}$ , and $N_{G}(x_{m})\subseteq V(C)\cup V(C^{\prime })$, or a tailed $m$-cycle $C_{m,\ell }$ in $G$ such that $y_{\ell }$ is adjacent to a vertex {of} $C$, say $z_{1}$, and $N_{G}(x_{m})\subseteq V(C)\cup V(C_{m,\ell })$. By assumption $m\equiv 1\pmod 3$. Let $G_{2}^{0}=C+C^{\prime }+x_{1}z_{1}$ or $ G_{2}^{0}=C+C_{m.\ell }+y_{\ell }z_{1}$ {(depending on which situation occurs, the first or the second one), }and let $G_{1}^{0}=G-G_{2}^{0}$. By Lemma \ref{MainLem}, $G_{2}^{0}$ has a 3-tuple of RDFs $\overrightarrow{f} =(f_{1},f_{2},f_{3})$ such that $\omega (\overrightarrow{f})\leq 2n(G_{2}^{0})+1$ and all vertices of $G_{2}^{0}$ but $x_{m}$ are $ \overrightarrow{f}$-strong. Considering our assumption and Lemma \ref{12}, { one of the following situations holds. }
\begin{description} \item[(a)] $G_{1}^{0}$ has a path $P=v_{1},\ldots ,v_{t}$ such that $v_{1}$ and $v_{t}$ are adjacent to some vertices in $V(G_{2}^{0})$, say $u,v$ (possibly $u=v$) and $N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(P)$. {We note that }$x_{m}\notin \{u,v\}$, {since }$ C^{\prime }$ or $C_{m,\ell }$ {has been chosen so that }$x_{m}$ satisfies $ N_{G}(x_{m})\subseteq V(C)\cup V(C_{m,\ell })$. {Hence }$u$ and $v$ are $ \overrightarrow{f}$-strong. \newline Let $G_{2}^{1}$ be {the graph }obtained from $G_{2}^{0}$ and the path $P$ by adding the edges $v_{1}u,v_{t}v$. By Lemma \ref{ear1}, $\overrightarrow{f}$ can be extended to a 3-triple $\overrightarrow{g}$ such that $\omega ( \overrightarrow{g})\leq 2n(G_{2}^{1})+1$ and all vertices of $ V(P)-\{v_{1},v_{t}\}$ are $\overrightarrow{g}$-strong.
\item[(b)] $G_{1}^{0}$ has a cycle $C^{\prime \prime }=v_{1},\ldots ,v_{t}v_{1}$ such that $v_{1}$ is adjacent to a vertex in $G_{2}^{0}$, say $ u $, and $N_{G}(v_{t})\subseteq V(G_{2}^{0})\cup V(C^{\prime \prime })$. {A same argument as in item (a) shows that }$u\neq x_{m}$, {and thus }$u$ is $ \overrightarrow{f}$-strong. \newline By assumption, we have $t\equiv 1\pmod 3$. Let $G_{2}^{1}$ be {\ the graph } obtained from $G_{2}^{0}$ and the cycle $C^{\prime \prime }$ by adding the edge $v_{1}u$. By Lemma \ref{tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple $\overrightarrow{g}$ such that $\omega ( \overrightarrow{g})\leq 2n(G_{2}^{1})+1$ and each vertex $V(C^{\prime \prime })-\{v_{t}\}$ is $\overrightarrow{g}$-strong.
\item[(c)] $G_{1}^{0}$ contains a tailed $m^{\prime }$-cycle $C_{m^{\prime },\ell ^{\prime }}$, such that $y_{\ell ^{\prime }}$ is adjacent to some vertex in $G_{2}^{0}$, say $u$, and $N_{G}(x_{2})\cup N_{G}(x_{m})\subseteq V(G_{2}^{0})\cup V(C_{m^{\prime },\ell ^{\prime }})$. {Note that }$u\neq x_{m}$ and $u$ is $\overrightarrow{f}$-strong. \newline As above $m^{\prime }\equiv 1\pmod 3$. Let $G_{2}^{1}$ be {the graph } obtained from $G_{2}^{0}$ and the tailed cycle $C_{m^{\prime },\ell ^{\prime }}$ by adding the edge $uy_{\ell ^{\prime }}$. By Lemma \ref {tailedcycle-3p+1}, $\overrightarrow{f}$ can be extended to a 3-triple $ \overrightarrow{g}$ such that $\omega (\overrightarrow{g})\leq 2n(G_{2}^{1})+1$ and all vertices of $V(C_{m^{\prime },\ell ^{\prime }})-\{x_{m^{\prime }}\}$ are $\overrightarrow{g}$-strong. \end{description}
Let $G_{1}^{1}=G-G_{2}^{1}$. By repeating {the }above process, we obtain a 3-tuple of RDFs $\overrightarrow{g^{\prime }}=(g_{1}^{\prime },g_{2}^{\prime },g_{3}^{\prime })$ such that $\omega (\overrightarrow{g^{\prime }})\leq 2n(G)+1\leq \frac{3n(4k+8)}{6k+8}$. It follows that $\omega (g_{j}^{\prime })\leq \frac{n(4k+8)}{6k+8}$ for some $j\in \{1,2,3\}$ as desired.
{Next we can }assume that there is {neither a }cycle $C^{\prime }=(x_{1}x_{2}\ldots x_{m}x_{1})$ such that $x_{1}$ is adjacent to a vertex in $C$ and $N_{G}(x_{m})\subseteq V(C)\cup V(C^{\prime })$, {nor a }tailed $ m $-cycle $C_{m,\ell }$ with $m\equiv 1\pmod 3$ in $G$ such that $y_{\ell }$ is adjacent to a vertex in $C$ and $N_{G}(x_{m})\subseteq V(C)\cup V(C_{m,\ell })$. Let $H_{2}^{0}=C$ and $H_{1}^{0}=G-H_{2}^{0}$. It follows from Lemma \ref{12} and the assumptions that $H_{1}^{0}$ has a path $ P=v_{1},\ldots ,v_{t}$ such that $v_{1},v_{t}$ are adjacent to some vertices in $V(H_{2}^{0})$, say $z_{1},z_{j}$ (possibly $j=1$) and $N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(H_{2}^{0})\cup V(P)$. We consider the following cases.
\noindent \textbf{Case 1.} $j=1$.\newline Let $G_{2}^{1}$ be {the graph }obtained from $H_{2}^{0}$ and the path $P$ by adding the edges $v_{1}z_{1},v_{t}z_{1}$ and let $G_{1}^{1}=G-G_{2}^{1}$. By Lemma \ref{MainLem}{-(1)}, $G_{2}^{1}$ has a triple $\overrightarrow{g}$ of RDFs such that $\omega (\overrightarrow{f})\leq 2n(G_{2}^{1})+1$ and all vertices of $G_{2}^{1}$ but $v_{1},v_{t}$ are $\overrightarrow{g}$-strong. If $V(G)=V(G_{2}^{1})$ {(and hence }$G_{1}^{1}$ is empty), then the result follows. {Hence, a}ssume that $V(G)\neq V(G_{2}^{1})$. By the assumptions and Lemma \ref{12}, we deduce that $G_{1}^{1}$ has a path $P^{\prime }=v_{1}^{\prime },\ldots ,v_{t^{\prime }}^{\prime }$ such that $ v_{1}^{\prime },v_{t^{\prime }}^{\prime }$ are adjacent to some vertices in $ V(G_{2}^{1})$, say $u,v$ (possibly $u=v$) and $N_{G}(v_{1}^{\prime })\cup N_{G}(v_{t^{\prime }}^{\prime })\subseteq V(G_{2}^{1})\cup V(P^{\prime })$. Let $G_{2}^{2}$ be obtained from $G_{2}^{1}$ and the path $P^{\prime }$ by adding the edges $v_{1}^{\prime }u,v_{t^{\prime }}^{\prime }v$ and let $ G_{1}^{2}=G-G_{2}^{2}$. {Note that }$v_{1},v_{t}\notin \{u,v\}$ {and thus }$ u,v$ are $\overrightarrow{g}$-strong. By Lemma \ref{ear1}, $\overrightarrow{g }$ can be extended to a 3-triple $\overrightarrow{g^{\prime }}$ such that $ \omega (\overrightarrow{f_{2}})\leq 2n(G_{2}^{2})+1$ and all vertices of $ V(P^{\prime })-\{v_{1}^{\prime },v_{t^{\prime }}^{\prime }\}$ are $ \overrightarrow{g^{\prime }}$-strong. By repeating {the }above process, we obtain a 3-tuple of RDFs $\overrightarrow{g^{\ast }}=(g_{1}^{\ast },g_{2}^{\ast },g_{3}^{\ast })$ such that $\omega (\overrightarrow{g^{\ast }} )\leq 2n(G)+1\leq \frac{3n(4k+8)}{6k+8}$. It follows that $\omega (g_{r}^{\ast })\leq \frac{n(4k+8)}{6k+8}$ for some $r\in \{1,2,3\}$ as desired.
\noindent \textbf{Case 2.} $j\neq 1$.\newline We distinguish the following {three }subcases.
\textbf{Subcase 2.1.} $t\equiv 1\pmod 3$.\newline Since $G$ has no cycle of length $\equiv 0\pmod 3$, we have $j\not\equiv 2 \pmod 3$. Let $G_{2}^{1}$ be {the graph }obtained from $H_{2}^{0} $ and the path $P$ by adding the edges $v_{1}z_{1},v_{t}z_{j}$ and let $ G_{1}^{1}=G-G_{2}^{1}$. By Lemma \ref{ear2}, $G_{2}^{1}$ has a triple $ \overrightarrow{f}$ of RDFs such that $\omega (\overrightarrow{f})\leq 2n(G_{2}^{1})+1$ and all vertices of $G_{2}^{1}$ but $v_{1},v_{t}$ are $ \overrightarrow{f_{1}}$-strong. As in Case 1, we can obtain a 3-tuple of RDFs $\overrightarrow{g^{\ast }}=(g_{1}^{\ast },g_{2}^{\ast },g_{3}^{\ast })$ such that $\omega (\overrightarrow{g^{\ast }})\leq 2n(G)+1\leq \frac{3n(4k+8) }{6k+8}$ {yielding the desired result}.
\textbf{Subcase 2.2.} $t\equiv 2\pmod 3$.\newline {Observe that if }$j\equiv 0\pmod 3$, {then }$ z_{1}v_{1}...v_{t}z_{j}z_{j+1}...z_{p}z_{1}$ would be a cycle of length $ \equiv 0\pmod 3$, a contradiction, {and if $j\equiv 1\pmod 3$, { then }$z_{1}v_{1}...v_{t}z_{j}z_{j-1}...z_{2}z_{1}$ would be a cycle of length $\equiv 0\pmod 3$, a contradiction again. Hence $j\equiv 2\pmod 3$}. { Now, as }in Subcase 2.1, we can get the result.
Considering Subcases 2.1 and 2.2, we may assume all ears of $C$ in $ G_{1}^{0} $ have length $\equiv 0\pmod 3$.
\textbf{Subcase 2.3.} $t\equiv 0\pmod 3$.\newline Considering the cycles generated by $C+P+v_{1}z_{1}+v_{t}z_{j}$ and that fact that $G$ has no cycle of length $\equiv 0\pmod 3$, we {\ deduce that }$ j\equiv 2\pmod 3$. Let $C_{1}=(z_{1}z_{2}\ldots z_{p}z_{1})$, $ C_{2}=(z_{1}v_{1}v_{2}\ldots v_{t}z_{j}z_{j-1}\ldots z_{2}z_{1})$ and $ C_{3}=(z_{1}v_{1}v_{2}\ldots v_{t}z_{j}z_{j+1}\ldots z_{p}z_{1})$. Clearly the cycles $C_{1},C_{2}$, $C_{3}$ are {all }of length $\equiv 2\pmod 3$.
{Assume first that }$C_{1}$ is not an induced cycle in $G$. Then by Lemma \ref{induced} and considering the ear we have $n(C_{1}\cup C_{2})\geq 6k+11$ . Let $G_{2}^{0}=C_{1}\cup C_{2}$ and $G_{1}^{0}=G-G_{2}^{0} $. It is not hard to see that $G_{2}^{0}$ has a 3-tuple $\overrightarrow{f}$ of RDFs such that $\omega (\overrightarrow{f})\leq 2n(G_{2}^{0})+2$ and all vertices of $ G_{2}^{0}$ but $v_{1},v_{t}$ are $\overrightarrow{f}$-strong. If $ V(G)=V(G_{2}^{0})$, then the result follows. {Hence }assume that $V(G)\neq V(G_{2}^{0})$. By the assumptions and Lemma \ref{12}, we deduce that $ G_{1}^{0}$ has a path $P^{\prime }=v_{1}^{\prime },\ldots ,v_{t^{\prime }}^{\prime }$ such that $v_{1}^{\prime },v_{t^{\prime }}^{\prime }$ are adjacent to some vertices in $V(G_{2}^{0})$, say $u,v$ (possibly $u=v$) and $ N_{G}(v_{1}^{\prime })\cup N_{G}(v_{t^{\prime }}^{\prime })\subseteq V(G_{2}^{0})\cup V(P^{\prime })$. {Note that }$u,v\notin \{v_{1},v_{t}\}$ { since }$N_{G}(v_{1})\cup N_{G}(v_{t})\subseteq V(H_{2}^{0})\cup V(P)$. Thus $ {u,v}$ are $\overrightarrow{f}$-strong. {Now, let }$G_{2}^{1}$ be {the graph }obtained from $G_{2}^{1}$ and the path $P^{\prime }$ by adding the edges $ v_{1}^{\prime }u,v_{t^{\prime }}^{\prime }v$ and let $G_{1}^{1}=G-G_{2}^{1}$ . By Lemma \ref{ear1}, $\overrightarrow{f} $ can be extended to a 3-triple $ \overrightarrow{g}$ such that $\omega (\overrightarrow{g})\leq 2n(G_{2}^{2})+1$ and all vertices\ of $P^{\prime }$ but $v_{1}^{\prime },v_{t^{\prime }}^{\prime }$ are $\overrightarrow{f_{2}}$-strong. By repeating above process, we obtain a 3-tuple of RDFs $\overrightarrow{ g^{\ast }}=(g_{1}^{\ast },g_{2}^{\ast },g_{3}^{\ast })$ such that $\omega ( \overrightarrow{g^{\ast }})\leq {2n(G)+2}\leq \frac{3n(4k+8)}{ 6k+8}$. It follows that $\omega (\overrightarrow{g_{j}^{\ast }})\leq \frac{ n(4k+8)}{6k+8}$ for some $j\in \{1,2,3\}$ as desired.
Assume now that $C_{1}$ is {an induced cycle}. By the choice of $C,$ we may assume that $G$ has no cycle of length $\equiv 2\pmod 3$ which is not induced. {Hence the cycle }$C_{2}$ {is also induced. }Let $ G_{2}^{0}=C_{1}\cup C_{2}$ and $G_{2}^{1}=G-G_{2}^{0}$. {There are the following two possibilities. }
\begin{itemize} \item $V(G)=V(G_{2}^{0})$.\newline Suppose $n(C_{1})=3t_{1}+2$ and $t=3t_{2}$. Using the fact that $n\geq 6k+9,$ we obtain \begin{equation*} \begin{array}{lll} n & = & n(C_{1})+t \\ & = & 3t_{1}+2+3t_{2} \\ & \geq & 3(2k+3) \end{array} \end{equation*} implying that $t_{1}+t_{2}\geq 2k+3-2/3$. {Since }$t_{1}+t_{2}$ {is integer, we deduce that }$t_{1}+t_{2}\geq 2k+3,$ and thus $n\geq 6k+11$. {Now, it }is easy to see that $\gamma _{R}(G)\leq \frac{2n+2}{3}\leq \frac{(4k+8)n}{6k+11} $.
\item $V(G_{2}^{0})\subsetneqq V(G)$.\newline Clearly $G_{2}^{0}$ has a triple $\overrightarrow{f^{0}}$ of RDFs such that $ \omega (\overrightarrow{f^{0}})\leq n(G_{2}^{0})+2$ and all vertices of $ G_{2}^{0}$ but $v_{1},v_{t}$ are $\overrightarrow{f^{0}}$-strong. By the assumptions and Lemma \ref{12}, we deduce that $G_{1}^{0}$ has a path $ P_{1}=v_{1}^{1},\ldots ,v_{q_{1}}^{1}$ such that $v_{1}^{1},v_{q_{1}}^{1}$ are adjacent to some vertices in $V(G_{2}^{0})$, say $u,v$ (possibly $u=v$) and $N_{G}(v_{1}^{1})\cup N_{G}(v_{q_{1}}^{1})\subseteq V(G_{2}^{1})\cup V(P_{1})$. {Recall that }$u,v\in \{v_{1},v_{t}\}$ {\ and thus they are }$ \overrightarrow{f^{0}}$-strong. {Moreover, }since every cycle of $G$ intersects $C_{1}$, we have $V(C_{1})\cap \{u,v\}\neq \emptyset $. {Hence vertices }$u,v$ {may belong to }$C_{1}$, $C_{2}$ or $C_{3}$. {Now, }seeing Case 1 and Subcase 2.1 and 2.2, we may assume that $q_{1}\equiv 0\pmod 3$. Let $q_{1}=3q_{1}^{\prime }$ and let $G_{2}^{1}$ be {the graph }obtained from $G_{2}^{0}$ and the path $P_{1}$ by adding the edges $ v_{1}^{1}u,v_{q_{1}}^{1}v$ and let $G_{1}^{1}=G-G_{2}^{1}$. By Lemma \ref {ear1}, $\overrightarrow{f^{0}}$ can be extended to a 3-triple $ \overrightarrow{f^{1}}$ {of }$G_{2}^{1}$ such that {$\omega (\overrightarrow{ f^{1}})\leq 2n(G_{2}^{1})+2$} and all vertices {\ of }$P_{1}$ but $ v_{1}^{1},v_{q_{1}}^{1}$ are $\overrightarrow{f^{1}}$-strong. If $ V(G)=V(G_{2}^{1})$, then $n=3t_{1}+3t_{2}+3q_{1}^{\prime }+2$. As above we can see that $n\geq 6k+11,$ {implying that }$\gamma _{R}(G)\leq \frac{2n+2}{3 }\leq \frac{(4k+8)n}{6k+11}$. {Hence assume that }$V(G_{2}^{1})\subsetneqq V(G)$. By the assumptions and Lemma \ref{12}, we deduce that $G_{1}^{1}$ has a path $P_{2}=v_{1}^{2},\ldots ,v_{q_{2}}^{2}$ such that $ v_{1}^{2},v_{q_{2}}^{2}$ are adjacent to some vertices in $V(G_{2}^{1})$, say $u^{\prime },v^{\prime }$ (possibly $u^{\prime }=v^{\prime }$) and $ N_{G}(v_{1}^{2})\cup N_{G}(v_{q_{2}}^{2})\subseteq V(G_{2}^{1})\cup V(P_{2})$ . Since every cycle of $G$ intersects $C_{1}$, we have $V(C_{1})\cap \{u^{\prime },v^{\prime }\}\neq \emptyset $. On the other hand, we note that $u^{\prime },v^{\prime }$ lies on a cycle of length $\equiv 2\pmod 3$. Seeing Case 1 and Subcase 2.1 and 2.2, we may assume that $q_{2}\equiv 0 \pmod 3$. Let $q_{2}=3q_{2}^{\prime }$ and let $G_{2}^{2}$ be {the graph } obtained from $G_{2}^{1}$ and the path $P_{2}$ by adding the edges $ v_{1}^{2}u,v_{q_{2}}^{2}v$ and let $G_{1}^{2}=G-G_{2}^{2}$. By Lemma \ref {ear1}, $\overrightarrow{f^{1}}$ can be extended to a 3-triple $ \overrightarrow{f^{2}}$ such that $\omega (\overrightarrow{f^{2}})\leq 2n(G_{2}^{2})+2$ and all {of }$P_{2}$ but $v_{1}^{2},v_{q_{2}}^{2}$ are $ \overrightarrow{f^{2}}$-strong. If $V(G)=V(G_{2}^{2})$, then $ n=3t_{1}+3t_{2}+3q_{1}^{\prime }+3q_{2}^{\prime }+2$. As above we can see that $n\geq 6k+11,$ {implying that }$\gamma _{R}(G)\leq \frac{2n+2}{3}\leq \frac{(4k+8)n}{6k+11}$. {\ Hence suppose that }$V(G_{2}^{k})\subsetneqq V(G)$ . ~ By repeating {the }above process, we obtain a subgraph $G_{2}^{k}$ with $ n(G_{2}^{k})\geq 6k+11$ and having a 3-tuple $\overrightarrow{f^{k}}$ of RDFs such that $\omega (\overrightarrow{f^{k}})\leq { 2n(G_{k}^{2})+2}$ and all vertices of $G_{2}^{k}$ with a neighbor outside of $G_{2}^{k}$ are $\overrightarrow{f^{k}}$-strong. If $V(G)=V(G_{2}^{k})$, then the result follows immediately. {Otherwise, let $ G_{1}^{k}=G-V(G_{2}^{k})$}. Now using Lemma \ref{ear1} we can obtain extend $ G_{2}^{k}$ to a subgraph $G_{2}^{k+1}$ by adding an ear in $G_{1}^{k}$ and extend $\overrightarrow{f^{k}}$ to a 3-tuple $\overrightarrow{f^{k+1}}$ of RDFs such that $\omega (\overrightarrow{f^{k+1}})\leq 2n(G_{k+1}^{2})+2$ and all vertices of $G_{2}^{k+1}$ with a neighbor outside of $G_{2}^{k+1}$ are $ \overrightarrow{f^{k+1}}$-strong. By repeating this process we obtain a 3-tuple $g$ of RDFs of $G$ such that $\omega (\overrightarrow{g})\leq 2n(G)+2 $ and this leads to the result as above. $
\Box $ \end{itemize}
\section{Some more lemmas}
Let $\mathcal{F}_{i}$ be the family of all cycles of length $\equiv i\pmod 3$ with $i\in \{0,1,2\}$. Let $\mathcal{F}_{0,2}$ be the family of all { connected }graphs obtained from a cycle $C$ of $\mathcal{F}_{0}$ and a cycle $C^{\prime }$ of $\mathcal{F}_{2}$ by {joining a vertex }$x$ {of }$C$ {\and a vertex }$y$ {of }$C^{\prime }$ {by either an edge }$xy${\ or by a nontrivial path that we add so that one of the envertices of the path is attached to }$x$ and the other one to $y$; $\mathcal{F}_{2,2}$ be the family of all {connected} graphs obtained from two cycles in $\mathcal{F}_{2}$ by adding an edge between them; and let $\mathcal{F}_{3}$ be the family of all graphs $G$ obtained from a graph $G^{\prime }$ in $\mathcal{F}_{0,2}$ and a graph $G^{\prime \prime }$ in $\mathcal{F}_{2,2}$ by adding {either an edge or a path joining a vertex of }$G^{\prime }$ to a vertex of $G^{\prime \prime }$ {so that all vertices of the path become of degree two in }$G.$
\
Let $\mathcal{B}_{r,s}\;(r+s\geq 2)$ be the family of {connected }graphs obtained from $r$ tailed cycles $C_{n_{1},\ell _{1}},\ldots ,C_{n_{r},\ell _{r}}$ and $s$ cycles $C_{m_{1}},\ldots ,C_{m_{s}}$, where $n_{i}\equiv 2 \pmod 3$ and $m_{j}\equiv 2\pmod 3$ for each $i,j$, by adding a new vertex $ z $ ({which we call }special vertex) and {\ joining by edges }$z$ to the { unique leaf }of each graph $C_{n_{i},\ell _{i}}$ and to one vertex of each cycle $C_{m_{j}}$. {Moreover, each of the }$s$ {cycles will be called a \textit{near cycle} of }$z.${\ }Let $\mathcal{E}=\cup _{r,s\geq 0;r+s\geq 2} \mathcal{B}_{r,s}$.
\begin{lemma}
\label{Lam3}\emph{Let $G$ be a connected graph {with at least two disjoint cycles and let }$\mathcal{F}$ be a family of pairwise disjoint cycles of length $\equiv 0,2\pmod 3$ in $G$ with $|\mathcal{F}|\geq 2$. Then $G$ has two disjoint subgraphs $G_{1}$ (possibly null) and $G_{2}$ such that $ V(G)=V(G_{1})\cup V(G_{2})$, $G_{1}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}$ is in $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}.$} \end{lemma}
\noindent \textbf{Proof.} The proof is by induction on the number of cycles in $\mathcal{F}$. First let $|\mathcal{F}|=2$ with $\mathcal{F} =\{C_{1},C_{2}\}$. {Since }$G$ {is connected, let }$P $ be a shortest path { joining a vertex of }$C_{1}$ {to a vertex of $C_{2}$. If both $C_{1},C_{2}$ have length $\equiv 0\pmod 3$, then let $G_{2}=C_{1}\cup C_{2}$ and if one of the {two }cycles has length $\equiv 2\pmod 3$, then let $ G_{2}=C_{1}+C_{2}+P$. Assume that $G_{1}=G-V(G_{1})$. Clearly $G_{1}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}$ belongs to $\mathcal{F} _{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}$, { establishing the base case.} }
\noindent Next let $|\mathcal{F}|=3$ and $\mathcal{F}=\{C_{1},C_{2},C_{3}\}$ . Assume that $C_{i}=x_{1}^{i}x_{2}^{i}\ldots x_{n_{i}}^{i}x_{1}^{i}$ for $ i\in \{1,2,3\}$. If {each cycle of }$\mathcal{F}$ has length $\equiv 0 \pmod 3 $, then let $G_{2}=C_{1}\cup C_{2}\cup C_{3}$ and $G_{1}=G-G_{2}$. {Clearly the result holds. Hence assume that }one of the {three }cycles has length $ \equiv 2\pmod 3$, say $C_{1}$. Let $P$ be a shortest path {joining a vertex of }$C_{1}$ {to a vertex in }$C_{2}$ or $C_{3}$. Assume, without loss of generality, that $P$ {joins }$C_{1}$ and $C_{2}$, where $ P=(x_{1}^{1}=)z_{0}z_{1}...z_{k}(=x_{1}^{2}).$ If $C_{3}$ has length $\equiv 0\pmod 3,$ {then by setting }$G_{2}=\left( C_{1}\cup C_{2}\cup P\right) \cup C_{3}$ and $G_{1}=G-G_{2},$ {it is clear that the result holds. Hence we assume that $C_{3}$ has length $\equiv 2\pmod 3.$ {Now, let }$ Q=(x_{1}^{3}=)y_{0}y_{1}...y_{s}$ be a shortest path {joining a vertex of }$ C_{3}$ {to a vertex }$y_{s}$ {belonging to }$V(C_{1})\cup V(C_{2})\cup V(P)$ . Assume that $y_{s}\in V(P)-\{x_{1}^{1},x_{1}^{2}\}$, say $y_{s}=z_{m}.$ If $C_{2}$ has length $\equiv 0\pmod 3,$ then {by setting }$G_{2}=\left( C_{1}\cup C_{3}\cup P^{\prime }\right) \cup C_{2},$ with $P^{\prime }=(x_{1}^{1}=)z_{0}z_{1}...z_{m},y_{s-1},\ldots,y_{0},$ and $G_{1}=G-G_{2},$ we get the desired result. Hence we assume that $C_{2}$ has length $\equiv 2 \pmod 3.$ In this case, the result holds by letting $G_{2}=C_{1}\cup C_{2}\cup C_{3}\cup P\cup Q$ and $G_{1}=G-G_{2}.$ Finally, assume, without loss of generality, that\ $y_{s}\in C_{2},$ say $y_{s}=x_{j}^{2}$ (possibly $ j=1$). Let $G_{2}=C_{1}Px_{1}^{2}\ldots x_{j}^{2}QC_{3}$ and $G_{1}=G-G_{2}$ . {Note that }$G_{2}$belongs to {$\mathcal{B}_{0,2}\cup \mathcal{B}_{1,1},$} {and clearly the desired result holds. } }
\noindent Assume now that $|\mathcal{F}|\geq 4$. If all cycles in $\mathcal{F
}$ have length $\equiv 0\pmod 3$, then the subgraphs $G_{2}=\cup _{i=1}^{|
\mathcal{F}|}C_{i}$ and $G_{1}=G-G_{2}$ satisfy the conditions {\ and the result holds. }Hence we assume that one of the cycles in $\mathcal{F} $, { say }$C_{0},${\ }has length $\equiv 2\pmod 3$. Let $\mathcal{F}^{\prime }= \mathcal{F}-\{C_{0}\}$ and let $G^{1}=G-V(C_{0})$. We consider two cases.
\noindent \textbf{Case 1.} $G^{1}$ is connected.\newline Then $\mathcal{F}^{\prime }$ is a family of disjoint cycles {of length }$
\equiv 0,2\pmod 3$ in $G^{1}$ {with }\emph{$|\mathcal{F}^{\prime }|\geq 2$}. By the induction hypothesis, $G^{1}$ has two disjoint subgraphs $ G_{1}^{\prime }$ (possibly null) and $G_{2}^{\prime }$ such that $ V(G^{1})=V(G_{1}^{\prime })\cup V(G_{2}^{\prime })$, $G_{1}^{\prime }$ has no cycle of $\mathcal{F}^{\prime }$, where each component of $G_{2}^{\prime } $ is in $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}.$ Let $H_{1},\ldots ,H_{p}$ be the components of $G_{2}^{\prime } $. {Suppose without loss of generality that $P:=(x_{1}^{0}=)v_{0}v_{1} \ldots v_{t}$ is a shortest path between $V(C_{0})$ and $V(G_{2}^{\prime })$ in $G$ where $v_{t}\in V(G_{2}^{\prime })$.} Without loss of generality, assume that $v_{t}\in V(H_{1}).$ If $H_{1}\in \mathcal{F}_{0}$, then let $ H_{1}^{\prime }=H_{1}+P+C_{0}$ and clearly {the }two subgraphs $ G_{2}=H_{1}^{\prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}=G-V(G_{2})$ satisfy the conditions {and result follows.
For the next, we can assume that $H_{1}$ contains at least two cycles. We distinguish the following. }
\textbf{Subcase 1.1.} $H_{1}\in \mathcal{F}_{2,2}$.\newline Suppose $H_{1}$ is obtained from two cycles $C_{1}=x_{1}^{1}\ldots x_{m_{1}}^{1}x_{1}^{1}$ and $C_{2}=x_{1}^{2}\ldots x_{m_{2}}^{2}x_{1}^{2}$ by adding a path $Q=(x_{1}^{1}=)z_{0}z_{1}\ldots z_{t}(=x_{1}^{2})$. {We further assume, without loss of generality, that }$v_{t}=x_{j}^{2}\in V(C_{2})$. {Let }$H_{1}^{\prime }$ {be the graph obtained from }$C_{0}\cup C_{1}$ {to which we add the path }$Px_{j-1}^{2}\ldots x_{2}^{2}Q$, {in other words, }$H_{1}^{\prime }$ {is obtained from }$C_{0}+P+H_{1} $ {by removing vertices }$x_{j+1}^{2},...,x_{m_{2}}^{2}.$ {Note that }$H_{1}^{\prime }$ { belongs to either $\mathcal{F}_{0,2}$ or {$\mathcal{B}_{0,2}\cup\mathcal{B} _{1,1} .$ } Now let $G_{2}^{\prime \prime }=H_{1}^{\prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}^{\prime \prime }=G-G_{2}^{\prime \prime }. $ Then the subgraphs $G_{2}^{\prime \prime }$ and $G_{1}^{\prime \prime }$ satisfy the conditions {and result follows}. }
\textbf{Subcase 1.2.} $H_{1}\in \mathcal{F}_{0,2}$.
\newline Using an argument similar to that described in the case $|\mathcal{F}|=3$, we can obtain two subgraphs $G_{2}^{\prime \prime }$ and $G_{1}^{\prime \prime }$ satisfying the conditions {and yielding the desired result}.
\textbf{Subcase 1.3.} $H_{1}\in \mathcal{B}_{r,s}$ where $r+s\geq 2$.\newline Let $z^{\ast }$ {be the special vertex of }$H_{1}.$ {If }$v_{t}=z^{\ast }$, then $H_{1}^{\prime }=H_{1}+P+C_{0}$ is a subgraph belonging to $\mathcal{E}$ and {thus }the subgraphs $G_{2}=H_{1}^{\prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}=G-V(G_{2})$ satisfy the conditions and the result follows. Hence we assume that $v_{t}\neq z^{\ast }.$ First let $r+s=2.$ Then $H_{1}$ is obtained from two cycles $C_{1}=x_{1}^{1}\ldots x_{m_{1}}^{1}x_{1}^{1}$ and $ C_{2}=x_{1}^{2}\ldots x_{m_{2}}^{2}x_{1}^{2}$ by adding a path $ Q=(x_{1}^{1}=)z_{0}z_{1}\ldots z_{t}(=x_{1}^{2}),$where $t\geq 2.$ If $ v_{t}\in \{z_{1},\ldots ,z_{t-1}\}$, then let $H_{1}^{\prime }=H_{1}+C_{0}+P. $ {Clearly, }$H_{1}^{\prime }\in \mathcal{B}_{r,s}$ where $ r+s=3$, and thus the subgraphs $G_{2}=H_{1}^{\prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}=G-V(G_{2})$ satisfy the conditions {and the result yields. Now, suppose, without loss of generality, that }$v_{t}=x_{j}^{2}\in V(C_{2}). $ {Let }$H_{1}$ be obtained $C_{0}\cup C_{1}$ {to which we add the path }$Px_{j-1}^{2}\ldots x_{2}^{2}Q,$ {and set }$G_{2}^{\prime \prime }=H_{1}^{\prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}^{\prime \prime }=G-G_{2}^{\prime \prime }.$ {Clearly, }$G_{1}^{\prime \prime }$ and $ G_{2}^{\prime \prime }$ {satisfy the condition and the desired result follows.}
{Now let } $r+s\geq 3.$ {Assume that }$v_{t}$ {belongs to one of the }${s+r}$ {cycles of }$H_{1},$ {say} $C^{\prime }.$ {\ Let }$H_{1}^{\prime }=C^{\prime }+C_{0}+P$ and $H_{1}^{\prime \prime }$ be the graph obtained from $H_{1}$ by deleting {the }vertices of $V(C^{\prime })$ and {the path \ (if any) joining }$z^{\ast }$ to $V(C^{\prime })$ in $H_{1}$. {\ Note that }$ H_{1}^{\prime }$ belongs to either {$\mathcal{F}_{2,2}$} or {$ \mathcal{B}_{0,2}\cup \mathcal{B}_{1,1}.$} {Now the subgraphs }$ G_{2}=H_{1}^{\prime }\cup H_{1}^{\prime \prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}=G-V(G_{2})$ satisfy the conditions and {the desired result holds. Assume that }$v_{t}$ belongs to a path on a tailed cycle $C_{m,\ell }$ of $ H_{1},$ and let $P^{\prime }$ be the subpath between $v_{t}$ and the cycle $ C^{\prime }$ of $C_{m,\ell }.$ {Let }$H_{1}^{\prime }=C^{\prime }+C_{0}+P+P^{\prime }$ and $H_{1}^{\prime \prime }$ be the graph obtained from $H_{1}$ by deleting {the }vertices of $V(C_{m,\ell }).$ {Note that }$ H_{1}^{\prime }$ belongs to either {$\mathcal{F}_{2,2}$} or $ \mathcal{B}_{0,2}.$ {Now the subgraphs }$G_{2}=H_{1}^{\prime }\cup H_{1}^{\prime \prime }\cup H_{2}\cup \ldots H_{p}$ and $G_{1}=G-V(G_{2})$ satisfy the conditions and {the desired result holds.}
\noindent \textbf{Case 2.} $G^{1}$ is disconnected.\newline Let $M_{1},\ldots ,M_{t}$ be the components of $G^{1}$. Assume first that a component $M_{i}$ contains all cycles of $\mathcal{F}-\{C_{0}\},$ say $M_{1}$ . Let $K$ be the subgraph of $G$ induced by $V(C_{0})\cup V(M_{1})$. Clearly $K$ is connected. Using an argument similar to that described in Case 1 on $ K-C_{0}$, we get the result.
Henceforth, we may assume that no $M_{i}$ contains all cycles of $\mathcal{F} -\{C_{0}\}$ for each $i$. Now, assume that a component $M_{i}$ contains at least two cycles of $\mathcal{F}$, say $M_{1}.$ Let $G^{2}=G-V(M_{1})$. Clearly $G^{2}$ is connected. Let $\mathcal{F}_{1}=\{C\mid C\in \mathcal{F}\; \mathrm{and}\;V(C)\subseteq V(M_{1})\}$ and $\mathcal{F}_{2}=\mathcal{F}- \mathcal{F}_{1}$. By the induction hypothesis, $M_{1}$ has two subgraphs $ K^{1},K^{2}$ such that $K^{1}$ does not contain any cycle of $\mathcal{F} _{1} $ and each component of $K^{2}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}.$ Moreover, $G^{2}$ has two subgraphs $K_{1}^{\prime }$ and $K_{2}^{\prime }$ such that $ K_{1}^{\prime }$ does not contain any cycle of $\mathcal{F}_{2}$ and each component of $K_{2}^{\prime } $ belongs to $\mathcal{F}_{0}\cup \mathcal{F} _{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}.$\textbf{\ }Now the two subgraphs $G_{1}=K^{1}\cup K_{1}^{\prime }$ and $G_{2}=K^{2}\cup K_{2}^{\prime }$ satisfies the conditions yielding the desired result.
{From now on, we can assume that }each $M_{i}$ contains at most one cycle of $\mathcal{F}$. {Suppose that only the }$s$ first $M_{i}$ contains exactly one cycle $C_{i}$ of $\mathcal{F}.$ Let {$C_{i}=x_{1}^{i}x_{2}^{i}\ldots x_{n_{i}}^{i}x_{1}^{i}$ for $0\leq i\leq s.$ In addition, let $ P_{i}:=(x_{1}^{i}=)w_{0}^{i}\ldots w_{\ell _{i}}^{i}$ be a shortest nontrivial path (possibly of order two) between $V(C_{i})$ and $V(C_{0})$ in $G$ for each $1\leq i\leq s,$ where $w_{\ell _{i}}^{i}\in V(C_{0})$. }If all cycles $C_{1},\ldots ,C_{s}$ have length $\equiv 0\pmod 3$, then the subgraphs $G_{2}=(C_{1}+P_{1}+C_{0})\cup C_{2}\cup \ldots \cup C_{s}$ and $ G_{1}=G-G_{2}$ satisfy the conditions {and the result follows}. {Hence, we assume that some }cycle $C_{i}\;(i\geq 1)$ has length $\equiv 2\pmod 3$. { Note that the }paths $P_{i}$'s {minus their end-vertices belonging to }$ V(C_{0})$ are disjoint. {If some }$C_{i}$ has length $\equiv 0\pmod 3,$ say $ C_{1}$, {then let }$L=C_{0}\cup (\cup _{i=2}^{s}C_{i})\cup (\cup _{i=2}^{s}P_{i})$. By the induction hypothesis, $L$ has two subgraphs $ L_{1},L_{2}$ such that $L_{1}$ has no cycle of $\mathcal{F}-\{C_{1}\}$ and each component of $L_{2}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F} _{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}$. Now $G_{2}=L_{2}\cup C_{1}$ and $G_{1}=G-G_{2}$ satisfy the conditions and the result holds. Hence we can assume that all cycles $C_{1},\ldots ,C_{s}$ have length $\equiv 2 \pmod 3 $. Let {$L=C_{0}\cup (\cup _{i=1}^{s}C_{i})\cup (\cup _{i=1}^{s}P_{i})$}. { Let $x_{i_{1}}^{0},\ldots ,x_{i_{t}}^{0}$ be the vertices of $C_{0}$ with degree at least three and assume, without loss of generality, that $ i_{1}<i_{2}<\ldots <i_{t}$.} Consider the following situations.
\textbf{Subcase 2.1.} $t=2$.\newline {If $\deg (x_{i_{1}}^{0}),\deg (x_{i_{2}}^{0})\geq 4$, then let $G_{2}$ be the graph obtained from $L$ by deleting all vertices of $V(C_{0})- \{x_{i_{1}}^{0},x_{i_{2}}^{0}\}$. Otherwise, let $G_{2}$ be the graph obtained from $L$ by deleting either the edge }$x_{i_{1}}^{0}x_{i_{2}}^{0}$ { (if any) or all the vertices $x_{i_{1}+1},\ldots ,x_{i_{2}-1}$. Then the subgraphs $G_{2}$ and $G_{1}=G-G_{2}$ satisfies the conditions and the result holds. }
\textbf{Subcase 2.2.} $t=3$.\newline {If $\deg (x_{i_{1}}^{0}),\deg (x_{i_{2}}^{0}),\deg (x_{i_{3}}^{0})\geq 4$, then let $G_{2}$ be the graph obtained from $L$ by deleting all vertices of $ V(C_{0})-\{x_{i_{1}}^{0},x_{i_{2}}^{0},x_{i_{3}}^{0}\}$. If $\deg (x_{i_{1}}^{0})=\deg (x_{i_{2}}^{0})=\deg (x_{i_{3}}^{0})=3$, then let $ G_{2} $ be the graph obtained from $L$ by deleting either the edge }$ x_{i_{2}}^{0}x_{i_{3}}^{0}$ (if any) or all the vertices {of $ \{x_{i_{2}+1}^{0},x_{i_{2}+2}^{0},\ldots ,x_{i_{3}-1}^{0}\}$. If, without loss of generality, $\deg (x_{i_{1}}^{0})=3$ and $\deg (x_{i_{3}}^{0})\geq 4$ . Let $G_{2}$ be the graph obtained from $L$ by deleting either the edge $ x_{i_{2}}^{0}x_{i_{3}}^{0}$ (if any) or {all vertices between of }$ x_{i_{2}}^{0}$ and $x_{i_{3}}^{0}$ {as well all vertices between $ x_{i_{3}}^{0}$ and $x_{i_{1}}^{0} $ {starting from }$x_{i_{3}+1}^{0}.$ {In either situation, the }subgraphs $G_{2}$ and $G_{1}=G-G_{2}$ satisfies the conditions {and the result follows}. }}
\textbf{Subcase 2.3.} $t\geq 4$.\newline {If $\deg (x_{i_{1}}^{0}),\deg (x_{i_{2}}^{0}),\deg (x_{i_{3}}^{0})\geq 4$, then let $G_{2}$ be the graph obtained from $L$ by deleting all vertices of $ V(C_{0})-\{x_{i_{1}}^{0},x_{i_{2}}^{0},\ldots ,x_{i_{t}}^{0}\}$. If $\deg (x_{i_{1}}^{0})=\deg (x_{i_{2}}^{0})=\ldots =\deg (x_{i_{t}}^{0})=3$, then let $G_{2}$ be the graph obtained from $L$ by deleting all vertices of $ \bigcup_{j=1}^{\lfloor t/2\rfloor }\{x_{i_{2j}+1}^{0},x_{i_{2j}+2}^{0},\ldots ,x_{i_{2j+1}-1}^{0}\}$. Assume without loss of generality that $\deg (x_{i_{1}}^{0})=3$ and $\deg (x_{i_{t}}^{0})\geq 4$. Let $L^{1}$ be the component of $L- \{x_{i_{1}-1}^{0}x_{i_{1}}^{0},x_{i_{2}}^{0}x_{i_{2}+1}^{0}\}$ containing $ x_{i_{1}}^{0}$, and let $L^{2}$ be the component of $L- \{x_{i_{3}-1}^{0}x_{i_{3}}^{0}\}$ containing $x_{i_{3}}^{0}$ if $\deg (x_{i_{3}}^{0})\geq 4$, and be the component of $L- \{x_{i_{3}}^{0}x_{i_{3}-1}^{0},x_{i_{4}}^{0}x_{i_{4}+1}^{0}\}$ containing $ x_{i_{3}}^{0}$ if $\deg (x_{i_{3}}^{0})=3$. Repeating this process we obtain a sequence $L^{1},\ldots ,L^{p}$ of subgraphs $L$ which contains all cycles of $L$ but $C_{0}$. Now the subgraphs $G_{2}=\cup _{i=1}^{p}L^{i}$ and $ G_{1}=G-G_{2}$ satisfies the conditions and the result follows.}$
\Box $
\begin{lemma} \label{mainn}Let $G$ be a connected graph with at least two disjoint cycles of length $\equiv 0,2\pmod 3$, and let $\mathcal{F}$ be the family of all cycles of $G$ with length $\equiv 0\;\mathrm{or}\;2\pmod 3$. Then there exists a maximal subfamily $\mathcal{T}$ of pairwise disjoint cycles of $
\mathcal{F}$ with $|\mathcal{T}|\geq 2$ and two disjoint subgraphs $G_{1}$ (possibly null) and $G_{2}$ of $G$ such that $V(G)=V(G_{1})\cup V(G_{2})$, $ G_{1}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{ E}$. \end{lemma}
\noindent \textbf{Proof.} {By Lemma \ref{Lam3}, for any maximal subfamily }$
\mathcal{T}$ of pairwise disjoint cycles of $\mathcal{F}$ with $|\mathcal{T}
|\geq 2$, $G$ has two disjoint subgraphs $G_{1}^{\mathcal{T}}$ {and }$G_{2}^{ \mathcal{T}}$ such that $V(G)=V(G_{1}^{\mathcal{T}})\cup V(G_{2}^{\mathcal{T} })$, $G_{1}^{\mathcal{T}}$ has no cycle of $\mathcal{T}$ {and }each component of $G_{2}^{\mathcal{T}}$ is in $\mathcal{F}_{0}\cup \mathcal{F} _{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}. $ {Now, let }$c_{\mathcal{T}}$ {denote the }number of cycles of $G_{2}^{\mathcal{T}}$, {and let }$s_{ \mathcal{T}}$ be the sum of the lengths of paths between two cycles in the components of $G_{2}^{\mathcal{T}}$ {that belong to }$\mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup (\cup _{r,s\geq 0;r+s=2}B_{r,s}).$ {\ Moreover, let } \begin{equation*} c_{\mathcal{F}}=\max \{c_{\mathcal{T}}\mid \mathcal{T}\;\mathrm{ is\;a\;maximal\;subfamily\;of\;pairwise\;disjoint\;cycles\;of}\;\mathcal{F}\;
\mathrm{with}\;|\mathcal{T}|\geq 2\}. \end{equation*} Choose a triple $(\mathcal{T},G_{1}^{\mathcal{T}},G_{2}^{\mathcal{T}})$ such that: (i) $c_{\mathcal{F}}=c_{\mathcal{T}}$; (ii) subject to (i): $s_{ \mathcal{T}}$ is maximized. {Notice that }$G_{2}^{\mathcal{T}}$ {may not contain all cycles of }$\mathcal{T}$. We claim that {the }two disjoint subgraphs $G_{1}^{\mathcal{T}}$ and $G_{2}^{\mathcal{T}}$ {chosen in this way yield the desired result. }
{It is clear that it suffices to show }that $G_{1}^{\mathcal{T}}$ has no cycle of $\mathcal{F}$. {Hence, suppose }to the contrary that $G_{1}^{ \mathcal{T}}$ contains at least one cycle of $\mathcal{F}$. Let $G_{2}^{1}$\ be obtained from $G_{2}^{\mathcal{T}}$ by adding { a maximum set of pairwise of cycles of $\mathcal{F}$ } with length $\equiv 0\pmod 3$ {\ belonging to }$G_{1}^{\mathcal{T}}$ and let $G_{1}^{1}=G-G_{2}^{1}$. Note that if $G_{1}^{\mathcal{T}}$ {contains no }cycle of $\mathcal{F} $ with length $\equiv 0\pmod 3$, then $G_{2}^{1}=G_{2}^{\mathcal{T}}$. {Now, l}et $ \mathcal{T}_{1}^{1}$ be the family of all cycles of $G_{2}^{1}$ {that belong }to $\mathcal{F}$ and {et }$\mathcal{T}^{1}$ be a maximal subfamily of $ \mathcal{F}$ such that $\mathcal{T}_{1}^{1}\subseteq \mathcal{T}^{1}$. If $ G_{1}^{1}$ does not contain any cycle of $\mathcal{F}$, then the family $ \mathcal{T}^{1}$ and the subgraphs $G_{1}^{1}$ and $G_{2}^{1}$ satisfy the conditions {which }leads to a contradiction because of $c_{\mathcal{T} ^{1}}>c_{\mathcal{T}}$. {Hence we }assume that $G_{1}^{1}$ contains at least one cycle of $\mathcal{F}$.
Let first $H^{1},\ldots H^{r}$ be the components of $G_{1}^{1}$ which contains at least two disjoint cycles of $\mathcal{F}$ (if any), and let $ \mathcal{F}^{i}$ be a maximal subfamily of pairwise disjoint cycles of $
\mathcal{F}$ {that are }in $H^{i}$ with $|\mathcal{F}^{i}|\geq 2$, for each $ 1\leq i\leq r$. By Lemma \ref{Lam3}, $H^{i}$ has two subgraphs $ H_{1}^{i},H_{2}^{i}$ such that $H_{1}^{i}$ has no cycles of $\mathcal{F}^{i}$ and each component of $H_{2}^{i}$ is in $\mathcal{F}_{0}\cup \mathcal{F} _{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}$. Let $G_{2}^{2}=G_{2}^{1}\cup (\cup _{i=1}^{r}H_{2}^{i})$, if $r\geq 1$ and $G_{1}^{2}=G-G_{2}^{2}$. Let $ \mathcal{T}_{2}^{1}$ be the family of all cycles of $G_{2}^{2}$ {\ that belong }to $\mathcal{F}$ and let $\mathcal{T}^{2}$ be a maximal subfamily of $\mathcal{F}$ such that $\mathcal{T}_{1}^{2}\subseteq \mathcal{T}^{2}$. If $ G_{1}^{2}$ does not contain any cycle of $\mathcal{F}$, then the family $ \mathcal{T}^{2}$ and the subgraphs $G_{1}^{2}$ and $G_{2}^{2}$ satisfy the conditions {which }leads to a contradiction because $c_{\mathcal{T}^{2}}>c_{ \mathcal{T}}$. {Hence, we }assume that $G_{1}^{2}$ contains at least one cycle of $\mathcal{F}$. If $G_{1}^{2}$ has a component with at least two disjoint cycles of $\mathcal{F}$, then we proceed as above.
Henceforth, we {can }assume that {each }component of $G_{1}^{2}$ has at most one cycle of $\mathcal{F}$. Let $C_{0}$ be a cycle of $G_{1}^{2}$ {belonging }to $\mathcal{F}$. Clearly, $C_{0}=x_{1}^{0}x_{2}^{0}\ldots x_{m_{0}}^{0}x_{1}^{0}$ is connected to a component of $G_{2}^{2}$ by {some } path (possibly an edge). Let $P=(x_{1}^{0}=)v_{0}v_{1}\ldots v_{t}$ be a shortest path between $V(C_{0})$ and $V(G_{2}^{2})$. Then $v_{t}$ belongs to {a component of }$G_{2}^{2}$, {say }$H_{1}.$ If $H_{1}\in \mathcal{F}_{0}$, then let $H_{1}^{\prime }=H_{1}+P+C_{0}$ and let $ G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime }$. Hence, assume that $H_{1}$ contains at least two cycles. We distinguish the following cases.
\textbf{Case 1.} $H_{1}\in \mathcal{F}_{2,2} $. \newline Suppose $H_{1}$ is obtained from two cycles $C_{1}=x_{1}^{1}\ldots x_{m_{1}}^{1}x_{1}^{1}$ and $C_{2}=x_{1}^{2}\ldots x_{m_{2}}^{2}x_{1}^{2}$ by adding an edge $x_{1}^{1}x_{1}^{2}$. We further assume, without loss of generality, that $v_{t}=x_{j}^{2}\in V(C_{2})$. {Let }$H_{1}^{\prime }$ be the graph obtained from $C_{0}\cup C_{1}$ to which we add the path $ Px_{j-1}^{2}\ldots x_{2}^{2}Q$, {in other words, $H_{1}^{\prime }$ {is obtained from }$C_{0}+P+H_{1}$ by removing vertices $ x_{j+1}^{2},...,x_{m_{2}}^{2}.$ Note that }$H_{1}^{\prime }$ belongs to either $\mathcal{B}_{0,2}\cup \mathcal{B}_{1,1}.$ Now let $ G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime} $ which we will discuss further below.
{
\textbf{Case 2.} $H_{1}\in \mathcal{F}_{0,2} $. \newline Suppose $H_{1}$ is obtained from two cycles $C_{1}=x_{1}^{1}\ldots x_{m_{1}}^{1}x_{1}^{1}$ and $C_{2}=x_{1}^{2}\ldots x_{m_{2}}^{2}x_{1}^{2}$ by adding a path $Q=(x_{1}^{1}=)z_{0}z_{1}\ldots z_{k}(=x_{1}^{2})$. Suppose without loss of generality that $m_1\equiv 0\pmod 3$ and $m_2\equiv \pmod 3$ . If $v_t=z_j$ for some $j$, then let $H_1^{\prime}$ be obtained from $ C_{0}\cup C_{2}$ to which we add the path $Pz_j\ldots z_{k}$, and let $ G_{2}^{3}=(G_{2}^{2}-H_{1})\cup (H_{1}^{\prime} \cup C_1)$. Suppose that $ v_t\in V(C_1)\cup V(C_2)$. {We further assume, without loss of generality, that }$v_{t}=x_{j}^{2}\in V(C_{2})$. {Let }$H_{1}^{\prime }$ {be the graph obtained from }$C_{0}\cup C_{1}$ {to which we add the path }$ Px_{j-1}^{2}\ldots x_{2}^{2}Q$, {in other words, }$H_{1}^{\prime }$ {is obtained from }$C_{0}+P+H_{1}$ {by removing vertices }$ x_{j+1}^{2},...,x_{m_{2}}^{2}.$ {Note that }$H_{1}^{\prime }$ belongs to either $\mathcal{F}_{0,2}$ or $\mathcal{B}_{0,2}\cup \mathcal{B}_{1,1}.$ Now let $G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime } $ which we will discuss further below. }
\textbf{Case 3.} $H_{1}\in \mathcal{B}_{r,s}$ where $r+s\geq 2$. \newline Let $z^{\ast }$ {be the special vertex of }$H_{1}.$ {If }$v_{t}=z^{\ast }$, then $H_{1}^{\prime }=H_{1}+P+C_{0}$ is a subgraph belonging to $\mathcal{E}$ . In this case, let $G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime }$ which we will discuss further below. Hence we assume that $v_{t}\neq z^{\ast }.$ First let $r+s=2.$ Then $H_{1}$ is obtained from two cycles $ C_{1}=x_{1}^{1}\ldots x_{m_{1}}^{1}x_{1}^{1}$ and $C_{2}=x_{1}^{2}\ldots x_{m_{2}}^{2}x_{1}^{2}$ by adding a path $Q=(x_{1}^{1}=)z_{0}z_{1}\ldots z_{t}(=x_{1}^{2}),$where $t\geq 2.$ If $v_{t}\in \{z_{1},\ldots ,z_{t-1}\}$, then let $H_{1}^{\prime }=H_{1}+C_{0}+P$ and $G_{2}^{3}=(G_{2}^{2}-H_{1}) \cup H_{1}^{\prime }$ . {Now, suppose, without loss of generality, that }$ v_{t}=x_{j}^{2}\in V(C_{2}). $ {Let }$H_{1}$ be obtained {from }$C_{0}\cup C_{1}$ {to which we add the path }$Px_{j-1}^{2}\ldots x_{2}^{2}Q.$ S{et }$ G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime }$ {which we will discuss further below}.
Now let $r+s\geq 3.$ {Assume that }$v_{t}$ {belongs to one of the }${s+r}$ cycles of $H_{1},$ {say} $C^{\prime }.$ Let $H_{1}^{\prime }=C^{\prime }+C_{0}+P$ and $H_{1}^{\prime \prime }$ be the graph obtained from $H_{1}$ by deleting the vertices of $V(C^{\prime })$ and the path (if any) joining $ z^{\ast }$ to $V(C^{\prime })$ in $H_{1}$. Note that $H_{1}^{\prime }$ belongs to either $\mathcal{F}_{0,2}$ or $\mathcal{B}_{0,2}\cup \mathcal{B} _{1,1}.$ Now let $G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime }$ . Assume that $v_{t}$ belongs to a path on a tailed cycle $C_{m,\ell }$ of $H_{1},$ and let $P^{\prime }$ be the subpath between $v_{t}$ and the cycle $ C^{\prime }$ of $C_{m,\ell }.$ {Let }$H_{1}^{\prime }=C^{\prime }+C_{0}+P+P^{\prime }$ and $H_{1}^{\prime \prime }$ be the graph obtained from $H_{1}$ by deleting {the }vertices of $V(C_{m,\ell }).$ {Note that }$ H_{1}^{\prime }$ {belongs to either }$\mathcal{F}_{0,2}$ or $\mathcal{B} _{0,2}.$ Suppose $G_{2}^{3}=(G_{2}^{2}-H_{1})\cup H_{1}^{\prime }\cup H_{1}^{\prime \prime }$. Obviously either the number of cycles of $G_{2}^{3}$ is greater than the number of cycles of $G_{2}^{\mathcal{T}}$ or the sum of lengths of paths between two cycles of $G_{2}^{3}$ that belong to $\mathcal{F }_{0,2}\cup \mathcal{F}_{2,2}\cup (\cup _{r,s\geq 0;r+s=2}B_{r,s})$ is greater than the corresponding sum of $G_{2}^{\mathcal{T}}$. Let $ G_{1}^{3}=G-G_{2}^{3}.$ Let $\mathcal{T}_{3}^{1}$ be the family of all cycles of $G_{2}^{3}$ which belongs to $\mathcal{F}$ and let $\mathcal{T} ^{3} $ be a maximal subfamily of $\mathcal{F}$ such that $\mathcal{T} _{1}^{3}\subseteq \mathcal{T}^{3}$. If $G_{1}^{3}$ does not contain any cycle of $\mathcal{F}$, then the family $\mathcal{T}^{3}$ and the subgraphs $ G_{1}^{3}$ and $G_{2}^{3}$ satisfy the conditions which leads to a contradiction because {of either }$c_{\mathcal{T}^{3}}>c_{\mathcal{T}}$ or $ s_{\mathcal{T}^{3}}>s_{\mathcal{T}}$. Hence we assume that $G_{1}^{3}$ contains at least one cycle of $\mathcal{F}$. We repeat the above precess. Since $G$ is finite, this process will stop and we obtain a maximal subfamily $\mathcal{T}^{\prime }$ of pairwise disjoint cycles of $\mathcal{F}
$ with $|\mathcal{T}^{\prime }|\geq 2 $ and two disjoint subgraphs $G_{1}$ (possibly null), $G_{2}$ of $G$ such that $V(G)=V(G_{1})\cup V(G_{2})$, $ G_{1}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{ E}$. $
\Box $
\begin{lemma} \label{mainnn}\emph{Let $G$ be a connected graph with at least two disjoint cycles of length $\equiv 0,2\pmod 3$, and let $\mathcal{F}$ be the family of all cycles of $G$ with length $\equiv 0\;\mathrm{or}\;2\pmod 3$. Then there exists a maximal subfamily $\mathcal{T}$ of pairwise disjoint cycles of $
\mathcal{F}$ with $|\mathcal{T}|\geq 2$ and two disjoint subgraphs $G_{1}$ (possibly null), $G_{2}$ of $G$ such that $V(G)=V(G_{1})\cup V(G_{2})$, $ G_{1}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{ F}_{3}\cup \mathcal{E}$.} \end{lemma}
\noindent \textbf{Proof.} Let $(\mathcal{T},G_{1},G_{2})$ {be the triple satisfying the conditions of Lemma \ref{mainn}. Hence }$G_{1}$ has no cycle of $\mathcal{F}$ {and each component of }$G_{2}$ {belongs to }$\mathcal{F} _{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{E}$. {If there are no two components }$H_{1}\in \mathcal{F}_{0,2},$ $H_{2}\in \mathcal{F} _{2,2}$ of $G_{2}$ {joined by a path }$P$ {in }$G$ {with all its vertices, except the end-vertices, belong to }$V(G_{1})$, then $G_{1}$ and $G_{2}$ are {the }desired subgraphs. {Hence we assume }that there {are two components }$ H_{1}\in \mathcal{F}_{0,2}$ and $H_{2}\in \mathcal{F}_{2,2}$ of $G_{2}$ { joined by a path }$P$ {in }$G$ {with all its vertices, except the end-vertices, belong to }$V(G_{1}).$ Let $G_{2}^{\prime }$ be {\ the graph } obtained from $G_{2}$ by adding the path $P$ and let $G_{1}^{\prime }=G-G_{2}^{\prime }$. Clearly $G_{1}^{\prime }$ and $G_{2}^{\prime }$ satisfy the conditions {and the result follows}. We can repeat this process { until we get two }subgraphs $G_{1}^{\ast }$ (possibly null) and $G_{2}^{\ast }$ such that $V(G)=V(G_{1}^{\ast })\cup V(G_{2}^{\ast })$, $G_{1}^{\ast }$ has no cycle of $\mathcal{F}$, each component of $G_{2}^{\ast }$ is in $ \mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{F} _{3}\cup \mathcal{E}$ and {such that no path in }$G$ {like to the one described above joins two components }$H^{\prime }\in \mathcal{F}_{0,2}$ and $H^{\prime \prime }\in \mathcal{F}_{2,2}$ of $G_{2}^{\ast }.$ $
\Box $
{From now on, }a graph in $\mathcal{(}\cup _{r+s\geq 2;s\leq 2}\mathcal{B} _{r,s})\cup \mathcal{F}_{3}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{0}$ {will be called }\emph{strong}. {Also, the }special vertex of each graph in $ \mathcal{B}_{r,s}$ {will be }called a \textit{strong vertex}.
\begin{lemma} \label{strong}\emph{Let $k\geq 1$ be an integer and let $G$ be a graph of order $n$ and minimum degree $\delta \geq 2$, which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles. If $G$ {is strong, }then $ G$ has a 3-tuple $\overrightarrow{f}$ of {RDFs} such that $\omega ( \overrightarrow{f})\leq \frac{(4k+8)3n}{6k+11}$ and all vertices of $G$ are $ \overrightarrow{f}$-strong. } \end{lemma}
\noindent \textbf{Proof.} Let $G\in (\cup _{r+s\geq 2;s\leq 2}\mathcal{B} _{r,s})\cup \mathcal{F}_{3}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{0}.$ Assume first that $G\in \mathcal{F}_{0}$. Then $\gamma _{R}(G)=\frac{2n}{3}< \frac{(4k+8)n}{6k+11}$. Let $G=x_{1}x_{2}\ldots x_{3t}x_{1}$ and define for $ j\in \{1,2,3\}$ the functions $f_{j}$ on $V(G)$ as follows: $ f_{j}(x_{3i+j})=2$ for $0\leq i\leq t-1$ and $f_{j}(x)=0$ otherwise. Clearly $f_{j}$ is an $\gamma _{R}(G)$-function for each $j\in \{0,1,2\}$ and the triple $\overrightarrow{f}=(f_{0},f_{1},f_{2})$ satisfies the desired result.
Assume now that $G\in \mathcal{F}_{0,2}$. {Since }$G$ has no induced $ \{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles, {we deduce that cycle of lentgh }$ \equiv 2\pmod 3$ in $\mathcal{F}_{0,2}${\ has order at least }$3k+5$, and thus $G$ has order at least $3k+8.$ {Now b}y Lemma \ref{MainLem}, $G$ has a 3-tuple $\overrightarrow{f}$ of RDFs such that $\omega (\overrightarrow{f} )\leq 2n+1$ and all vertices of $G$ are $\overrightarrow{f}$-strong. {A simple calculation shows that }$\omega (\overrightarrow{f})\leq \frac{ (4k+8)3n}{6k+11}$.
Next assume that\textbf{\ }$G\in \mathcal{F}_{3}$. {By definition, }$G$ is obtained from a graph $G_{1}\in \mathcal{F}_{0,2}$ and a graph in $G_{2}\in \mathcal{F}_{2,2}$ by adding either an edge $vw$ {or a path }$Q$ {joining a vertex of }$G_{1}$ to a vertex of $G_{2}$ {so that all vertices of }$Q${\ become of degree two in }$G.$ Let $G_{1}$ be obtained from two cycles $ C_{1}=x_{1}^{1}x_{2}^{1}\ldots x_{n_{1}}^{1}x_{1}^{1}\in \mathcal{F}_{0}$ and $C_{2}=x_{1}^{2}x_{2}^{2}\ldots x_{n_{2}}^{2}x_{1}^{2}\in \mathcal{F} _{2} $ by adding {\ either the edge }$x_{1}^{1}x_{1}^{2}$ {or a }path $P$ between $x_{1}^{1}$ and $x_{1}^{2}.$ {By Lemma \ref{MainLem} (items 8,9,10), $G_{1}$ has a a 3-tuple $\overrightarrow{f}=(f_{1},f_{2},f_{3})$ of RDFss such that $\omega (\overrightarrow{f},G_{1})\leq 2n(G_{1})+1$ and all vertices of $G_{1} $ are $\overrightarrow{f}$-strong. Moreover, }let $G_{2}$ be obtained from two cycles $C_{3}=x_{1}^{3}x_{2}^{3}\ldots x_{n_{1}}^{3}x_{1}^{3}\in \mathcal{F}_{2}$ and $C_{4}=x_{1}^{4}x_{2}^{4} \ldots x_{n_{4}}^{4}x_{1}^{4}\in \mathcal{F}_{2}$ by adding the edge $ x_{1}^{3}x_{1}^{4}$. Without loss of generality, we assume that the added edge $uv$ or the path $Q$ is between $V(C_{3})$ and $V(G_{1})$. {By sequentially applying Lemmas \ref{tailedcycle-3p+1} (items 3,4) (once on }$ uv $ or $Q$ and $C_{3},$ {and then on the resulting graph with }$C_{4}),$ {$ \overrightarrow{f}$ can be extended to a triple $\overrightarrow{g}$ of RDFs of $G$ such that $\omega (\overrightarrow{g},G_{2})\leq 2n(G_{2})+2$ and each vertex of $G_{2}$ is $\overrightarrow{g}$-strong. Since }$G$ has no induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles, {we deduce that order each cycle of lentgh }$\equiv 2\pmod 3$ in $G$ is at least $3k+5.$ {Using the fact that }$G$ has three cycles {of length }$\equiv 2\pmod 3$ and one cycle {of lentgh }$\equiv 0\pmod 3,$ we have {$n(G)\geq 9k+18.$ Therefore $ \omega (\overrightarrow{g})\leq 2n(G)+3\leq \frac{(4k+8)3n(G)}{6k+11}.$ }
Using a similar argument we can show that for any graph $G\in \mathcal{\cup } _{r+s\geq 2;s\leq 2}B_{r,s}$ the result is {also }true. $
\Box $
\begin{lemma} \label{non-strong1}\emph{Let $k\geq 1$ be an integer and let $G$ be a graph of order $n$, minimum degree $\delta \geq 2$, which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles. If $G\in \mathcal{B} _{r,s} $ with $s\geq 3$, then $G$ has a 3-tuple $\overrightarrow{f}$ of RDFs such that $\omega (\overrightarrow{f})\leq \frac{(4k+8)3n}{6k+11}$ and the special vertex as well as all vertices on tailed cycles of $G$ are $ \overrightarrow{f}$-strong. } \end{lemma}
\noindent \textbf{Proof.} Suppose $G$ be obtained from $r\geq 0$ graphs $ C_{n_{1},\ell _{1}},\ldots ,C_{n_{r},\ell _{r}}$ and $s\geq 3$ cycles $ C_{m_{1}},\ldots ,C_{m_{s}}$, where $n_{i}\equiv 2\pmod 3$ and $m_{j}\equiv 2 \pmod 3$ for each $i,j$, by adding a new vertex $z$ (special vertex) attached to {endvertices of the }$C_{n_{i},\ell _{i}}$'s and to one vertex of each cycle $C_{m_{j}}$. {We first note that each of the }$r+s\geq 3$ { cycles has order at least }$3k+5$, {and thus each tailed cycle contains at least }${(3k+5)+1}$ vertices. {Hence }${n(G)\geq (3k+5)(s+r)+r+1.}$ {\ Now, if }$r=0$, then the result follows from Lemmas \ref{MainLem}-(11) {and the previous fact}. {Hence assume }that $r\geq 1$. Let $H$ be obtained from $G$ by deleting all vertices of $C_{n_{i},\ell _{i}}$'s. By Lemma \ref{MainLem} (item 11), $H$ has a triple $\overrightarrow{f}$ such that $\omega ( \overrightarrow{f})\leq 2n(H)-s+4$ and $z$ is $\overrightarrow{f}$-strong. { Since }${n(H)\geq (3k+5)s+1,}$ we deduce that $\omega (\overrightarrow{f} )\leq \frac{(4k+8)3n(H)}{6k+11}$. {Now, by applying repeatedly }Lemma \ref {tailedcycle-3p+1}-(4) {on }$C_{n_{1},\ell _{1}},\ldots ,C_{n_{r},\ell _{r}}$ , we can extend $\overrightarrow{f}$ to a triple $\overrightarrow{g}$ of $G$ such that $\omega (g,\cup _{i=1}^{r}C_{n_{1},\ell _{1}})\leq \sum_{i=1}^{r}(2n(C_{n_{1},\ell _{1}})+1)$ and all newly added vertices are $ \overrightarrow{g}$-strong. {Therefore, }$\omega (\overrightarrow{g})\leq 2n+r+4-s.$ {Now by the previous fact on the order and the calculation, we can see that }$2n+r+4-s\leq \frac{(4k+8)3n}{6k+11}$, {which proves the result.}$
\Box ${\ }
\begin{lemma} \label{non-strong2} Let $k\geq 1$ be an integer and let $G\in \mathcal{F} _{2,2}$ be a graph of order $n$, minimum degree $\delta \geq 2$, which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$-cycles. Then
\begin{enumerate} \item $G$ has a $3$-tuple $f$ of RDFs such that $\overrightarrow{f}\leq \ 2n(G) + 1 \leq \frac{(4k+8)3n(G)}{(6k+11)}.$
\item If H is a graph obtained from $G$ and a cycle $ C_{3p+1}=x_{1}...x_{3p+1}x_1$ by adding an edge between them, then H has a 3-tuple $f$ of RDFs such that $\omega(\overrightarrow{f}) \leq \frac{ (4k+8)3n(G)}{(6k+11)}$ and all vertices of H but $x_{3p+1}$ are $ \overrightarrow{f}$-strong.
\item If $H$ is a graph obtained from $G$ and a tailed cycle $C_{3p+1,\ell }$ with vertex set $x_1,\ldots ,x_{3p+1}$, $y_{1},\ldots ,y_{\ell }$ by joining $y_{\ell }$ to a vertex of G, then H has a 3-tuple $f $ of RDFs such that $ \omega(\overrightarrow{f}) \leq \frac{\emph{(4k+8)3n(G)}}{(6k+11)}$ and all vertices of $H$ but $x_{3p+1}$ are $\overrightarrow{f}$-strong. \end{enumerate} \end{lemma}
\noindent \textbf{Proof.} (1) is easy to show and so we prove only (2) and (3). Let $G\in \mathcal{F}_{2,2}$ be formed from two cycles $C_{1}$ and $ C_{2}$ by adding an edge between them, and let $H$ be obtained from $G$ and the cycle $C_{3p+1}$ (resp. tailed cycle $C_{3p+1,\ell }$) by adding an edge $xy$ (resp. $xy_{\ell }$), where without loss of generality $x\in V(C_{2})$. Let $K$ be the graph obtained from $H$ by deleting all vertices of $V(C_{1})$ . By Lemma \ref{MainLem} (items 2,3 and 4), $K$ has a 3-tuple $ \overrightarrow{g}$ of RDFs of $K$ such that $\omega (\overrightarrow{g} )\leq 2n(K)+1$ and all vertices of $K$ except $x_{3p+1}$ are $ \overrightarrow{g}$-strong. Now by Lemma \ref{tailedcycle-3p+1}, we can extend $\overrightarrow{g}$ to a 3-tuple $\overrightarrow{f}$ of RDFs of $H$ such that $\omega (\overrightarrow{f})\leq 2n(H)+2$ and all vertices of $H$ except $x_{3p+1}$ are $\overrightarrow{g}$-strong. By assumption we have $ n(H)\geq 6k+14$ and {thus one can check that }$\omega (\overrightarrow{f} )\leq \frac{(4k+8)3n(H)}{6k+11}$. $
\Box $
\section{\protect
Proof of Conjecture \protect\ref{conj}}
Now we are ready to state our main result.{\ }
\begin{theorem} \label{Theorem} \emph{Let $G$ be a graph of order $n\ge 6k + 9$, minimum degree $\delta\ge 2$, which does not contain any induced $ \{C_5,C_8,\ldots,C_{3k+2}\}$-cycles. Then $\gamma_R(G)\le \frac{(4k+8)n}{ 6k+11}.$} \end{theorem}
\noindent \textbf{Proof.} Let $\mathcal{F}$ be the family of all cycles of $
G $ with length $\equiv 0\;\mathrm{or}\;2\pmod 3$. If $|\mathcal{F}|=0$, then the result follows from Theorem \ref{Th1} and if $|\mathcal{F}|\geq 1$ and $\mathcal{F}$ contains a cycle which intersect any cycle of $\mathcal{F}$ , then the result follows from Theorems \ref{Th2} and \ref{Th3}. Henceforth, we assume that each cycle of $\mathcal{F}$ belongs to a maximal subfamily $
\mathcal{T}$ of pairwise disjoint cycles of $\mathcal{F}$ with $|\mathcal{T}
|\geq 2$. Let $(G_{1}^{1},G_{2}^{1}),\ldots ,(G_{1}^{m},G_{2}^{m})$ be all pairs of subgraph such that $V(G)=V(G_{1}^{i})\cup V(G_{2}^{i})$, $G_{1}^{i}$ has no cycle of $\mathcal{F}$ and each component of $G_{2}^{i}$ belongs to $ \mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{F} _{3}\cup \mathcal{E}$. Let{\ }$s_{(G_{1}^{i},G_{2}^{i})}$ be the sum of the lengths of paths between two cycles in the components of $G_{2}^{\mathcal{T} } $ {that belong to }$\mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup (\cup _{r,s\geq 0;r+s=2}B_{r,s}).$ {Among all pairs }$(G_{1}^{i},G_{2}^{i}),$ let $ (G_{1},G_{2})$ {be one chosen so that: }
\begin{description} \item[(C$_{1}$)] the {number of }strong components of $G_{2}$ is maximized.
\item[(C$_{2}$)] subject to Condition (C$_{1}$): the number of cycles of $ G_{2}$ belonging to $\mathcal{F}$ is maximized.
\item[(C$_{3}$)] subject to Conditions (C$_{1}$) and (C$_{2}$): the number of components of $G_{2}$ in $\mathcal{F}_{2,2}$ is minimized.
\item[(C$_{4}$)] subject to Conditions (C$_{1}$), (C$_{2}$) and (C$_{3}$): $ s_{(G_{1},G_{2})}$ is maximized. \end{description}
We proceed with some {further claims that are needed for our proof. }
\noindent \textbf{Claim 1.} Let $M$ be a component of $G_{2}$ such that $M\in \mathcal{F}_{2,2}$. Then there is no path $v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G$ such that $v_{0},v_{t+1}\in M$, $v_{1},\ldots ,v_{t}\in V(G_{1})$ and $v_{0}$ and $v_{t+1}$ belong to different cycles of $ M$.\newline \noindent \textbf{Proof of Claim 1.} Suppose, to the contrary, that there is path $P=v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G$ such that $ v_{0},v_{t+1}\in M$, $v_{1},\ldots ,v_{t}\in V(G_{1})$ and $v_{0}$ and $ v_{t+1}$ belong to different cycles of $M$. {Let }$e^{\ast }$ {be the edge joining the two cycles of }$M${\ and let }$M^{\prime }$ be obtained from $M$ by deleting $e^{\ast }$ and adding path $P$. {Set }$G_{2}^{\prime }=(G_{2}-M)\cup M^{\prime }$ and $G_{1}^{\prime }=G-G_{2}$. Clearly $ V(G)=V(G_{1}^{\prime })\cup V(G_{2}^{\prime })$, $G_{1}$ has no cycle of $ \mathcal{F}$ and each component of $G_{2}^{\prime }$ {is in }$\mathcal{F} _{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{F}_{3}\cup \mathcal{E}$. But $G_{2}^{\prime }$ has one more strong component than $ G_{2},$ {contradicting our choice of }$(G_{1},G_{2}).$ $\ \ \ \ \ \ \ \ \ \ \ \hspace{3cm}\Box $ \ \ \
\
\noindent \textbf{Claim 2.} {For any two components }$M_{1}$ and $ M_{2}$ of $G_{2}$ {belonging to }$\mathcal{F}_{2,2}$, {there }is no path $ v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G$ such that $v_{0}\in M_{1}$, $ v_{t+1}\in M_{2}$ and $v_{1},\ldots ,v_{t}\in V(G_{1})$.\newline \noindent \textbf{Proof of Claim 2.} Suppose, to the contrary, that {for } two components $M_{1},M_{2}$ of $G_{2}$ {\ belonging to }$\mathcal{F}_{2,2},$ there is a path $v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G_{1}$ such that $ v_{0}\in M_{1}$ and $v_{t+1}\in M_{2}$. Suppose {that } $M_{1}$ is obtained from two cycles $C_{1}=u_{1}^{1}\ldots u_{m_{1}}^{1}u_{1}^{1}$ and $ C_{2}=u_{1}^{2}\ldots u_{m_{2}}^{2}u_{1}^{2}$ by adding {the edge }$ u_{1}^{1}u_{1}^{2}$, and let $M_{2}$ {be }obtained from two cycles $ C_{3}=u_{1}^{3}\ldots u_{m_{3}}^{3}u_{1}^{3}$ and $C_{4}=u_{1}^{4}\ldots u_{m_{4}}^{4}u_{1}^{4}$ by adding the edge $u_{1}^{3}u_{1}^{4}$. {Moreover, assume, without loss of generality, that }$v_{0}=u_{j}^{2}\in V(C_{2})$ where $j\geq m_{2}/2$ ({by relabeling the vertices if necessary) }and $ v_{t+1}=u_{b}^{3}\in V(C_{3})$ where $b\geq m_{3}/2$ ({by relabeling the vertices if necessary)}. {Now, let }$M$ be the subgraph obtained from $C_{1}$ and $C_{4} $ by adding the path $u_{1}^{1}u_{1}^{2}u_{2}^{2}\ldots u_{j}^{2}v_{1}v_{2}\ldots v_{t}u_{b}^{3}u_{b-1}^{3}\ldots u_{1}^{3}u_{1}^{4}$ . {Set }$G_{2}^{\prime }=(G_{2}-(M_{1}\cup M_{2}))\cup M$ and $G_{1}^{\prime }=G-G_{2}^{\prime }$. If $G_{1}^{\prime }$ has no cycle of $\mathcal{F}$, then by considering the pair $(G_{1}^{\prime },G_{2}^{\prime }) $ we get { one more strong component in }$G_{2}^{\prime }$ {than in }$G_{2},$ { contradicting our choice of }$(G_{1},G_{2}).$ {Hence we }assume that $ G_{1}^{\prime }$ has some cycles of $\mathcal{F}$.
First let $G_{1}^{\prime }$ has exactly one cycle $C$ of $\mathcal{F}$. If $ C $ has length $\equiv 0\pmod 3$, then as above we get a contradiction by considering the subgraphs $G_{2}^{\prime \prime }=G_{2}^{\prime }\cup C$ and $G_{1}^{\prime \prime }=G-G_{2}^{\prime \prime }$. Hence suppose $C$ has length $\equiv 2\pmod 3$. Since $G_{1}$ has no cycle of $\mathcal{F}$, we may assume that $C$ contains one of the vertices $u_{j+1}^{2},\ldots ,u_{m_{2}}^{2}$. {Let }$\ell \in \{j+1,\ldots ,m_{2}\}$ be the smallest index {such that }$u_{\ell }^{2}\in V(C)$. {Let }$M^{\prime }=(M\cup C)+u_{j}^{2}u_{j+1}^{2}\ldots u_{\ell }^{2}$. Clearly $M^{\prime }$ is strong {because it belongs to }$\mathcal{B}_{r,s},$ {with }$r+s\geq 3$ and $ s\leq 2$. {By considering the subgraphs }$G_{2}^{\prime \prime }=(G_{2}-(M_{1}\cup M_{2}))\cup M^{\prime }$ and $G_{1}^{\prime \prime }=G-G_{2}^{\prime \prime },$ {the pair }$(G_{1}^{\prime \prime },G_{2}^{\prime \prime })$ {leads to a contradiction on the choice of }$ (G_{1},G_{2}).$
Now let $G_{1}^{\prime }$ has at least two disjoint cycles $C$ and $ C^{\prime }$ of $\mathcal{F}$. Using an argument similar to that described in the proof of Lemma \ref{mainn}, we can obtain a pair $(G_{1}^{\prime \prime },G_{2}^{\prime \prime })$ such that $G_{1}^{\prime \prime }$ has no cycle of $\mathcal{F}$ and each component of $G_{2}^{\prime \prime }$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{F}_{3}\cup \mathcal{E}$, {where either }$G_{2}^{\prime \prime }$ { has more strong components than }$G_{2}$ or the number of cycles of $ G_{2}^{\prime \prime }$ {belonging to }$\mathcal{F}$ is greater than the\ number of cycles of $G_{2}$ {\ belonging to}$\mathcal{F}$ or $ s_{(G_{1}^{\prime \prime },G_{2}^{\prime \prime })}>s_{(G_{1},G_{2})}.$ {In either case, we obtain a }contradiction. $
\Box $
Recall that a component of $\mathcal{B}_{r,s}$ is not strong when $s\geq 3.$
\noindent \textbf{Claim 3.} Let $M_{1}$ and $M_{2}$ be two non-strong components of $G_{2}$ such that $M_{1}\in \mathcal{F}_{2,2}$ and $ M_{2}\in \mathcal{B}_{r,s}$. Then there is no path $v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G$ such that $v_{1},\ldots ,v_{t}\in V(G_{1})$, $ v_{0}\in M_{1}$, $v_{t+1}\in M_{2}$ and $v_{t+1}$ is\ not {the }special vertex of $M_{2}$.\newline \noindent \textbf{Proof of Claim 3.} Suppose, to the contrary, {\ that } there is a path $v_{0}v_{1}\ldots v_{t+1}\;(t\geq 1)$ in $G$ such that $ v_{1},\ldots ,v_{t}\in V(G_{1})$, $v_{0}\in M_{1}$, $v_{t+1}\in M_{2}$ and $ v_{t+1}$ is not\textbf{\ }special vertex of $M_{2}$. Suppose $M_{1}$ is obtained from two cycles $C^{1}=u_{1}^{1}\ldots u_{m_{1}}^{1}u_{1}^{1}$ and $ C^{2}=u_{1}^{2}\ldots u_{m_{2}}^{2}u_{1}^{2}$ by adding the edge $ u_{1}^{1}u_{1}^{2}$ and let $M_{2}$ obtained from $r\geq 0$ tailed-cycle $ C_{n_{1},\ell _{1}}$, $\ldots ,C_{n_{r},\ell _{r}}$ and $s\geq 3$ cycles $ C_{m_{1}},\ldots ,C_{m_{s}}$, where $n_{i}\equiv 2\pmod 3$ and $m_{j}\equiv 2 \pmod 3$ for each $i,j$, by adding a new vertex $z$ (special vertex) and attaching $z$ to the leaf of each tailed cycle $C_{n_{i},\ell _{i}}$ and to one vertex of each cycle $C_{m_{j}}$. Without loss of generality, that we may assume that $v_{1}$ is adjacent to the vertex $u_{j}^{2}\in V(C_{2})$ where $j\geq m_{2}/2$ ({by relabeling the vertices if necessary)}.
First let $v_{t+1}$ belongs to a cycle $C_{m_{i}}=w_{1}^{i}w_{2}^{i}\ldots w_{m_{i}}^{i}w_{1}^{i}$ for some $i.$ {Without loss of generality, let }$i=1$ and {$v_{t+1}=w_{q}^{1}$.} Let $M_{2}^{\prime }$ be obtained from $M_{2}$ by deleting the vertices of $C_{m_{1}}$, and $M_{1}^{\prime }=(C^{1}\cup C_{m_{1}})\cup u_{1}^{1}u_{1}^{2}u_{2}^{2}\ldots u_{j}^{2}v_{1}\ldots v_{t}w_{q}^{1}$. {In this case, consider the subgraphs } $G_{2}^{1}=(G_{2}-(M_{1}\cup M_{2}))\cup (M_{1}^{\prime }\cup M_{2}^{\prime })$ and $G_{1}^{1}=G-G_{2}^{1}$ {{which we will be discussing later}. }
Now {assume that }$v_{t+1}$ {belongs to a tailed cycle }$C_{n_{i},\ell _{i}}$ for some $i$, say $i=1$. Let $C=w_{1}^{1}w_{2}^{1}\ldots w_{n_{1}}^{1}w_{1}^{1}$ be the cycle of $C_{n_{1},\ell _{1}}$ and $ P=y_{1}^{1}\ldots y_{\ell _{1}}^{1}$ be the tail of $C_{n_{1},\ell _{1}}$ such that $w_{1}^{1}y_{1}^{1}\in E(G)$. {Consider the two situations depending on whether }$v_{t+1}$ {is on the cycle or the tail. If }$ v_{t+1}\in V(C)$, say $v_{t+1}=w_{q}^{1}$, {then let }$M_{2}^{\prime }$ be obtained from $M_{2}$ by deleting the vertices of $C_{n_{1},\ell _{1}}$, and $M_{1}^{\prime }=(C^{1}\cup C)\cup u_{1}^{1}u_{1}^{2}u_{2}^{2}\ldots u_{j}^{2}v_{1}\ldots v_{t}w_{q}^{i}$. {In this case, consider the subgraphs } $G_{2}^{1}=(G_{2}-(M_{1}\cup M_{2}))\cup (M_{1}^{\prime }\cup M_{2}^{\prime })$ and $G_{1}^{1}=G-G_{2}^{1}.$ {If }$v_{t+1}\in V(P),$ say $ v_{t+1}=y_{q}^{1}$, {then let }$M_{2}^{\prime }$ be obtained from $M_{2}$ by deleting the vertices of $C_{n_{1},\ell _{1}}$, and $M_{1}^{\prime }=(C^{1}\cup C)\cup u_{1}^{1}u_{1}^{2}u_{2}^{2}\ldots u_{j}^{2}v_{1}\ldots v_{t}y_{q}^{1}y_{q-1}^{1}\ldots y_{1}^{1}w_{1}^{1}$. {In this case, consider the subgraphs }$G_{2}^{1}=(G_{2}-(M_{1}\cup M_{2}))\cup (M_{1}^{\prime }\cup M_{2}^{\prime })$ and $G_{1}^{1}=G-G_{2}^{1}$.
{Observe that in any situation, either }the number of cycles of $G_{2}^{1}$ { belonging to }$\mathcal{F}$ is greater than {the one of }$G_{2}$ {that are in }$\mathcal{F}$ or $s_{(G_{1}^{1},G_{2}^{1})}>s_{(G_{1},G_{2})}$. {Now, if }$G_{1}^{1}$ has no cycle of $\mathcal{F}$, then the pair $ (G_{1}^{1},G_{2}^{1})$ leads to a contradiction. Otherwise, by repeating above process we can obtain a pair $(G_{1}^{\prime },G_{2}^{\prime })$ such that $G_{1}^{\prime}$ has no cycle of $\mathcal{F}$ and each component of $ G_{2}^{\prime}$ belongs to $\mathcal{F}_{0}\cup \mathcal{F}_{0,2}\cup \mathcal{F}_{2,2}\cup \mathcal{F}_{3}\cup \mathcal{E}$, {where either the number of }strong components of $G_{2}^{\prime }$ is greater than the {one of }$G_{2}$ or the number of cycles of $G_{2}^{\prime }$ {that are in }$ \mathcal{F}$ is greater than the number of cycles of $G_{2}$ {belonging to }$ \mathcal{F}$ or $s_{(G_{1}^{\prime },G_{2}^{\prime })}>s_{(G_{1},G_{2})}.$ { In either case, we have a contradiction and the desired claim follows. }$
\Box $
\noindent \textbf{Claim 4.} {If }$M\in \mathcal{B}_{r,s}\;${is a non-strong component of }$G_{2}$ {\ with a special vertex }$z,$ then there is no path $v_{0}v_{1}\ldots v_{t}v_{t+1}$ $(t\geq 1)$ in $G$ such that $ v_{1},\ldots ,v_{t}\in V(G_{1})$, $v_{0},v_{t+1}\in V(M)-\{z\}$ and $ v_{0},v_{t+1}$ belong to different {near cycles of }$z.$ \newline \noindent \textbf{Proof of Claim 4.} Let $M$ be obtained from $r\geq 0$ tailed-cycle $C_{n_{1},\ell _{1}}$, $\ldots ,C_{n_{r},\ell _{r}}$ and $s\geq 3$ cycles $C_{m_{1}},\ldots ,C_{m_{s}}$, where $n_{i}\equiv 2\pmod 3$ and $ m_{j}\equiv 2\pmod 3$ for each $i,j$, by adding a new vertex $z$ (special vertex) and attaching $z$ to the leaf of each tailed cycle $C_{n_{i},\ell _{i}}$ and to one vertex of each cycle $C_{m_{j}}$. {Moreover, let }$C_{m_{i}}=z_{1}^{i}z_{2}^{i} \ldots z_{m_{i}}^{i}z_{1}^{i}$ for {\ each }$i\in \{1,...,s\}$ and let $ V(C_{n_{i},\ell _{i}})=\{x_{1}^{i},\ldots ,x_{n_{i}}^{i},y_{1}^{i},\ldots ,y_{\ell _{i}}^{i}\},$ where $x_{1}^{i},\ldots ,x_{n_{i}}^{i}$ {induce in order the cycle }of $C_{n_{i},\ell _{i}}$ and $y_{1}^{i},\ldots ,y_{\ell _{i}}^{i}$ {induce in order the }tail of $C_{n_{i},\ell _{i}}$.
Suppose, to the contrary, that there is a path $P=v_{0}v_{1}\ldots v_{t}v_{t+1}$ in $G$ such that $v_{1},\ldots ,v_{t}\in V(G_{1}),$ $ v_{0},v_{t+1}\in V(M)-\{z\}$ and $v_{0},v_{t+1}$ belong to different {near cycles of }$z.$
First let $r+s=3$. Then $r=0$ and $s=3$. Assume, without loss of generality, that $v_{0}=u_{k}^{1}$ and $v_{t+1}=u_{j}^{2}$ where $j\leq m_{1}/2$ and $ k\leq m_{2}/2$. Let $M^{\prime }$ be obtained from $C_{1},C_{3}$ by adding the path $Pu_{j-1}^{2}\ldots u_{1}^{2}zu_{1}^{3}$. Note that if $ v_{t+1}=u_{1}^{2}$, then the added path will be {simply }$Pzu_{1}^{3}$. { Consider the subgraph }$G_{2}^{\prime }=(G_{2}-M)\cup M^{\prime }$. If $ G-G_{2}^{\prime }$ has no cycle of $\mathcal{F}$, then the pair $ (G-G_{2}^{\prime },G_{2}^{\prime })$ {provides a number of strong components in }$G_{2}^{\prime }$ {greater than the one of }$G_{2},$ {contradicting our choice of the pair }$(G_{1},G_{2}).$ {Assume now that }$G-G_{2}^{\prime }$ has exactly\textbf{\ }one cycle $C$ of $\mathcal{F}$. Then $V(C)$ meets at least a vertex of $\{u_{j+1}^{2},\ldots ,u_{m_{2}}^{2}\} $ and let $p$ be the largest integer that $u_{p}^{2}\in V(C) $. Let $M^{\prime \prime }$ be obtained from $M^{\prime }\cup C$ by adding the path $u_{1}^{2}\ldots u_{p}^{2}$. {Consider the subgraph }$G_{2}^{\prime \prime }=(G_{2}-M^{\prime })\cup M^{\prime \prime } $. Then, {as above, }the pair $(G-G_{2}^{\prime \prime },G_{2}^{\prime \prime })$ leads to a contradiction. {Hence we can assume that }$G-G_{2}^{\prime }$ has at least two disjoint cycles of $ \mathcal{F}$. Clearly each of these cycles meets at least a vertex of $ \{u_{j+1}^{2},\ldots ,u_{m_{2}}^{2}\}$. Consider the subgraph $G^{\prime }$ of $G-G_{2}^{\prime }$ induced by the vertices of these cycles and the vertices of $\{u_{j+1}^{2},\ldots ,u_{m_{2}}^{2}\}$, {and let }$K_{1}$ and $ K_{2}$ be two disjoint subgraphs of {$G^{\prime }$} satisfying the conditions of Lemma \ref{mainn}. Then the pair $(G_{2}^{\prime }\cup K_{2},G-(G_{2}^{\prime }\cup K_{2}))$ leads to a contradiction because the number of strong components of $G_{2}^{\prime }\cup K_{2}$ is greater than the number of strong components of $G_{2}$.
Now let $r+s\geq 4$, {and assume that }$P$ connects two cycles $C$ and $ C^{\prime }$ of $M$ {that are at distance one from }$z.$ Let $M^{\prime }$ be obtained from $M$ by deleting the vertices of $C\cup C^{\prime }$, and let $M^{\prime \prime }=C+C^{\prime }+P$. {Now, if we consider the subgraphs }$G_{2}^{\prime \prime }=(G_{2}-M)\cup (M^{\prime }\cup M^{\prime \prime })$ and $G_{1}^{\prime \prime }=G-G_{2}^{\prime \prime }$, {then one can see, as above, that the pair }$(G_{1}^{\prime \prime },G_{2}^{\prime \prime })$ leads to a contradiction. $
\Box $
\noindent \textbf{Claim 5.} Let $M_{1},M_{2}\in \mathcal{E}$ be two non-strong components of $G_{2}$ and let $z_{i}$ be the special vertex of $M_{i}$. Then there is no path $P=v_{0}v_{1}\ldots v_{t}v_{t+1}\;(t\geq 1) $ in $G$ such that $v_{1},\ldots v_{t}\in V(G_{1})$, $v_{0}$ belongs to a {near cycle }$C_{1}$ {of }$z_{1}$ and $v_{t+1}$ belongs to {near cycle }$ C_{2}$ {of }$z_{2}$.\newline \noindent \textbf{Proof of Claim 5.} {Suppose to the contrary that such a path }$P$ exists. Let $M_{i}^{\prime }$ be obtained from $M_{i}$ by deleting the vertices of $V(C_{i})$ for {each }$i\in \{1,2\} $ and {let }$ M=(C_{1}\cup C_{2})+P$. {\ Consider the subgraphs }$G_{2}^{\prime }=(G_{2}-(M_{1}\cup M_{2}))\cup (M_{1}^{\prime }\cup M_{2}^{\prime }\cup M)$ and $G_{1}^{\prime }=G-G_{2}^{\prime }$. {Since }$G_{2}^{\prime }${{\ {has more }}}strong components than $G_{2},$ {the pair }$(G_{1}^{\prime },G_{2}^{\prime })$ contradicts the choice of the {pair }$(G_{1},G_{2}).$ $
\Box $
{Now, let }$K_{0}$ be the subgraph of $G_{2}$ that consists of all non-strong components of $G_{2}$ and {let }$H_{0}$ be the subgraph of $G_{2}$ that consists of all strong components of $G_{2}$. By Lemma \ref{strong}, each component $M$ of $H_{0}$ has a $3$-tuple $\overrightarrow{f_{M}}$ of RDFs of $M$ such that $\omega (\overrightarrow{f_{M}})\leq \frac{(4k+8)3n(M) }{6k+11}$ and all vertices of $M$ are $\overrightarrow{f_{M}}$-strong. { Therefore, by combining }these $3$-tuples we obtain a 3-tuple $ \overrightarrow{f_{0}}$ of RDFs of $H_{0}$ such that $\omega ( \overrightarrow{f_{0}})\leq \frac{(4k+8)3n(H_{0})}{6k+11}$ and all vertices of $H_{0}$ are $\overrightarrow{f_{0}}$-strong. {Moreover, }by Lemmas \ref {non-strong1} and \ref{non-strong2}, each component $M$ of $K_{0}$ has a 3-tuple $\overrightarrow{g_{M}}$ of RDFs of $M$ such that $\omega ( \overrightarrow{g_{M}})\leq \frac{(4k+8)3n(M)}{6k+11},$ and {if further }$ M\in \mathcal{B}_{r,s}\;(s\geq 3)$, then its special vertex {as well as all all vertices }on tailed cycle are $\overrightarrow{g_{M}}$-strong. { Therefore, by combining }these $3$-tuples we obtain a $3$-tuple $ \overrightarrow{g_{0}}$ of RDFs of $K_{0}$ such that $\omega ( \overrightarrow{g_{0}})\leq \frac{(4k+8)3n(K_{0})}{6k+11}$ and all vertices of $K_{0}$ are $\overrightarrow{g_{0}}$-strong {except vertices on near cycles of some }special vertex or vertices on the component in $\mathcal{F} _{2,2}$.
If {there is a path }$P_{1}=v_{0}v_{1}\ldots v_{t}v_{t+1}$ in $G$ such that $ v_{1},\ldots ,v_{t}\in V(G_{1})$, $N(v_{1})\cup N(v_{t})\subseteq V(H_{0}\cup K_{0})\cup V(P_{1})$ and both $v_{0},v_{t+1}$ belong to a component $M\in \mathcal{F}_{2,2}$ of $K_{0}$, then we deduce from Claim 1 that both of $v_{0},v_{t+1}$ belong to {same }cycle of $M$. {Let }$M^{\prime }=M+P$. It follows from Lemma \ref{MainLem}-(5) {and Lemmas }\ref{ear1} and \ref{induced} that $M^{\prime }$ has a 3-tuple $\overrightarrow{f^{\prime }}$ of RDFs of $M^{\prime }$ such that $\omega (\overrightarrow{f^{\prime }} )\leq 2n(M^{\prime })+2\leq \frac{(4k+8)3n(M^{\prime })}{6k+11}$ and all vertices {of }$M^{\prime }$ are $\overrightarrow{f^{\prime }}$-strong { except }$v_{1}$ and $v_{t}$. {In this case, let }$K_{0}^{1}=K_{0}-M$, $
\overrightarrow{g_{0}^{1}}=\overrightarrow{g_{0}}|_{K_{0}^{1}}$ {(the restriction of }$\overrightarrow{g_{0}}$ on $K_{0}^{1})$ and $ H_{0}^{1}=H_{0}\cup (M+P)=H_{0}\cup M^{\prime }.$ {Let }$\overrightarrow{ f_{0}^{1}}$ be a 3-tuple {\ of RDFs }obtained by combining the 3-tuples $ f_{0}$ and $\overrightarrow{f^{\prime }}$. Clearly all vertices of $ H_{0}^{1} $ which have {\ a }neighbor outside $H_{0}^{1}\cup K_{0}^{1}$ are $ \overrightarrow{f_{0}^{1}}$-strong. By repeating this process we obtain two { \ sequences }of subgraphs $K_{0}\supseteq K_{0}^{1}\supseteq \ldots \supseteq K_{0}^{d}$ and $H_{0}\subseteq H_{0}^{1}\subseteq \ldots \subseteq H_{0}^{d}$ so that: \textrm{(i)} {there is no path }$P=w_{0}w_{1}\ldots w_{r}w_{r+1}$ in $G$ with $w_{1},\ldots ,w_{r}\in V(G_{1})-(\cup _{i=1}^{d}V(P_{i}))$, $N(w_{1})\cup N(w_{r})\subseteq V(H_{0}^{d}\cup K_{0}^{d})\cup (\cup _{i=1}^{d}V(P_{i}))$ and both $v_{0},v_{t+1}$ belong to a component $M\in \mathcal{F}_{2,2}$ of $K_{0}^{d}$, and \textrm{(ii)} $ H_{0}^{d}$ has a 3-tuple $\overrightarrow{f_{0}^{d}}$ such that all {its } vertices which have{a} neighbor outside $V(H_{0}^{d}\cup K_{0}^{d})$ are $ \overrightarrow{f_{0}^{d}}$-strong. Let $H_{1}=H_{0}^{d}$, $K_{1}=K_{0}^{d}$
, $\overrightarrow{g_{1}}=\overrightarrow{g_{0}}|_{K_{0}^{d}}$ and $ \overrightarrow{f_{1}}=f_{0}^{d}$. {Observe that $\omega ( \overrightarrow{f_{1}})\leq \frac{(4k+8)3n(H_{1})}{6k+11}$ and $\omega ( \overrightarrow{g_{1}})\leq \frac{(4k+8)3n(K_{1})}{6k+11}$. If $ V(G)=V(H_{1}\cup K_{1})$, then by combining 3-tuple $\overrightarrow{f_{1}}$ of $H_{1}$ and 3-tuple $\overrightarrow{g_{1}}$ of $K_{1}$, we get a 3-tuple $\overrightarrow{h}$ of $G$ such that $\omega (\overrightarrow{h})\leq \frac{ (4k+8)3n(G)}{6k+11}$ which will prove the theorem.} {Hence assume }that $ V(G)\neq V(H_{1}\cup K_{1})$, {and let }$G_{2}^{1}=H_{1}\cup K_{1}$ and $ G_{1}^{1}=G-G_{2}^{1}$.
If {there is a path }$P_{1}=v_{0}v_{1}\ldots ,v_{t}v_{t+1}$ in $G$ such that $v_{0},v_{t+1}\in V(H_{1})$, $v_{1},\ldots ,v_{t}\in V(G_{1}^{1})$ and $ N(v_{1})\cup N(v_{t})\subseteq V(H_{1})\cup V(P_{1})$, then let $ H_{1}^{1}=H_{1}+P$. By Lemma \ref{ear1}, we can extend $\overrightarrow{f_{1} }$ to a 3-tuple $\overrightarrow{f_{1}^{1}}$ of RDFs of $H_{1}^{1}$ such that $\omega (\overrightarrow{f_{1}^{1}})\leq \frac{(4k+8)3n(H_{1}^{1})}{ 6k+11}$, {where }all vertices of $H_{1}^{1}$ but $v_{1}$ {and }$v_{t}$ are $ \overrightarrow{f_{1}^{1}}$-strong. {Now, if there is a path }$ P_{2}=z_{0}z_{1}\ldots ,z_{m}z_{m+1}\;(m\geq 1)$ in $G$ such that $ z_{0},z_{m+1}\in V(H_{1}^{1})$, $z_{1},\ldots ,z_{m}\in V(G_{1}^{1})-V(P_{1}) $, $N(z_{1})\cup N(z_{m})\subseteq V(H_{1}^{1})\cup V(P_{2})$, then let $H_{1}^{2}=H_{1}^{1}+P_{2}$. By Lemma \ref{ear1}, we can extend $\overrightarrow{f_{1}^{1}}$ to a 3-tuple $\overrightarrow{f_{1}^{2}}$ of RDFs of $H_{1}^{2}$ such that $\omega (\overrightarrow{f_{1}^{2}})\leq \frac{(4k+8)3n(H_{1}^{2})}{6k+11}$ and all new vertices but $z_{1},z_{m}$ are $\overrightarrow{f_{1}^{2}}$-strong. By repeating this process we obtain a {sequence }of subgraphs $H_{1}\subseteq H_{1}^{1}\subseteq \ldots \subseteq H_{1}^{q}$ so that {there is no path }$P=w_{0}w_{1}\ldots ,w_{r}w_{r+1}$ in $G$ such that $w_{0},w_{r+1}\in V(H_{1}^{q})$, $ w_{1},\ldots ,w_{r}\in V(G_{1}^{1})-(\cup _{i=1}^{q}V(P_{i}))$ and $ N(w_{1})\cup N(w_{r})\subseteq V(H_{1}^{q})\cup V(P)$. Moreover, $H_{1}^{q}$ has a 3-tuple $\overrightarrow{f_{1}^{q}}$ of RDFs of $H_{1}^{q}$ such that $ \omega (\overrightarrow{f_{1}^{q}})\leq \frac{(4k+8)3n(H_{1}^{q})}{6k+11}$ and all vertices are $\overrightarrow{f_{1}^{q}}$-strong unless the vertices which have no neighbors outside of $H_{1}^{q}\cup K_{1}$. If $ V(G)=V(H_{1}^{q}\cup K_{1})$, {then as above, by combining 3-tuple $\overrightarrow{f_{1}^{q}}$ and 3-tuple $\overrightarrow{g_{2}}= \overrightarrow{g_{1}}$, the result follows.} {Hence }assume that $V(G)\neq V(H_{1}^{q}\cup K_{1})$, {{{and let }}}$H_{2}=H_{1}^{q}$, $ \overrightarrow{f_{2}}=\overrightarrow{f_{1}^{q}}$, $K_{2}=K_{1}$, $ \overrightarrow{g_{2}}=\overrightarrow{g_{1}}$, $G_{2}^{2}=H_{2}\cup K_{2}$ and $G_{1}^{2}=G-G_{2}^{2}$. {In the following we will use Lemma \ref{12} by applying its three items, one by one (in any order), starting with the subgraph }$G_{1}^{2}$ {and obtaining each time (when the item occurs) a sequence of subgraphs. The last subgraph of the sequence will be used for the next item. }
\noindent \textbf{Case 1.} {$G_{1}^{2}$} contains a tailed $m$-cycle $C_{m,\ell }\;({m\equiv 1\pmod 3})$, with vertex set $\{x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{\ell }\}$, such that $ y_{\ell }$ is adjacent to some vertex $x$ of $G_{2}^{2}$ and $ N_{G}(x_{m})\subseteq {V(G_{2}^{2})}\cup V(C_{m,\ell })$.\newline First {assume that }$x\in V(H_{2})$. Then $x$ is $\overrightarrow{f_{2}}$ -strong. {Let }$H_{2}^{1}$ be obtained from $H_{2}$ by adding the tailed cycle $C_{m,\ell }$ and the edge $xy_{\ell }$. By Lemma \ref {tailedcycle-3p+1}, $\overrightarrow{f_{2}}$ can be extended to a 3-tuple $ \overrightarrow{f_{2}^{1}}$ or RDFs of $H_{2}^{1}$ such that $\omega ( \overrightarrow{f_{2}^{1}})\leq \frac{(4k+8)3n(H_{2}^{1})}{6k+11}$ and all new vertices but $x_{m}$ are $\overrightarrow{f_{2}^{1}}$-strong. Set also $ K_{2}^{1}=K_{2}$ and $\overrightarrow{g_{2}^{1}}=\overrightarrow{g_{1}}$.
\noindent Now {assume that }$x$ belongs to a component $M$ of $K_{2}$ such that $M\in \mathcal{F}_{2,2}$. {Let }$M^{\prime }$ be obtained from $M$ and $ C_{m,\ell }$ by adding the edge $xy_{\ell }$. By Lemma \ref{MainLem}{-(5) and Lemma }\ref{tailedcycle-3p+1}, one can see that $M^{\prime }$ has a 3-tuple $f_{M^{\prime }}$ of RDFs such that $\omega (\overrightarrow{ f_{M^{\prime }}})\leq 2n(M^{\prime })+2\leq \frac{(4k+8)3n(M^{\prime })}{ 6k+11}$ and all of its vertices but $x_{m}$ are $\overrightarrow{ f_{M^{\prime }}}$-strong. {Let }$H_{2}^{1}=H_{2}\cup M^{\prime }$, $ K_{2}^{1}=K_{2}-M$, $g_{1}^{2}$ {be }the restriction of $g$ on $K_{2}^{1}$ and 3-tuple $\overrightarrow{f_{2}^{1}}$ is obtained from combining $ \overrightarrow{f_{M^{\prime }}}$ and $\overrightarrow{f_{2}}$. Note that all vertices of $H_{2}^{1}$ which have neighbor in {$ G_{1}^{2}-V(C_{m,\ell})$} are $\overrightarrow{f_{2}^{1}}$-strong.
\noindent Next assume that $x$ belongs to a component $M$ of $K_{2}$ such that $M\in \mathcal{B}_{r,s}\;({s\geq 3})$ and $x$ is $ \overrightarrow{g}$-strong. {{{Let }}}$M^{\prime }$ {{{ be }}}obtained {{{from }}}$M$ and $C_{m,\ell }$ by adding the edge $xy_{\ell }$ and let $H_{2}^{1}=H_{2}$ and $ K_{2}^{1}=(K_{2}-M)\cup M^{\prime }$. By Lemma \ref{tailedcycle-3p+1}, $ \overrightarrow{g_{2}}$ can be extended to a 3-tuple $g_{2}^{1}$ of RDFs of $ K_{2}^{1}$ such that $\omega (\overrightarrow{g_{2}^{1}})\leq \frac{ (4k+8)3n(K_{2}^{1})}{6k+11}$ and all newly added vertices but $x_{m}$ are $ \overrightarrow{g_{2}^{1}}$-strong.
Finally, {assume that }$x$ belongs to a component $M$ of $K_{2}$ such that $ M\in \mathcal{B}_{r,s}\;({s\geq 3})$ and $x$ is not $ \overrightarrow{g}$-strong. Then $x$ belongs to a {near }cycle $C$ {from the }special vertex of $M$. Let $M^{\prime }$ be obtained from $M$ by deleting the vertices of $C$ and {let }$M^{\prime \prime }$ be obtained from $C$ and $ C_{m,\ell }$ by adding the edge $xy_{\ell }$. {In this case, let }$ H_{2}^{1}=H_{2}\cup M^{\prime \prime }$, $K_{2}^{1}=K_{2}-V(C)$ and $ g_{1}^{2}$ is the restriction of $\overrightarrow{g_{2}}$ on $K_{2}^{1}$. By Lemma \ref{MainLem} (items 2,3,4), $f_{2}$ can be extended to a 3-tuple of RDFs $f_{2}^{1}$ of $H_{2}^{1}$ such that $\omega (\overrightarrow{f_{2}^{1}} )\leq \frac{(4k+8)3n(H_{2}^{1})}{6k+11}$ and all newly added vertices but $ x_{m}$ are $\overrightarrow{f_{2}^{1}}$-strong.
By repeatedly applying the above argument we obtain two\ {{{ sequences }}}of subgraphs $H_{2}\supseteq H_{2}^{1}\supseteq \ldots \supseteq H_{2}^{r_{1}}$ and {$K_{2},K_{2}^{1},\ldots ,K_{2}^{s_{1}}$} such that there is no tailed cycle $C_{m,\ell }\;({ m\equiv 1\pmod 3})$ in $G-(H_{2}^{r_{1}}\cup K_{2}^{s_{1}})$ whose end-vertex is adjacent to a vertex of $H_{2}^{r_{1}}\cup K_{2}^{s_{1}}$ . Let $H_{3}=H_{2}^{r_{1}}$, $K_{3}=K_{2}^{s_{1}}$, $f_{3}$ be a 3-tuple of RDFs of $H_{3}$ such that all vertices of $H_{3}$ which have a neighbor outside $H_{3}\cup K_{3}$ are $\overrightarrow{f_{3}}$-strong, and $g_{3}$ be a 3-tuple of RDFs of $K_{3}$ such that all newly added vertices of $K_{3}$ which have no neighbor outside $H_{3}\cup K_{3}$ are $\overrightarrow{g_{3}}$ -strong. {Let }$G_{2}^{3}=H_{3}\cup K_{3}$ and $G_{1}^{3}=G-G_{2}^{3}$.
\noindent \textbf{Case 2.} $G_{1}^{3}$ contains a cycle $ C_{m}=x_{1}x_{2}\ldots x_{m}x_{1}\;({m\equiv 1\pmod 3})$ such that $N_{G}(x_{m})\subseteq V(G_{2}^{3})\cup V(C_{m})$ and there is an edge $ x_{1}y$ with $y\in V(G_{3}^{2})$ Applying an argument similar to that described in Case 1, we obtained subgraphs $H_{4}$ and $K_{4}$ such that $ H_{3}\subseteq H_{4}$, and a 3-tuple $f_{4}$ of RDFs of $H_{4}$ so that all vertices of $H_{4}$ {having a }neighbor outside $H_{4}\cup K_{4}$ are $ \overrightarrow{f_{4}}$-strong, and a 3-tuple $\overrightarrow{g_{4}}$ of RDFs of $K_{4}$ so that all newly added vertices of $K_{3}$ {that have a } neighbor outside $H_{4}\cup K_{4}$ are $\overrightarrow{g_{4}}$-strong. Assume that $G_{2}^{4}=H_{4}\cup K_{4}$ and $G_{1}^{4}=G-G_{2}^{4}$.
Let $\overrightarrow{h}$ be a 3-tuple defined on $G_{2}^{4}$ {\ obtained by combining }$\overrightarrow{f_{4}}$ and $\overrightarrow{g_{4}} $.
\noindent \textbf{Case 3.} $G_{1}^{4}$ has a path $ P=v_{0}v_{1},\ldots ,v_{t}v_{t+1}\;(t\geq 1)$ such that $v_{0},v_{t+1}\in G_{2}^{4}$, $v_{1},\ldots ,v_{t}\in V(G_{1}^{4})$ and $N_{G}(v_{1})\cup N_{G}(v_{t+1})\subseteq V(G_{2}^{4})\cup V(P)$.\newline By Claims 1,2,3,4 {and }5, at least one of the vertices $v_{0},v_{t+1}$ is $ \overrightarrow{h}$-strong. First {assume that each of }$v_{0}$ and $v_{t+1}$ is $\overrightarrow{h}$-strong. {Let }$G_{2}^{5}=G_{2}^{4}+P$ and $ G_{1}^{5}=G-G_{2}^{5}$. By Lemma \ref{ear1}, we can extend $\overrightarrow{h }$ to a 3-tuple $\overrightarrow{h^{1}}$ of RDFs of $G_{2}^{5}$ such that $ \omega (\overrightarrow{h^{1}})\leq \frac{(4k+8)3n(G_{2}^{5})}{6k+11}$ and all vertices of $G_{2}^{5}$ which have neighbor in $G_{1}^{5}$ are $ \overrightarrow{h^{1}}$-strong. {Assume now, without loss of generality, that }$v_{0}$ is $\overrightarrow{h}$-strong and $v_{t+1}$ is not $ \overrightarrow{h}$-strong. It follows that $v_{t+1}$ is on {a near }cycle from a special vertex of a component of $K_{4}$ or is in a component of $ K_{4}$ {that belongs }to $\mathcal{F}_{2,2}$. If $v_{t+1}$ is on a {near } cycle $C$ from a special vertex, then let $G_{2}^{5}=G_{2}^{4}+P$. By Lemma
\ref{tailedcycle-3p+1}, $h|_{G_{2}^{4}-C}$ can be extended to a 3-tuple $ \overrightarrow{h^{1}}$ of RDFs of $G_{2}^{5}$ such that $\omega ( \overrightarrow{h^{1}})\leq \frac{(4k+8)3n(G_{2}^{5})}{6k+11}$ and all vertices of $G_{2}^{5}$ which have neighbors in $G_{1}^{5}-V(P)$ are $ \overrightarrow{h^{1}}$-strong. If $v_{t_{1}}$ is in a component $M$ of $
K_{4}$ {belonging }to $\mathcal{F}_{2,2}$, then let $G_{2}^{5}=G_{2}^{4}+P$. By applying Lemma \ref{tailedcycle-3p+1} twice, $h|_{G_{2}^{4}-M}$ can be extended to a 3-tuple $\overrightarrow{h^{1}}$ of RDFs of $G_{2}^{5}$ such that $\omega (\overrightarrow{h^{1}})\leq \frac{(4k+8)3n(G_{2}^{5})}{6k+11}$ and all vertices of $G_{2}^{5}$ which have neighbors in $G_{1}^{5}-V(P)$ are $\overrightarrow{h^{1}}$-strong.
By repeating this process we obtain a 3-tuple $\overrightarrow{h}$ of RDFs $ G $ such that $\omega (\overrightarrow{h})\leq \frac{(4k+8)3n(G)}{6k+11}$, { \ implying that }$\gamma _{R}(G)\leq \frac{(4k+8)n(G)}{6k+11}$ as desired. $ \ \ \
\Box $
{{{Now, the next result settling }}}Conjecture \ref{conj} is an immediate consequence of Theorem \ref{Theorem} and the Gallai-type result $ \gamma _{R}(G)+\partial (G)=n$ which is valid for every graph $G$ of order $ n.$
\begin{corollary} \emph{Let $G$ be a graph of order $n\ge 6k + 9$, minimum degree $\delta\ge 2$ , which does not contain any induced $\{C_5,C_8,\ldots,C_{3k+2}\} $-cycles. Then $\partial(G)\ge \frac{(2k+3)n}{6k+11}.$} \end{corollary}
\section{Acknowledgment} This work was supported by the National Key R \& D Program of China (Grant No. 2019YFA0706402) and the Natural Science Foundation of Guangdong Province under grant 2018A0303130115.
\end{document} | arXiv |
If the roots of the quadratic equation $\frac32x^2+11x+c=0$ are $x=\frac{-11\pm\sqrt{7}}{3}$, then what is the value of $c$?
By the quadratic formula, the roots of the equation are $$x=\frac{-(11)\pm\sqrt{(11)^2-4(\frac32)c}}{2(\frac32)},$$which simplifies to $$x=\frac{-11\pm\sqrt{121-6c}}{3}.$$This looks exactly like our target, except that we have to get the $121-6c$ under the square root to equal $7$. So, we solve the equation $121-6c=7$, which yields $c=\boxed{19}$. | Math Dataset |
\begin{document}
\title [Universal inner inverses] {Adjoining a universal inner inverse to a ring element} \thanks{ Archived at \url{http://arxiv.org/abs/1505.02312}\,. After publication of this note, updates, errata, related references etc., if found, will be recorded at \url{http://math.berkeley.edu/~gbergman/papers} }
\subjclass[2010]{Primary: 16S10, 16S15.
Secondary: 16D40, 16D70, 16E50, 16U99.}
\keywords{Universal adjunction of an inner inverse to an element of a $\!k\!$-algebra; normal forms in rings and modules }
\author{George M. Bergman} \address{University of California\\ Berkeley, CA 94720-3840, USA} \email{[email protected]}
\begin{abstract} Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\lang q\mid pqp=\nolinebreak p\rang.$ We obtain normal forms for elements of $R',$ and for elements of $\!R'\!$-modules arising by extension of scalars from $\!R\!$-modules. The details depend on where in the chain $pR\cap Rp \subseteq pR\cup Rp \subseteq pR + Rp \subseteq R$ the unit $1$ of $R$ first appears.
This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von~Neumann regular ring; but that goal remains distant.
We end with a normal form result for the algebra obtained by tying together a $\!k\!$-algebra $R$ given with a nonzero element $p$ satisfying $1\notin pR+Rp$ and a $\!k\!$-algebra $S$ given with a nonzero $q$ satisfying $1\notin qS+Sq,$ via the pair of relations $p=pqp,$ $q=qpq.$ \end{abstract} \maketitle
\section{Motivation: monoids of projective modules}\label{S.motivation} It is known that the abelian monoid of isomorphism classes of finitely generated projective modules over a general ring is subject to no nonobvious restrictions -- the obvious restrictions being
\begin{equation}\begin{minipage}[c]{35pc}\label{d.conical} no two nonzero elements of the monoid have sum zero, \end{minipage}\end{equation} and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.o-} the monoid has an element $u$ such that every element is a summand in $nu$ for some positive integer $n.$ \end{minipage}\end{equation}
(Namely, $u$ is the isomorphism class of the free module of rank~$1.)$
Indeed, every abelian monoid $M$ satisfying~\eqref{d.conical} and~\eqref{d.o-} is known to be the monoid of finitely generated projective modules of some hereditary $\!k\!$-algebra, for any field $k.$ (This was proved for finitely generated $M$ in \cite[Theorem~6.2]{cPu}, while Theorem~6.4 of that paper claimed to show that if one weakened `hereditary' to `semihereditary', the assumption that $M$ was finitely generated could be dropped. The argument indeed gave a $\!k\!$-algebra $R$ having $M$ as its monoid of finitely generated projectives, but the proof that $R$ was semihereditary was incorrect. However, in \cite[Theorem~3.4]{u_deriv&} it is shown that the $R$ so constructed is not merely semihereditary, but hereditary. For a similar result, see \cite[Corollary~4.5]{A+G3}.)
Recall that a ring $R$ is called {\em von~Neumann regular} if every element $p\in R$ has an {\em inner inverse}, that is, an element $q\in R$ satisfying $pqp=p.$ The monoid of isomorphism classes of finitely generated projective modules over a von~Neumann regular ring is known to satisfy not only~\eqref{d.conical} and~\eqref{d.o-}, but a strong additional restriction, the {\em Riesz refinement property} \cite{separative}:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.refinement} If $A_0\oplus A_1\cong B_0\oplus B_1,$ then there exist $C_{ij}$ $(i,j\in\{0,1\})$\\ such that $A_i\cong C_{i0}\oplus C_{i1}$ and $B_i\cong C_{0i}\oplus C_{1i};$ \end{minipage}\end{equation}
that is, any such isomorphism $A_0\oplus A_1\cong B_0\oplus B_1$ can be written in the trivial form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.trivO+} $(C_{00}\oplus C_{01})\oplus (C_{10}\oplus C_{11})\ \cong \ (C_{00}\oplus C_{10})\oplus (C_{01}\oplus C_{11}).$ \end{minipage}\end{equation}
Until a couple of decades ago, it was an open question whether~\eqref{d.conical}-\eqref{d.refinement} completely characterized the monoids of finitely generated projectives of von~Neumann regular rings. Then F.\,Wehrung \cite{FW_card} constructed a monoid of cardinality $\aleph_2$ satisfying~\eqref{d.conical}-\eqref{d.refinement} which cannot occur as such a monoid of projectives. More recently, he has given an example of a {\em countable} monoid satisfying~\eqref{d.conical}-\eqref{d.refinement} which does not occur in this way for any von~Neumann regular algebra over an {\em uncountable} field \cite[\S4]{Ara}. It remains open whether every {\em countable} monoid satisfying~\eqref{d.conical}-\eqref{d.refinement} is the monoid of finitely generated projectives of {\em some} von~Neumann regular ring.
But there is in fact a strong condition, not implied by~\eqref{d.conical}-\eqref{d.refinement}, which is not known to fail in any von~Neumann regular ring:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.separative} $A\oplus A\cong A\oplus B\cong B\oplus B\ \implies\ A\cong B.$ \end{minipage}\end{equation}
An abelian monoid satisfying~\eqref{d.separative}. is called {\em separative}. A positive answer to the question of whether the monoid of finitely generated projectives of every von~Neumann regular ring is separative would solve several other questions about such rings \cite{separative}. We remark that it is known \cite{FW}, \cite[\S4]{A+E} that every monoid satisfying~\eqref{d.conical} and \eqref{d.o-} can be embedded in one that also satisfies~\eqref{d.refinement} (which can be taken countable if the original monoid was). Hence, applying this to monoids for which~\eqref{d.separative} fails, one sees that there do exist abelian monoids satisfying~\eqref{d.conical}-\eqref{d.refinement} but not~\eqref{d.separative}. For more on these questions, see \cite{Ara}, \cite{A+E}, \cite{A+G}, \cite{A+G2},~\cite{separative}.
Now it is known that many universal constructions on $\!k\!$-algebras make only ``obvious'' changes in the structure of the monoid of finitely generated projectives \cite{cP}, \cite{cPu}, \cite{u_deriv&}. This suggests that to investigate the possible structures of those monoids for von~Neumann regular $\!k\!$-algebras, we could start with a general $\!k\!$-algebra, recursively adjoin universal inner inverses to its elements till it becomes von~Neumann regular, and see what conditions this process forces on the monoid of projectives.
That plan has not proved as easy as I hoped. We obtain below normal forms for elements of the $\!k\!$-algebra $R'=R\lang q\mid pqp=p\rang$ and for elements of modules $M\otimes_R\,R';$ but it is not clear whether these can be used to get useful results on isomorphism classes of modules.
The descriptions of the algebra $R'$ will show surprising differences, depending on how near to invertible the element $p\in R$ to which we adjoin a universal inner inverse is. Below, we begin with a case that is challenging enough to illustrate our method without being excessively difficult, the case where $p$ is farthest from invertible, namely, where $1\notin pR+Rp$~(\S\ref{S.norm}). We then quickly cover the easy cases where $1\in pR$ and/or $1\in Rp,$ i.e., where $p$ is left or right invertible, or both (\S\ref{S.1-sided}). Finally, we treat the surprisingly difficult intermediate case where $1\in pR+Rp,$ but $1\notin pR\cup Rp$~(\S\ref{S.1=:norm}). We also examine the particular instance of this construction where $R$ is the Weyl algebra~(\S\ref{S.Weyl}). The last main results of the paper~(\S\ref{S.mutual}) concern a variant of the above constructions, in which the pair of relations $pqp=p,$ $qpq=q,$ is used to join together two given $\!k\!$-algebras.
For reasons to be noted in~\S\ref{S.enough?}, the difficult results of~\S\ref{S.1=:norm} (and the easy results of~\S\ref{S.1-sided}) may be less useful than the results of~\S\ref{S.norm}; so some readers may wish to skip or skim them. A list of the sections of this note containing the most important material, in this light, along with some others, noted in curly brackets, that are less essential but not very difficult, is: \S\S\ref{S.defs} \ref{S.norm} \{\ref{S.digress}\} \ref{S.modules} \{\ref{S.enough?} \ref{S.1-sided}\} \ref{S.mutual} \{\ref{S.further}\}.
Incidentally, though, as noted above, there exist monoids satisfying conditions~\eqref{d.conical}-\eqref{d.refinement} but not condition~\eqref{d.separative}, no ``concrete'' examples of such monoids appear to be known, but only constructions which obtain them by starting with a monoid satisfying neither~\eqref{d.refinement} nor~\eqref{d.separative}, universally adjoining elements $C_{ij}$ as required by~\eqref{d.refinement}, and repeating this construction transfinitely -- i.e., the analog of the way non-separative von~Neumann regular rings might be constructed if the plan suggested above is successful. It would, of course, be of interest to have concrete examples in both the monoid and the algebra situations.
I am indebted to P.\,Ara, T.\,Y.\,Lam, N.\,Nahlus, and, especially, to K.\,O'Meara for helpful comments, corrections and suggestions regarding this note.
\section{Generalities}\label{S.defs} All rings will here be associative with unit; and the rings for which we will study the construction of universal inner inverses will be algebras over a fixed field $k.$ If $R$ is a nonzero $\!k\!$-algebra, we will identify the $\!k\!$-subspace of $R$ spanned by $1$ with $k.$
I am using the term ``inner inverse'' (at the advice of T.\,Y.\,Lam) for what I had previously known as a ``quasi-inverse'', since the latter term also has a different, better-established sense. (Elements $x$ and $y$ of a not necessarily unital ring $R$ are called quasi-inverses in that sense if $xy=yx=-x-y;$ in other words, if on adjoining a unit to $R,$ one gets mutually inverse elements $1+x$ and $1+y.$ We will not consider that concept here. On the other hand, the choice of the letter $q$ for universal inner inverses below is based on my having used ``quasi-inverse'' in early drafts of this note, while the element whose inner inverse we are adjoining will be denoted $p$ because of the visual matching of the shapes of these two letters.)
Note that if $R$ is a ring of endomorphisms of an abelian group $A,$ then an inner inverse of an element $p\in R$ is an endomorphism $q$ that takes every member of the image of $p$ to some inverse image under $p$ of that element, with no restriction on what it does to elements not in the image of $p.$ From this it is easy to show that in the algebra of endomorphisms of any $\!k\!$-vector space, every element has an inner inverse; so such algebras are examples of von~Neumann regular rings.
The relation ``is an inner inverse of'' is not symmetric: if $q$ is an inner inverse of $p,$ $p$ need not be an inner inverse of $q.$ For example, any element of any ring is an inner inverse of $0,$ but $0$ is not an inner inverse of any nonzero element. However, if an element $p$ has an inner inverse $q,$ we find that $q'=qpq$ is an inner inverse of $p$ such that $p$ {\em is} an inner inverse of $q'.$ Thus, the condition that an element of a ring have an inner inverse is equivalent to the condition that it have a ``mutual inner inverse''. Even when both relations $pqp=p$ and $qpq=q$ hold, however, $p$ does not uniquely determine $q.$ For instance, in the ring $M_2(R)$ of $2\times 2$ matrices over any ring $R,$ any two members of $\{e_{11}+r e_{12}\mid r\in R\}$ are inner inverses of one another.
Our normal form results for algebras constructed by adjoining universal inner inverses will be proved using the ring-theoretic version of the Diamond Lemma, as developed in \cite[\S1]{<>}. However, where in \cite{<>} I formalized reduction rules as ordered pairs $(W,f),$ with $W$ a word in our given generators, and $f$ a linear combination of words, to be substituted for occurrences of $W$ as subwords of other words, I here use the more informal notation ``$W\mapsto f$''. (Another formulation of the Diamond Lemma appears as \cite[Proposition~1]{LB}. Bokut' \cite{LB+2}, \cite{LB+PK} refers to it as ``the method of Gr\"{o}bner-Shirshov bases''.)
Given a $\!k\!$-algebra $R$ and an element $p\in R,$ our construction of a normal form for elements of $R\lang q\mid pqp=p\rang$ will start with a $\!k\!$-basis for $R,$ which we shall want to choose in a way that allows us to see which elements of $R$ are left and/or right multiples of $p.$ In describing such a basis, it will be convenient to use
\begin{definition}\label{D.basis_rel} If $U\subseteq V$ are $\!k\!$-vector-spaces, then a {\em $\!k\!$-basis of $V$ relative to $U$} will mean a subset $B\subseteq V$ with the property that every element of $V$ can be written uniquely as the sum of an element of $U$ and a $\!k\!$-linear combination of elements of $B.$ \end{definition}
Thus, the general basis of $V$ relative to $U$ can be obtained by choosing a basis $B'$ of $V/U,$ and selecting one inverse image in $V$ of each element of $B';$ or, alternatively, by choosing any direct-sum complement to $U$ in $V,$ and taking a basis of that complement. Clearly, the union of any $\!k\!$-basis of $U$ and any $\!k\!$-basis of $V$ relative to $U$ is a $\!k\!$-basis of $V.$
\begin{lemma}\label{L.V_1+V_2} Suppose $V_1,$ $V_2$ are subspaces of a vector space $W,$ and let $B_0$ be a basis of $V_1\cap V_2,$ $B_1$ a basis of $V_1$ relative $V_1\cap V_2,$ and $B_2$ a basis of $V_2$ relative $V_1\cap V_2.$
Then $B_0,$ $B_1,$ and $B_2$ are disjoint, and their union is a basis of $V_1+V_2.$ \textup{(}Hence $B_1$ is also a basis of $V_1+V_2$ relative to $V_2,$ and $B_2$ a basis of $V_1+V_2$ relative to $V_1.)$
Hence if, further, $B_3$ is a basis of $W$ relative to $V_1+V_2,$ then $B_0\cup B_1\cup B_2\cup B_3$ is a basis of $W.$ \end{lemma}
\begin{proof} The disjointness of $B_0,$ $B_1,$ and $B_2$ is immediate. The fact that $B_2$ is a basis of $V_2$ relative $V_1\cap V_2$ means that its image in $V_2/(V_1\cap V_2)$ is a basis thereof. But $V_2/(V_1\cap V_2)\cong (V_1+V_2)/V_1,$ so $B_2$ is also a basis of $V_1+V_2$ relative to $V_1,$ hence its union with the basis $B_0\cup B_1$ of $V_1$ is a basis of $V_1+V_2,$ giving the first assertion, and, in the process, the parenthetical note that follows it. The final assertion is then immediate. \end{proof}
\section{A normal form for $R\lang q\mid pqp=p\rang$ when $1\notin pR+Rp.$}\label{S.norm}
Here is the situation we will consider first:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1_notin_Rp+pR} In this section, $R$ will be a $\!k\!$-algebra, and $p$ a fixed element of $R$ such that $1\notin pR+Rp.$ (So in particular, $R$ is nonzero.) \end{minipage}\end{equation}
Under this assumption, I claim we can take a $\!k\!$-basis of $R$ of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.B=} $B\cup\{1\}\ =\ B_{++}\cup B_{+-} \cup B_{-+}\cup B_{--}\cup \{1\},$ \end{minipage}\end{equation}
where
\begin{equation}\begin{minipage}[c]{35pc}\label{d.B_} $B_{++}$ is any $\!k\!$-basis of $pR\cap Rp$ which, if $p\neq 0,$ contains $p,$
$B_{+-}$ is any $\!k\!$-basis of $pR$ relative to $pR\cap Rp,$
$B_{-+}$ is any $\!k\!$-basis of $Rp$ relative to $pR\cap Rp,$
$B_{--}$ is any $\!k\!$-basis of $R$ relative to $pR+Rp+k.$ \end{minipage}\end{equation}
(Mnemonic: a $+$ on the left signals left divisibility by $p,$ a $+$ on the right, right divisibility.)
Indeed, let $B_{++},$ $B_{+-},$ $B_{-+},$ $B_{--}$ be sets as in~\eqref{d.B_}. By Lemma~\ref{L.V_1+V_2}, $B_{++}\cup B_{+-} \cup B_{-+}$ will be a $\!k\!$-basis of $pR+Rp.$ By assumption, $1\notin pR+Rp,$ so $B_{++}\cup B_{+-} \cup B_{-+}\cup\{1\}$ is a $\!k\!$-basis of $pR+Rp+k.$ Hence bringing in the $\!k\!$-basis $B_{--}$ of $R$ relative to that subspace gives us a $\!k\!$-basis of $R.$
Below, we will typically denote an element of $B$ by a letter such as $x.$ However, when such an element is specified as belonging to $B_{++}\cup B_{+-}$ (respectively, to $B_{++}\cup B_{-+}),$ we shall often find it useful to write it in a form such as $px$ (respectively, $xp).$ Note that if $p$ is a zero-divisor in $R,$ the $x$ in such an expression will not be uniquely determined. We could assume one such representation fixed for each member of $B_{++}\cup B_{+-},$ but we shall not find this necessary; rather, the uses to which we shall put such expressions will not depend on the choice of $x.$ In particular, note that given elements $xp\in Rp$ and $py\in pR,$ the value of $xpy$ depends only on the elements $xp$ and $py,$ not on the choices of $x$ and $y.$ For if $xp=x'p$ and $py=py',$ then $xpy=x'py=x'py'.$
In the case of elements specified as belonging to $B_{++},$ we will often use three representations, $x=x'p=\nolinebreak px''.$
The construction of a normal form for $R\lang q\mid pqp=p\rang$ in this section, and of similar normal forms in subsequent sections, involves considerations both of {\em elements} of $\!k\!$-algebras, and of {\em expressions} for such elements. We shall tread the thin line between ambiguity and cumbersome notation by making
\begin{convention}\label{Cv.exprs} Throughout this note, when we consider a $\!k\!$-algebra $S$ generated by a set $G,$ an {\em expression} for an element $s\in S$ will mean an element of the free $\!k\!$-algebra $k\lang G\rang$ which maps to $s$ under the natural homomorphism $k\lang G\rang\to S.$ A {\em word} or {\em monomial} will mean a member of the free monoid generated by $G$ in $k\lang G\rang.$ Thus, in descriptions of reductions $W\mapsto f,$ the word $W$ and the expression $f$ are understood to lie in $k\lang G\rang.$
A family of words will be said to {\em give a $\!k\!$-basis for $S$} if the $\!k\!$-subspace of $k\lang G\rang$ spanned by that family maps bijectively to $S$ under the above natural homomorphism; in other words, if the family is mapped one-to-one into $S,$ and its image is a $\!k\!$-basis of $S.$
We shall use the same symbols for elements of $k\lang G\rang$ and their images in $S,$ distinguishing these by context: in descriptions of normal forms and reductions, our symbols will denote elements of $k\lang G\rang,$ while in statements that a {\em relation} holds in $S,$ they will denote elements of $S.$ \end{convention}
In the situation at hand, the outputs of our reductions for $R\lang q\mid pqp=p\rang$ will often have to be expressed in terms of the operations of $R.$ For this purpose, we make the notational convention that for any $\!k\!$-algebra expression $f$ for an element of $R,$ we shall write $f_R$ for the unique $\!k\!$-linear combination of elements of $B\cup\{1\}$ which gives the value of $f$ in $R.$
(Thus, when we come to reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->} below, the inputs will be words of lengths~$2$ and $3$ respectively, while the outputs, by this notational convention, are $\!k\!$-linear combinations of words of lengths~$\leq 1.)$
Note also that since the monomials that span the free algebra $k\lang B\cup\{q\}\rang$ include the empty word $1,$ and none of the reductions we will give has $1$ as its input, $1$ will belong to the $\!k\!$-basis described in the theorem.
We can now state and prove our normal form.
\begin{theorem}\label{T.1_notin} Let $R$ be a $\!k\!$-algebra, $p$ an element of $R$ such that $1\notin pR+Rp,$ $B\cup\{1\}$ a $\!k\!$-basis of $R$ as in~\eqref{d.B=} and~\eqref{d.B_} above, and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.R'} $R'\ =\ R\,\lang q\mid pqp=p\rang,$ \end{minipage}\end{equation}
the $\!k\!$-algebra gotten by adjoining to $R$ a universal inner inverse $q$ to $p.$
Then $R'$ has a $\!k\!$-basis given by the set of those words in the generating set $B\cup\{q\}$ that contain no subwords of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xy} $xy$ \quad with $x,y\in B$ \end{minipage}\end{equation}
nor
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqpy} $(xp)\,q\,(py)$ \quad with $xp\in B_{++}\cup B_{-+}$ and $py\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
The reduction to the above normal form may be accomplished by the systems of reductions
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xy|->} $xy\ \mapsto\ (xy)_R$ \quad for all $x,y\in B,$ \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqpy|->} $(xp)\,q\,(py)\ \mapsto\ (xpy)_R$ \quad for all $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
\end{theorem}
\begin{proof} Clearly, $R'$ is generated as a $\!k\!$-algebra by $B\cup\{q\},$ and we see that the relations
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xy=}
$xy\ =\ (xy)_R$ \quad for $x,y$ as in~\eqref{d.xy|->} \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqpy=}
$(xp)\,q\,(py)\ =\ (xpy)_R$ \quad for $xp,\ py$ as in~\eqref{d.xpqpy|->} \end{minipage}\end{equation}
do hold in $R'.$ Moreover, these relations are sufficient to define $R'$ in terms of our generators. Indeed the relations~\eqref{d.xy=} constitute a presentation of $R;$ to get the additional relation $pqp=p$ of~\eqref{d.R'}, note that if $p=0$ this is vacuous, while if $p\neq 0,$ it is the case of~\eqref{d.xpqpy=} where $xp=p=py.$
Since~\eqref{d.xy=} and~\eqref{d.xpqpy=} give a presentation of $R',$ the statement of the Diamond Lemma in \cite[Theorem~1.2]{<>} tells us that the reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->} will yield a normal form for $R'$ if, first of all, they satisfy an appropriate condition guaranteeing that repeated applications of these reductions to any expression eventually terminate, and if, moreover, every ``ambiguity'', in the sense of \cite{<>}, is ``resolvable''.
The first of these conditions is immediate, since each of our reductions replaces a word by a linear combination of shorter words; so the partial ordering on the set of all words which makes shorter words ``$<$'' longer words, and distinct words of equal length incomparable, is, in the language of \cite{<>}, a semigroup partial ordering that is compatible with our reduction system, and has descending chain condition.
To show that all ambiguities are resolvable, we note that there are four sorts of ambiguously reducible words (notation explained below):
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xyz} $x\cdot y\cdot z,$ \quad where $x,y,z\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqpyz} $(xp)\,q \cdot (py)\cdot z,$ \quad where $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-},$ $z\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xypqpz} $x\cdot (yp) \cdot q\,(pz),$ \quad where $x\in B,$ $yp\in B_{++}\cup B_{-+},$ $pz\in B_{++}\cup B_{+-},$\quad and \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqyqpz} $(xp)\,q \cdot y\cdot q\,(pz),$ \quad where $xp\in B_{++}\cup B_{-+},$ $y=py'=y''p\in B_{++},$ $pz\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
In each of these words, I have placed dots so as to indicate the two competing reductions applicable to the word in question, namely, the application of one of the reductions~\eqref{d.xy|->} or~\eqref{d.xpqpy|->} to the product of the two strings of generators surrounding the first dot, and the application of another such reduction to the product of the two strings surrounding the second dot. For example, in~\eqref{d.xpqpyz} we can either reduce
$(xp)\,q\,(py)$ using~\eqref{d.xpqpy|->}, or reduce
$(py)z$ using~\eqref{d.xy|->}.
In each case, each of our two competing reductions will, as noted, turn the indicated expression into a $\!k\!$-linear combination of shorter words. Most of these new words are in turn subject to a second reduction. (The exceptions are those that arise from an occurrence of the empty word, $1,$ in the output of the first reduction.) I claim that for each of \eqref{d.xyz}-\eqref{d.xpqyqpz}, after these reductions are complete, the two resulting expressions are equal; namely, that we get $(xyz)_R,$ $(xpyz)_R,$ $(xypz)_R$ and $(xyz)_R,$ respectively.
I will show this first, in detail, for the simplest case,~\eqref{d.xyz}, then in outline for the most complicated case,~\eqref{d.xpqyqpz}, then note briefly what happens in the intermediate cases~\eqref{d.xpqpyz} and~\eqref{d.xypqpz}.
In the case of~\eqref{d.xyz}, let
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xy_R} $(xy)_R\ =\ \sum_{u\in B\cup\{1\}} \alpha_u u$ $(\alpha_u\in k).$ \end{minipage}\end{equation}
Thus, the result of the ``left-hand'' reduction of $x\cdot y\cdot z$ is $\sum_{u\in B\cup\{1\}} \alpha_u u\,z.$ Now for $u=1,$ the empty string, we have $uz=z,$ which we can write $(uz)_R,$ while for all other $u,$ the string $uz$ can be reduced to
$(uz)_R$ by an application of~\eqref{d.xy|->}. Hence the expression $\sum \alpha_u u\,z$ can be reduced using~\eqref{d.xy|->} to $\sum \alpha_u (uz)_R=(\sum \alpha_u uz)_R,$ which by~\eqref{d.xy_R} equals $(xyz)_R,$ as claimed. By symmetry, the calculation beginning with the right-hand reduction of $x\cdot y\cdot z$ likewise yields $(xyz)_R,$ showing that, in the language of \cite{<>}, the ambiguity corresponding to~\eqref{d.xyz} is resolvable.
Let us now look at the case of~\eqref{d.xpqyqpz}, but without explicitly writing expressions $f_R$ as linear combinations of basis elements, merely understanding that they represent such linear combinations, and that the analogs of the reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->}
for such linear combinations can be achieved by applying~\eqref{d.xy|->}
or~\eqref{d.xpqpy|->} respectively to each word in the linear expression.
Writing $y$ in~\eqref{d.xpqyqpz} as $py',$ we see that the result of applying~\eqref{d.xpqpy|->} to $(xp)\,q\,(py')$ is $(xpy')_R,$ so the left-hand reduction of $(xp)\,q\,y\,q\,(pz)$ gives $(xpy')_R\,q\,(pz).$ Using now the fact that in~\eqref{d.xpqyqpz}, $py'=y''p,$ we can rewrite this as $(xy''p)_R\,q\,(pz).$ Since $xy''p$ is right-divisible by $p,$
$(xy''p)_R$ is a $\!k\!$-linear combination of elements of $B_{++}\cup B_{-+},$ so we can apply~\eqref{d.xpqpy|->} to each term of this expression, and get $(xy''pz)_R,$ in other words, $(xyz)_R.$ Again, by symmetry the calculation beginning with the right-hand reduction gives the same result.
The cases~\eqref{d.xpqpyz} and~\eqref{d.xypqpz} combine features of the above two. In the former, for instance, the reader is invited to verify that whether we begin with the reduction of $(xp)\,q\,(py)$ or of $(py)\,z,$ a following application of reductions of the other sort brings us to the common answer $(xpyz)_R.$ In this case, the two parts of the verification are not left-right dual to each other; rather, the verification of~\eqref{d.xpqpyz} is left-right dual to that of~\eqref{d.xypqpz}.
Since all our ambiguities are resolvable, \cite[Theorem~1.2]{<>} tell us that the words in $B$
which do not have as subwords any words appearing as inputs of reductions~\eqref{d.xy|->} or~\eqref{d.xpqpy|->} form a $\!k\!$-basis of $R',$ as claimed. \end{proof}
\section{A digression on algebras over non-fields.}\label{S.digress}
An immediate consequence of the above theorem is that $R$ can be embedded in a $\!k\!$-algebra in which $p$ has an inner inverse. However, this can be more easily seen from the fact that $R$ embeds in the algebra of all endomorphisms of its underlying $\!k\!$-vector-space, which is von~Neumann regular.
On the other hand, letting $K$ be a general commutative ring (so as not to violate our convention that $k$ denotes a field), a $\!K\!$-algebra $R$ with a specified element $p$ need not be embeddable in a $\!K\!$-algebra in which $p$ has an inner inverse. For instance, if $K$ is an integral domain, $p$ a nonzero nonunit of $K,$ and $R=K/(p^2),$ we see that in $R,$ the image of $p$ is nonzero, but if an inner inverse $q$ to $p$ is adjoined, then since $p\in K$ must remain central, we get $p=p\,q\,p=p^2q=0$ in $R'.$ The following result (which will not be used in the sequel) shows, inter alia, that for $K$ a general commutative ring, such problems occur if and only if $K$ is not itself von~Neumann regular.
\begin{proposition}\label{P.KvN} For $K$ a commutative ring, the following conditions are equivalent.\\[.5em]
\textup{(a)} \ $K$ is von~Neumann regular.\\[.5em]
\textup{(b)} \ Every $\!K\!$-algebra $R$ can be embedded in a von~Neumann regular $\!K\!$-algebra.\\[.5em]
\textup{(c)} \ For every ideal $I$ of $K$ and element $p\in K/I,$ the $\!K\!$-algebra $K/I$ can be embedded in a $\!K\!$-algebra in which $p$ has an inner inverse.\\[.5em]
\textup{(d)} \ For every $\!K\!$-module $M$ and nonzero $x\in M,$ one has $x\notin PM$ for some maximal ideal $P$ of $K.$ \end{proposition}
\begin{proof} We shall show that (a)$\implies$(d)$\implies$(b)$\implies$(c)$\implies$(a).
(a)$\implies$(d): Given $M$ and $x$ as in (d), let $P$ be maximal among proper ideals of $K$ containing the annihilator of $x,$ and suppose by way of contradiction that $x\in PM,$ so that $x=\sum_1^{n} a_i x_i$ with $a_i\in P,$ $x_i\in M.$ The ideal of $K$ generated by the $a_i$ will be generated by an idempotent $e,$ since $K$ is von~Neumann regular \cite[Theorem~1.1(a)$\!\implies\!$(b)]{KG}, so $e\in P,$ and since each $a_i$ lies in $eK,$ we have $ex=x.$ This says that $(1-e)x=0,$ so $1-e\in P$ (since $P$ contains the annihilator of $x),$ so $1=e+(1-e)\in P,$ contradicting the assumption that $P$ is proper.
(d)$\implies$(b): Assuming (d), we shall show that for every nonzero $x\in R,$ there is a homomorphism from $R$ to a von~Neumann regular $\!K\!$-algebra which does not annihilate $x.$ Hence $R$ embeds in a direct product of such algebras, which will itself be von~Neumann regular.
Given $x\in R-\{0\},$ if we regard $R$ as a $\!K\!$-module,~(d) says that $x\notin PR$ for some maximal ideal $P$ of $K.$ Regarding $K/P$ as a field, this tells us that $x$ has nonzero image in the $\!K/P\!$-algebra $R/PR.$ And as noted at the beginning of this section, every algebra over a field $k$ embeds in a von~Neumann regular $\!k\!$-algebra.
(b)$\implies$(c): Apply (b) with $R=K/I.$
(c)$\implies$(a): Take any $p\in K,$ and apply (c) with $I=p^2 K,$ and with the image $\overline{p}$ of $p$ in $K/I$ in the role of $p.$ This gives us a $\!K\!$-algebra containing $K/I$ in which $\overline{p}$ has an inner inverse $\overline{q},$ and we compute $\overline{p}= \overline{p}\,\overline{q}\,\overline{p}= \overline{p}^2\,\overline{q}= 0\,\overline{q}=0.$ Thus $p\in I=p^2 K,$ i.e., in $K$ we can write $p=p^2 q,$ and since $K$ is commutative, this equals $pqp.$ Thus every $p\in K$ has an inner inverse, so $K$ is von~Neumann regular. \end{proof}
\section{$R'\!$-modules}\label{S.modules}
Returning to the situation of $R$ a $\!k\!$-algebra, and $p\in R$ with $1\notin pR+Rp,$ for which we have described the extension $R'=R\lang q\mid p=pqp\rang,$ we now want to describe the $\!R'\!$-module $M\otimes_R R'$ for an arbitrary right $\!R\!$-module $M,$ and examine such questions as whether an inclusion of $\!R\!$-modules $M\subseteq N$ induces an embedding of $M\otimes_R R'$ in $N\otimes_R R'.$
Our normal form for $R'$ will generalize easily to a normal form for $M\otimes_R R',$ but we shall find that an inclusion of $\!R\!$-modules does not necessarily induce an embedding of $\!R'\!$-modules. The reason is that the relation $p=pqp$ in $R'$ makes $1-qp$ right annihilate $p,$ hence $1-qp$ also annihilates all elements of the form $xp$ in any right $\!R'\!$-module. We shall in fact see that in $M\otimes_R R',$ the set of elements of $M$ annihilated by $1-qp$ is precisely $Mp.$ Hence if $M$ is a submodule of an $\!R\!$-module $N,$ and there is an element $y\in M$ which is not a multiple of $p$ in $M,$ but becomes one in $N,$ then the map of $\!R'\!$-modules induced by the inclusion $M\subseteq N$ will kill the nonzero element $y(1-qp).$
However, we shall find that we can describe the structure of the $\!R'\!$-submodule of $N\otimes_R R'$ generated by $M$ wholly in terms of the $\!R\!$-module structure of $M,$ and the set of elements of $M$ which become multiples of $p$ in $N.$ Let us set up language and notation to handle this. In the next definition, we do not assume $1\notin pR+Rp,$ since we will be calling on it again in sections where that assumption does not apply.
\begin{definition}\label{D.tempered} Let $k$ be a field, $R$ a $\!k\!$-algebra, and $p$ an element of $R.$
By a {\em $\!p\!$-tempered} right $\!R\!$-module, we shall mean a pair $(M,M_+)$ where $M$ is a right $\!R\!$-module, and $M_+$ is any $\!k\!$-vector-subspace of $M$ which contains the subspace $Mp,$ is annihilated by the right annihilator of $p$ in $R,$ and is closed under multiplication by the subring $\{x\in R\mid px\in Rp\}\subseteq R.$
A {\em morphism} of $\!p\!$-tempered right $\!R\!$-modules $h:(M,M_+)\to (N,N_+)$ will mean an $\!R\!$-module homomorphism $h:M\to N$ such that $h(M_+)\subseteq N_+.$ Such a morphism will be called an {\em embedding} of $\!p\!$-tempered right $\!R\!$-modules if it is one-to-one, and satisfies $M_+=h^{-1}(N_+).$
Finally, let $R'=R\lang q\mid p=pqp\rang.$ Then for any $\!p\!$-tempered $\!R\!$-module $(M,M_+),$ we shall denote by $(M,M_+)\otimes_{(R,p)} R'$ the quotient of $M\otimes_R R'$ by the submodule generated by all elements
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xqp-x} $xqp-x$ \quad for $x\in M_+.$ \end{minipage}\end{equation}
\end{definition}
Observe that if $M_+=Mp,$ then $(M,M_+)\otimes_{(R,p)} R'$ is simply $M\otimes_R R'.$
For $B\cup\{1\}$ a $\!k\!$-basis of $R,$ and $f$ an expression representing an element of $R,$ we shall continue to write $f_R$ for the $\!k\!$-linear expression in elements of $B\cup\{1\}$ that gives the value of $f.$ Likewise, if we are given a $\!k\!$-basis $C$ of $M,$ then for any expression $f$ representing an element of $M$ \textup{(}for example, any $\!k\!$-linear combination of words each given by an element of $C$ followed by a \textup{(}possibly empty\textup{)} string of elements of $B),$ we shall write $f_M$ for the $\!k\!$-linear expression in elements of $C$ giving the value of $f$ in $M.$
Using the version of the Diamond Lemma for modules in \cite[\S9.5]{<>}, let us now prove
\begin{proposition}\label{P.M_norm} Let $k,$ $R,$ $p,$ $B,$ and $R'=R\lang q\mid p=pqp\rang$ be as in Theorem~\ref{T.1_notin}. Let $(M,M_+)$ be a $\!p\!$-tempered right $\!R\!$-module, let $C_+$ be a $\!k\!$-basis of $M_+,$ and let $C_-$ be a $\!k\!$-basis of $M$ relative to $M_+,$ so that $C=C_+\cup C_-$ is a $\!k\!$-basis of $M.$
Then $(M,M_+)\otimes_{(R,p)} R'$ has $\!k\!$-basis given by all words $w$ that are composed of an element of $C$ followed by a \textup{(}possibly empty\textup{)} string of elements of $B\cup\{q\},$ such that $w$ contains no subwords~\eqref{d.xy} or~\eqref{d.xpqpy} as in Theorem~\ref{T.1_notin}, nor any subwords
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xy} $xy$ \quad with $x\in C$ and $y\in B$ \end{minipage}\end{equation}
or
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xqpy} $x\,q\,(py)$ \quad with $x\in C_+$ and $py\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
The reduction to the above normal form may be accomplished by the system of reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->} given in Theorem~\ref{T.1_notin}, together with
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xy|->} $xy\ \mapsto\ (xy)_M$ \quad for $x\in C,$ $y\in B$ \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xqpy|->} $x\,q\,(py)\ \mapsto\ (xy)_M$ \quad for $x\in C_+,$ $py\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
\end{proposition}
\begin{proof}[Sketch of proof]
Let us first observe that in~\eqref{d.M:xqpy|->}, though the basis-element $py$ may not uniquely determine $y,$ the element $(xy)_M$ is nonetheless well-defined, since if $py$ can also be written $py',$ then $y$ and $y'$ differ by a member of the right annihilator of $p,$ so by the definition of $\!p\!$-tempered $\!R\!$-module, their difference annihilates $x\in M_+.$
The relations corresponding to the reductions~\eqref{d.xy|->}, \eqref{d.xpqpy|->},
\eqref{d.M:xy|->} and~\eqref{d.M:xqpy|->} all hold in $(M,M_+)\otimes_{(R,p)} R'.$
Indeed, those corresponding to applications of~\eqref{d.xy|->} and~\eqref{d.xpqpy|->}
hold by the structure of $R';$ those corresponding to \eqref{d.M:xy|->} by the
$\!R\!$-module structure of $M,$ and those corresponding to~\eqref{d.M:xqpy|->} because in defining $(M,M_+)\otimes_{(R,p)} R',$ we have divided out by the submodule generated by all elements~\eqref{d.M:xqp-x}. And in fact, we see that the relations corresponding to these reductions constitute a presentation of the $\!R'\!$-module $(M,M_+)\otimes_{(R,p)} R'.$ As before, our reductions decrease the lengths of words, so if all ambiguities of our reduction system are resolvable, it will yield a normal form for the $\!R'\!$-module $(M,M_+)\otimes_{(R,p)} R'.$
The ambiguities are of two sorts: the four given by~\eqref{d.xyz}-\eqref{d.xpqyqpz}, which are resolvable by Theorem~\ref{T.1_notin}, and the four analogous ones in which the leftmost factor comes from $C$ rather than $B:$
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xyz} $x\cdot y\cdot z,$ where $x\in C,$ $y,z\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xqpyz} $x\,q \cdot (py)\cdot z,$ where $x\in C_+,$ $py\in B_{++}\cup B_{+-},$ $z\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xypqpz} $x\cdot (yp) \cdot q\,(pz),$ where $x\in C,$ $yp\in B_{++}\cup B_{-+},$ $pz\in B_{++}\cup B_{+-},$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M:xqyqpz} $x\,q \cdot y\cdot q\,(pz),$ where $x\in C_+,$ $y=py'=y''p\in B_{++},$ $pz\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
I claim~\eqref{d.M:xyz}-\eqref{d.M:xqyqpz} are resolvable by computations analogous to those we used for~\eqref{d.xyz}-\eqref{d.xpqyqpz}, the common forms to which the results of the two possible reductions lead now being $(xyz)_M$ for~\eqref{d.M:xyz} and~\eqref{d.M:xqpyz}, $(xypz)_M$ for~\eqref{d.M:xypqpz}, and $(xy'z)_M$ for~\eqref{d.M:xqyqpz}. Let us sketch the verifications.
The resolvability of~\eqref{d.M:xyz} follows from the fact that $M$ is an $\!R\!$-module.
The case of \eqref{d.M:xqpyz} is like that of~\eqref{d.xpqpyz}, the one difference being that where there we wrote the leftmost basis element as $xp,$
here it is a general element $x\in M_+;$ but in either case, our reduction~\eqref{d.M:xqpy|->} allows us (roughly speaking) to drop a following ``$qp$''.
In~\eqref{d.M:xypqpz}, if we begin by reducing $x\cdot (yp)$
using~\eqref{d.M:xy|->}, that product becomes $(xyp)_M,$ the representation of a member of $Mp\subseteq M_+,$ hence its expression in terms of $C$ involves only members of $C_+.$
Hence by~\eqref{d.M:xqpy|->}, when we multiply it by $q\,(pz),$ each of the resulting products reduces to the value we would have gotten if we had simply multiplied by $z,$ so the result is indeed $(xypz)_M.$ If instead we first reduce $(yp)\,q\cdot (pz)$ to $(ypz)_R$
using~\eqref{d.xpqpy|->}, then multiply $x$ by this, applying~\eqref{d.M:xy|->} to each term occurring, we get the same result $(xypz)_M.$
The calculation for~\eqref{d.M:xqyqpz} combines the features of the two preceding cases. The reduction of $xq\cdot y$ works as in the case of~\eqref{d.M:xqpyz} once we rewrite $y$ as $py',$ and gives $(xy')_M.$ Moreover, because $py'=y''p,$ the subspace $M_+\subseteq M$ is carried into itself by multiplication by $y'$ (see end of second paragraph of Definition~\ref{D.tempered}), so $xy'\in M_+;$ hence multiplication of $(xy')_M$ by $q\,(pz)$ is the same as multiplication by $z,$ and gives $(xy'z)_M.$ On the other hand, if we begin by reducing $y\cdot q\,(pz)=(y''p)\,q\,(pz)$
to $(y''pz)_R=(py'z)_R,$ then the result of multiplying $x\,q$ by this is again $(xy'z)_M,$ by application of~\eqref{d.M:xqpy|->} to each term occurring. \end{proof}
Here are some easy consequences.
\begin{corollary}\label{C.M->MOX} For $R,$ $p,$ $R'$ and $(M,M_+)$ as in Proposition~\ref{P.M_norm}, the canonical $\!R\!$-module homomorphism $M\to(M,M_+)\otimes_{(R,p)} R'$ is an embedding; and identifying $M$ with its image under this map, we have
\begin{equation}\begin{minipage}[c]{35pc}\label{d.M+=} $M_+\ =\ M\ \cap\ ((M,M_+)\otimes_{(R,p)} R')\,p\ = \ \{x\in M\mid x(qp-1)=0$ in $(M,M_+)\otimes_{(R,p)} R'\}.$ \end{minipage}\end{equation}
In particular, for any $\!p\!$-tempered $\!R\!$-module $(M,M_+),$ the module $M$ can be embedded in an $\!R\!$-module $N$ so that $M_+=M\cap Np.$ \end{corollary}
\begin{proof} The map $M\to(M,M_+)\otimes_{(R,p)} R'$ takes elements of the $\!k\!$-basis $C$ of $M$ to themselves as elements of the $\!k\!$-basis of $(M,M_+)\otimes_{(R,p)} R'$ described in Proposition~\ref{P.M_norm}; hence it is one-to-one.
In~\eqref{d.M+=}, it is easy to see that the leftmost and rightmost subspaces are equal, since for a $\!k\!$-linear combination $x$ of the elements of $C,$ the reduction rules reduce $xqp$ to $x$ if and only if all the basis elements occurring in $x$ belong to $C_+,$ i.e., if and only if $x\in M_+.$ To see the equality of the middle and rightmost subspaces, note that in any right $\!R'\!$-module, and so in particular, in $(M,M_+)\otimes_{(R,p)} R',$ every right multiple of $p$ is annihilated by $qp-1,$ and conversely, any element $x$ satisfying $x(qp-1)=0$ satisfies $x=xqp,$ and so is a right multiple of~$p.$
The final statement is seen on taking $N=(M,M_+)\otimes_{(R,p)} R',$ regarded as an $\!R\!$-module. \end{proof}
\begin{corollary}\label{C.MinN} Let $R,$ $p$ and $R'$ be as in Proposition~\ref{P.M_norm}, and let $h:(M,M_+)\to(N,N_+)$ be a morphism of $\!p\!$-tempered $\!R\!$-modules. Then the induced homomorphism of $\!R'\!$-modules $h\otimes_{(R,p)}R': (M,M_+)\otimes_{(R,p)} R'\to (N,N_+)\otimes_{(R,p)} R'$ is one-to-one if and only if $h$ is an embedding of $\!p\!$-tempered $\!R\!$-modules in the sense of Definition~\ref{D.tempered}. \end{corollary}
\begin{proof} Suppose $h$ is an embedding of $\!p\!$-tempered $\!R\!$-modules. Then without loss of generality, we can assume that $M$ is a submodule of $N,$ and $M_+=M\cap N_+.$ Let us take a $\!k\!$-basis $C^{(0)}_+\cup C^{(0)}_-$ of $M$ as in the statement of Proposition~\ref{P.M_norm}, and extend $C^{(0)}_+$ to a $\!k\!$-basis $C^{(0)}_+\cup C^{(1)}_+$ of $N_+.$ By Lemma~\ref{L.V_1+V_2}, $C^{(0)}_+\cup C^{(0)}_- \cup C^{(1)}_+$ is a $\!k\!$-basis of $M+N_+,$ and we can extend this to a $\!k\!$-basis $C^{(0)}_+\cup C^{(0)}_- \cup C^{(1)}_+\cup C^{(1)}_-$ of $N.$ If we now write this basis as $(C^{(0)}_+\cup C^{(1)}_+)\cup(C^{(0)}_-\cup C^{(1)}_-)$ and use it to define a normal form in $(N,N_+)\otimes_{(R,p)} R',$ we see that $(M,M_+)\otimes_{(R,p)} R'$ forms a submodule thereof; so the induced homomorphism is one-to-one.
Conversely, if that induced homomorphism is one-to-one, then restricting it to the embedded copies of $M$ and $N$ in those modules, we see that $h$ is one-to-one. Moreover, the elements of $M$ that are annihilated by $qp-1$ in $(M,M_+)\otimes_{(R,p)} R'$ will be those whose images are annihilated by that element in $(N,N_+)\otimes_{(R,p)} R',$ i.e., $M_+=h^{-1}(N_+).$ Thus the homomorphism is indeed an embedding of $\!p\!$-tempered $\!R\!$-modules. \end{proof}
\section{Do we need to go beyond the case $1\notin pR+Rp$?}\label{S.enough?}
Above, we have studied the properties of $R'=R\lang q\mid p=pqp\rang$ when $1\notin pR+Rp.$ In the next five sections we examine cases where $1\in pR+Rp.$ But it may well be that for attacking the problem of whether the monoid of finitely generated projectives of a von~Neumann regular $\!k\!$-algebra is always separative, the case considered above is all that matters.
Indeed, Pere Ara (personal communication) notes that the separativity question for unital von~Neumann regular algebras is equivalent to the same question for nonunital von~Neumann regular algebras. For if $R$ were a unital example with non-separative monoid, then regarding it as a nonunital algebra, its (slightly larger) monoid of projectives would still have that property; while conversely, if we had a nonunital example $R,$ then the algebra $R^1$ gotten by adjoining a unit to $R$ would be a unital example. Note, moreover, that if $R$ is any nonunital $\!k\!$-algebra, then the process of adjoining a universal inner inverse to an element $p\in R$ can be carried out by passing to $R^1,$ universally adjoining an inner inverse to $p$ in $R^1$ as a unital algebra, then dropping the adjoined unit (i.e., passing to the nonunital subalgebra generated by $R\cup\{q\}).$ In this construction, $1\notin pR^1 + R^1p,$ since $pR^1 + R^1p\subseteq R;$ hence the construction of universally adjoining an inner inverse to $p$ in $R^1$ falls under the case considered in the preceding sections.
Kevin O'Meara (personal communication) has likewise suggested that the study of the separativity question can be reduced to the case $1\notin pR+Rp.$
So the reader mainly interested in tackling that question using our normal form may wish to skip or skim~\S\S\ref{S.1-sided}-\S\ref{S.Weyl}.
However, the cases considered in those sections seem interesting; especially the case $1\in pR+Rp-(pR\cup Rp),$ where the elaborate complexity of the normal form we shall discover suggests some strange territory to be explored; so we include them.
Let us first get the easy case out of the way.
\section{Normal forms when $1\in pR$ and/or $1\in Rp.$}\label{S.1-sided}
Since the two cases $1\in pR$ and $1\in Rp$ are left-right dual, let us assume without loss of generality that $1\in pR.$ This says $p$ has a right inverse; let us fix such a right inverse $q_0\in R.$ It will clearly be an inner inverse to $p,$ so our motivation for adjoining a universal inner inverse (to move our ring a step toward being von~Neumann regular) is not relevant here; but for the sake of our general understanding of the adjunction of inner inverses, we are including this case.
If $q$ is any other inner inverse to $p,$ then right multiplying the relation $pqp=p$ by $q_0,$ we get $pq=1;$ in other words, once $p$ has a right inverse, every inner inverse to $p$ is a right inverse. Moreover, subtracting the equations $pq_0=1$ and $pq=1,$ we get $p(q_0-q)=0;$ so all right inverses to $p$ are obtained by adding to $q_0$ arbitrary elements that right annihilate~$p.$
Note that if both $1\in pR$ and $1\in Rp$ hold, then $p$ will be invertible, and if we adjoin a universal inner inverse, it will have to be an inverse to $p,$ hence will fall together with the existing inverse; so in that case, the adjunction of a universal inner inverse to $p$ leaves $R$ unchanged. Hence we will assume below that $1\in pR$ but $1\notin Rp.$ (In particular, $1\neq 0,$ equivalently, $R\neq\{0\}.)$
Since $1\in pR,$ we have $pR=R,$ so the analog of the sort of basis of $R$ that we used in the preceding sections becomes simpler. Namely, we take a basis
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:B=} $B\cup\{1\}\ =\ B_{++}\cup B_{+-}\cup \{1\},$ \end{minipage}\end{equation}
where
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:B_} $B_{++}$ is any $\!k\!$-basis of $Rp=pRp$ containing $p,$
$B_{+-}$ is any $\!k\!$-basis of $R=pR$ relative to $Rp+k.$ \end{minipage}\end{equation}
Our extension
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:R'} $R'\ =\ R\lang q\mid pqp=p\rang\ =\ R\lang q\mid pq=1\rang$ \end{minipage}\end{equation}
is clearly spanned by words in $B\cup\{q\}$ which contain no subwords either of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xy} $xy$ \quad with $x,y\in B$ \end{minipage}\end{equation}
or of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xpq} $(xp)\,q$ \quad with $xp\in B_{++},$ \end{minipage}\end{equation}
and any expression in our generators can be reduced to a linear combination of such words via the system of reductions
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xy|->} $xy\ \mapsto\ (xy)_R$ \quad for all $x,y\in B$ \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xpq|->} $(xp)\,q\ \mapsto\ x_R$ \quad for all $xp\in B_{++}\,.$ \end{minipage}\end{equation}
In contrast to the situation of the preceding sections, the element $xp$ of~\eqref{d.RI:xpq|->} {\em does} determine $x:$ if $q_0\in R$ is a right inverse to $p,$ we see that $x=(xp)\,q_0;$ so the expression $x_R$
in~\eqref{d.RI:xpq|->} is well-defined.
We find that the only ambiguities of this reduction system are
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xyz} $x\cdot y\cdot z,$ where $x,y,z\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.RI:xypq} $x\cdot (yp) \cdot q,$ where $x\in B,$ $yp\in B_{++},$ \end{minipage}\end{equation}
and it is straightforward to verify, by the approach used in~\S\ref{S.norm}, that these are both resolvable.
Note that in the resulting normal form, elements of $B_{++}$ can appear nowhere but in the last position in a reduced word.
(We would get the same ring $R'$ if we adjoined to $R$ an element $u$ subject to the relation $pu=0;$ that extension is isomorphic to the one constructed above via the identification of $q$ with $q_0+u.$ The construction using $u$ would be simpler to study on its own, but the construction using $q$ lends itself better to comparison with the other cases.)
We can likewise look at extension of scalars from $\!R\!$-modules to $\!R'\!$-modules. Since our assumption that $p$ has a right inverse is not left-right symmetric, right and left modules need to be considered separately.
If $M$ is a right $\!R\!$-module, we take, as in the preceding section, a $\!k\!$-basis $C_+\cup C_-$ for $M,$ where $C_+$ is a $\!k\!$-basis for $Mp.$ In this situation, we do not have to think about a more general $\!k\!$-subspace $M_+,$ consisting of elements that might become the multiples of $p$ in an overmodule, because the upper and lower bounds for such an $M_+$ given in Definition~\ref{D.tempered} coincide: $x$ is a multiple of $p$ in $M$ if and only if $x=x q_0 p,$ i.e., if and only if $x$ is annihilated by the element $1-q_0 p$ of the right annihilator of $p$ in $R.$
It is straightforward to verify that $M\otimes_R R'$ is spanned by words in $C\cup B\cup \{q\}$
in which elements of $C$ occur in the leftmost position and only there, and which are irreducible under the reductions~\eqref{d.RI:xy|->}
and~\eqref{d.RI:xpq|->}, and also the corresponding reductions in which the leftmost element of $B,$ respectively $B_{++},$ is replaced by an element of $C,$ respectively $C_+.$ In this case, we see that if the leftmost factor of a reduced word is in $C_+,$ then that factor is the whole word.
Turning to left $\!R\!$-modules $M,$ we find that we do not have to distinguish a subspace $pM$ or $M_+$ at all, since $pM=M.$ Again, we get reduced words having the same formal descriptions as for reduced words of $R'.$ In this case, the analog of the fact that elements of $B_{++}$ and $C_+$ can only appear in final position is that elements of $B_{++}$ never appear. (Whatever such an element might be followed by -- a member of $B,$ a $q,$ or a member of $C$ -- leads to a reducible word.)
Returning to our development of the structure of $R'=R\lang q\mid pqp=p\rang,$ we remark that the uninteresting case that we referred to briefly at the start of this section and then put aside, where $p$ is invertible in $R,$ so that $R'=R,$ is the one case where the subalgebra of $R'$ generated by $q$ may fail to be a polynomial ring $k[q].$ Rather, it will, necessarily, fall together with $k[p^{-1}]\subseteq R,$ which, if $p$ is algebraic over~$k,$ is the finite-dimensional subalgebra of $R$ generated by $p.$
\section{The case where $1\in pR+Rp - (pR\cup Rp):$ groping toward a normal form.}\label{S.1=}
We now consider the most difficult case, that in which $1\in pR+Rp,$ but where $1$ does not lie in $pR$ or $Rp.$ In this section we illustrate the process of trying to find a normal form, discovering more and more reductions as we go. In the next section, we shall make precise the pattern that these show, and prove that the resulting set of reductions does lead to a normal form for $R'.$
Let us begin with a general observation, and a slight digression.
In any ring $R$ with an element $p$ that has an inner inverse $q,$ so that
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pqp} $pqp\ =\ p,$ \end{minipage}\end{equation}
note that $p$ is left-annihilated by $pq-1$ and right-annihilated by $qp-1.$ Consequently,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pq-1*qp-1} $(pq-1)(pR+Rp)(qp-1)\ =\ \{0\}.$ \end{minipage}\end{equation}
Hence in the situation we are now interested in, where $1\in pR+Rp,$ we get $(pq-1)1(qp-1)\ =\ 0,$ i.e.,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pqqp} $pqqp\ =\ pq + qp - 1.$ \end{minipage}\end{equation}
In the $\!k\!$-algebra $R$ presented simply by two generators $p$ and $q$ and the relations~\eqref{d.1=:pqp} and~\eqref{d.1=:pqqp}, we find that the reduction system
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pqp|->} $pqp\ \mapsto\ p,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pqqp|->} $pqqp\ \mapsto\ pq+qp-1$ \end{minipage}\end{equation}
satisfies the conditions of the Diamond Lemma: there are just four ambiguities, corresponding to the words
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:pq_only} $pq\cdot p\cdot qp,$ \quad $pq\cdot p\cdot qqp,$ \quad $pqq\cdot p\cdot qp,$ \quad $pqq\cdot p\cdot qqp,$ \end{minipage}\end{equation}
and straightforward computations show that these are all resolvable. So this algebra has a normal form with basis the set of all strings of $\!p\!$'s and $\!q\!$'s that contain no substrings $pqp$ or $pqqp.$ Curiously, this algebra itself satisfies $1\in pR+Rp,$ by~\eqref{d.1=:pqqp}. Consequently, it is universal among $\!k\!$-algebras $R$ given with elements $p$ and $q$ satisfying~\eqref{d.1=:pqp} and such that $1\in pR+Rp.$ (It is not, however, universal among $\!k\!$-algebras given with elements $p$ and $q$ satisfying~\eqref{d.1=:pqp} together with specified elements $s$ and $t$ such that $1=ps+tp,$ i.e., it is not $k\lang p, q, s, t\mid p=pqp,\,1=ps+tp\rang,$ the universal example one would first think of.)
The above algebra might be worthy of study, but it is not one of the algebras we are preparing to investigate here. Those are the algebras $R'$ gotten by starting with a $\!k\!$-algebra $R$ given with an element $p$ such that
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=_but} $1\in pR+Rp$ but $1\notin pR,$ $1\notin Rp,$ \end{minipage}\end{equation}
and adjoining a universal inner inverse $q$ to $p.$
As in the preceding sections, we shall start by taking an appropriate $\!k\!$-basis of $R.$ A problem is that since $1\in pR+Rp,$ we can't take a $\!k\!$-basis containing sets $B_{++},$ $B_{+-}$ and $B_{-+}$ as in~\eqref{d.B=} and~\eqref{d.B_}, and {\em also} the unit $1$ (which we want to represent by the empty word in our normal form for $R').$ What we shall do instead is choose a spanning set for $R$
rather like that of~\eqref{d.B=} and~\eqref{d.B_}, but which is not quite $\!k\!$-linearly independent, then handle the one linear relation it satisfies as an extra reduction,~\eqref{d.1=:ps|->} below.
(Naively we might, instead, think of using a normal form for $R'$ in which $1$ is not represented by the empty monomial, but by the sum of a basis element from $pR$ and a basis element from $Rp.$ However, the version of the Diamond Lemma we are using requires that we regard $1$ as the empty monomial in our generators, so that won't work. It {\em would} work if we use the version of the Diamond Lemma for nonunital rings. But then, to restore unitality, we would have to throw in reductions that force our new generator $q$ to be fixed under left and right multiplication by the sum-of-generators that gives the $1$ of $R,$ and this seems messier than the path we shall follow.)
So given $R$ satisfying~\eqref{d.1=_but}, let us fix elements $s,$ $t\in R$ such that
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=} $1\ =\ ps+tp$ in $R,$ \end{minipage}\end{equation}
and choose a spanning set for $R$ as a $\!k\!$-vector-space, of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:B=} $B\cup\{1\}\ =\ B_{++}\cup B_{+-} \cup B_{-+}\cup B_{--}\cup \{1\},$ \end{minipage}\end{equation}
where
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:B_} $B_{++}$ is any $\!k\!$-basis of $pR\cap Rp$ containing the element $p,$
$B_{+-}$ is any $\!k\!$-basis of $pR$ relative to $pR\cap Rp$ containing the element $ps,$
$B_{-+}$ is any $\!k\!$-basis of $Rp$ relative to $pR\cap Rp$ containing the element $tp,$
$B_{--}$ is any $\!k\!$-basis of $R$ relative to $pR+Rp.$ \end{minipage}\end{equation}
Note that the condition above that $B_{+-}$ contain $ps$ can be achieved because $ps$ does not lie in $pR\cap Rp;$ if it did,~\eqref{d.1=} would imply $1\in Rp,$ contrary to our assumptions. By the dual observation, the condition that $B_{-+}$ contain $tp$ can also be achieved. Since $pR+Rp$ contains $1,$ we don't have to throw a ``$+k$'' onto the $pR+Rp$ in the description of $B_{--}$ as in~\eqref{d.B_}. Thus the above $B$ will be a $\!k\!$-basis of $R$ by Lemma~\ref{L.V_1+V_2}. But this means that $B\cup\{1\}$ will not. Rather, by~\eqref{d.1=}, $B\cup\{1\}-\{ps\}$ will be a $\!k\!$-basis of $R.$
For any $\!k\!$-algebra expression $f$ in the elements of $B,$ let $f_R$ denote the unique $\!k\!$-linear combination of elements of $B\cup\{1\}-\{ps\}$ representing the value of $f$ in $R.$ Then we see that $R$ can be presented using the generating set $B,$ the relations corresponding to the reductions
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xy|->} $xy\ \mapsto\ (xy)_R$ \quad for all $x,y\in B,$ \end{minipage}\end{equation}
and the relation corresponding to the single additional reduction
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:ps|->} $(ps)\ \mapsto\ 1-(tp).$ \end{minipage}\end{equation}
We now construct $R'$ by adjoining an additional generator $q,$ and imposing the relation $pqp=p.$ As in \S\ref{S.norm}, this leads to the further reductions
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xpqpy|->} $(xp)\,q\,(py)\ \mapsto\ (xpy)_R$ \quad for all $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-}\,.$ \end{minipage}\end{equation}
But in view of~\eqref{d.1=:pqqp}, these reductions cannot be sufficient to give a normal form for $R',$ so they must have non-resolvable ambiguities.
And indeed, note that for any $xp\in B_{++}\cup B_{-+},$ the word $(xp)\,q\,(ps)$
is ambiguously reducible, using~\eqref{d.1=:xpqpy|->} on the one hand or~\eqref{d.1=:ps|->} on the other. Equating the results gives the relation $(xps)_R = (xp)\,q-(xp)\,q\,(tp).$ Regarding this as a formula for reducing the longest monomial that it involves, $(xp)\,q\,(tp),$ we get a new family of reductions,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xpqtp|->} $(xp)\,q\,(tp)\ \mapsto\ (xp)\,q-(xps)_R$\quad for all $xp\in B_{++}\cup B_{-+}\,.$ \end{minipage}\end{equation}
These, in turn, lead to an ambiguity in the reduction of any word $(xp)\,q\cdot(tp)\cdot q\,(py):$
we can apply~\eqref{d.1=:xpqtp|->}, getting $(xp)\,qq\,(py)-(xps)_R\,q\,(py),$
or~\eqref{d.1=:xpqpy|->}, getting $(xp)\,q\,(tpy)_R.$ So let us again make the relation equating these expressions into a reduction affecting the longest word occurring, which is now $(xp)\,qq\,(py).$ With a view to what is to come, I will number this
\begin{equation}\tag{\xppy{2}}\begin{minipage}[c]{35pc} $(xp)\,qq\,(py)\ \mapsto\ (xps)_R\,q\,(py)+(xp)\,q\,(tpy)_R$\quad for all $xp\in B_{++}\cap B_{-+}$ and $py\in B_{++}\cap B_{+-}\,.$ \end{minipage}\end{equation}
(Digression: If we rewrite the factor $(xps)_R$ in the first term of the output of the above reduction as $(x(1\,{-}\,tp))_R=x_R-(xtp)_R,$ and inversely, rewrite the factor $(tpy)_R$ at the end of the last term as $((1\,{-}\,ps)y)_R=y_R-(psy)_R,$ then the resulting terms of~(\xppy{2}) include
$(xtp)_R\,q\,(py)$ and $(xp)\,q\,(psy)_R,$ which by~\eqref{d.1=:xpqpy|->} reduce to $(xtpy)_R$ and $(xpsy)_R,$ which then sum to $(x(ps+tp)y)_R=(xy)_R.$ This turns~(\xppy{2}) into
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xpqqpyalt} $(xp)\,qq\,(py)\ \mapsto\ x_R\,q\,(py)+(xp)\,q\,y_R-(xy)_R.$ \end{minipage}\end{equation}
This can be seen as embodying~\eqref{d.1=:pqqp}; it represents the result of multiplying that equation on the left by $x$ and on the right by $y.$ The form~\eqref{d.1=:xpqqpyalt} has the nice feature of not depending on the choice of $s$ and $t$ in~\eqref{d.1=}, but it has the downside that it involves expressions $x_R,$ $y_R$ and $(xy)_R$ which are not uniquely determined by the given basis elements $xp$ and $py,$ in contrast to the situation we had in \S\ref{S.norm}, where expressions occurring in our reductions, such as $(xpy)_R,$ were shown to depend only on the basis elements $xp$ and $py.$ For this reason we will use~(\xppy{2}) rather than~\eqref{d.1=:xpqqpyalt}.)
The five families of reductions~\eqref{d.1=:xy|->}, \eqref{d.1=:ps|->},
\eqref{d.1=:xpqpy|->}, \eqref{d.1=:xpqtp|->}, (\xppy{2}) that we have accumulated at this point admit $20$ families of ambiguities! Namely, the {\em final} factor in $B$ of the input-monomials of each of these sorts of reductions can coincide with the {\em initial} factor in $B$
of the input-monomials of most of these sorts of reduction, the exceptions being that the lone factor $(ps)$ of the input of~\eqref{d.1=:ps|->} cannot coincide with the initial factors of the inputs of~\eqref{d.1=:xpqpy|->},
\eqref{d.1=:xpqtp|->} or (\xppy{2}), nor with the final factor of the input of~\eqref{d.1=:xpqtp|->}
(thus eliminating four of the $25$ potential pairings); nor does one get an ambiguity by overlapping~\eqref{d.1=:ps|->} with itself.
Many of these $20$ sorts of ambiguities are already resolvable, either because of the way they incorporate the structure of the associative ring $R,$ or because some of the later reductions were introduced precisely to make earlier ambiguities resolvable. Summarizing long and tedious hand computations (which we will be able to circumvent in the next section), one finds that of those $20$ sorts of ambiguities, 17 are resolvable, the three exceptions being
\begin{equation}\begin{minipage}[c]{35pc}\label{d.three.ambig} $(xp)\,q\cdot (tp)\cdot q\,(tp),\qquad (xp)\,q\cdot (tp)\cdot qq\,(py),\qquad (xp)\,qq\cdot(ps).$ \end{minipage}\end{equation}
Of these, the first and third turn out to yield a common relation. Selecting, as usual, the longest monomial in that relation, and writing the result as a formula reducing that monomial to a combination of the others, this takes the form
\begin{equation}\tag{\xptp{2\,}}\begin{minipage}[c]{35pc} $(xp)\,qq\,(tp)\ \mapsto\ (xp)\,qq - (xps)_R\,q + (xps)_R\,q\,(tp) - (xp)\,q\,(tps)_R.$ \end{minipage}\end{equation}
The middle ambiguity shown in~\eqref{d.three.ambig} yields a different reduction:
\begin{equation}\tag{\xppy{3}}\begin{minipage}[c]{35pc} $(xp)\,qqq\,(py)\ \mapsto\ (xps)_R\,qq\,(py) + (xp)\,q\,(tps)_R\,q\,(py) + (xp)\,qq\,(tpy)_R - (xps)_R\,q\,(tpy)_R.$ \end{minipage}\end{equation}
Examining the reductions we have been getting (after~\eqref{d.1=:xy|->}
and~\eqref{d.1=:ps|->}, which describe $R$ itself), they appear to fall into two series (as indicated in numbering I have given them), one starting with~\eqref{d.1=:xpqpy|->}, (\xppy{2}),
(\xppy{3}), the other with~\eqref{d.1=:xpqtp|->}, (\xptp{2}). Examining which ambiguities turned out to yield which new reductions, one can guess which should yield the next term in each series. In this way one finds, for instance, the next reduction in the \eqref{d.1=:xpqpy|->}-series:
\begin{equation}\tag{\xppy{4}}\begin{minipage}[c]{35pc} $(xp)\,qqqq\,(py)\ \mapsto\ (xps)_R\,qqq\,(py) + (xp)\,q\,(tps)_R\,qq\,(py) + (xp)\,qq\,(tps)_R\,q\,(py) + (xp)\,qqq\,(tpy)_R\\ \hspace*{2em}- (xps)_R\,q\,(tps)_R\,q\,(py) - (xps)_R\,qq\,(tpy)_R - (xp)\,q\,(tps)_R\,q\,(tpy)_R.$ \end{minipage}\end{equation}
The pattern of the inputs of~\eqref{d.1=:xpqpy|->}, (\xppy{2}), (\xppy{3}), (\xppy{4}) is clear; but what about the outputs? It appears that (ignoring signs, for the moment), each term in the output of a reduction~``(\xppy{n})'' is obtained from the input monomial $(xp)\,q^n\,(py)$ by replacing or not replacing the initial $(xp)\,q$ with $(xps)_R,$ replacing or not replacing the final $q\,(py)$ with $(tpy)_R,$ and replacing or not replacing some of the remaining $\!q\!$'s with $(tps)_R.$ But not every possible combination of such changes and non-changes shows up in our reductions; only those where no two successive $\!q\!$'s are changed.
Can we make sense of this?
\section{The normal form, described and proved.}\label{S.1=:norm}
The relations in $R'$ that yield the reductions~(\xppy{n}) can in fact be derived from scratch in roughly the way we obtained the relation~\eqref{d.1=:pqqp}; namely, by inserting terms $1=ps+tp$
between certain factors of the input monomial, partly expanding the result, and then simplifying using~\eqref{d.1=} and~\eqref{d.1=:xpqpy|->}. For example, the relation corresponding to~(\xppy{2}) can be gotten as follows:
\begin{equation}\begin{minipage}[c]{25pc}\label{d.insert-2} \hspace*{-5.8em}$(xp)\,qq\,(py)\ = \ (xp)\,q\,(ps+tp)\,q\,(py)\ \\[.3em] =\ (xp)\,q\,(ps)\,q\,(py)+ (xp)\,q\,(tp)\,q\,(py)\ \\[.3em] =\ (xps)_R\,q\,(py)+(xp)\,q\,(tpy)_R.$ \end{minipage}\end{equation}
However, not every string of insertions of terms $(ps)$ and $(tp)$ between $\!q\!$'s in a word $(xp)\,q\dots q\,(py)$ admits a simplification of the sort used above. We could not, for instance, simplify a string $\dots (tp)\,q\,(tp)\dots$
using~\eqref{d.1=:xpqpy|->}, because the second $(tp)$ does not begin with a $p,$ and so does not give us a ``$pqp$'' to reduce.
So the equations on which we should perform the simplifications that will yield the reductions~(\xppy{n}) for general $n$ are not obvious. For instance, the next case,~(\xppy{3}), can be obtained similarly by writing $(xp)\,qqq\,(yp)$ as $(xp)\,q\,(ps+tp)\,q\,(ps+tp)\,q\,(py),$ then using the expansion
\begin{equation}\begin{minipage}[c]{35pc}\label{d.insert-3} $(xp)\,q\,(ps+tp)\,q\,(ps+tp)\,q\,(py)\ = \ (xp)\,q\,(ps)\,q\,(ps+tp)\,q\,(py)\\[.3em] \hspace*{2em}+ (xp)\,q\,(tp)\,q\,(ps)\,q\,(py) + (xp)\,q\,(ps+tp)\,q\,(tp)\,q\,(py) - (xp)\,q\,(ps)\,q\,(tp)\,q\,(py),$ \end{minipage}\end{equation}
and reducing these terms. That~\eqref{d.insert-3} is an identity of associative rings is easy to check. (Clearly, before checking it we can drop the initial $(xp)$ and final $(py)$ of each term.) But it is not obvious how we would have come up with that identity to use. In the next lemma we shall describe and prove a sequence of identities to which the result of deleting the initial $(xp)$ and final $(py)$ from each term of~\eqref{d.insert-3} belongs, and the $\!n\!$-th step of which will similarly allow us to obtain~(\xppy{n}).
(Though I believe in the principle of stating results in their abstractly most natural form, since they may prove useful in contexts very different from the ones for which they were devised, the lemma below is unabashedly rigged to be used in the specific context we will apply it in, for the sake of smoothing that application. I will re-state it in a more general form as Corollary~\ref{C.multilin}, when we are through with the work of this section.)
\begin{lemma}\label{L.identity} Let $n\geq 2$ be an integer, $F$ the free associative $\!k\!$-algebra on generators $p,$ $s,$ $t,$ $q,$ and $S(n)$ the set of elements of $F$ which can be obtained by the following procedure:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.insert} Starting with the monomial $q^n,$ insert between each pair of successive $\!q\!$'s either $(ps+tp),$ or $(ps)$ alone, or $(tp)$ alone, in such a way that every $q$ that is immediately preceded by $(tp)$ is either immediately followed by $(ps)$ or is the final $q,$ and every $q$ that is followed immediately by $(ps)$ is either immediately preceded by $(tp)$ or is the initial $q.$
\hspace*{1em}Then multiply the resulting element by $(-1)^d,$ where $d$ is the number of $\!q\!$'s in its expression which are preceded by $(tp)$ and/or followed by $(ps).$ \textup{(}I.e., which are initial and followed by $(ps),$ or are simultaneously preceded by $(tp)$ and followed by $(ps),$ or are final and preceded by $(tp)).$ \end{minipage}\end{equation}
Then the sum in $F$ of the set $S(n)$ is $0.$ \end{lemma}
\begin{proof} Let us multiply out each element of the set $S(n)$ described above to get a sum of monomials; i.e., wherever a factor $(ps+tp)$ occurs in such a product, write the product as the sum of a product having $(ps)$ and a product having $(tp)$ in that position. Thus, each of the resulting monomials will contain $n$ $\!q\!$'s, with every pair of successive $\!q\!$'s having either a $(tp)$ or a $(ps)$ between them. Let $W(n)$ be the set of all monomials of this form. We must prove that for every $w\in W(n),$ the sum of the coefficients with which it occurs in members of $S(n)$ is $0.$
Within a monomial $w\in W(n),$ let us call an occurrence of $q$ ``marked'' if it is initial and followed by $(ps),$ or preceded by $(tp)$ and followed by $(ps),$ or final and preceded by $(tp).$ Every $w\in W(n)$ has at least one marked $q;$ for if there is at least one factor $(ps),$ then the $q$ preceding the first such factor (whether it is initial or preceded by a $(tp))$ will be marked, while if there are no factors $(ps),$ then the final $q$ will be preceded by a $(tp),$ and hence marked. On the other hand, two successive $\!q\!$'s can never both be marked, since if they have a $(tp)$ between them, the left-hand $q$ won't be marked, while if they have a $(ps)$ between them, the right-hand $q$ won't be marked.
Let $e\geq 1$ be the number of marked $\!q\!$'s in $w.$ I claim that there are exactly $2^e$ elements $v\in S(n)$ which contain a $\pm w$ in their expansion, half of them with a plus sign and half with a minus sign. Indeed, given $w,$ all such elements $v\in S(n)$ can be found by a construction that makes the following binary choice at each marked $q$ of $w:$ If the marked $q$ in question is neither initial nor final, so that it is preceded by a $(tp)$ and followed by a $(ps),$ the choice is between keeping these factors $(tp)$ and $(ps)$ unchanged in $v,$ or replacing both with $(ps+tp).$ (The definition of $S(n)$ doesn't allow any other possibilities.) If the $q$ in question is initial, the choice is simply between keeping the following $(ps)$ unchanged or replacing it with $(ps+tp),$ while if it is final, the choice is between keeping the preceding $(tp)$ unchanged or replacing it with $(ps+tp).$ (Since successive $\!q\!$'s cannot be marked, the effects of choices at different marked $\!q\!$'s will not conflict with each other.) Finally, for factors $(tp)$ of $w$ that do not precede marked $\!q\!$'s, and factors $(ps)$ that do not follow marked $\!q\!$'s, there is no choice: we replace these with $(ps+tp).$
It is not hard to see from the definition of $S(n)$ that these $2^e$ ways of modifying $w$ indeed give precisely the elements $v\in S(n)$ that have $w$ in their expansion. Moreover, by the second paragraph of~\eqref{d.insert}, such an element of $S(n)$ will bear a plus sign if the number of marked $\!q\!$'s around which we did not choose to change the adjacent factor(s) of $w$ to $(ps+tp)$ is even, a minus sign if that number is odd. Hence half of the resulting occurrences of $w$ have a plus sign and half have a minus sign, so they sum to zero; and since this is true for each $w,$ we get $\sum_{v\in S(n)} v =0,$ as claimed. \end{proof}
Now returning to the $\!k\!$-algebra $R'=R\lang q\mid p=pqp\rang,$ where $1=ps+tp$ in $R,$ let us map the free algebra of the above lemma into $R'$ by sending each indeterminate to the element of $R'$ denoted by the same letter. The lemma then tells us that in $R',$ a certain sum of signed products is zero. Hence if we choose any $(xp)\in B_{++}\cup B_{-+}$ and $(py)\in B_{++}\cup B_{+-},$ and multiply that sum of products on the left by $(xp)$ and on the right by $(py),$ the resulting sum of products is still zero. In the expressions for these products, we now can cross out each factor $(ps+tp),$ since it equals $1,$ and replace occurrences of $(xp)\,q\,(ps),$ $(tp)\,q\,(ps),$ and $(tp)\,q\,(py)$ by $(xps)_R,$ $(tps)_R,$ and $(tpy)_R$ respectively. The one element of $(xp)\,S(n)\,(py)$ in which no reduction of these three sorts is made is the one that had $(ps+tp)$ in all $n{-}1$ positions, and is now simply $(xp)\,q^n\,(py).$ Using the relation we have obtained to express that product as a linear combination of products with fewer remaining $\!q\!$'s, we get,
\begin{corollary}\label{C.identity} For $R,$ $p,$ $B$ as in the preceding section, any $(xp)\in B_{++}\cup B_{-+}$ and $(py)\in B_{++}\cup B_{+-},$ and any $n\geq 2,$ let $T((xp),n,(py))$ be the set of $\!k\!$-linear combinations of words in $B$ formed by modifying the word $(xp)\,q^n\,(py)$ as follows:
Choose any {\em nonempty} subset of the string of $n$ $\!q\!$'s in that word, to be called ``marked'' $\!q\!$'s, such that no two adjacent $\!q\!$'s are both marked. If the first $q$ in the string is marked, replace the initial term $(xp)\,q$ with $(xps)_R.$ If the last $q$ is marked, replace the final term $q\,(py)$ with $(tpy)_R.$ Replace every marked $q$ that is neither initial nor final with $(tps)_R.$ Finally, multiply the result by $-1$ if the number of marked $\!q\!$'s was~{\em even}.
Then the reduction
\begin{equation}\tag{\xppy{n}}\begin{minipage}[c]{35pc} $(xp)\,q^n\,(py)\ \mapsto\ \sum_{v\in T((xp),n,(py))}\ v$ \end{minipage}\end{equation}
corresponds to a relation holding in $R'.$ \textup{(}I.e., the elements of $R'$ represented by the input and the output of~\textup{(\xppy{n})} are equal.\textup{)}\qed \end{corollary}
Some remarks before we go further:
The number of terms in the set $S(n)$ of Lemma~\ref{L.identity} (and hence in the reduction~(\xppy{n}), counting the input term as well as the terms in $T((xp),n,(py))),$ is the $\!n{+}2\!$'nd Fibonacci number, $F_{n+2},$ since this is known to be the number of subsets of a sequence of $n$ elements containing no two successive elements \cite[p.\,14, Problem~1(b)]{comb}.
The reduction~\eqref{d.1=:xpqpy|->} clearly deserves to be called~(\xppy{1}); but we assumed $n\geq 2$ in the preceding lemma and corollary because the $n=1$ case differs from the general case in a couple of ways. On the one hand, when $n=1,$ the initial $q$ is also the final $q,$ so we get an output term $(xpy)_R,$ which is not one of the three sorts that occur when $n\geq 2.$
More important, the two sides of~\eqref{d.1=:xpqpy|->} do not differ as a result of where factors $(ps+tp),$ $(ps)$ or $(tp)$ appeared in a term $v,$ but simply as to whether or not one reduces the product $(xp)\,q\,(py)$ in the tautology $(xp)\,q\,(py)=(xp)\,q\,(py).$
Nevertheless,~\eqref{d.1=:xpqpy|->} has precisely the right form to be described as reduction~(\xppy{1}), and we will so consider it when we describe our normal form for~$R'.$
(We might consider the lone $q$ ``unmarked'' in the input of~\eqref{d.1=:xpqpy|->} and ``marked'' in the output.)
Let us note, finally, that monomials occurring in the output of~(\xppy{n}) may admit further reductions. For instance, in the output term $(xps)_R\,q\,(py)$ of~(\xppy{2}), some of the elements of $B$ appearing in $(xps)_R$ may be of the form $(x'p),$ allowing reductions $(x'p)\,q\,(py)\mapsto (x'py)_R.$ (Indeed, all of them will have this form if $x$ is a right multiple of $p$ in $R,$ since then $xps=x(1-tp)$ will be a right multiple of $p.)$ However, this does not interfere with our application of the Diamond Lemma. The formulation of that lemma does not require that the output of each reduction not admit further reductions, but simply that it be a linear combination of words smaller than the word one started with, under an appropriate partial ordering.
We now turn to the other family of reductions we encountered, beginning with~\eqref{d.1=:xpqtp|->}. Since, as just noted, it is not essential that all the terms of the outputs of our reductions be, themselves, reduced, we can make a slight simplification in the form of~(\xptp{2}), replacing the final $(tp)$ in the third output term by $(1-ps),$ to which it is equal in $R.$ Two terms then cancel, after which~(\xptp{2}) takes the form
\begin{equation}\tag{\xpTP{2}}\begin{minipage}[c]{35pc} $(xp)\,qq\,(tp)\ \mapsto\ (xp)\,qq - (xps)_R\,q\,(ps) - (xp)\,q\,(tps)_R.$ \end{minipage}\end{equation}
This leads to a version of the~\eqref{d.1=:xpqtp|->}-series of reductions that is easily deduced from Corollary~\ref{C.identity}.
\begin{corollary}\label{C.identity_tp} For every $n\geq 2$ and $(xp)\in B_{++}\cup B_{-+},$ the reduction
\begin{equation}\tag{\xpTP{n}}\begin{minipage}[c]{35pc} $(xp)\,q^n\,(tp)\ \mapsto\ (xp)\,q^n - \sum_{v\in T((xp),n,(ps))} v,$ \end{minipage}\end{equation}
where $\sum_{v\in T((xp),n,(ps))} v$ is defined as in Corollary~\ref{C.identity}, corresponds to a relation holding in $R'.$ \end{corollary}
\begin{proof} Applying Corollary~\ref{C.identity} with $py=ps$ (which is allowable, since $ps\in B_{+-}),$ we get $(xp)\,q^n\,(ps)=\sum_{v\in T((xp),n,(ps))} v$ in $R'.$ Rewriting the factor $(ps)$ on the left-hand side as $1-(tp),$ multiplying out, moving the shorter of the two resulting terms to the right-hand side, and changing all signs, we get $(xp)\,q^n\,(tp)=(xp)\,q^n - \sum_{v\in T((xp),n,(ps))} v,$ the desired relation. \end{proof}
We now have four families of reductions,~\eqref{d.1=:xy|->},
\eqref{d.1=:ps|->}, (\xppy{n}) and~(\xpTP{n}), where in the last two, we from now on allow all $n\geq 1,$
counting~\eqref{d.1=:xpqpy|->} as (\xppy{1}), and~\eqref{d.1=:xpqtp|->} as~(\xpTP{1}); and we wish to show that these together determine a normal form for elements of $R'.$ We have established that they correspond to relations holding in $R'.$ Moreover, they imply the defining relations for that $\!k\!$-algebra in terms of our generating set $B\cup\{q\},$
since~\eqref{d.1=:xy|->} and~\eqref{d.1=:ps|->} determine the structure of $R,$ while the imposed relation $pqp=p$ is the case of~(\xppy{1}) where both $xp$ and $py$ are $p.$ It remains to find a partial ordering on words in $B\cup\{q\}$ respecting multiplication and having descending chain condition, with respect to which all of these reductions are strictly decreasing, and to prove that the ambiguities of the resulting reduction system are resolvable.
The required partial ordering can be obtained by associating to every word $w$ in $B\cup\{q\}$ the $\!3\!$-tuple with first entry the number of $\!q\!$'s in $w,$ second entry the number of occurrences of members of $B$ in $w,$ and third entry the number of occurrences of the particular element $(ps)\in B$ in $w,$ and considering one word greater than another if the corresponding $\!3\!$-tuples are so related under lexicographic order, while considering distinct words which correspond to the same $\!3\!$-tuple incomparable. It is easy to see that this ordering has descending chain condition and respects formal multiplication of words (juxtaposition), and that in each of our reductions, all words of the output are strictly less than the input word.
(The first coordinate of the $\!3\!$-tuple is enough to show this last property for the reductions~(\xppy{n}); the second coordinate is needed for the reductions~\eqref{d.1=:xy|->}, and for the first term of the output of~(\xpTP{n}), while the third coordinate is only needed for~\eqref{d.1=:ps|->}.)
Proving resolvability of ambiguities will, of course, be the hard task.
The ambiguities among the cases of~\eqref{d.1=:xy|->} and~\eqref{d.1=:ps|->} are, as usual, resolvable because they describe the structure of the associative $\!k\!$-algebra~$R.$
I claim that ambiguities based on the fact that a word can be reduced either by~(\xppy{n}) or by~\eqref{d.1=:xy|->}, i.e., those involving words of the forms $(xp)\,q^n\,\cdot (py)\cdot z$ and $x\cdot (yp)\cdot q^n\,(pz),$ are also easily shown to be resolvable. As in the case of the ambiguities~\eqref{d.xpqpyz} and~\eqref{d.xypqpz} considered in \S\ref{S.norm}, the reason will be, in the former case, that right multiplication by $z$ carries $pR$ left $\!R\!$-linearly into itself, and in the latter, that left multiplication by $x$ carries $Rp$ into itself right $\!R\!$-linearly; so that whether we apply the reduction~(\xppy{n}) before or after that operation, we get the same result. For more detail, let us, in the case of $(xp)\,q^n\,\cdot (py)\cdot z,$ subdivide the summands in $\sum_{v\in T((xp),n,(py))} v$ in~(\xppy{n}) according to whether they end with $(py)$ or $(tpy)_R,$ writing that reduction as
\begin{equation}\begin{minipage}[c]{35pc}\label{d.A,B} $(xp)\,q^n\,(py)\ \mapsto \ \sum_{v\in T((xp),n,(py))}\ v\ = \ A((xp),n)\,(py) + B((xp),n)\,(tpy)_R.$ \end{minipage}\end{equation}
The reader can now easily verify that whichever of the two competing reductions we perform first on $(xp)\,q^n\,\cdot (py)\cdot z,$ the output can be reduced to $A((xp),n)(pyz)_R + B((xp),n)(tpyz)_R.$
The case of $x\cdot (yp)\cdot q^n\,(pz)$ is handled similarly, using a decomposition of the elements of $T((xp),n,(py))$ by initial rather than final factors, which we write down for later reference as
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C,D} $(xp)\,q^n\,(py)\ \mapsto\ \sum_{v\in T((xp),n,(py))}\ v\ = \ (xp)\,C(n,(py)) + (xps)_R\,D(n,(py)),$ \end{minipage}\end{equation}
(though in the present application, the roles of the $(xp)$ and $(py)$ in the above formula are played by the elements $(yp)$ and $(pz)).$
The resolution of ambiguities arising from words $x\cdot (yp)\cdot q^n\,(tp),$ which can be reduced using either~\eqref{d.1=:xy|->} on the left or~(\xpTP{n}) on the right, is verified similarly.
A little more complicated is the case of $(xp)\,q^n\cdot (tp)\cdot y,$
which can be reduced either by applying~(\xpTP{n}) on the left, or~\eqref{d.1=:xy|->} on the right. We recall that the result of reducing $(xp)\,q^n(tp)$ by~(\xpTP{n}) is $(xp)\,q^n$ minus the result of reducing $(xp)\,q^n(ps)$ by~(\xppy{n}); so applying this reduction in
$(xp)\,q^n (tp)\,y,$ and then making appropriate applications of~\eqref{d.1=:xy|->}, we get $(xp)\,q^n y$ minus the result of reducing
$(xp)\,q^n(psy)_R$ by~(\xppy{n}). On the other hand, if we begin by applying~\eqref{d.1=:xy|->}, we get $(xp)\,q^n (tpy)_R.$ Since $tp=1-ps$ in $R,$ we have $(tpy)_R = y - (psy)_R,$ and this leads to a decomposition of $(xp)\,q^n (tpy)_R$ as the difference of two terms, one of which is $(xp)\,q^n y,$ while the other, $(xp)\,q^n(psy)_R,$ can be reduced as just mentioned. So both reductions lead to $(xp)\,q^n y$ minus the result of reducing $(xp)\,q^n(psy)_R$ using~(\xppy{n}), showing that this ambiguity is also resolvable.
And the ambiguities coming from words $(xp)\,q^n\cdot(ps),$ which can be reduced either by applying~(\xppy{n}) to the whole expression, or~\eqref{d.1=:ps|->} to the final factor, are resolvable because of the reductions~(\xpTP{n}), which were introduced precisely to handle them. (These ambiguities are, incidentally, what are called in~\cite{<>} ``inclusion ambiguities'', where the input-word of one reduction is a subword of the input-word of another reduction. All other ambiguities occurring in this note are ``overlap ambiguities''.)
We are left with the ambiguities resulting from the overlap of two words both of which admit reductions in our \eqref{d.1=:xpqpy|->}-series and/or our \eqref{d.1=:xpqtp|->}-series.
Again, some of these are fairly straightforward to show resolvable. Consider first a word $(xp)\,q^m\cdot y \cdot q^n (pz)$ where $y=py'=y''p\in B_{++},$ with $m,n\geq 2,$ to which we can apply either~(\xppy{m}) on the left, or~(\xppy{n}) on the right. It is not hard to verify that whichever of those operations we apply first, the other will then be applicable to all the words in the resulting expression. (For instance, though the result of first applying~(\xppy{m}) will include some terms in which the factor $y=py'$ has been absorbed into a product $(tpy')_R,$ the relation $py'=y''p$ allows us to rewrite this as $(ty''p)_R,$ so it lies in $Rp,$ hence is a $\!k\!$-linear combination of generators in $B_{++}\cup B_{-+},$ allowing a subsequent application of~(\xppy{n}) to each term.) One finds that the result of either order of reductions is the sum of a set of terms which can be constructed as follows: Starting with the word $(xp)\,q^m\cdot y \cdot q^n (pz),$ ``mark'' an arbitrary subset of the $\!q\!$'s, subject to the condition that the marked subset of the first $m$ $\!q\!$'s be nonempty and contain no pair of adjacent $\!q\!$'s, and that the marked subset of the last $n$ $\!q\!$'s likewise be nonempty and contain no adjacent pair. Now, as before, we make appropriate replacements involving the marked $\!q\!$'s. What most of these should be are clear from the statement of Corollary~\ref{C.identity}. For instance, if the first $q$ is marked, replace $(xp)\,q$ by $(xps)_R;$ if the last $q$ is marked, replace $q\,(pz)$ by $(tpz)_R;$ if a $q$ that is neither initial nor final in the string $q^m$ or $q^n$ is marked, replace it with $(tps)_R.$ Likewise, if the last $q$ before the factor $y$ is marked, but the $q$ following the $y$ is not, then we replace $q\,y=q\,py'$ by $(tpy')_R,$ while if the $q$ following the $y$ is marked but not the one that precedes it, we replace $y\,q=(y''p)\,q$ by $(y''ps)_R.$ But what if both of those $\!q\!$'s are marked? Then we find that the results of performing either of those two replacements, followed by the reductions corresponding to the other, give the same result. Indeed, the replacements correspond to two ways of reducing $(tp)\,q\,y\,q\,(ps)\,,$ which is an instance of~\eqref{d.xpqyqpz}, the resolvability of which was verified in~\S\ref{S.norm}; both reductions of that factor give $(tys)_R.$ So this ambiguity is also resolvable.
The corresponding ambiguities with $m$ and/or $n$ equal to $1$ are resolved in the same way, with the obvious adjustments; e.g., if $m=1,$ then instead of $q(py')$ being replaced by $(tpy')_R$ in some terms of~\eqref{d.A,B}, we will have $(xp)\,q\,(py')$ replaced by $(xpy')_R.$ (The case $m=n=1$ is precisely~\eqref{d.xpqyqpz}.)
And the ambiguities $(xp)\,q^m\cdot y \cdot q^n (tp),$ which can be reduced by applying~(\xppy{m}) on the left, or~(\xpTP{n}) on the right, are handled like the above, {\em mutatis mutandis}.
There remain two sorts of ambiguities, which get a bit more interesting. These will be given as~\eqref{d.mspn} and~\eqref{d.mspn+} below, but let us prepare for them with the following observation. Up to this point, ambiguities involving reductions~(\xppy{n}) and/or~(\xpTP{n}) for certain values of $n$ were resolved using only reductions indexed by the same value(s) of $n$ and the value $1.$ Now if this were the case for the remaining sorts of ambiguities as well, then for any set $N$ of positive integers containing~$1,$
the system of reductions given by~\eqref{d.1=:xy|->},
\eqref{d.1=:ps|->}, and the~(\xppy{n}) and~(\xpTP{n}) for $n\in N$
would have all ambiguities resolvable, and so would determine a basis of monomials for the $\!k\!$-algebra presented by the generating set $B\cup\{q\}$ and the relations corresponding to those reductions. But we have seen that the relations corresponding to~\eqref{d.1=:xy|->},
\eqref{d.1=:ps|->}, and~(\xppy{1}) are sufficient to present $R',$ in which all of the~(\xppy{n}) and~(\xpTP{n}) are satisfied; so all such subsystems of our system of reductions would yield $\!k\!$-bases for $R'.$ Yet some of these bases of $R'$ (those arising from larger sets $N)$ would be properly contained in others (arising from smaller sets $N),$ since a larger set of reductions would make more words reducible.
Since a proper inclusion among bases of $R'$ is impossible, it {\em must} be true that in resolving some of the ambiguities we have not yet considered, reductions with larger subscripts than those involved in the ambiguities themselves must be used. And indeed, we saw in the preceding section that trying to resolve the ambiguity $(xp)\,q\cdot(tp)\cdot q\,(py),$ arising from the reductions~(\xpTP{1}) and~(\xppy{1}), required us to introduce the new reduction~(\xppy{2}). So with the expectation that this will happen, let us look at the two remaining families of ambiguities.
Consider first an ambiguously reducible word of the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn} $(xp)\,q^m\cdot(tp)\cdot q^n\,(py).$ \end{minipage}\end{equation}
In analyzing the effects of our reductions on this expression, we shall use both of the notations introduced in~\eqref{d.A,B} and~\eqref{d.C,D}. Observe that in~\eqref{d.A,B}, the term $A((xp),n)(py)$ arises from those ways of marking $q^n$ such that at least one $q$ is marked, but the last $q$ is {\em not} marked (since $(py)$ has not been absorbed in a term $(tpy)_R),$ while $B((xp),n)(tpy)_R$ arises from those ways of marking $q^n$ under which a set of $\!q\!$'s {\em including} the last $q$ is marked. Similarly, the two terms of~\eqref{d.C,D} arise from ways of marking $q^n$ so that the {\em first} $q$ is not, respectively, is, marked. Note, finally that in the notation of~\eqref{d.A,B}, the output of the reduction~(\xpTP{m}) is
\begin{equation}\begin{minipage}[c]{35pc}\label{d.spA,B} $(xp)\,q^m - A((xp),m)(ps) - B((xp),m)(tps)_R.$ \end{minipage}\end{equation}
Now starting with the monomial~\eqref{d.mspn}, if we apply, on the one hand,~(\xpTP{m}) with its output written as~\eqref{d.spA,B} to the terms surrounding the first dot, and, on the other hand,~(\xppy{n}) with output written as in~\eqref{d.C,D} to the terms surrounding the second dot, then the equation equating the results, which we hope to show can be established by further reductions in our system, is
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspnABCD} $(xp)\,q^{m+n}\,(py)\ -\ A((xp),m)\,(ps)\,q^n\,(py)\ - \ B((xp),m)\,(tps)_R\,q^n\,(py)\\[.3em] \hspace*{5em}=\ (xp)\,q^m (tp)\,C(n,(py))\ + \ (xp)\,q^m (tps)_R\,D(n,(py)).$ \end{minipage}\end{equation}
The very first term above is the input of the reduction~(\xppy{m+n}), so our ambiguity will be resolvable if, on moving the other terms of~\eqref{d.mspnABCD} to the right-hand side, the value we end up with there, namely
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn_1} $A((xp),m)\,(ps)\,q^n\,(py)\ +\ B((xp),m)\,(tps)_R\,q^n\,(py)\ +\\[.3em] \hspace*{5em}(xp)\,q^m (tp)\,C(n,(py))\ +\ (xp)\,q^m (tps)_R\,D(n,(py))$ \end{minipage}\end{equation}
can be reduced to the output of~(\xppy{m+n}). We can, in fact, recognize two of the terms of~\eqref{d.mspn_1} as parts of that output. Namely, $B((xp),m)\,(tps)_R\,q^n\,(py)$ can be seen to be the sum of all terms gotten from $(xp)\,q^{m+n}\,(py)$ by marking some subset of the first $m$ $\!q\!$'s which includes the $\!m\!$-th, and then making the replacements described in Corollary~\ref{C.identity}. Likewise, $(xp)\,q^m (tps)_R\,D(n,(py))$ is the sum of the terms we get on marking some subset of the last $n$ $\!q\!$'s which includes the first of these, and making the appropriate replacements. So it suffices to show that the first and third terms of~\eqref{d.mspn_1} can be reduced to the sum of the other terms in the output of~(\xppy{m+n}).
(As they stand, the do not consist of such terms, since terms in the outputs of our \eqref{d.1=:xpqpy|->}-series reductions have no internal factors $(ps)$ or $(tp).)$
The term $A((xp),m)\,(ps)\,q^n\,(py)$ can be reduced by applying~\eqref{d.1=:ps|->} to the factor $(ps),$ while the term $(xp)\,q^m\,(tp)\,C(n,(py))$ can be reduced by applying~(\xpTP{m}) to its initial factors. Then these two remaining terms of~\eqref{d.mspn_1} become
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn_2} $A((xp),m)\,q^n\,(py)\ -\ A((xp),m)\,(tp)\,q^n\,(py)\ +\\[.3em] \hspace*{2em}(xp)\,q^m\,C(n,(py)) -\ A((xp),m)\,(ps)\,C(n,(py))\ -\ B((xp),m)\,(tps)_R\,C(n,(py)).$ \end{minipage}\end{equation}
Of these terms, the first can be seen to consist of those summands in the output of~(\xppy{m+n}) in which, again, only a subset of the first $m$ $\!q\!$'s have been marked, this time a subset which does {\em not} include the $\!m\!$-th $q;$ and the third term likewise consists of those summands in which a subset of the last $n$ $\!q\!$'s have been marked, but that subset does not include the first of these. Dropping these two terms from our calculation, the first of the remaining terms can be reduced using~(\xppy{n}). After performing that reduction, and again using~\eqref{d.C,D} to describe the output, the terms that remain to be considered take the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn_3} $- A((xp),m)\,(tp)\,C(n,(py)) - A((xp),m)\,(tps)_R\,D(n,(py))\\[.3em] \hspace*{2em}- A((xp),m)\,(ps)\,C(n,(py)) - B((xp),m)\,(tps)_R\,C(n,(py)).$ \end{minipage}\end{equation}
The first and third of these can be combined using the relation $ps+tp=1$ (or, formally, by applying the reduction~\eqref{d.1=:ps|->} to the latter, and adding the result to the former), giving
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn_4} $- A((xp),m)\,C(n,(py)) - A((xp),m)\,(tps)_R\,D(n,(py)) - B((xp),m)\,(tps)_R\,C(n,(py)).$ \end{minipage}\end{equation}
I claim that this expression gives precisely the remaining terms of the output of~(\xppy{m+n}), i.e., those in which there are marked $\!q\!$'s among both the first $m$ and the last $n.$ Indeed, the first summand in~\eqref{d.mspn_4} contains the terms of this sort in which neither the $\!m\!$-th nor the $\!(m{+}1)\!$-st $q$ is marked; the next gives those in which the $\!m\!$-th $q$ is again not marked, but the $\!(m{+}1)\!$-st is, and the last gives those in which the $\!m\!$-th is marked, but not the $\!(m{+}1)\!$-st. Since adjacent $\!q\!$'s cannot both be marked, these are all the possibilities.
But are the negative signs in~\eqref{d.mspn_4} what we want? I agonized over this till I finally saw that they are. In the description of the output of our reductions in the \eqref{d.1=:xpqpy|->}-series, a term is assigned a minus sign if it involves an {\em even} number of marked $\!q\!$'s, a plus sign if it involves an {\em odd} number. (This is the result of moving these terms of the expression proved to sum to zero in Lemma~\ref{L.identity} to the opposite side of the equation from $(xp)\,q^n (py).)$ Hence when we form a product such as $A((xp),m)\,C(n,(py))$ in~\eqref{d.mspn_4}, the terms involving an even total number of marked $\!q\!$'s will end up with a plus sign and those with an odd total number will have a minus sign. So this and the other terms of~\eqref{d.mspn_4} indeed need negative signs to give the corresponding summands in the output of~(\xppy{m+n}).
Once again, the verifications of the cases where $m$ and/or $n$ is $1$ differ only in minor formal details from the above.
And the resolvability of the one remaining sort of ambiguity,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.mspn+} $(xp)\,q^m\cdot(tp)\cdot q^n\,(tp)$ \end{minipage}\end{equation}
reduces to the resolvability of the case of~\eqref{d.mspn} where $(py)$ is $(ps),$ via Corollary~\ref{C.identity_tp}. These computations establish
\begin{theorem}\label{T.1=} Let $R$ be a $\!k\!$-algebra, let $p$ be an element of $R$ such that $1\in pR+Rp$ but $1\notin pR$ and $1\notin Rp,$ choose $s,t\in R$ such that $1=ps+tp$ as in~\eqref{d.1=}, let $B\cup\{1\}$ be a spanning set for $R$ satisfying~\eqref{d.1=:B=} and~\eqref{d.1=:B_}, and let
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:R'} $R'\ =\ R\lang q\mid pqp=p\rang.$ \end{minipage}\end{equation}
Then $R'$ has a $\!k\!$-basis given by all words in the generating set $B\cup\{q\}$ that contain no subwords of any of the following forms:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xy} $xy$ \quad with $x,y\in B,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:ps} $(ps),$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xpqpy} $(xp)\,q^n\,(py)$ \quad with $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-},$ and $n\geq 1,$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1=:xpqtp} $(xp)\,q^n\,(tp)$ \quad with $xp\in B_{++}\cup B_{-+}$ and $n\geq 1.$ \end{minipage}\end{equation}
The reduction to the above normal form may be accomplished by the systems of reductions \eqref{d.1=:xy|->},
\eqref{d.1=:ps|->},
\textup{(\xppy{n})}~\textup{(}as shown in Corollary~\ref{C.identity}, but with~\eqref{d.1=:xpqpy|->} also included as \textup{(\xppy{1}))} and
\textup{(\xpTP{n})} \textup{(}as shown in Corollary~\ref{C.identity_tp}, but with~\eqref{d.1=:xpqtp|->} also included as \textup{(\xpTP{1}))}.\qed \end{theorem}
Combining this with the results of \S\ref{S.norm} and~\S\ref{S.1-sided}, we see that we have determined the structure of \mbox{$R\lang q\mid pqp=p\rang$} for all cases of a $\!k\!$-algebra $R$ and an element $p\in R.$
\section{Some consequences, and a couple of loose ends}\label{S.&}
The proof of Theorem~\ref{T.1=} extends without difficulty to give the analog of Proposition~\ref{P.M_norm}, with ``$\!p\!$-tempered $\!R\!$-module'' still defined as in Definition~\ref{D.tempered}. As in that proposition, we take a $\!k\!$-basis $C=C_+\cup C_-$ for $M,$ and supplement the reductions we have used in the normal form for $R'$ with corresponding reductions in which the leftmost factor, $x$ or $(xp),$ is replaced with an element of $C;$ an arbitrary element in the case of the $x$ of~\eqref{d.1=:xy}, a member of $C_+$ in the case of the $(xp)$ of~\eqref{d.1=:xpqpy} and~\eqref{d.1=:xpqtp}.
We will not write the result out in detail; but let us note a common feature of this and our other results on the extension of $\!p\!$-tempered $\!R\!$-modules to $\!R'\!$-modules.
\begin{corollary}\label{C.tempered} If $R$ is a $\!k\!$-algebra, $p$ any element of $R,$ and $(M,M_+)$ a $\!p\!$-tempered $\!R\!$-module, as defined in Definition~\ref{D.tempered}, then the canonical map $M\to (M,M_+)\otimes_{(R,p)} R'$ is an embedding such that \textup{(}identifying $M$ with its image under that map\textup{)} we have $M_+= M\cap(M\otimes_{(R,p)} R')\,p.$\qed \end{corollary}
We can also ask when an inclusion of $\!k\!$-algebras leads to an embedding of extensions of these algebras by universal inner inverses. This is answered by
\begin{proposition}\label{P.R0,R1} Let $R^{(0)}\subseteq R^{(1)}$ be an inclusion of $\!k\!$-algebras, and $p$ an element of $R^{(0)}.$ Then the following conditions are equivalent.\\[.5em]
\textup{(a)} The induced map $R^{(0)}\lang q\mid pqp=p\rang \ \to\ R^{(1)}\lang q\mid pqp=p\rang$ is an embedding.\\[.5em]
\textup{(b)} $R^{(0)}\cap(R^{(1)} p) = R^{(0)} p,$ and $R^{(0)}\cap(pR^{(1)}) = pR^{(0)},$ and $R^{(0)}\cap (p R^{(1)}{+}R^{(1)} p) = p R^{(0)}{+}R^{(0)} p.$
\end{proposition}
\begin{proof} The direction (a)$\implies$(b) will not use our normal form results, and, indeed, holds without the assumption that $k$ is a field. Observe that in any ring, if an element $p$ has an inner inverse $q,$ then an element $x$ is right divisible by $p$ if and only if $x(1-qp)=0:$ ``only if'' is clear, while ``if'' holds because the indicated equation makes $x=xqp.$ (We used the same idea in the proof of Corollary~\ref{C.M->MOX}.) Under the assumption~(a), this immediately gives the first equality of~(b); the second is seen dually. The third holds by the similar criterion saying that $x\in pR+Rp$ if and only if $(1-pq)\,x\,(1-qp)=0,$ where ``if'' holds because that equation can be written $x=pqx+xqp-pqxqp.$
The proof that (b)$\implies$(a) will use our normal form results. Under the assumptions of~(b), note that whichever of the cases ``$\!1\notin pR+Rp$'', ``$\!1\in pR+Rp-(pR\cup Rp)$'', ``$\!1\in pR$'', ``$\!1\in Rp$'' apply to $R^{(1)}$ will also apply to $R^{(0)}.$
Let us now take a generating set $B^{(0)}= B_{++}^{(0)}\cup B_{+-}^{(0)} \cup B_{-+}^{(0)}\cup B_{--}^{(0)}$ for $R^{(0)}$ as in our development of the case under which $R^{(0)}$ and $R^{(1)}$ both fall. (Some of these sets will be empty if we are in a case where $p$ is right and/or left invertible.) It follows from~(b) that we can extend each of $B_{++}^{(0)},$ $B_{-+}^{(0)},$ $B_{+-}^{(0)},$ $B_{--}^{(0)}$ to a subset $B_{++}^{(1)},$ $B_{-+}^{(1)},$ $B_{+-}^{(1)},$ $B_{--}^{(1)}$ of $R^{(1)}$ satisfying the corresponding conditions, so as to yield a generating set $B^{(1)}$ for $R^{(1)}$ with each component containing the corresponding component of the generating set $B^{(0)}$ for $R^{(0)}.$ Using these generating sets, the normal form expression for each element of $R^{(0)}\lang q\mid pqp=p\rang$ is also the normal form of its image in $R^{(1)}\lang q\mid pqp=p\rang.$ Hence, if an element is nonzero in the former ring, so is its image in the latter, establishing~(a). \end{proof}
(Incidentally, the first two equalities of~(b) above do not imply the third. For a counterexample, let $R^{(1)}$ be the Weyl algebra, written as in~\eqref{d.Weyl} below, and $R^{(0)}$ its subalgebra $k[p].$ Then $1\notin pR^{(0)}+R^{(0)}p$ but $1\in pR^{(1)}+ R^{(1)}p,$ so the third condition of~(b) fails, though the first two clearly hold. More on the algebra gotten by adjoining an inner inverse to $p$ in the Weyl algebra in the next section.)
Is there a generalization of the above proposition based on a concept of a ``$\!p\!$-tempered $\!k\!$-algebra'' $R,$ in which certain $\!k\!$-subspaces of $R$ are specified whose elements are to be treated like right and/or left multiples of $p$? A difficulty is that although, when we are dealing with genuine left and right multiples of $p,$ reductions $(xp)\,q\,(py)\mapsto (xpy)_R$ turn out to be well-defined, there is no evident reduction of $xqy$ when $x$ and $y$ are elements ``to be regarded as'' a right and a left multiple of $p$ respectively. But I have not looked closely at the question.
Let's clear up a couple of loose ends. I mentioned in the preceding section that the formulation of Lemma~\ref{L.identity} used there was rigged for quick application. Here is the promised more abstract version. Note that the $n$ of the result below corresponds to $n{-}1$ in Lemma~\ref{L.identity}, since there are $n{-}1$ places in which to insert factors between $n$ $q$'s. The proof is essentially as before.
\begin{corollary}[to proof of Lemma~\ref{L.identity}]\label{C.multilin} Let $n\geq 1,$ let $A$ and $A'$ be abelian groups, let $\mu: A^n\to A'$ be an $\!n\!$-linear map, and let $x$ and $y$ be elements of $A.$ Let $S(n)$ be the family of elements $\pm\,\mu(a_1,\dots,a_n)\in A'$ which arise from all ways of choosing each $a_i$ from the $\!3\!$-element set $\{x,y,x{+}y\},$ and also choosing the sign plus or minus, so as to satisfy the following conditions.
\begin{equation}\begin{minipage}[c]{35pc}\label{d.insert'} If $(a_1,\dots,a_n)$ has an $x$ in a nonfinal position, it has a $y$ in the next position.
If $(a_1,\dots,a_n)$ has a $y$ in a noninitial position, it has an $x$ in the preceding position.
If the number of occurrences in $(a_1,\dots,a_n)$ of the substring ``$x,y\!$'', plus the number of occurrences of initial $y$ and/or final $x,$ is odd, then the sign appended to $\mu(a_1,\dots,a_n)$ is $-;$ otherwise it is $+.$ \end{minipage}\end{equation}
Then the sum of the resulting set $S(n)$ of elements $\pm\,\mu(a_1,\dots,a_n)\in A'$ is $0.$
\textup{(}Above, if two of $x,\,y,\,x{+}y\in A$ happen to be equal, we treat them as formally distinct in interpreting~\eqref{d.insert'}.\textup{)}\qed \end{corollary}
Is the above the nicest version of the result? A ``cleaner'' form would be the special case where $A$ is the free abelian group on $\{x,y\},$ and $A'$ the $\!n\!$-fold tensor power of $A,$ since the form given above can be obtained from that case by composing with maps into general $A$ and $A';$ and that case would avoid distractions when studying the combinatorics of the result. On the other hand, the form given above simplifies applications such as we are making here.
Turning back to the proof of resolvability of the ambiguities~\eqref{d.mspn} and~\eqref{d.mspn+}, it might be possible to make this cleaner by first obtaining identities involving the families $S(n)\subseteq k\lang p,s,t,q\rang.$ If we let $S'(n)$ denote the subset of $S(n)$ consisting of those elements, in the construction of which the final $q$ was {\em not} marked (and, for convenience, define $S'(1)=\{q\}),$ then we can express the $S(n)$ in terms of these sets:
\begin{equation}\begin{minipage}[c]{35pc}\label{d.S=S'-} $S(n)\ =\ S'(n)\ \cup\ -S'(n{-}1)\,(tp)\,q \quad (n\geq 2),$ \end{minipage}\end{equation}
and give a recursive construction of the $S'(n):$
\begin{equation}\begin{minipage}[c]{35pc}\label{d.S'12} $S'(1)=\{q\},\qquad S'(2)=\{q\,(ps{+}tp)\,q,\ q\,(ps)\,q\},$ \end{minipage}\end{equation}
\begin{equation}\begin{minipage}[c]{35pc}\label{d.S'_rec} $S'(n)\ =\ S'(n{-}1)\,(ps{+}tp)\,q\ \cup\ -S'(n{-}2)\,(tp)\,q\,(ps)\,q \quad (n\geq 3).$ \end{minipage}\end{equation}
We would likewise let $S''(n)\subseteq S(n)$ be the subset determined by the condition the that {\em initial} $q$ not be marked, and give the corresponding formulas for these sets; and we could probably develop formulas which, mapped to our ring $R',$ would be equivalent to the resolvability of our ambiguities. If the method of the preceding section should prove useful beyond the particular results we obtain there, this approach might be worth pursuing.
\section{What if $R$ is the Weyl algebra? Don't ask!}\label{S.Weyl}
A well-known example of a ring with an element $p$ that is neither left nor right invertible, but which satisfies $1\in pR+Rp,$ is the Weyl algebra. This is usually denoted $A=k\lang x,y\mid yx=xy+1\rang$ or $A=k\lang x,\,d/dx\rang;$ but for consistency with the notation in the rest of this note, let us write it
\begin{equation}\begin{minipage}[c]{35pc}\label{d.Weyl} $R\ =\ k\lang\,p,s\mid ps + (-s)p = 1\rang\,.$ \end{minipage}\end{equation}
It is natural to ask whether we can get a nice normal form for the extension
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:R'=} $R'\ =\ k\lang\,p,s,q\mid ps + (-s)p = 1,\ pqp=p\rang\,.$ \end{minipage}\end{equation}
If we want to apply the construction of Theorem~\ref{T.1=}, we first need to determine the $\!k\!$-subspaces $pR$ and $Rp$ of $R,$ and their intersection. It is a standard result that a $\!k\!$-basis of the Weyl algebra is given by
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:s^mp^n} $\{s^m p^n\mid m,n\geq 0\}.$ \end{minipage}\end{equation}
Indeed, every element of $R$ can be reduced to a linear combination of members of this basis by repeated application of the reduction
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:ps|->} $ps\ \mapsto\ sp+1,$ \end{minipage}\end{equation}
which has no ambiguities.
Since right multiplying a $\!k\!$-linear combination of elements of~\eqref{d.W:s^mp^n} by $p$ gives a $\!k\!$-linear combination of such elements having $n>0,$ it follows that $Rp$ is precisely the $\!k\!$-subspace of $R$ spanned by the elements $s^m\,p^n$ with $n>0.$
One can characterize $pR$ similarly using the basis $\{p^n s^m\mid m,n\geq 0\},$ but this does not help if we want to study both subspaces at the same time. So let us, for now, represent elements of $R$ using the basis~\eqref{d.W:s^mp^n}, and investigate what linear combinations of these basis elements lie in $pR.$
If we apply~\eqref{d.W:ps|->} repeatedly starting with $p\,s^m,$ we get
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:ps^m} $p\,s^m\ =\ s^m\,p\ +\ m\,s^{m-1}.$ \end{minipage}\end{equation}
Thus, $s^m p\equiv -\,m s^{m-1}\pmod{pR},$ and right multiplying this congruence by $p^{n-1},$ we get
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:s^mp^n==} $s^m p^n\ \equiv\ -m\,s^{m-1}p^{n-1}\hspace*{-.6em}\pmod{pR}$\quad for $m,n\geq 1.$ \end{minipage}\end{equation}
We can iterate~\eqref{d.W:s^mp^n==}, decreasing the exponents of $s$ and $p$ until one of them goes to zero. So if $n>m,$ we conclude that $s^m p^n$ is congruent modulo $pR$ to an integer multiple of a positive power of $p;$ hence it lies in $pR;$ and since it also lies in $Rp,$ we get
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:n>m} $s^m p^n\in pR\cap Rp$\quad if $n>m.$ \end{minipage}\end{equation}
On the other hand, if $m\geq n > 1,$ it is convenient to iterate~\eqref{d.W:s^mp^n==} only to the point of bringing the exponent of $p$ down to $1.$ That gives us $s^m p^n\equiv (-1)^{n-1}\,m(m-1)\dots(m-n+2)\,s^{m-n+1}\,p\pmod{pR},$ so again, since both expressions lie in $Rp,$ we get
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:m_geq_n} $s^m p^n\ \equiv\ (-1)^{n-1}\,m(m{-}1)\dots(m{-}n{+}2)\,s^{m-n+1}\,p \pmod{pR\cap Rp}$\quad if $m\geq n > 1.$ \end{minipage}\end{equation}
Combining~\eqref{d.W:n>m},~\eqref{d.W:m_geq_n}, and the vacuous relation $s^m\ \equiv\ s^m \pmod{pR\cap Rp},$ we get
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:pR+Rp_big} Every element of $R$ is congruent modulo $pR\cap Rp$ to a linear combination of words $s^m$ and~$s^m p.$ \end{minipage}\end{equation}
Since the family of words $\{s^m,\,s^m p\mid m\geq 0\}$ is ``small'' compared with the full $\!k\!$-basis~\eqref{d.W:s^mp^n}, we see that when we form our desired spanning set $B,$ ``most of'' that set can be expected to lie in the component $B_{++}\,.$
Further details depend on the characteristic of $k.$ We shall consider the case where $\mathrm{char}(k)=0.$
In this case, solving~\eqref{d.W:ps^m} for $s^{m-1},$ we see that every power of $s$ lies in $pR+Rp.$ Hence,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:R=pR+Rp} If $\mathrm{char}(k)=0,$ then $R=pR+Rp.$ \end{minipage}\end{equation}
It is now easy to verify that
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:B=} \hspace*{0em}If $\mathrm{char}(k)=0,$ then a spanning set $B$ for $R$ with the properties of~\eqref{d.1=:B=}, \eqref{d.1=:B_} is given by
$B_{++}\ =\ \{s^m p^n\mid n>m\geq 0\}\ \cup \ \{s^m p^n + ms^{m-1}\,p^{n-1}\mid m\geq n>1\}$\quad (cf.~\eqref{d.W:n>m} and~\eqref{d.W:s^mp^n==}),
$B_{-+}\ =\ \{-s^m\,p\mid m>0\},$
$B_{+-}\ =\ \{p\,s^m\mid m>0\} \ =\ \{s^m\,p + m\,s^{m-1}\mid m>0\}$\quad (cf.~\eqref{d.W:ps^m}),
$B_{--}\ =\ \emptyset.$ \end{minipage}\end{equation}
(I have put a minus sign into the entries of $B_{-+}$ to conform with the convention made in~\eqref{d.1=:B=}, that $B_{-+}$ contain $tp,$ which, in writing~\eqref{d.Weyl}, we have taken to be $(-s)p.)$
Using the above basis, we can obtain by Theorem~\ref{T.1=} a normal form for $R'=k\lang p,s,q\mid ps+(-s)p= 1,\ \linebreak[1] pqp=p\rang.$
But what we would really like is a normal form in terms of the generators $p,$ $s$ and $q.$ When first exploring the case $1\in pR+Rp -(pR\cup Rp)$ of the subject of this note, I took the Weyl algebra as a sample case, and tried to find such a normal form; but the ambiguities among reductions I obtained kept spawning new reductions, without apparent pattern. This, along with calculations showing that the forms of these reductions must depend on the characteristic of $k,$ led me to doubt for a long time that any reasonable normal form could be found when $1\in pR+Rp -(pR\cup Rp).$ It was only when I dropped the case of the Weyl algebra, and returned to consideration of a general $\!k\!$-algebra, that I was able to get anywhere.
However, with the results of \S\ref{S.1=:norm} now at hand, we can develop a normal form for this algebra $R'$ in terms of $p,$ $s$ and $q,$ and shall do so below (still assuming $\mathrm{char}(k)=0).$
(Let me here moderate the semi-facetious title of this section, to merely say that if, at some point the reader chooses not to slog further through the lengthy argument for the sake of a normal form whose value is not evident, I will not argue with his or her choice.)
In preparation for the result, let us note that the normal form that we would get by simply applying Theorem~\ref{T.1=} to the basis~\eqref{d.W:B=} for $R$ is somewhat atypical among applications of that theorem. In the general situation of Theorem~\ref{T.1=}, if we take from our spanning set $B$ three elements $xp\in B_{++}\cup B_{-+},$ $y\in B_{--},$ $pz\in B_{++}\cup B_{+-}$ (note the choice of $y$ here, the opposite of what we considered when looking for ambiguities!), then products $(xp)\,q^m\,y\,q^n\,(pz)$ are irreducible: the presence of $y\in B_{--}$ between $(xp)$ and $(pz)$ blocks any reductions. However, with a basis like~\eqref{d.W:B=}, where $B_{--}$ is empty, no such blockage is possible; and we find that in any string of elements of $B\cup\{q\}$ that is reduced with respect to the normal form of Theorem~\ref{T.1=}, no element of $B_{++}\cup B_{-+}$ can occur anywhere to the left of an element of $B_{++}\cup B_{+-}.$ So the elements of $B$ (if any) occurring interspersed among the $\!q\!$'s in our word will begin with a sequence (possibly empty) of members of $B_{+-},$ and end with a sequence (possibly empty) of members of $B_{-+},$ with at most a single member of $B_{++}$ between these.
This will prepare us for the fact that words in the normal form based on $p,$ $s$ and $q$ that we shall obtain will typically have a sort of singularity in the middle. To prepare us in a more detailed way for the form they will have, let us note that where in \S\ref{S.1=:norm}, a key tool was to apply, between various pairs of $\!q\!$'s in a string $q\dots q,$ the relation $1=ps+tp,$ in the present situation we can, more generally, whenever an $s^m$ $(m\geq 0)$ appears between $\!q\!$'s, apply the result of putting $m{+}1$ for $m$ in~\eqref{d.W:ps^m} and solving for~$s^m:$
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:s^m=} $s^m\ =\ (m+1)^{-1}(ps^{m+1} - s^{m+1}p).$ \end{minipage}\end{equation}
Using these ideas, we shall now prove
\begin{theorem}\label{T.W} Let $k$ be a field of characteristic~$0.$ Then the algebra
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:R'=.} $R'\ =\ k\lang p,s,q\mid ps=sp+1,\ pqp=p\rang$ \end{minipage}\end{equation}
has a $\!k\!$-basis consisting of all words in $p,$ $s$ and $q$ in which no $p$ is immediately followed by an $s,$ and the $\!p\!$'s that occur \textup{(}if any\textup{)} form a single consecutive string. In other words, every such word has the form
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:n_form} $s^{a_0}\,q\,s^{a_1}\,q\,\dots\,q\,s^{a_{m-1}}\,q\,s^{a_m}\,p^b\,q\, s^{a_{m+1}}\,q\,\dots\,q\,s^{a_n},$ \end{minipage}\end{equation}
where $0\leq m\leq n,$ $b\geq 0,$ and all $a_i\geq 0.$ \textup{(}Remark: If $b=0,$ then $m$ is, of course, not uniquely defined.\textup{)} \end{theorem}
\begin{proof} We shall first show that every monomial in our generators can be reduced to a linear combination of monomials~\eqref{d.W:n_form}, so that these span $R',$ then that the set of such monomials is $\!k\!$-linearly independent. We will not follow the formalism of the Diamond Lemma, though some of the ideas will be similar. In particular, in the first part of our proof, we shall associate to every monomial a $\!4\!$-tuple of natural numbers, and show that every monomial not of the form~\eqref{d.W:n_form} is equal in $R'$ to a $\!k\!$-linear combination of monomials each of which has smaller associated $\!4\!$-tuple, under lexicographic ordering. This is enough to show that every monomial is a $\!k\!$-linear combination of monomials~\eqref{d.W:n_form}. (If not every monomial were so expressible, there would be a least $\!4\!$-tuple associated with a counterexample monomial $w,$ and applying a reduction of the indicated sort to $w$ would give a contradiction.)
The $\!4\!$-tuple we shall associate with a word $w$ is
\begin{equation}\begin{minipage}[c]{35pc}\label{d.aaab} $h(w)\ =\ (a_q(w),\,a_p(w),\,a_s(w),\,b_{p,s}(w)),$ \end{minipage}\end{equation}
where the first three coordinates are the numbers of $\!q\!$'s, $\!p\!$'s and $\!s\!$'s in $w,$ and the last is the number of occurrences of a $p$ anywhere before an $s,$ i.e., the number of ordered pairs $(i,j)$ with $i<j$ such that the $\!i\!$-th factor of $w$ is a $p$ and the $\!j\!$-th is an $s.$ This refinement of the coordinate ``number of occurrences of the element $(ps)$'' that we used in \S\ref{S.1=:norm} is needed here: if we simply counted occurrences of the string $ps,$ calling this number $a_{ps}(w),$ then inequalities involving this function would not respect formal multiplication of monomials: clearly, $a_{ps}(sp)<a_{ps}(ps),$ yet multiplying these monomials on the left by $p$ we find that $a_{ps}(psp)\not<a_{ps}(pps).$ However, I claim that for $h$ defined by~\eqref{d.aaab}, if $h(u)\leq h(u')$ and $h(v)\leq h(v'),$ with at least one of these inequalities strict, then $h(uv)<h(u'v').$ Indeed, this is obvious except in the case where the first three coordinates of $h(u)$ agree with those of $h(u')$ and the first three coordinates of $h(v)$ agree with those of $h(v'),$ so that the comparison depends on the $\!4\!$-th coordinate, $b_{p,s}.$ Now it is easy to see that in general, $b_{p,s}(uv) = b_{p,s}(u) + b_{p,s}(v) + a_p(u)\,a_s(v),$ so when the $\!a\!$-coordinates are the same for $u$ and $u',$ and likewise for $v$ and $v',$ the $\!b_{p,s}\!$-coordinate of our product depends additively on the $\!b_{p,s}\!$-coordinates of the factors, from which the desired inequality follows.
So let us assume $w$ is a monomial not of the form~\eqref{d.W:n_form}, and prove that it is a linear combination of monomials with smaller values of $h.$
If $w$ contains a sequence $ps,$ then applying the relation $ps=sp+1,$ we get a sum of two monomials on each of which $h$ clearly has value $<h(w).$
If $w$ has no subsequence $ps,$ then to fail to have the form~\eqref{d.W:n_form}, it must have two $\!p\!$'s with a nonempty string of non-$\!p\!$ terms between them. Writing $u$ and $v$ for the (possibly empty) segments before and after these two $\!p\!$'s, we can write
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:up...pv} $w\ =\ u\ p\,q\,s^{m_1}\,q\,\dots\,q\,s^{m_{n-1}}\,q\,s^{m_n}\,p\ v,$ \quad where $n\geq 1$ and $s_{m_1},\dots,s_{m_n}\geq 0.$ \end{minipage}\end{equation}
It will now suffice to show that the string between $u$ and $v,$
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:p...p} $p\,q\,s^{m_1}\,q\,\dots\,q\,s^{m_n}\,q\,s^{m_n}\,p$ \end{minipage}\end{equation}
is equal in $R'$ to a $\!k\!$-linear combination of words on which $h$ has lower values.
As a first step, let us use the relation~\eqref{d.W:ps^m} in reverse, to replace the final $s^{m_n}\,p$ of~\eqref{d.W:p...p} with $p\,s^{m_n} - m_n s^{m_n-1}$ if $m_n>0,$ turning~\eqref{d.W:p...p} into a linear combination of two monomials. One of these, the one arising from the $s^{m_n-1}$ term, has a strictly lower value of $h,$ so we can ignore it. The other,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:p...ps^m} $p\,q\,s^{m_1}\,q\,\dots\,q\,s^{m_{n-1}}\,q\,p\,s^{m_n}$ \end{minipage}\end{equation}
has value of $h$ that is higher than~\eqref{d.W:p...p}, but only in its $b_{p,s}$ coordinate. I claim now that we can further rewrite~\eqref{d.W:p...ps^m} so as to turn it into a $\!k\!$-linear combination of monomials all involving fewer $\!q\!$'s, and hence all having lower values of $h$ than~\eqref{d.W:p...p} has. If $m_n=0$ (the case we temporarily excluded at the start of this paragraph),~\eqref{d.W:p...ps^m} is the same as~\eqref{d.W:p...p}; so in either case we have the latter expression to consider.
If $n=1,$ then~\eqref{d.W:p...ps^m} has the form $p\,q\,p\,s^{m_1},$ which clearly equals $p\,s^{m_1},$ giving a decreased value of $a_q,$ as desired. For $n>1,$ the idea will be, as indicated, to apply~\eqref{d.W:s^m=} to each of the factors $s^{m_i}$ nestled between the $\!q\!$'s, and then treat these as we treated the factors $1=ps+tp$ in~\S\ref{S.1=:norm}. Now replacing $s^m_i$ by $(m_i+1)^{-1}(ps^{m_i+1} - s^{m_i+1}p)$ gives terms with larger values of $a_p,$ $a_s$ and (usually) $b_{p,s}$ than~\eqref{d.W:p...ps^m} had; but it does not affect the value of $a_q,$ so we will still be safe if the result can be reduced to a linear combination of terms all having lower values of $a_q.$
To formalize this process, we shall apply Corollary~\ref{C.multilin}, with $n{-}1$ for the $n$ of that corollary, taking for $A$ a $\!2\!$-dimensional $\!k\!$-vector-space with basis written $\{x,y\},$ and for $A'$ the underlying $\!k\!$-vector-space of $R'.$ Let us define $\!k\!$-linear maps $\mu_1,\dots,\mu_{n-1}: A\to R'$ by letting $\mu_i$ carry $x$ to $(m_i+1)^{-1}s^{m_i+1}\,p,$ and $y$ to $(m_i+1)^{-1}p\,s^{m_i+1},$ so that it carries $x{+}y$ to $s^{m_i}.$ Define $\mu:A^{n-1}\to A'$ by
\begin{equation}\begin{minipage}[c]{35pc}\label{d.W:*m} $\mu(a_1,\dots,a_{n-1})\ = \ p\,q\,\mu_1(a_1)\,q\,\dots\,q\,\mu_{n-1}(a_{n-1})\,q\,p\,s^{m_n}.$ \end{minipage}\end{equation}
By Corollary~\ref{C.multilin}, the sum of the set $S(n{-}1)$ defined in that corollary using the map~\eqref{d.W:*m} equals zero. We find that the term of that sum in which all of $a_1,\dots,a_{n-1}$ are $x{+}y$ is exactly~\eqref{d.W:p...ps^m}, while in every other term, at least one of the $n$ $\!q\!$'s has a $p$ before it and a $p$ after it, so that an application of the relation $pqp=p$ allows us to reduce the number of $\!q\!$'s. So~\eqref{d.W:p...ps^m} is equal to a linear combination of monomials each involving fewer $\!q\!$'s, as claimed. This completes our proof that the elements~\eqref{d.W:n_form} span $R'.$
How shall we now show these elements $\!k\!$-linearly independent? One approach would be to formalize the above argument as giving a reduction system in the sense of the Diamond Lemma, and verify that all its ambiguities are reducible. But that verification was already tedious in the simpler context of Theorem~\ref{T.1=}.
Rather, let us apply Theorem~\ref{T.1=} to the generating set~\eqref{d.W:B=} of $R',$ and then show that when the monomials~\eqref{d.W:n_form} are expressed in terms of the basis given by that theorem, they have distinct leading terms, proving them $\!k\!$-linearly independent.
Of course, to define ``leading term'', we need a total ordering on the basis of $R'$ in question. To describe the ordering we will use, let the ``weight'' of a member of the basis $B$ of~\eqref{d.W:B=} be the highest exponent of $s$ appearing in its expression. (E.g., the weight of $s^m p + m s^{m-1}\in B_{+-}$ is $m.)$ We now define a word $w$ in the elements of $B\cup\{q\}$ to be larger than a word $w'$ if it involves more $\!q\!$'s; or if it involves the same number of $\!q\!$'s but the total weight of the factors from $B$ is higher, or if we have equality of both of these, but it has more terms from $B_{+-};$ while when all of these are equal, let the total ordering be chosen in an arbitrary fashion.
We now consider a word $w$ of the form~\eqref{d.W:n_form}, and the operation of expressing it in the normal form of Theorem~\ref{T.1=} determined by the basis~\eqref{d.W:B=} of our Weyl algebra; and ask what its leading term with respect to the above ordering will be.
First, suppose that $b,$ the exponent of $p$ in~\eqref{d.W:n_form}, is zero. Then to write $w$ as an expression (not reduced, to start with) in the elements of in $B\cup\{q\},$ we may replace every term $s^{a_i}$ with $a_i>0$ by $(a_i+1)^{-1}((ps^{a_i+1}) - (s^{a_i+1}p)),$ while writing any factors $s^{a_i}$ with $a_i=0$ as $1,$ the empty word. When we multiply this expression out, every pair of successive $\!q\!$'s are either adjacent, or have between them a generator $(ps^{a_i+1})\in B_{+-}$ or $(s^{a_i+1}p)\in B_{-+}.$ Those of the resulting words that have a member of $B_{-+}$
anywhere to the left of a member of $B_{+-}$ can be reduced by one of the reductions in our \eqref{d.1=:xpqpy|->}-series to a linear combination of words involving smaller numbers of $\!q\!$'s. Of those that remain, we see that the one that will be largest under our ordering will be (by the stipulation regarding elements of $B_{+-}$ in our description of that ordering), the one with the greatest number of factors from $B_{+-};$ i.e., the one in which $(ps^{a_i+1})\in B_{+-}$ has been used in each position where $a_i>0.$ Clearly this leading reduced word determines the sequence of exponents $a_i,$ hence it uniquely determines $w.$
Next, suppose $b=1.$ The first step in expressing $w$ in terms of the generators~\eqref{d.W:B=} is the same as before, except that the factor $s^{a_m} p,$ unlike the factors $s^{a_i},$ is not modified, since it is, as it stands, a member of $B_{-+}.$ In this case, all the words we get that have a member of $B_{+-}$ {\em after} that term again have a member of $B_{-+}$ to the left of a member of $B_{+-},$ and so can be reduced to terms with fewer $\!q\!$'s, so the terms that cannot be so reduced must have factors $(s^{a_i+1}p)\in B_{-+}$ in those positions. On the other hand, of the terms before $s^{a_m}\,p,$ the largest one under our ordering will again have all factors from $B$ of the form $(ps^{a_i+1})\in B_{+-}.$ So the largest term occurring determines both the sequence of $a_i$ and the position where the $p$ occurs in $w$ (namely, the position where the first element of $B_{-+}$ appears). Moreover, that leading term is not equal to the leading term of an expression with $b=0,$ since as we have seen, the latter have no factors in $B_{-+}.$
Finally, if $b>1,$ we have behavior similar to the case $b=1,$ except that the factor $s^{a_m}\,p^b$ now reduces to the sum of an element of $B_{++}$ and possibly an expression lower under our ordering. (By the description of $B_{++}$ in~\eqref{d.W:B=}, such a lower summand will appear if $b\leq a_m.)$ Only the former summand need be looked at; and we see again that the unique term having members of $B_{+-}$ before that element of $B_{++},$ and members of $B_{-+}$ after it, will be irreducible under the normal form of Theorem~\ref{T.1=}, and will give the leading term of our reduced expression. This leading term now determines both the value of $b$ and, as before, the values of $m$ and of the $a_i,$ and so again determines $w.$
This completes the proof of the Theorem. \end{proof}
When $\mathrm{char}(k)=e>0,$ things are somewhat different. On the one hand,~\eqref{d.W:m_geq_n} simplifies pleasantly whenever
$e\,|\,m(m{-}1)\dots(m{-}n{+}2).$ On the other hand, I claim that the elements $s^m$ with $m\equiv -1\pmod{e}$ are $\!k\!$-linearly independent modulo $pR+Rp.$ Indeed, since $R$ is spanned over $k$ by elements $s^m p^n,$ the space $pR$ is spanned by elements $p\,s^m p^n,$ and using~\eqref{d.W:ps^m} we see that in the expansions of these elements in terms of the basis~\eqref{d.W:s^mp^n}, basis elements $s^m$ with $m\equiv -1\pmod{e}$ never appear with nonzero coefficients. Since they also certainly do not appear with nonzero coefficients in the expressions in that basis for elements of $Rp,$ they do not appear in the expressions for elements of $pR+Rp.$ One finds that $\{s^m\mid m\equiv -1\pmod{e}\}$ can be taken as a basis of $B_{--}.$ Probably one can get a normal form for $R'$ somewhat like the above; but with multiple clusters of $\!p\!$'s allowed, separated by strings $q\,s^m\,q$ with $m\equiv -1\pmod{e}.$ However, I have not looked into this.
\section{Late addendum: mutual inner inverses}\label{S.mutual}
At about the time this paper was accepted for publication, I received a preprint of~\cite{A+KOM}, in which P.\,Ara and K.\,O'Meara used results in the preprint version of this note to answer an open question on nilpotent regular elements in rings. Their method required them to extend the result of Theorem~\ref{T.1_notin}, for a certain $R,$ to get a description of the $\!k\!$-algebra generated over that $R$ by a universal {\em mutual} inner inverse of $p,$ $R''=R\lang q\mid pqp=p,\,qpq=q\rang.$ This led me to wonder whether I could save them that awkwardness, and get some useful general results, by extending some of the material of this paper to mutual inner inverses. (Incidentally, what I am calling ``mutual inner inverses'' are more often called ``generalized inverses'', and are so called in \cite{A+KOM}. But I prefer to use here a term that highlights their relation with inner inverses.)
The symmetry of the property of being mutually inner inverse suggests that, just as $p$ is taken in Theorem~\ref{T.1_notin} to be an element of a fairly general $\!k\!$-algebra $R,$ so $q$ might be taken from another such $\!k\!$-algebra $S.$ And, indeed, it turns out that if such $p$ and $q$ are {\em nonzero} and satisfy $1\notin pR+Rp,$ $1\notin qS+Sq,$ then we can build on Theorem~\ref{T.1_notin} to get a very similar normal form for this construction. In this normal form, we will, on the one hand, use a $\!k\!$-basis $B$ for $R$ as in Theorem~\ref{T.1_notin} (but note that in the present situation, the qualifying phrase ``if $p\neq 0$'' can be removed from the condition that $B_{++}$ contain $p,$ in the first line of~\eqref{d.B_}, since, as noted above, $p$ is here assumed nonzero). Likewise, we will use a $\!k\!$-basis for $S$ of the analogous form,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C=} $C\cup\{1\}\ =\ C_{++}\cup C_{+-} \cup C_{-+}\cup C_{--}\cup \{1\},$ \end{minipage}\end{equation}
where
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C_} $C_{++}$ is any $\!k\!$-basis of $qS\cap Sq$ which contains $q,$
$C_{+-}$ is any $\!k\!$-basis of $qS$ relative to $qS\cap Sq,$
$C_{-+}$ is any $\!k\!$-basis of $Sq$ relative to $qS\cap Sq,$
$C_{--}$ is any $\!k\!$-basis of $S$ relative to $qS+Sq+k.$ \end{minipage}\end{equation}
We can now state and prove
\begin{theorem}\label{T.mutual} Suppose $R$ and $S$ are $\!k\!$-algebras \textup{(}which for notational simplicity we will assume are disjoint except for the common subfield $k),$ and let $p\in R-\{0\},$ $q\in S-\{0\}$ satisfy
\begin{equation}\begin{minipage}[c]{35pc}\label{d.1_notinx2} $1\notin pR+Rp,\qquad 1\notin qS+Sq.$ \end{minipage}\end{equation}
Let $B\cup\{1\}$ be a $\!k\!$-basis for $B$ as in~\eqref{d.B=} and~\eqref{d.B_}, and $C\cup\{1\}$ a $\!k\!$-basis for $S$ as in~\eqref{d.C=} and~\eqref{d.C_}.
Then the $\!k\!$-algebra $T$ freely generated by the two $\!k\!$-algebras $R$ and $S,$ subject to the two additional relations
\begin{equation}\begin{minipage}[c]{35pc}\label{d.pqp,qpq} $pqp\ =\ p,\quad qpq\ =\ q,$ \end{minipage}\end{equation}
has a $\!k\!$-basis given by all words in $B\cup C$ which contain no subwords as in~\eqref{d.xy} or~\eqref{d.xpqpy} \textup{(}that is, no subwords of the form $xy$ with $x,y\in B,$ or $(xp)\,q\,(py)$ with $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-}),$ nor any subwords of the analogous forms
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C:xy} $xy$ \quad with $x,y\in C,$ \end{minipage}\end{equation}
or
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C:xqpqy} $(xq)\,p\,(qy)$ \quad with $xq\in C_{++}\cup C_{-+}$ and $qy\in C_{++}\cup C_{+-}\,.$ \end{minipage}\end{equation}
The reduction to the above normal form may be accomplished by the reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->} of Theorem~\ref{T.1_notin}, together with the analogous reductions,
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C:xy|->} $xy\ \mapsto\ (xy)_S$ \quad for all $x,y\in C,$ \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.C:xqpqy|->} $(xq)\,p\,(qy)\ \mapsto\ (xqy)_S$ \quad for all $xq\in C_{++}\cup C_{-+},$ $qy\in C_{++}\cup C_{+-}\,.$ \end{minipage}\end{equation}
\end{theorem}
\begin{proof}
It is clear that the reductions~\eqref{d.xy|->}, \eqref{d.xpqpy|->},
\eqref{d.C:xy|->} and~\eqref{d.C:xqpqy|->} correspond to relations holding in $T,$ and include enough relations to present that algebra, and that they all reduce the lengths of their input-words. So it suffices to check that all ambiguities of the resulting reduction system are resolvable.
Note that the input-word of each of the reductions~\eqref{d.xy|->},~\eqref{d.xpqpy|->}
begins and ends with generators from $B,$ while the input-words of~\eqref{d.C:xy|->} and~\eqref{d.C:xqpqy|->} begin and end with generators from $C.$
Hence, if an ambiguity in our reduction system involves an overlap of only one letter, the two words must either both come from~\eqref{d.xy|->} and/or~\eqref{d.xpqpy|->}, or both come from~\eqref{d.C:xy|->} and/or~\eqref{d.C:xqpqy|->}. In the former case, that ambiguity will be resolvable by Theorem~\ref{T.1_notin}, and in the latter case, by that same theorem applied with $S,$ $q$ and $p$ in the roles of $R,$ $p$ and $q.$
It remains to consider two-letter overlaps. We implicitly noted in the proof of Theorem~\ref{T.1_notin}
that there were no such overlaps involving only reductions~\eqref{d.xy|->} and/or~\eqref{d.xpqpy|->}; so there are likewise none involving only~\eqref{d.C:xy|->} and/or~\eqref{d.C:xqpqy|->}. Hence two-letter overlaps must involve one reduction from the former family and one from the latter. However, the only generators appearing in both families of reductions are $p$ and $q.$ From this it is easy to check that the remaining ambiguously reducible monomials are precisely
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xpqpqy} $(xp)\cdot q\,p\cdot (qy),$\quad where\quad $xp\in B_{++}\cup B_{-+},$ $qy\in C_{++}\cup C_{+-},$ \end{minipage}\end{equation}
and
\begin{equation}\begin{minipage}[c]{35pc}\label{d.xqpqpy} $(xq)\cdot p\,q\cdot (py),$\quad where\quad $xq\in C_{++}\cup C_{-+},$ $py\in B_{++}\cup B_{+-}.$ \end{minipage}\end{equation}
I claim that the two competing reductions applicable to~\eqref{d.xpqpqy} each reduce it to $(xp)(qy).$
Indeed, to reduce the initial string $(xp)\,q\,p$ in~\eqref{d.xpqpqy}, we write the factor $p$ as $(p1)\in B_{++}$ and apply~\eqref{d.xpqpy|->}, getting $(xp)\,q\,(p1)\mapsto (xp1)_R=(xp);$ which reduces the product~\eqref{d.xpqpqy} to $(xp)(qy).$
The other reduction similarly applies~\eqref{d.C:xqpqy|->} to the final string $q\,p\,(qy),$ and gives the same result.
Likewise, the two reductions applicable to~\eqref{d.xqpqpy} both reduce it to $(xq)(py).$
Hence all the ambiguities of our reduction system are resolvable, so $T$ has a normal form given by the words irreducible under that system; that is, those having no subwords~\eqref{d.xy}, \eqref{d.xpqpy}, \eqref{d.C:xy} or \eqref{d.C:xqpqy}, as required. \end{proof}
The construction needed for \cite{A+KOM} can now be gotten as a special case.
\begin{corollary}\label{C.mutual} As in Theorem~\ref{T.1_notin}, let $R$ be a $\!k\!$-algebra, $p$ an element of $R$ such that $1\notin pR+Rp,$ and $B\cup\{1\}$ a basis of $R$ as in~\eqref{d.B=} and~\eqref{d.B_}; and let us also assume $p\neq 0.$ Let
\begin{equation}\begin{minipage}[c]{35pc}\label{d.R''} $R''\ =\ R\,\lang q\mid pqp=p,\,qpq=q\rang,$ \end{minipage}\end{equation}
i.e., the $\!k\!$-algebra gotten by adjoining to $R$ a universal mutual inner inverse $q$ to $p.$
Then $R''$ has a $\!k\!$-basis given by all words in the generating set $B\cup\{q\}$ which contain no subwords as in~\eqref{d.xy} or~\eqref{d.xpqpy} \textup{(}that is, no subwords of the form $xy$ with $x,y\in B$ or $(xp)\,q\,(py)$ with $xp\in B_{++}\cup B_{-+},$ $py\in B_{++}\cup B_{+-}),$ nor any subwords
\begin{equation}\begin{minipage}[c]{35pc}\label{d.qpq} $q\,p\,q.$ \end{minipage}\end{equation}
The reduction to the above normal form may be accomplished by the reductions~\eqref{d.xy|->} and~\eqref{d.xpqpy|->} of Theorem~\ref{T.1_notin}, together with the reduction
\begin{equation}\begin{minipage}[c]{35pc}\label{d.qpq|->} $q\,p\,q\ \mapsto\ q.$ \end{minipage}\end{equation}
\end{corollary}
\begin{proof} The normal form described is essentially that of the case of Theorem~\ref{T.mutual} where $S$ is the polynomial ring $k[q],$ and $C=C_{++}=\{q^n\mid n>0\}.$ There is the formal difference that words in the basis described in this corollary may contain strings of the generator $q,$ while each such string is represented in the basis gotten from Theorem~\ref{T.mutual} as a single generator $(q^n);$
however, the systems of elements of $R''$ described by the resulting words are clearly the same. Likewise, in the indicated case of Theorem~\ref{T.mutual}, the reduction~\eqref{d.qpq|->} is supplemented by the reductions $(q^m)\,p\,(q^n)\mapsto (q^{m+n-1})$ for all $m,n>0;$
but the reduction~\eqref{d.qpq|->} applied to the subword $qpq$ of the length-$\!m{+}n{+}1\!$ string $q^m p\,q^n$ clearly has the corresponding effect.
We remark that it would have been no harder -- but also not significantly easier -- to verify directly that adding~\eqref{d.qpq|->} to the reductions of Theorem~\ref{T.1_notin} yields a reduction system for $R''$ with all ambiguities resolvable. \end{proof}
It is easy to supplement Theorem~\ref{T.mutual} with a normal form result paralleling Proposition~\ref{P.M_norm} for the $\!T\!$-module induced by a $\!p\!$-tempered right $\!R\!$-module, or by a $\!q\!$-tempered right $\!S\!$-module, defined analogously.
\section{Further questions and observations}\label{S.further}
Do results paralleling Theorem~\ref{T.mutual} and Corollary~\ref{C.mutual} hold without the hypotheses $1\notin pR+Rp$ and $1\notin qS+Sq$?
For the analog of Corollary~\ref{C.mutual}, where we are only free to modify $R,$ we can say ``yes'' in the situation of \S\ref{S.1-sided}, and ``probably'' in that of \S\ref{S.1=:norm}. In former situation, taking $p$ right invertible in $R,$ we saw in \S\ref{S.1-sided} that in $R'=R\lang q\mid pqp=p\rang$ our adjoined element $q$ also became a right inverse to $p.$ But this makes $p$ and $q$ mutually inner inverse; so $R''=R';$ so the additional relation $qpq=q$ and the reduction $qpq\mapsto q$ have no additional effect, nor does exclusion of the string $qpq$ from words in our basis. Thus, for this case the analog of Corollary~\ref{C.mutual} is trivially true.
For $R$ and $p$ as in \S\ref{S.1=:norm}, hand calculations I have made suggest that the analog of Corollary~\ref{C.mutual} also holds:
All the ambiguities arising from overlaps between~\eqref{d.qpq|->}
and the reductions \eqref{d.1=:xpqpy|->}, (\xppy{2}),
(\xppy{3}), \eqref{d.1=:xpqtp|->} and~(\xptp{2}) appear to be resolvable, so it is likely that computations like those of \S\ref{S.1=:norm} can prove the same for ambiguities involving~\eqref{d.qpq|->} and any of the reductions~(\xppy{n}) and~(\xptp{n}).
On the other hand, for Theorem~\ref{T.mutual}, the obvious generalization with $R$ no longer assumed to satisfy $1\notin pR+Rp,$ while $S$ is still assumed to satisfy $1\notin qS+Sq,$ but not restricted to be $k[q],$ definitely does not hold. For an extreme example, if $p\in R$ is a nonzero element generating within $R$ a finite-dimensional field extension $F$ of $k,$ then $F$ will also be generated by $p^{-1},$ hence if an element $q\in S$ is to become an inner inverse of $p$ in an algebra containing (embedded copies of) both $R$ and $S,$ the subalgebra of $S$ generated by $q$ must have the same structure $F;$ which cannot be true if $1\notin qS\cap Sq$ (and is very restrictive even if this is not assumed). To see that there are also obstructions to the analog of Theorem~\ref{T.mutual} when $1\in pR+Rp-(pR\cup Rp),$ take $S=k[q\mid q^2=0]$ (which clearly satisfies $1\notin qS+Sq).$ We saw in \S\ref{S.1=} that the relations $pqp=p$ and $1\in pR+Rp$ together imply $pqqp=pq+qp-1$~\eqref{d.1=:pqqp}. Combining this with the relation $q^2=0$ holding in $S,$ we get $pq+qp=1.$ But $pq,$ $qp$ and $1$ are distinct words not containing any subwords \eqref{d.1=:xy}-\eqref{d.1=:xpqtp}, \eqref{d.C:xy} or~\eqref{d.C:xqpqy}; so if the analog of Theorem~\ref{T.mutual} held, they would be $\!k\!$-linearly independent. Nor does it help to assume, instead, that $S$ and $q$ satisfy $1\in qS+Sq-(qS\cup Sq);$ for if we take for $S$ the $2\times 2$ matrix ring over $k,$ and for $q$ the square-zero matrix $e_{12},$ we get the same problem just described.
But perhaps others will be able to find useful normal form results for some cases of this construction.
We end this note by recording an alternative way to construct the algebra $R''=R\,\lang q\mid pqp=p,\,qpq=q\rang$ from $R'=R\,\lang q\mid pqp=p\rang,$ implicitly noted in the original version of~\cite{A+KOM}. This does not require that our algebras be over a field, so we assume an arbitrary commutative base ring $K.$
\begin{lemma}[after P.\,Ara and K.\,O'Meara, original version of \cite{A+KOM}]\label{L.retract} Let $R$ be an algebra over a commutative ring $K,$ let $p$ be any element of $R,$ and let $R'$ be the $\!K\!$-algebra $R\,\lang q\mid pqp=p\rang.$
Then $R'$ admits a retraction \textup{(}idempotent $\!K\!$-algebra endomorphism\textup{)} $\varphi$ that fixes the image of $R,$ and takes $q$ to $qpq.$ The retract $\varphi(R')$ is naturally isomorphic to $R''=R\,\lang q\mid pqp=p,\,qpq=q\rang,$ via an isomorphism $\psi$ that carries $q\in R''$ to $\varphi(q)=qpq\in\varphi(R').$ \end{lemma}
\begin{proof} The defining relation $pqp=p$ of $R'$ clearly implies the two relations
\begin{equation}\begin{minipage}[c]{35pc}\label{d.p(qpq)p&} $p\cdot qpq\cdot p=p$\quad and\quad $qpq\cdot p\cdot qpq=qpq.$ \end{minipage}\end{equation}
The first shows that $qpq$ satisfies the relation over $R$ that is imposed on $q$ in $R';$ hence $R'$ admits an endomorphism $\varphi$ over $R$ taking $q$ to $qpq,$ and by the second relation, $\varphi$ is idempotent. Moreover, the relations of~\eqref{d.p(qpq)p&} together show that the image of $q$ in $\varphi(R')$ satisfies the relations imposed on $q$ in the definition of $R'';$ so we get a homomorphism $\psi:R''\to\varphi(R')$ taking $q$ to $\varphi(q)=qpq.$ On the other hand, the factor-map $\theta:R'\to R''$ takes $qpq\in R'$ to $qpq=q\in R'',$ from which it is easily seen that the restriction of $\theta$ to $\varphi(R')$ is a $\!2\!$-sided inverse to $\psi,$ establishing the asserted isomorphism. \end{proof}
So if we know the structure of $R',$ the above lemma gives us a way of studying $R''.$ However, I have not found it easy to apply this to the description of $R'$ that we obtained in~\S\ref{S.1=:norm} for the case $1\in pR+Rp - (pR\cup Rp),$ because substituting $qpq$ for $q$ in normal-form expressions for elements of $R'$ gives expressions that are in general not in normal form. E.g., for $n>1$ the image $\varphi(q^n)$ can be reduced repeatedly using~\eqref{d.1=:pqqp}, and it is hard to see just what relations such reductions lead to.
\end{document} | arXiv |
Optimal selection of cleaner products in a green supply chain with risk aversion
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On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality
April 2015, 11(2): 529-547. doi: 10.3934/jimo.2015.11.529
A new method for strong-weak linear bilevel programming problem
Yue Zheng 1, , Zhongping Wan 2, , Shihui Jia 3, and Guangmin Wang 4,
College of Mathematics and Physics, Huanggang Normal University, Huanggang 438000, China
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072
School of Science, Wuhan University of Science and Technology, Wuhan 430081, China
School of Economics and Management, China University of Geosciences, Wuhan 430074, China
Received October 2012 Revised April 2014 Published September 2014
We first propose an exact penalty method to solve strong-weak linear bilevel programming problem (for short, SWLBP) for every fixed cooperation degree from the follower. Then, we prove that the solution of penalized problem is also that of the original problem under some conditions. Furthermore, we give some properties of the optimal value function (as a function of the follower's cooperation degree) of SWLBP. Finally, we develop a method to acquire the critical points of the optimal value function without enumerating all values of the cooperation degree from the follower, and thus this function is also achieved. Numerical results show that the proposed methods are feasible.
Keywords: partial cooperation, pessimistic formulation, penalty method, critical point., optimistic formulation, Bilevel programming.
Mathematics Subject Classification: Primary: 90C26, 90C3.
Citation: Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial & Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529
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Yue Zheng Zhongping Wan Shihui Jia Guangmin Wang | CommonCrawl |
On a heated incompressible magnetic fluid model
Rough solutions for the periodic Korteweg--de~Vries equation
On the blow-up boundary solutions of the Monge -Ampére equation with singular weights
Haitao Yang 1, and Yibin Chang 1,
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Received July 2010 Revised July 2011 Published October 2011
We consider the Monge-Ampére equations det$D^2 u = K(x) f(u)$ in $\Omega$, with $u|_{\partial\Omega}=+\infty$, where $\Omega$ is a bounded and strictly convex smooth domain in $R^N$. When $f(u) = e^u$ or $f(u)= u^p$, $p>N$, and the weight $K(x)\in C^\infty (\Omega )$ grows like a negative power of $d(x)=dist(x, \partial \Omega)$ near $\partial \Omega$, we show some results on the uniqueness, nonexistence and exact boundary blow-up rate of strictly convex solutions for this problem. Existence of such solutions will be also studied in a more general case.
Keywords: Monge-Ampére equation, uniqueness, singular weight, blow-up solution, boundary behavior..
Mathematics Subject Classification: Primary: 35j05, 35j25; Secondary: 35B4.
Citation: Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
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Haitao Yang Yibin Chang | CommonCrawl |
\begin{document}
\title[On a conjecture of L\"u, Li and Yang]{On a conjecture of L\"u, Li and Yang} \date{} \author[I. Lahiri and S. Majumder]{Indrajit Lahiri and Sujoy Majumder} \address{Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India.} \email{[email protected]} \address{Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India.} \email{[email protected], [email protected]} \renewcommand{\arabic{footnote}}{} \subjclass[2010]{30D35} \keywords{ Meromorphic functions, derivative, small function.} \renewcommand{\arabic{footnote}}{\arabic{footnote}} \setcounter{footnote}{0} \begin{abstract} In connection to a conjecture of W. L\"u. Q. Li and C. Yang we prove a result on small function sharing by a power of a meromorphic function with few poles and its derivative. Our results improve a number of known results. \end{abstract} \thanks{Typeset by \AmS -\LaTeX} \maketitle \section{Introduction Definitions and Results} In the paper a meromorphic function means it is meromorphic in the open complex plane $\mathbb{C}$. we use the standard notations of Nevanlinna theory e.g., $N(r, f)$, $m(r, f)$, $T(r, f)$, N(r, a; f)$, \overline N(r, a; f)$, $m(r, a;f)$ etc.\{see \cite{4}\}. We denote by $S(r, f)$ a quantity, not necessarily the same at each of its occurrence, that satisfies the condition $S(r, f) = o\{T(r, f)\}$ as $r \to \infty$ except possibly a set of finite linear measure.
A meromorphic function $a = a(z)$ is called a small function of a meromorphic function $f$, if $T(r, a) = S(r, f)$. Let us denote by $S(f)$ the class of all small functions of $f$. Clearly $\mathbb{C} \subset S(f)$ and if $f$ is a transcendental function, then every polynomial is a member of $S(f)$.
Let $f$ and $g$ be two non-constant meromorphic functions and $a \in S(f) \cap S(g)$. If $f - a$ and $g - a$ have the same zeros with the same multiplicities, then we say that $f$ and $g$ share the small function $a$ CM (counting multiplicities) and if we do not consider the multiplicities, then we say that $f$ and $g$ share the small function $a$ IM (ignoring multiplicities).
Let $k$ be a positive integer and $a\in S(f)$. We use $N_{k)}(r,a;f)$ to denote the counting function of zeros of $f - a$ with multiplicity not greater than $k$ , $N_{(k+1}(r,a;f)$ to denote the counting function of zeros of $f - a$ with multiplicity greater than $k$. Similarly we use $\overline N_{k)}(r,a;f)$ and $\overline N_{(k+1}(r,a;f)$ are their respective reduced functions.
In 1996, Br\"{u}ck \cite{2} studied the relation between $f$ and $f'$ if an entire function $f$ shares only one finite value CM with it's derivative $f'$. In this direction an interesting conjecture was proposed by Br\"{u}ck \cite{2}, which is still open in its full generality. \begin{conjA} Let $f$ be a non-constant entire function. Suppose \begin{eqnarray*} \rho_{1}(f):=\limsup\limits_{r \to \infty}\frac{\log \log T(r,f)}{\log r},\end{eqnarray*} the hyper-order of $f$, is not a positive integer or infinity. If $f$ and $f'$ share a finite value $a$ CM, then \begin{eqnarray}\label{s} \frac{f'-a}{f-a}=c\end{eqnarray} for some non-zero constant $c$.\end{conjA} The Conjecture for the special cases $(1)$ $a=0$ and $(2)$ $N(r,0;f')=S(r,f)$ had been established by Br\"{u}ck \cite{2}. From the differential equations \begin{eqnarray}\label{1} \frac{f'-a}{f-a}=e^{z^{n}},\;\;\; \frac{f'-a}{f-a}=e^{e^{z}},\end{eqnarray} we see that when $\rho_{1}(f)$ is a positive integer or infinity, the conjecture does not hold.\par The conjecture for the case that $f$ is of finite order had been proved by Gundersen and Yang \cite{4aa}, the case that $f$ is of infinite order with $\rho_{1}(f)<\frac{1}{2}$ had been proved by Chen and Shon \cite{2a}. Recently Cao \cite{1} proved that the Br\"{u}ck conjecture is also true when $f$ is of infinite order with $\rho_{1}(f)=\frac{1}{2}$. But the case $\rho_{1}(f)> \frac{1}{2}$ is still open. However, the corresponding conjecture for meromorphic functions fails in general (see \cite{4aa}). For example, if \begin{eqnarray*} f(z)=\frac{2e^{z}+z+1}{e^{z}+1},\end{eqnarray*} then $f$ and $f'$ share $1$ CM, but (\ref{s}) does not hold.\par It is interesting to ask what happens if $f$ is replaced by a power of it, say, $f^{n}$ in Br\"{u}ck's conjecture. From (\ref{1}) we see that the conjecture does not hold without any restriction on the hyper-order when $n=1$. So we only need to focus on the problem when $n\geq 2$.\par Perhaps Yang and Zhang \cite{16a} were the first to consider the uniqueness of a power of an entire function $F=f^{n}$ and its derivative $F'$ when they share certain value and that leads to a specific form of the function $f$. \par Yang and Zhang \cite{16a} proved that the Br\"{u}ck conjecture holds for the function $f^{n}$ and the order restriction on $f$ is not needed if $n$ is relatively large. Actually they proved the following result. \begin{theoA}\cite{16a} Let $f$ be a non-constant entire function, $n(\geq 7)$ be an integer and let $F=f^{n}$. If $F$ and $F'$ share $1$ CM, then $F\equiv F'$, and $f$ assumes the form $f(z)=ce^{\frac{1}{n}z}$, where $c$ is a non-zero constant.\end{theoA}
Improving all the results obtained in \cite{16a}, Zhang \cite{16aa} proved the following theorem. \begin{theoB}\cite{16aa} Let $f$ be a non-constant entire function, $n$, $k$ be positive integers and $a(\not\equiv 0,\infty)$ be a meromorphic small function of $f$. If $f^{n}-a$ and $(f^{n})^{(k)}-a$ share $0$ CM and $n\geq k+5$, then $f^{n}\equiv (f^{n})^{(k)}$, and $f$ assumes the form $f(z)=ce^{\frac{\lambda}{n}z}$, where $c$ is a non-zero constant and $\lambda^{k}=1$.\end{theoB} In 2009, Zhang and Yang \cite{17} further improved the above result in the following manner. \begin{theoC}\cite{17} Let $f$ be a non-constant entire function, $n$, $k$ be positive integers and $a(\not\equiv 0,\infty)$ be a meromorphic small function of $f$. Suppose $f^{n}-a$ and $(f^{n})^{(k)}-a$ share $0$ CM and $n\geq k+2$. Then conclusion of \textrm{Theorem B} holds.\end{theoC} In 2010, Zhang and Yang \cite{18} further improved the above result in the following manner. \begin{theoD}\cite{18} Let $f$ be a non-constant entire function, $n$ and $k$ be positive integers. Suppose $f^{n}$ and $(f^{n})^{(k)}$ share $1$ CM and $n\geq k+1$. Then conclusion of \textrm{Theorem B} holds.\end{theoD} In 2011, L\"{u} and Yi \cite{11} proved the following extension of Theorem D. \begin{theoE} \cite{11} Let $f$ be a transcendental entire function, $n$, $k$ be two integers with $n\geq k+1$, $F=f^{n}$ and $Q\not\equiv 0$ be a polynomial. If $F-Q$ and $F^{(k)}-Q$ share $0$ CM, then $F\equiv F^{(k)}$ and $f(z)=ce^{wz/n}$, where $c$ and $w$ are non-zero constants such that $w^{k}=1$.\end{theoE}
\begin{rem} It is easy to see that the condition $n\geq k+1$ in \textrm{Theorem E} is sharp by the following example.\end{rem} \begin{exm} Let $f(z)=e^{e^{z}}\int\limits_{0}^{z} e^{-e^{t}}(1-e^{t})t\; dt$ and $n=1$, $k=1$. Then \begin{eqnarray*} \frac{f'(z)-z}{f(z)-z}=e^{z}\end{eqnarray*} and $f'(z)-z$ and $f(z)-z$ share $0$ CM, but $f'\not\equiv f$. \end{exm}
In \cite{LLY} W. L\"u, Q. Li and C. Yang asked the question of considering two shared polynomials in Theorem E instead of a single shared polynomial. They answered the question for the first derivative of the power of a transcendental entire function and further proposed the following conjecture: \begin{conB} Let $f$ be a transcendental entire function, $n$ be a positive integer. If $f^{n} - Q_{1}$ and $(f^{n})^{(k)} - Q_{2}$ share $0$ CM and $n \geq k + 1$, then $\displaystyle (f^{n})^{(k)} = \frac{Q_{2}}{Q_{1}}f^{n}$, where $Q_{1}$ and $Q_{2}$ are polynomials with $Q_{1}Q_{2} \not\equiv 0$. If, further, $Q_{1} \equiv Q_{2}$, then $\displaystyle f = ce^{\frac{\omega z}{n}}$, where $c$ and $\omega$ are nonzero constants such that $\omega^{k} = 1$. \end{conB}
Recently the second author \cite{SM} fully resolved {\bf Conjecture B}. Thus giving rise to a further investigation of the possibility of replacing in {\bf Conjecture B} the shared polynomials by shared small functions. In the paper we, in one hand solve this problem and also in the other hand we try to relax the nature sharing of small functions, thereby improve a number of known results including that in \cite{SM}.
Extending the idea of weighted sharing \{\cite{IL1, IL2}\}, Lin and Lin \cite{7a} introduced the notion of weakly weighted sharing which is defined as follows. \begin{defi}\cite{7a} Let $f$ and $g$ be two non-constant meromorphic functions sharing a ``IM", for $a\in S(f)\cap S(g)$, and $k$ be a positive integer or $\infty$.\begin{enumerate}\item[(i)] $\overline N_{k)}^{E}(r,a)$ denotes the counting function of those zeros of $f - a$ whose multiplicities are equal to the corresponding zeros of $g - a$, both of their multiplicities are not greater than $k$, where each zero is counted only once. \item[(ii)] $\overline N_{(k}^{0}(r,a)$ denotes the reduced counting function of those zeros of $f - a$ which are zeros of $g - a$, both of their multiplicities are not less than $k$, where each zero is counted only once.\end{enumerate} \end{defi} \begin{defi}\cite{7a} For $a\in S(f) \cap S(g)$, if $k$ is a positive integer or $\infty$ and \begin{eqnarray*} \overline N_{k)}(r,a;f)-\overline N_{k)}^{E}(r,a)=S(r,f),\;\;\;\overline N_{k)}(r,a;g)-\overline N_{k)}^{E}(r,a)=S(r,g);\end{eqnarray*} \begin{eqnarray*} \overline N_{(k+1}(r,a;f)- \overline N_{(k+1}^{0}(r,a)=S(r,f),\;\;\;\overline N_{(k+1}(r,a;g)- \overline N_{(k+1}^{0}(r,a)=S(r,g); \end{eqnarray*} or if $k=0$ and \begin{eqnarray*} \overline N(r,a;f)-\overline N_{0}(r,a)=S(r,f),\;\;\;\overline N(r,a;g)-\overline N_{0}(r,a)=S(r,g),\end{eqnarray*} then we say $f$ and $g$ weakly share $a$ with weight $k$. Here we write $f$, $g$ share $``(a,k)"$ to mean that $f$, $g$ weekly share $a$ with weight $k$.\end{defi} Obviously, if $f$ and $g$ share $``(a,k)"$, then $f$ and $g$ share $``(a,p)"$ for any $p\;\;(0\leq p\leq k)$. Also we note that $f$ and $g$ share $a$ $``IM"$ or $``CM"$ if and only if $f$ and $g$ share $``(a,0)"$ or $``(a,\infty)"$, respectively (for the definitions of $``IM"$ and $``CM"$ see pp. 225 - 226 \cite{12a}). \par
We note that a rational function $f$ with $\overline N(r, \infty ; f) = S(r, f)$ must be a polynomial. Also a small function of a polynomial must be a constant. Since $k \geq 1$, clearly if $f$ is a polynomial, then the relation $(f^{n})^{(k)} = cf^{n}$ does not hold for any nonzero constant $c$ and $n \geq k$. Therefore in the following theorems we assume $f$ to be transcendental. \begin{theo}\label{t1} Let $f$ be a transcendental meromorphic function such that $ N(r,\infty;f)=S(r,f)$ and $a_{i} = a_{i}(z)(\not\equiv 0,\infty)$ be small functions of $f$, where $i=1,2$. Let $n$ and $k$ be two positive integers such that $n\geq k+1$. If $f^{n}-a_{1}$ and $(f^{n})^{(k)}-a_{2}$ share $``(0,1)"$, then $(f^{n})^{(k)}\equiv \frac{a_{2}}{a_{1}}f^{n}$. Furthermore, if $a_{1} \equiv a_{2}$, then $f(z)=ce^{\frac{\lambda}{n}z}$ where $c$ and $\lambda$ are non-zero constants such that $\lambda^{k}=1$. \end{theo} \begin{theo}\label{t2} Let $f$ be a transcendental meromorphic function such that $\overline N(r,\infty;f)=S(r,f)$ and $a_{i} = a_{i}(z)(\not\equiv 0,\infty)$ be small functions of $f$, where $i=1,2$. Let $n$ and $k$ be two positive integers such that $n\geq k$. If $f^{n}-a_{1}$ and $(f^{n})^{(k)}-a_{2}$ share $``(0,0)"$ and $\overline N_{2)}(r,0;f)=S(r,f)$, then $(f^{n})^{(k)}\equiv \frac{a_{2}}{a_{1}}f^{n}$. Furthermore, if $a_{1} \equiv a_{2}$, then $f^{n}\equiv (f^{n})^{(k)}$ and $f$ assumes the form $f(z)=ce^{\frac{\lambda}{n}z}$, where $c$ is a non-zero constant and $\lambda^{k}=1$. \end{theo} \begin{note}\label{n1} If $k \geq 2$, then in Theorem \ref{t2} instead of $\overline N_{2)}(r,0;f)=S(r,f)$ we can assume $N_{1)}(r,0;f)=S(r,f)$.\end{note} \begin{rem} It is easy to see that the condition $n\geq k+1$ in \textrm{Theorem \ref{t1}} is sharp by the following examples.\end{rem} \begin{exm} Let $f(z)=e^{2z}+z$. Then $f-a_{1}$ and $f'-a_{2}$ share $0$ CM and $N(r,\infty;f)=0$, but $f'\not\equiv \frac{a_{2}}{a_{1}}f$, where $a_{1}(z)=z+1$ and $a_{2}(z)=3$. \end{exm} \begin{exm} Let $f(z)=e^{2z}+z^{2}+z$. Then $f-a_{1}$ and $f'-a_{2}$ share $0$ CM and $N(r,\infty;f)=0$, but $f'\not\equiv \frac{a_{2}}{a_{1}}f$, where $a_{1}(z)=z^{2}+z+1$ and $a_{2}(z)=2z+3$. \end{exm} \begin{exm} Let \begin{eqnarray*} f(z)=e^{e^{z^{2}}}+1,\;\;a_{1}(z)=\frac{1}{1+e^{-z^{2}}},\;\;a_{2}(z)=-\frac{2z}{1+e^{-z^{2}}}.\end{eqnarray*}
We note that \begin{eqnarray*} f(z)-a_{1}(z)=\frac{1}{e^{z^{2}}+1}\Big[\Big(e^{z^{2}}+1\Big)e^{e^{z^{2}}}+1\Big]\end{eqnarray*} and \begin{eqnarray*} f'(z)-a_{2}(z)=\frac{2z}{1+e^{-z^{2}}}\Big[\Big(e^{z^{2}}+1\Big)e^{e^{z^{2}}}+1\Big].\end{eqnarray*}
Then $f-a_{1}$ and $f'-a_{2}$ share $``(0,\infty)"$ and $ N(r,\infty;f)=0$, but $f\not\equiv \frac{a_{2}}{a_{1}}f'$. \end{exm} \begin{exm} Let \begin{eqnarray*} f(z)=1-5(z+1)+ze^{z}\end{eqnarray*} and $a_{1}(z)=a_{2}(z)=-(4 + 4z + 5z^{2})$. We note that \begin{eqnarray*} f(z)-a_{1}(z)=z(e^{z}+5z-1)\end{eqnarray*} and \begin{eqnarray*} f'(z)-a_{2}(z)=(z+1)(e^{z}+5z-1).\end{eqnarray*} Then $f-a_{1}$ and $f'-a_{2}$ share $``(0,\infty)"$ and $N(r,\infty;f)=0$, but $f\not\equiv f'$.
\end{exm} \begin{rem} It is easy to see that the conditions $\overline N_{2)}(r,0;f)=S(r,f)$ and $\overline N(r,\infty;f)=S(r,f)$ in \textrm{Theorem \ref{t2}} are essential by the following examples.\end{rem} \begin{exm} Let \begin{eqnarray*} f(z)=z^{2}+\frac{1}{2}e^{(z-1)^{2}},\;\;a_{1}(z)=z^{2}+\frac{1}{2}\;\; and\;\;a_{2}(z)=3z-1.\end{eqnarray*}
We note that \begin{eqnarray*} f(z)-(z^{2}+\frac{1}{2})=\frac{1}{2}\Big[e^{(z-1)^{2}}-1\Big]\end{eqnarray*} and \begin{eqnarray*} f'(z)-(3z-1)=(z-1)\Big[e^{(z-1)^{2}}-1\Big]. \end{eqnarray*}
Obviously $f-a_{1}$ and $f'-a_{2}$ share $0$ IM, and $\overline N_{2)}(r,0;f)\not=S(r,f)$ and $\overline N(r, \infty; f) = 0$, but $f'\not\equiv \frac{a_{2}}{a_{1}}f$. \end{exm} \begin{exm} Let \begin{eqnarray*} f(z)=\frac{2}{1-e^{-2z}}.\end{eqnarray*} Clearly $f'(z)=-\frac{4e^{-2z}}{(1-e^{-2z})^{2}}$.
We note that \begin{eqnarray*} f(z)-1=\frac{1+e^{-2z}}{1-e^{-2z}}\;\;\; and\;\;\; f'(z)-1=-\frac{(1+e^{-2z})^{2}}{(1-e^{-2z})^{2}}. \end{eqnarray*} Obviously $f$ and $f'$ share $1$ IM, $\overline N(r,\infty;f)\neq S(r,f)$ and $\overline N_{2)}(r,0;f) = 0$, but $f'\not\equiv f$. \end{exm}
\begin{exm} Let $f(z) = 1 + \tan z$. Since $\tan z$ does not assume the values $\pm i$, it follows that $f(z)$ does not assume the values $1 \pm i$. So by the second fundamental theorem, $\overline N(r, 0; f) = \overline N_{2)}(r, 0; f) = T(r, f) + S(r, f)$ and $\overline N(r, \infty ; f) = T(r, f) + S(r, f)$. Also we see that $f'(z) - 1 = (f(z) - 1)^{2}$ and so $f$ and $f'$ share the value $1$ IM, but $f \not\equiv f'$. \end{exm}
\section {Lemmas} In this section we present the lemmas which will be needed in the sequel. \begin{lem}\label{l1}\cite{3} Suppose that $f$ is a transcendental meromorphic function and that \begin{eqnarray*} f^{n}(z)P(f(z))=Q(f(z)),\end{eqnarray*} where $P(f(z))$ and $Q(f(z))$ are differential polynomials in $f$ with functions of small proximity related to $f$ as the coefficients and the degree of $Q(f(z))$ is at most $n$. Then $m(r,P)=S(r,f).$ \end{lem} \begin{lem}\label{l5} \cite{4} Let $f$ be a non-constant meromorphic function and let $a_{1}(z)$, $a_{2}(z)$ be two meromorphic functions such that $T(r,a_{i})=S(r,f)$, $i=1,2$. Then \begin{eqnarray*} T(r,f)\leq \overline N(r,\infty;f)+\overline N(r,a_{1};f)+\overline N(r,a_{2};f)+S(r,f).\end{eqnarray*} \end{lem} \begin{lem}\label{l7}\cite{6a} Let $f(z)$ be a non-constant entire function and $k(\geq 2)$ be an integer. If $f(z)f^{(k)}(z)\not=0$, then $f(z)=e^{az+b}$, where $a\not=0, b$ are constant.\end{lem}
\begin{lem}\label{l9} Let $f$ be a non-constant meromorphic function such that $(f^{n})^{(k)}\equiv f^{n}$, where $k, n\in\mathbb{N}$. If $n\geq k$, then $f$ assumes the form $f(z)=ce^{\frac{\lambda}{n}z}$, where $c\in\mathbb{C}\setminus\{0\}$ and $\lambda^{k}=1$. \end{lem} \begin{proof} First we suppose \begin{eqnarray}\label{r1} (f^{n})^{(k)}\equiv f^{n}.\end{eqnarray} We claim that $f$ does not have any pole. In fact, if $z_{0}$ is a pole of $f$ with multiplicity $p$, then $z_{0}$ is a pole of $f^{n}$ with multiplicity $np$ and a pole of $(f^{n})^{(k)}$ with multiplicity $np+k$, which is impossible by (\ref{r1}). Hence $f$ is a non-constant entire function. From (\ref{r1}), it is clear that $f$ can not be a polynomial. Therefore $f$ is a transcendental entire function.\\ We now consider the following two cases.\\ {\bf Case 1.} Let $n>k$.\\ If $z_{1}$ is a zero of $f$ with multiplicity $q$, then $z_{1}$ is a zero of $f^{n}$ with multiplicity $nq$ and a zero of $(f^{n})^{(k)}$ with multiplicity $nq-k$ , which is impossible by (\ref{r1}). Therefore from (\ref{r1}), we conclude that $f^{n}(z)(f^{n}(z))^{(k)}\not=0$. If $k\geq 2$, then by \textrm{Lemma \ref{l7}} we have $f(z)=ce^{\frac{\lambda}{n}z}$, where $c\in\mathbb{C}\setminus\{0\}$ and $\lambda^{k}=1$. Next we suppose $k=1$. Since $f(z)\not=0, \infty$, it follows that $f(z)=e^{\alpha(z)}$, where $\alpha(z)$ is a non-constant entire function. Now from (\ref{r1}) we have $\alpha'(z)=\frac{1}{n}$, i.e., $\alpha(z)=\frac{1}{n}z+c_{0}$, where $c_{0}\in\mathbb{C}$. Consequently $f(z)=ce^{\frac{1}{n}z}$, where $c=e^{c_{0}}$.\\ {\bf Case 2.} Let $n=k$.\\ First we suppose $n=k=1$. Then from (\ref{r1}) we have $f(z)\equiv f'(z)$ and so $f(z)=ce^{z}$, where $c\in\mathbb{C}\setminus\{0\}$.\\ Next we suppose $n=k\geq 2$. Let $F=f^{n}$.
Then we have \begin{eqnarray}\label{r3} F^{(k)}&=&\frac{d^{k}}{dz^{k}}\Big\{f^{k}\Big\}\\&=& \frac{d^{k-1}}{dz^{k-1}}\Big\{kf^{k-1}f'\Big\}\nonumber\\&=& k\frac{d^{k-2}}{dz^{k-2}}\Big\{(k-1)f^{k-2}(f')^{2}+f^{k-1}f''\Big\}\nonumber\\&=& k(k-1)\frac{d^{k-2}}{dz^{k-2}}\Big\{f^{k-2}(f')^{2}\Big\} +k\frac{d^{k-2}}{dz^{k-2}}\Big\{f^{k-1}f''\Big\}\nonumber\\&=& k(k-1)\frac{d^{k-3}}{dz^{k-3}}\Big\{(k-2)f^{k-3}(f')^{3}\Big\}+k(k-1)\frac{d^{k-3}}{dz^{k-3}}\Big\{2f^{k-2}f'f''\big\}\nonumber\\&&+k\frac{d^{k-3}}{dz^{k-3}}\Big\{(k-1)f^{k-2}f'f''\Big\}+k\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-1}f''\Big\}\nonumber\\&=&k(k-1)(k-2)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-3}(f')^{3}\Big\}+2k(k-1)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-2}f'f''\big\}\nonumber\\&&+k(k-1)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-2}f'f''\Big\}+k\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-1}f''\Big\}\nonumber\\&=& \ldots\ldots\nonumber \\&=& k!(f')^{k}+R(f)\nonumber, \end{eqnarray} where $R(f)$ is a differential polynomial in $f$ such that each term of $R(f)$ contains $f^{m}$ for some $m (1\leq m\leq n-1)$ as a factor. \par From (\ref{r1}), we observe that $f$ can not have any multiple zero. Let $z_{2}$ be a simple zero of $f$. Clearly $z_{2}$ is a zero of $F$ of multiplicity $k$. From (\ref{r1}), it is clear that $z_{2}$ is also a zero of $F^{(k)}$. On the other hand $z_{2}$ is a zero of $R(f)$. Now from (\ref{r3}), we observe that $z_{2}$ is a zero of $f'$, which is impossible. Therefore $f$ can not have any simple zero. Hence $f$ does not have any zero. Since from (\ref{r1}) we see that $(f^{n}(z))^{(k)}f^{n}(z) \neq 0$, by Lemma \ref{l7} we have $f(z)=ce^{\frac{\lambda}{n}z}$, where $c\in\mathbb{C}\setminus\{0\}$ and $\lambda^{k}=1$. This completes the proof. \end{proof}
\section {Proofs of the theorems} \begin{proof}[Proof of Theorem \ref{t1}] Let \begin{eqnarray}\label{e4} F=f^{n}.\end{eqnarray} Since $S(r,f^{n})=S(r,f)$, from \textrm{Lemma \ref{l5}} we see that \begin{eqnarray*} nT(r,f)\leq \overline N(r,0;F)+\overline N(r,a_{1};F)+S(r,f^{n})=\overline N(r,0;f)+\overline N(r,a_{1};F)+S(r,f).\end{eqnarray*} Since $n\geq k+1$, it follows that $\overline N(r,a_{1};F)\not=S(r,f)$. As $F-a_{1}$ and $F^{(k)}-a_{2}$ share $``(0,1)"$, it follows that $\overline N(r,a_{2};F^{(k)})\not=S(r,f)$.\\ Let $z_{0}$ be a common zero of $F-a_{1}$ and $F^{(k)}-a_{2}$ such that $a_{i}(z_{0})\not=0,\infty$ (otherwise the reduced counting functions of those zeros of $F-a_{1}$ and $F^{(k)}-a_{2}$ which are the zeros or poles of $a_{1}(z)$ and $a_{2}(z)$ respectively are equal to $S(r,f)$), where $i=1,2$. Clearly $F(z_{0}),\; F^{(k)}(z_{0})\not=0$. Suppose $z_{0}$ is a zero of $F-a_{1}$ of multiplicity $p_{0}$. Since $F-a_{1}$ and $F^{(k)}-a_{2}$ share $``(0,1)"$, it follows that $z_{0}$ must be a zero of $F^{(k)}-a_{2}$ of multiplicity $q_{0}$. Then in some neighbourhood of $z_{0}$, we get by Taylor's expansion \begin{eqnarray*} F(z)=a_{10}+a_{1r_{0}}(z-z_{0})^{r_{0}}+a_{1r_{0}+1}(z-z_{0})^{r_{0}+1}+\ldots, a_{10}\not=0\end{eqnarray*} \begin{eqnarray*} a_{1}(z)=b_{10}+b_{1s_{0}}(z-z_{0})^{s_{0}}+b_{1s_{0}+1}(z-z_{0})^{s_{0}+1}+\ldots, b_{10}\not=0.\end{eqnarray*} Since $z_{0}$ is a zero of $F-a_{1}$ of multiplicity $p_{0}$, it follows that $a_{10}=b_{10}$ and $p_{0} \geq \min\{r_{0}, s_{0}\}$. Let us assume that \begin{eqnarray*} F(z)-a_{1}(z)=c_{1p_{0}}(z-z_{0})^{p_{0}}+c_{1p_{0}+1}(z-z_{0})^{p_{0}+1}+\ldots, c_{1p_{0}}\not=0.\end{eqnarray*} Therefore $\frac{F(z)-a_{1}(z)}{a_{1}(z)}=O((z-z_{0})^{p_{0}})$ and so $\frac{F(z)}{a_{1}(z)}-1=O((z-z_{0})^{p_{0}})$.
Similarly $\frac{F^{(k)}(z)-a_{2}(z)}{a_{2}(z)}=O((z-z_{0})^{q_{0}})$ and $\frac{F^{(k)}(z)}{a_{2}(z)}-1=O((z-z_{0})^{q_{0}})$.\\ Finally we conclude that $F-a_{1}$ and $F^{(k)}-a_{2}$ share $``(0,1)"$ if and only if $\frac{F}{a_{1}}$ and $\frac{F^{(k)}}{a_{2}}$ share $``(1, 1)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$ respectively.\par Let $F_{1}=\frac{f^{n}}{a_{1}}$ and $G_{1}=\frac{(f^{n})^{(k)}}{a_{2}}$. Clearly $F_{1}$ and $G_{1}$ share $``(1, 1)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$ respectively and so $\overline N(r,1;F_{1})=\overline N(r,1;G_{1})+S(r,f)$. Let \begin{eqnarray}\label{e1} \Phi=\frac{F_{1}'(F_{1}-G_{1})}{F_{1}(F_{1}-1)} = \frac{F'_{1}}{F_{1} -1}\left(1 - \frac{G_{1}}{F_{1}}\right) = \frac{F_{1}'}{F_{1} - 1}\left(1 - \frac{a_{1}}{a_{2}}\cdot \frac{F^{(k)}}{F}\right).\end{eqnarray} We now consider the following two cases.\\ {\bf Case 1.} Let $\Phi\not\equiv 0$. Then clearly $G_{1}\not\equiv F_{1}$, i.e., $(f^{n})^{(k)}\not\equiv \frac{a_{2}}{a_{1}} f^{n}$. Now from (\ref{e1}) we get $m(r,\infty;\Phi)=S(r,f)$.
Let $z_{1}$ be a zero of $f$ of multiplicity $p$ such that $a_{i}(z_{1})\not=0, \infty$, where $i=1,2$. Then $z_{1}$ will be a zero of $F_{1}$ and $G_{1}$ of multiplicities $np$ and $np-k$ respectively and so from (\ref{e1}) we get \begin{equation}\label{e11a} \Phi(z)=O((z-z_{1})^{np-k-1}).\end{equation}
Since $n\geq k+1$, it follows that $\Phi$ is holomorphic at $z_{1}$.
Let $z_{2}$ be a common zero of $F_{1}-1$ and $G_{1}-1$ such that $a_{i}(z_{2})\not=0, \infty$, where $i=1,2$. Suppose $z_{2}$ is a zero of $F_{1}-1$ of multiplicity $q$. Since $F_{1}$ and $G_{1}$ share $``(1,1)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$ respectively, it follows that $z_{2}$ must be a zero of $G_{1}-1$ of multiplicity $r$. Then in some neighbourhood of $z_{2}$, we get by Taylor's expansion \begin{eqnarray*} F_{1}(z)-1=b_{q}(z-z_{2})^{q}+b_{q+1}(z-z_{2})^{q+1}+\ldots, b_{q}\not=0\end{eqnarray*} \begin{eqnarray*} G_{1}(z)-1=c_{r}(z-z_{2})^{r}+c_{r+1}(z-z_{2})^{r+1}+\ldots, c_{r}\not=0.\end{eqnarray*}
Clearly \begin{eqnarray*} F_{1}'(z)=qb_{q}(z-z_{2})^{q-1}+(q+1)b_{q+1}(z-z_{2})^{q}+\ldots.\end{eqnarray*} Note that \begin{eqnarray*} F_{1}(z)-G_{1}(z)=\left\{\begin{array}{clcr} & b_{q}(z-z_{2})^{q}+\ldots,&\;\;\;\;\;\;\;\;{\text {if}}\; q<r \\ & -c_{r}(z-z_{2})^{r}- \ldots,&\;\;\;\;\;\;\; {\text {if}}\; q>r\\ & (b_{q}-c_{q})(z-z_{2})^{q}+\ldots,&\;\;\;\;\;\;\; {\text {if}}\; q=r.\end{array}\right.\end{eqnarray*} Clearly from (\ref{e1}) we get \begin{equation}\label{e11as} \Phi(z)=O\big((z-z_{2})^{t-1}\big),\end{equation} where $t\geq\min\{ q, r\}$. Now from (\ref{e11as}), it follows that $\Phi$ is holomorphic at $z_{2}$. \par We note from (\ref{e1}) that if $z_{*}$ is a zero of $F_{1} - 1$ that is also a zero of $a_{2}$ with multiplicity $p_{1}$, then $z_{*}$ is a possible pole of $\Phi$ with multiplicity at most $1 + p_{1}$. Again if $z^{*}$ is a zero of $f$ that is also a zero of $a_{2}$ with multiplicity $p_{2}$, then $z^{*}$ is a possible pole of $\Phi$ with multiplicity at most $k + p_{2}$. So from (\ref{e1}), above discussion and the hypothesis of \textrm{Theorem \ref{t1}} we note that \begin{eqnarray*} N(r, \infty; \Phi) &\leq & (k + 1)N(r, \frac {a_{1}}{a_{2}}) + (k + 1)N(r,0; a_{1}) + (k + 1)N(r, 0; a_{2})\\ && + (k + 1)\overline N(r, F_{1}) + (k + 1)\overline N(r, f) \\ & = & (k + 1)\overline N(r, F_{1}) + S(r, f) \\ & = & S(r, f). \end{eqnarray*} Consequently $T(r,\Phi)=S(r,f)$.\par Let $q\geq 2$. Since $F_{1}$ and $G_{1}$ share $``(1,1)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$, it follows that $r\geq 2$. Therefore from (\ref{e11as}) we see that \begin{eqnarray*} \overline N_{(2}(r,1;F_{1})\leq N(r,0;\Phi) + S(r, f)\leq T(r,\Phi) + S(r, f) = S(r,f).\end{eqnarray*}
Since $F_{1}$ and $G_{1}$ share $``(1,1)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$, it follows that $\overline N_{(2}(r,1;G_{1})=S(r,f)$. Again from (\ref{e1}) we get \begin{eqnarray*} \frac{1}{F_{1}}=\frac{1}{\Phi}\left(\frac{F_{1}'}{F_{1} - 1} -\frac{F_{1}'}{F_{1}}\right)\Big[1-\frac{a_{1}}{a_{2}}\frac{(f^{n})^{(k)}}{f^{n}}\Big] \end{eqnarray*} and so $m(r,\frac{1}{F_{1}})=S(r,f)$. Hence \begin{eqnarray}\label{e2} m(r, 0; f) = m(r,\frac{1}{f})=S(r,f).\end{eqnarray} We consider the following two sub-cases.\\ {\bf Sub-case 1.1.} Let $n>k+1$.\\ From (\ref{e11a}) we see that $N(r,0;f)\leq N(r,0;\Phi)\leq T(r,\Phi)+O(1)=S(r,f)$. Then from (\ref{e2}) we get $T(r,f)=S(r,f)$, which is a contradiction.\\ {\bf Sub-case 1.2.} Let $n=k+1$.\\ Since for $p \geq 2$, we have $np - k - 1 = (k + 1)p - k - 1 \geq p$, from (\ref{e11a}) we see that $$N_{(2}(r,0;f)\leq N(r,0;\Phi)\leq T(r,\Phi)+O(1)=S(r,f).$$ Then (\ref{e2}) gives \begin{eqnarray}\label{e14} T(r,f)=N_{1)}(r,0;f)+S(r,f).\end{eqnarray} Note that $\overline N_{(2}(r, a_{1};F) = \overline N_{(2}(r,1; F_{1}) + S(r, f) = S(r,f)$, $\overline N_{(2}(r, a_{2};F^{(k)}) = \overline N_{(2}(r,1; G_{1}) + S(r, f) = S(r,f)$ and $\overline N(r,\infty;F) = S(r,f)$. Let \begin{eqnarray}\label{e5} \beta=\frac{F^{(k)}-a_{2}}{F-a_{1}},\;\text{i.e.},\; F^{(k)}-a_{2}=\beta (F-a_{1}).\end{eqnarray} We claim that $\beta\not\equiv 0$. If not, suppose $\beta\equiv 0$. Then from (\ref{e5}) we have $(f^{n})^{(k)}\equiv a_{2}$. Since $n=k+1$, we immediately have $N_{1)}(r,0,f)=S(r,f)$ and so from (\ref{e14}) we arrive at a contradiction. Hence $\beta\not\equiv 0$. We now consider following two sub-cases.\\ {\bf Sub-case 1.2.1.} Suppose $T(r,\beta)\not=S(r,f)$.\\ Let $z_{11}$ be a zero of $F-a_{1}$ such that $F^{(k)}(z_{11})-a_{2}(z_{11})\not=0$. Then obviously $\beta$ has a pole at $z_{11}$. Let $z_{12}$ be a zero of $F^{(k)}-a_{2}$ such that $F(z_{12})-a_{1}(z_{12})\not=0$. In that case $\beta$ has a zero at $z_{12}$. Let $z_{13}$ be a common zero of $F-a_{1}$ and $F^{(k)}-a_{2}$. Since $F-a_{1}$ and $F^{(k)}-a_{2}$ share $``(0,1)"$, it follows that $\beta$ has a zero at $z_{13}$ if $z_{13}$ is a zero of $F-a_{1}$ and $F^{(k)}-a_{2}$ with multiplicities $p_{13}(\geq 2)$ and $q_{13}(\geq 2)$ respectively such that $p_{13}<q_{13}$ and $\beta$ has a pole at $z_{13}$ if $q_{13}<p_{13}$. Therefore \begin{eqnarray*} \overline N(r,0;\beta)\leq \overline N_{(2}(r,a_{2};F^{(k)})+S(r,f)=S(r,f)\end{eqnarray*} and \begin{eqnarray*} \overline N(r,\infty;\beta)\leq \overline N_{(2}(r,a_{1};F)+S(r,f)=S(r,f).\end{eqnarray*} Let $\xi=\frac{\beta'}{\beta}$. Clearly \begin{eqnarray*} T(r,\xi)= N(r,\infty;\frac{\beta'}{\beta})+ m(r,\frac{\beta'}{\beta})=\overline N(r,0;\beta)+ \overline N(r,\infty;\beta)+S(r,\beta)=S(r,f)+S(r,\beta).\end{eqnarray*} Note that \begin{eqnarray*} T(r,\beta)&\leq& T(r, F^{(k)}-a_{2})+T(r,F-a_{1})\\&\leq & T(r,F^{(k)})+T(r,F)+ S(r,F)+ S(r,G)\\&\leq& (k+1)T(r,f^{n})+ nT(r,f)+S(r,f)\\&=& n(k+2)T(r,f)+S(r,f),\end{eqnarray*} which implies that $S(r,\beta)$ can be replaced by $S(r,f)$. Consequently $T(r,\xi)=S(r,f)$. By logarithmic differentiation we get from (\ref{e5})
\begin{eqnarray}\label{e7} F^{(k+1)}F-\xi F^{(k)}F-F^{(k)}F'&=&a_{1}F^{(k+1)}-(\xi a_{1}+a_{1}')F^{(k)}-a_{2}F'\\&&+(a_{2}'-\xi a_{2})F+\xi a_{1}a_{2}+a_{2}a_{1}'-a_{1}a_{2}'\nonumber.\end{eqnarray} We deduce from (\ref{e4}) that \begin{eqnarray}\label{e8} F^{(k)}&=&\frac{d^{k}}{dz^{k}}\Big\{f^{k+1}\Big\}\\&=& \frac{d^{k-1}}{dz^{k-1}}\Big\{(k+1)f^{k}f'\Big\}\nonumber\\&=& (k+1)\frac{d^{k-2}}{dz^{k-2}}\Big\{k f^{k-1}(f')^{2}+f^{k}f''\Big\}\nonumber\\&=& (k+1)k\;\frac{d^{k-2}}{dz^{k-2}}\Big\{f^{k-1}(f')^{2}\Big\} +(k+1)\frac{d^{k-2}}{dz^{k-2}}\Big\{f^{k}f''\Big\}\nonumber\\&=& (k+1)k\;\frac{d^{k-3}}{dz^{k-3}}\Big\{(k-1)f^{k-2}(f')^{3}\Big\}+(k+1)k\;\frac{d^{k-3}}{dz^{k-3}}\Big\{2f^{k-1}f'f''\big\}\nonumber\\&&+(k+1)\frac{d^{k-3}}{dz^{k-3}}\Big\{k f^{k-1}f'f''\Big\}+(k+1)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k}f'''\Big\}\nonumber\\&=&(k+1)k(k-1)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-2}(f')^{3}\Big\}+2(k+1)k\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-1}f'f''\big\}\nonumber\\&&+(k+1)k\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k-1}f'f''\Big\}+(k+1)\frac{d^{k-3}}{dz^{k-3}}\Big\{f^{k}f'''\Big\}\nonumber\\&=& \ldots\ldots\nonumber \\&=& (k+1)!f(f')^{k}+\frac{k(k-1)}{4}(k+1)!f^{2}(f')^{k-2}f''+\ldots +(k+1)f^{k}f^{(k)}\nonumber. \end{eqnarray} Therefore \begin{eqnarray}\label{e8sm} \frac{f'}{f}F^{(k)}&=&(k+1)!(f')^{k+1}+\frac{k(k-1)}{4}(k+1)!f(f')^{k-1}f''+\ldots+\\&&(k+1)f^{k-1}f'f^{(k)}\nonumber\end{eqnarray} and \begin{eqnarray}\label{e9} F^{(k+1)}&=&(k+1)!(f')^{k+1}+\frac{k(k+1)}{2}(k+1)!f(f')^{k-1}f''+\ldots+\\&&(k+1)f^{k}f^{(k+1)}\nonumber.\end{eqnarray} Substituting (\ref{e4}), (\ref{e8}), (\ref{e8sm}) and (\ref{e9}) into (\ref{e7}), we have \begin{eqnarray}\label{e10} f^{n}(z)P(z)=Q(z),\end{eqnarray} where $Q(z)$ is a differential polynomial in $f$ of degree $n$ and \begin{eqnarray}\label{e11} P(z)&=& F^{(k+1)}-\xi F^{(k)}-n\frac{f'}{f} F^{(k)}\\&=& -k(k+1)!(f')^{k+1}-(k+1)!\xi f(f')^{k}\nonumber\\&&+\frac{k(k+1)(3-k)(k+1)!}{4} f(f')^{k-1}f''+\ldots+(k+1)f^{k}f^{(k+1)}\nonumber\\&&-(k+1)\xi f^{k}f^{(k)}-(k+1)^{2}f^{k-1}f'f^{(k)}=-k(k+1)!(f')^{k+1}+R_{1}(f)\nonumber,\end{eqnarray} is a differential polynomial in $f$ of degree $k+1$, where $R_{1}(f)$ is a differential polynomial in $f$ such that each term of $R_{1}(f)$ contains $f^{m}$ for some $m (1\leq m\leq n-1)$ as a factor.\par
We suppose that $P\equiv 0$. Then from (\ref{e11}) we get $\displaystyle F^{(k + 1)} - \xi F^{(k)} - n\frac{f'}{f}F^{(k)} \equiv 0$ and so $\displaystyle \frac{F^{(k + 1)}}{F^{(k)}} = \xi + n\frac{f'}{f} = \frac{\beta'}{\beta} + \frac{F'}{F}.$ By integration we have $F^{(k)}=D\beta F$, where $D\in\mathbb{C}\setminus\{0\}$. Since $n=k+1$ and $\overline N(r,\infty;\beta)=S(r,f)$, it follows that $\overline N(r,0;f)=S(r,f)$. Then from (\ref{e14}) we have $T(r,f)=S(r,f)$, which is a contradiction. So $P\not\equiv 0$. Then by \textrm{Lemma \ref{l1}} we get $m(r,P)=S(r,f)$. Since $N(r,f) = S(r, f)$ we have \begin{eqnarray}\label{e12} T(r,P)=S(r,f)\;\; \text{and} \;\; T(r,P')=S(r,f).\end{eqnarray} Note that from (\ref{e11}) we get \begin{eqnarray}\label{e15} P'(z)= A_{1}(f')^{k}f''+B_{1}(f')^{k+1}+S_{1}(f),\end{eqnarray} is a differential polynomial in $f$, where $A_{1}=-\frac{1}{4}k(k+1)^{2}(k+1)!$, $B_{1}=-(k+1)!\xi $ and $S_{1}(f)$ is a differential polynomial in $f$ such that each term of $S_{1}(f)$ contains $f^{m}$ for some $m(1\leq m\leq n-1)$ as a factor.\par Let $z_{3}$ be a simple zero of $f$ such that $\xi(z_{3})\not= 0, \infty $. Then from (\ref{e11}) and (\ref{e15}) we have \begin{eqnarray*} P(z_{3})=-k(k+1)!(f'(z_{3}))^{k+1},\;\; P'(z_{3})= A_{1}(f'(z_{3}))^{k}f''(z_{3})+B_{1}(z_{3})(f'(z_{3}))^{k+1}.\end{eqnarray*} This shows that $z_{3}$ is a zero of $Pf''-[K_{1}P'-K_{2}P]f'$, where $K_{1}=\frac{-k(k+1)!}{A_{1}}$ and $K_{2}=\frac{B_{1}}{A_{1}}$. Also $T(r,K_{1})=S(r,f)$ and $T(r,K_{2})=S(r,f)$. Let \begin{eqnarray}\label{e16} \Phi_{1}=\frac{Pf''-[K_{1}P'-K_{2}P]f'}{f}.\end{eqnarray} Then clearly $m(r, \Phi_{1}) = S(r, f)$ and since $N_{(2}(r, 0; f) + N(r, f) = S(r, f)$, we have $T(r,\Phi_{1})=S(r,f)$. From (\ref{e16}) we obtain \begin{eqnarray}\label{e17} f''(z)=\alpha_{1}(z)f(z)+\beta_{1}(z)f'(z),\end{eqnarray} where \begin{eqnarray}\label{e18} \alpha_{1}=\frac{\Phi_{1}}{P}\;\text{and}\;\beta_{1}=K_{1}\frac{P'}{P}-K_{2}.\end{eqnarray}
Differentiating (\ref{e17}) and using it repeatedly we have \begin{eqnarray}\label{e17sm} f^{(i)}(z)=\alpha_{i-1}(z)f(z)+\beta_{i-1}(z)f'(z),\end{eqnarray} where $i\geq 2$ and $T(r,\alpha_{i-1})=S(r,f)$, $T(r,\beta_{i-1})=S(r,f)$.
Also (\ref{e18}) yields \begin{eqnarray}\label{e19}P'=\Big(\frac{\beta_{1}}{K_{1}}+\frac{K_{2}}{K_{1}}\Big)P\end{eqnarray} and \begin{eqnarray*} \beta_{1}=K_{1}\frac{P'}{P}-K_{2}=\frac{-k(k+1)!}{A_{1}}\frac{P'}{P}-\frac{B_{1}}{A_{1}},\end{eqnarray*} so that \begin{eqnarray}\label{e1911} A_{1}\beta_{1}+B_{1}+k(k+1)!\frac{P'}{P}=0.\end{eqnarray} Now we consider following two sub-cases.\\ {\bf Sub-case 1.2.1.1.} Let $k=1$.\\ Now from (\ref{e11}) and (\ref{e17}) we have \begin{eqnarray*} \label{s301}P=-2(f')^{2}-2\xi ff'+2ff''=-2(f')^{2}+(2\beta_{1}-2\xi)ff'+2\alpha_{1}f^{2}\end{eqnarray*} and so \begin{eqnarray*} \label{s302}P'=(-2\beta_{1}-2\xi)(f')^{2}+(2\beta_{1}'-2\xi'+2\beta_{1}^{2}-2\beta_{1}\xi)ff'+(2\alpha_{1}\beta_{1}-2\alpha_{1}\xi+2\alpha_{1}')f^{2}.\end{eqnarray*} Note that $K_{1}=1$ and $K_{2}=\xi$ and so from (\ref{e19}) we have \begin{eqnarray}\label{s3022} \big(\beta_{1}'-\xi'-\beta_{1}\xi+\xi^{2}\big)f'+\big(-2\alpha_{1}\xi+\alpha_{1}'\big)f\equiv 0.\end{eqnarray} If $-2\alpha_{1}\xi + \alpha_{1}' \equiv 0$, then from (\ref{s3022}) we get, because $f f' \not\equiv 0$, \begin{eqnarray}\label{s3022a}\beta_{1}'-\xi'-\beta_{1}\xi+\xi^{2} \equiv 0.\end{eqnarray} Let $\beta_{1} \equiv \xi$. Then a simple calculation gives $\displaystyle 2\frac{\beta'}{\beta} =\frac{P'}{P}$ and so on integration we get $\displaystyle \beta^{2} = d_{0} P$, where $d_{0}\in\mathbb{C}\setminus\{0\}$. This contradicts the fact that $T(r, \beta ) \neq S(r, f)$. So $\beta_{1} \not \equiv \xi$. Now from (\ref{s3022a}) we get $\displaystyle \frac{\beta_{1}' - \xi'}{\beta_{1} - \xi} = \xi = \frac{\beta'}{\beta}$. So on integration we get $\beta = d_{1}(\beta_{1} - \xi)$, where $d_{1}\in\mathbb{C}\setminus\{0\}$. This contradicts the fact that $T(r, \beta) \neq S(r, f)$. So we conclude that $-2\alpha_{1}\xi+\alpha_{1}'\not\equiv 0$.
Then from (\ref{s3022}) we see that if $z_{4}$ is a simple zero of $f$, then $z_{4}$ is either a pole of $-2\alpha_{1}\xi + \alpha_{1}'$ or a zero of $\beta_{1}' - \xi' - \beta_{1}\xi + \xi^{2}$. Hence $$ N_{1)}(r, 0; f) \leq N(r, \infty; -2\alpha_{1}\xi + \alpha_{1}') + N(r, 0; \beta_{1}' - \xi' - \beta_{1}\xi + \xi^{2}) = S(r, f).$$ So we arrive at a contradiction by (\ref{e14}).\\ {\bf Sub-case 1.2.1.2.} Let $k\geq 2$. \\ From (\ref{e8}) and (\ref{e9}) we have $F^{(k)}=T_{1}(f)$, $F^{(k+1)}=(k+1)!(f')^{k+1}+T_{2}(f)$ and $F^{(k+2)}=\frac{(k+1)(k+2)}{2}(k+1)!(f')^{k}f''+T_{3}(f)$, where $T_{1}(f)$, $T_{2}(f)$ and $T_{3}(f)$ are differential polynomials in $f$ such that each term of $T_{1}(f)$, $T_{2}(f)$ and $T_{3}(f)$ contain $f$ as a factor.\\ Comparing (\ref{e7}) and (\ref{e10}) and noting that $F = f^{n} = f^{k + 1}$ we have \begin{eqnarray}\label{s304} Q &=& a_{1}F^{(k+1)}-\big(\xi a_{1}+a_{1}'\big)F^{(k)}-a_{2}F' +\big(a_{2}'-\xi a_{2}\big)F+\gamma\\&=& a_{1}\{(k+1)!(f')^{k+1}+ T_{2}(f)\}-(\xi a_{1}+a_{1}')T_{1}(f)-(k+1)a_{2}f^{k}f'\nonumber\\&&+(a_{2}'-\xi a_{2})f^{k+1}+\gamma\nonumber,\end{eqnarray} where $\gamma=\xi a_{1}a_{2}+a_{2}a_{1}'-a_{1}a_{2}'$.
Now suppose $\gamma(z)\equiv 0$. Then by integration we obtain $\beta=d_{2}\frac{a_{2}}{a_{1}}$, where $d_{2}\in\mathbb{C}\setminus\{0\}$ and so $T(r,\beta)=S(r,f)$, which is a contradiction. Consequently $\gamma(z)\not\equiv 0$. Similarly we can verify that $\xi a_{1}+a_{1}'\not\equiv 0$ and $a_{2}' - \xi a_{2} \not\equiv 0$. We further note that $T(r,\gamma)= S(r,f)$. Differentiating (\ref{s304}) we have \begin{eqnarray}\label{s305} \;\;\;\;Q' &=& a_{1}'F^{(k+1)}+a_{1}F^{(k+2)}-(\xi a_{1}+a_{1}')F^{(k+1)}-(\xi a_{1}+a_{1}')'F^{(k)}-a_{2}'F'-a_{2}F''\\&&+(a_{2}'-\xi a_{2})'F+(a_{2}'-\xi a_{2}) F'+\gamma'\nonumber\\&=& a_{1}'\Big\{(k+1)!(f')^{k+1}+T_{2}(f)\Big\}+a_{1}\Big\{\frac{(k+1)(k+2)}{2}(k+1)!(f')^{k}f''+T_{3}(f)\Big\}\nonumber\\&&-(\xi a_{1}+a_{1}')\Big\{(k+1)!(f')^{k+1}+T_{2}(f)\Big\}-(\xi a_{1}+a_{1}')'T_{1}(f)-(k+1)a_{2}'f^{k}f'\nonumber\\&&-a_{2}\Big\{k(k+1)f^{k-1}(f')^{2}+(k+1)f^{k}f''\Big\}+(a_{2}'-\xi a_{2})'f^{k+1}\nonumber\\&& +(k+1)(a_{2}'-\xi a_{2})f^{k}f'+\gamma'\nonumber.\end{eqnarray}
Let $z_{5}$ be a simple zero of $f(z)$ such that $z_{5}$ is not a zero or a pole of $a_{1}$, $a_{2}$ and $\xi$. Then from (\ref{e10}), (\ref{s304}) and (\ref{s305}) we have \begin{eqnarray*} \gamma(z_{5})=A(z_{5})(f'(z_{5}))^{k+1},\;\; \gamma'(z_{5})=A_{2}(z_{5})(f'(z_{5}))^{k}f''(z_{5})+B_{2}(z_{5})(f'(z_{5}))^{k+1},\end{eqnarray*} where $A(z)=-(k+1)!a_{1}(z)$, $A_{2}(z)= -\frac{(k+1)(k+2)}{2}(k+1)!a_{1}(z)$ and $B_{2}(z)=(k+1)!\xi(z)a_{1}(z)$. This shows that $z_{5}$ is a zero of $\gamma f''-[K_{3}\gamma'-K_{4}\gamma]f'$, where $K_{3}=\frac{A}{A_{2}}$ and $K_{4}=\frac{B_{2}}{A_{2}}$. Also $T(r,K_{3})=S(r,f)$ and $T(r,K_{4})=S(r,f)$.
Let \begin{eqnarray}\label{s306} \Phi_{2}=\frac{\gamma f''-[K_{3}\gamma'-K_{4}\gamma]f'}{f}.\end{eqnarray} Then clearly $T(r,\Phi_{2})=S(r,f)$. From (\ref{s306}) we obtain \begin{eqnarray}\label{s307} f''=\phi_{1}f+\psi_{1}f',\end{eqnarray} where \begin{eqnarray}\label{s308} \phi_{1}=\frac{\Phi_{2}}{\gamma}\;\text{and}\;\psi_{1}=K_{3}\frac{\gamma'}{\gamma}-K_{4}.\end{eqnarray}
Now we show that $\psi_{1}\not\equiv\beta_{1}$. If $\psi_{1}\equiv\beta_{1}$ then from (\ref{e18}) and (\ref{s308}) we have \begin{eqnarray*} \frac{2}{(k+1)(k+2)}\frac{\gamma'}{\gamma}+\frac{2}{(k+1)(k+2)}\xi \equiv \frac{4}{(k+1)^{2}}\frac{P'}{P}-\frac{4}{k(k+1)^{2}}\xi,\end{eqnarray*} i.e., \begin{eqnarray*} 2k(k+2)\frac{P'}{P}-k(k+1)\frac{\gamma'}{\gamma}\equiv (k^{2}+3k+4)\frac{\beta'}{\beta}.\end{eqnarray*}
On integration we have \begin{eqnarray*} \beta^{k^{2}+3k+4}\equiv \frac{d_{3} P^{2k(k+2)}}{\gamma^{k(k+1)}},\end{eqnarray*} where $d_{3}\in\mathbb{C}\setminus\{0\}$ and so from (\ref{e12}) we have $T(r,\beta)=S(r,f)$, a contradiction.
Now from (\ref{s307}) we have \begin{eqnarray}\label{s309} f^{(i)}=\phi_{i-1}f+\psi_{i-1}f',\end{eqnarray} where $i\geq 2$ and $T(r,\phi_{i-1})=S(r,f)$, $T(r,\psi_{i-1})=S(r,f)$. Also from (\ref{e11}), (\ref{e15}) and (\ref{s309}) we have respectively \begin{eqnarray}\label{s310} P=-k(k+1)!(f')^{k+1}+\sum\limits_{j=1}^{k+1}T_{j}f^{j}(f')^{k+1-j},\end{eqnarray} \begin{eqnarray}\label{s312} &&P'=(A_{1}\psi_{1}+B_{1})(f')^{k+1}+\sum\limits_{j=1}^{k+1}S_{j}f^{j}(f')^{k+1-j},\end{eqnarray} where $T(r,T_{j})=S(r,f)$ and $T(r,S_{j})=S(r,f)$.
Multiplying (\ref{s310}) by $P'$ and (\ref{s312}) by $P$ and then subtracting we get \begin{eqnarray}\label{e211} H_{0}(f')^{k+1}+H_{1}f(f')^{k}+\ldots+H_{k+1}f^{k+1}\equiv 0,\end{eqnarray} where \begin{eqnarray}\label{e1912} H_{0}=P\Big[A_{1}\psi_{1}+B_{1}+k(k+1)!\frac{P'}{P}\Big]\end{eqnarray} and $H_{j}=PS_{j}-P'T_{j}$ for $j=1,2,\ldots, k+1$. Since $\beta_{1}\not\equiv \psi_{1}$ and $P\not\equiv 0$, it follows from (\ref{e1911}) and (\ref{e1912}) that $H_{0}\not\equiv 0$. Again since $H_{0}(f')^{k+1}\not\equiv 0$, from (\ref{e211}) we conclude that $H_{i}\not\equiv 0$ for at least one $i\in\{1,2,\ldots,k+1\}$. Let $S=\{1,2,\ldots, k+1\}$ and $S_{1}=\{i\in S: H_{i}\not\equiv 0\}$. Note that $T(r,H_{0})=S(r,f)$ and $T(r,H_{j})=S(r,f)$ for $j\in S_{1}$.\\ Now from (\ref{e211}) we see that a simple zero of $f$ must be either a zero of $H_{0}$ or a pole of at least one $H_{i}$'s, where $i\in S_{1}$. Therefore \[N_{1)}(r,0;f) \leq N(r, 0; H_{0})+\sum\limits_{\substack{j\\j\in S_{1}}} N(r,\infty;H_{j}) + S(r, f) = S(r,f).\] So we arrive at a contradiction by (\ref{e14}).\\ {\bf Sub-case 1.2.2.} Suppose $T(r,\beta)=S(r,f)$.\\ Then from (\ref{e5}) we have \begin{eqnarray}\label{e23a} F^{(k)}-\beta F\equiv a_{2}-\beta a_{1}.\end{eqnarray} If $a_{2}-\beta a_{1}\equiv 0$, then from (\ref{e23a}) we get $(f^{n})^{(k)}\equiv \frac{a_{2}}{a_{1}} f^{n}$, which contadicts the fact that $\Phi\not\equiv 0$. So we suppose that $a_{2}-\beta a_{1}\not\equiv 0$. Let $z_{6}$ be a simple zero of $f$. If $z_{6}$ is not a pole of $\beta$, then from (\ref{e23a}) we see that $z_{6}$ is a zero of $a_{2} - a_{1}\beta$. Therefore \[N_{1)}(r, 0; f) \leq N(r, 0; a_{2} - a_{1}\beta) + N(r, \infty ; \beta) = S(r, f).\] So by (\ref{e14}) we arrive at a contradiction.\\ {\bf Case 2.} Let $\Phi\equiv 0$. Now from (\ref{e1}) we get $F_{1}\equiv G_{1}$, i.e., $ (f^{n})^{(k)}\equiv \frac{a_{2}}{a_{1}}f^{n}$.
Furthermore if $a_{1} \equiv a_{2}$, then $f^{n}\equiv (f^{n})^{(k)}$, and by \textrm{Lemma \ref{l9}}, $f$ assumes the form $f(z)=ce^{\frac{\lambda}{n}z}$, where $c\in\mathbb{C}\setminus\{0\}$ and $\lambda^{k}=1$.\end{proof}
\begin{proof}[Proof of Theorem \ref{t2}] Let $F_{1}=\frac{f^{n}}{a_{1}}$ and $G_{1}=\frac{(f^{n})^{(k)}}{a_{2}}$. Clearly $F_{1}$ and $G_{1}$ share $``(1, 0)"$ except for the zeros and poles of $a_{1}(z)$ and $a_{2}(z)$ and so $\overline N(r,1;F_{1})=\overline N(r,1;G_{1})+S(r,f)$. We now consider following two cases.\\ {\bf Case 1.} Let $F_{1}\not \equiv G_{1}$.\\ Then \begin{eqnarray}\label{e24} \overline N(r,1;F_{1})& \leq& \overline N(r,0;G_{1}-F_{1}\mid F_{1}\neq 0)+S(r,f)\\&\leq & \overline N(r,0;\frac{G_{1}-F_{1}}{F_{1}})+S(r,f) \nonumber\\&\leq& T(r,\frac{G_{1}-F_{1}}{F_{1}})+S(r,f)\nonumber\\&\leq& T(r,\frac{G_{1}}{F_{1}})+S(r,f)\nonumber\\&\leq& N(r,\infty;\frac{G_{1}}{F_{1}})+m(r, \infty;\frac{G_{1}}{F_{1}})+S(r,f)\nonumber\\&=& N(r,\infty;\frac{a_{1}}{a_{2}}\frac{(f^{n})^{(k)}}{f^{n}})+m(r,\infty;\frac{a_{1}}{a_{2}}\frac {(f^{n})^{(k)}}{f^{n}})+S(r,f)\nonumber\\
&\leq &k\;\overline N(r,\infty;f)+ k\;\overline N(r,0;f^{n})+S(r,f) \nonumber \\ &=& k\;\overline N(r,0;f)+S(r,f).\nonumber\end{eqnarray} Now using (\ref{e24}) and $\overline N_{2)}(r, 0; f) = S(r, f)$, we get from the second fundamental theorem that \begin{eqnarray}\label{e25} n\;T(r,f)&=& T(r,f^{n})+S(r,f)\\&\leq& T(r,F_{1})+S(r,f)\nonumber\\ &\leq & \overline N(r,\infty;F_{1})+\overline N(r,0;F_{1})+\overline N(r,1;F_{1})+S(r,F)\nonumber\\ &\leq & \;\overline N(r,\infty;f)+\overline N(r,0;f^{n})+\overline N(r,1;F_{1})+S(r,f)\nonumber\\ &\leq &(k+1) \;\overline N(r,0;f) + S(r,f) \nonumber \\ & \leq & \frac{k + 1}{3} N(r, 0; f) + S(r, f) \nonumber \\ &\leq & \frac{k + 1}{3} T(r, f) + S(r, f).\nonumber\end{eqnarray} Since $n\geq k$, (\ref{e25}) leads to a contradiction.\\ {\bf Case 2.} $F_{1} \equiv G_{1}$. Then $ (f^{n})^{(k)}\equiv \frac{a_{2}}{a_{1}}f^{n}$. Furthermore if $a_{1} \equiv a_{2}$, then $f^{n}\equiv (f^{n})^{(k)}$, and by \textrm{Lemma \ref{l9}}, $f$ assumes the form $f(z)=ce^{\frac{\lambda}{n}z}$, where $c\in\mathbb{C}\setminus\{0\}$ and $\lambda^{k}=1$. \end{proof}
\end{document} | arXiv |
The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook
The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University.
Generalized Anxiety Disorder:
Examination of Fear Generalization and Anticipation of Affective Stimuli A Dissertation Presented
Tsafrir Greenberg to
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in
Biopsychology Stony Brook University
Stony Brook University The Graduate School
Tsafrir Greenberg
We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend
acceptance of this dissertation
Lilianne R. Mujica-Parodi, Ph.D., Dissertation Advisor Associate Professor, Biomedical Engineering Turhan Canli, Ph.D., Chairperson of Defense
Associate Professor, Psychology
Brenda Anderson, Ph.D., Associate Professor, Psychology Greg Hajcak, Ph.D., Associate Professor, Psychology
Tor Wager, Ph.D., Associate Professor, Psychology and Neuroscience, University of Colorado at Boulder
This dissertation is accepted by the Graduate School
Charles Taber
Abstract of the Dissertation Generalized Anxiety Disorder:
Examination of Fear Generalization and Anticipation of Affective Stimuli by
Tsafrir Greenberg Doctor of Philosophy
in Biopsychology
Stony Brook University 2012
Despite increasing interest in generalized anxiety disorder (GAD) it remains one of the least studied anxiety disorders and little is known about its neurobiology. The aim of the current research study was to investigate the neural basis of GAD from two perspectives: fear
generalization and anticipation of affective outcomes. Thirty-two women with GAD (17 with and 15 without comorbid major depression disorder) and 25 healthy women underwent
functional magnetic resonance imaging (fMRI) while completing two tasks: a fear generalization task in which participants were presented with a conditioned stimulus (CS) that co-terminates with a brief electric shock, and generalization stimuli (GS) that range in perceptual similarity to the CS, and an anticipation task in which predetermined cues warned participants of upcoming aversive, positive, 'uncertain' (either aversive or positive) or neutral movie clips. For the
area and caudate followed a generalization gradient, with a peak response to the CS that declines with greater perceptual dissimilarity of the GS to the CS; consistent with participants' self-report and autonomic responses for each stimulus. In contrast, reactivity in the ventromedial prefrontal cortex (vmPFC), a region associated with fear inhibition, showed an opposite response pattern (i.e., largest response to the GS most dissimilar to the CS). Patients with GAD, compared to healthy individuals, exhibited a flatter vmPFC gradient suggestive of deficient recruitment of vmPFC during fear inhibition. Anticipation of all affective clips engaged a common set of regions – the insula, dorsal anterior cingulate cortex, thalamus, caudate, prefrontal cortex and inferior parietal area – involved in various preparatory processes such as sustained attention, increased arousal, appraisal, and regulation. The nucleus accumbens (NAcc) and medial prefrontal cortex (mPFC), regions implicated in reward processing, were selectively engaged during anticipation of positive clips and the mid-insula was selectively engaged during aversive anticipation. Anticipation during the 'uncertain' condition reflected a preparatory response for both aversive and positive stimuli showing activation in the NAcc, mPFC and mid-insula. Patients with GAD, compared to healthy individuals, exhibited enhanced reactivity during anticipation of neutral clips and a selective increase in visual cortical activation during affective anticipation consistent with a less discriminant anticipatory response. Findings underscore two processes – deficits in fear inhibition of stimuli that resemble conditioned danger cues and excessive anticipation of innocuous stimuli – which may contribute to symptoms of worry and anxiety in GAD.
List of Tables ... viii
List of Figures. ... ix
List of Abbreviations... xi
Chapter 1. General Introduction ... 1
Chapter 2: General Methods for the Study ... 9
Chapter 3: Demographics/Descriptives and Self-Report Measures... 18
Chapter 4: Neural Reactivity Tracks Fear Generalization Gradients ... 21
4.1 Introduction ... 21 4.2 Method (experiment 1) ... 23 4.3 Results (experiment 1) ... 26 4.4 Method (experiment 2) ... 26 4.5 Results (experiment 2) ... 28 4.6 Discussion ... 31
Chapter 5: Ventromedial Prefrontal Cortex Reactivity is Altered in Generalized Anxiety Disorder during Fear Generalization ... 42
5.1 Introduction ... 42
5.2 Method ... 44
5.3 Results ... 46
5.4 Discussion ... 49
Chapter 6: Anticipation of High Arousal Aversive and Positive Movie Clips Engages Common and Distinct Neural Substrates ... 60
Chapter 7: Individuals with Generalized Anxiety Exhibit Enhanced Anticipatory Response to Neutral Movie Clips ... 80
Chapter 8: General Discussion... 93
Table 1. Mean scores and standard deviations for self-report questionnaires……….... ...20 Table 2. Correlations between self-report questionnaires……….…...………... .20 Table 3. Brain activation associated with response to the CSunpaired and generalization
stimuli (Healthy group) ………... ...36 Table 4. Brain activation during aversive, positive, and 'uncertain' anticipation
versus neutral anticipation (Healthy group) ……….…...………... ...74 Table 5. Brain activation during aversive, positive, and 'uncertain' anticipation
versus neutral anticipation (GAD group) ……….…...………... ...88 Table 6. Comparison of brain activation for the GAD group versus Healthy group
Figure 1. Illustration of a single trial from the anticipation task (positive condition)……... 12 Figure 2. Activation map and neural gradients for the right and left insula
(Pilot study)………..………..………... .. 37 Figure 3. Post-task ratings of shock likelihood and pupillary response as a function of
stimulus type (Healthy group) ………..………... ... 38 Figure 4. Activation maps and neural gradients for the right and left insula
(Healthy group) ………….……….………... ... 39 Figure 5. Activation maps and neural gradients for the anterior cingulate cortex, right
supplementary motor area and right caudate (Healthy group)………... ... 40 Figure 6. Activation maps and neural gradients for the ventromedial prefrontal cortex
and somatosensory area (Healthy group) ……... ... 41 Figure 7.Post-task ratings of shock likelihood for the GAD and Healthy groups as a
function of stimulus type………... ... 54 Figure 8. Activation maps and neural gradients for the right insula, anterior cingulate
cortex, right supplementary motor area and right caudate for the GAD
and Healthy groups……..………..…... ... 55 Figure 9. Activation maps and neural gradients in the ventromedial prefrontal cortex
and somatosensory area for the GAD and Healthy groups ………... ... 56 Figure 10. Correlations between STAI-T and BDI scores and slope coefficients of individual
vmPFC gradients for the GAD and Healthy groups……..………..…... ... 57 Figure 11. Psychophysiological interactions (N=50) for the right anterior insula
Figure 12. Psychophysiological interactions (N=50) for the right anterior insula
seed during presentation of the CS relative to all GS……….………..…... 59 Figure 13. Mean anticipatory ratings for the aversive, positive, 'uncertain' and neutral
conditions (Healthy group) … ... ..76 Figure 14. Mean valence scores for the aversive, positive, 'uncertain' aversive,
'uncertain' positive and neutral clips (Healthy group) … ... ..77 Figure 15. Mean arousal scores for the aversive, positive, 'uncertain' aversive,
'uncertain' positive and neutral clips (Healthy group) … ... ..77 Figure 16. Brain activation for the conjunction of aversive, positive, and 'uncertain'
anticipation versus neutral anticipation contrasts (Healthy group) ……... ... 78 Figure 17. Selective brain activation during aversive anticipation versus positive
anticipation (Healthy group) …….………..…... ... 78 Figure 18. Selective brain activation during anticipation versus clip anticipation
(Healthy group)………..…... .. 79 Figure 19. Mean anticipatory ratings for the aversive, positive, 'uncertain' and neutral
conditions for the GAD-only, Comorbid and Healthy groups … ... ..90 Figure 20. Mean valence scores for the aversive, positive, 'uncertain' aversive, 'uncertain'
positive and neutral clips for the GAD-only, Comorbid and Healthy groups … ..91 Figure 21. Mean arousal scores for the aversive, positive, 'uncertain' aversive, 'uncertain'
positive and neutral clips for the GAD-only, Comorbid and Healthy groups… . ..91 Figure 22. Comparisons of brain activation for the GAD group (N=32) versus Healthy
List of Abbreviations ACC anterior cingulate cortex
aINS anterior insula ANOVA analysis of variance ASI anxiety sensitivity index
BA Brodmann area
BDI Beck depression inventory
BDNF brain-derived neurotrophic factor BOLD blood oxygen level-dependent CBT cognitive behavioral therapy
CS conditioned stimulus
dACC dorsal anterior cingulate cortex dlPFC dorsolateral prefrontal cortex
DPSS-R disgust propensity and sensitivity scale-revised DSM diagnostic and statistical manual of mental disorders
EPI echo planar imaging
fMRI functional magnetic resonance imaging
FOV field of view
FWE family-wise error
GAD generalized anxiety disorder
GS generalization stimuli
IAPS the international affective picture system IPA inferior parietal area
IUS intolerance of uncertainty scale lPFC lateral prefrontal cortex
MASQ mood and anxiety symptom questionnaire
MASQ-AA mood and anxiety symptom questionnaire - anxious arousal subscale MASQ-AD mood and anxiety symptom questionnaire - anhedonic depression subscale MASQ-GDA mood and anxiety symptom questionnaire - general distress anxiety subscale MASQ-GDD mood and anxiety symptom questionnaire - general distress depression subscale
MDD major depression disorder mPFC medial prefrontal cortex
MRI magnetic resonance imaging
NAcc nucleus accumbens
PE prediction error
PFC prefrontal cortex pINS posterior insula
PPI psycho-physiological interaction PSWQ Penn state worry questionnaire PTSD post-traumatic stress disorder
ROI region of interest
SAD social anxiety disorder
SCID structured clinical interview for the diagnostic and statistical manual of mental disorders
SD standard deviation
SMA supplementary motor area
SPM statistical parametric mapping
STAI-T trait version of the state-trait anxiety inventory SVC small volume correction
TE echo time
TR repetition time
UCS unconditioned stimulus
Chapter 4 of this dissertation is a reprint of the manuscript titled 'Neural reactivity tracks fear generalization gradients' as it appears in the journal of Biological Psychology (Greenberg, Carlson, Cha, Hajcak, & Mujica-Parodi, 2011). Chapter 5 is a reprint of the manuscript titled 'Ventromedial prefrontal cortex reactivity is altered in generalized anxiety disorder during fear generalization' as it appears in Depression and Anxiety (Greenberg, Carlson, Cha, Hajcak, & Mujica-Parodi, 2012).
Chapter 1: General Introduction
Generalized anxiety disorder (GAD) is characterized by excessive and uncontrollable worry about a number of events or activities, accompanied by at least three of the following symptoms: restlessness, fatigue, difficulty concentrating, irritability, muscle tension and
difficulty sleeping (American Psychiatric Association, 1994). Estimates for lifetime prevalence of GAD range from 1.9% to 5.4% (Brown, O'Leary, & Barlow, 2001) with women twice as likely to be diagnosed with the disorder (Wittchen, Zhao, Kessler, & Eaton, 1994). GAD is often comorbid with other anxiety or mood disorders with rates ranging from 45% to 98% (Goisman, Goldenberg, Vasile, & Keller, 1995). Comorbidity with major depression disorder (MDD) is of particular interest because of high rates (Kessler, Chiu, Demler, Merikangas, & Walters, 2005) and some overlap in symptoms between the two disorders.
Starting with the diagnostic and statistical manual of mental disorders, third edition revised (DSM-III-R; American Psychiatric Association, 1987) worry was established as the key diagnostic symptom of GAD. Worry is a verbal-semantic process involving uncontrollable thinking about the possibility of future negative events. In the short run, worry reduces
threatening imagery and somatic arousal but overtime may interfere with emotional processing and help maintain anxiety-producing cognitions (Borkovec, 1994). The content of worry in GAD is similar to that of other psychiatric disorders and healthy individuals (Abel & Borkovec, 1995). However, GAD patients worry with greater frequency, typically have worries in more than one domain and report more difficulties controlling their worry behavior (Borkovec, 1994; Borkovec, Shadick, & Hopkins, 1991).
Although GAD is associated with substantial human (encompassing social, family and occupational functioning as well as perceived emotional and physical well-being) and economic
burden (Hoffman, Dukes, & Wittchen, 2008) it remains one of the least studied anxiety disorders (Dugas, Anderson, Deschenes, & Donegan, 2010). A common explanation for this paucity of research is that it has taken a long time for the diagnosis of GAD to establish validity, given its early residual status and substantial revisions in diagnostic criteria over the years (Mennin, Heimberg, & Turk, 2004). Investigation of the neurobiology of GAD has lagged behind other anxiety disorders in particular. Findings from the few available neuroimaging studies have been mixed regarding the brain regions underlying its psychopathology (Shin & Liberzon, 2010). Enhanced amygdala activation in GAD has been observed during aversive anticipation (Nitschke et al., 2009) and in response to fearful (E. B. McClure, Monk, et al., 2007) and masked angry faces (Monk et al., 2008) but not in two other studies (Blair et al., 2008; Whalen et al., 2008). Furthermore, contrasting results have been reported with respect to prediction of treatment outcome based on pre-treatment levels of amygdala response (E. B. McClure, Adler, et al., 2007; Whalen et al., 2008). The role of the frontal cortex in GAD requires clarification as well.
Enhanced activation in dorsal and rostral ACC has been reported in response to fearful faces (E. B. McClure, Monk, et al., 2007) and in rostral ACC during a decision-making task (Krain et al., 2008). In contrast, patients exhibited a diminished response in pregenual ACC during implicit regulation of emotional conflict (Etkin, Prater, Schatzberg, Menon, & Greicius, 2009).
The aim of the present research was to investigate the neural basis of GAD from two perspectives: fear generalization and anticipation of affective outcomes.
1.1.Fear generalization: Fear learning mechanisms such as fear conditioning and extinction are central to etiological accounts and treatment of pathological anxiety (Mineka & Oehlberg, 2008). Paradigms designed to test these mechanisms have been developed and studied extensively in
animals and adapted for human studies (Delgado, Olsson, & Phelps, 2006). A related learning mechanism that received less attention in humans is fear generalization, which describes the transfer of a conditioned fear response to new stimuli that are similar to the conditioned stimulus (CS). Fear generalization has gained interest in recent years due to its hypothesized relevance to the transfer of fear responses from threat-related stimuli to potentially innocuous cues – a process that appears to be prevalent in most anxiety disorders. Aside from clinical observation, this hypothesis was motivated by evidence that anxiety patients, compared to healthy individuals, show enhanced fear-responding to safety cues during fear-conditioning (Lissek et al., 2005).
Recent studies have validated laboratory-based procedures for testing fear generalization using fear-potentiated startle (Hajcak et al., 2009; Lissek et al., 2008) and skin conductance (Dunsmoor, Mitroff, & LaBar, 2009; Vervliet, Kindt, Vansteenwegen, & Hermans, 2010) to quantify fear responses to a CS and to generalization stimuli (GS) that vary in perceptual similarity to the CS. Such paradigms generate generalization gradients with a peak response to the CS that declines with increasing dissimilarity of the GS to the CS. In healthy individuals, this generalization gradient is characterized by a steep linear slope with sharp decreases across consecutive GS. In patients with panic disorder, on the other hand, the generalization gradient has a flatter linear slope with gradual decreases across stimuli, reflecting greater generalization (Lissek et al., 2010). Overgeneralization of the fear response to stimuli that resemble an initial danger cue may help maintain symptoms by increasing the number of cues in the environment that can trigger additional panic attacks. Such cues can be external and/or internal (e.g., autonomic changes) underscoring the range of stimuli that are potentially involved in the generalization process. In GAD, a similar process of overgeneralization may contribute to an increase in the number of cues/events that elicit worry behavior. This prediction of
overgeneralization in GAD is supported by findings of increased sensitivity to uncertainty (Dugas, Marchand, & Ladouceur, 2005) and a proclivity to interpret neutral/ambiguous stimuli as threatening (Eysenck, Mogg, May, Richards, & Mathews, 1991; Mathews, Richards, & Eysenck, 1989) among GAD patients (Lissek, 2012).
The neurocircuitry supporting fear generalization is unknown. Of relevance, however, are brain regions involved in acquisition and inhibition of fear responses identified with paradigms of fear conditioning and extinction. In human neuroimaging studies, the amygdala, ACC and insula are most consistently activated during fear acquisition showing stronger responses to a conditioned stimulus compared to an unconditioned stimulus (e.g., Büchel, Morris, Dolan, & Friston, 1998; LaBar, Gatenby, Gore, LeDoux, & Phelps, 1998; Morris & Dolan, 2004). Additional regions involved in acquisition and expression of fear are the thalamus, striatum, hippocampus, premotor areas and stimulus-relevant sensory areas (Sehlmeyer et al., 2009). During extinction, activation in the ventromedial prefrontal cortex (vmPFC) has been observed (e.g., Gottfried & Dolan, 2004) consistent with its role in extinction recall and fear inhibition. The interaction between the vmPFC and hippocampus is of potential importance for fear generalization. Both regions are jointly activated in response to an extinguished versus
unextinguished CS during the extinction phase (Milad et al., 2007). The hippocampus is involved in contextual modulation of fear expression (Corcoran & Quirk, 2007) and is thought to mediate stimulus representation (Gluck & Myers, 1993). Notably, hippocampally lesioned rats show increased generalization (Wild & Blampied, 1972). In the context of fear generalization, the hippocampus may affect vmPFC modulation of the fear response such that reactivity to the CS is enhanced and reactivity to the GS is inhibited.
For the current research study we optimized a previously validated generalization paradigm (Hajcak et al., 2009) for functional imaging. The aims were: (1) to identify brain regions involved in fear generalization; (2) to determine whether neural reactivity in these regions shows a similar generalization gradient to that reported with other psychophysiological measures of fear (i.e., fear-potentiated startle and skin conductance), and (3) to test whether the slope of such neural gradients, indexing generalization, differs in GAD and healthy individuals. In addition, we examined connectivity of primary regions engaged by the generalization task using psychophysiological interaction analyses. Regions of interest were informed by previous studies of fear conditioning and extinction (Gottfried & Dolan, 2004; LaBar et al., 1998;
Sehlmeyer et al., 2009) and included the amygdala, insula, thalamus, caudate, ACC and vmPFC. We hypothesized that reactivity in one or more of these regions would demonstrate a similar gradient response to the pattern reported in laboratory-based studies. Based on evidence for overgeneralization in anxious individuals (Dunsmoor, White, & LaBar, 2011; Lissek et al., 2008), we predicted that patients with GAD would demonstrate greater generalization as evidenced by flatter neural gradient slopes compared to healthy individuals.
1.2.Anticipation: Exaggerated anticipation is a central feature of anxiety disorders. Besides feeling anxious in the presence of certain stimuli/events, individuals with anxiety disorders often experience severe anxiety in expectation of confronting these stimuli/events (Barlow, 2000). Importantly, anticipatory anxiety leads to avoidance behavior that could further contribute to the maintenance of symptoms by interfering with extinction. Most neuroimaging studies of anxiety examined brain activation during stimuli presentation – findings showing differential reactivity between anxious individuals and healthy comparisons to disorder-specific and generally aversive
stimuli (e.g., Etkin & Wager, 2007). In addition, there is accumulating evidence for group differences in brain reactivity during the anticipatory phase.
Anticipatory anxiety has been typically assessed for brief periods preceding aversive images (Nitschke, Sarinopoulos, Mackiewicz, Schaefer, & Davidson, 2006; Simmons, Matthews, Stein, & Paulus, 2004; Simmons, Strigo, Matthews, Paulus, & Stein, 2006). Such paradigms most consistently engage the insula. Other regions implicated are the ACC, amygdala and prefrontal cortex. Anxiety-prone individuals (defined on the basis of high trait anxiety) show greater insula activation during aversive anticipation, relative to healthy comparisons (Simmons et al., 2006). In GAD, enhanced amygdala activation has been observed during anticipation of both aversive and neutral pictures, suggesting that patient's anticipatory response is less discriminate and triggered by a broader range of stimuli (Nitschke et al., 2009).
For the current study we used a modified anticipation task designed to better characterize the anticipatory response and facilitate investigation of potential differences in this response between GAD patients and healthy individuals. The task included multiple conditions of anticipation: aversive anticipation, positive anticipation, 'uncertain' anticipation and neutral anticipation. Target stimuli were short movie clips depicting a medical procedure (aversive condition), sexual situations (positive condition) and a person driving (neutral condition). For the 'uncertain' condition, either positive or aversive clips were presented with unknown
probability. We used longer anticipation periods (16 seconds), compared to previous studies, to test whether the same regions are engaged when anticipated events are more distal and to examine the temporal dynamics of the anticipatory response. We also collected self-report measures at the end of each trial to gauge participants' subjective experience of anticipatory anxiety and its association with neural reactivity, as well as autonomic measures of arousal
during the anticipation phase.
The aims were: (1) to establish normative sustained anticipatory responses for high arousal aversive and positive stimuli and a condition of uncertainty, and (2) to investigate whether these responses deviate in patients with GAD. We hypothesized that anticipation of aversive, positive and 'uncertain' clips would recruit common regions previously reported in studies of aversive anticipation (Nitschke et al., 2006) and that additional regions associated with appetitive processing such as the nucleus accumbens (NAcc) and mPFC would be preferentially engaged during anticipation of positive clips. The insula, which has been implicated in
processing disgust, was expected to be preferentially engaged during aversive anticipation. Based on previous findings demonstrating an enhanced anticipatory response in anxious individuals (Simmons et al., 2011) and heightened amygdala activation in GAD during anticipation of both aversive and neutral images(Nitschke et al., 2009) , we hypothesized that GAD patients would exhibit greater anticipatory reactivity and reduced discriminability across conditions compared to healthy individuals. Finally, based on evidence for greater intolerance of uncertainty (Ladouceur et al., 1999) and a tendency to interpret ambiguous/neutral stimuli as threatening (Eysenck et al., 1991; Mathews et al., 1989) among GAD patients, we expected greater disparity in anticipatory reactivity between groups for the 'uncertain' and neutral conditions.
1.3 Overview of chapters
In the current study we investigated the neural correlates of fear generalization and anticipation in a group of women diagnosed with GAD (N=32) and a group of healthy women (N=25). Chapter 2 presents the methods for the study. Chapter 3 presents a summary of
the manuscript 'Neural reactivity tracks fear generalization gradients' (Greenberg, Carlson, Cha, Hajcak, & Mujica-Parodi, 2011) and presents findings for the generalization task in the Healthy group and a pilot sample (N =8). Chapter 5 is a reprint of the manuscript 'Ventromedial prefrontal cortex reactivity is altered in generalized anxiety disorder during fear generalization' (Greenberg, Carlson, Cha, Hajcak, & Mujica-Parodi, 2012) and presents findings for the
generalization task in the GAD group and between-group comparisons with the Healthy group. Chapter 6 presents findings for the anticipation task in the Healthy group. Chapter 7 presents findings for the anticipation task in the GAD group and between-group comparisons with the Healthy group. Finally, Chapter 8 presents a general discussion for the entire study.
Chapter 2: General Methods for the Study 2.1 Participants
Fifty-seven women participated in the study, 15 with a current DSM-IV diagnosis of GAD (Mean age = 22.1; SD = 4.3), 17 with a diagnosis of GAD and comorbid major depression disorder (MDD; Mean age = 22.4; SD = 4.8) and 25 healthy individuals (Mean age = 21.6; SD = 5.1); ages for the entire sample ranged from 18 to 44. The study included only female
participants to reflect biases in diagnoses and research participation for GAD. Participants were recruited from the Stony Brook University campus, Suffolk community, and online websites. The Structured Clinical Interview for DSM-IV Axis I Disorders -Patient Edition, Version 2 (SCID-I/P; First, Spitzer, Gibbon, & Williams, 2002) was administered to confirm diagnoses of GAD in the patient groups and absence of Axis I diagnoses in the Healthy group. None of the participants were currently using any psychotropic medications. Forty-nine participants were right handed, four were left handed and four were ambidextrous. The study was approved by the Stony Brook University Institutional Review Board; all participants provided informed consent. 2.2 Study procedure
Potential participants completed a phone screening to determine whether they met general requirements for the study which included: age between 18-50, no history of psychosis or substance dependence, no intake of any psychiatric medication for at least two months, eligibility for MRI (i.e., no metal in body, not pregnant or lactating, no history of brain injury or neurological disorder, no claustrophobia) and likelihood to fulfill criteria for the GAD,
Comorbid, or Healthy groups based on responses to a modified version of the Mini-International Neuropsychiatric Interview (Sheehan et al., 1998) and a semi-structured diagnostic interview designed to screen for 17 Axis I disorders. Those who met these requirements were scheduled for
a SCID interview. The inclusion criteria for the GAD group were: 1) current diagnosis of GAD with no history of MDD in the past 12 months and 2) absence of any other Axis I disorder except for specific phobia. For the Comorbid group they were: 1) current diagnosis of GAD and MDD, and 2) absence of any other Axis I disorder except for specific phobia, and for the Healthy group they were absence of any current or past Axis I diagnosis. Individuals who met criteria for any of the experimental groups were scheduled for an fMRI session at the Social, Cognitive and
Affective Neuroscience (SCAN) imaging center at Stony Brook University. The fMRI session consisted of a structural scan and two functional tasks: generalization and anticipation. We used an eye tracker (Eyelink 1000; SR Research Ltd., Ontario, Canada) to monitor participants' compliance during the tasks. Following the fMRI session, participants completed self-report questionnaires.
2.3 fMRI tasks
The generalization and anticipation tasks were programmed with Experiment Builder (SR Research Ltd.) and presented with an MRI compatible 60 Hz projector with 1024 × 768
resolution (Psychology Software Tools, Inc., Sharpsburg, PA). During the tasks we collected pupil response measures (Eyelink 1000; SR Research Ltd.) to assess participants' arousal levels.
2.3.1 Generalization task 2.3.1.1 Procedure
Prior to the scan, an electric shock, delivered to the left wrist (Constant Voltage
Stimulator STM 200; Biopac Systems Inc., Goleta, CA), was individually set for each participant to a level that was "uncomfortable but not painful". Shock levels were comparable across
middle-sized rectangle (CS) indicated a 50% probability that they would receive a subsequent electric shock, but that shocks would never follow rectangles of greater or lesser size. A conditioning phase was administered next, which included five presentations of the CS with electric shock (i.e., CSpaired) and one presentation of each of the other six rectangles. The task
immediately followed. 2.3.1.2 Task design
The task consisted of three blocks presented consecutively. Each block included 40 trials (5 trials × 8 conditions) for a total of 120 trials. The stimuli were seven red rectangles with identical height (56 pixels) and varying width (112 to 448 pixels) presented against a black background. The middle-sized rectangle (280 pixels) was the conditioned stimulus (CS). Half of the time the CS co-terminated with a 500 ms electric shock (CSpaired), while half of the time it did
not (CSunpaired). The six remaining rectangles differed by ±20%, ±40% or ±60% in width from the
CS and served as the generalization stimuli (GS). Stimuli were presented pseudorandomly for 2 seconds with a jittered interstimulus interval ranging from 4 to 10 seconds, during which a white fixation cross was shown on a black background. The duration of the task was 15 minutes and 24 seconds. Following the task, participants rated the likelihood of having been shocked for each rectangle, on a Likert-type scale of 1 ("certainly not shocked") to 5 ("certainly shocked").
2.3.2 Anticipation task 2.3.2.1 Task design
The Anticipation task consisted of 40 trials presented pseudorandomly: 10 aversive trials (anticipation of an aversive movie clip), 10 positive trials (anticipation of a positive movie clip), 10 neutral trials (anticipation of a neutral movie clip), and 10 'uncertain' trials (anticipation of
either a positive or aversive movie clip; 50% each). As shown in figure 1, each trial began with a white fixation crosshair presented in the center of a black screen (jittered 10 - 14 seconds). The fixation crosshair was followed by an anticipation period during which a countdown from 16 to 1 was numerically presented in the center of the screen with a specific cue associated with each trial type presented above it (16 seconds duration; the cues were: '▼' for an aversive clip, '▲' for a positive clip, '?' for an 'uncertain' clip and '─' for a neutral clip. A movie clip was presented next (6 seconds) and then participants rated the strength of their emotional response during the countdown period on a 4-point scale (4 seconds). The duration of the task was 25 minutes and 30 seconds.
Figure 1. Illustration of a single trial (anticipation of a positive clip).
2.3.2.2 Stimuli
The aversive clips were taken from a non-commercial surgery film of arm amputation which has been shown to reliably elicit disgust and, to a lesser extent, fear (Gross & Levenson, 1995), and an additional film of a thigh surgery found online. The positive clips were sexually
explicit and taken from two scenes featured in Eyes of Desire 2 (Femme Productions) and Island Fever 2 (Digital Playground), which were pilot rated high on arousal and positive valence by women (N = 8). The neutral clips were taken from the movie Quiet Earth (1985) and consisted of a scene showing a man driving.
2.4 Post-task rating of clips
After completion of the anticipation task, participants rated each clip for valence and arousal using two 9-point scales. The valence scale ranged from 1 = "negative" to 9 = "positive". The arousal scale ranged from 1 = "calm" to 9 = "excited" (participants were instructed that a high score on this scale could reflect either negative or positive arousal).
2.5 Image acquisition
Participants were scanned with a 3 Tesla Siemens Trio scanner at the Social, Cognitive, and Affective Neuroscience center at Stony Brook University. First, we obtained T1-weighted structural scans with repetition time (TR) = 1900 ms, echo time (TE) = 2.53, Flip angle = 9°, field of view (FOV) = 176 × 250 × 250 mm, and Matrix = 176 × 256 × 256 mm. For the Generalization task we acquired a total of 440 T2*-weighted echoplanar images (EPI) with an oblique coronal angle with a TR = 2100 ms, TE = 23 ms, Flip angle = 83°, FOV = 224 × 224 mm, Matrix = 96 × 96 mm, Slices = 37, and slice thickness = 3.5 mm. For the Anticipation task, we acquired 760 T2*-weighted EPI images with an oblique coronal angle with a TR = 2100 ms, TE = 23 ms, Flip angle = 83°, FOV = 224 × 224 mm, Matrix = 96 × 96 mm, Slices = 37, and slice thickness = 3.5 mm.
2.6 Image analysis
Preprocessing procedures were performed in SPM8 (www.fil.ion.ucl.ac.uk/spm) and included slice time correction, motion correction, normalization and smoothing with a 6-mm full
width at half maximum Gaussian kernel. Specific procedures for statistical analyses are provided in the relevant chapters.
2.7 Psychophysiological interaction analysis (PPI)
PPI is a connectivity technique based on regression models; it identifies which voxels within the entire brain (or within regions of interest) show increased coupling with a seed region, in response to specific conditions of a task (see Friston et al., 1997 for details). The design matrix in PPI includes three variables: 1) A variable that reflects the experimental paradigm. 2) A time-series extracted from the seed region and 3) an interaction variable that represents the interaction between variables 1 and 2. Importantly, the variance associated with the interaction term is not accounted for by the main effects of the task and physiological correlations. Specific procedures for the PPI analysis are provided in the relevant chapters.
2.8 Pupillary response
Pupil data was processed using custom MATLAB codes (MathWorks). Specific procedures for preprocessing and analysis are provided in the relevant chapters.
2.9 Self-report questionnaires
After the fMRI session participants completed the following self-report questionnaires: Penn State Worry Questionnaire (PSWQ)
A 16-item scale that assesses the frequency and intensity of worry (Meyer, Miller, Metzger, & Borkovec, 1990) – the primary diagnostic symptom of GAD (American Psychiatric Association, 1994). Each item is rated by participants on a scale of 1 ("not at all typical of me") to 5 ("very typical of me"). The total score of the questionnaire ranges from 16 to 80. The PSWQ has high internal consistency, α ranges from .86 - .95, and good test-retest reliability, r ranges from .74 - .93 (Molina & Borkovec, 1994).
Intolerance of Uncertainty Scale (IUS)
A 27-item scale designed to assess how individuals respond to uncertainty on cognitive, emotional and behavioral levels (Freeston, Rhéaume, Letarte, & Dugas, 1994). Individuals with GAD are thought to be especially sensitive to uncertainty regarding future events which can contribute to their worry behavior (Ladouceur, Gosselin, & Dugas, 2000). Each item is rated on a scale of 1 ("not at all characteristic of me") to 5 ("entirely characteristic of me"). The total score ranges from 27 to 135. The IUS has good test-retest reliability; r = .74 (Dugas, Freeston, & Ladouceur, 1997) and excellent internal consistency; α = .91 (Freeston et al., 1994). Anxiety Sensitivity Index (ASI)
A 16-item scale designed to assess the extent to which an individual believes that anxiety may lead to harmful physical, psychological, or social consequences (Peterson & Reiss, 1992). Each item is rated on a scale of 0 ("very little") to 4 ("very much"). The total score ranges from 0 to 64. A recent meta-analysis found that of all the anxiety disorders anxiety sensitivity is most strongly associated with panic disorder and GAD (Naragon-Gainey, 2010). The ASI has good internal consistency, α ranging between .82 - .91 and satisfactory test-retest reliability with r ranging from .71 to .75 (Peterson & Reiss, 1993).
Trait version of the State-Trait Anxiety Inventory (STAI-T)
A 20-item scale that measures stable individual differences in proneness to respond with tension, apprehension, and heightened autonomic activity (Spielberger, Gorsuch, Lushene, Vagg, & Jacobs, 1983). Each item is rated on a scale of 1 ("almost never") to 4 ("almost always"). The total score ranges from 20 to 80. The STAI-T has been widely used in both clinical and
nonclinical populations. While trait anxiety varies across individuals, those with anxiety disorders reliably exhibit the highest levels (Zinbarg & Barlow, 1996). The STAI-T has high
internal consistency, α ranging between .86-.95 (Spielberger et al., 1983) and high test-retest reliability, average r = .88 (Barnes, Harp, & Jung, 2002).
Beck Depression Inventory Second Edition (BDI-II)
A 21-item self-report measure that assesses the presence and severity of depressive symptoms (Beck, Steer, & Brown, 1996). Participants are asked to choose one of four
statements, rated from 0 to 3, which best describes how they have been feeling during the past two weeks. The total score ranges from 0 to 63. A total score of 0 - 13 represents minimal levels of depressive symptoms, 14 - 19 mild levels, 20 - 28 moderate levels, and 29 - 63 severe levels. The scale has high internal consistency, α = .91, and test-retest reliability, r = 0.93 (Beck et al., 1996).
Disgust Propensity and Sensitivity Scale - Revised (DPSS-R)
A 16-item measure designed to assess the extent to which individuals experience disgust (i.e., disgust propensity) and their appraisal of these experiences (i.e., disgust sensitivity; van Overveld, de Jong, Peters, Cavanagh, & Davey, 2006). The two subscales consist of 8 items each. Each item is rated on a scale of 1 ("never") to 5 ("always"). The total score for each subscale ranges from 8 to 40. The DPSS-R has good test-retest reliability with alpha coefficients of 0.78 for the Disgust Propensity subscale and 0.77 for the Disgust Sensitivity subscale (van Overveld et al., 2006). Both disgust propensity and disgust sensitivity can modulate neural responses to aversive/disgusting stimuli (Mataix-Cols et al., 2008) therefore scores from this scale were included as covariates for analysis of the aversive clips presented in the anticipation task.
The Mood and Anxiety Symptom Questionnaire (MASQ)
A 90-item measure that assesses discrete dimensions of anxiety and depression and mixed symptomatology. The scale is based on the tripartite model of affect and was designed to measure its three factors (i.e., anxiety-specific, depression-specific and shared symptoms; Watson & Clark, 1991). The MASQ consists of three general distress subscales: anxious
symptoms (11 items), depressive symptoms (12 items) and mixed symptoms (28 items), and two specific subscales: an anxiety-specific subscale – anxious arousal (17 items) – that measures
physiological hyperarousal or somatic arousal and a depression-specific subscale – anhedonic
depression (22 items) – that measures positive effect and loss of interest/motivation. Participants indicate on a five-point scale, ranging from 1 ("not at all") to 5 ("extremely") to what extent they experienced various symptoms during the past week. Total scores for the various subscales are: 11 to 55 for the general distress anxiety subscale (MASW-GDA), 12 to 60 for the general distress depression subscale GDD), 17 to 85 for the anxious arousal subscale (MASQ-AA) and 22 to 110 for the anhedonic depression subscale (MASQ-AD). Due to high correlations, observed in previous studies (Watson et al., 1995), between the mixed symptoms subscale and both the MASW-GDA and MASQ-GDD, scores for the mixed symptoms subscale were not calculated. MASQ scales have good internal consistency (Watson et al., 1995).
Chapter 3: Summary of Demographics/Descriptives and Self-report Measures 3.1 Demographics
The racial representation across study groups was: 26 Caucasians, 14 Asians, 13 African Americans, 3 multiracial (i.e., two or more races) and 1 other. The highest level of education attained by participants was as follows: 42 had one or more years of college, 9 had one or more years of graduate/professional school, 3 attained a Bachelor's degree, 2 attained an Associate's degree, and 1 attained a Master's degree. Chai square tests indicated that the GAD group and Healthy group did not differ on these two variables (both ps > .2).
3.2 Stage of menstrual cycle
We approximated participants' menstrual phase on the day of testing based on self-report of the first and last date of their last menstrual period prior to the testing session (assuming a 28-day cycle). Based on this information 26 participants were in the Menstrual (1-4 28-days) or Follicular phase (1-12 days), 5 were in the Ovulation phase (13-15 days) and 22 were in the Luteal phase (16-28 days). In addition, one participant was in the menopause phase and three participants had irregular cycles. A Chai square test for all remaining participants (N= 53) indicated that women in the GAD group and Healthy group did not differ with respect to the phase of their menstrual cycle during testing (p >.5).
Table 1 shows mean scores and standard deviations for the Penn State Worry
Questionnaire (PSWQ), Intolerance of Uncertainty (IUS), trait anxiety scale (STAI-T), Anxiety Sensitivity Index (ASI), Beck Depression Inventory (BDI) and four subscales from the Mood and Anxiety Symptom Questionnaire (the general distress anxiety subscale; MASQ-GD, the general distress depression subscale; MASQ-GDD, the anxious arousal subscale; MASQ-AA
and the anhedonic depression subscale; MASQ-AD) for the GAD-only, Comorbid and Healthy groups. The GAD-only and Comorbid groups scored higher than the Healthy group on all measures consistent with greater severity of symptoms in patients (all ps < .001; except for the anxious arousal subscale for which the difference in scores between the Comorbid and Healthy groups was marginally significant p = .066). Mean scores for the PSWQ, STAI-T and BDI reflected clinical levels of worry and anxiety (Antony, Orsillo, & Roemer, 2001) and mild to moderate levels of depression in the GAD-only and Comorbid groups. For the Disgust Propensity and Sensitivity Scale-Revised (DPSS-R), scores in the GAD group were higher (p =.003) and in the Comorbid group marginally higher (p = .076) compared to the Healthy group.
Between-group comparisons for the two patient groups revealed higher scores in the Comorbid group for the MASQ- GDD (p = .048) and the MASQ-AD (p = .003), and a trend for higher scores on the BDI (p = .084), consistent with greater symptoms of depression. All pairwise comparisons were made using the Bonferroni adjustment for multiple comparisons. 3.4 Correlations between self-report questionnaires
As shown in Table 2 correlations between the various questionnaires were all significant (ps ≤ .001). Correlations between the STAI-T, MASQ-GDA and MASQ-AA and between the BDI, MASQ-GDD and MASQ-AD demonstrate good convergent validity for measures of anxiety and depression. However, moderate to high correlations across all questionnaires suggest that in general their discriminant validity is low, particularly for the STAI-T and BDI. The MASQ-AA and MASQ-AD were specifically designed to distinguish anxiety and depressive symptoms and do perform better in that regard. The correlation between the two subscales was the second lowest when examined for the entire sample (r = .42; p = .001) and higher in the Comorbid group (r = .47; p = .056) than in the GAD-only group (r = .33; p = .23).
Table 1. Mean scores and standard deviations for self-report questionnaires for the GAD-only, Comorbid and Healthy groups.
Que GAD-only (N=15) Comorbid (N=17) Healthy (N=25)
Mean SD Mean SD Mean SD
PSWQ 63.67 9.96 65.29 8.21 47.60 10.39 IUS 88.73 17.89 85.12 19.06 52.56 11.47 STAI-T 52.67 10.98 57.24 8.99 37.52 6.52 ASI 34.13 11.85 33.18 13.86 16.72 5.97 BDI (affective) 6.87 4.45 9.29 5.43 1.88 1.86 BDI (somatic) 11.47 9.09 16.59 6.94 2.88 2.49 BDI (total) 18.33 13.07 25.88 11.53 4.76 3.54 MASQ-GDA 25.80 9.64 26.47 5.17 16.80 4.03 MASQ-GDD 30.53 12.63 38.59 11.07 17.88 3.60 MASQ-AA 31.13 10.52 26.41 7.47 21.04 4.08 MASQ-AD 67.07 15.85 82.76 11.14 53.44 11.36
Table 2. Bivariate correlations between self-report questionnaires (N = 57).
PSWQ IUS STAI-T ASI BDI
total MASQ GDA MASQ GDD MASQ AA MASQ AD PSWQ -- .799* .758* .681* .622* .632* .603* .412* .603* IUS -- .846* .766* .753* .701* .676* .562* .660* STAI-T -- .725* .864* .721* .825* .543* .858* ASI -- .704* .683* .585* .696* .500* BDI total -- .694* .832* .614* .833* MASQ GDA -- .685* .722* .603* MASQ GDD -- .456* .779* MASQ AA -- .415* MASQ AD --
PSWQ = Penn State Worry Questionnaire; IUS = intolerance of uncertainty; STAI-T = Trait Anxiety
inventory; ASI = Anxiety Sensitivity index; BDI affective = total affective symptoms from the Beck Depression Inventory; BDI somatic = total somatic symptoms from the Beck Depression Inventory; BDI total = total score from the Beck Depression Inventory; MASQ-GDA = Mood and Anxiety Symptom Questionnaire-General Distress scale Anxiety; MASQ-GDD = Mood and Anxiety Symptom Questionnaire- General Distress scale Depression; MASQ-AA = Mood and Anxiety Symptom Questionnaire-Anxious Arousal; MASQ-AD = Mood and Anxiety Symptom Questionnaire-Anhedonic Depression; *p ≤ .001
Chapter 4: Neural Reactivity Tracks Fear Generalization Gradients 4.1 Introduction
Paradigms that assess fear learning have provided valuable translational tools for understanding the etiology, maintenance and treatment of anxiety disorders (Milad, Rauch, Pitman, & Quirk, 2006; Mineka & Oehlberg, 2008). The acquisition and extinction of conditioned fear responses involve a common neurocircuitry across species that includes the amygdala, insula, anterior cingulate cortex, hippocampus, sensory areas, and ventromedial prefrontal cortex (Büchel & Dolan, 2000; LeDoux, 2000; Phelps, Delgado, Nearing, & LeDoux, 2004). In addition to acquisition and extinction, there is increasing interest in fear generalization, which describes the transfer of a conditioned fear response to stimuli that are perceptually similar to the conditioned stimulus (CS). Insofar as the transfer of fear responses from threat-related stimuli to potentially innocuous cues is a common feature in anxiety disorders (Lissek et al., 2008), fear generalization may be a key learning process in the development and maintenance of pathological anxiety.
Recent studies have validated laboratory-based procedures for testing fear generalization, which involves the assessment of fear responses to a CS and to generalization stimuli (GS) that vary in perceptual similarity to the CS (Hajcak et al., 2009; Lissek et al., 2008). In these
paradigms, fear responses were quantified with the fear-potentiated startle reflex, which followed a generalization gradient: the strongest startle reflex was elicited during the CS, with a steep decline corresponding to the relative decrease in similarity of the GS to the CS1 (Hajcak et al., 2009; Lissek et al., 2008). Lissek and colleagues assessed fear generalization in a paradigm in which participants had to learn which stimulus was the CS and which were the GS. On the other
1 Comparable results have also been obtained using skin conductance (Dunsmoor et al., 2009; Vervliet et al., 2010), which is a more general measure of arousal that is not specific to defensive motivation.
hand, Hajcak and colleagues found comparable results even when participants were explicitly instructed regarding the identity of the CS and the reinforcement contingencies to the CS and GS. Despite being told explicitly which stimulus was the CS, and never being shocked following a GS, participants in the Hajcak et al. study had larger startle responses and reported greater shock likelihood as GS were more perceptually similar to the CS.
Fear generalization paradigms could be useful for assessing pathological fear and risk for anxious psychopathology. For instance, patients with panic disorder exhibit a flatter fear gradient with more gradual decreases in fear response to the GS (Lissek et al., 2010). Hajcak and
colleagues (2009) reported fear generalization deficits in a generalization paradigm as a function of variation in the brain-derived neurotrophic factor (BDNF) genotype, which has been related to both learning and anxiety-related behaviors.
In the current study, we sought to extend this work by examining neural activity using fMRI in a fear generalization paradigm that we previously employed (Hajcak et al., 2009). The aim was to elucidate the brain regions associated with generalization, which have received little attention in the literature, and to examine whether reactivity in these regions exhibit a similar generalization gradient to that reported with peripheral measures of fear. These neural gradients may be useful in identifying deficits in the generalization process and may be relevant to future work on pathological anxiety (e.g., Lissek et al., 2010). In the current study, the CS was a middle-sized rectangle, and the GS were six additional rectangles varying in width from the CS by ±20%, ±40% or ±60%.
In an initial experiment (N = 8), we examined regions of interest (ROIs) based on neuroimaging studies of fear learning that have implicated key areas in the expression and inhibition of autonomic and behavioral fear responses.(Dunsmoor, Prince, Murty, Kragel, &
LaBar, 2011; Sehlmeyer et al., 2009). These ROIs included the amygdala, insula, thalamus, caudate, anterior cingulate cortex (ACC), and ventromedial prefrontal cortex (vmPFC). We hypothesized that reactivity in one or more of these regions would demonstrate a similar gradient response to the pattern reported in previous laboratory-based studies. In a second experiment (N = 25), we conducted a whole-brain analysis and obtained additional self-report ratings and physiological measures.
4.2 Method: Experiment 1 (Pilot study) 4.2.1 Participants
Eight individuals (6 females and 2 males) participated in the study (Mean age = 23.2; SD = 4.7). All reported being right handed. Potential participants were screened for prescription and recreational drug usage, as well as neurological and psychological histories. The study was approved by the Stony Brook University Institutional Review Board; all participants provided informed consent.
4.2.2 Procedure
Stimulator STM 200; Biopac Systems Inc.), was individually set for each participant to a level that was "uncomfortable but not painful". Instructions for the task were then provided.
Participants were told that the middle-sized rectangle (CS) indicated a 50% probability that they would receive a subsequent electric shock, but that shocks would never follow rectangles of greater or lesser size. A conditioning phase was administered next, which included five presentations of the CS with electric shock (i.e., CSpaired) and one presentation of each of the
generalization within the context of a paradigm that combined instructed and associative fear learning.
4.2.3 Task
The task consisted of three blocks presented consecutively. Each block included 40 trials (5 trials 8 conditions) for a total of 120 trials. The stimuli were seven red rectangles with identical height (56 pixels) and varying width (112 to 448 pixels) presented against a black background. The middle-sized rectangle (280 pixels) was the conditioned stimulus (CS). Half of the time the CS co-terminated with a 500 ms electric shock (CSpaired), while half of the time it did
CS and served as the generalization stimuli (GS). Stimuli were presented pseudorandomly for 2 seconds with a jittered interstimulus interval ranging from 4 to 10 seconds, during which a white fixation cross was shown on a black background. The duration of the task was 15 minutes and 12 seconds.
4.2.4 Image Acquisition
Participants were scanned with a 3 Tesla Siemens Trio scanner at the Stony Brook Social, Cognitive, and Affective Neuroscience (SCAN) center. A total of 456 T2*-weighted echoplanar images were acquired with an oblique coronal angle and TR = 2000 ms, TE = 22 ms, Flip Angle = 83º, Matrix = 96 96, FOV = 224 224 mm, Slices = 36 and Slice Thickness = 3.5 mm. In addition, we obtained T1-weighted structural scans with TR = 1900 ms, TE = 2.53, Flip angle = 9°, FOV = 176 250 250 mm, and Matrix = 176 256 256 mm.
4.2.5 Image Analysis
Preprocessing procedures were performed in SPM8 and included slice time correction, motion correction, normalization and smoothing with a 6-mm full width at half maximum
Gaussian kernel. Preprocessed images were entered into a general linear model in which each rectangle was modeled as an event with no duration; CSpaired and CSunpaired were modeled
separately. The six motion parameters estimated during realignment were included as regressors of no interest and serial autocorrelations were modeled using an AR (1) process. First-level single subject statistical parameter maps were created for the 'CSpaired - Baseline' (i.e., fixation),
'CSunpaired - Baseline' and each of the 'GS - Baseline' contrasts. These contrasts, except for
'CSpaired - Baseline', were used in a second-level random effects repeated measures analysis.
4.2.6 Gradients of neural reactivity
Individual bilateral masks were created for the amygdala, insula, thalamus, caudate nucleus, anterior cingulate cortex (ACC) and ventromedial prefrontal cortex (vmPFC) using the Masks for Regions of Interest Analysis software (Walter et al., 2003). A region of interest (ROI) analysis for the F-contrast (main group effect) was performed using an initial threshold of α = .01
(uncorrected) and extent threshold = 20 contiguous voxels, and a small volume familywise error rate
corrected α = .05, for each mask.
Neural gradients were generated for the right and left insula (which were the only regions that showed significant activation with these thresholds) by extracting the first eigenvariate (i.e., the principal component) from a 6 mm sphere centered on the local maxima within each region, for each of the 'CSunpaired - Baseline' and 'GS - Baseline' contrasts, across all participants. Mean
values for CSunpaired, as well as GS ±20%, GS ±40% and GS ±60%, were plotted as a four-point
4.3.1 Generalization gradients of neural activation
Generalization gradients for the right and left insula are shown in Figure 2(c) and Figure 2(b), respectively. Reactivity in the right (F(3,21) = 18, p < .001) and left (F(3,21) = 13.3, p < .001)
insula varied as a function of stimulus type. For the right insula, pairwise comparisons revealed higher reactivity for the CSunpaired versus GS ±40% (p = .004) and GS ±60% (p =.01), and for the
GS ±20% versus GS ±40% (p = .02). A comparison of the GS ±20% to GS ±60% was
marginally significant (p =.053). For the left insula, reactivity was higher for the CSunpaired versus
GS ±40% (p =.007) and GS ±60% (p =.03), and for the GS ±20% versus both GS ±40% (p = .03) and GS ±60% (p =.04).
4.4Method:Experiment 2 4.4.1 Participants
Twenty-five women participated in the study (Mean age = 21.6; SD = 5.1). All reported being right-handed except for one participant, who reported being ambidextrous. Participants were screened for psychiatric illness with the Structured Clinical Interview for DSM-IV Axis I Disorders-Patient Edition, Version 2 (SCID-I/P; First et al., 2002). All other screening
procedures were identical to Experiment 1. The study was approved by the Stony Brook University Institutional Review Board; all participants provided informed consent. 4.4.2 Experimental paradigm
The experimental paradigm was identical to Experiment 1 except for the addition of post-task ratings of shock likelihood for each rectangle, obtained on a Likert-type scale of 1
Eyelink-1000 (SR Research Ltd., Ontario, Canada) as a measure of activation of the sympathetic nervous system, as well as a 12 second increase in task length to accommodate a change in TR and TE due to scanner requirements.
4.4.3 Image Acquisition and Analysis
A total of 440 T2*-weighted echoplanar images were acquired with an oblique coronal angle and TR = 2100 ms, TE = 23 ms, flip Angle = 83º, matrix = 96 96, FOV = 224 224 mm, slices = 37 and slice thickness = 3.5 mm. Parameters for acquisition of structural images, as well as preprocessing procedures and statistical analysis were identical to Experiment 1.
Gradients of neural reactivity were generated for all brain regions for which we found significant clusters for the main effect group F-contrast using a whole brain threshold of α = .001uncorrected and extent threshold of 20 contiguous voxels.
4.4.5 Preprocessing of pupil data
Pupil data was processed using custom MATLAB codes (MathWorks). First, we excluded periods of eye blinks detected by an on-line parsing system (Eyelink; SR Research Ltd., Ontario). We used a window of 100 ms prior to onsets of eye blinks and 300 ms following their offset in order to minimize after-blink constriction effects. Missing values were linearly interpolated. We adopted pre-processing procedures from Hupé et al. (2009). Specifically, a baseline for each trial was calculated by averaging data points from 500 ms immediately preceding the onset of the stimulus and then subtracting this mean from each trial. The baseline corrected values were z-scored to allow comparison across participants and filtered using a low-pass filter (4 Hz cutoff frequency) to reduce measurement noise. After preprocessing, trials were examined and excluded if they had outliers, exceeding two standard deviations, and no pupillary
light reflex in response to luminosity changes of presented stimuli. The average number of trials excluded for each participant was 19.1% (SD = 9.1). For statistical analysis, the pupillary
response was defined as the overall pupil diameter change (i.e., area under curve) within a 1000 ms window, beginning 1 second after stimulus onset. Data for one participant was excluded due to technical problems.
Self-report and neural measures were evaluated with repeated measures analysis of variance. Pupillary response was evaluated with a mixed linear model in order to account for missing trials across conditions and to increase sensitivity of the analysis. Pairwise comparisons for all measures were made using the Bonferroni adjustment for multiple comparisons.
4.5.1 Self-reported shock likelihood
Shock likelihood ratings varied as a function of stimulus type (F(3,72) = 77.9, p < .001;
Figure 3a). Pairwise comparisons showed that shocks were rated as more likely following the CS (M = 3.9, SD = 1) compared to GS ±20% (M = 3.02, SD = 0.88; p = .007), GS ±40% (M = 1.66, SD = 0.67; p < .001) or GS ±60% (M = 1, SD = 0.14; p < .001). In addition, shocks were rated as more likely following the GS ±20% compared to both GS ±40% (p < .001) and GS ±60% (p < .001), and the GS ±40% compared to GS ±60% (p < .001).
4.5.2 Pupillary response
A generalization gradient of pupillary response is presented in Figure 3(b). Pupillary response varied as a function of stimulus type (F(4,44) = 30.4, p < .001). Pairwise comparisons
showed that pupillary response was larger for the CSunpaired versus GS ±40% (p < .001) and GS
significant (p = .078). Comparisons for GS ±20% versus GS ±40% and GS ±60% and for GS ±40% versus GS ±60% were not significant (all ps > .10).
4.5.3 Gradient of neural reactivity
Generalization gradients for the right and left insula are presented in Figure 4(c) and Figure 4(b), respectively. Reactivity in both regions varied as a function of stimulus type (right insula: F(3,72) = 21.8, p < .001; left insula: F(3,72) = 9.2, p < .001). For the right insula, there was
higher reactivity for the CSunpaired versus GS ±40% (p < .001) and GS ±60% (p <.001), and for
the GS ±20% versus GS ±40% (p < .001). A comparison of the GS ±20% to GS ±60% was marginally significant (p =.055). For the left insula, reactivity was higher for the CSunpaired versus
GS ±40% (p =.002) and GS ±60% (p =.04), and for the GS ±20% versus GS ±40% (p < .001) and GS ±60% (p <.03). In addition to the insula, the F-contrast revealed significant clusters for the right supplementary motor area (SMA), anterior cingulate cortex (ACC), somatosensory cortex, caudate, ventromedial (vmPFC) and primary visual cortex. We therefore, generated gradients for these brain regions in order to compare their response patterns to that of the insula. Reactivity in the ACC (Figure 5a; F(3,72) = 9.7, p < .003), right SMA (Figure 5b; F(3,72) = 15.7, p
< .001), and the right and left caudate (Figure 5c; right caudate: F(3,72) = 11.52, p < .001; left
caudate: (F(3,72) = 9.39, p < .001) showed a similar response pattern with higher reactivity
associated with increased similarity of GS to CS. For the ACC, reactivity was higher for the CSunpaired and for the GS ±20 relative to GS ±40% (p < .001and p = .001, respectively). For the
right SMA, reactivity was higher for the CSunpaired versus GS ±40 (p < .001) and GS ±60% (p =
.001), and for the GS ±20% versus GS ±40 (p = .003) and GS ±60% (p = .03). For the right caudate, reactivity was higher for the CSunpaired versus GS ±40 (p < .001) and GS ±60% (p = .04),
higher for the CSunpaired versus GS ±40 (p < .001) and GS ±60% (p = .03), and for the GS ±20%
versus GS ±40 (p = .01). In contrast, reactivity in the vmPFC (Figure 6a; F(3,72) = 13, p < .001)
and somatosensory cortex (Figure 6b; F(3,72) = 13, p < .001) showed a reverse response pattern.
For the vmPFC, reactivity was higher for the GS ±60% versus GS ±40 (p = .01), GS ±20% (p < .001) and CSunpaired ( p < .001), and for the GS ±40 versus CSunpaired ( p = .02). Similarly, for the
somatosensory cortex, reactivity was higher for the GS ±60% versus GS ±40 (p = .03), GS ±20% (p = .007) and CSunpaired (p < .001), and for the GS ±40 versus CSunpaired (p = .02). Finally, there
was no significant effect of stimulus type for the right (F(3,72) = 2.5, p = .08) and left (F(3,72) = .5,
p =.66) visual cortex.
4.5.4 Direct comparisons of CSunpaired versus GS
Areas of activation for the CSunpaired versus GS ±40% and GS ±60% are presented in
Table 1. This direct comparison showed enhanced activation in the anterior insula, SMA, cingulate gyrus, caudate, thalamus, and frontal areas. Reactivity in these regions is commonly reported in neuroimaging studies of fear learning (Sehlmeyer et al., 2009) and suggests that generalization engages the same circuitry. There was no significant activation for the CSunpaired
versus GS ±20% contrast.
Examination of the reverse contrast (i.e., GS ±40% and GS ±60% versus CSunpaired; Table
2) showed increased activation in the vmPFC, somatosensory and motor areas, and subcallosal and posterior cingulate. Both the vmPFC and cingulate are associated with modulation of the fear response. There was no significant activation for the GS ±20% versus CSunpaired contrast.
4.5.5 Amygdala reactivity
The CSunpaired versus GS comparisons did not reveal significant activation in the
the amygdala (Büchel et al., 1998; LaBar et al., 1998), we conducted a linear time modulation analysis to test whether amygdala reactivity showed a decline in reactivity over time. For this analysis, we included linear regressors in a new statistical model to account for time effects in reactivity to the CSunpaired, GS ±20%, GS ±40% and GS ±60%. There was a significant decrease
in amygdala reactivity over time for all four trial groups (CSunpaired:Right: x = 32, y = -2, z = -12,
t(96 ) = 4.9, p < .001, Left: x = -26, y = -2, z = -12, t(96 ) = 3.82, p = .01; GS ±20%: Right: x = 32,
y = -2, z = -18, t(96 ) = 4.12, p = .005; GS ±40%: Right: x = 18, y = 4, z = -18, t(96 ) = 4.07, p =
.03, Left: x = -26, y = 0, z = -16, t(96 ) = 4.07, p = .006; GS ±60%: Right: x = 26, y = -2, z = -22,
t(96 ) = 3.56, p = .03; small volume corrected with a bilateral amygdala mask and α = .05). There
was no interaction of time and stimulus type.
4.6 Discussion
Across two experiments, the right and left insula showed increased activation to the CS and decreases in response amplitude as a function of increasing dissimilarity between the GS and CS. In addition to the insula, the anterior cingulate cortex (ACC) the right supplementary motor area (SMA), and caudate showed a similar reactivity pattern in the second experiment2.
Reactivity in the ventromedial prefrontal cortex (vmPFC) and primary somatosensory cortex showed the opposite pattern (i.e., largest response to the GS most dissimilar to the CS), and reactivity in the visual cortex was sensitive to increases in stimuli size. Thus, a pattern consistent with autonomic quantifications of generalization was restricted to the insula, ACC, right SMA, and caudate, and was not universally present across the brain. The inverse reactivity pattern in the vmPFC, an area linked to extinction recall and safety learning (Corcoran & Quirk, 2007;
2 Reactivity patterns for the ACC and SMA in experiment 1 also showed a linear trend but only with a less stringent whole-brain threshold of α = .01.
l=http://search.proquest.com/docview/304520113?accountid=14172
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Puzzling Meta
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Middle weight puzzle
There are 100 white and 101 black balls. All the balls have different weights. Balls of the same colour look the same. White balls are numbered from lighter to heavier (from 1 to 100). The same is true for black balls (they are numbered from 1 to 101).
What is the minimal number of weightings on a balance scale to find the ball with the middle weight? (That is, the ball which will have the number 101 if all the balls would be sorted and numbered from lighter to heavier.)
Additional explanations:
1. On a balance scale you can compare X balls on one arm to X balls on another arm and tell whether they are lighter, heavier or equal in relation to each other.
2. Since the balls are numbered, you don't have to compare balls of the same colour to each other using the scale as you know the result already.
I know that there is a solution with 8 weightings (though I do not know it). I am especially interested in the proof of minimality, since I want to be sure that 8 (or any other number) is the minimum.
logical-deduction
Gamow
klm123klm123
$\begingroup$ Are we weighing the balls on a numerical or two-sided scale? Also, is the colour of the balls relevant to the question at all? (If not, then it would be easier just to say there are 201 balls.) $\endgroup$
– user88
$\begingroup$ @JoeZ., usual for this type of puzzle scales:They are called balance scale if I am not wrong. Colour is definitely relevant, please reread the question. $\endgroup$
– klm123
$\begingroup$ Oh, they're all numbered. Right, that makes things much easier. $\endgroup$
$\begingroup$ @PerManne, may be, may be not. This is not obvious for me. $\endgroup$
$\begingroup$ Step 1: Weigh W100 vs. B1. If B1 is heavier, you're done! If not... Steps 2-7: ???. Step 8: Profit! $\endgroup$
Proof of minimality
8 is the minimum, with the following claims:
In the worst case, the result of comparison can only be one of the two possibilities (left heavier or right heavier), since equality might not be present and hence we can't rely on that.
Therefore one weighting gives 1 bit of information.
There are 201 possible outcomes (each ball is a candidate of the median), which can't be encoded in 7 bits (7 bits give at most 128 possible outcomes)
Therefore the minimum possible number of weightings is 8.
Now it's left for us to prove that there is a solution with 8 weightings.
The 8 weightings
Per Manne almost gets it correct, it's just that things get trickier after the sixth weighting.
So, assuming the same thing as what Per Manne has said (using left sack to store balls which are certainly lighter than the middle ball, and right sack to store those which are heavier), the procedure to find the $(k+1)$-th ball for $k$ white balls ($w_1, w_2, \ldots, w_k$) and $k'$ black balls ($b_1, b_2, \ldots, b_{k'}$), where $k'$ is either $k$ or $k+1$, is as follows.
Let $m = \lceil\frac{k+1}{2}\rceil$ and $n = \lceil\frac{k'+1}{2}\rceil$
Compare $w_m$ with $b_n$
If $w_m > b_n$ then put $b_1, b_2, \ldots, b_{n-1}$ on left sack and $w_m, w_{m+1}, \ldots, w_{k}$ on right sack
If $w_m < b_n$ then put $w_1, w_2, \ldots, w_{m-1}$ on left sack and $b_n, w_{n+1}, \ldots, b_{k'}$ on right sack
Rename the balls, repeat to step 1 until we have 4 balls left (follow Per Manne's answer for more details) - you can actually continue this procedure until you get the result; the following steps are only for clarity.
At 4 balls at two balls each (we have done 6 weightings up to this point), we have these 6 possible combinations:
$w_1, w_2, b_1, b_2$
$w_1, b_1, w_2, b_2$
$b_1, w_1, w_2, b_2$
$w_1, b_1, b_2, w_2$
$b_1, w_1, b_2, w_2$
$b_1, b_2, w_1, w_2$
Note that at this time we have one more ball in right sack compared to left sack, so we are looking for the third element here.
Compare $w_2$ with $b_2$ (actually you would do this also if you follow step 1-4 above, this is just for clearer explanation)
If $w_2 > b_2$ then we have either of the case (4), (5), or (6)
If $w_2 < b_2$ then we have either of the case (1), (2), or (3)
At both case, we have three possibilities left. Since the two cases are analogous, we take the first three (1), (2), and (3).
Compare $w_2$ and $b_1$.
If $w_2 > b_1$ then the third element is $w_2$
If $w_2 < b_1$ then the third element is $b_1$
Note that after step 5 (that is, after 6 weightings), we only do two more weightings (step 6 and step 8), totaling to 8 weightings.
Therefore 8 is the minimum.
justhalfjusthalf
$\begingroup$ Actually you can skip 1,2 in the proof. $\endgroup$
$\begingroup$ I was just trying to be as rigorous as possible, because in the problem statement it is said On balance scale you can compare X balls on one arm to X balls on another arm and tell whether they are lighter, heavier or equal to each other. =) $\endgroup$
– justhalf
$\begingroup$ I am unconvinced by claim 1. It is certainly possible to infer things from multi-ball weighings; for example, if we weigh black ball 1 against all the white balls at once and find the black ball is heavier, we can deduce that black ball 1 is the median. We need a more sophisticated argument to show that multi-ball weighings give us no advantage. $\endgroup$
$\begingroup$ Hmm, now that you say it, actually we don't need step 1, because the outcome will still be 1 bit (equality might not appear in the worst case, so we can't rely on that). $\endgroup$
$\begingroup$ @HemantAgarwal if 50th black ball is heavier than 50th white ball, it doesn't necessarily mean that the 51st black ball is the 101st ball, since it can still be heavier than the 51st white ball, or the 52nd white ball, etc. In other words, it is also possible that the lightest black ball is heavier than the heaviest white ball, which in that case the first black ball is the median. Also, the method I wrote above (which comes from Per Manne's answer below) is just one possible solution. There could be other solutions. $\endgroup$
Edit: The first version contained a silly mistake (after the sixth weighing), which justhalf has pointed out. I have corrected the solution, as well as the general comment at the end.
Number the balls W1, W2, ..., W100 and B1, B2, ..., B101 in order of increasing weight. Bring out one table and two empty sacks. Put all the balls on the table in two ordered rows, white and black. Put one sack on the left side and one on the right side of the table. After each weighing, we will put all balls that we know are lighter than the middle ball in the left sack, and all balls that we know are heavier than the middle ball in the right sack. When we are done, there should be 100 balls in either sack and one ball remaining on the table.
We will always compare one white ball against one black ball, and we will pick each ball from the middle of its row. There will be an odd/even theme here, but let's just start and see what happens.
We begin by comparing W50 and B51. (The choice W51 and B51 gives similar results.) There are two possibilities.
(1) If W50 > B51 then the 51 black balls B1,...,B51 go into the left sack and the 50 white balls W51,...,W100 go into the right sack.
(2) If W50 < B51 then the 50 white balls W1,...,W50 go into the left sack and the 50 black balls B52,...,B101 go into the right sack.
The worst case is (2), so we will focus on that. (In case (1), it is possible to leave B51 on the table and get the same situation as in case (2), if desired.) Rename the balls W1,...,W50 and B1,...,B51, and compare W25 with B26. The worst case is when W25 < B26, then W1,...,W25 go into the left sack and B27,...,B51 go into the right sack. We have 75 balls in each sack, and 25 white and 26 black balls on the table. (Note that this time it is the larger of these two numbers which is even!)
Again, we rename the balls W1,...,W25 and B1,...,B26, and we compare W13 with B14. The worst case is when W13 < B14, then W1,...,W12 go into the left sack and B14,...,B26 go into the right sack. We now have 87 balls in the left sack, 88 balls in the right sack, and 13 white and 13 black balls on the table.
Rename tha balls W1,...,W13 and B1,...,B13, and compare W7 with B7. In either case, seven balls go into the left sack and six into the right sack. We have 94 balls in either sack, and 6 balls of one color (say white) and 7 balls of the other color (say black) on the table.
After the next weighing (the fifth) we will in the worst case have 97 balls in each sack, and 3 white and 4 black balls on the table.
After the sixth weighing we will in the worst case have 98 balls in the left sack, 99 balls in the right sack, and 2 balls of each color on the table.
Rename the balls W1, W2 and B1, B2. Compare W2 with B2. If W2 < B2 then put W1 in the left sack and B2 in the right sack. If W2 > B2 then put B1 in the left sack and W2 in the right sack. We are left with only two balls on the table, and in the eight and last weighing we compare them against each other.
Some general considerations:
With this method, if there are 2k white balls and 2k+1 black balls on the table, with equally many balls in each sack, one will get k white balls and k+1 black balls on the table after one weighing (as always in the worst case). We can write this as $(2k,2k+1)\mapsto (k,k+1)$. Similarly, one can consider the other possibilities for even and odd numbers of balls, and get the following four possibilities for each weighing:
$$(2k,2k+1)\mapsto(k,k+1), \qquad (2k+1,2k+2)\mapsto(k+1,k+1),$$ $$(2k+1,2k+1)\mapsto(k,k+1), \qquad (2k,2k)\mapsto (k,k).$$
(If the number of white and black balls on the table is equal, then one sack has one ball more than the other, otherwise the sacks have the same number of balls.)
Per MannePer Manne
$\begingroup$ I was looking for a solution similar to yours, but couldn't figure out how to put it in words. $\endgroup$
Thanks for contributing an answer to Puzzling Stack Exchange!
Twelve balls and a scale
Twelve balls on a scale, where one ball is lighter and another is heavier
$N_{max}$ balls in 3 weightings
Different Ball Puzzle
A balance with three pans, detecting the lightest pan (find the one lighter ball)
A balance with three pans, detecting the lightest pan (find the one heavier ball)
A balance with three pans, detecting the lightest pan (find the two heavier balls)
A balance with three pans, detecting the lightest pan (find the one lighter/heavier ball, for a given number of balls) | CommonCrawl |
1(a) Give the comparison of DFS, BFS, Iterative deeping and Bidirectional search.
1(c) Explain modus ponen with suitable example.
1(d) Draw and Explain general model of Learning Agent.
1(e) Explain the Limitation of propositional logic with suitable example.
2(a) Explain Hill climbing and simulated Annealing with suitable example.
2(b) Explain Goal based and utility based agent with block diagram.
3(a) Consider the given game tree. Apply $\alpha$-$\beta$ pruning where $\square$-max node, 0- min node.
3(b) Explain Role learning and Inductive learning with suitable examples.
4(a) Consider the following sentence.
Prove that Tom is mortal using modus ponen and Resolution.
4(b) Draw an explain the expert system Architecture.
5(a) Consider the given tree, apply breadth first search algorithm and also write the order in which nodes are expanded.
5(b) Write the Planning algorithm for spare tyre problem.
Q6) Write the short note on any four. | CommonCrawl |
\begin{document}
\title{Quantum probes to assess correlations in a composite system}
\author{Andrea Smirne $^{1,2}$, Simone Cialdi $^{1,2}$, Giorgio Anelli $^1$, Matteo G.A. Paris $^{1,3}$, Bassano Vacchini $^{1,2}$}
\affiliation{ $^1$ \mbox{Dipartimento di Fisica, Universit{\`a} degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy}\\ $^2$ \mbox{INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy}\\ $^3$ CNISM, Udr Milano, I-20133 Milan, Italy }
\begin{abstract}
We suggest and demonstrate experimentally a strategy to obtain relevant information about a composite system by only performing measurements on a small and easily accessible part of it, which we call quantum probe. We show in particular how quantitative information about the angular correlations of couples of entangled photons generated by spontaneous parametric down conversion is accessed through the study of the trace distance between two polarization states evolved from different initial conditions. After estimating the optimal polarization states to be used as quantum probe, we provide a detailed analysis of the connection between the increase of the trace distance above its initial value and the amount of angular correlations. \end{abstract}
\pacs{03.65.Yz,03.65.Ta,42.50.Dv}
\maketitle
The control of quantum systems plays a basic role in the experimental investigation of the predictions of quantum theory as well as in the development of quantum technologies for applications. Indeed, great attention has been recently payed to engineering the dynamics of quantum systems in order to properly generate, manipulate and exploit significant quantum features
\cite{Myatt2000,Kraus2004,Chou2005,Verstraete2009,Barreiro2011,Krauter2011}.
Consider a large quantum system whose full characterization is only partially feasible or requires complex measurement schemes.
In such a case, it is crucial to develop effective strategies in order to assess relevant pieces of information about the overall system by only monitoring a small part, which then acts as a probe. A natural procedure is to control the interaction of the small subsystem with the rest of the total system in such a way that the former can encode the information of interest. Here, we provide an explicit example of this strategy in an all-optical setup, where the system under study consists of entangled couples of photons generated by spontaneous parametric down conversion (SPDC) \cite{Hong1985,Joobeur1994,Kwiat1999,Gerry2005}. By properly engineering the interaction between polarization and momentum degrees of freedom of the photons via a 1D spatial light modulator (SLM), we can access some information regarding the momentum correlations between the two photons by simply performing visibility measurements on the polarization degrees of freedom.
As specific figure of merit, we exploit the trace distance between polarization states. As we shall see, an increase of the trace distance above its initial value allows to detect some information on momentum correlations, which has moved to the polarization degrees of freedom thanks to the engineered interaction. The trace-distance analysis of quantum dynamics has been recently introduced, leading to important results concerning the non-Markovianity of a quantum dynamics \cite{Breuer2009,Apollaro2011,Liu2011,Vacchini2011,Breuer2012,Smirne2013}, the characterization of the presence of initial correlations between the quantum system and its environment \cite{Laine2010b,Smirne2011,Dajka2011,Gessner2011}, the relevance of non-local memory effects \cite{Laine2012,Liu2012,Laine2012b} and the reservoir engineering in ultracold gases \cite{Haikka2011,Haikka2012,Haikka2013}.
The paper is structured as follows. In the next Section we describe the physical system we are going to investigate and present the details of the experimental apparatus. In Section II we illustrate the trace-distance approach to the dynamics of an open system and present the details of the calculations of its evolution for different angular and polarization states. Section III is devoted to illustrate the experimental results about the optmization of the probe and the link between the behavior of trace distance and the initial correlations in the angular degrees of freedom. Finally, Section IV closes the paper with some concluding remarks.
\section{The physical system and the experimental apparatus}\label{sec:exs} Our overall system consists of couples of entangled photons generated by SPDC in a two-crystal geometry \cite{Kwiat1999}. The couples are detected along two beams, named signal and idler, which are centered around the directions fixed by the phase matching condition. The two-photon state generated by SPDC can be written \begin{eqnarray} \label{eq:totall}
|\psi\rangle = && \int d \omega_p d \omega_s d \theta_s d \theta_i A(\omega_p) \tilde{F}(\Delta k_{\perp}) {\mbox{Sinc}}( \Delta k_{\parallel} L/2) \nonumber\\ &&\times\left[\cos{\alpha}\ket{{\small{H,\theta_s, \omega_s}}}\ket{{\small{H,\theta_i,\omega_p-\omega_s}}}\right.\nonumber\\ && \left.+e^{i \Phi(\omega_p, \theta_s, \theta_i)}\sin{\alpha} \ket{{\small{V,\theta_s, \omega_s}}}\ket{{\small{V,\theta_i,\omega_p-\omega_s}}}\right], \end{eqnarray} where up to first order in frequency and angle, \begin{eqnarray} \Delta k_{\parallel} &=& -\frac{\omega_p^0 \theta^0}{2 c}(\theta_s+\theta_i)\nonumber\\ \Delta k_{\perp} &=& \frac{\omega_p^0}{2 c} (\theta_s-\theta_i) + \frac{2 \theta^0 \omega_s}{c}.\label{eq:kk} \end{eqnarray} Here, $\omega_p$ is the shift of the pump frequency with respect to the central frequency $\omega^0_p (405nm)$, $\theta_s$ and $\omega_s$ ($\theta_i$ and $\omega_i=\omega_p-\omega_s$) are the signal (idler) angle and frequency shift with respect to the phase matching condition, $\theta_s^0=\theta_i^0\equiv\theta^0=3^{\circ}$ and $\omega^0_s=\omega^0_i= \omega^0_p/2$, while $\ket{{\small{P,\theta,\omega}}}$ denotes the single-photon state with polarization ${\small{P = H,V}}$, angle $\theta$ and frequency $\omega$. Moreover, $A(\omega_p)$ is the spectral amplitude of the pump, $\tilde{F}(\Delta k_{\perp})$ is the Fourier transform of its spatial amplitude
and the ${\mbox{Sinc}}(\Delta k_{\parallel} L/2)$ function arises due to the finite crystal size ($L=1mm$) along the longitudinal direction.
The two-crystal geometry implies that the polarization degrees of freedom of the two photons are entangled and it further introduces the phase term $\Phi(\omega_p, \theta_s, \theta_i)$, which is due to the different optical paths followed by the couples of photons generated in the first and in the second crystal \cite{Rangarajan2009,Cialdi2010a,Cialdi2012}. To first order this term reads $\Phi(\omega_p, \theta_s, \theta_i) \approx \Phi_0 + \Delta \tau \omega_p + \kappa \theta_s +\eta \theta_i$. Finally, the probabilities of generating $\ket{{\hbox{\small VV}}}$ or $\ket{{\hbox{\small HH}}}$ photons, $\sin^2{\alpha}$ and $\cos^2{\alpha}$ respectively, are determined by the polarization of the incident laser.
The overall state in Eq.(\ref{eq:totall}) fixes in particular the correlations between signal and idler angular degrees of freedom. By properly engineering the two-photon evolution, relevant information about these angular correlations gets encoded into the polarization degrees of freedom and then can be easily accessed. In fact, through the SLM we can impose an arbitrary polarization- and position-dependent phase shift to the two-photon state in Eq.(\ref{eq:totall}). On the one hand, a linear phase $\overline{\Phi} \equiv -\Phi_0 - \kappa \theta_s -\eta \theta_i$ is set to offset the corresponding terms in the first-order expansion of $\Phi(\omega_p, \theta_s, \theta_i)$ \cite{Rangarajan2009,Cialdi2010a,Cialdi2012,Cialdi2010b}. On the other hand, a further linear phase on both signal and idler beams emulates a time evolution of the two-photon state \cite{Cialdi2011}.
The experimental setup is shown in Fig. \ref{fig:1}. A linearly polarized cw 405 nm diode laser (Newport LQC405-40P) passes through two cylindrical lenses, which compensate beam astigmatism (AC), then through a spatial filter (SF) composed by two lenses and a pin-hole in the Fourier plane to obtain a Gaussian profile by removing the multimode spatial structure of the laser pump. Finally, a telescopic system (TS) prepares a beam with the proper radius and divergence. A couple of 1 mm beta-barium borate crystals (C), cut for type-I downconversion, with optical axis aligned in perpendicular planes, are used as a source of polarization and momentum entangled photon pairs with $\theta^0=3^{\circ}$. We use a compensation crystal on the pump (DC) \cite{Cialdi2008}, which acts on the delay time between the vertical and horizontal polarization, and a couple of thin crystals ($0.5 mm$) for the spatial walk-off compensation (WO). An interference filter or a long pass filter (F) is put on the signal path to select the spectral width of the radiation (10nm or 45nm). In order to obtain different spectral widths or a particular spectral profile, we use a 4f optical system after the coupler on the signal path. The 4f system consists of two gratings (G1 and G2) of $1200 lines/mm$ and two achromatic lenses (L1 and L2) with $f=35mm$. The distance between the lenses and the grating is $f$ and the distance between the two lenses is 2f. In this configuration, in between the two lenses the spectral components are focalized and well separated, so that it is possible to put here a slit to select the wanted spectral width. An SLM, which is a liquid crystal phase mask $(64 \times— 10 mm)$ divided in 640 horizontal pixels each $d=100 \mu m$ wide, is set before the detectors, at 310 mm from the generating crystals, in order to introduce the spatial phase function. When the mirror (M) is switched on the radiation path a cylindrical lens (L) generates the Fourier Transform profile of the pump at his focal distance (1m), where a CCD camera is located. A couple of polarizer (P) is used to measure the visibility of the entangled state. \begin{figure}
\caption{(Color online) Schematic diagram of the experimental setup, as described in Sect.\ref{sec:exs}. }
\label{fig:1}
\end{figure}
\section{Trace-distance analysis} \subsection{General strategy} The trace distance between two quantum states $\rho^1$ and $\rho^2$ is defined as \begin{equation}\label{eq:tdd}
D(\rho^1, \rho^2) = \frac{1}{2} \mbox{Tr}\left|\rho^1-\rho^2\right|=\frac{1}{2}\sum_k|x_k|, \end{equation} with $x_k$ eigenvalues of the traceless operator $\rho^1-\rho^2$, and it is a metric on the space of physical states such that it holds $0\leq D(\rho^1, \rho^2) \leq1$. The physical meaning of the trace distance lies in the fact that it measures the distinguishability between two quantum states \cite{Fuchs1999}. As a consequence, given an open quantum system $S$ interacting with an environment $E$ \cite{Breuer2002}, any variation of the trace distance of two open system's states $D(\rho^1_{\scriptscriptstyle S}(t), \rho^2_{\scriptscriptstyle S}(t))$ can be read in terms of an exchange of information between the open system and the environment \cite{Breuer2009,Laine2010b,Breuer2012}. Here, $\rho^1_{\scriptscriptstyle S}(t)$ and $\rho^2_{\scriptscriptstyle S}(t)$ are open system's states evolved from different initial total states $\rho^1_{\scriptscriptstyle S\!E}(0)$ and $\rho^2_{\scriptscriptstyle S\!E}(0)$ through the relation $\rho^k_{\scriptscriptstyle S}(t)=\mbox{tr}_{\scriptscriptstyle E}\left\{U(t) \rho^k_{\scriptscriptstyle S\!E}(0) U^{\dag}(t)\right\}$, $k=1,2$, where the total system $SE$ is assumed to be closed and hence evolves through a unitary dynamics $U(t)$ \cite{Breuer2002}. In particular, if there are no initial system-environment correlations, $ \rho^1_{\scriptscriptstyle S\!E}(0)=\rho^1_{\scriptscriptstyle S}(0)\otimes\rho^1_{\scriptscriptstyle E}(0)$ and $\rho^2_{\scriptscriptstyle S\!E}(0)=\rho^2_{\scriptscriptstyle S}(0)\otimes\rho^2_{\scriptscriptstyle E}(0)$, an increase of the trace distance above its initial value, \begin{equation}\label{eq:ris}
D(\rho^1_{\scriptscriptstyle S}(t), \rho^2_{\scriptscriptstyle S}(t)) > D(\rho^1_{\scriptscriptstyle S}(0), \rho^2_{\scriptscriptstyle S}(0)), \end{equation} witnesses the difference of the two initial environmental states, i.e. $\rho^1_{\scriptscriptstyle E}(0) \neq \rho^2_{\scriptscriptstyle E}(0)$ \cite{Laine2010b,Smirne2010}. This relation already shows how the trace distance between open system's states allows to access nontrivial information regarding the environment. More specifically, in the following we present a quantitative link between the trace distance behavior and the environmental correlations.
In view of the trace-distance analysis, our physical system can be characterized as follows. The polarization degrees of freedom are the open system $S$ and the angular degrees of freedom the corresponding environment $E$. The latter are in turn manipulated by varying the divergence of the pump, as well as by selecting the frequency-spectrum width of the two-photon state generated by SPDC.
We therefore study the evolution of the trace distance between two polarization states evolved from different initial $SE$ states, which can be considered product states thanks to the compensation of the phase term introduced by the SLM. In particular, we investigate how the trace-distance evolution of the polarization degrees of freedom, which are a small and easily accessible component of the total system, is sensitive to the different angular correlations within $\rho^1_{\scriptscriptstyle E}$ and $\rho^2_{\scriptscriptstyle E}$, thus allowing to assess this characteristic feature of the overall two-photon state. A logical scheme of the experiment is depicted in Fig.\ref{fig:2}.
Let us emphasize that our apparatus exploits all the degrees of freedom of the photons generated by SPDC: polarization degrees of freedom as the open system, angles as the environment and frequencies, together with the spatial properties of the pump, as a tool to vary the correlations within the environment.
\begin{figure}
\caption{(Color online) Logical scheme of the experiment. In the first stage system and environment are uncorrelated, and the environmental states ($E^1$ and $E^2$) differ due to correlations. The wiring represents the information about these different correlations. In the second stage system and environment are coupled through the SLM, so that information on the environmental correlations is transferred to the couple of system states ($S^1$ and $S^2$), making them more distinguishable, and it is finally read out through the detector (D) acting on the system only, in the third and final stage. Note that the two states refer to two distinct runs of the experiment.}
\label{fig:2}
\end{figure}
\subsection{Trace distance evolution for different angular and polarization states}
In our apparatus, the angular state after the compensation of the phase through the SLM can be described, setting $\bm{\theta}\equiv (\theta_s,\theta_i)$, as \begin{equation}
\label{eq:rhoe}
\rho_{\scriptscriptstyle E} = \int d\bm{\theta}d\bm{\theta}' h(\bm{\theta};\bm{\theta}') \ket{\small{\bm{\theta}}}\bra{\small{\bm{\theta}'}}, \end{equation} with \begin{equation}\label{eq:rhoee} h(\bm{\theta}; \bm{\theta}') \equiv {\mbox{Sinc}}(\bm{\theta}){\mbox{Sinc}}(\bm{\theta}') \int_{\Omega_s} d \omega_s \tilde{F}(\Delta k_{\perp})\tilde{F}^*(\Delta k_{\perp}'). \end{equation} The influence of the pump spectrum on the angular state can be neglected \cite{Cialdi2010a,Cialdi2010b} and the integration over $\omega_s$ is performed on the frequency interval $\Omega_s$ selected by the filter or the $4f$ setup on the signal path. The joint probability distribution $P(\bm{\theta})\equiv h(\bm{\theta}; \bm{\theta})$ determines the angular correlations, which can be quantified as \begin{equation}\label{eq:corr}
C=\frac{\langle \theta_s \theta_i\rangle-\langle \theta_s\rangle\langle\theta_i\rangle}{\sqrt{V_{s} V_i}}, \end{equation} with $V_{j}=\langle \theta^2_j\rangle-\langle \theta_j\rangle^2$ variance of the angular distribution $P(\theta_j)$, $j=s,i$. In particular, given a collimated beam with a large pump waist, so that the transverse momentum is nearly conserved, the signal and idler angles are the more correlated the less the selected frequency spectrum is wide. On a similar footing, for a $10 nm$-spectrum and a fixed pump waist, the correlation of the angular degrees of freedom grows with decreasing pump divergence. Thus, we can control the initial correlations of the environment by selecting the frequency spectrum of the two-photon state or the divergence of the pump.
The polarization state after the purification trough the SLM reads \begin{equation}\label{eq:rhos}
\rho_{\scriptscriptstyle S} = \gamma \ket{\psi}\bra{\psi} + (1-\gamma)\rho^m, \end{equation} where \begin{equation}\label{eq:rhoss}
\ket{\psi}=\cos{\alpha}\ket{{\hbox{\small HH}}}+\sin{\alpha}\ket{{\hbox{\small VV}}}, \end{equation} see Eq.(\ref{eq:totall}), is a pure state and $\rho^m = \cos^2{\alpha}\ket{{\hbox{\small HH}}}\bra{{\hbox{\small HH}}}+\sin^2{\alpha}\ket{{\hbox{\small VV}}}\bra{{\hbox{\small VV}}}$ the corresponding mixture. In our setting, we can control $\alpha$ via the polarization of the pump, while $\gamma$ can be modified by changing the crystal along the pump which precompensates the delay time due to the two-crystal geometry \cite{Cialdi2008}. The purity $p=\mbox{Tr}\rho_{\scriptscriptstyle S}^2$ of $\rho_{\scriptscriptstyle S}$ is \begin{equation}\label{eq:pur} p=1-\frac 12 (1-\gamma^2) \sin^2(2\alpha), \end{equation}
whereas its concurrence $\mathcal{C}$ \cite{Wootters1997} reads \begin{equation}\label{eq:conc}
\mathcal{C}=\gamma|\sin(2\alpha)|. \end{equation}
In the following, we will compare the evolution of polarization states evolved from different initial states $\rho^k_{\scriptscriptstyle S}(0)\otimes\rho^k_{\scriptscriptstyle E}(0)$, $k=1,2$. The two initial open system's states $\rho^k_{\scriptscriptstyle S}(0)$ have polarization parameters $\alpha_k$ and $\gamma_k$, see Eqs.(\ref{eq:rhos}) and (\ref{eq:rhoss}), while the two initial environmental states $\rho^k_{\scriptscriptstyle E}(0)$ have angular amplitudes $h_k(\bm{\theta}; \bm{\theta}')$, see Eqs.(\ref{eq:rhoe}) and (\ref{eq:rhoee}), and then joint angular probability distributions $P_k(\bm{\theta})=h_k(\bm{\theta}; \bm{\theta})$. In particular, we impose through the SLM a linear phase which can be described through the unitary operators \begin{equation}\label{eq:U}
U(\beta)\ket{{\small{V \theta_s}}}\ket{{\small{V \theta_i}}} = e^{i \beta (\theta_s-\theta_i)} \ket{{\small{V \theta_s}}}\ket{{\small{V \theta_i}}}, \end{equation} where $\beta$ is the evolution parameter. The polarization states for a generic value of $\beta$ are then \begin{equation}\label{eq:rhoskb}
\rho^k_{\scriptscriptstyle S}(\beta) = \frac{\epsilon_k(\beta)}{\sin(2\alpha_k)} \ket{\psi_k}\bra{\psi_k} + \left(1- \frac{\epsilon_k(\beta)}{\sin(2\alpha_k)}\right)\rho^m_k, \end{equation} where $\ket{\psi_k}=\cos{\alpha_k}\ket{{\hbox{\small HH}}}+\sin{\alpha_k}\ket{{\hbox{\small VV}}}$ and \begin{equation}\label{eq:epsk}
\epsilon_k(\beta) =\gamma_k \sin(2\alpha_k) \int d \theta_s d \theta_i e^{i \beta (\theta_s-\theta_i)} P_k(\theta_s,\theta_i), \end{equation} which is a real function of $\beta$ since the joint probability distribution is symmetric under the exchange $\theta_s \leftrightarrow \theta_i$. It is worth emphasizing that the absolute value of $\epsilon_k(\beta)$ equals the concurrence as well as the interferometric visibility of the state $\rho^k_{\scriptscriptstyle S}(\beta)$. In particular, we measure the visibility by counting the coincidences with polarizers set at $45^{\circ},45^{\circ}$ and at $45^{\circ},-45^{\circ}$, see \cite{Cialdi2009} for further details. Moreover, by virtue of the specific evolution obtained through the SLM,
$\epsilon_k(\beta)$ is fixed by the Fourier transform of the spatial profile $|\tilde{F}(\Delta k_{\perp})|^2$, which at first order is a function of $\theta_s-\theta_i$ and $\omega_s$, see Eq.(\ref{eq:kk}), thus depending on both the pump divergence and the selected frequency spectrum. Thus the engineered evolution, see Eq.(\ref{eq:U}), guarantees that the interferometric visibility is sensitive to the different angular correlations in the environment.
Finally, the trace distance $D(\beta)\equiv D(\rho^1_{\scriptscriptstyle S}(\beta), \rho^2_{\scriptscriptstyle S}(\beta))$ between the polarization states is simply given by, see Eqs. (\ref{eq:tdd}) and (\ref{eq:rhoskb}), \begin{equation}\label{eq:dbeta} D(\beta) =\sqrt{(\cos^2\alpha_1-\cos^2\alpha_2)^2+\left(\epsilon_1(\beta)-\epsilon_2(\beta)\right)^2/4}. \end{equation}
\section{Experimental results} \subsection{Characterization of the probe} As a first step, we show how the choice of the initial polarization states, $\rho^1_{\scriptscriptstyle S}(0)$ and $\rho^2_{\scriptscriptstyle S}(0)$, influences in a critical way whether the subsequent trace-distance evolution is an effective probe of the different angular correlations in the two initial angular states. To this aim, we fix $\rho^1_{\scriptscriptstyle E}(0)$ and $\rho^2_{\scriptscriptstyle E}(0)$ as the states corresponding to $\Delta \lambda_1 = 45 nm$ and $\Delta \lambda_2 = 10 nm$, respectively, that is a weakly and a strongly correlated angular state, while we consider different couples of initial polarization states. To this aim we set $\alpha=\pi/4$ for both $\rho^1_{\scriptscriptstyle S}(0)$ and $\rho^2_{\scriptscriptstyle S}(0)$, and keep $\gamma_1$ fixed, while we vary $\gamma_2$ by inserting different precompensation crystals. Specifically, we exploit a $3 mm$ crystal to fully compensate the delay time $\Delta \tau$ \cite{Cialdi2008}, a $1 mm$ crystal to partially compensate it and we also consider the case without any precompensation crystal. The experimental data, together with the theoretical prediction obtained by Eq.(\ref{eq:dbeta}), are shown in Fig.\ref{fig:3}.(a). For high values of $\gamma_2$, the trace distance between polarization states actually satisfies Eq.(\ref{eq:ris}) and then witnesses the different initial conditions in the angular degrees of freedom. The information due to the differences in $\rho^1_{\scriptscriptstyle E}(0)$ and $\rho^2_{\scriptscriptstyle E}(0)$ flows to the polarization degrees of freedom because of the engineered interaction. Thus, one can access through simple visibility measurements on the open system some information which was initially outside it. On the other hand, the revival of the trace distance above its initial value decreases with the decreasing of $\gamma_2$, and for low enough values of $\gamma_2$ the trace distance remains below its initial value for the whole evolution. The loss of purity and entanglement due to a decrease of the parameter $\gamma$ in the initial polarization states can prevent the subsequent trace distance from being an effective probe of the different correlations in the angular states.
The relative weight of vertically and horizontally polarized photons generated by SPDC is determined by the parameter $\alpha$, which can be controlled by properly rotating a half-wave plate set on the pump beam. In Fig.\ref{fig:3}.(b) we report the experimental data and theoretical predictions of the trace-distance behavior for a given value of $\alpha_1$ as well as fixed $\gamma_1$ and $\gamma_2$, while considering different values of $\alpha_2$.
One can see that, even if the growth of the trace distance above its initial value decreases with the decreasing of $\alpha_2$, it is still visible also for a sensible imbalance between vertically and horizontally polarized photons. Indeed, for a fixed $\gamma<1$ the decrease of $\sin(2\alpha)$ in the polarization states $\rho^k_{\scriptscriptstyle S}(0)$ corresponds to a decrease of the concurrence, but to an increase of the purity, see Eqs.(\ref{eq:pur}) and (\ref{eq:conc}). Contrary to what happens for a decrease of the parameter $\gamma_2$, see Fig.\ref{fig:3}.(a), the open system always recovers the information initially outside it from the very beginning of its evolution and the trace-distance maximum increases with the increasing of the initial distinguishability between the two polarization states.
\begin{figure}
\caption{(Color online) Trace distance $D(\beta)$ versus the evolution parameter $\beta$ for different couples of initial system states and different environmental correlations, see Eq.(\ref{eq:corr}), showing how the trace-distance growth is sensitive to the environmental correlations. (a) and (b): $\rho^1_{\scriptscriptstyle E}(0)$ and $\rho^2_{\scriptscriptstyle E}(0)$ are kept fixed ($\Delta k_1=\Delta k_2=18 mm^{-1}$, $\Delta \lambda_1 = 45 nm$ and $\Delta \lambda_2= 10 nm$), $\rho^1_{\scriptscriptstyle S}(0)$ is fixed with $\alpha_1=\pi/4$ and $\gamma_1=0.91$ (mainly due to contributions to the phase in Eq.(\ref{eq:totall}) which are not compensated to the first order), $\rho^2_{\scriptscriptstyle S}(0)$ corresponds to $\alpha_2=\pi/4$ and $\gamma_2 = 0.96$ (blue line), $0.73$ (red line), $0.52$ (green line) in (a), while $\gamma_2=0.96$ and $\alpha_2 = \pi/4$ (blue line), $0.675$ (red line), $0.575$ (green line) in (b). (c) and (d): $\rho^1_{\scriptscriptstyle S}(0)$ and $\rho^2_{\scriptscriptstyle S}(0)$ are fixed ($\alpha_1=\alpha_2=\pi/4$, $\gamma_1 = 0.91$ and $\gamma_2=0.96$), $\rho^1_{\scriptscriptstyle E}(0)$ is fixed with $\Delta \lambda_1 = 45 nm$ and $\Delta k_1 =18 mm^{-1}$, while $\rho^2_{\scriptscriptstyle E}(0)$ corresponds to $\Delta \lambda_2 = 10 nm$ and $\Delta k_2= 18 mm^{-1}$ (blue line), $24 mm^{-1}$ (red line), $29 mm^{-1}$ (green line) in (c) and to $\Delta k_2 =18 mm^{-1}$ and $\Delta \lambda_2 = 10 nm$ (blue line), $20 nm$ (red line), $30 nm$ (green line) in (d). The insets in (c) and (d) show the angular correlations (red line) and the purity (blue line) of $\rho^2_{\scriptscriptstyle E}(0)$ as a function of $\Delta k_2$ (in (c)) or $\Delta \lambda_2$ (in (d)). Experimental data are reported with their error bars, the solid lines represent the theoretical predictions.}
\label{fig:3}
\end{figure}
\subsection{Trace distance as a witness of initial correlations in the angular degrees of freedom} The analysis of the previous paragraph shows that the optimal probe of the angular correlations is achieved by exploiting the highest amount of purity and entanglement of the polarization degrees of freedom available within our setting. Now, we study how this optimal probe reveals changes in the angular correlations. Hence, we fix $\rho^1_{\scriptscriptstyle S}(0)$, $\rho^2_{\scriptscriptstyle S}(0)$ and we investigate the trace-distance evolution $D(\rho^1_{\scriptscriptstyle S}(\beta),\rho^2_{\scriptscriptstyle S}(\beta))$ for different couples of initial angular states. In particular, we take as reference environmental state $\rho^1_{\scriptscriptstyle E}(0)$ the state with weak angular correlations, which is obtained by means of a collimated beam and a $45nm$-spectrum. We compare the evolution of the subsequent polarization state $\rho^1_{{\scriptscriptstyle S}}(\beta)$ with the evolution of a state $\rho^2_{\scriptscriptstyle S}(\beta)$ evolved in the presence of strong initial angular correlations in $\rho^2_{\scriptscriptstyle E}(0)$. We repeat this procedure by changing the amount of correlations $C$ in $\rho^2_{\scriptscriptstyle E}(0)$, see Eq.(\ref{eq:corr}), thus studying the connection between $C$ and the effectiveness of the quantum probe of the angular correlations quantified by the increase of the trace distance above its initial value.
\begin{figure}
\caption{(Color online) Experimental values of the maximum increase of the trace distance above its initial value as a function of the angular correlations $C$, see Eq.(\ref{eq:corr}). The experimental data are referred to the behavior of the trace distance for different beam divergences (blue line, compare with Fig.\ref{fig:3}.(c)) or different widths of the frequency spectrum (red line, compare with Fig.\ref{fig:3}.(d)).}
\label{fig:4}
\end{figure}
In Fig.\ref{fig:3}.(c), one can see the experimental data and theoretical prediction concerning the different trace-distance evolutions $D(\rho^1_{\scriptscriptstyle S}(\beta),\rho^2_{\scriptscriptstyle S}(\beta))$ which correspond to the different beam divergences exploited, together with a $10nm$-spectrum,
in the preparation of the environmental state $\rho^2_{{\scriptscriptstyle E}}(0)$. The divergence is enlarged by suitably setting a telescopic system of lenses, so that the FWHM $\Delta k$ of $|\tilde{F}(\Delta k_{\perp})|^2$ is increased, while the $220 \mu m$ spot on the generating crystals is kept fixed \cite{Cialdi2012}. The increase of the trace distance above its initial value grows with the angular correlations $C$ in $\rho^2_{{\scriptscriptstyle E}}(0)$. The behavior of $D(\rho^1_{\scriptscriptstyle S}(\beta),\rho^2_{\scriptscriptstyle S}(\beta))$ indicates that, for the specific choice of $\rho^1_{\scriptscriptstyle S}(0)$ and $\rho^2_{\scriptscriptstyle S}(0)$, the trace distance can actually witness even a small difference in the angular correlations of $\rho^1_{\scriptscriptstyle E}(0)$ and $\rho^2_{\scriptscriptstyle E}(0)$. The direct connection between the trace distance and the correlations in the environment is further shown in Fig.\ref{fig:4}, where the difference between the maximum and the initial value of the trace distance is plotted as a function of the angular correlations. The experimental data point out that the probe represented by the trace distance is sensitive to the different amount of correlations within the environment, which is indeed not a priori entailed by Eq.(\ref{eq:ris}).
As a further check of the connection between angular correlations and the increase of the trace distance above its initial value, we take into account environmental states $\rho^2_{\scriptscriptstyle E}(0)$ in which the angular correlations are modified by selecting different frequency spectra of the two-photon state. Besides affecting the angular correlations, this also influences the purity of $\rho^2_{\scriptscriptstyle E}(0)$. As it may be inferred from Eqs.(\ref{eq:rhoe}) and (\ref{eq:rhoee}), a wider frequency spectrum implies a lower angular purity, as a consequence of the fact that the pure state generated by SPDC in Eq.(\ref{eq:totall}) also involves the frequencies. However, the trace-distance evolution does not keep track of the purity of the environmental state. In particular, the growth of the trace distance above its initial value is not affected by the different purities of $\rho^2_{\scriptscriptstyle E}(0)$ in the two situations, but is determined by the amount of angular correlations $C$, see the insets in Fig.\ref{fig:3}.(c), (d) and Fig.\ref{fig:4}. Indeed, this can be explained through Eqs.(\ref{eq:epsk}) and (\ref{eq:dbeta}): the trace distance solely depends on the angular probability distribution $P(\theta_s,\theta_i)$, while it is independent of angular coherences.
\section{Conclusion} We have theoretically described and experimentally demonstrated a strategy to assess relevant information about a composite system by only observing a small and easily accessible part of it. By exploiting couples of entangled photons generated by SPDC and engineering a proper interaction by means of a SLM, we could reveal correlations within the angular degrees of freedom of the photons by monitoring the trace distance evolution between couples of polarization states. After estimating the optimal probe, we have shown that the increase of the trace distance between system states above its initial value provides a signature of the amount of angular correlations in the environmental states.
\acknowledgments This work has been supported by the COST Action MP1006 and the MIUR Project FIRB LiCHIS-RBFR10YQ3H.
\end{document} | arXiv |
\begin{document}
\title{Parabolic automorphisms of projective surfaces \ (after M. H. Gizatullin)}
\begin{abstract} In 1980, Gizatullin classified rational surfaces endowed with an automorphism whose action on the Neron-Severi group is parabolic: these surfaces are endowed with an elliptic fibration invariant by the automorphism. The aim of this expository paper is to present for non-experts the details of Gizatullin's original proof, and to provide an introduction to a recent paper by Cantat and Dolgachev. \end{abstract}
\section{Introduction}
Let $X$ be a projective complex surface. The Neron-Severi group $\mathrm{NS}\,(X)$ is a free abelian group endowed with an intersection form whose extension to $\mathrm{NS}_{\R}(X)$ has signature $(1, \mathrm{h}^{1,1}(X)-1)$. Any automorphism of $f$ acts by pullback on $\mathrm{NS}\,(X)$, and this action is isometric. The corresponding isometry $f^*$ can be of three different types: elliptic, parabolic or hyperbolic. These situations can be read on the growth of the iterates of $f^*$. If $|| \, . \, ||$ is any norm on $\mathrm{NS}_{\R}(X)$, they correspond respectively to the following situations: $||(f^*)^n||$ is bounded, $||(f^*)^n|| \sim C n^2$ and $||(f^*)^n|| \sim \lambda^n$ for $\lambda >1$. This paper is concerned with the study of parabolic automorphisms of projective complex surfaces. The initial motivation to their study was that parabolic automorphisms don't come from $\mathrm{PGL}(N, \C)$ via some projective embedding $X \hookrightarrow \P^N$. Indeed, if $f$ is an automorphism coming from $\mathrm{PGL}(N, \C)$, then $f^*$ must preserve an ample class in $\mathrm{NS}\,(X)$, so $f^*$ is elliptic. The first known example of such a pair $(X, f)$, due to initially to Coble \cite{Coble} and popularised by Shafarevich, goes as follows: consider a generic pencil of cubic curves in $\mathbb{P}^2$, it has $9$ base points. Besides, all the curves in the pencil are smooth elliptic curves except $12$ nodal curves. After blowing up the nine base points, we get a elliptic surface $X$ with $12$ singular fibers and $9$ sections $s_1, \ldots, s_9$ corresponding to the exceptional divisors, called a Halphen surface (of index $1$). The section $s_1$ specifies an origin on each smooth fiber of $X$. For $2 \leq i \leq 8 $, we have a natural automorphism $\sigma_i$ of the generic fiber of $X$ given by the formula $\sigma_i(x)=x+s_i-s_1$. It is possible to prove that the $\sigma_i$'s extend to automorphisms of $X$ and generate a free abelian group of rank $8$ in $\mathrm{Aut}\,(X)$. In particular, any nonzero element in this group is parabolic since the group of automorphisms of an elliptic curve fixing the class of an ample divisor is finite. \par
In many aspects, thisexample is a faithful illustration of parabolic automorphisms on projective surfaces. A complete classification of pairs $(X, f)$ where $f$ is a parabolic automorphism of $X$ is given in \cite{GIZ}.
In his paper, Gizatullin considers not only parabolic automorphisms, but more generally groups of automorphisms containing only parabolic or elliptic\footnote{Gizatullin considers only parabolic elements, but most of his arguments apply to the case of groups containing elliptic elements as well as soon an they contain at least \textit{one} parabolic element.} elements. We call such groups of moderate growth, since the image of any element of the group in $\mathrm{GL}(\mathrm{NS}(X))$ has polynomial growth. Gizatullin's main result runs as follows:
\begin{theorem}[\cite{GIZ}] \label{Main} Let $X$ be a smooth projective complex surface and $G$ be an infinite subgroup of $\mathrm{Aut}\, (X)$ of moderate growth. Then there exists a unique elliptic $G$-invariant fibration on $X$. \end{theorem} Of course, if $X$ admits one parabolic automorphism $f$, we can apply this theorem with the group $G=\ensuremath{\mathbb Z}$, and we get a unique $f$-invariant elliptic fibration on $X$. It turns out that it is possible to reduce Theorem \ref{Main} to the case $G=\ensuremath{\mathbb Z}$ by abstract arguments of linear algebra.
\par
In all cases except rational surfaces, parabolic automorphisms come from minimal models, and are therefore quite easy to understand. The main difficulty occurs in the case of rational surfaces. As a corollary of the classification of relatively minimal elliptic surfaces, the relative minimal model of a rational elliptic surface is a Halphen surface of some index $m$. Such surfaces are obtained by blowing up the base points of a pencil of curves of degree $3m$ in $\mathbb{P}^2$. By definition, $X$ is a Halphen surface of index $m$ if the divisor $-mK_X$ has no fixed part and $|-mK_X|$ is a pencil without base point giving the elliptic fibration. \begin{theorem}[\cite{GIZ}] \label{Second} Let $X$ be a Halphen surface of index $m$, $S_1, \ldots, S_{\lambda}$ the reducible fibers and $\mu_i$ the number of reducible components of $S_i$, and $s=\sum_{i=1}^{\lambda} \{\mu_i-1\}$. Then $s \leq 8$, and there exists a free abelian group $G_X$ of rank $s-8$ in $\mathrm{Aut}\,(X)$ such that every nonzero element of this group is parabolic and acts by translation along the fibers. If $\lambda \geq 3$, $G$ has finite index in $\mathrm{Aut}\,(X)$. \end{theorem} The number $\lambda$ of reducible fibers is at least two, and the case $\lambda=2$ is very special since all smooth fibers of $X$ are isomorphic to a fixed elliptic curve. Such elliptic surfaces $X$ are now called Gizatullin surfaces, their automorphism group is an extension of $\C^{\times}$ by a finite group, $s=8$, and the image of the representation $\rho \colon \mathrm{Aut}\,(X) \rightarrow \mathrm{GL}\, (\mathrm{NS}\,(X))$ is finite.
\par
Let us now present applications of Gizatullin's construction. The first application lies in the theory of classification of birational maps of surfaces, which is an important subject both in complex dynamics and in algebraic geometry. One foundational result in the subject is Diller-Favre's classification theorem \cite{DF}, which we recall now. If $X$ is a projective complex surface and $f$ is a birational map of $X$, then $f$ acts on the Neron-Severi group $\mathrm{NS}\,(X)$. The conjugacy types of birational maps can be classified in four different types, which can be detected by looking at the growth of the endomorphisms $(f^*)^n$. The first type corresponds to birational maps $f$ such that $|| (f^*)^n || \sim \alpha n$. These maps are never conjugate to automorphisms of birational models on $X$ and they preserve a rational fibration. The three other remaining cases are $|| (f^*)^n ||$ bounded, $|| (f^*)^n || \sim Cn^2$ and $|| (f^*)^n || \sim C \lambda^n$. In the first two cases, Diller and Favre prove that $f$ is conjugate to an automorphism of a birational model of $X$. The reader can keep in mind the similarity between the last three cases and Nielsen-Thurston's classification of elements in the mapping class group into three types: periodic, reducible and pseudo-Anosov. The first class is now well understood (see \cite{BD2}), and constructing automorphisms in the last class is a difficult problem (see \cite{BKv}, \cite{MM} for a systematic construction of examples in this category, as well as \cite{BK}, \cite{BD} and \cite{DG} for more recent results). The second class fits exactly to Gizatullin's result: using it, we get that $f$ preserves an elliptic fibration. \par
One other feature of Gizatullin's theorem is to give a method to construct hyperbolic automorphisms on surfaces. This seems to be paradoxal since Gizatullin's result only deals with parabolic automorphisms. However, the key idea is the following: if $f$ and $g$ are two parabolic (or even elliptic) automorphisms of a surface generating a group $G$ of moderate growth, then $f^*$ and $g^*$ share a common nef class in $\mathrm{NS}\,(X)$, which is the class of any fiber of the $G$-invariant elliptic fibration. Therefore, if $f$ and $g$ don't share a fixed nef class in $\mathrm{NS}\, (X)$, some element in the group $G$ must be hyperbolic. \par
Let us describe the organization of the paper. \S \ref{3} is devoted to the theory of abstract isometries of quadratic forms of signature $(1, n-1)$ on $\R^n$. In \S \ref{3.1}, we recall their standard classification into three types (elliptic, parabolic and hyperbolic). The next section (\S \ref{3.2}) is devoted to the study of special parabolic isometries, called parabolic translations. They depend on an isotropic vector $\theta$, the direction of the translation, and form an abelian group $\mathcal{T}_{\theta}$. We prove in Proposition \ref{ray} and Corollary \ref{wazomba} one of Gizatullin's main technical lemmas: if $u$ and $v$ are two parabolic translations in different directions, then $uv$ or $u^{-1}v$ must be hyperbolic. Building on this result, we prove in \S \ref{3.3} a general structure theorem (Theorem \ref{ptfixe}) for groups of isometries fixing a lattice and containing no hyperbolic elements. In \S \ref{4}, we recall classical material in birational geometry of surfaces which can be found at different places of \cite{DF}. In particular, we translate the problem of the existence of an $f$-invariant elliptic fibration in terms of the invariant nef class $\theta$ (Proposition \ref{nefnef}), and we also prove using the fixed point theorem of \S \ref{3.3} that it is enough to deal with the case $G=\Z f$ in Theorem \ref{Main}. Then we settle this theorem for all surfaces except rational ones. In \S \ref{5} and \S \ref{6}, we prove Gizatullin's theorem. \par
Roughly speaking, the strategy goes as follows: the invariant nef class $\theta$ is always effective, we represent it by a divisor $C$. This divisor behaves exactly as a fiber of a minimal elliptic surface, we prove this in Lemmas \ref{base} and \ref{genre}. The conormal bundle $N^*_{C/X}$ has degree zero on each component of $C$, but is not always a torsion point in $\mathrm{Pic}\, (C)$. If it is a torsion point, it is easy to produce the elliptic fibration by a Riemann-Roch type argument. If not, we consider the trace morphism $\mathfrak{tr} \colon \mathrm{Pic}\, (X) \rightarrow \mathrm{Pic}\, (C)$ and prove in Proposition \ref{torsion} that $f$ acts finitely on $\mathrm{ker}\, (\mathfrak{tr})$. In Proposition \ref{elliptic}, we prove that $f$ also acts finitely on a large part of $\mathrm{im}\, (\mathfrak{tr})$. By a succession of clever tricks, it is possible from there to prove that $f$ acts finitely on $\mathrm{Pic}\, (X)$; this is done in Proposition \ref{chic}. \par
In \S 6 we recall the classification theory of relatively minimal rational elliptic surfaces; we prove in Proposition \ref{primitive} that they are Halphen surfaces. In Proposition \ref{sept} and Corollary \ref{sympa}, we prove a part of Theorem \ref{Second}: the existence of parabolic automorphisms imposes a constraint on the number of reducible components of the fibration, namely $s \leq 7$. We give different characterisations of Gizatullin surfaces (that is minimal elliptic rational surfaces with two singular fibers) in Proposition \ref{waza}. Then we prove the converse implication of Theorem \ref{Second}: the numerical constraint $s \leq 7$ is sufficient to guarantee the existence of parabolic automorphisms. Lastly, we characterize minimal elliptic surfaces carrying no parabolic automorphisms in Proposition \ref{hapff}: the generic fiber must have a finite group of automorphisms over the function field $\mathbb{C}(t)$. At the end of the paper, we carry out the explicit calculation of the representation of $\mathrm{Aut}\, (X)$ on $\mathrm{NS}\, (X)$ for unnodal Halphen surfaces (that is Halphen surfaces with irreducible fibers) in Theorem \ref{classieux}. These surfaces are of crucial interest since their automorphism group is of maximal size in some sense, see \cite{CD} for a precise statement. \par
Throughout the paper, we work over the field of complex numbers. However, Gizatullin's arguments can be extended to any field of any characteristic with minor changes. We refer to the paper \cite{CD} for more details. \par
\textbf{Acknowledgements} I would like to thank Charles Favre for pointing to me Gizatullin's paper and encouraging me to write this survey, as well as Jeremy Blanc, Julie D\'eserti and Igor Dolgachev for very useful comments.
\tableofcontents
\section{Notations and conventions} \label{2} Throughout the paper, $X$ denotes a smooth complex projective surface, which will always assumed to be rational except in \S \ref{4}. \par
By divisor, we will always mean $\Z$-divisor. A divisor $D=\sum_i a_i\, D_i$ on $X$ is called primitive if $\mathrm{gcd}(a_i)=1$. \par
If $D$ and $D'$ are two divisors on $X$, we write $D \sim D'$ (resp. $D \equiv D'$) if $D$ and $D'$ are linearly (resp. numerically) equivalent.
\par
For any divisor $D$, we denote by $|D|$ the complete linear system of $D$, that is the set of effective divisors linearly equivalent to $D$; it is isomorphic to $\mathbb{P}\, \bigl( \mathrm{H}^0(X, \oo_X(D) \bigr)$. \par
The group of divisors modulo numerical equivalence is the Neron-Severi group of $X$, we denote it by $\mathrm{NS} (X)$. By Lefschetz's theorem on $(1, 1)$-classes, $\mathrm{NS}\,(X)$ is the set of Hodge classes of weight $2$ modulo torsion, this is a $\Z$-module of finite rank. We also put $\mathrm{NS}\,(X)_{\R}=\mathrm{NS}\,(X) \otimes_{\Z} \R$. \par
If $f$ is a biregular automorphism of $X$, we denote by $f^*$ the induced action on $\mathrm{NS}\,(X)$. We will always assume that $f$ is \textit{parabolic}, which means that the induced action $f^*$ of $f$ on $\mathrm{NS}_{\R}(X)$ is parabolic. \par
The first Chern class map is a surjective group morphism $\mathrm{Pic}\,(X) \xrightarrow{\mathrm{c}_1} \mathrm{NS}\,(X)$, where $\mathrm{Pic}\,(X)$ is the Picard group of $X$. This morphism is an isomorphism if $X$ is a rational surface, and $\mathrm{NS}\,(X)$ is isomorphic to $\Z^r$ with $r=\chi(X)-2$. \par
If $r$ is the rank of $\mathrm{NS}\,(X)$, the intersection pairing induces a non-degenerate bilinear form of signature $(1, r-1)$ on $X$ by the Hodge index theorem. Thus, all vector spaces included in the isotropic cone of the intersection form are lines. \par
If $D$ is a divisor on $X$, $D$ is called a nef divisor if for any algebraic curve $C$ on $X$, $D.C \geq 0$. The same definition holds for classes in $\mathrm{NS}\,(X)_{\R}$. By Nakai-Moishezon's criterion, a nef divisor has nonnegative self-intersection.
\section{Isometries of a Lorentzian form} \label{3}
\subsection{Classification} \label{3.1} Let $V$ be a real vector space of dimension $n$ endowed with a symmetric bilinear form of signature $(1, n-1)$. The set of nonzero elements $x$ such that $x^2 \geq 0$ has two connected components. We fix one of this connected component and denote it by $\mathfrak{N}$. \par
In general, an isometry maps $\mathfrak{N}$ either to $\mathfrak{N}$, either to $- \mathfrak{N}$. The index-two subgroup $\mathrm{O}_+ (V)$ of $\mathrm{O}(V)$ is the subgroup of isometries leaving $\mathfrak{N}$ invariant. \par
There is a complete classification of elements in $\mathrm{O}_+ (V)$. For nice pictures corresponding to these three situations, we refer the reader to Cantat's article in \cite{Milnor}. \begin{proposition} \label{classification} Let $u$ be in $\mathrm{O}_+ (V)$. Then three distinct situations can appear: \par
\begin{enumerate} \item \textbf{u is hyperbolic} \par \noindent There exists $\lambda>1$ and two distinct vectors $\theta_{+}$ and $\theta_-$ in $\mathfrak{N}$ such that $u(\theta_+)=\lambda \, \theta_+$ and $u(\theta_-)=\lambda^{-1} \theta_-$. All other eigenvalues of $u$ are of modulus $1$, and $u$ is semi-simple. \item \textbf{u is elliptic} \par \noindent All eigenvalues of $u$ are of modulus $1$ and $u$ is semi-simple. Then $u$ has a fixed vector in the interior of $\mathfrak{N}$. \item \textbf{u is parabolic} \par \noindent All eigenvalues of $u$ are of modulus $1$ and $u$ fixes pointwise a unique ray in $\mathfrak{N}$, which lies in the isotropic cone. Then $u$ is not semi-simple and has a unique non-trivial Jordan block which is of the form $\begin{pmatrix} 1&1&0\\ 0&1&1\\ 0&0&1 \end{pmatrix}$ where the first vector of the block directs the unique invariant isotropic ray in $\mathfrak{N}$. \end{enumerate} \end{proposition}
\begin{proof} The existence of an eigenvector in $\mathfrak{N}$ follows from Brouwer's fixed point theorem applied to the set of positive half-lines in $\mathfrak{N}$, which is homeomorphic to a closed euclidian ball in $\mathbb{R}^{n-1}$. Let $\theta$ be such a vector and $\lambda$ be the corresponding eigenvalue. \par
$*$ If $\theta$ lies in the interior of $\mathfrak{N}$, then $V=\R\, \theta \oplus {\theta}^{\perp}$. Since the bilinear form is negative definite on ${\theta}^{\perp}$, $u$ is elliptic. \par
$*$ If $\theta$ is isotropic and $\lambda \neq 1$, then $\mathrm{im}\, (u-\lambda^{-1} \mathrm{id}) \subset \theta^{\perp}$ so that $\lambda^{-1}$ is also an eigenvalue of $u$. Hence we get two isotropic eigenvectors $\theta_+$ and $\theta_-$ corresponding to the eigenvalues $\lambda$ and $\lambda^{-1}$. Then $u$ induces an isometry of $\theta_+^{\perp} \cap \theta_-^{\perp}$, and $u$ is hyperbolic. \par
$*$ If $\theta$ is isotropic and $\lambda=1$, and if no eigenvector of $u$ lies in the interior of $\mathfrak{N}$, we put $v=u-\textrm{id}$. If $\theta'$ is a vector in $\mathrm{ker} \, (v)$ outside $\theta^{\perp}$, then $\theta' + t \theta$ lies in the interior of $\mathfrak{N}$ for large values of $t$ and is fixed by $u$, which is impossible. Therefore $\mathrm{ker}\, (v) \subset \theta^{\perp}$. In particular, we see that $\mathbb{R \theta}$ is the unique $u$-invariant isotropic ray. \par
Since $\theta$ is isotropic, the bilinear form is well-defined and negative definite on $\theta^{\perp}/{\ensuremath{\mathbb R}\theta}$, so that $u$ induces a semi-simple endomorphism $\overline{u}$ on $\theta^{\perp}/{\ensuremath{\mathbb R}\theta}$. Let $P$ be the minimal polynomial of $\overline{u}$, $P$ has simple complex roots. Then there exists a linear form $\ell$ on $\theta^{\perp}$ such that for any $x$ orthogonal to $\theta$, $P(u)(x)=\ell(x) \, \theta$. Let $E$ be the kernel of $\ell$. Remark that \[ \ell(x)\, \theta= u\{\ell(x)\, \theta \}=u \,\{P(u)(x)\}=P(u)(u(x))=\ell(u(x))\, \theta \]
so that $\ell \circ u=\ell$, which implies that $E$ is stable by $u$. Since $P(u_{|E})=0$, $u_{|E}$ is semi-simple. \par
Assume that $\theta$ doesn't belong to $E$. Then the quadratic form is negative definite on $E$, and $V=E \oplus E^{\perp}$. On $E^{\perp}$, the quadratic form has signature $(1,1)$. Then the situation becomes easy, because the isotropic cone consists of two lines, which are either preserved or swapped. If they are preserved, we get the identity map. If they are swapped, we get a reflexion along a line in the interior of the isotropic cone, hence an elliptic element. In all cases we get a contradiction.
\par
Assume that $u_{| \theta^{\perp}}$ is semi-simple. Since $\mathrm{ker}\, (v) \subset \theta^{\perp}$, we can write $\theta^{\perp}=\mathrm{ker}\, (v)\, \oplus W$ where $W$ is stable by $v$ and $v_{|W}$ is an isomorphism. Now $\mathrm{im}\,(v) =\mathrm{ker}\,(v)^{\perp}$, and it follows that $\mathrm{im}\, (v)=\mathbb{R} \theta \oplus W$. Let $\zeta$ be such that $v(\zeta)=\theta$. Then $u(\zeta)=\zeta+\theta$, so that $u(\zeta)^2=\zeta^2+2 (\zeta. \theta)$. It follows that $\zeta. \theta=0$, and we get a contradiction. In particular $\ell$ is nonzero. \par
Let $F$ be the orthogonal of the subspace $E$, it is a plane in $V$ stable by $u$, containing $\theta$ and contained in $\theta^{\perp}$. Let $\theta'$ be a vector in $F$ such that $\{\theta, \theta'\}$ is a basis of $F$ and write $u(\theta')=\alpha \theta + \beta \theta'$. Since $\theta$ and $\theta'$ are linearly independent, $\theta'^2 <0$. Besides, $u(\theta')^2=\theta'^2$ so that $\beta^2=1$. Assume that $\beta=-1$. If $x=\theta'-\frac{\alpha}{2} \theta$, then $u(x)=-x$, so that $u_{\theta^{\perp}}$ is semi-simple. Thus $\beta=1$. Since $\alpha \neq 0$ we can also assume that $\alpha=1$. \par
Let $v=u-\textrm{id}$. We claim that $\mathrm{ker}\,(v) \subset E$. Indeed, if $u(x)=x$, we know that $x\in \theta^{\perp}$. If $x \notin E$, then $P(u)(x) \neq 0$. But $P(u)(x)=P(1) \, x$ and since $\theta \in E$, $P(1)=0$ and we get a contradiction. This proves the claim. \par
Since $\mathrm{im}\,(v) \subseteq \mathrm{ker}\,(v)^{\perp}$, $\mathrm{im}\,(v)$ contains $F$. Let $\theta''$ be such that $v(\theta'')=\theta'$. Since $v(\theta^{\perp}) \subset E$, $\theta'' \notin \theta^{\perp}$. The subspace generated with $\theta$, $\theta'$ and $\theta''$ is a $3 \times 3$ Jordan block for $u$. \end{proof}
\begin{remark} \label{bof} Elements of the group $\mathrm{O}_{+}(V)$ can be distinguished by the growth of the norm of their iterates. More precisely: \begin{enumerate}
\item[--] If $u$ is hyperbolic, $||u^n|| \sim C\lambda^{n}$. \par
\item[--] If $u$ is elliptic, $||u^n||$ is bounded. \par
\item[--] If $u$ is parabolic, $||u^n|| \sim C {n}^2$. \end{enumerate} \end{remark} We can sum up the two main properties of parabolic isometries which will be used in the sequel:
\begin{lemma} \label{lemmenef} Let $u$ be a parabolic element of $\mathrm{O}_{+}(V)$ and $\theta$ be an isotropic fixed vector of $u$. \begin{enumerate} \item If $\alpha$ is an eigenvector of $u$, $\alpha^2 \leq 0$. \par
\item If $\alpha$ is fixed by $u$, then $\alpha \, . \,\theta=0$. Besides, if $\alpha^2=0$, $\alpha$ and $\theta$ are proportional. \end{enumerate} \end{lemma}
\subsection{Parabolic isometries} \label{3.2} The elements which are the most difficult to understand in $\mathrm{O}_{+}(V)$ are parabolic ones. In this section, we consider a distinguished subset of parabolic elements associated with any isotropic vector.
\par
Let $\theta$ be an isotropic vector in $\mathfrak{N}$ and $Q_{\theta}=\theta^{\perp}/ \ensuremath{\mathbb R}\theta$. The quadratic form is negative definite on $Q_{\theta}$. Indeed, if $x\, . \, \theta=0$, $x^2 \leq 0$ with equality if and only if $x$ and $\theta$ are proportional, so that $x=0$ in $Q_{\theta}$. If \[ \mathrm{O}_{+}(V)_{\theta}=\{ u \in \mathrm{O}_+(V) \, \, \textrm{such that} \, \, u(\theta)=\theta\} \] we have a natural group morphism \[ \chi_{\theta} \colon \mathrm{O}_{+}(V)_{\theta} \rightarrow \mathrm{O}(Q_{\theta}), \] and we denote by $\mathcal{T}_{\theta}$ its kernel. Let us fix another isotropic vector $\eta$ in $\mathfrak{N}$ which is not collinear to $\theta$, and let $\pi \colon V \rightarrow \theta^{\perp} \cap \eta^{\perp}$ be the orthogonal projection along the plane generated by $\theta$ and $\eta$. \begin{proposition} \label{commutatif}$ $ \par \begin{enumerate} \item The map $\varphi \colon \mathcal{T}_{\theta} \rightarrow \theta^{\perp} \cap \eta^{\perp}$ given by $\varphi(u)=\pi \{ u(\eta) \}$ is a group isomorphism. \item Any element in $\mathcal{T}_{\theta} \setminus \{ \textrm{id} \}$ is parabolic. \end{enumerate} \end{proposition}
\begin{proof} We have $V=\{\theta^{\perp} \cap \eta^{\perp} \oplus \ensuremath{\mathbb R}\theta\} \oplus \ensuremath{\mathbb R}\eta=\theta^{\perp} \oplus \ensuremath{\mathbb R}\eta$. Let $u$ be in $G_{\theta}$, and denote by $\zeta$ the element $\varphi(u)$. Let us decompose $u(\eta)$ as $a \theta + b \eta + \zeta$. Then $0=u(\eta)^2=2ab\, (\theta\, . \, \eta)+ \zeta^2$ and we get \[ ab=-\dfrac{\zeta^2}{2\, (\theta. \eta)} \cdot \] Since $u(\theta)=\theta, \theta\, .\, \eta=\theta \, u(\eta)=b\, (\theta \, . \, \eta)$ so that $b=1$. This gives \[a=-\dfrac{\zeta^2}{2\, (\theta. \eta)} \cdot \] By hypothesis, there exists a linear form $\lambda \colon \theta^{\perp}\cap \eta^{\perp} \rightarrow \R$ such that for any $x$ in $\theta^{\perp} \cap \eta^{\perp}$, $u(x)=x+\lambda(x) \, \theta$. Then we have \[ 0=x\, . \, \eta = u(x)\,.\, u(\eta)=x \, . \, \zeta+ \lambda(x) \, \theta\, . \, \eta \] so that \[ \lambda(x)=-\dfrac{(x\, . \, \zeta)}{(\theta\, . \, \eta)} \cdot \] This proves that $u$ can be reconstructed from $\zeta$. For any $\zeta$ in $\theta^{\perp} \cap \eta^{\perp}$, we can define a map $u_{\zeta}$ fixing $\theta$ by the above formul\ae, and it is an isometry. This proves that $\varphi$ is a bijection. To prove that $\varphi$ is a morphism, let $u$ and $u'$ be in $G_{\theta}$, and put $u''=u' \circ u$. Then \[ \qquad \zeta''=\pi \{u' (u(\eta))\}=\pi \{u' (\zeta +a \theta + \eta) \}=\pi \{ \zeta + \lambda(\zeta) \theta + a \theta + \zeta' + a' \theta + \eta\}=\zeta + \zeta'. \] It remains to prove that $u$ is parabolic if $\zeta \neq 0$. This is easy: if $x=\alpha \theta + \beta \eta + y$ where $y$ is in $\theta^{\perp} \cap \eta^{\perp}$, then $u(x)=\{\alpha+ \lambda(y) \} \theta + \{\beta \zeta + y\}$. Thus, if $u(x)=x$, we have $\lambda(y)=0$ and $\beta=0$. But in this case, $x^2=y^2 \leq 0$ with equality if and only if $y=0$. It follows that $\R_{+}\theta$ is the only fixed ray in $\mathfrak{N}$, so that $u$ is parabolic. \end{proof} \begin{definition} Nonzero elements in $\mathcal{T}_{\theta}$ are called parabolic translations along $\theta$. \end{definition} This definition is justified by the fact that elements in the group $\mathcal{T}_{\theta}$ act by translation in the direction $\theta$ on $\theta^{\perp}$. \begin{proposition} \label{ray} Let $\theta$, $\eta$ be two isotropic and non-collinear vectors in $\mathfrak{N}$, and $\varphi \colon \mathcal{T}_{\theta} \rightarrow \theta^{\perp} \cap \eta^{\perp}$ and $\psi \colon \mathcal{T}_{\eta} \rightarrow \theta^{\perp} \cap \eta^{\perp}$ the corresponding isomorphisms. Let $u$ and $v$ be respective nonzero elements of $\mathcal{T}_{\theta}$ and $\mathcal{T}_{\eta}$, and assume that there exists an element $x$ in $\mathfrak{N}$ such that $u(x)=v(x)$. Then there exists $t > 0$ such that $\psi(v)=t\, \varphi(u)$. \end{proposition} \begin{proof} Let us write $x$ as $\alpha \theta + \beta \eta + y$ where $y$ is in $\theta^{\perp} \cap \eta^{\perp}$. Then \[ \qquad u(x)=\alpha \, \theta + \beta \zeta + y + \lambda(y) \, \theta \quad \textrm{and} \quad v(x)= \alpha \, \zeta' + \beta \eta + y + \mu(y) \, \eta. \] \noindent Therefore, if $u(x)=v(x)$, \[ \qquad \{\alpha + \lambda(y)\} \, \theta - \{\beta + \mu(y) \}\, \eta + \{\beta \zeta - \alpha \zeta' \} =0 \] Hence $\beta \zeta - \alpha \zeta' =0$. We claim that $x$ doesn't belong to the two rays $\ensuremath{\mathbb R}\theta$ and $\ensuremath{\mathbb R}\eta$. Indeed, if $y=0$, $\alpha=\beta=0$ so that $u(x)=0$. Thus, since $x$ lies in $\mathfrak{N}$, $x \, . \, \theta >0$ and $x \, . \, \eta >0$ so that $\alpha >0$ and $\beta >0$. Hence $\zeta'=\dfrac{\beta}{\alpha}\, \zeta$ and $\dfrac{\beta}{\alpha}>0$. \end{proof} \begin{corollary} \label{wazomba} Let $\theta$, $\eta$ two isotropic and non-collinear vectors in $\mathfrak{N}$ and $u$ and $v$ be respective nonzero elements of $\mathcal{T}_{\theta}$ and $\mathcal{T}_{\eta}$. Then $u^{-1}v$ or $uv$ is hyperbolic. \end{corollary} \begin{proof} If $u^{-1}v$ is not hyperbolic, then there exists a nonzero vector $x$ in $\mathfrak{N}$ fixed by $u^{-1} v$. Thus, thanks to Proposition \ref{ray}, there exists $t>0$ such that $\psi(v)=t\, \varphi(u)$. By the same argument, if $uv$ is not hyperbolic, there exists $s>0$ such that $\psi(v)=s\, \varphi(u^{-1})=-s\, \varphi(u)$. This gives a contradiction. \end{proof} \subsection{A fixed point theorem} \label{3.3} In this section, we fix a lattice $\Lambda$ of rank $n$ in $V$ and assume that the bilinear form on $V$ takes integral values on the lattice $\Lambda$. We denote by $\mathrm{O}_{+}(\Lambda)$ the subgroup of $\mathrm{O}_{+}(V)$ fixing the lattice. We start by a simple characterisation of elliptic isometries fixing $\Lambda$: \begin{lemma} \label{fini} $ $ \par \begin{enumerate} \item An element of $\mathrm{O}_{+}(\Lambda)$ is elliptic if and only if it is of finite order. \par
\item An element $u$ of $\mathrm{O}_{+}(\Lambda)$ is parabolic if and only if it is quasi-unipotent (which means that there exists an integer $k$ such that $u^k-1$ is a nonzero nilpotent element) and of infinite order. \end{enumerate} \end{lemma} \begin{proof} $ $ \par \begin{enumerate} \item A finite element is obviously elliptic. Conversely, if $u$ is an elliptic element of $\mathrm{O}_{+}(\Lambda)$, there exists a fixed vector $\alpha$ in the interior of $\mathfrak{N}$. Since $\ker\, (u-\mathrm{id})$ is defined over $\Q$, we can find such an $\alpha$ defined over $\Q$. In that case, $u$ must act finitely on $\alpha^{\perp} \cap \Lambda$ and we are done. \par
\item A quasi-unipotent element which is of infinite order is parabolic (since it is not semi-simple). Conversely, if $g$ is a parabolic element in $\mathrm{O}_{+}(\Lambda)$, the characteristic polynomial of $g$ has rational coefficients and all its roots are of modulus one. Therefore all eigenvalues of $g$ are roots of unity thanks to Kronecker's theorem. \end{enumerate} \end{proof}
One of the most important properties of parabolic isometries fixing $\Lambda$ is the following: \begin{proposition} \label{gauss} Let $u$ be a parabolic element in $\mathrm{O}_{+}(\Lambda)$. Then \emph{:} \begin{enumerate} \item There exists a vector $\theta$ in $\mathfrak{N} \cap \Lambda$ such that $u(\theta)=\theta$. \item There exists $k>0$ such that $u^k$ belongs to $\mathcal{T}_{\theta}$. \end{enumerate} \end{proposition} \begin{proof}$ $ \par \begin{enumerate} \item Let $W= \mathrm{ker}\,(f-\mathrm{id})$, and assume that the line $\ensuremath{\mathbb R}\theta$ doesn't meet ${\Lambda}_{\Q}$. Then the quadratic form $q$ is negative definite on $\theta^{\perp} \cap W_{\Q}$. We can decompose $q_{W_{\Q}}$ as $-\sum_i \ell_i^2$ where the $\ell_i$'s are linear forms on $W_{\Q}$. Then $q$ is also negative definite on $W$, but $q(\theta)=0$ so we get a contradiction. \par
\item By the first point, we know that we can choose an isotropic invariant vector $\theta$ in $\Lambda$. Let us consider the free abelian group $\Sigma:=(\theta^{\perp} \cap \Lambda)/ \ensuremath{\mathbb Z}\theta$, the induced quadratic form is negative definite. Therefore, since $u$ is an isometry, the action of $u$ is finite on $\Sigma$, so that an iterate of $u$ belongs to $\mathcal{T}_{\theta}$.
\end{enumerate} \end{proof}
The definition below is motivated by Remark \ref{bof}. \begin{definition} A subgroup $G$ of $\mathrm{O}_+ (V)$ is of \textit{moderate growth} if it contains no hyperbolic element. \end{definition} Among groups of moderate growth, the most simple ones are finite subgroups of $\mathrm{O}_+ (V)$. Recall the following well-known fact: \begin{lemma} \label{burnside} Any torsion subgroup of $\mathrm{GL}(n, \Q)$ is finite. \end{lemma}
\begin{proof} Let $g$ be an element in $G$, and $\zeta$ be an eigenvalue of $g$. If $m$ is the smallest positive integer such that $\zeta^m=1$,
then $\varphi(m)=\mathrm{deg}_{\Q} (\zeta) \leq n$ where $\varphi(m)=\sum_{d |m} d $. Since $\varphi(k)\underset{k \rightarrow + \infty}{\longrightarrow} + \infty$, there are finitely many possibilities for $m$. Therefore, there exists a constant $c(n)$ such that the order of any $g$ in $G$ divides $c(n)$. This means that $G$ has finite exponent in $\mathrm{GL}(n, \C)$, and the Lemma follows from Burnside's theorem. \end{proof} As a consequence of Lemmas \ref{fini} and \ref{burnside}, we get:
\begin{corollary} \label{burnside} A subgroup of $\mathrm{O}_+ (\Lambda)$ is finite if and only if all its elements are elliptic. \end{corollary} We now concentrate on infinite groups of moderate growth. The main theorem we want to prove is Gizatullin's fixed point theorem: \begin{theorem} \label{ptfixe} Let $G$ be an infinite subgroup of moderate growth in $\mathrm{O}_+ (\Lambda)$. Then \emph{:} \begin{enumerate} \item There exists an isotropic element $\theta$ in $\mathfrak{N} \cap \Lambda$ such that for any element $g$ in $G$, $g(\theta)=\theta$. \par
\item The group $G$ can be written as $G=\Z^{r} \rtimes H$ where $H$ is a finite group and $r>0$. \end{enumerate}
\end{theorem} \begin{proof} $ $ \par \begin{enumerate} \item Thanks to Corollary \ref{burnside}, $G$ contains parabolic elements. Let $g$ be a parabolic element in $G$ and $\theta$ be an isotropic vector. Let $\Lambda^*=(\theta^{\perp} \cap \Lambda)/ \ensuremath{\mathbb Z}\theta$. Since the induced quadratic form on $\Lambda^*$ is negative definite, and an iterate of $g$ acts finitely on $\Lambda^*$; hence $g^k$ is in $\mathcal{T}_{\theta}$ for some integer $k$. \par
\noindent Let $\tilde{g}$ be another element of $G$, and assume that $\tilde{g}$ doesn't fix $\theta$. We put $\eta=\tilde{g}(\theta)$. If $u={g}^k$ and $v=\tilde{g} g^k \tilde{g}^{-1}$, then $u$ and $v$ are nonzero elements of $\mathcal{T}_{\theta}$ and $\mathcal{T}_{\eta}$ respectively. Thanks to Corollary \ref{waza}, $G$ contains hyperbolic elements, which is impossible since it is of moderate growth. \par
\item Let us consider the natural group morphism \[ \varepsilon \colon G \hookrightarrow \mathrm{O}_{+}(V) \rightarrow \mathrm{O}(\Lambda^*). \] The image of $\varepsilon$ being finite, $\ker \, (\varepsilon)$ is a normal subgroup of finite index in $G$. This subgroup is included in $\mathcal{T}_{\theta}$, so it is commutative. Besides, it has no torsion thanks to Proposition \ref{commutatif} (1), and is countable as a subgroup of $\mathrm{GL}_n(\Z)$. Thus it must be isomorphic to $\Z^r$ for some $r$. \end{enumerate}
\end{proof}
\section{Background material on surfaces} \label{4}
\subsection{The invariant nef class} \label{4.1} Let us consider a pair $(X, f)$ where $X$ is a smooth complex projective surface and $f$ is an automorphism of $X$ whose action on $\mathrm{NS}(X)_{\R}$ is a parabolic isometry.
\begin{proposition} \label{tropmalin} There exists a unique non-divisible nef vector $\theta$ in $\mathrm{NS}\,(X) \cap \ker \left( f^* - \mathrm{id} \right)$. Besides, $\theta$ satisfies $\theta^2=0$ and ${K}_X. \theta=0$. \end{proposition} \begin{proof} Let $\mathcal{S}$ be the space of half-lines $\R_{+} \mu$ where $\mu$ runs through nef classes in $\mathrm{NS}\,(X)$. Taking a suitable affine section of the nef cone so that each half-line in $\mathcal{S}$ is given by the intersection with an affine hyperplane, we see that $\mathcal{S}$ is bounded and convex, hence homeomorphic to a closed euclidian ball in $\R^{n-1}$. By Brouwer's fixed point theorem, $f^*$ must fix a point in $\mathcal{S}$. This implies that $f^* \theta=\lambda \,\theta$ for some nef vector $\theta$ and some positive real number $\lambda$ which must be one as $f$ is parabolic. \par
Since $\theta$ is nef, $\theta^2 \geq 0$. By Lemma \ref{lemmenef} (1), $\theta^2=0$ and by Lemma \ref{lemmenef} (2), $K_X . \theta=0$. It remains to prove that $\theta$ can be chosen in $\mathrm{NS}\,(X)$. This follows from Lemma \ref{gauss} (1). Since $\ensuremath{\mathbb R}\theta$ is the unique fixed isotropic ray, $\theta$ is unique up to scaling. It is completely normalized if it is assumed to be non-divisible. \end{proof} \begin{proposition} \label{mg} Let $G$ be an infinite group of automorphisms of $X$ having moderate growth. Then there exists a $G$-invariant nef class $\theta$ in $\mathrm{NS}\,(X)$. \end{proposition} \begin{proof} This follows directly from Theorem \ref{ptfixe} and Proposition \ref{tropmalin}. \end{proof} \subsection{Constructing elliptic fibrations} \label{4.2} In this section, our aim is to translate the question of the existence of $f$-invariant elliptic fibrations in terms of the invariant nef class $\theta$. \begin{proposition} \label{nefnef}
If $(X, f)$ is given, then $X$ admits an invariant elliptic fibration if and only if a multiple $N \theta$ of the $f$-invariant nef class can be lifted to a divisor $D$ in the Picard group $\mathrm{Pic}\,(X)$ such that $\mathrm{dim} \, |D| =1$. Besides, such a fibration is unique. \end{proposition} \begin{proof} Let us consider a pair $(X, f)$ and assume that $X$ admits a fibration $X \xrightarrow{\pi} {C}$ invariant by $f$ whose general fiber is a smooth elliptic curve, where $C$ is a smooth algebraic curve of genus $g$. Let us denote by $\beta$ the class of a general fiber $X_z=\pi^{-1}(z)$ in $\mathrm{NS}\,(X)$. Then $f^* \beta=\beta$. The class $\beta$ is obviously nef, so that it is a multiple of $\theta$. This implies that the fibration $(\pi, C)$ is unique: if $\pi$ and $\pi'$ are two distinct $f$-invariant elliptic fibrations, then $\beta . \,\beta' >0$; but $\theta^2=0$. \par
Let $C \xrightarrow{\varphi} \mathbb{P}^1$ be any branched covering (we call $N$ its degree), and let us consider the composition $X\xrightarrow{\varphi \,\circ\, \pi} \mathbb{P}^1$. Let $D$ be a generic fiber of this map. It is a finite union of the fibers of $\pi$, so that the class of $D$ in $\mathrm{NS}\,(X)$ is $N \beta$. Besides, $\mathrm{dim} \, |D| \geq 1$. In fact $\mathrm{dim} \, |D|=1$, otherwise $D^2$ would be positive. This yields the first implication in the proposition.
\par
To prove the converse implication, let $N$ be a positive integer such that if $N \theta$ can be lifted to a divisor $D$ with $\mathrm{dim} \, |D|=1$. Let us decompose $D$ as $F+M$, where $F$ is the fixed part (so that $|D|=|M|$). Then $0=D^2=D.F+D.M$ and since $D$ is nef, $D.M=0$. Since $|M|$ has no fixed component, $M^2 \geq 0$ so that the intersection pairing is semi-positive on the vector space generated by $D$ and $M$. It follows that $D$ and $M$ are proportional, so that $M$ is still a lift of a multiple of $\theta$ in $\mathrm{Pic}\,(X)$.
\par
Since $M$ has no fixed component and $M^2=0$, $|M|$ is basepoint free. By the Stein factorisation theorem, the generic fiber of the associated Kodaira map $X \rightarrow |M|^*$ is the disjoint union of smooth curves of genus $g$. The class of each of these curves in the Neron-Severi group is a multiple of $\theta$. Since $\theta^2=\theta . K_X=0$, the genus formula implies $g=1$. To conclude, we take the Stein factorisation of the Kodaira map to get a true elliptic fibration. \par
It remains to prove that this fibration is $f$-invariant. If $\mathcal{C}$ is a fiber of the fibration, then $f(\mathcal{C})$ is numerically equivalent to $\mathcal{C}$ (since $f^* \theta=\theta$), so that $\mathcal{C}. f(\mathcal{C})=0$. Therefore, $f(\mathcal{C})$ is another fiber of the fibration. \end{proof} \begin{remark} \label{malin} The unicity of the fibration implies that any $f^N$-elliptic fibration (for a positive integer $N$) is $f$-invariant. \end{remark} In view of the preceding proposition, it is natural to try to produce sections of $D$ by applying the Riemann-Roch theorem. Using Serre duality, we have \begin{equation} \label{RR} \mathrm{h}^0(D)+\mathrm{h}^0(K_X-D) \geq \chi(\mathcal{O}_X)+\frac{1}{2} D.(D-K_X)=\chi({\mathcal{O}_X}). \end{equation} In the next section, we will use this inequality to solve the case where the minimal model of $X$ is a $K3$-surface. \begin{corollary} \label{reduction} If Theorem \ref{Main} holds for $G=\mathbb{Z}$, then it holds in the general case. \end{corollary}
\begin{proof} Let $G$ be an infinite subgroup of $\mathrm{Aut}\,(X)$ of moderate growth, $f$ be a parabolic element of $X$, and assume that there exists an $f$-invariant elliptic fibration $\mathcal{C}$ on $X$. If $\theta$ is the invariant nef class of $X$, then $G$ fixes $\theta$ by Proposition \ref {mg}. This proves that $\mathcal{C}$ is $G$-invariant. \end{proof}
\subsection{Kodaira's classification} \label{4.3} Let us take $(X, f)$ as before. The first natural step to classify $(X, f)$ would be to find what is the minimal model of $X$. It turns out that we can rule out some cases without difficulties. Let $\kappa(X)$ be the Kodaira dimension of $X$. \par
-- If $\kappa(X)=2$, then $X$ is of general type so its automorphism group is finite. Therefore this case doesn't occur in our study. \par
-- If $\kappa(X)=1$, we can completely understand the situation by looking at the Itaka fibration $X \dasharrow |mK_X|^*$ for $m >> 0$, which is $\mathrm{Aut}\,(X)$-invariant. Let $F$ be the fixed part of $|mK_X|$ and $D=mK_X-F$.
\begin{lemma} The linear system $|D|$ is a base point free pencil, whose generic fiber is a finite union of elliptic curves. \end{lemma} \begin{proof}
If $X$ is minimal, we refer the reader to \cite[pp. 574-575]{GH}. If $X$ is not minimal, let $Z$ be its minimal model and $X \xrightarrow{\pi} Z$ the projection. Then $K_X=\pi^*K_Z+E$, where $E$ is a divisor contracted by $\pi$, so that $|mK_X|=|mK_Z|=|D|$. \end{proof} We can now consider the Stein factorisation $X \rightarrow Y \rightarrow Z$ of $\pi$. In this way, we get an $\mathrm{Aut} (X)$-invariant elliptic fibration $X \rightarrow Y$. \par
-- If $\kappa(X)=0$, the minimal model of $X$ is either a $K3$ surface, an Enriques surface, or a bielliptic surface. We start by noticing that we can argue directly in this case on the minimal model: \begin{lemma} If $\kappa(X)\!=\!0$, every automorphism of $X$ is induced by an automorphism of its minimal model. \end{lemma} \begin{proof}
Let $Z$ be the minimal model of $X$ and $\pi$ be the associated projection. By classification of minimal surfaces of Kodaira dimension zero, there exists a positive integer $m$ such that $mK_Z$ is trivial. Therefore, $mK_X$ is an effective divisor $\mathcal{E}$ whose support is exactly the exceptional locus of $\pi$, and $|mK_X|=\{\mathcal{E}\}$. It follows that $\mathcal{E}$ is invariant by $f$, so that $f$ descends to $Z$. \end{proof} \par
$*$ Let us deal with the K3 surface case. We pick any lift $D$ of $\theta$ in $\mathrm{Pic}\,(X)$. Since $\chi(\oo_X)=2$, we get by (\ref{RR}) \[ \mathrm{h}^0(D)+\mathrm{h}^0(-D) \geq 2. \] Since $D$ is nef, $-D$ cannot be effective, so that $\mathrm{h}^0(-D)=0$. We conclude using Proposition \ref{nefnef}. \par
$*$ This argument doesn't work directly for Enriques surfaces, but we can reduce to the K3 case by arguing as follows: if $X$ is an Enriques surface, its universal cover $\widetilde{X}$ is a K3 surface, and $f$ lifts to an automorphism $\tilde{f}$ of $\widetilde{X}$. Besides, $\tilde{f}$ is still parabolic. Therefore, we get an $\tilde{f}$-invariant elliptic fibration $\pi$ on $\widetilde{X}$. \par
If $\sigma$ is the involution on $\widetilde{X}$ such that $X=\widetilde{X}/\sigma$, then $\tilde{f}=\sigma \circ \tilde{f} \circ \sigma^{-1}$, by the unicity of the invariant fibration, $\pi \circ \sigma=\pi$. Thus, $\pi$ descends to $X$. \par
$*$ The case of abelian surfaces is straightforward: an automorphism of the abelian surface $\C^2/ \Lambda$ is given by some matrix $M$ in $\mathrm{GL}(2; \Lambda)$. Up to replacing $M$ by an iterate, we can assume that this matrix is unipotent. If $M= \mathrm{id} + N$, then the image of $N \colon \Lambda \rightarrow \Lambda$ is a sub-lattice $\Lambda^*$ of $\Lambda$ spanning a complex line $L$ in $\mathbb{C}^2$. Then the elliptic fibration $\mathbb{C}^2/ \Lambda \xrightarrow{N} L / \Lambda^*$ is invariant by $M$. \par
$*$ It remains to deal with the case of bi-elliptic surfaces. But this is easy because they are already endowed with an elliptic fibration invariant by the whole automorphism group. \par
-- If $\kappa(X)=-\infty$, then either $X$ is a rational surface, or the minimal model of $X$ is a ruled surface over a curve of genus $g \geq 1$. The rational surface case is rather difficult, and corresponds to Gizatullin's result; we leave it apart for the moment. For blowups of ruled surfaces, we remark that the automorphism group must preserve the ruling. Indeed, for any fiber $\mathcal{C}$, the projection of $f(\mathcal{C})$ on the base of the ruling must be constant since $\mathcal{C}$ has genus zero. Therefore, an iterate of $f$ descends to an automorphism of the minimal model $Z$. \par
We know that $Z$ can be written as $\mathbb{P}(E)$ where $E$ is a holomorphic rank $2$ bundle on the base of the ruling. By the Leray-Hirsh theorem, $\mathrm{H}^{1,1}(Z)$ is the plane generated by the first Chern class $\mathrm{c}_1(\mathcal{O}_E(1))$ of the relative tautological bundle and the pull-back of the fundamental class in $\mathrm{H}^{1, 1}(\mathbb{P}^1)$. Thus, $f^*$ acts by the identity on $\mathrm{H}^{1,1}(Z)$, hence on $\mathrm{H}^{1,1}(X)$. \section{The rational surface case} \label{5} \subsection{Statement of the result} \label{5.1} From now on, $X$ will always be a rational surface, so that $\mathrm{h}^1(X, \mathcal{O}_X)=\mathrm{h}^2(X, \mathcal{O}_X)=0$. It follows that $\mathrm{Pic}\,(X) \simeq \mathrm{NS}\,(X) \simeq \mathrm{H}^2(X, \Z)$, which imply that numerical and linear equivalence agree. In this section, we prove the following result: \begin{theorem}[\cite{GIZ}] \label{Gizzz}
Let $X$ be a rational surface and $f$ be a parabolic automorphism of $X$. If $\theta$ is the nef $f$-invariant class in $\mathrm{NS}\,(X)$, then there exists an integer $N$ such that $\mathrm{dim}\, |N \theta|=1$. \end{theorem} Thanks to Proposition \ref{mg} and Corollary \ref{reduction}, this theorem is equivalent to Theorem \ref{Main} for rational surfaces and is the most difficult result in Gizatullin's paper.
\subsection{Properties of the invariant curve} \label{5.2}
The divisor ${K}_X-\theta$ is never effective. Indeed, if $H$ is an ample divisor, $K_X. H <0$ so that $(K_X-\theta).H <0$. Therefore, we obtain by (\ref{RR}) that $|\theta| \neq \varnothing$, so that $\theta$ can be represented by a possibly non reduced and non irreducible curve $C$. We will write the curve $C$ as the divisor $\sum_{i=1}^d a_i \,C_i$ where the $C_i$ are irreducible. Since $\theta$ is non divisible in $\mathrm{NS}\,(X)$, $C$ is primitive. \par
In the sequel, we will make the following assumptions, and we are seeking for a contradiction: \par
\textbf{Assumptions} \begin{itemize}
\item[(1)] We have $|N \theta|=\{NC\}$ for all positive integers $N$. \item[(2)] For any positive integer $k$, the pair $(X, f^k)$ is minimal. \end{itemize} \par
Let us say a few words on $(2)$. If for some integer $k$ the map $f^k$ descends to an automorphism $g$ of a blow-down $Y$ of $X$, then we can still argue with $(Y, g)$. The corresponding invariant nef class will satisfy $(1)$. Thanks to Remark \ref{malin}, we don't lose anything concerning the fibration when replacing $f$ by an iterate. \par
We study thoroughly the geometry of $C$. Let us start with a simple lemma. \begin{lemma} \label{ttsimple} If $D_1$ and $D_2$ are two effective divisors whose classes are proportional to $\theta$, then $D_1$ and $D_2$ are proportional (as divisors). \end{lemma}
\begin {proof} There exists integers $N$, $N_1$, and $N_2$ such that $N_1 D_1 \equiv N_2 D_2 \equiv N \theta$. Therefore, $N_1 D_1$ and $N_2 D_2$ belong to $|N \theta|$ so they are equal. \end{proof} The following lemma proves that $C$ looks like a fiber of a minimal elliptic surface. \begin{lemma} \label{base} $ $ \begin{enumerate} \item For $1 \leq i \leq d$, $K_X. C_i=0$ and $C.C_i=0$. If $d \geq 2$, $C_i^2<0$. \item The classes of the components $C_i$ in $\mathrm{NS}\,(X)$ are linearly independent. \item The intersection form is nonpositive on the $\Z$-module spanned by the $C_i$'s. \item If $D$ is a divisor supported in $C$ such that $D^2=0$, then $D$ is a multiple of $C$. \end{enumerate} \end{lemma} \begin{proof} $ $ \begin{enumerate}
\item Up to replacing $f$ by an iterate, we can assume that all the components $C_i$ of the curve $C$ are fixed by $f$. By Lemma \ref{lemmenef}, $C_i^2\leq 0$ and $C.K_X=C.C_i=0$ for all $i$. Assume that $d \geq 2$. If $C_i^2=0$, then $C$ and $C_i$ are proportional, which would imply that $C$ is divisible in $\mathrm{NS}\,(X)$. Therefore $C_i^2<0$. If $K_X.C_i<0$, then $C_i$ is a smooth and $f$-invariant exceptional rational curve. This contradicts Assumption (2). Thus $K_X.C_i \geq 0$. Since $K_X.C=0$, it follows that $K_X.C_i=0$ for all $i$. \par
\item If there is a linear relation among the curves $C_i$, we can write it as $D_1 \equiv D_2$, where $D_1$ and $D_2$ are linear combinations of the $C_i$ with positive coefficients (hence effective divisors) having no component in common. We have $D_1^2=D_1.D_2 \geq 0$. On the other hand $C. D_1=0$ and $C^2=0$, so by the Hodge index theorem $C$ and $D_1$ are proportional. This contradicts Lemma \ref{ttsimple}. \par
\item Any divisor $D$ in the span of the $C_i$'s is $f$-invariant, so that Lemma \ref{lemmenef} (1) yields $D^2 \leq 0$. \par
\item If $D^2=D.C=0$, then $D$ and $C$ are numerically proportional. Therefore, there exists two integers $a$ and $b$ such that $aD-bC \equiv 0$. By Lemma \ref{ttsimple}, $aD=bC$ and since $C$ is primitive, $D$ is a multiple of $C$. \end{enumerate} \end{proof}
\begin{lemma} \label{genre} $ $ \begin{enumerate} \item The curve $C$ is $1$-connected \emph{(}see \cite[pp. 69]{BPVDV}\emph{)}. \item We have $\mathrm{h}^0(C, \oo_C)=\mathrm{h}^1(C, \oo_C)=1$. \item If $d=1$, then $C_1$ has arithmetic genus one. If $d \geq2$, all the curves $C_i$ are rational curves of self-intersection $-2$. \end{enumerate} \end{lemma} \begin{proof} $ $ \begin{enumerate} \item Let us write $C=C_1+C_2$ where $C_1$ and $C_2$ are effective and supported in $C$, with possible components in common. By Lemma \ref{base} (3), $C_1^2\leq 0$ and $C_2^2\leq 0$. Since $C^2=0$, we must have $C_1.C_2 \geq 0$. If $C_1.C_2=0$, then $C_1^2=C_2^2=0$ so that by Lemma \ref{base} (4), $C_1$ and $C_2$ are multiples of $C$, which is impossible. \par
\item By $(1)$ and \cite[Corollary 12.3] {BPVDV}, $\mathrm{h}^0(C, \oo_C)=1$. The dualizing sheaf $\omega_C$ of $C$ is the restriction of the line bundle $K_X+C$ to the divisor $C$. Therefore, for any integer $i$ between $1$ and $d$, $\mathrm{deg}\, ({\omega_C})_{| C_i}=(K_X+C).C_i=0$ by Lemma \ref{base} (1). Therefore, by \cite[Lemma 12.2] {BPVDV}, $\mathrm{h}^0(C, \omega_C)\leq 1$ with equality if and only if $\omega_C$ is trivial. We can now apply the Riemann-Roch theorem for singular embedded curves \cite[Theorem 3.1]{BPVDV}: since $\omega_C$ has total degree zero, we have $\chi(\omega_C)=\chi(\oo_C)$. But using Serre duality \cite[Theorem 6.1]{BPVDV}, $\chi(\omega_C)=-\chi(\oo_C)$ so that $\chi(\oo_C)=\chi(\omega_C)=0$. It follows that $\mathrm{h}^1(C, \oo_C)=1$. \par
\item This follows from the genus formula: $2p_a(C_i)-2=C_i^2+K_X.C_i=C_i^2<0$ so that $p_a(C_i)=0$ and $C_i^2=-2$. Now the geometric genus is always smaller than the arithmetic genus, so that the geometric genus of $C_i$ is $0$, which means that $C_i$ is rational.
\end{enumerate} \end{proof}
We can now prove a result which will be crucial in the sequel: \begin{proposition} \label{crucial} Let $D$ be a divisor on $X$ such that $D. C=0$. Then there exists a positive integer $N$ and a divisor $S$ supported in $C$ such that for all $i$, $(ND-S).\, C_i=0$. \end{proposition} \begin{proof} Let $V$ be the $\Q$-vector space spanned by the $C_i$'s in $\mathrm{NS}_{\Q}(X)$, by Lemma \ref{base} (3), it has dimension $r$. We have a natural morphism $\lambda \colon V \rightarrow \Q^r$ defined by $\lambda(x)=(x.C_1, \ldots, x.C_r)$. The kernel of this morphism are vectors in $V$ orthogonal to all the $C_i$'s. Such a vector is obviously isotropic, and by Lemma \ref{base} (4), it is a rational multiple of $D$. Therefore the image of $\lambda$ is a hyperplane in $\Q^r$, which is the hyperplane $\sum_i a_i x_i=0$. Indeed, for any element $x$ in $V$, we have $\sum_i a_i \, (x.C_i)=x.C=0$. \par
Let us consider the element $w=(D.C_1, \ldots, D.C_r)$ in $\Q^r$. Since $\sum_i a_i \, (D.C_i)=D.C=0$, we have $w=\lambda(S)$ for a certain $S$ in $V$. This gives the result. \end{proof} \subsection{The trace morphism} \label{5.3} In this section, we introduce the main object in Gizatullin's proof: the \textit{trace morphism}. For this, we must use the Picard group of the embedded curve $C$. It is the moduli space of line bundles on the complex analytic space $\mathcal{O}_C$, which is $\mathrm{H}^1(C, \oo_C^{\,\times})$. \par
Recall \cite[Proposition 2.1]{BPVDV} that $\mathrm{H}^1(C, \Z_C)$ embeds as a discrete subgroup of $\mathrm{H}^1(C, \oo_C)$. The connected component of the line bundle $\oo_C$ is denoted by $\mathrm{Pic}^0 (C)$, it is the abelian complex Lie group $\mathrm{H}^1(C, \oo_C)/ \mathrm{H}^1(C, \Z_C)$. We have an exact sequence \[ 0 \rightarrow \mathrm {Pic}^0(C) \rightarrow \mathrm{Pic}\,(C) \xrightarrow{\mathrm{c}_1} \mathrm{H}^1(C, \Z) \] and $\mathrm{H}^1 (C, \mathbb{Z})\simeq \Z^d$. Therefore, connected components of $\mathrm{Pic}\,(C)$ are indexed by sequences $(n_1, \ldots, n_d)$ corresponding to the degree of the line bundle on each irreducible component of $C$. By Lemma \ref{genre} (2), $\mathrm{Pic}^0(C)$ can be either $\C$, $\C^{\times}$, or an elliptic curve.
\par
The trace morphism is a group morphism $\mathfrak{tr} \colon \mathrm{Pic}\,(X) \rightarrow \mathrm{Pic}\, (C)$ defined by $ \mathfrak{tr}\, (\mathcal{L})=\mathcal{L}_{| C}$ . Remark that $C.C_i=0$ for any $i$, so that the line bundle $\oo_X\left(C\right)$ restricts to a line bundle of degree zero on each component $a_i \,C_i$. \begin{proposition} $ $ \label{torsion} \begin{enumerate} \item The line bundle $\mathfrak{tr}\left(\oo_X (C ) \right)$ is not a torsion point in $\mathrm{Pic}^0(C)$. \item The intersection form is negative definite on $\mathrm{ker}\, (\mathfrak{tr})$. \end{enumerate} \end{proposition} \begin{proof} $ $ \begin{enumerate} \item Let $N$ be an integer such that $N\mathfrak{tr}\left(\oo_X (C ) \right)=0$ in $\mathrm{Pic}\,(C)$. Then we have a short exact sequence \[ 0 \rightarrow \oo_X((N-1)C) \rightarrow \oo_X (NC) \rightarrow \oo_C \rightarrow 0. \] \par
\noindent Now $\mathrm{h}^2(X, \oo_X((N-1)C))=\mathrm{h}^0(\oo_X(-(N-1)C+K_X)=0$, so that the map \[ \mathrm{H}^1(X, \oo_X(NC)) \rightarrow \mathrm{H}^1(C, \oo_C) \] is onto. It follows from Lemma \ref{genre} (2) that $\mathrm{h}^1(X, \oo_X(NC)) \geq 1$ so that by Riemann-Roch \[ \mathrm{h}^0(X, \oo_X(NC)) \geq \mathrm{h}^1(X, \oo_X(NC)) + \chi(\oo_X) \geq 2. \]
This yields a contradiction since we have assumed that $|N\theta|=\{NC\}$. \par
\item Let $D$ be a divisor in the kernel of $\mathfrak{tr}$. By the Hodge index theorem $D^2 \leq 0$. Besides, if $D^2=0$, then $D$ and $C$ are proportional. In that case, a multiple of $C$ would be in $\ker \,(\mathfrak{tr})$, hence $\mathfrak{tr}\, (\oo_X(C))$ would be a torsion point in $\mathrm{Pic}\,(C)$. \end{enumerate} \end{proof} \section{Proof of Gizatullin's theorem} \label{6} \subsection{The general strategy} \label{6.1} The strategy of the proof is simple in spirit. Let $\mathfrak{P}$ be the image of $\mathfrak{tr}$ in $\mathrm{Pic}\, (C)$, so that we have an exact sequence of abelian groups \[ 1 \rightarrow \ker \, (\mathfrak{tr}) \rightarrow \mathrm{Pic}\, (X) \rightarrow \mathfrak{P} \rightarrow 1 \] By Proposition \ref{torsion}, the intersection form is negative definite on $\ker \,(\mathfrak{tr})$, so that $f^*$ is of finite order on $\ker \,(\mathfrak{tr})$. In the first step of the proof, we will prove that for any divisor $D$ on $X$ orthogonal to $C$, $f^*$ induces a morphism of finite order on each connected component of any element $\mathfrak{tr} (D)$ in $\mathrm{Pic}\,(C)$. \par
In the second step, we will prove that the action of $f^*$ on $\mathrm{Pic} (X)$ is finite. This will give the desired contradiction. \subsection{Action on the connected components of $\mathfrak{P}$} \label{6.2} In this section, we prove that $f^*$ acts finitely on "many" connected components of $\mathfrak{P}.$ More precisely:\begin{proposition} \label{elliptic} Let $D$ be in $\mathrm{Pic}\, (X)$ such that $D.C=0$, and let $\mathfrak{X}_D$ be a connected component of $\mathfrak{tr} (D)$ in $\mathrm{Pic}\, (C)$. Then the restriction of $f^*$ to $\mathfrak{X}_D$ is of finite order. \end{proposition} \begin{proof} We start with the case $D=0$ so that $\mathfrak{X}=\mathrm{Pic}^0(C)$. Then three situations can happen: \par
-- If $\mathrm{Pic}^0 (C)$ is an elliptic curve, then its automorphism group is finite (by automorphisms, we mean group automorphisms). \par
-- If $\mathrm{Pic}^0 (C)$ is isomorphic to $\C^{\times}$, its automorphism group is $\{\mathrm{id}, z \rightarrow z^{-1}\}$, hence of order two, so that we can also rule out this case. \par
-- Lastly, if $\mathrm{Pic}^0 (C)$ is isomorphic to $\C$, its automorphism group is $\C^{\times}$. We know that $C$ is a non-zero element of $\mathrm{Pic}^0(C)$ preserved by the action of $f^*$. This forces $f^*$ to act trivially on $\mathrm{Pic}^0 (C)$. \par
Let $D$ be a divisor on $X$ such that $D.C=0$. By Proposition \ref{crucial}, there exists a positive integer $N$ and a divisor $S$ supported in $C$ such that $N \mathfrak{tr}\,(D)- \mathfrak{tr}\,(S) \in \mathrm{Pic}^0(C)$. Let $m$ be an integer such that $f^m$ fixes the components of $C$ and acts trivially on $\mathrm{Pic}\,(C)$. We define a map $\lambda \colon \ensuremath{\mathbb Z}\rightarrow \mathrm{Pic}^0(C)$ by the formula \[ \lambda(k)=(f^{km})^* \{\mathfrak{tr} (D)\}-\mathfrak{tr} (D) \] \par
\begin{enumerate} \item[] \textbf{Claim 1}: $\lambda$ does not depend on $D$. \par Indeed, if $D'$ is in $\mathfrak{X}_D$, then $\mathfrak{tr}\,(D'-D) \in \mathrm{Pic}^0(C)$ so that \[ (f^{km})^*(D'-D)=D'-D. \] This gives $(f^{km})^* \{\mathfrak{tr} (D')\}-\mathfrak{tr} (D')=(f^{km})^* \{\mathfrak{tr} (D)\}-\mathfrak{tr} (D)$ \par
\item[] \textbf{Claim 2}: $\lambda$ is a group morphism. \begin{alignat*}{3} \lambda(k+l) & = (f^{km})^*(f^{lm})^* \{\mathfrak{tr} (D)\}-\mathfrak{tr} (D) \\ & = \begin{aligned}[t]
(f^{km})^*\left\{(f^{lm})^* \{\mathfrak{tr} (D)\} \right\}
& -\left\{(f^{lm})^* \{\mathfrak{tr} (D)\} \right\} \\
& + (f^{lm})^* \{\mathfrak{tr} (D)\}-\mathfrak{tr} (D)
\end{aligned} \\ &= \lambda(k) + \lambda(l) \quad \textrm{by Claim 1}. \end{alignat*} \par
\item[] \textbf{Claim 3}: $\lambda$ has finite image. \par For any integer $k$, since $N \,\mathfrak{tr}\,(D)- \mathfrak{tr}\,(S) \in \mathrm{Pic}^0(C)$, $(f^{km})^* \{N\, \mathfrak{tr} (D) \}= N \,\mathfrak{tr} (D)$. Therefore, we see that $(f^{km})^* \{ \mathfrak{tr} (D) \} - \mathfrak{tr} (D)=\lambda(k)$ is a $N$-torsion point in $\mathrm{Pic}^0(C)$. Since there are finitely many $N$-torsion points, we get the claim. \end{enumerate} \par
We can now conclude. By claims $2$ and $3$, there exists an integer $s$ such that the restriction of $\lambda$ to $s \Z$ is trivial. This implies that $D$ is fixed by $f^{ms}$. By claim 1, all elements in $\mathfrak{X}_D$ are also fixed by $f^{ms}$. \end{proof} \subsection{Lift of the action from $\mathfrak{P}$ to the Picard group of $X$} \label{6.3} By Proposition \ref{torsion} (2) and Proposition \ref{elliptic}, up to replacing $f$ with an iterate, we can assume that $f$ acts trivially on all components $\mathfrak{X}_D$, on $\mathrm{ker}\, (\mathfrak{tr})$, and fixes the components of $C$. \par
Let $r$ be the rank of $\mathrm{Pic}\, (X)$, and fix a basis $E_1, \ldots, E_r$ of $\mathrm{Pic}\, (X)$ composed of irreducible reduced curves. Let $n_i=E_i.C$. If $n_i=0$, then either $E_i$ is a component of $C$, or $E_i$ is disjoint from $C$. In the first case $E_i$ is fixed by $f$. In the second case, $E_i$ lies in the kernel of $\mathfrak{tr}$, so that it is also fixed by $f$. \par
Up to re-ordering the $E_i$'s, we can assume that $n_i>0$ for $1 \leq i \leq s$ and $n_i=0$ for $i>s$. We put $m=n_1 \ldots n_s$, $m_i=\frac{m}{n_i}$ and $L_i=m_iE_i$.
\begin{proposition} \label{chic} For $1 \leq i \leq s$, $L_i$ is fixed by an iterate of $f$. \end{proposition} \begin{proof} For $1 \leq i \leq s$, we have $L_i.C=m$, so that for $1\leq i, j \leq s $, $(L_i-L_j).C=0$. Therefore, by Proposition \ref{elliptic}, an iterate of $f$ acts trivially on $\mathfrak{X}_{L_i-L_j}$. Since there are finitely many couples $(i,j)$, we can assume (after replacing $f$ by an iterate) that $f$ acts trivially on all $\mathfrak{X}_{L_i-L_j}$. \par
Let us now prove that $f^* L_i$ and $L_i$ are equal in $\mathrm{Pic}\,(X)$. Since $f^*$ acts trivially on the component $\mathfrak{X}_{L_i-L_j}$, we have $\mathfrak{tr}\,(f^*L_i-L_i)=\mathfrak{tr}\,(f^*L_j-L_j)$. Let $D=f^*L_1-L_1$. Then for any $i$, we can write $f^*L_i-L_i=D+D_i$ where $\mathfrak{tr}\, (D_i)$=0. \par
Let us prove that the class $D_i$ in $\mathrm{Pic}\, (X)$ is independent of $i$. For any element $A$ in $\mathrm{ker}\, (\mathfrak{tr})$, we have \[ D_i. \,A=(f^*L_i-L_i-D). \,A=f^*L_i . f^*A-L_i . \,A-D. \,A=-D. \,A \] since $f^*A=A$. Now since the intersection form in non-degenerate on $\mathrm{ker}\,(\mathfrak{tr})$, if $(A_k)_k$ is an orthonormal basis of $\mathrm{ker}\,(\mathfrak{tr})$, \[ D_i=-\sum_k (D_i . \, A_k) \,A_k=\sum_k (D . \, A_k) \, A_k. \] Therefore, all divisors $D_i$ are linearly equivalent. Since $D_1=0$, we are done. \end{proof} We can end the proof of Gizatullin's theorem. Since $L_1, \ldots, L_s, E_{s+1}, \ldots, E_{r}$ span $\mathrm{Pic}\, (X)$ over $\Q$, we see that the action of $f$ on $\mathrm{Pic}\,(X)$ is finite. This gives the required contradiction.
\section{Minimal rational elliptic surfaces} \label{7} Throughout this section, we will assume that $X$ is a rational elliptic surface whose fibers contain no exceptional curves; such a surface will be called by a slight abuse of terminology a minimal elliptic rational surface. \subsection{Classification theory} \label{7.1} The material recalled in this section is more or less standard, we refer to \cite[Chap. II \S 10.4]{DS} for more details. \begin{lemma} \label{hehehe}
Let $X$ be a rational surface with $K_X^2=0$. Then $|-K_X| \neq \varnothing$. Besides, for any divisor $\mathfrak{D}$ in $|-K_X|$ \emph{:} \begin{enumerate} \item $\mathrm{h}^1(\mathfrak{D}, \oo_{\mathfrak{D}})=1$. \item For any divisor $D$ such that $0< D < \mathfrak{D}$, $\mathrm{h}^1({D}, \oo_{D})=0$. \item $\mathfrak{D}$ is connected and its class is non-divisible in $\mathrm{NS}\,(X)$. \end{enumerate} \end{lemma} \begin{proof} $ $
The fact that $|-K_X| \neq \varnothing$ follows directly from the Riemann-Roch theorem. \begin{enumerate} \item We write the exact sequence of sheaves \[ 0 \rightarrow \oo_X(-\mathfrak{D}) \rightarrow \oo_X \rightarrow \oo_{\mathfrak{D}} \rightarrow 0. \] Since $X$ is rational, $\mathrm{h}^1(X, \oo_X)=\mathrm{h}^2(X, \oo_X)=0$; and since $\mathfrak{D}$ is an anticanonical divisor, we have by Serre duality \[ \mathrm{h^2(X, -\mathfrak{D})}=\mathrm{h^0(X, K_X)}=1. \] \item We use the same proof as in (1) with $D$ instead of $\mathfrak{D}$. We have \[ \mathrm{h^2(X, -{D})}=\mathrm{h^0(X, K_X+D)}=\mathrm{h^0(X, D-\mathfrak{D})}=0. \] \item The connectedness follows directly from $(1)$ and $(2)$: if $\mathfrak{D}$ is the disjoint reunion of two divisors $\mathfrak{D}_1$ and $\mathfrak{D}_2$, then $\mathrm{h}^0(\mathfrak{D}, \oo_\mathfrak{D})=\mathrm{h}^0(\mathfrak{D}_1, \oo_{\mathfrak{D}_1})+\mathrm{h}^0(\mathfrak{D}_2, \oo_{\mathfrak{D}_2})=0$, a contradiction. \par
\noindent Assume now that $\mathfrak{D}=m \mathfrak{D}'$ in $\mathrm{NS}\,(X)$, where $\mathfrak{D}'$ is not necessarily effective and $m \geq 2$. Then, using Riemann-Roch, \[ \qquad\mathrm{h}^0(X, {\mathfrak{D}'})+\mathrm{h}^0(X, -(m+1) \mathfrak{D}')\geq 1. \]
If $-(m+1) \mathfrak{D}'$ is effective, then $|NK_X|\neq \varnothing$ for some positive integer $N$, which is impossible. Therefore the divisor $\mathfrak{D}'$ is effective; and $\mathfrak{D}-\mathfrak{D'}=(m-1) \mathfrak{D'}$ is also effective. Using Riemann-Roch one more time, \[ \begin{aligned} \qquad\mathrm{h}^0(\mathfrak{D}', \oo_{\mathfrak{D}'})-\mathrm{h}^1(\mathfrak{D}', \oo_{\mathfrak{D}'})=\chi(\oo_{\mathfrak{D}'})&=\chi(\oo_X)-\chi(\mathcal{O}_X(-\mathfrak{D}'))\\ &=-\frac{1}{2} \mathfrak{D}' .(\mathfrak{D}'+K_X)=0. \end{aligned} \] Thanks to $(2)$, since $0 < \mathfrak{D'} < \mathfrak{D}$, $\mathrm{h}^1(\mathfrak{D}', \oo_{\mathfrak{D}'})=0$, so that $\mathrm{h}^0(\mathfrak{D}', \oo_{\mathfrak{D}'})=0$. This gives again a contradiction.
\end{enumerate}
\end{proof}
\begin{proposition} \label{dix} Let $X$ be a rational minimal elliptic surface and $C$ be a smooth fiber. \begin{enumerate} \item $K_X^2=0$ and $\mathrm{rk} \, \{\mathrm{Pic}\,(X)\}=10.$ \item For any irreducible component $E$ of a reducible fiber , $E^2 <0$ and $E.K_X=0.$ \item There exists a positive integer $m$ such that $-mK_X=C$ in $\mathrm{Pic}\,(X)$. \end{enumerate} \end{proposition} \begin{proof} Let $C$ be any fiber of the elliptic fibration. Then for any reducible fiber $D=\sum_{i=1}^s a_i D_i$, $D_i . C=C^2=0$. By the Hodge index theorem, $D_i^2 \leq 0$. If $D_i^2=0$, then $D_i$ is proportional to $C$. Let us write $D=a_i D_i+(D-a_i D_i)$. On the one hand, $a_i D_i .(D-a_i D_i)=0$ since $D_i$ and $D-D_i$ are proportional to $C$. On the other hand, $a_i D_i .(D-a_i D_i)>0$ since $D$ is connected. This proves the first part of (2). \par
We have $K_X.C=C.C=0$. By the Hodge index theorem, $K_X^2 \leq 0$. We have an exact sequence \[ 0 \rightarrow K_X \rightarrow K_X+C \rightarrow \omega_C \rightarrow 0. \] \par
Since $\mathrm{h}^0(C, \omega_C)=1$ and $\mathrm{h^0(X, K_X)}=\mathrm{h}^1(X, K_X)=\mathrm{h}^1(X, \oo_X)=0$, $\mathrm{h}^0(X, K_X+C)=1$. Thus, the divisor $D=K_X+C$ is effective. Since $D.C=0$, all components of $D$ are irreducible components of the fibers of the fibration. The smooth components cannot appear, otherwise $K_X$ would be effective. Therefore, if $D=\sum_{i=1}^{s} a_i D_i$, we have $D_i^2<0$. Since $X$ is minimal, $K_X.D_i \geq 0$ (otherwise $D_i$ would be exceptional). Thus, $K_X.D \geq 0$. \par
Since $C$ is nef, we have $D^2=(K_X+C).D \geq K_X.D\geq0$. On the other hand, $D.C=0$ so that $D^2=0$ by the Hodge index theorem. Thus $K_X^2=0$. Since $X$ is rational, it follows that $\mathrm{Pic}\, (X)$ has rank $10$. This gives (1).
\par
Now $K_X^2=C^2=C.K_X=0$ so that $C$ and $K_X$ are proportional. By Lemma \ref{hehehe}, $K_X$ is not divisible in $\mathrm{NS}\,(X)$, so that $C$ is a multiple of $K_X$. Since $|dK_X|=0$ for all positive $d$, $C$ is a negative multiple of $K_X$. This gives (3). \par
The last point of (2) is now easy: $E.K_X=-\frac{1}{m} E.C=0$. \end{proof} We can be more precise and describe explicitly the elliptic fibration in terms of the canonical bundle. \begin{proposition} \label{primitive}
Let $X$ be a minimal rational elliptic surface. Then for $m$ large enough, we have $\mathrm{dim}\, |-mK_X| \geq 1$. For $m$ minimal with this property, $|-mK_X|$ is a pencil without base point whose generic fiber is a smooth and reduced elliptic curve. \end{proposition} \begin{proof}
The first point follows from Proposition \ref{dix}. Let us prove that $|-mK_X|$ has no fixed part. As usual we write $-mK_X=F+D$ where $F$ is the fixed part. Then since $C$ is nef and proportional to $K_X$, $C.F=C.D=0$. Since $D^2\geq 0$, by the Hodge index theorem $D^2=0$ and $D$ is proportional to $C$. Thus $D$ and $F$ are proportional to $K_X$.
\par
By Lemma \ref{hehehe}, the class of $K_X$ is non-divisible in $\mathrm{NS}\,(X)$. Thus $F=m' \mathfrak{D}$ for some integer $m'$ with $0 \leq m' <m$. Hence $D=(m-m') \,\mathfrak{D}=-(m-m') \,K_X$ and $\mathrm{dim}\, |D| \geq 1$. By the minimality of $m$, we get $m'=0$.
\par
Since $K_X^2=0$, $-mK_X$ is basepoint free and $|-mK_X|=1$. Let us now prove that the divisors in $|-mK_X|$ are connected. If this is not the case, we use the Stein decomposition and write the Kodaira map of $-mK_X$ as \[
X \rightarrow S \xrightarrow{\psi} |-mK_X|^* \]
where $S$ is a smooth compact curve, and $\psi$ is finite. Since $X$ is rational, $S=\mathbb{P}^1$ and therefore we see that each connected component $D$ of a divisor in $|-mK_X|$ satisfies $\mathrm{dim}\, |D| \geq 1$. Thus $\mathrm{dim}\, |D| \geq 2$ and we get a contradiction.
\par
We can now conclude: a generic divisor in $|-mK_X|$ is smooth and reduced by Bertini's theorem. The genus formula shows that it is an elliptic curve. \end{proof} \begin{remark}$ $ \par \begin{enumerate}
\item Proposition \ref{primitive} means that the relative minimal model of $X$ is a \textit{Halphen surface} of index $m$, that is a rational surface such that $|-mK_X|$ is a pencil without fixed part and base locus. Such a surface is automatically minimal.
\item The elliptic fibration $X \rightarrow |-mK_X|^*$ doesn't have a rational section if $m \geq 2$. Indeed, the existence of multiple fibers (in our situation, the fiber $m \mathfrak{D}$) is an obstruction for the existence of such a section. \end{enumerate} \end{remark} \subsection{Reducible fibers of the elliptic fibration} \label{6.2} We keep the notation of the preceding section: $X$ is a Halphen surface of index $m$ and $\mathfrak{D}$ is an anticanonical divisor. \begin{lemma} \label{woodloot}
All the elements of the system $|-mK_X|$ are primitive, except the element $m \mathfrak{D}$. \end{lemma} \begin{proof}
Since $K_X$ is non-divisible in $\mathrm{NS}\,(X)$, a non-primitive element in $|-mK_X|$ is an element of the form $k D$ where $D \in |m' \mathfrak{D}|$ and $m=k m'$. But $\mathrm{dim} \,|m' \mathfrak{D}|=0$ so that $|D|=|m' \mathfrak{D}|=\{m' \mathfrak{D}\}$. \end{proof}
In the sequel, we denote by $S_1, \ldots, S_\lambda$ the reducible fibers of $|-mK_X|$. We prove an analog of Lemma \ref{base}, but the proofs will be slightly different. \begin{lemma} \label{libre} $ $ \begin{enumerate}
\item Let $S=\alpha_1 E_1 + \ldots + \alpha_{\nu} E_{\nu}$ be a reducible fiber of $|-mK_X|$. Then the classes of the components $E_i$ in $\mathrm{NS}\,(X)$ are linearly independent. \item If $D$ is a divisor supported in $S_1 \cup \ldots \cup S_{\lambda}$ such that $D^2=0$, then there exists integers $n_i$ such that $D=n_1S_1 + \ldots + n_{\lambda} S_{\lambda}$. \end{enumerate} \end{lemma} \begin{proof} If there is a linear relation among the curves $E_i$, we can write it as $D_1 \equiv D_2$, where $D_1$ and $D_2$ are linear combinations of the $E_i$ with positive coefficients (hence effective divisors) having no component in common. We have $D_1^2=D_1. \, D_2 \geq 0$. On the other hand $S.\, D_1=0$ and $D^2=0$, so by the Hodge index theorem $S$ and $D_1$ are proportional. Let $E$ be a component of $S$ intersecting $D_0$ but not included in $D_0$. If $a \,D_1 \sim b \,S$, then $0=b \,S.\,E=a \,D_1.\, E>0$, and we are done. \par
For the second point, let us write $D=D_1+ \ldots +D_{\lambda}$ where each $D_i$ is supported in $S_i$. Then the $D_i$'s are mutually orthogonal. Besides, $D_i. C=0$, so that by the Hodge index theorem $D_i^2 \leq 0$. Since $D^2=0$, it follows that $D_i^2=0$ for all $i$. \par
We pick an $i$ and write $D_i=D$ and $S_i=S$. Then there exists integers $a$ and $b$ such that $a D \sim b S$. Therefore, if $D=\sum \beta_q \,E_q$, $\sum_q (a \alpha_q-b \beta_q) \,E_q=0 $ in $\mathrm{NS}\,(X)$. By Lemma \ref{libre}, $a \alpha_q-b \beta_q=0$ for all $q$, so that $b$ divides $a \alpha_q$ for all $q$. By Lemma \ref{woodloot}, $b$ divides $a$. If $b=ac$, then $\beta_q=c \alpha_q$ for all $q$, so that $D=cS$. \end{proof}
\par
Let $\rho \colon X \rightarrow \mathbb{P}^1$ be the Kodaira map of $|-mK_X|$, and $\xi$ be the generic point of $\mathbb{P}^1$. We denote by $\mathfrak{X}$ the algebraic variety $\rho^{-1}(\xi)$, which is a smooth elliptic curve over the field $\mathbb{C}(t)$. \par
Let $\mathcal{N}$ be the kernel of the natural restriction map ${\mathfrak{t}} \colon \mathrm{Pic}\,(X) \rightarrow\mathrm{Pic}\,(\mathfrak{X})$. The image of $\mathfrak{t}$ is the set of divisors on $\mathfrak{X}$ defined over the field $\mathbb{C}(t)$, denoted by $\mathrm{Pic}\, (\mathfrak{X}/\C(t))$. \par
The algebraic group $\mathrm{Pic}_0(\mathfrak{X})$ acts naturally on $\mathfrak{X}$, and this action is simple and transitive over any algebraic closure of $\C(t)$.
\begin{proposition} \label{sept}
If $S_1, \ldots, S_{\lambda}$ are the reducible fibers of the pencil $|-mK_X|$ and $\mu_j$ denotes the number of components of each curve $S_j$, then $ \mathrm{rk}\, \mathcal{N}=1+\sum_{i=1}^{\lambda} \,\{\mu_i- 1\}. $ \end{proposition}
\begin{proof}
The group $\mathcal{N}$ is generated by $\mathfrak{D}$ and the classes of the reducible components of $|-mK_X|$. \par
We claim that the module of relations between these generators is generated by the relations
$\alpha_1 [E_1] + \ldots + \alpha_{\nu} [E_{\nu}]=m[\mathfrak{D}]$ where $\alpha_1 E_1 + \cdots + \alpha_s E_s$ is a reducible member of $|-mK_X|$. \par
Let $D$ be of the form $a \mathfrak{D}+D_1+ \cdots + D_{\lambda}$ where each $D_i$ is supported in $S_i$, and assume that $D \sim 0$. Then $(D_1+ \cdots + D_{\lambda})^2=0$. Thanks to Lemma \ref{libre} (2), each $D_i$ is equal to $n_i S_i$ for some $n_i$ in $\Z$. Then $a+m \,\{\sum_{i=1}^{\lambda} n_i \}=0$, and \[ a \mathfrak{D}+D_1+ \cdots + D_{\lambda}=\sum_{i=1}^{\lambda} n_i \,(S_i-m \mathfrak{D}). \] We also see easily that these relations are linearly independent over $\Z$. Thus, since the number of generators is $1+\sum_{i=1}^{\lambda} \mu_i$, we get the result. \end{proof} \begin{corollary} \label{sympa} We have the inequality $\sum_{i=1}^{\lambda} \,\{\mu_i-1\} \leq 8$. Besides, if $\sum_{i=1}^{\lambda} \,\{\mu_i-1\}=8$, every automorphism of $X$ acts finitely on $\mathrm{NS}\,(X)$. \end{corollary} \begin{proof} We remark that $\mathcal{N}$ lies in ${K}_X^{\perp}$, which is a lattice of rank $9$ in $\mathrm{Pic}\,(X)$. This yields the inequality $\sum_{i=1}^{\lambda} \,(\mu_i-1) \leq 8$. \par
Assume $\mathcal{N}=K_X^{\perp}$, and let $f$ be an automorphism of $X$. Up to replacing $f$ by an iterate, we can assume that $\mathcal{N}$ is fixed by $f$. Thus $f^*$ is a parabolic translation leaving the orthogonal of the isotropic invariant ray $\mathbb{R} K_X$ pointwise fixed. It follows that $f$ acts trivially on $\mathrm{Pic}\,(X)$. \end{proof} Lastly, we prove that there is a major dichotomy among Halphen surfaces. Since there is no proof of this result in Gizatullin's paper, we provide one for the reader's convenience. \par
Let us introduce some notation: let $\mathrm{Aut}_0(X)$ be the connected component of $\mathrm{id}$ in $\mathrm{Aut}\, (X)$ and $\widetilde{\mathrm{Aut}}\, (X)$ be the group of automorphisms of $X$ preserving fiberwise the elliptic fibration. \begin{proposition}[see {\cite[Prop. B]{GIZ}}] \label{waza} Let $X$ be a Halphen surface. Then $X$ has at least two degenerate fibers. The following are equivalent: \begin{enumerate} \item[(i)] $X$ has \textit{exactly} two degenerate fibers. \item[(ii)] $\mathrm{Aut}_0(X)$ is an algebraic group of positive dimension. \item[(iii)] $\widetilde{\mathrm{Aut}}\, (X)$ has infinite index in $\mathrm{Aut}\, (X)$. \end{enumerate} Under any of these conditions, $\mathrm{Aut}_0(X) \simeq\C^{\times}$, and $\widetilde{\mathrm{Aut}}\, (X)$ is finite, and $\mathrm{Aut}_0(X)$ has finite index in $\mathrm{Aut}\, (X)$. \end{proposition}
\begin{proof} Let $\mathcal{Z}$ be the finite subset of $\mathbb{P}^1$ consisting of points $z$ such that $\pi$ is not smooth at some point of the fiber $X_z$, and $U$ be the complementary set of $\mathcal{Z}$ in $\mathbb{P}^1$. The points of $\mathcal{Z}$ correspond to the degenerate fibers of $X$.
\par
Let $\mathcal{M}_1$ be the moduli space of elliptic curves, considered as a complex orbifold. It is the quotient orbifold $\mathfrak{h} / \mathrm{SL}(2; \mathbb{Z})$ and its coarse moduli space $|\mathcal{M}_1|$ is $\C$. The elliptic surface over $U$ yields a morphism of orbifolds $\phi \colon U \rightarrow \mathcal{M}_{1}$, hence a morphism $| \phi | \colon U \rightarrow \mathbb{C}$. The orbifold universal cover of $\mathcal{M}_{1}$ is $\mathfrak{h}$, so that $|\phi|$ induces a holomorphic map $\widetilde{U} \rightarrow \mathfrak{h}$.
\par
If $\# \mathcal{Z} \in \{0, 1, 2\}$, then $\widetilde{U}=\mathbb{P}^1$ or $\widetilde{U}=\C$ and $|\phi|$ is constant. This means that all fibers of $X$ over $U$ are isomorphic to a fixed elliptic curve $E$. \par
Let $H$ be the isotropy group of $\mathcal{M}_1$ at $E$, it is a finite group of order $2$, $4$ or $6$. Then $\phi$ factorizes as the composition $U \rightarrow {B}H \rightarrow \mathcal{M}_1$ where ${B} H$ is the orbifold $\bullet_{H}$. The stack morphisms from $U$ to ${B}H$ are simply $H$-torsors on $U$, and are in bijection with $\mathrm{H}^1(U, H)$. \par
In the case $\# \mathcal{Z} \in \{0, 1\}$, that is $U=\mathbb{P}^1$ or $U=\mathbb{C}$, then $\mathrm{h}^1(U, H)=0$. Thus $X$ is birational to $E \times \mathbb{P}^1$ which is not possible for rational surfaces. This proves the first part of the theorem. \par
(iii) $\Rightarrow$ (i)
We have an exact sequence \[ 0 \rightarrow \widetilde{\mathrm{Aut}}\,(X) \rightarrow \mathrm{Aut}\,(X) \xrightarrow{\kappa} \mathrm{Aut}\, (\mathbb{P}^1) \] The image of $\kappa$ must leave the set $\mathcal{Z}$ globally fixed. If $\# \mathcal{Z} \geq 3$, then the image of $\kappa$ is finite, so that $\widetilde{\mathrm{Aut}}\,(X)$ has finite index in $\mathrm{Aut}\,(X)$. \par
(i) $\Rightarrow$ (ii) In this situation, we deal with the case $U=\C^{\times}$. The group $\mathrm{H}^1(\mathbb{C}^{\times}, H)$ is isomorphic to $H$. For any element $h$ in $H$, let $n$ be the order of $h$ and $\zeta$ be a $n$-th root of unity. The cyclic group $\ensuremath{\mathbb Z}/ n \Z$ acts on $\mathbb{C}^{\times} \times E$ by the formula $p.(z, e)=(\zeta^{p} z, h^p . e)$. The open elliptic surface over $\mathbb{C}^{\times}$ associated with the pair $(E, h)$ is the quotient of $\mathbb{C}^{\times} \times E$ by $\Z/n\Z$. We can compactify everything: the elliptic surface associated with the pair $(E, h)$ is obtained by desingularizing the quotient of $\mathbb{P}^1 \times E$ by the natural extension of the $\ensuremath{\mathbb Z}/ n\Z$-action defined formerly. By this construction, we see that the $\mathbb{C}^{\times}$ action on $\pi^{-1}(U)$ extends to $X$. Thus $\mathrm{Aut}_0(X)$ contains $\C^{\times}$. \par
(i) $\Rightarrow$ (iii) We have just proven in the previous implication that if $X$ has two degenerate fibers, then the image of $\kappa$ contains $\C^{\times}$. Therefore $\widetilde{\mathrm{Aut}}\,(X)$ has infinite index in $\mathrm{Aut}\,(X)$. \par
(ii) $\Rightarrow$ (i) We claim that $\widetilde{\mathrm{Aut}}\, (X)$ is countable. Indeed, $\widetilde{\mathrm{Aut}}\, (X)$ is a subgroup of $\mathrm{Aut}\, (\mathfrak{X}/ \C(t))$ which contains $\mathrm{Pic}\, (\mathfrak{X}/ \C(t))$ as a finite index subgroup; and $\mathrm{Pic}\, (\mathfrak{X}/ \C(t))$ is a quotient of $\mathrm{Pic}\,(X)$ which is countable since $X$ is rational. Therefore, if $\mathrm{Aut}_0(X)$ has positive dimension, then the image of $\kappa$ is infinite. The morphism $|\phi| \colon U \rightarrow \mathbb{C}$ is invariant by the action of $\mathrm{im}\, (\kappa)$, so it must be constant. As we have already seen, this implies that $X$ has two degenerate fibers. \par
It remains to prove the last statement of the Proposition. Since $\widetilde{\mathrm{Aut}}\,(X)$ is a countable group, $\widetilde{\mathrm{Aut}}\,(X) \cap \mathrm{Aut}_0 (X)=\{ \mathrm{id} \}$. Thus, $\mathrm{Aut}_0(X) \simeq \kappa \left(\mathrm{Aut}_0(X) \right) \simeq \mathbb{C}^{\times}$. Let $\varepsilon$ denote the natural representation of $\mathrm{Aut}\,(x)$ in $\mathrm{NS}(X)$. Since $\mathrm{Aut}_0(X) \subset \mathrm{ker}\,(\varepsilon)$, $\mathrm{ker}\,(\varepsilon)$ is infinite. Thanks to \cite{HH}, $\mathrm{im}(\varepsilon)$ is finite. To conclude, it suffices to prove that $\mathrm{Aut}_0(X)$ has finite index in $\mathrm{ker}\,(\varepsilon)$. Any smooth curve of negative self-intersection must be fixed by $\mathrm{ker}\,(\varepsilon)$. Let $\mathbb{P}^2$ be the minimal model of $X$ (which is either $\mathbb{P}^2$ or $\mathbb{F}_n$) and write $X$ as the blowup of $\mathbb{P}^2$ along a finite set $Z$ of (possibly infinitly near) points. Since $\mathrm{Aut}_0(\mathbb{P}^2)$ is connected, $\mathrm{ker}\,(\varepsilon)$ is the subgroup of elements of $\mathrm{Aut}\,(\mathbb{P}^2)$ fixing $Z$. This is a closed algebraic subgroup of $\mathrm{Aut}\,(\mathbb{P}^2)$, so $\mathrm{ker}\,(\varepsilon)_0$ has finite index in $\mathrm{ker}\,(\varepsilon)$. Since $\mathrm{ker}\,(\varepsilon)_0=\mathrm{Aut}_0(X)$, we get the result.
\end{proof}
\begin{remark} Minimal elliptic surfaces with two degenerate fibers are called Gizatullin surfaces, they are exactly the rational surfaces possessing a nonzero regular vector field. They are Halphen surfaces of index $1$, their detailed construction is given in \cite[\S 2]{GIZ}. They have two reducible fibers $S_1$ and $S_2$ which satisfy $\mu_1+ \mu_2=10$, and $\mathrm{Aut}_0(X)$ has always finite index in $\mathrm{Aut}\,(X)$. \end{remark}
\subsection{The main construction} \label{7.3} In this section, we construct explicit parabolic automorphisms of Halphen surfaces. \begin{theorem} \label{penible} Let $X$ be a Halphen surface such that $\sum_{i=1}^{\lambda} \,\{\mu_i-1\} \leq 7$. Then there exists a free abelian group $G$ of finite index in ${\mathrm{Aut}}\,(X)$ of rank $8-\sum_{i=1}^{\lambda} \,\{\mu_i-1\}$ such that any non-zero element in $G$ is a parabolic automorphism acting by translation on each fiber of the fibration. \end{theorem} \begin{proof}Let $\widetilde{\mathrm{Aut}}(X)$ be the subgroup of $\mathrm{Aut}\,(X)$ corresponding to automorphisms of $X$ preserving the elliptic fibration fiberwise. By \cite[Chap. II \S 10 Thm.1]{DS}, any automorphism of $\mathfrak{X}$ defined over $\mathbb{C}(t)$ extends to an automorphism of $X$. Thus $\widetilde{\mathrm{Aut}}\,(X)=\mathrm{Aut} (\mathfrak{X}/ \mathbb{C}(t))$. \par
Since $\mathfrak{X}$ is a smooth elliptic curve, $\mathrm{Pic}_0 \{\mathfrak{X} / \mathbb{C}(t)\}$ has finite index in $\mathrm{Aut} (\mathfrak{X}/ \mathbb{C}(t))$, so that $\mathrm{Pic}_0 \{\mathfrak{X} / \mathbb{C}(t)\}$ has finite index in $\widetilde{\mathrm{Aut}}\,(X)$. \par
The trace morphism $\mathfrak{t} \colon \mathrm{Pic}\,(X) \rightarrow \mathrm{Pic} \{\mathfrak{X} / \mathbb{C}(t)\}$ is surjective and for any divisor $D$ in $\mathrm{Pic}\,(X)$ we have $\mathrm{deg}\, \mathfrak{t}(D)=D.C$. Therefore \[ K_X^{\perp}/ \mathcal{N} \simeq \mathrm{Pic}_0 \{\mathfrak{X} / \mathbb{C}(t)\} \hookrightarrow \widetilde{\mathrm{Aut}}\,(X) \] where the image of the last morphism has finite index. By Proposition \ref{sept}, the rank of $\mathcal{N}$ is $\sum_{i=1}^{\lambda} (\mu_i-1) +1$, which is smaller that $8$. Let $G$ be the torsion-free part of ${K_X^{\perp}}/{\mathcal{N}}$; the rank of $G$ is at least one. Any $g$ in $G$ acts by translation on the generic fiber $\mathfrak{X}$ and this translation is of infinite order in $\mathrm{Aut}\, (\mathfrak{X})$. Beside, via the morphism $\mathrm{Pic}\,(X) \rightarrow \mathrm{Pic}\,(\mathfrak{X})$, $g$ acts by translation by $\mathfrak{tr}\,(g)$ on $\mathrm{Pic}\,(\mathfrak{X})$, so that the action of $g$ on $\mathrm{Pic}\,(X)$ has infinite order.
\par
Let $g$ in $G$, and let $\lambda$ be an eigenvalue of the action of $g$ on $\mathrm{Pic}\, (X)$, and assume that $|\lambda| > 1$. If $g^*v=\lambda v$, then $v$ is orthogonal to $K_X$ and $v^2=0$. It follows that $v$ is collinear to $K_X$ and we get a contradiction. Therefore, $g$ is parabolic. \par
To conclude the proof it suffices to prove that $ \widetilde{\mathrm{Aut}}\,(X)$ has finite index in ${\mathrm{Aut}}\,(X)$. Assume the contrary. Then Proposition \ref{waza} implies that $X$ has two degenerate fibers, that is $X$ is a Gizatullin surface. In that case $\mu_1+\mu_2=10$ (by the explicit description of Gizatullin surfaces) and we get a contradiction.
\end{proof} \begin{corollary} \label{hapff} Let $X$ be a Halphen surface. The following are equivalent: \begin{enumerate} \item[(i)] $\sum_{i=1}^{\lambda} \{\mu_i-1\}=8$. \item[(ii)] The group $\widetilde{\mathrm{Aut}}(X)$ is finite. \item[(iii)] The image of $\mathrm{Aut}\, (X)$ in $\mathrm{GL}\!\left(\mathrm{NS}\,(X)\right)$ is finite. \end{enumerate} \end{corollary} \begin{proof} (i) $\Leftrightarrow$ (ii) Recall (see the proof of Proposition \ref{penible}) that $K_X^{\perp}/\mathcal{N}$ has finite index in $\widetilde{\mathrm{Aut}}\, (X)$. This gives the equivalence between (i) and (ii) since $K_X^{\perp}/\mathcal{N}$ is a free group of rank $8-\sum_{i=1}^{\lambda} \{\mu_i-1\}$. \par
(i) $\Rightarrow$ (iii) This is exactly Corollary \ref{sympa}. \par
(iii) $\Rightarrow$ (i) Assume that $\sum_{i=1}^{\lambda} \{\mu_i-1\} \leq 7$. Then $X$ carries parabolic automorphisms thanks to Theorem \ref{penible}. This gives the required implication. \end{proof} Let us end this section with a particular but illuminating example: \textit{unnodal Halphen surfaces}. By definition, an unnodal Halphen surface is a Halphen surface without reducible fibers. In this case, $\mathcal{N}$ is simply the rank one module $\Z K_X$, so that we have an exact sequence \[ 0 \rightarrow \Z K_X \rightarrow K_X^{\perp} \underset{\lambda}{\hookrightarrow} {\mathrm{Aut}}\, (X) \] where the image of the last morphism has finite index. Then:
\begin{theorem} \label{classieux} For any $\alpha$ in $K_X^{\perp}$ and any $D$ in $\mathrm{NS}\, (X)$, \[ \lambda_{\alpha}^*(D)=D-m\,(D.K_X)\, \alpha+\left\{m\,(D. \alpha)-\frac{m^2}{2} (D.K_X)\, \alpha^2 \right\} K_X. \] \end{theorem} \begin{proof} We consider the restriction map $\mathfrak{t} \colon \mathrm{Pic}\, (X) \rightarrow \mathrm{Pic}(\mathfrak{X}/\C(t))$ sending $K_X^{\perp}$ to $\mathrm{Pic}_0(\mathfrak{X}/\C(t))$. Then $\mathfrak{t}({\alpha})$ acts on the curve $\mathfrak{X}$ by translation, and also on the Picard group of $\mathfrak{X}$ by the standard formula \[ \mathfrak{t}({\alpha})^* (\mathfrak{Z})=\mathfrak{Z}+ \mathrm{deg}\,(\mathfrak{Z})\, \mathfrak{t}({\alpha}). \] \par
Applying this to $\mathfrak{Z}=\mathfrak{t}(D)$ and using the formula $\mathrm{deg}\, \mathfrak{t}(D)=-m\,(D.K_X)$, we get \[ \mathfrak{t}\left(\mathfrak{\lambda}_{\alpha}^* (D)\right)=\mathfrak{t}(D)-m\, (D.K_X) \, \mathfrak{t}(\alpha). \] Hence there exists an integer $n$ such that \[ \mathfrak{\lambda}_{\alpha}^* (D)=D-m\, (D.K_X)\, \alpha + n\, K_X. \] Then \[ \mathfrak{\lambda}_{\alpha}^* (D)^2=D^2-2m \,(D.K_X)\,(D. \alpha)+m^2\, (D.K_X)^2\,\alpha^2+2n\, (D.K_X). \] \par
We can assume without loss of generality that we have $(D.K_X)\neq 0$ since $\mathrm{Pic}\,(X)$ is spanned by such divisors $D$. Since $\mathfrak{\lambda}_{\alpha}^* (D)^2=D^2$, we get \[ n=m\, (D. \alpha) -\frac{m^2}{2} \,(D.K_X) \,\alpha^2. \] \end{proof}
\end{document} | arXiv |
Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise
Algebraic limit cycles for quadratic polynomial differential systems
August 2018, 23(6): 2487-2498. doi: 10.3934/dcdsb.2018066
A new flexible discrete-time model for stable populations
Eduardo Liz
Departamento de Matemática Aplicada Ⅱ, Universidade de Vigo, 36310 Vigo, Spain
Received May 2017 Published August 2018 Early access February 2018
Fund Project: This research has been supported by the Spanish Government and FEDER, under grant MTM2013-43404-P.
Figure(5)
We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.
Keywords: Discrete population model, global stability, density dependence, Allee effects, bifurcations of fixed points, extinction.
Mathematics Subject Classification: 39A10, 92D25.
Citation: Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066
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Figure 1. Different graphs of the map $f$ defined in (1.1). (a): $f$ is unimodal for $\gamma<1$; (b): $f$ is increasing for $\gamma = 1$, with a unique positive fixed point if $\beta>1$; (c) and (d): $f$ is increasing for $1<\gamma<2$, and can have 0, 1, or $2$ positive fixed points; (e): $f$ is increasing and convex for $\gamma = 2$, with linear growth at infinity; it has a unique positive fixed point if $\beta>\delta$ and no positive fixed points if $\beta\leq\delta$; (f): $f$ is increasing and convex, with superlinear growth at infinity, if $\gamma>2$. In all cases, the red dashed line represents the graph of $y = x$
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Figure 2. Graph of the map $\beta = F_{\delta}(\gamma)$ showing the survival/extinction switches for (1.1), which only occur if $\beta<1+\delta$
Figure 3. Relative position of the graphs of $f_1(x) = \beta x^{\gamma-1}$ (red color) and $f_2(x) = 1+\delta x$ (blue color) when equation $f_1(x) = f_2(x)$ has two positive solutions
Figure 4. Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria, which, in case of bistability, establish the boundary between the basins of attraction of the extinction equilibrium 0 and the nontrivial attractor $p$. (a): $\beta = 3, \delta = 1$; (b): $\beta = 2, \delta = 1$; (c): $\beta = 2, \delta = 1.5$; (b): $\beta = 2, \delta = 2.5$. Each case is an example of the corresponding case in Theorem 5.1
Figure 5. Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria. (a): $\beta = 0.9, \delta = 1.5$; (b): $\beta = 0.9, \delta = 0.5$
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Evaluation of body-surface-area adjusted dosing of high-dose methotrexate by population pharmacokinetics in a large cohort of cancer patients
Usman Arshad1,2 na1,
Max Taubert1 na1,
Tamina Seeger-Nukpezah3,
Sami Ullah1,2,
Kirsten C. Spindeldreier4,
Ulrich Jaehde2,
Michael Hallek3,
Uwe Fuhr1,
Jörg Janne Vehreschild3,5,6 &
Carolin Jakob3
The aim of this study was to identify sources of variability including patient gender and body surface area (BSA) in pharmacokinetic (PK) exposure for high-dose methotrexate (MTX) continuous infusion in a large cohort of patients with hematological and solid malignancies.
We conducted a retrospective PK analysis of MTX plasma concentration data from hematological/oncological patients treated at the University Hospital of Cologne between 2005 and 2018. Nonlinear mixed effects modeling was performed. Covariate data on patient demographics and clinical chemistry parameters was incorporated to assess relationships with PK parameters. Simulations were conducted to compare exposure and probability of target attainment (PTA) under BSA adjusted, flat and stratified dosing regimens.
Plasma concentration over time data (2182 measurements) from therapeutic drug monitoring from 229 patients was available. PK of MTX were best described by a three-compartment model. Values for clearance (CL) of 4.33 [2.95–5.92] L h− 1 and central volume of distribution of 4.29 [1.81–7.33] L were estimated. An inter-occasion variability of 23.1% (coefficient of variation) and an inter-individual variability of 29.7% were associated to CL, which was 16 [7–25] % lower in women. Serum creatinine, patient age, sex and BSA were significantly related to CL of MTX. Simulations suggested that differences in PTA between flat and BSA-based dosing were marginal, with stratified dosing performing best overall.
A dosing scheme with doses stratified across BSA quartiles is suggested to optimize target exposure attainment. Influence of patient sex on CL of MTX is present but small in magnitude.
Methotrexate (MTX) is considered an efficacious, cost-effective and acceptably safe drug for the treatment of many hematological/oncological disorders and autoimmune diseases [1]. The folate analogue MTX acts as an antineoplastic agent via competitive inhibition of dihydrofolate dehydrogenase, resulting in depletion of purines and thymidylate leading to impairment of DNA synthesis [2, 3]. The drug can be administered via multiple routes of administrations and has a wide variation in dosing regimens including low (< 50 mg/m2), intermediate (50–500 mg/m2) and high (> 500 mg/m2) dose regimens [1, 4]. The pronounced inter-individual variability (IIV) of PK and toxicity of MTX [5,6,7] renders individualization of dosing regimens difficult.
Hepatic metabolism accounts for a considerably lower fraction of its clearance (CL) compared to renal elimination, as the main fraction (80–90%) of the drug is primarily eliminated via glomerular filtration and active tubular secretion [8, 9]. Nephrotoxicity associated with MTX impairs its CL, leading to further aggravation of toxicity such as myelosuppression and mucositis. In subjects with extracellular fluid accumulations, the drug has been shown to undergo delayed elimination [10]. A recent in vitro study by Euteneuer et al. [11] showed a sex-dependent regulation of renal transport proteins, which might play a role in the CL of MTX. To handle the variability associated with MTX exposure, monitoring of its plasma concentrations (therapeutic drug monitoring, TDM) and serum creatinine (SCr) is recommended to safeguard a relatively constant drug exposure with an acceptable risk/benefit ratio particularly in patients with impaired renal function [12]. Furthermore, MTX dosing is often guided by body surface area (BSA) estimates to account for body size-related differences in CL and volume of distribution (V). However, concerns regarding potential under- and over-exposure in certain patient groups, such as with obesity, have been expressed [13]. BSA is furthermore a highly variable measure that depends on the arbitrary choice of a BSA equation [14]. Thus, further clarification of the clinical implications of BSA based dosing for MTX is required.
Modeling of PK data has the potential to optimize TDM, where tailored dose adjustments can be made according to model-predicted concentrations of a drug [15]. Bayesian population PK analysis has been used to assist TDM guided dose adjustments for MTX [15]. In addition, population PK analysis provides the possibility to identify and quantify covariate effects on drug exposure [16, 17]. This may provide a better understanding of drug's pharmacology and assist adjustments in dosage regimen according to patient's individual characteristics e.g., renal/hepatic function, genotype of drug metabolizing enzymes or transporters, and/or anthropometric characteristics. Models capturing covariate relationships have been found useful in oncology for individualized dose adaptations such as in case of busulfan, topotecan and docetaxel [16].
The current study was aimed to identify and evaluate covariates influencing PK of MTX, particularly patient sex and body surface area (BSA), by developing a population PK model using the TDM data collected from patients with hematological and solid malignancies. The model was further aimed to be used for the evaluation of the ongoing clinical practice of administering MTX based on individual BSA via a simulation study.
Patients, treatment and sampling
MTX plasma concentration and covariate data was obtained from the Cologne Cohort of Neutropenic Patients (CoCoNut) [18]. Experimental protocols were approved by the local ethics committee (name and email address: Ethics Committee of the Faculty of Medicine, University of Cologne, Cologne, Germany, [email protected]; date of approval: 14.01.2014, approval file number: 13–108). All methods were performed in accordance with the local and international guidelines and regulations. Data from neutropenic patients (neutrophils < 500 /mm3) with hematological malignancies or solid tumors and treated with high-dose MTX at the Department I of Internal Medicine, University Hospital of Cologne, between January 2005 and February 2018 were considered. The data from clinical laboratory was imported via Health Level Seven from the laboratory information system. The dosing information was imported from the integrated software for chemotherapy using a csv export. Further patient characteristics were documented manually in the CoCoNut database.
MTX was administered via 4 h or 24 h intravenous infusions depending on underlying malignancy. TDM was routinely performed at 42 h and 48 h post-dose for both the 4 h and 24 h protocols, while an additional sample was scheduled for 4 h MTX infusion at 24 h. If target plasma concentration exceeded the desired thresholds (> 1 μmol/L at 42 h and > 0.3 μmol/L at 48 h), TDM was performed at least every 6 h. These thresholds reflect the internal guidance document developed to translate the available heterogenous evidence [10, 19] to an actionable recommendation also appropriate for the organisational conditions in our hospital. On the same basis, for the 24 h MTX infusion, leucovorin rescue was routinely performed with 30 mg/m2 (after 42 h and 48 h) and 15 mg/m2 (after 54 h and 60 h). If the desired plasma concentration of MTX was not reached, leucovorin was administered every 6 h at a dose (mg) equivalent to the product of MTX plasma concentration (μmol/L) and body patient weight (kg).
MTX plasma concentrations were quantified using competitive immunoassays with 0.009 μmol/L as the lower limit of quantification (LLOQ). Demographic covariates included patient's age, sex, weight and height. Covariate data from clinical chemistry analysis included SCr, plasma total bilirubin (BT), γ-glutamyltransferase (GGT), uric acid concentrations, absolute leukocyte counts (WBC), and BSA.
Dosing, concentration and covariate data was subjected to screening prior to PK analysis. R (version 3.5.1) was used to prepare the dataset for model development. Dataset preparation was assisted by visual inspection of individual concentration time profiles. Patients with missing dosing information at treatment initiation were identified for exclusion from subsequent analysis. Subjects with missing dosing information during the treatment were flagged and concentration measurements at time points subsequent to the missing dosing information were excluded. Due to the significant amount of missing covariate data throughout the treatment course, the covariate evaluation was based on baseline covariate data for the start of treatment.
PK model development
Data were analyzed by the nonlinear mixed effects modeling approach using NONMEM 7.4.3 (ICON, Development Solutions, Elliot City, MD, USA). Perl speaks NONMEM (PsN), Pirana and Xpose4 were used to assist model development, evaluation and post processing [20,21,22]. Structural model development. A combination of iterative two-stage (ITS) and first order conditional estimation with interaction (FOCE-I) methods was applied for parameter estimation. Likelihood ratio tests (LRT) or the Akaike information criterion (AIC) were used for the evaluation of nested and non-nested models, respectively. A nested model with fewer parameters or a decrease in objective function value (OFV) by 3.84 (i.e., p < 0.05, one degree of freedom) was given preference. The model with a lower AIC value in case of non-nested models was preferred.
Model evaluation criteria comprised of plausibility of parameter estimates, reduction in unexplained and residual variability, shrinkage and precision in parameter estimates. Visual inspection through goodness of fit (GOF) plots included observed versus individual/population predicted concentrations (IPRED/PRED) over time. Residual error models were evaluated with the help of conditional weighted residuals (CWRES) versus observed concentrations and versus time after first dose (TAFD). Numerical predictive checks (NPCs) were used for further assessment by comparing the empirical cumulative distribution function of the observed concentrations with the theoretical cumulative distribution, computed from simulated data.
Compartmental analysis was performed in a stepwise manner. IIV was incorporated using exponential terms (ηiiv) which describes the deviation of PK parameter values of an individual from the population estimate [17]. Interoccasion variability (IOV), defined as the variability between individual cycles of MTX therapy, was incorporated in the model via random effects (ηiov) [23]. The PK parameter P in a specific subject was parametrized as shown in Eq. 1.
$$ \mathrm{P}=\uptheta \times {\mathrm{e}}^{\upeta_{\mathrm{iiv}}+{\upeta}_{\mathrm{iov}}} $$
Where θ is a fixed effect, representing the median PK parameter in the population. Additive, proportional and combined error models were tested to estimate the residual unexplained variability (RUV).
Covariate model development
Covariate data was analyzed to identify covariate-parameter relationships. Covariate preselection was performed considering scientific plausibility as an essential criterion. Graphical evaluation of covariates was performed including CWRES vs covariate, empirical Bayes estimates (EBEs) versus covariate, and covariate versus covariate plots. Significance of covariate relationship was principally guided by decrement in OFV and/or unexplained variability. A stepwise covariate evaluation was carried out as follows. At each step, the covariate providing the largest reduction in OFV was included (forward inclusion) or the covariate providing the lowest increase in OFV was eliminated (backward elimination). Selection criteria were a ∆OFV of 3.84 (p < 0.05) for forward inclusion and a ∆OFV of 6.63 (p < 0.01) for backward elimination.
Continuous covariates were included as linear relationships (Eq. 2) or power relationships (Eq. 3) centered around their median values. BSA effect was centered around the typical value of 1.73 m2.
$$ {\mathrm{Covariate}}_{\mathrm{effect}}=1+\left({\mathrm{Covariate}}_{\mathrm{i}}-{\mathrm{Covariate}}_{\mathrm{median}}\right)\times {\uptheta}_{\mathrm{Covariate}} $$
$$ {\mathrm{Covariate}}_{\mathrm{effect}}={\left(\raisebox{1ex}{${\mathrm{Covariate}}_{\mathrm{i}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{Covariate}}_{\mathrm{median}}$}\right.\right)}^{\uptheta_{\mathrm{Covariate}}} $$
Categorical relationships were given as Covariateeffect = 1 + Covariatei × θCovariate, where Covariatei is the individual covariate value in the ith subject and θCovariate represents the effect size of the covariate relationship to a PK parameter. Covariate inclusion and evaluation criteria are presented in the supplementary material.
Evaluation of BSA-based, flat and stratified dosing regimens
Stochastic simulations were designed using the final model, including covariates, for the comparative evaluation of drug exposure under BSA-based (linear scaling using BSA), flat and stratified dosing 24 h infusion regimens. Stratified dosing regimens comprised of 3 BSA-based stratifications i.e., subjects at lower and upper BSA extremes (< 25th and > 75th BSA percentiles) as well as the middle (25th–75th percentile) proportion of population. No further differences in virtual patient characteristics were part of the simulated populations. The three sets of simulated populations differed only in the administered dosing regimens. Target for an adequate dosing regimen included two criteria. First, plasma concentrations should not exceed 1.0 and 0.3 μmol/L at 42 h and 48 h after the start of infusion, respectively, based on the current TDM protocol at the University Hospital, Cologne. Second, the achieved AUC should be in the range of ±30% of the AUC of a subject with a typical BSA of 1.73m2.
Patient and treatment characteristics
In total, 229 cancer patients (83 females) with 2182 plasma concentration measurements were included in the PK analysis. The majority of patients received 4 h and 24 h infusions, while 18 patients occasionally received 12 h and 48 h infusions. Only a single patient received a 72 h infusion. A median of 3 dosing cycles (range, 1–9) per patient were part of the available data. The number of plasma concentration measurements per patient ranged from 1 to 65 with a median of 7 measurements. Patient and clinical laboratory parameters are summarized in Tables 1 and 2.
Table 1 Population characteristics. Median and range for measured values are shown
Table 2 Population disease characteristics
PK model
A three-compartment model with linear elimination adequately described MTX plasma concentrations (Supplementary Figure 1 & 2). We decided to use a linear CL model instead of a model with an additive nonlinear CL component (combined model) for the subsequent evaluations, although the latter provided a better fit with (∆OFV of − 70 points). The fraction of CL contributed by the linear component in the combined model was 4.77 L/h, whereas nonlinear CL solely contributed 0.42 L/h at median MTX concentrations (2.20 μmol/L). Furthermore, run times were distinctly longer (~ 60 h compared to ~ 1 h), preventing from a proper covariate analysis, and parameter estimation was unstable. Estimates from the combined model with linear and nonlinear CL components are presented in the Supplementary Table.
RUV was appropriately described by a combined (additive and exponential) error model. Mean PK parameters with 95% CI and RSE obtained from the bootstrap analysis (1000 samples) are presented in Table 3.
Table 3 Population PK parameter estimates from bootstrap analysis
Covariate analysis
SCr was found to be a significant covariate on CL with an OFV reduction by 191. Inclusion of patient's sex and age on CL further improved the model fit (∆OFVs of 32.0 and 13.0, respectively). Inclusion of BSA provided a significant reduction in OFV by 4.40 on CL. A 16% [7–25%] lower CL was estimated in females. Reduction in IIV of individual parameters was limited, with a decrease in 2.40, 0.56 and 1.44 (%) after inclusion of SCr, age and sex, respectively. IIV and IOV on CL in the covariate model were 29.7 and 23.1%, respectively. The resulting equation for the individual CL (CLi) is shown in Eq. 4.
$$ {\mathrm{CL}}_{\mathrm{i}}=4.52{\left(\raisebox{1ex}{${\mathrm{SCr}}_{\mathrm{i}}$}\!\left/ \!\raisebox{-1ex}{$0.74$}\right.\right)}^{-0.49}{\left(\raisebox{1ex}{${\mathrm{Age}}_{\mathrm{i}}$}\!\left/ \!\raisebox{-1ex}{$58$}\right.\right)}^{-0.18}{\left(\raisebox{1ex}{${\mathrm{BSA}}_{\mathrm{i}}$}\!\left/ \!\raisebox{-1ex}{$1.73$}\right.\right)}^{-0.23}\left(1+{\mathrm{Sex}}_{\mathrm{i}}\times -0.16\right) $$
Where, sex was coded as 0 for males and 1 for females. Estimates for covariate relationships are summarized in Table 3.
BSA-based versus flat and stratified dosing regimens
Figures 1 and 2 presents the distribution of AUC and plasma concentrations respectively, in the virtual population stratified by BSA quartiles for BSA-based, flat and stratified dosing regimens. A gradual increase in MTX AUC with increase in BSA was associated with BSA-based regimen, while the contrary was observed with flat dosing regimen. Stratified dosing displayed a consistent AUC across all the BSA quartiles. Concerning the decline of MTX concentrations until 42 and 48 h postdose, the higher clearance for higher BSA values more than compensated for the concentration differences between BSA-based and stratified dosing just at the end of the infusions.
Distribution of simulated area under the curve (AUC; median with 95% CI) across body surface area (BSA) quartiles for BSA-based, flat and stratified dosing. Description of doses under each regimen is presented in Table 4
Distribution of simulated plasma concentrations (median with 95% CI) across body surface area (BSA) quartiles for BSA-based, flat and stratified dosing. Dashed horizontal lines represent the desired threshold plasma concentrations (< 1 μmol/L at 42 h post-dose and < 0.3 μmol/L at 48 h post-dose). Description of doses under each regimen is presented in Table 4
The percentage of subjects attaining both the target criteria (probability of target attainment; PTA) was calculated for dose levels of 500, 1000 and 2000 mg/m2 (reference dose). An optimized flat dosing regimen, i.e. a regimen in which each subject received the same dose, was identified by simulating a range of doses and choosing the dose that provided the highest PTA. Subsequently, the procedure was repeated with doses stratified according to the BSA groups (lower extreme: < 25%, middle proportion: 25–75% and upper extreme > 75%) and the above-mentioned dose optimization was repeated for each of the three BSA regions separately. Thus, the stratified dosing approach resulted in three separate doses, corresponding to the three defined BSA groups. Figure 3 presents the PTA across the BSA groups for respective dose levels under BSA-based, flat and stratified dosing regimens. Stratified dosing provided marginally higher PTA for both the upper and lower BSA extremes compared to BSA-based and flat dosing, respectively. Based on simulation results, selection of doses with the highest PTA (comparable to BSA-based doing regimen) identified under flat and stratified dosing regimens are presented in Table 4.
Probability of target attainment (PTA) across body surface area (BSA) groups under BSA-based, flat and stratified dosing regimens. Reference BSA-based dose levels range from 500 to 2000 mg/m2. Numbers in the bars represent respective PTA values. (0 = < 25%, 1 = 25–50%, 2 = 50–75%, 3 = > 75%). Description of doses under each regimen is presented in Table 4
Table 4 Selection of stratified doses with highest probability of target attainment (PTA) compared to that of the body surface area (BSA)-based and flat doses of MTX administered as 24 h continuous infusion
High-dose MTX is essential in cancer therapies despite its high toxicity. However, the management of delayed MTX elimination challenges clinicians to prevent potentially life-threatening MTX-associated toxicities. Further, the high toxicity can cause a premature termination of the MTX administration, which decreases its potential efficiency [10]. In this study, we investigated the optimization of MTX dose adjustment as a potential factor to reduce MTX toxicity.
A three-compartment PK model of MTX is presented. Patient sex, age, BSA and SCr were related to CL. A 16% lower CL was estimated for females compared to males. Simulations using the final covariate model support dosing stratified for BSA quartiles.
The identification of clinically relevant covariates has been the main objective of numerous population PK evaluations of MTX, providing inconsistent findings on covariate relationships [24,25,26,27,28,29,30,31,32,33,34] . In contrast to our study, several previously published studies did not support a sex effect [24, 30,31,32,33,34]. Apart from differences in sample sizes, the particular combination of covariates in the model might have contributed to this inconsistency. For example, the inclusion of sex in a model containing SCr effect provided a distinct model improvement (OFV reduced by 32.0 points). In comparison, only a marginal improvement (OFV reduced by 4.40 points) resulted in the univariate evaluation (i.e., without considering any other covariates). This finding supports that a sex effect should be considered to account for differences in creatinine generation rates in male and female subjects. To quantify the contribution of sex effects beyond renal function, data on urinary excretion might be useful. However, such data was not available for patients in our database. The effect of sex on CL needs further investigation.
Age was related to MTX CL in a few studies [30, 32] while inconsistencies exist in the majority of studies [25, 34,35,36,37,38]. Some studies presented the influence of body weight and patient's age on both the CL and V of MTX [31, 33]. Mei et al. showed that V of MTX increased with increasing age and supported the preference of age over body weight as a covariate influencing V. A relationship between weight and V was reported by some other studies as well [30, 31, 34, 39, 40]. Age was found to be significant on CL in our study with a ∆OFV of 13.0.
SCr was the most significant covariate with a ∆OFV of 191. This is in line with other studies, where MTX elimination was significantly correlated with SCr [34, 41, 42]. The observed effect is physiologically plausible as MTX is primarily eliminated by the kidney [24]. Nevertheless, the covariate relationship between SCr concentrations and MTX CL faces disagreements in other studies [30, 31, 33, 37].
The covariate analysis in the present evaluation was based on baseline covariate information due to missing covariate data during the treatment time course for a significant number of patients. The development of covariate models incorporating time-varying covariate data is a useful approach in general, as it may provide a better explanation of IIV and IOV of PK parameters and thereby improve the predictive ability of the model. This might be of particular interest if a population PK model is used for Bayesian TDM over a prolonged treatment course. However, we intended to use covariate data mainly to generate an initial idea on expected concentrations before administering the first dose. During the treatment course, concentration data becomes available, and covariate data is of less clinical relevance. If improper imputation methods are applied, a misspecified model and distorted predictions might result. Thus, we believe that a model comprising solely baseline covariate data provides advantages and might therefore be preferable over a model with time-varying covariate data.
Preference of BSA-based dosing over flat dosing or based on other measures, such as patient genotype / phenotype, is an ongoing debate. In contrast to the BSA-based dosing, flat dosing is proposed for several anticancer drugs where BSA has been shown not to reduce the random PK variability to a clinically relevant degree [43,44,45]. Apart from the simplified clinical handling of flat dosing, BSA-based dosing introduces additional uncertainties which are difficult to assess due to the arbitrary choice of BSA equation [46]. Therefore, BSA as a body size measure should ideally be avoided if precise dose calculations are intended. Furthermore, scaling doses with BSA is likely to provide implausibly low or high doses in subjects with exceptionally low or high BSA. Owing to the simplicity needed to implement body size-based dosing regimens in clinical practice, a direct, proportional relationship between BSA and dose is often assumed. This is contradictory to our current knowledge on physiology and PK and further adds to the uncertainties. Although MTX PK is linear, i.e. exhibits a proportional increase in exposure (in terms of AUC) with the increase in dose, it does not imply that MTX exposure is reciprocally proportional to BSA. In our study, the change of exposure attributable to BSA was smaller and we only observed relevant differences between BSA and flat dosing for patients with either very low or very high BSA. A stratified approach is a reasonable alternative to BSA-based dosing with individuals in the upper and lower BSA quartiles. A stratified approach is a reasonable alternative to BSA-based dosing with individuals in the upper and lower BSA quartiles. It is important to mention that the current findings are based on retrospective data and need to be further validated in a prospective study. It should be noted that these findings are conditional on the defined TDM target. The TDM target is a concentration threshold associated with overexposure, while no threshold for underexposure is currently available. To avoid possible underexposure, achieving 70–130% of the AUC for a subject with a typical BSA of 1.73m2 was used as an additional criterion in the simulation analysis. No pharmacokinetic/pharmacodynamic (PK/PD) target related to efficacy is part of the TDM at the University Hospital Cologne, and, to the best of our knowledge, no validated PK/PD target related to efficacy is currently available.
Apart from the covariate and BSA evaluation, a non-linear CL component was identified in this study. Non-linear elimination has been reported before and might be attributable to the transporter-mediated tubular secretion of MTX [47,48,49]. Despite the significant improvement of the model after inclusion of non-linear CL, the impact of the non-linear component on estimated exposure and the excess of TDM thresholds was negligible. Thus, non-linearity seems to be of minor clinical relevance in the current cohort of patients. This might change if additional targets, such as PK/PD targets related to efficacy, become available. In this case, the model with the non-linear component as presented in the Supplement might be re-evaluated. Furthermore, the non-linear CL component might have a more pronounced impact on PK in presence of genetic polymorphisms and when MTX is co-administered with substrates, inducers, or inhibitors of the relevant membrane transporters.
A major limitation of the current investigation is that the optimal target exposure regarding the efficacy of MTX in various malignancies is unknown. Data on minimum drug exposure needed to achieve a positive therapeutic outcome with minimal toxicity is currently scarce. Dedicated efforts are needed to draw conclusions based on the efficacy profile of the drug with respect to the underlying disease. Furthermore, TDM data is generally obtained from clinical practice and therefore provides a reduced data quality compared to clinical trial data. Although the data was checked carefully for inconsistencies, it cannot be precluded that errors in TDM procedures translate into model misspecifications.
A three-compartment model described PK of MTX. A lower CL estimated for the female patients needs to be investigated in future studies. Plasma SCr, patient age, sex and BSA were found additionally as statistically significant covariates on CL. Stratified MTX dosing can be a reasonable alternative to BSA guided dosing.
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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Open Access funding enabled and organized by Projekt DEAL.
Usman Arshad and Max Taubert are co-first authors.
Department I of Pharmacology, Faculty of Medicine and University Hospital Cologne, Center for Pharmacology, University of Cologne, Gleueler Str 24, 50931, Cologne, Germany
Usman Arshad, Max Taubert, Sami Ullah & Uwe Fuhr
Institute of Pharmacy, Clinical Pharmacy, University of Bonn, Bonn, Germany
Usman Arshad, Sami Ullah & Ulrich Jaehde
Department I of Internal Medicine, Faculty of Medicine and University Hospital Cologne, University of Cologne, Cologne, Germany
Tamina Seeger-Nukpezah, Michael Hallek, Jörg Janne Vehreschild & Carolin Jakob
Hospital Pharmacy, University Hospital Cologne, Cologne, Germany
Kirsten C. Spindeldreier
German Center for Infection Research (DZIF), Partner site Bonn-Cologne, Cologne, Germany
Jörg Janne Vehreschild
Department of Internal Medicine, Hematology and Oncology, Faculty of Medicine and University Hospital of Frankfurt, Goethe University Frankfurt, Frankfurt am Main, Germany
Usman Arshad
Max Taubert
Tamina Seeger-Nukpezah
Sami Ullah
Ulrich Jaehde
Michael Hallek
Uwe Fuhr
Carolin Jakob
U.A. and M.T. contributed in data analysis, developed the model and simulations, and finally wrote the manuscript. C.J. and K.S. curated the database. C.J., T.S., J.J.V. and U.F. contributed towards the planning of the study and reviewed/edited the manuscript. S.U. contributed in the simulation analysis. U.J. and M.H. contributed intellectual input during the analysis and manuscript development. The author(s) read and approved the final manuscript.
Correspondence to Usman Arshad.
In the study, data from the Cologne Cohort of Neutropenic Patients (CoCoNut) were used. CoCoNut was approved by the local ethics committee (name and email address: Ethics Committee of the Faculty of Medicine, University of Cologne, Cologne, Germany, [email protected]; date of approval: 14.01.2014, approval file number: 13–108). CoCoNut is registered as data integration center at ClinicalTrials.gov (ClinicalTrials.gov identifier: NCT01821456). All methods were performed in accordance with the local and international guidelines and regulations. No interventions were performed as part of this study. Data collection and storage were carried out on site using current techniques of privacy assurance. In this scenario, according to legal regulations patient informed and verbal consent is waived in the state of North Rhine-Westphalia, Germany, which is confirmed by the decision of the Ethics Committee of the Faculty of Medicine, University of Cologne, Cologne, Germany.
There are no financial and non-financial competing interests related to this study.
Supplementary Table. Bootstrap population PK parameter estimates of the combined linear and nonlinear model obtained from bootstrap analysis. Supplementary Figure 1. Goodness of fit plots; A: observed vs individual predicted (IPRED) concentration (mg/L); B: observed vs population predicted (PRED) concentrations; C: conditional weighted residuals (CWRES) vs population predicted concentrations; D: conditional weighted residuals vs time after first dose (TAFD). Concentrations are presented on log scale in the upper panel. Supplementary Figure 2. Numerical predictive check comparing each observation with its own simulated distribution: Continuous line is the empirical cumulative distribution function of the observed concentrations. Dashed line with shaded area is the predicted cumulative distribution with 95% prediction interval computed from simulated data.
Arshad, U., Taubert, M., Seeger-Nukpezah, T. et al. Evaluation of body-surface-area adjusted dosing of high-dose methotrexate by population pharmacokinetics in a large cohort of cancer patients. BMC Cancer 21, 719 (2021). https://doi.org/10.1186/s12885-021-08443-x
Received: 08 February 2021
Covariates | CommonCrawl |
\begin{document}
\title{On the Obata Theorem in a weighted Sasakian manifold} \author{$^{\ast }$Shu-Cheng Chang$^{1}$} \address{$^{1}$Department of Mathematics and Taida Institute for Mathematical Sciences (TIMS), National \ Taiwan University, Taipei 10617, Taiwan} \email{[email protected] } \author{$^{\ast \ast }$Daguang Chen$^{2}$} \address{$^{2}$Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China} \email{[email protected]} \author{$^{\ast }$Chin-Tung Wu$^{3}$} \address{$^{3}$Department of applied Mathematics, National Pingtung University, Pingtung 90003, Taiwan} \email{[email protected]} \thanks{$^{\ast }$Research supported in part by the MOST of Taiwan\\ $^{\ast \ast }$Research supported in part by NSFC grant No. 11831005/11571360.}
\begin{abstract} In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obtain the CR\ Obata theorem in a compact weighted Sasakian manifold with or without boundary. \end{abstract}
\subjclass{Primary 32V05, 32V20; Secondary 53C56.} \keywords{Weighted CR Obata Theorem, Sasakian manifold, CR Dirichlet eigenvalue, Weighted CR Reilly formula, Bakry-Emery pseudohermitian Ricci curvature, Witten sub-Laplacian.} \maketitle
\section{Introduction}
Let $(M,J,\theta ,d\sigma )$ be a compact strictly pseudoconvex CR $(2n+1)$ -manifold with a weighted volume measure $d\sigma =e^{-\phi (x)}\theta \wedge \left( d\theta \right) ^{n}$ for a given smooth weighted function $ \phi $. In this paper, we first derive the weighted CR Reilly formula (\ref {0}) associated with the Witten sub-Laplacian (\cite{cckl}) \begin{equation*} \begin{array}{c} \mathcal{L}=\Delta _{b}-\nabla _{b}\phi \cdot \nabla _{b} \end{array} \end{equation*} in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold with or without boundary. Here $\nabla _{b}$ is the subgradient and $\Delta _{b}$ is the sub-Laplacian as in section $2$. Secondly, we obtain the corresponding first eigenvalue estimate for the Witten sub-Laplacian in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold with or without smooth boundary. With its applications, we obtain the CR\ Obata theorem in a compact weighted Sasakian manifold with or without boundary which is served as a generalization of results in \cite{cc1}, \cite{cc2} and \cite{lw}.
Note that the Witten sub-Laplacian $\mathcal{L}$ satisfies the following integration by parts equation \begin{equation*} \begin{array}{c} \int_{M}g\left( \mathcal{L}f\right) d\sigma =-\int_{M}\langle \nabla _{b}f,\nabla _{b}g\rangle d\sigma =\int_{M}f\left( \mathcal{L}g\right) d\sigma , \end{array} \end{equation*} for all smooth functions $f,$ $g$ in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold $M$\ without boundary.\ As in \cite{cckl} the ($\infty $-dimensional) Bakry-Emery pseudohermitian Ricci curvature $Ric( \mathcal{L})$ and the corresponding torsion $Tor(\mathcal{L})$ are defined by \begin{equation} \begin{array}{l} Ric(\mathcal{L})(W,W)=R_{\alpha \overline{\beta }}W^{\alpha }W^{\overline{ \beta }}+(n+2){\func{Re}}[\phi _{\alpha \overline{\beta }}W^{\alpha }W^{ \overline{\beta }}], \\ Tor(\mathcal{L})(W,W)=2{\func{Re}}[(\sqrt{-1}A_{\overline{\alpha }\overline{ \beta }}-\frac{n+2}{n+1}\phi _{\overline{\alpha }\overline{\beta }})W^{ \overline{\alpha }}W^{\overline{\beta }}], \end{array} \label{2019} \end{equation} for all $W=W^{\alpha }Z_{\alpha }+W^{\overline{\alpha }}Z_{\overline{\alpha } }\in T^{1,0}(M)\oplus T^{0,1}(M).$
Now we recall the weighted CR Paneitz operator $P_{0}^{\phi }$ (Definition \ref{d2}) \begin{equation} \begin{array}{c} P_{\beta }^{\phi }f=P_{\beta }f-\frac{1}{2}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle _{,\beta }+\frac{n}{2}\sqrt{-1}f_{0}\phi _{\beta }. \end{array} \label{40} \end{equation} Here $P^{\phi }f=\sum_{\beta =1}^{n}(P_{\beta }^{\phi }f)\theta ^{\beta }$ and $\overline{P}^{\phi }f=\sum_{\beta =1}^{n}(\overline{P}_{\beta }^{\phi }f)\theta ^{\overline{\beta }}$ (Definition \ref{d1}). Then the weighted CR Paneitz operator $P_{0}^{\phi }$ is defined by \begin{equation*} \begin{array}{c} P_{0}^{\phi }f:=4e^{\phi }[\delta _{b}(e^{-\phi }P^{\phi }f)+\overline{ \delta }_{b}(e^{-\phi }\overline{P}^{\phi }f)]. \end{array} \end{equation*}
By using integrating by parts to the CR Bochner formula (\ref{11}) for $ \mathcal{L}$ with respect to the given weighted volume measure $d\sigma $, we derive the following weighted CR Reilly formula.
\begin{theorem} \label{Reilly'sformula} Let $(M,J,\theta ,d\sigma )$ be a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold with or without boundary $\Sigma $ . Then for any real smooth function $f$, we have \begin{equation} \begin{array}{ll} & \frac{n+1}{n}\int_{M}[(\mathcal{L}f)^{2}-\frac{2n}{n+1}\sum_{\beta ,\gamma
}|f_{\beta \gamma }-\frac{1}{2}f_{\beta }\phi _{\gamma }|^{2}]d\sigma \\ = & \frac{n+2}{4n}\int_{M}fP_{0}^{\phi }fd\sigma +\int_{M}[Ric(\mathcal{L})- \frac{n+1}{2}Tor(\mathcal{L})](\nabla _{b}f,\nabla _{b}f)d\sigma \\ & +\frac{n+1}{2}\int_{M}\left( \mathcal{L}f\right) \langle \nabla _{b}f,\nabla _{b}\phi \rangle d\sigma -\frac{n+2}{4}\int_{M}[\mathcal{L}\phi
+\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}]|\nabla _{b}f|^{2}d\sigma \\ & +\frac{3}{4n}C_{n}\int_{\Sigma }(\mathcal{L}f)f_{e_{_{2n}}}d\Sigma _{p}^{\phi }-\frac{n+2}{2n}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{ \overline{n}}^{\phi }f)d\Sigma _{p}^{\phi } \\ & +C_{n}\int_{\Sigma }f_{e_{2n}}\Delta _{b}^{t}fd\Sigma _{p}^{\phi }+\frac{1 }{2}\sqrt{-1}C_{n}\int_{\Sigma }(f^{\overline{\beta }}B_{n\overline{\beta } }f-f^{\beta }B_{\overline{n}\beta }f)d\Sigma _{p}^{\phi } \\ & +\frac{1}{4}C_{n}\int_{\Sigma }H_{p.h}f_{e_{2n}}^{2}d\Sigma _{p}^{\phi }- \frac{1}{2}C_{n}\int_{\Sigma }\alpha f_{e_{n}}f_{e_{2n}}d\Sigma _{p}^{\phi }+ \frac{3}{4}C_{n}\int_{\Sigma }f_{0}f_{e_{n}}d\Sigma _{p}^{\phi } \\ & -\frac{n^{2}+3n-1}{4n}C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle f_{e_{2n}}d\Sigma _{p}^{\phi }+\frac{n-1}{2n} C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle _{e_{2n}}fd\Sigma _{p}^{\phi } \\
& +\frac{n+2}{8}C_{n}\int_{\Sigma }|\nabla _{b}f|^{2}\phi _{e_{_{2n}}}d\Sigma _{p}^{\phi }+\frac{1}{4}C_{n}\int_{\Sigma }\sum_{j,k=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{2n},e_{k}\right\rangle f_{e_{j}}f_{e_{k}}d\Sigma _{p}^{\phi } \\ & -\frac{1}{4}C_{n}\int_{\Sigma }\sum_{j=1}^{2n-1}[\left\langle \nabla _{e_{j}}e_{n},e_{j}\right\rangle f_{e_{n}}+f_{e_{j}}\phi _{e_{j}}]f_{e_{2n}}d\Sigma _{p}^{\phi }. \end{array} \label{0} \end{equation} Here $P_{0}^{\phi }$ is the weighted CR Paneitz operator on $M,\ C_{n}:=2^{n}n!;$ $B_{\beta \overline{\gamma }}f:=f_{\beta \overline{\gamma } }-\frac{1}{n}f_{\sigma }$ $^{\sigma }h_{\beta \overline{\gamma }},$ $\Delta _{b}^{t}:=\frac{1}{2}\sum_{j=1}^{2n-1}[\left( e_{j}\right) ^{2}-(\nabla _{e_{j}}e_{j})^{t}]$ is the tangential sub-Laplacian of $\Sigma $ and $ H_{p.h}$ is the $p$-mean curvature of $\Sigma $ with respect to the Legendrian normal $e_{2n},$ $\alpha e_{2n}+T\in T\Sigma $ for some function $ \alpha $ on $\Sigma \backslash S_{\Sigma },$ the singular set $S_{\Sigma }$ consists of those points where the contact bundle $\xi =\ker \theta $ coincides with the tangent bundle $T\Sigma $ of $\Sigma ,$ and $d\Sigma _{p}^{\phi }=e^{-\phi }\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{n-1}\wedge e^{2n-1}\wedge e^{n}$ is the weighted $p$-area element on $ \Sigma .$ \end{theorem}
In Lemma \ref{lemma}, we observe that \begin{equation} \begin{array}{lll} P_{0}^{\phi }f & = & 2\left( \mathcal{L}^{2}+n^{2}T^{2}\right) f-4n{\func{Re} }Q^{\phi }f-2n^{2}\phi _{0}f_{0} \\ & = & 2\square _{b}^{\phi }\overline{\square }_{b}^{\phi }f-4nQ^{\phi }f-2n^{2}\phi _{0}f_{0}-2n\sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle \\ & = & 2\overline{\square }_{b}^{\phi }\square _{b}^{\phi }f-4n\overline{Q} ^{\phi }f-2n^{2}\phi _{0}f_{0}+2n\sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle , \end{array} \label{4c} \end{equation} for the weighted Kohn Laplacian $\square _{b}^{\phi }f=(-\mathcal{L}+n\sqrt{ -1}T)f.$ This implies that $P_{0}^{\phi }$ is a self-adjoint operator in a compact weighted pseudohermitian $(2n+1)$-manifold without boundary. However, in order for the CR Paneitz $P_{0}^{\phi }$ to be self-adjoint when $M$ with the nonempty smooth boundary $\Sigma ,$ one needs all smooth functions on $M$ satisfy some\ suitable boundary conditions on $\Sigma $. That is, one can consider the following Dirichlet eigenvalue problem for $ P_{0}^{\phi }$: \begin{equation} \begin{array}{c} \left\{ \begin{array}{c} P_{0}^{\phi }\varphi =\mu _{_{D}}\varphi \ \ \mathrm{on\ }M, \\ \varphi =0=\mathcal{L}\varphi \ \mathrm{on\ }\Sigma . \end{array} \right. \end{array} \label{1} \end{equation} Hence \begin{equation} \begin{array}{c} \int_{M}\varphi P_{0}^{\phi }\varphi d\sigma \geq \mu _{_{D}}^{1}\int_{M}\varphi ^{2}d\sigma \end{array} \label{2} \end{equation} for the first Dirichlet eigenvalue $\mu _{_{D}}^{1}$ and all smooth functions on $M$ with $\varphi =0=\mathcal{L}\varphi $ on $\Sigma .$ We refer \cite{ccw} for some details in case that $\phi $ is constant.
In general, $\mu _{_{D}}^{1}$ is not always nonnegative. It is related to the nonnegativity of the weighted CR Paneitz operator $P_{0}^{\phi }$ in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold.
\begin{definition} Let $(M,J,\theta ,d\sigma )$ be a compact weighted strictly pseudoconvex CR $ (2n+1)$-manifold with smooth boundary $\Sigma $. We say that the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative if \begin{equation*} \begin{array}{c} \int_{M}\varphi P_{0}^{\phi }\varphi d\mu \geq 0 \end{array} \end{equation*} for all smooth functions $\varphi $ with suitable boundary conditions (\ref {1}) as in Dirichlet eigenvalue problem. \end{definition}
\begin{remark} \label{r1} Let $(M,J,\theta ,d\sigma )$ be a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold of vanishing torsion with or without smooth boundary $\Sigma $ and $\phi _{0}$ vanishes on $M$. It follows from ( \ref{4c}) that the weighted Kohn Laplacian $\square _{b}^{\phi }$ and $ \overline{\square }_{b}^{\phi }$ commute and they are diagonalized simultaneously with \begin{equation*} \begin{array}{c} P_{0}^{\phi }=\square _{b}^{\phi }\overline{\square }_{b}^{\phi }+\overline{ \square }_{b}^{\phi }\square _{b}^{\phi }. \end{array} \end{equation*} Then the corresponding weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative (Lemma \ref{l41}). That is $\mu _{_{D}}^{1}\geq 0.$ \end{remark}
With its applications, we first derive the first eigenvalue estimate and weighted CR\ Obata theorem in a closed weighted strictly pseudoconvex CR $ (2n+1)$-manifold.
\begin{theorem} \label{Thm}Let $(M,J,\theta ,d\sigma )$ be a closed weighted strictly pseudoconvex CR $(2n+1)$-manifold with the nonnegative weighted CR Paneitz operator $P_{0}^{\phi }$. Suppose that \begin{equation} \begin{array}{c} \lbrack Ric(\mathcal{L})-\frac{n+1}{2}Tor(\mathcal{L})](Z,Z)\geq k\left\langle Z,Z\right\rangle \end{array} \label{2019B} \end{equation} for all $Z\in T_{1,0}$ and a positive constant $k.$ Then the first eigenvalue of the Witten sub-Laplacian $\mathcal{L}$ satisfies the lower bound \begin{equation} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}, \end{array} \label{01a} \end{equation} where $\omega =\underset{M}{\mathrm{osc}}\phi =\underset{M}{\sup }\phi - \underset{M}{\inf }\phi $ and for nonnegative constant $l$ with\ $0\leq l< \frac{k}{n+2}$ such that \begin{equation} \begin{array}{c}
\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l \end{array} \label{2019A} \end{equation} on $M.$ Moreover, if the equality (\ref{01a}) holds, then $M$\ is CR isometric to a standard CR $(2n+1)$-sphere. \end{theorem}
Furthermore, $P_{0}^{\phi }$ is nonnegative if the torsion is zero (i.e. Sasakian) and $\phi _{0}$ vanishes (Lemma \ref{l41}). Then we have the following CR\ Obata theorem in a closed weighted Sasakian $(2n+1)$-manifold.
\begin{corollary} \label{C1} Let $(M,J,\theta ,d\sigma )$ be a closed weighted Sasakian $ (2n+1) $-manifold. Suppose that \begin{equation*} \begin{array}{c} \lbrack Ric(\mathcal{L})-\frac{n+1}{2}Tor(\mathcal{L})](Z,Z)\geq k\left\langle Z,Z\right\rangle \end{array} \end{equation*} and \begin{equation*} \phi _{0}=0, \end{equation*} for all $Z\in T_{1,0},$ a positive constant $k.$ Then the first eigenvalue of the Witten sub-Laplacian $\mathcal{L}$ satisfies the lower bound \begin{equation} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}, \end{array} \label{01b} \end{equation} where $\omega =\underset{M}{\mathrm{osc}}\phi =\underset{M}{\sup }\phi - \underset{M}{\inf }\phi $ and for nonnegative constant $l$ with\ $0\leq l< \frac{k}{n+2}$ such that \begin{equation*} \begin{array}{c}
\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l \end{array} \end{equation*} on $M.$ Moreover, if the equality (\ref{01b}) holds, then $M$\ is CR isometric to a standard CR $(2n+1)$-sphere. \end{corollary}
\begin{remark} 1. Theorem \ref{Thm} and Corollary \ref{C1} are done as in \cite{gr}, \cite {ch} and \cite{cc3} in case that the weighted function $\phi $ is constant in which $l=0$.
2. Note that (\ref{2019A}) is equivalent to \begin{equation*} \begin{array}{c}
\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}=\Delta _{b}\phi -
\frac{2n+3}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l<\frac{4k}{n+2}. \end{array} \end{equation*} Then by comparing (\ref{2019}), (\ref{2019A}) and (\ref{2019B}), it has a plenty of rooms for the choice of the weighted function $\phi $. For example, it is the case by a small perturbation of the subhessian of the weighted function $\phi .$ \end{remark}
Secondly, we consider the following Dirichlet eigenvalue problem of the Witten sub-Laplacian $\mathcal{L}$ in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold $M$ with smooth boundary $\Sigma $: \begin{equation} \left\{ \begin{array}{ccll} \mathcal{L}f & = & -\lambda _{1}f & \mathrm{on\ }M, \\ f & = & 0 & \mathrm{on\ }\Sigma . \end{array} \right. \label{1b} \end{equation} Then we have the following CR first\ Dirichlet eigenvalue estimate and its weighted Obata Theorem.
\begin{theorem} \label{TB} Let $(M,J,\theta ,d\sigma )$ be a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold with the smooth boundary $\Sigma $ and the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative. Suppose that \begin{equation*} \begin{array}{c} \lbrack Ric(\mathcal{L})-\frac{n+1}{2}Tor(\mathcal{L})](Z,Z)\geq k\left\langle Z,Z\right\rangle \end{array} \end{equation*} for all $Z\in T_{1,0},$ and the pseudohermitian mean curvature and connection $1$-form satisfies \begin{equation*} \begin{array}{c} H_{p.h}-\tilde{\omega}_{n}^{\;n}(e_{n})-\frac{n+2}{2}\phi _{e_{2n}}\geq 0 \end{array} \end{equation*} on $\Sigma $ and $H_{p.h}+\tilde{\omega}_{n}^{\;n}(e_{n})$ is also nonnegative on $\Sigma $ for $n\geq 2$. Then the first Dirichlet eigenvalue of the Witten sub-Laplacian $\mathcal{L}$ satisfies the lower bound \begin{equation} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}, \end{array} \label{01c} \end{equation} where $\omega =\underset{M}{\mathrm{osc}}\phi =\underset{M}{\sup }\phi - \underset{M}{\inf }\phi $ and for nonnegative constant $l$ with\ $0\leq l< \frac{k}{n+2}$ such that \begin{equation*} \begin{array}{c}
\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l \end{array} \end{equation*} on $M.\ $Moreover, if the equality (\ref{01c}) holds, then $M$\ is isometric to a hemisphere in a standard CR $(2n+1)$-sphere. \end{theorem}
\begin{corollary} Let $(M,J,\theta ,d\sigma )$ be a compact weighted Sasakian $(2n+1)$ -manifold with smooth boundary $\Sigma $. Suppose that \begin{equation*} \begin{array}{c} \lbrack Ric(\mathcal{L})-\frac{n+1}{2}Tor(\mathcal{L})](Z,Z)\geq k\left\langle Z,Z\right\rangle \end{array} \end{equation*} for all $Z\in T_{1,0},$ and \begin{equation*} \phi _{0}=0. \end{equation*} Furthermore, assume that the pseudohermitian mean curvature and connection $ 1 $-form satisfies \begin{equation*} \begin{array}{c} H_{p.h}-\tilde{\omega}_{n}^{\;n}(e_{n})\geq 0 \end{array} \end{equation*} on $\Sigma $ and $H_{p.h}+\tilde{\omega}_{n}^{\;n}(e_{n})$ is also nonnegative on $\Sigma $ if $n\geq 2.$ Then the first Dirichlet eigenvalue of the Witten sub-Laplacian $\mathcal{L}$ satisfies the lower bound \begin{equation*} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}, \end{array} \end{equation*} where $\omega =\underset{M}{\mathrm{osc}}\phi =\underset{M}{\sup }\phi - \underset{M}{\inf }\phi $ and for nonnegative constant $l$ with\ $0\leq l< \frac{k}{n+2}$ such that \begin{equation*} \begin{array}{c}
\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l \end{array} \end{equation*} on $M.\ $Moreover, if the equality holds then $M$\ is isometric to a hemisphere in a standard CR $(2n+1)$-sphere. \end{corollary}
We briefly describe the methods used in our proofs. In section $2$, we introduce the weighted CR Paneitz operator $P_{0}^{\phi }$. In section $3$, by using integrating by parts to the weighted CR Bochner formula (\ref{11}), we can derive the CR version of weighted Reilly's formula. By applying the weighted CR Reilly's formula, we are able to obtain the first eigenvalue estimate of the Witten sub-Laplacian as in section $4$ in a closed weighted strictly pseudoconvex CR $(2n+1)$-manifold and its weighted Obata Theorem. In section $5,$ we derive the first Dirichlet eigenvalue estimate in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold with boundary $ \Sigma $ and its corresponding weighted Obata-type Theorem.
\textbf{Acknowledgements} This work was partially done while the second author visited Taida Institute of Mathematical Sciences (TIMS), Taiwan. He would like to thank the institute for its hospitality.
\section{The weighted CR Paneitz Operator}
We first introduce some basic materials in a strictly pseudoconvex CR $ (2n+1) $-manifold $(M,J,\theta )$. Let $(M,J,\theta )$ be a $(2n+1)$ -dimensional, orientable, contact manifold with contact structure $\xi =\ker \theta $. A CR structure compatible with $\xi $ is an endomorphism $J:\xi \rightarrow \xi $ such that $J^{2}=-1$. We also assume that $J$ satisfies the following integrability condition: If $X$ and $Y$ are in $\xi $, then so is $[JX,Y]+[X,JY]$ and $J([JX,Y]+[X,JY])=[JX,JY]-[X,Y]$. A CR structure $J$ can extend to $\mathbb{C}\mathbf{\otimes }\xi $ and decomposes $\mathbb{C} \mathbf{\otimes }\xi $ into the direct sum of $T_{1,0}$ and $T_{0,1}$ which are eigenspaces of $J$ with respect to eigenvalues $\sqrt{-1}$ and $-\sqrt{-1 }$, respectively. A manifold $M$ with a CR structure is called a CR manifold. A pseudohermitian structure compatible with $\xi $ is a $CR$ structure $J$ compatible with $\xi $ together with a choice of contact form $ \theta $. Such a choice determines a unique real vector field $T$ transverse to $\xi $, which is called the characteristic vector field of $\theta $, such that ${\theta }(T)=1$ and $\mathcal{L}_{T}{\theta }=0$ or $d{\theta }(T, {\cdot })=0$. Let $\left\{ T,Z_{\beta },Z_{\overline{\beta }}\right\} $ be a frame of $TM\otimes \mathbb{C}$, where $Z_{\beta }$ is any local frame of $ T_{1,0},\ Z_{\overline{\beta }}=\overline{Z_{\beta }}\in T_{0,1}$ and $T$ is the characteristic vector field. Then $\{\theta ,\theta ^{\beta },\theta ^{ \overline{\beta }}\}$, which is the coframe dual to $\left\{ T,Z_{\beta },Z_{ \overline{\beta }}\right\} $, satisfies \begin{equation} \begin{array}{c} d\theta =\sqrt{-1}h_{\beta \overline{\gamma }}\theta ^{\beta }\wedge \theta ^{\overline{\gamma }}, \end{array} \label{dtheta} \end{equation} for some positive definite Hermitian matrix of functions $(h_{\beta \overline{\gamma }})$. Actually we can always choose $Z_{\beta }$ such that $ h_{\beta \overline{\gamma }}=\delta _{\beta \gamma }$; hence, throughout this note, we assume $h_{\beta \overline{\gamma }}=\delta _{\beta \gamma }$.
The Levi form $\left\langle \ ,\ \right\rangle $ is the Hermitian form on $ T_{1,0}$ defined by \begin{equation*} \begin{array}{c} \left\langle Z,W\right\rangle =-\sqrt{-1}\left\langle d\theta ,Z\wedge \overline{W}\right\rangle . \end{array} \end{equation*} We can extend $\left\langle \ ,\ \right\rangle $ to $T_{0,1}$ by defining $ \left\langle \overline{Z},\overline{W}\right\rangle =\overline{\left\langle Z,W\right\rangle }$ for all $Z,W\in T_{1,0}$. The Levi form induces naturally a Hermitian form on the dual bundle of $T_{1,0}$, also denoted by $ \left\langle \ ,\ \right\rangle $, and hence on all the induced tensor bundles. Integrating the Hermitian form (when acting on sections) over $M$ with respect to the volume form $d\mu =\theta \wedge (d\theta )^{n}$, we get an inner product on the space of sections of each tensor bundle.
The pseudohermitian connection of $(J,\theta )$ is the connection $\nabla $ on $TM\otimes \mathbb{C}$ (and extended to tensors) given in terms of a local frame $Z_{\beta }\in T_{1,0}$ by \begin{equation*} \nabla Z_{\beta }=\theta _{\beta }{}^{\gamma }\otimes Z_{\gamma },\quad \nabla Z_{\overline{\beta }}=\theta _{\overline{\beta }}{}^{\overline{\gamma }}\otimes Z_{\overline{\gamma }},\quad \nabla T=0, \end{equation*} where $\theta _{\beta }{}^{\gamma }$ are the $1$-forms uniquely determined by the following equations: \begin{equation} \begin{split} d\theta ^{\beta }& =\theta ^{\gamma }\wedge \theta _{\gamma }{}^{\beta }+\theta \wedge \tau ^{\beta }, \\ \tau _{\beta }\wedge \theta ^{\beta }& =0=\theta _{\beta }{}^{\gamma }+\theta _{\overline{\beta }}{}^{\overline{\gamma }}. \end{split} \label{structure equs} \end{equation} We can write (by Cartan lemma) $\tau _{\beta }=A_{\beta \gamma }\theta ^{\gamma }$ with $A_{\beta \gamma }=A_{\gamma \beta }$. The curvature of the Tanaka-Webster connection, expressed in terms of the coframe $\{\theta =\theta ^{0},\theta ^{\beta },\theta ^{\overline{\beta }}\}$, is \begin{equation*} \begin{split} \Pi _{\beta }{}^{\gamma }& =\overline{\Pi _{\bar{\beta}}{}^{\overline{\gamma }}}=d\theta _{\beta }{}^{\gamma }-\theta _{\beta }{}^{\sigma }\wedge \theta _{\sigma }{}^{\gamma }, \\ \Pi _{0}{}^{\beta }& =\Pi _{\beta }{}^{0}=\Pi _{0}{}^{\bar{\beta}}=\Pi _{ \bar{\beta}}{}^{0}=\Pi _{0}{}^{0}=0. \end{split} \end{equation*} Webster showed that $\Pi _{\beta }{}^{\gamma }$ can be written \begin{equation*} \begin{array}{c} \Pi _{\beta }{}^{\gamma }=R_{\beta }{}^{\gamma }{}_{\rho \bar{\sigma}}\theta ^{\rho }\wedge \theta ^{\bar{\sigma}}+W_{\beta }{}^{\gamma }{}_{\rho }\theta ^{\rho }\wedge \theta -W^{\gamma }{}_{\beta \bar{\rho}}\theta ^{\bar{\rho} }\wedge \theta +\sqrt{-1}(\theta _{\beta }\wedge \tau ^{\gamma }-\tau _{\beta }\wedge \theta ^{\gamma }) \end{array} \end{equation*} where the coefficients satisfy \begin{equation*} \begin{array}{c} R_{\beta \overline{\gamma }\rho \bar{\sigma}}=\overline{R_{\gamma \bar{\beta} \sigma \bar{\rho}}}=R_{\overline{\gamma }\beta \bar{\sigma}\rho }=R_{\rho \overline{\gamma }\beta \bar{\sigma}},\ \ W_{\beta \overline{\gamma }\rho }=W_{\rho \overline{\gamma }\beta }. \end{array} \end{equation*}
We will denote components of covariant derivatives with indices preceded by comma; thus write $A_{\rho \beta ,\gamma }$. The indices $\{0,\beta , \overline{\beta }\}$ indicate derivatives with respect to $\{T,Z_{\beta },Z_{ \overline{\beta }}\}$. For derivatives of a scalar function, we will often omit the comma, for instance, $u_{\beta }=Z_{\beta }u,\ u_{\gamma \bar{\beta} }=Z_{\bar{\beta}}Z_{\gamma }u-\theta _{\gamma }{}^{\rho }(Z_{\bar{\beta} })Z_{\rho }u,\ u_{0}=Tu$ for a smooth function $u$ .
For a real function $u$, the subgradient $\nabla _{b}$ is defined by $\nabla _{b}u\in \xi $ and $\left\langle Z,\nabla _{b}u\right\rangle =du(Z)$ for all vector fields $Z$ tangent to contact plane. Locally $\nabla _{b}u=u^{\beta }Z_{\beta }+u^{\overline{\beta }}Z_{\overline{\beta }}$. We can use the connection to define the subhessian as the complex linear map \begin{equation*} \begin{array}{c} (\nabla ^{H})^{2}u:T_{1,0}\oplus T_{0,1}\rightarrow T_{1,0}\oplus T_{0,1} \text{\ \textrm{by}\ }(\nabla ^{H})^{2}u(Z)=\nabla _{Z}\nabla _{b}u. \end{array} \end{equation*} In particular, \begin{equation*} \begin{array}{c}
|\nabla _{b}u|^{2}=2\sum_{\beta }u_{\beta }u^{\beta },\quad |\nabla _{b}^{2}u|^{2}=2\sum_{\beta ,\gamma }(u_{\beta \gamma }u^{\beta \gamma }+u_{\beta \overline{\gamma }}u^{\beta \overline{\gamma }}). \end{array} \end{equation*} Also the sub-Laplacian is defined by \begin{equation*} \begin{array}{c} \Delta _{b}u=Tr\left( (\nabla ^{H})^{2}u\right) =\sum_{\beta }(u_{\beta }{}^{\beta }+u_{\overline{\beta }}{}^{\overline{\beta }}). \end{array} \end{equation*} The pseudohermitian Ricci tensor and the torsion tensor on $T_{1,0}$ are defined by \begin{equation*} \begin{array}{l} Ric(X,Y)=R_{\gamma \bar{\beta}}X^{\gamma }Y^{\bar{\beta}} \\ Tor(X,Y)=\sqrt{-1}\sum_{\gamma ,\beta }(A_{\overline{\gamma }\bar{\beta}}X^{ \overline{\gamma }}Y^{\bar{\beta}}-A_{\gamma \beta }X^{\gamma }Y^{\beta }), \end{array} \end{equation*} where $X=X^{\gamma }Z_{\gamma },\ Y=Y^{\beta }Z_{\beta }$.
Let $M$ be a compact strictly pseudoconvex CR $(2n+1)$-manifold with a weighted volume measure $d\sigma =e^{-\phi (x)}d\mu $ for a given smooth function $\phi $. In this section, we define the weighted CR Paneitz operator $P_{0}^{\phi }$. First we recall the definition of the CR Paneitz operator $P_{0}.$
\begin{definition} \label{d1} (\cite{gl}) Let $(M,J,\theta )$ be a compact strictly pseudoconvex CR $(2n+1)$-manifold. We define \begin{equation*} \begin{array}{c} Pf=\sum_{\gamma ,\beta =1}^{n}(f_{\overline{\gamma }\;\ \beta }^{\,\overline{ \text{ }\gamma }}+\sqrt{-1}nA_{\beta \gamma }f^{\gamma })\theta ^{\beta }=\sum_{\beta =1}^{n}(P_{\beta }f)\theta ^{\beta }, \end{array} \end{equation*} which is an operator that characterizes CR-pluriharmonic functions. Here \begin{equation*} \begin{array}{c} P_{\beta }f=\sum_{\gamma =1}^{n}(f_{\overline{\gamma }\;\ \beta }^{\,\text{\ }\overline{\gamma }}+\sqrt{-1}nA_{\beta \gamma }f^{\gamma }),\text{ \ }\beta =1,\cdots ,n, \end{array} \end{equation*} and $\overline{P}f=\sum_{\beta =1}^{n}\left( \overline{P}_{\beta }f\right) \theta ^{\overline{\beta }}$, the conjugate of $P$. The CR Paneitz operator $ P_{0}$ is defined by \begin{equation*} \begin{array}{c} P_{0}f=4[\delta _{b}(Pf)+\overline{\delta }_{b}(\overline{P}f)], \end{array} \end{equation*} where $\delta _{b}$ is the divergence operator that takes $(1,0)$-forms to functions by $\delta _{b}(\sigma _{\beta }\theta ^{\beta })=\sigma _{\beta }^{\;\ \beta }$, and similarly, $\overline{\delta }_{b}(\sigma _{\overline{ \beta }}\theta ^{\overline{\beta }})=\sigma _{\overline{\beta }}^{\;\ \overline{\beta }}$. \end{definition}
One can define (\cite{gl}) the purely holomorphic second-order operator $Q$ by \begin{equation*} \begin{array}{c} Qf:=2\sqrt{-1}(A^{\alpha \beta }f_{\alpha }),_{\beta }. \end{array} \end{equation*} Note that $[\Delta _{b},T]f=2{\func{Im}}Qf$ and observe that \begin{equation} \begin{array}{lll} P_{0}f & = & 2(\Delta _{b}^{2}+n^{2}T^{2})f-4n\mathrm{Re}Qf \\ & = & 2\square _{b}\overline{\square }_{b}f-4nQf\text{ }=\text{ }2\overline{ \square }_{b}\square _{b}f-4n\overline{Q}f, \end{array} \label{3} \end{equation} for $\square _{b}f=(-\Delta _{b}+n\sqrt{-1}T)f=-2f_{\overline{\beta }}$ $^{ \overline{\beta }}$ be the Kohn Laplacian operator.
With respect to the weighted volume measure $d\sigma ,$ we define the purely holomorphic second-order operator $Q^{\phi }$ by \begin{equation*} \begin{array}{c} Q^{\phi }f:=Qf-2\sqrt{-1}(A^{\alpha \beta }f_{\alpha }\phi _{\beta })=2\sqrt{ -1}e^{\phi }(e^{-\phi }A^{\alpha \beta }f_{\alpha }),_{\beta }, \end{array} \end{equation*} and thus we have \begin{equation} \begin{array}{c} \lbrack \mathcal{L},T]f=2{\func{Im}}Q^{\phi }f+\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle . \end{array} \label{3a} \end{equation}
\begin{definition} \label{d2} We define the weighted CR Paneitz operator $P_{0}^{\phi }$ as follows \begin{equation*} \begin{array}{c} P_{0}^{\phi }f=4e^{\phi }[\delta _{b}(e^{-\phi }P^{\phi }f)+\overline{\delta }_{b}(e^{-\phi }\overline{P}^{\phi }f)]. \end{array} \end{equation*} Here $P^{\phi }f=\sum_{\beta =1}^{n}(P_{\beta }^{\phi }f)\theta ^{\beta }$ and $\overline{P}^{\phi }f=\sum_{\beta =1}^{n}(\overline{P}_{\beta }^{\phi }f)\theta ^{\overline{\beta }}$, the conjugate of $P^{\phi }$ with \begin{equation*} \begin{array}{c} P_{\beta }^{\phi }f=P_{\beta }f-\frac{1}{2}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle _{,\beta }+\frac{n}{2}\sqrt{-1}f_{0}\phi _{\beta }. \end{array} \end{equation*} \end{definition}
We explain why the weighted CR Paneitz operator $P_{0}^{\phi }$ to be defined in this way. Comparing with the Riemannian case, we have the extra term $\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle $ in the CR Bochner formula (\ref{A}) for $\Delta _{b}$, which is hard to deal. From \cite{cc2}, we can relate $\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle $ with $\langle \nabla _{b}f,\nabla _{b}\Delta _{b}f\rangle $ by \begin{equation*} \begin{array}{c} \langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle =\frac{1}{n}\langle \nabla _{b}f,\nabla _{b}\Delta _{b}f\rangle -\frac{2}{n}\langle Pf+\overline{P} f,d_{b}f\rangle -2Tor(\nabla _{b}f_{\mathbb{C}},\nabla _{b}f_{\mathbb{C}}), \end{array} \end{equation*} then by integral with respect to the volume measure $d\mu =\theta \wedge \left( d\theta \right) ^{n}$ yields \begin{equation} \begin{array}{c} n^{2}\int_{M}f_{0}^{2}d\mu =\int_{M}\left( \Delta _{b}f\right) ^{2}d\mu - \frac{1}{2}\int_{M}fP_{0}fd\mu +n\int_{M}Tor(\nabla _{b}f,\nabla _{b}f)d\mu . \end{array} \label{5} \end{equation} This integral says that the integral of the square of $f_{0}$ can be replace by the integral of the square of $\Delta _{b}f$ and the integral of the CR Paneitz operator $P_{0}.$ For the CR Bochner formula (\ref{Bochnerformula}) for $\mathcal{L}$, we also have the extra term $\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle -f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle $, which can be related by \begin{equation*} \begin{array}{c} \langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle -f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle =\frac{1}{n}\langle \nabla _{b}f,\nabla _{b} \mathcal{L}f\rangle -\frac{2}{n}\langle P^{\phi }f+\overline{P}^{\phi }f,d_{b}f\rangle -Tor(\nabla _{b}f,\nabla _{b}f). \end{array} \end{equation*} Then by integral with respect to the weighted volume measure $d\sigma =e^{-\phi }\theta \wedge \left( d\theta \right) ^{n},$ one gets \begin{equation*} \begin{array}{c} n^{2}\int_{M}f_{0}^{2}d\sigma =\int_{M}\left( \mathcal{L}f\right) ^{2}d\sigma -\frac{1}{2}\int_{M}fP_{0}^{\phi }fd\sigma +n\int_{M}Tor(\nabla _{b}f,\nabla _{b}f)d\sigma . \end{array} \end{equation*} This integral have the same type as (\ref{5}) when we replace $\mathcal{L},$ $P_{0}^{\phi }$ and $d\sigma $ by $\Delta _{b},$ $P_{0}$ and $d\mu ,$ respectively.
First we compare the relation between $P_{0}^{\phi }$ and $P_{0}.$
\begin{lemma} Let $(M,J,\theta )$ be a compact strictly pseudoconvex CR $(2n+1)$-manifold. We obtain \begin{equation} \begin{array}{c} P_{0}^{\phi }f=P_{0}f-4\langle Pf+\overline{P}f,d_{b}\phi \rangle -2\mathcal{ L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -2n\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\rangle -2n^{2}f_{0}\phi _{0}. \end{array} \label{4a} \end{equation} \end{lemma}
\begin{proof} By the definition of $P_{0}^{\phi }$ and (\ref{40}), we compute \begin{equation*} \begin{array}{lll} P_{0}^{\phi }f & = & 4[\delta _{b}(P^{\phi }f)+\overline{\delta }_{b}( \overline{P}^{\phi }f)]-4\langle P^{\phi }f+\overline{P}^{\phi }f,d_{b}\phi \rangle \\ & = & 4(P_{\beta }f-\frac{1}{2}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle _{,\beta }+\frac{n}{2}\sqrt{-1}f_{0}\phi _{\beta })^{,\beta } \\ & & +4(P_{\overline{\beta }}f-\frac{1}{2}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle _{,\overline{\beta }}-\frac{n}{2}\sqrt{-1} f_{0}\phi _{\overline{\beta }})^{,\overline{\beta }} \\ & & -4\langle Pf+\overline{P}f,d_{b}\phi \rangle -2\left\langle \nabla _{b}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle ,\nabla _{b}\phi \right\rangle \\ & = & P_{0}f-4\langle Pf+\overline{P}f,d_{b}\phi \rangle -2\mathcal{L} \left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle \\ & & -2n\left( \langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\rangle +nf_{0}\phi _{0}\right) . \end{array} \end{equation*} \end{proof}
Second, we define the weighted Kohn Laplacian operator as \begin{equation*} \begin{array}{c} \square _{b}^{\phi }f:=(-\mathcal{L}+n\sqrt{-1}T)f, \end{array} \end{equation*} then we have the similar formula for $P_{0}^{\phi }$ like the expression for $P_{0}$ (\ref{3}).
\begin{lemma} \label{lemma}Let $(M,J,\theta )$ be a compact strictly pseudoconvex CR $ (2n+1)$-manifold. We have \begin{equation} \begin{array}{lll} P_{0}^{\phi }f & = & 2\left( \mathcal{L}^{2}+n^{2}T^{2}\right) f-4n{\func{Re} }Q^{\phi }f-2n^{2}\phi _{0}f_{0} \\ & = & 2\square _{b}^{\phi }\overline{\square }_{b}^{\phi }f-4nQ^{\phi }f-2n^{2}\phi _{0}f_{0}-2n\sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle \\ & = & 2\overline{\square }_{b}^{\phi }\square _{b}^{\phi }f-4n\overline{Q} ^{\phi }f-2n^{2}\phi _{0}f_{0}+2n\sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle . \end{array} \label{4} \end{equation} \end{lemma}
\begin{proof} By the straightforward calculation, we have \begin{equation*} \begin{array}{lll} \square _{b}^{\phi }\overline{\square }_{b}^{\phi }f & = & (-\mathcal{L}+n \sqrt{-1}T)(-\mathcal{L}f-n\sqrt{-1}f_{0}) \\ & = & \left( \mathcal{L}^{2}f+n^{2}f_{00}\right) +n\sqrt{-1}[\mathcal{L},T]f \\ & = & \left( \mathcal{L}^{2}f+n^{2}f_{00}\right) +n\sqrt{-1}\left( 2{\func{Im }}Q^{\phi }f+\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle \right) \end{array} \end{equation*} and \begin{equation*} \begin{array}{lll} \square _{b}^{\phi }\overline{\square }_{b}^{\phi }f & = & (-\mathcal{L}+n \sqrt{-1}T)(-\mathcal{L}f-n\sqrt{-1}f_{0}) \\ & = & (-\mathcal{L}+n\sqrt{-1}T)(-\Delta _{b}f+\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -n\sqrt{-1}f_{0}) \\ & = & \mathcal{L}(\Delta _{b}f-\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}f_{0}) \\ & & -n\sqrt{-1}T(\Delta _{b}f-\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}f_{0}) \\ & = & \Delta _{b}(\Delta _{b}f-\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}f_{0}) \\ & & -\left\langle \nabla _{b}\phi ,\nabla _{b}(\Delta _{b}f-\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}f_{0})\right\rangle \\ & & -n\sqrt{-1}T(\Delta _{b}f-\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}f_{0}) \\ & = & \Delta _{b}^{2}f+n^{2}f_{00}-\mathcal{L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}[\mathcal{L},T]f-\left\langle \nabla _{b}\phi ,\nabla _{b}\Delta _{b}f\right\rangle \\ & = & \Delta _{b}^{2}f+n^{2}f_{00}-\mathcal{L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle +n\sqrt{-1}\mathcal{L}f_{0}-\left\langle \nabla _{b}\phi ,\nabla _{b}\Delta _{b}f\right\rangle \\ & & -n\sqrt{-1}[\Delta _{b}f_{0}-2{\func{Im}}Qf-f_{\overline{\beta }}\phi _{\beta 0}-f_{\beta }\phi _{\overline{\beta }0} \\ & & \text{ \ \ \ \ \ \ \ \ \ \ \ }-\left\langle \nabla _{b}\phi ,\nabla _{b}f_{0}\right\rangle +A^{\overline{\alpha }\overline{\beta }}f_{\overline{ \alpha }}\phi _{\overline{\beta }}+A^{\alpha \beta }f_{\alpha }\phi _{\beta } \mathcal{]} \\ & = & \Delta _{b}^{2}f+n^{2}f_{00}-\mathcal{L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -\left\langle \nabla _{b}\phi ,\nabla _{b}\Delta _{b}f\right\rangle \\ & & +n\sqrt{-1}[2{\func{Im}}Qf+(f_{\overline{\beta }}\phi _{0\beta }+f_{\beta }\phi _{0\overline{\beta }})-2(A^{\overline{\alpha }\overline{ \beta }}f_{\overline{\alpha }}\phi _{\overline{\beta }}+A^{\alpha \beta }f_{\alpha }\phi _{\beta })] \end{array} \end{equation*} By using the equation \begin{equation*} \begin{array}{c} n\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\rangle =\langle \nabla _{b}\phi ,\nabla _{b}\Delta _{b}f\rangle -2\langle Pf+\overline{P}f,d_{b}\phi \rangle +2n\sqrt{-1}(A^{\overline{\alpha }\overline{\beta }}f_{\overline{\alpha } }\phi _{\overline{\beta }}-A^{\alpha \beta }f_{\alpha }\phi _{\beta }), \end{array} \end{equation*} we deduce \begin{equation*} \begin{array}{lll} \square _{b}^{\phi }\overline{\square }_{b}^{\phi }f & = & \Delta _{b}^{2}f+n^{2}f_{00}-\mathcal{L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -n\left\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\right\rangle -2\langle Pf+\overline{P}f,d_{b}\phi \rangle \\ & & +2n\sqrt{-1}(A^{\overline{\alpha }\overline{\beta }}f_{\overline{\alpha }}\phi _{\overline{\beta }}-A^{\alpha \beta }f_{\alpha }\phi _{\beta })+2n{ \func{Im}}Qf \\ & & +n\sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle -2n\sqrt{-1}(A^{\overline{\alpha }\overline{\beta }}f_{\overline{\alpha } }\phi _{\overline{\beta }}+A^{\alpha \beta }f_{\alpha }\phi _{\beta }) \\ & = & \frac{1}{2}P_{0}f+2n{\func{Re}}Qf-\mathcal{L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -n\left\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\right\rangle -2\langle Pf+\overline{P}f,d_{b}\phi \rangle \\ & & +2n\func{Im}Qf-4n\sqrt{-1}A^{\alpha \beta }f_{\alpha }\phi _{\beta }+n \sqrt{-1}\left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle . \end{array} \end{equation*} Finally, we obtain \begin{equation*} \begin{array}{lll} 2\square _{b}^{\phi }\overline{\square }_{b}^{\phi }f & = & P_{0}f-2\mathcal{ L}\left\langle \nabla _{b}\phi ,\nabla _{b}f\right\rangle -2n\left\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\right\rangle -4\langle Pf+\overline{P} f,d_{b}\phi \rangle \\ & & +4n(Qf-2\sqrt{-1}A^{\alpha \beta }f_{\alpha }\phi _{\beta })+2n\sqrt{-1} \left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle \\ & = & P_{0}^{\phi }f+2n^{2}\phi _{0}f_{0}+nQ^{\phi }f+2n\sqrt{-1} \left\langle \nabla _{b}\phi _{0},\nabla _{b}f\right\rangle , \end{array} \end{equation*} as desired. \end{proof}
In the following we show that if the pseudohermitian torsion of $M$ is zero and $\phi _{0}$ vanishes, then the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative for all smooth functions with Dirichlet boundary condition (\ref{1}).
\begin{lemma} \label{l41} Let $(M,J,\theta ,d\sigma )$ be a compact weighted Sasakian $ (2n+1)$-manifold with boundary $\Sigma $. If $\phi _{0}$ vanishes, then the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative for all smooth functions with Dirichlet boundary condition (\ref{1}). In particular, the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative in a closed weighted Sasakian $(2n+1)$-manifold if $\phi _{0}$ vanishes. \end{lemma}
\begin{proof} The zero pseudohermitian torsion and $\phi _{0}=0$ implies that the weighted CR Paneitz operator becomes $P_{0}^{\phi }=\square _{b}^{\phi }\overline{ \square }_{b}^{\phi }+\overline{\square }_{b}^{\phi }\square _{b}^{\phi },$ and the weighted Kohn Laplacian $\square _{b}^{\phi }$ and $\overline{ \square }_{b}^{\phi }$ commute, so they are diagonalized simultaneously on the finite dimensional eigenspace of $\square _{b}^{\phi }$ with respect to any nonzero eigenvalue. And we know that the eigenvalues of $\square _{b}^{\phi }$ (and thus of $\overline{\square }_{b}^{\phi }$) are all nonnegative, since for any real function $\varphi $ with $\varphi =0$ on $ \Sigma $ \begin{equation*} \begin{array}{lll} \int_{M}\varphi \square _{b}^{\phi }\varphi d\sigma & = & \int_{M}\varphi (- \mathcal{L}\varphi +n\sqrt{-1}\varphi _{0})d\sigma \\ & = & \int_{M}[\left\vert \nabla _{b}\varphi \right\vert ^{2}+\frac{1}{2}n \sqrt{-1}\varphi ^{2}\phi _{0}]d\sigma -\frac{1}{2}C_{n}\int_{\Sigma }\varphi \lbrack \varphi _{e_{_{2n}}}+n\sqrt{-1}\alpha \varphi ]d\Sigma _{p}^{\phi } \\ & = & \int_{M}\left\vert \nabla _{b}\varphi \right\vert ^{2}d\sigma , \end{array} \end{equation*} here we used the condition $\phi _{0}$ vanishes on $M$. Therefore, $ P_{0}^{\phi }$ is nonnegative. \end{proof}
\begin{lemma} Let $(M,J,\theta )$ be a compact Sasakian $3$-manifold with the smooth boundary $\Sigma $. Then for the first eigenfunction $f$ of Dirichlet eigenvalue problem \begin{equation*} \left\{ \begin{array}{ccll} \Delta _{b}f & = & -\lambda _{1}f & \mathrm{on\ }M, \\ f & = & 0 & \mathrm{on\ }\Sigma , \end{array} \right. \end{equation*} we have \begin{equation*} \begin{array}{c} \int_{M}fP_{0}fd\mu =\frac{16}{\lambda _{1}}\int_{M}\left\vert P_{1}f\right\vert ^{2}d\mu \geq 0. \end{array} \end{equation*} \end{lemma}
\begin{proof} It follows from the formula $\left[ \Delta _{b},T\right] f=4{\func{Im}}[ \sqrt{-1}\left( A_{\overline{1}\overline{1}}f_{1}\right) ,_{1}]=0$ that $ \Delta _{b}f_{0}=-\lambda _{1}f_{0}.$ From the divergence theorem, we compute \begin{equation} \begin{array}{l} \int_{M}[\left\vert \nabla _{b}f\right\vert ^{2}-\lambda _{1}f^{2}]d\mu =\int_{M}\left\vert \nabla _{b}f\right\vert ^{2}+f\Delta _{b}fd\mu =\int_{\Sigma }ff_{e_{2}}\theta \wedge e^{1}=0 \end{array} \label{27} \end{equation} and \begin{equation} \begin{array}{c} \int_{M}[\left\vert \nabla _{b}f_{0}\right\vert ^{2}-\lambda _{1}f_{0}^{2}]d\mu =\int_{M}\left\vert \nabla _{b}f_{0}\right\vert ^{2}+f_{0}\Delta _{b}f_{0}d\mu =\int_{\Sigma }f_{0}f_{0e_{2}}\theta \wedge e^{1}=0, \end{array} \label{28} \end{equation} here we used the last equation is zero which will be showed later. From the identity $P_{0}f=2\left( \Delta _{b}^{2}f+f_{00}\right) =2\left( \lambda _{1}^{2}f+f_{00}\right) $ on $M$ and from (\ref{c}), we have \begin{equation} \begin{array}{c} \int_{M}fP_{0}fd\mu =2\int_{M}\left( \lambda _{1}^{2}f^{2}-f_{0}^{2}\right) d\mu -4\int_{\Sigma }\alpha ff_{0}\theta \wedge e^{1}=2\int_{M}\left( \lambda _{1}^{2}f^{2}-f_{0}^{2}\right) d\mu . \end{array} \label{29} \end{equation} Also from the identity $P_{1}f=f_{\overline{1}11}=\frac{1}{2}\left( \Delta _{b}f-\sqrt{-1}f_{0}\right) ,_{1}=-\frac{1}{2}\left( \lambda _{1}f_{1}+\sqrt{ -1}f_{01}\right) ,$ we get \begin{equation*} \begin{array}{c} \left\vert P_{1}f\right\vert ^{2}=\frac{1}{8}[\lambda _{1}^{2}\left\vert \nabla _{b}f\right\vert ^{2}+\left\vert \nabla _{b}f_{0}\right\vert ^{2}+2\lambda _{1}\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle ]. \end{array} \end{equation*} Thus, from (\ref{27}), (\ref{28}), (\ref{d}) and (\ref{29}), we have \begin{equation*} \begin{array}{lll} 8\int_{M}\left\vert P_{1}f\right\vert ^{2}d\mu & = & \int_{M}[\lambda _{1}^{2}\left\vert \nabla _{b}f\right\vert ^{2}+\left\vert \nabla _{b}f_{0}\right\vert ^{2}-2\lambda _{1}f_{0}^{2}]d\mu \\ & = & \int_{M}[\lambda _{1}^{3}f^{2}-\lambda _{1}f_{0}^{2}]d\mu =\frac{1}{2} \lambda _{1}\int_{M}fP_{0}fd\mu , \end{array} \end{equation*} as desired.
In the following we claim that $\int_{\Sigma }f_{0}f_{0e_{2}}\theta \wedge e^{1}=0.$ By using the equation \begin{equation} \begin{array}{c} \int_{M}\psi _{0}d\mu =\int_{M}\mathrm{div}_{b}\left( J\nabla _{b}\psi \right) d\mu =\int_{\Sigma }\psi _{e_{1}}\theta \wedge e^{1}=0 \end{array} \label{30} \end{equation} for any real function $\psi $ which vanishes on $\Sigma $ to get that $ \int_{M}\left( f^{2}\right) _{0}d\mu =0,$ and from \begin{equation*} \begin{array}{c} \int_{M}[f\Delta _{b}f_{0}+\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle ]d\mu =\int_{\Sigma }ff_{0e_{2}}\theta \wedge e^{1}=0, \end{array} \end{equation*} one obtains $\int_{M}\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle d\mu =0$. Thus \begin{equation} \begin{array}{c} 0=\int_{M}[f_{0}\Delta _{b}f+\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle ]d\mu =\int_{\Sigma }f_{0}f_{e_{2}}\theta \wedge e^{1}. \end{array} \label{32} \end{equation} On the other hand, since $f=0$ on $\Sigma $ and $\alpha e_{2}+T$ is tangent along $\Sigma $ from the definition of $\alpha ,$ so on $\Sigma $ we have \begin{equation*} \begin{array}{c} f_{0}=-\alpha f_{e_{2}} \end{array} \end{equation*} and \begin{equation*} \begin{array}{c} -\alpha \theta \wedge e^{1}=e^{1}\wedge e^{2}=\frac{1}{2}d\theta \end{array} \end{equation*} which is a nonnegative $2$-form on $\Sigma $. It follows from (\ref{32}) that \begin{equation*} \begin{array}{c} 0=\int_{\Sigma }f_{0}f_{e_{2}}\theta \wedge e^{1}=-\int_{\Sigma }\alpha f_{e_{2}}^{2}\theta \wedge e^{1}=\int_{\Sigma }f_{e_{2}}^{2}e^{1}\wedge e^{2} \end{array} \end{equation*} to imply that \begin{equation*} \begin{array}{l} \int_{\Sigma }f_{0}f_{0e_{2}}\theta \wedge e^{1}=-\int_{\Sigma }\alpha f_{e_{2}}f_{0e_{2}}\theta \wedge e^{1}=\int_{\Sigma }f_{e_{2}}f_{0e_{2}}e^{1}\wedge e^{2}=0, \end{array} \end{equation*} here we used the H\H{o}lder's inequality in the last equation. \end{proof}
Finally, we recall the Lemma\ 4.1 in \cite{ccw}.
\begin{lemma} \label{lemma 3.1}Let $(M,J,\theta )$ be a compact strictly pseudoconvex CR $ (2n+1)$-manifold with the smooth boundary $\Sigma $ if $n\geq 2$. Then for the first eigenfunction $f$ of Dirichlet eigenvalue problem \begin{equation*} \left\{ \begin{array}{ccll} \Delta _{b}f & = & -\lambda _{1}f & \mathrm{on\ }M, \\ f & = & 0 & \mathrm{on\ }\Sigma , \end{array} \right. \end{equation*} we have \begin{equation*} \begin{array}{l}
\frac{n-1}{8n}\int_{M}fP_{0}fd\mu =\int_{M}\sum_{\beta ,\gamma }|f_{\beta \overline{\gamma }}-\frac{1}{n}f_{\sigma }{}^{\sigma }h_{\beta \overline{
\gamma }}|^{2}d\mu +\frac{1}{16}C_{n}\int_{\Sigma }(H_{p.h}+\tilde{\omega} _{n}^{\;n}(e_{n}))f_{e_{2n}}^{2}d\Sigma _{p} \end{array} \end{equation*} which implies \begin{equation*} \begin{array}{c} \int_{M}fP_{0}fd\mu \geq 0 \end{array} \end{equation*} if $H_{p.h}+\tilde{\omega}_{n}^{\;n}(e_{n})$ is nonnegative on $\Sigma $. \end{lemma}
\section{The Weighted CR Reilly Formula}
Let $M$ be a compact strictly pseudoconvex CR $(2n+1)$-manifold with boundary $\Sigma $. We write $\theta _{\gamma }^{\;\text{\ }\beta }=\omega _{\gamma }^{\;\text{\ }\beta }+\sqrt{-1}\tilde{\omega}_{\gamma }^{\;\text{\ } \beta }$ with $\omega _{\gamma }^{\;\text{\ }\beta }=\mathrm{Re}(\theta _{\gamma }^{\;\text{\ }\beta })$, $\tilde{\omega}_{\gamma }^{\;\text{\ } \beta }=\mathrm{Im}(\theta _{\gamma }^{\;\text{\ }\beta })$ and $Z_{\beta }= \frac{1}{2}(e_{\beta }-\sqrt{-1}e_{n+\beta })$ for real vectors $e_{\beta }$ , $e_{n+\beta }$, $\beta =1,\cdots ,n$. It follows that $e_{n+\beta }=Je_{\beta }$. Let $e^{\beta }=\mathrm{Re}(\theta ^{\beta })$, $e^{n+\beta }=\mathrm{Im}(\theta ^{\beta })$, $\beta =1,\cdots ,n$. Then $\{\theta ,e^{\beta },e^{n+\beta }\}$ is dual to $\{T,e_{\beta },e_{n+\beta }\}$. Now in view of (\ref{dtheta}) and (\ref{structure equs}), we have the following real version of structure equations: \begin{equation*} \left\{ \begin{array}{l} d\theta =2\sum_{\beta }e^{\beta }\wedge e^{n+\beta }, \\ \nabla e_{\gamma }=\omega _{\gamma }^{\;\text{\ }\beta }\otimes e_{\beta }+ \tilde{\omega}_{\gamma }^{\;\text{\ }\beta }\otimes e_{n+\beta },\text{ } \nabla e_{n+\gamma }=\omega _{\gamma }^{\;\text{\ }\beta }\otimes e_{n+\beta }-\tilde{\omega}_{\gamma }^{\;\text{\ }\beta }\otimes e_{\beta }, \\ de^{\gamma }=e^{\beta }\wedge \omega _{\beta }^{\;\text{\ }\gamma }-e^{n+\beta }\wedge \tilde{\omega}_{\beta }^{\;\text{\ }\gamma }\text{ \textrm{mod} }\theta ;\text{ }de^{n+\gamma }=e^{\beta }\wedge \tilde{\omega} _{\beta }^{\text{ \ }\gamma }+e^{n+\beta }\wedge \omega _{\beta }^{\text{ \ } \gamma }\text{ \textrm{mod} }\theta . \end{array} \right. \end{equation*}
Let $\Sigma $ be a surface contained in $M$. The singular set $S_{\Sigma }$ consists of those points where $\xi $ coincides with the tangent bundle $ T\Sigma $ of $\Sigma $. It is easy to see that $S_{\Sigma }$ is a closed set. On $\xi ,$ we can associate a natural metric $\langle $ $,$ $\rangle = \frac{1}{2}d\theta (\cdot ,J\cdot )$ call the Levi metric. For a vector $ v\in \xi ,$ we define the length of $v$ by $\left\vert v\right\vert ^{2}=\langle v,v\rangle .$ With respect to the Levi metric, we can take unit vector fields $e_{1},\cdots ,e_{2n-1}\in \xi \cap T\Sigma $ on $\Sigma \backslash S_{\Sigma }$, called the characteristic fields and $e_{2n}=Je_{n}$ , called the Legendrian normal. The $p$(pseudohermitian)-mean curvature $ H_{p.h}$ on $\Sigma \backslash S_{\Sigma }$ is defined by \begin{equation*} \begin{array}{c} H_{p.h}=\sum_{j=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{2n},e_{j}\right\rangle =-\sum_{j=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{j},e_{2n}\right\rangle . \end{array} \end{equation*} For $e_{1},\cdots ,e_{2n-1}$ being characteristic fields, we have the $p$ -area element \begin{equation*} \begin{array}{c} d\Sigma _{p}=\theta \wedge e^{1}\wedge e^{n+1}\wedge \cdots \wedge e^{n-1}\wedge e^{2n-1}\wedge e^{n} \end{array} \end{equation*} on $\Sigma $ and all surface integrals over $\Sigma $ are with respect to this $2n$-form $d\Sigma _{p}$. Note that $d\Sigma _{p}$ continuously extends over the singular set $S_{\Sigma }$ and vanishes on $S_{\Sigma }$.
We also write $f_{e_{j}}=e_{j}f$ and $\nabla _{b}f=\frac{1}{2}(f_{e_{\beta }}e_{\beta }+f_{e_{n+\beta }}e_{n+\beta })$. Moreover, $ f_{e_{j}e_{k}}=e_{k}e_{j}f-\nabla _{e_{k}}e_{j}f$ and $\Delta _{b}f=\frac{1}{ 2}\sum_{\beta }(f_{e_{\beta }e_{\beta }}+f_{e_{n+\beta }e_{n+\beta }})$. Next we define the subdivergence operator $div_{b}(\cdot )$ by $ div_{b}(W)=W^{\beta },_{\beta }+W^{\overline{\beta }},_{\overline{\beta }}$ for all vector fields $W=W^{\beta }Z_{\beta }+W^{\overline{\beta }}Z_{ \overline{\beta }}$ and its real version is $div_{b}(W)=\varphi _{\beta ,e_{\beta }}+\psi _{n+\beta ,e_{n+\beta }}$ for $W=\varphi _{\beta }e_{\beta }+\psi _{n+\beta }e_{n+\beta }$. We define the tangential subgradient $ \nabla _{b}^{t}$ of a function $f$ by $\nabla _{b}^{t}f=\nabla _{b}f-\langle \nabla _{b}f,e_{2n}\rangle e_{2n}$ and the tangent sub-Laplacian $\Delta _{b}^{t}$ of $f$ by $\Delta _{b}^{t}f=\frac{1}{2} \sum_{j=1}^{2n-1}[(e_{j})^{2}-(\nabla _{e_{j}}e_{j})^{t}]f,$ where $(\nabla _{e_{j}}e_{j})^{t}$ is the tangential part of $\nabla _{e_{j}}e_{j}$.
We first recall the following CR Bochner formula for $\Delta _{b}$.
\begin{lemma} Let $(M,J,\theta )$ be a strictly pseudoconvex CR $(2n+1)$-manifold. For a real function $f$, we have \begin{equation} \begin{array}{lll}
\frac{1}{2}\Delta _{b}|\nabla _{b}f|^{2} & = & |(\nabla
^{H})^{2}f|^{2}+\langle \nabla _{b}f,\nabla _{b}\Delta _{b}f\rangle \\ & & +[2Ric-(n-2)Tor](\nabla _{b}f_{\mathbb{C}},\nabla _{b}f_{\mathbb{C}}) \\ & & +2\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle , \end{array} \label{A} \end{equation} where $\nabla _{b}f_{\mathbb{C}}=f^{\beta }Z_{\beta }$ is the corresponding complex $(1,0)$-vector field of $\nabla _{b}f$. \end{lemma}
Now we derive the following CR Bochner formula for $\mathcal{L}$.
\begin{lemma} Let $(M,J,\theta )$ be a strictly pseudoconvex CR $(2n+1)$-manifold. For a real function $f$, we have \begin{equation} \begin{array}{lll}
\frac{1}{2}\mathcal{L}|\nabla _{b}f|^{2} & = & |(\nabla
^{H})^{2}f|^{2}+\langle \nabla _{b}f,\nabla _{b}\mathcal{L}f\rangle \\ & & +[Ric+(\nabla ^{H})^{2}\phi -\frac{n-2}{2}Tor](\nabla _{b}f,\nabla _{b}f) \\ & & +2\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle -f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle , \end{array} \label{Bochnerformula} \end{equation} where $(\nabla ^{H})^{2}\phi (\nabla _{b}f,\nabla _{b}f)=\phi _{\beta \gamma }f^{\beta }f^{\gamma }+\phi _{\overline{\beta }\overline{\gamma }}f^{ \overline{\beta }}f^{\overline{\gamma }}+\phi _{\beta \overline{\gamma } }f^{\beta }f^{\overline{\gamma }}+\phi _{\overline{\beta }\gamma }f^{ \overline{\beta }}f^{\gamma }.$ \end{lemma}
The proof of the above formula follows from the definition of $\mathcal{L}$ and the identity \begin{equation} \begin{array}{c} \langle \nabla _{b}f,\nabla _{b}\langle \nabla _{b}f,\nabla _{b}\phi \rangle
\rangle -\frac{1}{2}\langle \nabla _{b}\phi ,\nabla _{b}|\nabla _{b}f|^{2}\rangle =(\nabla ^{H})^{2}\phi (\nabla _{b}f,\nabla _{b}f)-f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle . \end{array} \label{10} \end{equation} Also we note that \begin{equation} \begin{array}{ccc} \langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle & = & \frac{1}{n}\langle \nabla _{b}f,\nabla _{b}\mathcal{L}f\rangle -\frac{2}{n}\langle P^{\phi }f+ \overline{P}^{\phi }f,d_{b}f\rangle \\ & & +f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle -Tor(\nabla _{b}f,\nabla _{b}f). \end{array} \label{10a} \end{equation} Then the CR Bochner formula for $\mathcal{L}$ becomes \begin{equation} \begin{array}{lll}
\frac{1}{2}\mathcal{L}|\nabla _{b}f|^{2} & = & |(\nabla ^{H})^{2}f|^{2}+ \frac{n+2}{n}\langle \nabla _{b}f,\nabla _{b}\mathcal{L}f\rangle \\ & & +[Ric+(\nabla ^{H})^{2}\phi -\frac{n+2}{2}Tor](\nabla _{b}f,\nabla _{b}f) \\ & & -\frac{4}{n}\langle P^{\phi }f+\overline{P}^{\phi }f,d_{b}f\rangle +f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle . \end{array} \label{11} \end{equation}
For the proof of the weighted CR Reilly formula, we need a series of formulae as we derived in \cite{ccw}.
\begin{lemma} Let $(M,J,\theta ,d\sigma )$ be a compact weighted strictly pseudoconvex CR $ (2n+1)$-manifold with boundary $\Sigma $. For real functions $f$ and $g$, we have \begin{equation} \begin{array}{c} \int_{M}\left( \mathcal{L}f\right) d\sigma =\frac{1}{2}C_{n}\int_{\Sigma }f_{e_{_{2n}}}d\Sigma _{p}^{\phi }, \end{array} \label{a} \end{equation} \begin{equation} \begin{array}{c} \int_{M}[g\mathcal{L}f+\langle \nabla _{b}f,\nabla _{b}g\rangle ]d\sigma = \frac{1}{2}C_{n}\int_{\Sigma }gf_{e_{_{2n}}}d\Sigma _{p}^{\phi }, \end{array} \label{b} \end{equation} \begin{equation} \begin{array}{c} \int_{M}[ff_{00}-ff_{0}\phi _{0}+f_{0}^{2}]d\sigma =-C_{n}\int_{\Sigma }\alpha ff_{0}d\Sigma _{p}^{\phi }, \end{array} \label{c} \end{equation} \begin{equation} \begin{array}{c} \int_{M}[\langle J\nabla _{b}f,\nabla _{b}f_{0}\rangle -f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle +nf_{0}^{2}]d\sigma =\frac{1}{2} C_{n}\int_{\Sigma }f_{0}f_{e_{n}}d\Sigma _{p}^{\phi }, \end{array} \label{d} \end{equation} \begin{equation} \begin{array}{c} \int_{M}[\langle P^{\phi }f+\overline{P}^{\phi }f,d_{b}f\rangle +\frac{1}{4} fP_{0}^{\phi }f]d\sigma =\frac{1}{2}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{\overline{n}}^{\phi }f)d\Sigma _{p}^{\phi }. \end{array} \label{e} \end{equation} Here$\ d\sigma =e^{-\phi }\theta \wedge (d\theta )^{n}$ is the weighted volume measure and $d\Sigma _{p}^{\phi }=e^{-\phi }d\Sigma _{p}$ is the weighted $p$-area element of $\Sigma ,$ and $C_{n}=2^{n}n!$. \end{lemma}
\begin{lemma} Let $(M,J,\theta ,d\sigma )$ be a compact weighted pseudohermitian $(2n+1)$ -manifold with boundary $\Sigma $. For any real-valued function $f$ on $ \Sigma ,$ we have \begin{equation} \begin{array}{c} \int_{\Sigma }[f_{e_{n}}+(2\alpha -\phi _{e_{n}})f]d\Sigma _{p}^{\phi }=0, \end{array} \label{f} \end{equation} \begin{equation} \begin{array}{c} \int_{\Sigma }[f_{\overline{\beta }}+(\sum_{\gamma \neq n}\theta _{\overline{ \beta }}^{\;\text{\ }\overline{\gamma }}(Z_{\overline{\gamma }})+\frac{1}{2} \theta _{\overline{\beta }}^{\;\text{\ }\overline{n}}(e_{n})-\phi _{ \overline{\beta }})f]d\Sigma _{p}^{\phi }=0\text{ \textrm{for any} }\beta \neq n. \end{array} \label{g} \end{equation} \end{lemma}
\textbf{The Proof of Theorem}\textup{\textbf{\ \ref{Reilly'sformula}:}}
\begin{proof} By integrating the CR Bochner formula (\ref{11}) for $\mathcal{L}$, from ( \ref{b}) and (\ref{e}), we have \begin{equation*} \begin{array}{lll}
\frac{1}{2}\int_{M}\mathcal{L}|\nabla _{b}f|^{2}d\sigma & = &
\int_{M}|(\nabla ^{H})^{2}f|^{2}d\sigma +\int_{M}[Ric+(\nabla ^{H})^{2}\phi - \frac{n+2}{2}Tor](\nabla _{b}f,\nabla _{b}f)d\sigma \\ & & -\frac{n+2}{n}\int_{M}(\mathcal{L}f)^{2}d\sigma +\frac{1}{n} \int_{M}fP_{0}^{\phi }fd\sigma +\int_{M}f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle d\sigma \\ & & +\frac{n+2}{2n}C_{n}\int_{\Sigma }(\mathcal{L}f)f_{e_{_{2n}}}d\Sigma _{p}^{\phi }-\frac{2}{n}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{ \overline{n}}^{\phi }f)d\Sigma _{p}^{\phi }. \end{array} \end{equation*} By combining (\ref{10a}), (\ref{d}) and (\ref{e}), we have \begin{equation} \begin{array}{lll} n^{2}\int_{M}f_{0}^{2}d\sigma & = & \int_{M}\left( \mathcal{L}f\right) ^{2}d\sigma -\frac{1}{2}\int_{M}fP_{0}^{\phi }fd\sigma +n\int_{M}Tor(\nabla _{b}f,\nabla _{b}f)d\sigma \\ & & +\frac{1}{2}C_{n}\int_{\Sigma }[nf_{0}f_{e_{n}}-(\mathcal{L} f)f_{e_{_{2n}}}]d\Sigma _{p}^{\phi }+\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{\overline{n}}^{\phi }f)d\Sigma _{p}^{\phi }. \end{array} \label{7} \end{equation} Also by (\ref{10}) and (\ref{b}), one gets \begin{equation} \begin{array}{ll} & \int_{M}f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle d\sigma \\ = & \int_{M}(\nabla ^{H})^{2}\phi (\nabla _{b}f,\nabla _{b}f)d\sigma
+\int_{M}\left[ \left( \mathcal{L}f\right) \langle \nabla _{b}f,\nabla _{b}\phi \rangle -\frac{1}{2}\left( \mathcal{L}\phi \right) |\nabla _{b}f|^{2}\right] d\sigma \\
& +\frac{1}{4}C_{n}\int_{\Sigma }[|\nabla _{b}f|^{2}\phi _{e_{_{2n}}}-2\langle \nabla _{b}f,\nabla _{b}\phi \rangle f_{e_{_{2n}}}]d\Sigma _{p}^{\phi }. \end{array} \label{8} \end{equation}
Note that $|(\nabla ^{H})^{2}f|^{2}=2\sum_{\beta ,\gamma }[|f_{\beta \gamma
}|^{2}+|f_{\beta \overline{\gamma }}|^{2}]$ and \begin{equation*} \begin{array}{c}
\sum_{\beta ,\gamma }|f_{\beta \overline{\gamma }}|^{2}=\sum_{\beta ,\gamma
}|f_{\beta \overline{\gamma }}-\frac{1}{n}f_{\sigma }{}^{\sigma }h_{\beta
\overline{\gamma }}|^{2}+\frac{1}{4n}\left( \Delta _{b}f\right) ^{2}+\frac{n }{4}f_{0}^{2} \end{array} \end{equation*} with $\Delta _{b}f=\mathcal{L}f+\langle \nabla _{b}f,\nabla _{b}\phi \rangle .$ It follows from (\ref{7}) and (\ref{8})\ that \begin{equation} \begin{array}{ll}
& \frac{1}{2}\int_{M}\mathcal{L}|\nabla _{b}f|^{2}d\sigma \\
= & 2\int_{M}\sum_{\beta ,\gamma }\left[ |f_{\beta \gamma }|^{2}+|f_{\beta \overline{\gamma }}-\frac{1}{n}f_{\sigma }{}^{\sigma }h_{\beta \overline{
\gamma }}|^{2}\right] d\sigma -\frac{n+1}{n}\int_{M}(\mathcal{L}f)^{2}d\sigma \\ & +\int_{M}[Ric+2(\nabla ^{H})^{2}\phi -\frac{n+1}{2}Tor](\nabla _{b}f,\nabla _{b}f)d\sigma +\frac{3}{4n}\int_{M}fP_{0}^{\phi }fd\sigma \\
& +\frac{1}{2n}\int_{M}[2(n+1)\left( \mathcal{L}f\right) \langle \nabla _{b}f,\nabla _{b}\phi \rangle -n\left( \mathcal{L}\phi \right) |\nabla _{b}f|^{2}+\langle \nabla _{b}f,\nabla _{b}\phi \rangle ^{2}]d\sigma \\ & +\frac{2n+3}{4n}C_{n}\int_{\Sigma }(\mathcal{L}f)f_{e_{_{2n}}}d\Sigma _{p}^{\phi }-\frac{3}{2n}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{ \overline{n}}^{\phi }f)d\Sigma _{p}^{\phi } \\
& +\frac{1}{4}C_{n}\int_{\Sigma }[f_{0}f_{e_{n}}+|\nabla _{b}f|^{2}\phi _{e_{_{2n}}}-2\langle \nabla _{b}f,\nabla _{b}\phi \rangle f_{e_{_{2n}}}]d\Sigma _{p}^{\phi }. \end{array} \label{9} \end{equation}
In the following, we deal with the term $\int_{M}\sum_{\beta ,\gamma
}|f_{\beta \overline{\gamma }}-\frac{1}{n}f_{\sigma }{}^{\sigma }h_{\beta
\overline{\gamma }}|^{2}d\sigma .$ The divergence formula for the trace-free part of $f_{\beta \overline{\gamma }}$: \begin{equation*} \begin{array}{c} B_{\beta \overline{\gamma }}f=f_{\beta \overline{\gamma }}-\frac{1}{n} f_{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}, \end{array} \end{equation*} is given by \begin{equation*} \begin{array}{lll} (B^{\beta \overline{\gamma }}f)(B_{\beta \overline{\gamma }}f) & = & (f^{\beta }B_{\beta \overline{\gamma }}f),^{\overline{\gamma }}-\frac{n-1}{n} (fP_{\beta }f),^{\beta }+\frac{n-1}{8n}fP_{0}f \\ & = & e^{\phi }(e^{-\phi }f^{\beta }B_{\beta \overline{\gamma }}f),^{ \overline{\gamma }}-\frac{n-1}{n}e^{\phi }(e^{-\phi }fP_{\beta }f),^{\beta }+ \frac{n-1}{8n}fP_{0}f \\ & & -\frac{n-1}{2n}f\langle Pf+\overline{P}f,d_{b}\phi \rangle +\frac{1}{2} (f^{\beta }\phi ^{\overline{\gamma }}B_{\beta \overline{\gamma }}f+f^{ \overline{\beta }}\phi ^{\gamma }B_{\overline{\beta }\gamma }f). \end{array} \end{equation*}
By using the identities $\frac{1}{2}\langle \nabla _{b}\phi ,\nabla _{b}|\nabla _{b}f|^{2}\rangle =f_{\overline{\beta }\gamma }f^{\overline{ \beta }}\phi ^{\gamma }+f_{\beta \overline{\gamma }}f^{\beta }\phi ^{ \overline{\gamma }}+f_{\beta \gamma }f^{\beta }\phi ^{\gamma }+f_{\overline{ \beta }\overline{\gamma }}f^{\overline{\beta }}\phi ^{\overline{\gamma }}$ and \begin{equation*} \begin{array}{c} f_{\sigma }{}^{\sigma }f_{\overline{\beta }}\phi ^{\overline{\beta }}+f_{ \overline{\sigma }}{}^{\overline{\sigma }}f_{\beta }\phi ^{\beta }=\frac{1}{2 }[\Delta _{b}f\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle +nf_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle ], \end{array} \end{equation*} and from (\ref{4a}), we get \begin{equation*} \begin{array}{ll}
& \sum_{\beta ,\gamma }|f_{\beta \overline{\gamma }}-\frac{1}{n}f_{\sigma
}{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}\text{ }=\text{ }(B^{\beta \overline{\gamma }}f)(B_{\beta \overline{\gamma }}f) \\ = & e^{\phi }(e^{-\phi }f^{\beta }B_{\beta \overline{\gamma }}f),^{\overline{ \gamma }}-\frac{n-1}{n}e^{\phi }(e^{-\phi }fP_{\beta }f),^{\beta }+\frac{n-1
}{8n}fP_{0}^{\phi }f+\frac{1}{4}\langle \nabla _{b}\phi ,\nabla _{b}|\nabla _{b}f|^{2}\rangle \\ & -\frac{1}{2}(f_{\beta \gamma }f^{\beta }\phi ^{\gamma }+f_{\overline{\beta }\overline{\gamma }}f^{\overline{\beta }}\phi ^{\overline{\gamma }})-\frac{1 }{4n}[\Delta _{b}f\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle +nf_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle ] \\ & +\frac{n-1}{4n}f[\mathcal{L}\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle +n\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\rangle +n^{2}f_{0}\phi _{0}]. \end{array} \end{equation*} By integrate both sides of the above equation, then apply the equation \begin{equation*} \begin{array}{c} \int_{M}f[\langle J\nabla _{b}\phi ,\nabla _{b}f_{0}\rangle +nf_{0}\phi _{0}]d\sigma =\int_{M}f_{0}\langle J\nabla _{b}f,\nabla _{b}\phi \rangle d\sigma +\frac{1}{2}C_{n}\int_{\Sigma }ff_{0}\phi _{e_{_{n}}}d\Sigma _{p}^{\phi }, \end{array} \end{equation*} use (\ref{40}) to get \begin{equation*} \begin{array}{c} \sqrt{-1}(P_{n}f-P_{\overline{n}}f)=\sqrt{-1}(P_{n}^{\phi }f-P_{\overline{n} }^{\phi }f)-\frac{1}{2}\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle _{e_{2n}}+\frac{n}{2}f_{0}\phi _{e_{n}} \end{array} \end{equation*} and (\ref{8}) again, we obtain \begin{equation} \begin{array}{ll}
& \int_{M}\sum_{\beta ,\gamma }|f_{\beta \overline{\gamma }}-\frac{1}{n}
f_{\sigma }{}^{\sigma }h_{\beta \overline{\gamma }}|^{2}d\sigma \\ = & \frac{n-1}{8n}\int_{M}fP_{0}^{\phi }fd\sigma +\frac{n-2}{4} \int_{M}(\nabla ^{H})^{2}\phi (\nabla _{b}f,\nabla _{b}f)d\sigma -\frac{n}{8}
\int_{M}\left( \mathcal{L}\phi \right) |\nabla _{b}f|^{2}d\sigma \\ & +\frac{(n-2)(n+1)}{4n}\int_{M}\left( \mathcal{L}f\right) \langle \nabla _{b}f,\nabla _{b}\phi \rangle d\sigma -\frac{1}{4n}\int_{M}\langle \nabla _{b}f,\nabla _{b}\phi \rangle ^{2}d\sigma \\ & -\frac{1}{2}\int_{M}(f_{\beta \gamma }f^{\beta }\phi ^{\gamma }+f_{ \overline{\beta }\overline{\gamma }}f^{\overline{\beta }}\phi ^{\overline{ \gamma }})d\sigma +\frac{1}{4}\sqrt{-1}C_{n}\int_{\Sigma }(f^{\overline{ \beta }}B_{n\overline{\beta }}f-f^{\beta }B_{\overline{n}\beta }f)d\Sigma _{p}^{\phi } \\ & -\frac{n-1}{4n}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{\overline{n} }^{\phi }f)d\Sigma _{p}^{\phi }+\frac{n-1}{4n}C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle _{e_{2n}}fd\Sigma _{p}^{\phi } \\
& +\frac{n}{16}C_{n}\int_{\Sigma }|\nabla _{b}f|^{2}\phi _{e_{_{2n}}}d\Sigma _{p}^{\phi }-\frac{n^{2}-n-1}{8n}C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle f_{e_{2n}}d\Sigma _{p}^{\phi }. \end{array} \label{6} \end{equation} Substituting these into the right hand side of (\ref{9}), we final get \begin{equation} \begin{array}{ll}
& \frac{1}{2}\int_{M}\mathcal{L}|\nabla _{b}f|^{2}d\sigma \\
= & 2\int_{M}\sum_{\beta ,\gamma }|f_{\beta \gamma }-\frac{1}{2}f_{\beta
}\phi _{\gamma }|^{2}d\sigma -\frac{n+1}{n}\int_{M}(\mathcal{L}f)^{2}d\sigma +\frac{n+2}{4n}\int_{M}fP_{0}^{\phi }fd\sigma \\ & +\int_{M}[Ric+2(\nabla ^{H})^{2}\phi -\frac{n+1}{2}Tor](\nabla _{b}f,\nabla _{b}f)d\sigma +\frac{n+1}{2}\int_{M}\left( \mathcal{L}f\right) \langle \nabla _{b}f,\nabla _{b}\phi \rangle d\sigma \\
& -\frac{n+2}{4}\int_{M}[\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi
|^{2}]|\nabla _{b}f|^{2}d\sigma +\frac{2n+3}{4n}C_{n}\int_{\Sigma }(\mathcal{ L}f)f_{e_{_{2n}}}d\Sigma _{p}^{\phi } \\ & +\frac{1}{4}C_{n}\int_{\Sigma }f_{0}f_{e_{n}}d\Sigma _{p}^{\phi }-\frac{n+2 }{2n}\sqrt{-1}C_{n}\int_{\Sigma }f(P_{n}^{\phi }f-P_{\overline{n}}^{\phi }f)d\Sigma _{p}^{\phi } \\ & +\frac{1}{2}\sqrt{-1}C_{n}\int_{\Sigma }(f^{\overline{\beta }}B_{n \overline{\beta }}f-f^{\beta }B_{\overline{n}\beta }f)d\Sigma _{p}^{\phi }+
\frac{n+2}{8}C_{n}\int_{\Sigma }|\nabla _{b}f|^{2}\phi _{e_{_{2n}}}d\Sigma _{p}^{\phi } \\ & +\frac{n-1}{2n}C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle _{e_{2n}}fd\Sigma _{p}^{\phi }-\frac{n^{2}+n-1}{4n} C_{n}\int_{\Sigma }\left\langle \nabla _{b}f,\nabla _{b}\phi \right\rangle f_{e_{2n}}d\Sigma _{p}^{\phi }. \end{array} \label{20} \end{equation}
On the other hand, the divergence theorem (\ref{a}) implies that \begin{equation*} \begin{array}{ll}
& 2\int_{M}\mathcal{L}|\nabla _{b}f|^{2}d\sigma \text{ }=\text{ }
C_{n}\int_{\Sigma }\left( |\nabla _{b}f|^{2}\right) _{e_{2n}}d\Sigma _{p}^{\phi } \\ = & C_{n}\int_{\Sigma }[\sum_{\beta \neq n}\left( f_{e_{\beta }}f_{e_{\beta }e_{2n}}+f_{e_{n+\beta }}f_{e_{n+\beta }e_{2n}}\right) +f_{e_{n}}f_{e_{n}e_{2n}}+f_{e_{2n}}f_{e_{2n}e_{2n}}]d\Sigma _{p}^{\phi }. \end{array} \end{equation*} Substituting the commutation relations $f_{e_{\beta }e_{n+\gamma }}=f_{e_{n+\gamma }e_{\beta }},\ f_{e_{n+\beta }e_{n+\gamma }}=f_{e_{n+\gamma }e_{n+\beta }}\ $for all ${\small \beta \neq \gamma }$, $ f_{e_{n}e_{2n}}=f_{e_{2n}e_{n}}+2f_{0}$ and \begin{equation} \begin{array}{c} \sum_{\beta \neq n}2(f_{\beta \overline{\beta }}+f_{\overline{\beta }\beta })+f_{e_{n}e_{n}}=\sum_{j=1}^{2n-1}f_{e_{j}e_{j}}=2\Delta _{b}^{t}f+H_{p.h}f_{e_{2n}} \\ f_{e_{2n}e_{2n}}=2\Delta _{b}f-\sum_{j=1}^{2n-1}f_{e_{j}e_{j}}=2\Delta _{b}f-2\Delta _{b}^{t}f-H_{p.h}f_{e_{2n}}, \end{array} \label{21} \end{equation} into the above equation, then integrating by parts from (\ref{f}) and (\ref {g}) yields \begin{equation*} \begin{array}{ll}
& 2\int_{M}\mathcal{L}|\nabla _{b}f|^{2}d\sigma \\ = & C_{n}\int_{\Sigma }\sum_{\beta \neq n}(f_{e_{\beta }}f_{e_{2n}e_{\beta }}+f_{e_{n+\beta }}f_{e_{2n}e_{n+\beta }})d\Sigma _{p}^{\phi }+C_{n}\int_{\Sigma }[f_{e_{n}}(f_{e_{2n}e_{n}}+2f_{0})+f_{e_{2n}}f_{e_{2n}e_{2n}}]d\Sigma _{p}^{\phi } \\ = & C_{n}\int_{\Sigma }[\sum_{\beta \neq n}2(f_{\overline{\beta } }f_{e_{2n}Z_{\beta }}+f_{\beta }f_{e_{2n}Z\overline{_{\beta }} })+f_{e_{n}}f_{e_{2n}e_{n}}]d\Sigma _{p}^{\phi }+C_{n}\int_{\Sigma }[2f_{e_{n}}f_{0}+f_{e_{2n}}f_{e_{2n}e_{2n}}]d\Sigma _{p}^{\phi } \\ = & C_{n}\int_{\Sigma }[\sum_{\beta \neq n}2(f_{\beta }\phi _{\overline{ \beta }}+f_{\overline{\beta }}\phi _{\beta }-f_{\beta \overline{\beta }}-f_{ \overline{\beta }\beta })+f_{e_{n}}\phi _{e_{n}}-f_{e_{n}e_{n}}]f_{e_{2n}}d\Sigma _{p}^{\phi } \\ & +C_{n}\int_{\Sigma }[2f_{e_{n}}(f_{0}-\alpha f_{e_{2n}})+f_{e_{2n}}(2\Delta _{b}f-2\Delta _{b}^{t}f-H_{p.h}f_{e_{2n}})]d\Sigma _{p}^{\phi } \\ & -C_{n}\int_{\Sigma }[f_{e_{n}}(\nabla _{e_{n}}e_{2n})f+f_{e_{2n}}(\nabla _{e_{n}}e_{n})f+2\sum_{\beta \neq n}(f_{\beta }(\nabla _{Z_{\overline{\beta } }}e_{2n})f+f_{\overline{\beta }}(\nabla _{Z_{\beta }}e_{2n})f)]d\Sigma _{p}^{\phi } \\ & +2C_{n}\int_{\Sigma }\sum_{\beta \neq n}[\theta _{\overline{n}}{}^{ \overline{\beta }}(Z_{\overline{\beta }})f_{n}-\frac{1}{2}\theta _{\overline{ \beta }}{}^{\overline{n}}(e_{n})f_{\beta }+\theta _{n}{}^{\beta }(Z_{\beta })f_{\overline{n}}-\frac{1}{2}\theta _{\beta }{}^{n}(e_{n})f_{\overline{ \beta }}]f_{e_{2n}}d\Sigma _{p}^{\phi } \\ = & 2C_{n}\int_{\Sigma }f_{e_{2n}}\left( \Delta _{b}f-2\Delta _{b}^{t}f\right) d\Sigma _{p}^{\phi }-C_{n}\int_{\Sigma }H_{p.h}f_{e_{2n}}^{2}d\Sigma _{p}^{\phi }+2C_{n}\int_{\Sigma }[\alpha f_{e_{n}}f_{e_{2n}}-f_{0}f_{e_{n}}]d\Sigma _{p}^{\phi } \\ & +C_{n}\int_{\Sigma }\sum_{j=1}^{2n-1}[\left\langle \nabla _{e_{j}}e_{n},e_{j}\right\rangle f_{e_{n}}+f_{e_{j}}\phi _{e_{j}}]f_{e_{2n}}d\Sigma _{p}^{\phi }-C_{n}\int_{\Sigma }\sum_{j,k=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{2n},e_{k}\right\rangle f_{e_{j}}f_{e_{k}}d\Sigma _{p}^{\phi }, \end{array} \end{equation*} here we use the equations \begin{equation*} \begin{array}{ll} & 2\sum_{\beta \neq n}[\theta _{\overline{n}}{}^{\overline{\beta }}(Z_{ \overline{\beta }})f_{n}-\frac{1}{2}\theta _{\overline{\beta }}{}^{\overline{ n}}(e_{n})f_{\beta }+\theta _{n}{}^{\beta }(Z_{\beta })f_{\overline{n}}- \frac{1}{2}\theta _{\beta }{}^{n}(e_{n})f_{\overline{\beta }}] \\ = & \sum_{j=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{n},e_{j}\right\rangle f_{e_{n}}+(\nabla _{e_{n}}e_{n})f+H_{p.h}f_{e_{2n}} \end{array} \end{equation*} and \begin{equation*} \begin{array}{c} \sum_{\beta \neq n}2[f_{\beta }(\nabla _{Z_{\overline{\beta }}}e_{2n})f+f_{ \overline{\beta }}(\nabla _{Z_{\beta }}e_{2n})f]+f_{e_{n}}(\nabla _{e_{n}}e_{2n})f=\sum_{j,k=1}^{2n-1}\left\langle \nabla _{e_{j}}e_{2n},e_{k}\right\rangle f_{e_{j}}f_{e_{k}}, \end{array} \end{equation*} the fact that (\ref{21}) holds only on $\Sigma \backslash S_{\Sigma }.$ However, $d\Sigma _{p}^{\phi }$ can be continuously extends over the singular set $S_{\Sigma }$ and vanishes on $S_{\Sigma }.$ Finally, by combining the above integral into (\ref{20}), we can then obtain (\ref{0}). This completes the proof of the Theorem. \end{proof}
\section{First Eigenvalue Estimate and Weighted Obata Theorem}
In this section, by applying the weighted CR Reilly formula, we give the first eigenvalue estimate and derive the Obata-type theorem in a closed weighted strictly pseudoconvex CR $(2n+1)$-manifold.
\textbf{The Proof of Theorem}\textup{\textbf{\ \ref{Thm}:}}
\begin{proof} Under the curvature condition $[Ric(\mathcal{L})-\frac{n+1}{2}Tor(\mathcal{L}
)](\nabla _{b}f_{\mathbb{C}},\nabla _{b}f_{\mathbb{C}})\geq k|\nabla _{b}f|^{2}$ for a positive constant $k$ and nonnegative weighted CR Paneitz operator $P_{0}^{\phi },$ the integral formula (\ref{0}) yields \begin{equation*} \begin{array}{ll} & \frac{n+1}{n}\lambda _{1}^{2}\int_{M}f^{2}d\sigma \\
\geq & k\int_{M}|\nabla _{b}f|^{2}d\sigma -\frac{n+1}{4}\lambda _{1}\int_{M}\langle \nabla _{b}f^{2},\nabla _{b}\phi \rangle d\sigma
-(n+2)l\int_{M}|\nabla _{b}f|^{2}d\sigma \\ = & (k-(n+2)l)\lambda _{1}\int_{M}f^{2}d\sigma +\frac{n+1}{4}\lambda _{1}\int_{M}\phi \mathcal{L}f^{2}d\sigma \\
= & (k-(n+2)l)\lambda _{1}\int_{M}f^{2}d\sigma +\frac{n+1}{2}\lambda _{1}\int_{M}\phi \lbrack |\nabla _{b}f|^{2}-\lambda _{1}f^{2}]d\sigma \\
\geq & (k-(n+2)l)\lambda _{1}\int_{M}f^{2}d\sigma +\frac{n+1}{2}\lambda _{1}\int_{M}[(\inf \phi )|\nabla _{b}f|^{2}-(\sup \phi )\lambda _{1}f^{2}]d\sigma \\ = & [(k-(n+2)l)\lambda _{1}-\frac{n+1}{2}\omega \lambda _{1}^{2}]\int_{M}f^{2}d\sigma , \end{array} \end{equation*}
here we assume $\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l$ for some nonnegative constant $l$ and let $\omega =\underset{M}{\mathrm{ osc}}\phi =\underset{M}{\sup }\phi -\underset{M}{\inf }\phi $. It implies\ that the first eigenvalue $\lambda _{1}$ will satisfies \begin{equation*} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}. \end{array} \end{equation*}
Moreover, if the above inequality becomes equality, then $\omega =\underset{M }{\mathrm{osc}}\phi =0$ and thus the weighted function $\phi $ will be constant, we have $\mathcal{L=}$ $\Delta _{b}$ and $P_{0}^{\phi }=P_{0}$. In this case we can let $l=0,$ then the first eigenvalue of the sub-Laplacian $ \Delta _{b}$ achieves the sharp lower bound \begin{equation*} \begin{array}{c} \lambda _{1}=\frac{nk}{n+1}, \end{array} \end{equation*} and it reduces to the original Obata-type Theorem for the sub-Laplacian $ \Delta _{b}$ in a closed strictly pseudoconvex CR $(2n+1)$-manifold $ (M,J,\theta ).$ It following from Chang-Chiu \cite{cc1}, \cite{cc2} and Li-Wang \cite{lw} that $M$\ is CR isometric to a standard CR $(2n+1)$-sphere. \end{proof}
\section{First Dirichlet Eigenvalue Estimate and Weighted Obata Theorem}
In this section, we derive the first Dirichlet eigenvalue estimate in a compact weighted strictly pseudoconvex CR $(2n+1)$-manifold $(M,J,\theta ,d\sigma )$ with boundary $\Sigma $ and its corresponding weighted Obata-type Theorem.
\textbf{The Proof of Theorem}\textup{\textbf{\ \ref{TB}:}}
\begin{proof} Since $f=0$ on $\Sigma $ and $e_{j}$ is tangent along $\Sigma $ for $1\leq j\leq 2n-1$, then $f_{e_{j}}=0$ for $1\leq j\leq 2n-1$ and $\Delta _{b}^{t}f= \frac{1}{2}\sum_{j=1}^{2n-1}[\left( e_{j}\right) ^{2}-(\nabla _{e_{j}}e_{j})^{t}]f=0$ on $\Sigma .$ Furthermore, since $\mathcal{L} f=-\lambda _{1}f\ $on$\mathrm{\ }M$ and $f=0$ on $\Sigma ,$ then $\mathcal{L} f=0$ on $\Sigma $. It follows from (\ref{21}) and $\Delta _{b}f=\mathcal{L} f+\langle \nabla _{b}f,\nabla _{b}\phi \rangle $ that \begin{equation*} \begin{array}{ll} & 4\sqrt{-1}C_{n}\int_{\Sigma }(f^{\overline{\beta }}B_{n\overline{\beta } }f-f^{\beta }B_{\overline{n}\beta }f)d\Sigma _{p}^{\phi } \\ = & C_{n}\int_{\Sigma }\sum_{\beta \neq n}[f_{e_{\beta }}(f_{e_{\beta }e_{2n}}-f_{e_{n+\beta }e_{n}})+f_{e_{n+\beta }}(f_{e_{\beta }e_{n}}+f_{e_{n+\beta }e_{2n}})]d\Sigma _{p}^{\phi } \\ & +C_{n}\int_{\Sigma }f_{e_{2n}}[(f_{e_{n}e_{n}}+f_{e_{2n}e_{2n}})-\frac{2}{n }\Delta _{b}f]d\Sigma _{p}^{\phi } \\ = & C_{n}\int_{\Sigma }f_{e_{2n}}\{[\left( e_{n}\right) ^{2}-(\nabla _{e_{n}}{}^{e_{n}})]f+(\frac{2n-2}{n}\Delta _{b}f-2\Delta _{b}^{t}f-H_{p.h}f_{e_{2n}})\}d\Sigma _{p}^{\phi } \\ = & \frac{n-1}{n}C_{n}\int_{\Sigma }\phi _{e_{2n}}f_{e_{2n}}^{2}d\Sigma _{p}^{\phi }-C_{n}\int_{\Sigma }(H_{p.h}+\tilde{\omega}_{n}^{ \;n}(e_{n}))f_{e_{2n}}^{2}d\Sigma _{p}^{\phi }. \end{array} \end{equation*} Also under the curvature condition $[Ric(\mathcal{L})-\frac{n+1}{2}Tor(
\mathcal{L})](\nabla _{b}f_{\mathbb{C}},\nabla _{b}f_{\mathbb{C}})\geq k|\nabla _{b}f|^{2}$ for a positive constant $k,$ the weighted CR Paneitz operator $P_{0}^{\phi }$ is nonnegative and $H_{p.h}-\tilde{\omega} _{n}^{\;n}(e_{n})-\frac{n+2}{2}\phi _{e_{2n}}\geq 0$ on $\Sigma $, then the integral formula (\ref{0}) yields \begin{equation*} \begin{array}{ll} & \frac{n+1}{n}\lambda _{1}^{2}\int_{M}f^{2}d\sigma \\
\geq & k\int_{M}|\nabla _{b}f|^{2}d\sigma -\frac{n+1}{4}\lambda _{1}\int_{M}\langle \nabla _{b}f^{2},\nabla _{b}\phi \rangle d\sigma
-(n+2)l\int_{M}|\nabla _{b}f|^{2}d\sigma \\ & +\frac{1}{8}C_{n}\int_{\Sigma }[H_{p.h}-\tilde{\omega}_{n}^{\;n}(e_{n})- \frac{n+2}{2}\phi _{e_{2n}}]f_{e_{2n}}^{2}d\Sigma _{p}^{\phi } \\ = & [k-(n+2)l]\lambda _{1}\int_{M}f^{2}d\sigma +\frac{n+1}{4}\lambda _{1}\int_{M}\phi \mathcal{L}f^{2}d\sigma \\
= & [k-(n+2)l]\lambda _{1}\int_{M}f^{2}d\sigma +\frac{n+1}{2}\lambda _{1}\int_{M}\phi \lbrack |\nabla _{b}f|^{2}-\lambda _{1}f^{2}]d\sigma \\ \geq & [(k-(n+2)l)\lambda _{1}-\frac{n+1}{2}\omega \lambda _{1}^{2}]\int_{M}f^{2}d\sigma , \end{array} \end{equation*}
here we assume $\mathcal{L}\phi +\frac{1}{2(n+2)}|\nabla _{b}\phi |^{2}\leq 4l$ on $M$ for some nonnegative constant $l$ and $\omega =\underset{M}{ \mathrm{osc}}\phi =\underset{M}{\sup }\phi -\underset{M}{\inf }\phi $. It implies\ that the first eigenvalue of $\mathcal{L}$ will satisfy \begin{equation*} \begin{array}{c} \lambda _{1}\geq \frac{2n[k-(n+2)l]}{(n+1)(2+n\omega )}. \end{array} \end{equation*}
Moreover, if the above inequality becomes equality, then $\omega =\underset{M }{\mathrm{osc}}\phi =0$ and thus the weighted function $\phi $ will be constant, we get $\mathcal{L=}$ $\Delta _{b}$ and $P_{0}^{\phi }=P_{0},$ which is nonnegative for $n\geq 2$ by Lemma \ref{lemma 3.1}. Note that we need to assume that $P_{0}$ is nonnegative for $n=1$. Also the corresponding eigenfunction $f$ will satisfy \begin{equation*} \begin{array}{c} f_{\alpha \beta }=0\text{ \ \textrm{for all} }\alpha ,\beta , \end{array} \end{equation*} \begin{equation*} \begin{array}{c} \lbrack Ric-\frac{n+1}{2}Tor](\nabla _{b}f_{\mathbb{C}},\nabla _{b}f_{
\mathbb{C}})=k|\nabla _{b}f|^{2}, \end{array} \end{equation*} and \begin{equation*} \begin{array}{c} P_{0}f=0 \end{array} \end{equation*} on $M,$ and $f=0$ on $\Sigma .$ We also have $H_{p.h}=0$ and $\tilde{\omega} _{n}^{\;n}(e_{n})=0$ on $\Sigma .$ In this case we can let $l=0,$ then the first eigenvalue of the sub-Laplacian $\Delta _{b}$ achieves the sharp lower bound \begin{equation*} \begin{array}{c} \lambda _{1}=\frac{nk}{n+1}. \end{array} \end{equation*}
By applying the same method of Li-Wang \cite{lw}, it can be showed that the pseudohermitian torsion vanishes on $M$ with boundary $\Sigma $. Then we follow from Chang-Chiu \cite{cc2} to define the Webster (adapted) Riemannian metric $g_{\varepsilon }$ of $(M,J,\theta )$ by \begin{equation*} \begin{array}{c} g_{\varepsilon }=\varepsilon ^{2}\theta ^{2}+\frac{1}{2}d\theta (\cdot ,J\cdot )\text{ }\mathrm{for}\text{ }\varepsilon >0\text{ }\mathrm{with} \text{ }(n+1)\varepsilon ^{2}=k. \end{array} \end{equation*} Since $Ric(Z,Z)\geq k\left\langle Z,Z\right\rangle $ and free torsion, the Theorem 4.9 in \cite{cc2} says that the Ricci curvature of $g_{\varepsilon }$ satisfies \begin{equation*} \begin{array}{c} Rc_{g_{\varepsilon }}\geq \lbrack (2n+1)-1]\frac{k}{n+1}. \end{array} \end{equation*} And the mean curvature $H_{\varepsilon }$ is zero on $\Sigma $ with respect to the metric $g_{\varepsilon },$ which will be presented in the next. Then Theorem 1.2 in \cite{cc2} will yield that the eigenfunction $f$ of $\Delta _{b}$\ achieves the sharp lower bound for the first Dirichlet eigenvalue of the Laplace $\Delta _{\varepsilon }$ with respect to the Riemannian metric $ g_{\varepsilon }$. Therefore, by Theorem $4$ in \cite{Re}, $M$ is isometric to a hemisphere in a standard CR $(2n+1)$-sphere.
In the following we show that the mean curvature $H_{\varepsilon }$ is zero on $\Sigma $ with respect to the Webster metric $g_{\varepsilon }$. We choose $\{l_{j}=e_{j},l_{2n}=\frac{\alpha e_{2n}+T}{\sqrt{\alpha ^{2}+\varepsilon ^{2}}}\}_{j=1}^{2n-1},$ here $e_{n+\beta }=Je_{\beta }$ for $1\leq \beta \leq n,$ to form an orthonormal tangent frame and $\nu =\frac{1 }{\sqrt{\alpha ^{2}+\varepsilon ^{2}}}\left( \varepsilon e_{2n}-\frac{\alpha }{\varepsilon }T\right) $ be an unit normal vector on the non-singular set $ \Sigma \backslash S_{\Sigma }.$ The second fundamental form is then defined by $h_{ij}^{\varepsilon }=-\left\langle \nu ,\nabla _{l_{i}}^{R}l_{j}\right\rangle _{g_{\varepsilon }}$ and the mean curvature is defined by $H_{\varepsilon }=\sum_{j=1}^{2n}h_{jj}^{\varepsilon },$ here $ \nabla ^{R}$ is the corresponding Riemannian connection. By the Lemma 4.3 in \cite{cc2}, we have \begin{equation*} \begin{array}{lll} \nabla ^{R}e_{\beta } & = & \omega _{\beta }^{\;\gamma }\otimes e_{\gamma }+( \tilde{\omega}_{\beta }^{\;\gamma }+\varepsilon ^{2}\delta _{\beta \gamma }\theta )\otimes e_{n+\gamma }+e^{n+\beta }\otimes T, \\ \nabla ^{R}e_{n+\beta } & = & -(\tilde{\omega}_{\beta }^{\;\gamma }+\varepsilon ^{2}\delta _{\beta \gamma }\theta )\otimes e_{\gamma }+\omega _{\beta }^{\;\gamma }\otimes e_{n+\gamma }-e^{\beta }\otimes T, \\ \nabla ^{R}T & = & -\varepsilon e^{n+\gamma }\otimes e_{\gamma }+\varepsilon e^{\gamma }\otimes e_{n+\gamma }, \end{array} \end{equation*} for $\beta =1,2,\cdots ,n.$ Then the mean curvature $H_{\varepsilon }$ is given explicitly by \begin{equation*} \begin{array}{lll} H_{\varepsilon } & = & -\sum_{j=1}^{2n-1}\left\langle \nu ,\nabla _{l_{j}}^{R}l_{j}\right\rangle _{g_{\varepsilon }}-\left\langle \nu ,\nabla _{l_{2n}}^{R}l_{2n}\right\rangle _{g_{\varepsilon }} \\ & = & -\sum_{j=1}^{2n-1}\frac{1}{\sqrt{\alpha ^{2}+\varepsilon ^{2}}} \left\langle \varepsilon e_{2n}-\frac{\alpha }{\varepsilon }T,\nabla _{e_{j}}e_{j}\right\rangle _{g_{\varepsilon }} \\ & & -\frac{1}{\left( \alpha ^{2}+\varepsilon ^{2}\right) ^{2}}\left\langle \varepsilon e_{2n}-\frac{\alpha }{\varepsilon }T,l_{2n}(\alpha )\left( \varepsilon ^{2}e_{2n}-\alpha T\right) \right\rangle _{g_{\varepsilon }} \\ & = & \frac{\varepsilon }{\sqrt{\alpha ^{2}+\varepsilon ^{2}}}H_{p.h}-\frac{ \varepsilon }{\alpha ^{2}+\varepsilon ^{2}}l_{2n}(\alpha ), \end{array} \end{equation*} and thus \begin{equation*} H_{\varepsilon }=H_{p.h}=0 \end{equation*} on $\Sigma ,$ if $\alpha $ vanishes on $\Sigma .$
In the following we show that $\alpha $ vanishes identically on $\Sigma .$ It follows from the formula $\left[ \Delta _{b},T\right] f=4{\func{Im}}[ \sqrt{-1}\left( A^{\beta \gamma }f_{\beta }\right) ,_{\gamma }]=0$ that $ \Delta _{b}f_{0}=-\lambda _{1}f_{0}.$ Again from the equation \begin{equation*} \begin{array}{c} \int_{M}\psi _{0}d\mu =\int_{M}\mathrm{div}_{b}\left( J\nabla _{b}\psi \right) d\mu =\frac{1}{2}C_{n}\int_{\Sigma }\psi _{e_{n}}d\Sigma _{p}=0 \end{array} \end{equation*} for any real function $\psi $ which vanishes on $\Sigma $ to get that $ \int_{M}\left( f^{2}\right) _{0}d\mu =0$ for $f=0$ on $\Sigma $ and also \begin{equation*} \begin{array}{c} \int_{M}[f\Delta _{b}f_{0}+\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle ]d\mu =\frac{1}{2}C_{n}\int_{\Sigma }ff_{0e_{2n}}d\Sigma _{p}=0, \end{array} \end{equation*} which implies that $\int_{M}\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle d\mu =0$. Thus \begin{equation} \begin{array}{c} 0=\int_{M}[f_{0}\Delta _{b}f+\left\langle \nabla _{b}f,\nabla _{b}f_{0}\right\rangle ]d\mu =\frac{1}{2}C_{n}\int_{\Sigma }f_{0}f_{e_{2n}}d\Sigma _{p}. \end{array} \label{22} \end{equation} On the other hand, since $f=0$ on $\Sigma $ and $\alpha e_{2n}+T$ is tangent along $\Sigma $ from the definition of $\alpha ,$ so on $\Sigma $ we have \begin{equation*} \begin{array}{c} f_{0}=-\alpha f_{e_{2n}} \end{array} \end{equation*} and \begin{equation*} \begin{array}{c} -\alpha \theta \wedge e^{n}\wedge \left( d\theta \right) ^{n-1}=e^{n}\wedge e^{2n}\wedge \left( d\theta \right) ^{n-1}=\frac{1}{2n}\left( d\theta \right) ^{n} \end{array} \end{equation*} which is a nonnegative $2n$-form on $\Sigma $. It follows from (\ref{22}) that \begin{equation*} \begin{array}{c} 0=\frac{1}{2}C_{n}\int_{\Sigma }f_{0}f_{e_{2n}}d\Sigma _{p}=-\int_{\Sigma }\alpha f_{e_{2n}}^{2}\theta \wedge e^{n}\wedge \left( d\theta \right) ^{n-1}=\frac{1}{2n}\int_{\Sigma }f_{e_{2n}}^{2}\left( d\theta \right) ^{n}. \end{array} \end{equation*} This will yield that $\alpha f_{e_{2n}}^{2}=0$ on $\Sigma $ and therefore $ f_{0}=0$ on $\Sigma ,$ so $T$ is tangent to $\Sigma $ and thus $\alpha $ vanishes identically on $\Sigma \backslash S_{\Sigma }$ and continuously extends over the singular set $S_{\Sigma }$ and the same constant on $ S_{\Sigma }$. \end{proof}
\end{document} | arXiv |
Assessing the quality of Smilacis Glabrae Rhizoma (Tufuling) by colormetrics and UPLC-Q-TOF-MS
Xicheng He1,2,
Tao Yi1,
Yina Tang1,
Jun Xu1,
Jianye Zhang1,
Yazhou Zhang1,
Lisha Dong2 &
Hubiao Chen1
Chinese Medicine volume 11, Article number: 33 (2016) Cite this article
The quality of the materials used in Chinese medicine (CM) is generally assessed based on an analysis of their chemical components (e.g., chromatographic fingerprint analysis). However, there is a growing interest in the use of color metrics as an indicator of quality in CM. The aim of this study was to investigate the accuracy and feasibility of using color metrics and chemical fingerprint analysis to determine the quality of Smilacis Glabrae Rhizoma (Tufuling) (SGR). The SGR samples were divided into two categories based on their cross-sectional coloration, including red SGR (R-SGR) and white SGR (W-SGR).
Forty-three samples of SGR were collected and their colors were quantized based on an RGB color model using the Photoshop software. An ultra-performance liquid chromatography/quadrupole time-of-flight mass spectrometry (UPLC/QTOF MS) system was used for chromatographic fingerprint analysis to evaluate the quality of the different SGR samples. Hierarchical cluster analysis and dimensional reduction were used to evaluate the data generated from the different samples. Pearson correlation coefficient was used to evaluate the relationship between the color metrics and the chemical compositions of R-SGR and W-SGR.
The SGR samples were divided into two different groups based on their cross-sectional color, including color A (CLA) and B (CLB), as well as being into two separate classes based on their chemical composition, including chemical A (CHA) and B (CHB). Standard fingerprint chromatograms were for CHA and CHB. Statistical analysis revealed a significant correlation (Pearson's r = −0.769, P < 0.001) between the color metrics and the results of the chemical fingerprint analysis.
The SGR samples were divided into two major clusters, and the variations in the colors of these samples reflected differences in the quality of the SGR material. Furthermore, we observed a statistically significant correlation between the color metrics and the quality of the SGR material.
The quality of Chinese medicine (CM) continues to attract considerable attention. The quality control procedures traditionally used in CM are based on the color, smell and texture characteristics of the raw materials, as well as several other sensory properties. Color in particular is an essential method of quality control for evaluating the materials used in CM, but the effectiveness of this quality indictor is limited by its subjective nature. However, recent technological advances mean that it is now possible to measure and quantify colors using instrumental methods. For example, the major colors of an image can be represented by 8 bits in a computer, meaning that each color axis can be represented by 8 bits or 28 = 256 different values in the Red–Green–Blue (RGB) color space [1, 2]. Using imaging software, it is possible to read the RGB values of every pixel of a sample image. These data can subsequently be used to generate average RGB values for the different areas of an image, which can be used to represent color information.
Today, the quality of CM is mainly determined based on an analysis of their chemical composition using sophisticated analytical techniques, such as chromatographic fingerprint analysis, which can be used to provide an indication of their intrinsic quality. Instrument-based analytical techniques have therefore been widely accepted for quality evaluation and species identification in CM, supplementing traditional techniques based on the sensory properties of these materials [3–5]. Establishing a connection between the color and the chemical composition of the materials used in CM would provide solid evidence for establishing a new method of quality control, which could validate the correlation between the sensory properties and quality of CM. In this study, we have selected Smilacis Glabrae Rhizoma (Tufuling) (SGR) as a model medicinal herb to evaluate the relationship between the quality of this material and its color metrics and chemical composition data.
SGR, which is the dried rhizome of Smilax glabra Roxb., can be red or white (Fig. 1). Generally, the therapeutic effects of white SGR are believed to be more pronounced than those of the red material [6–8], suggesting that the color of this material could be used as an indicator of its quality. SGR samples can therefore be divided into two groups depending on their cross-sectional color, including red SGR (R-SGR) and white SGR (W-SGR) [6].
The samples of SGR (a) Red cross-sectional SGR. b White cross-sectional SGR
However, the results of another study revealed that most SGR samples contain both red and white cross-sections, and that the quality of these materials can vary considerably [9]. Despite the potential implications of this discovery, there has been other work conducted in this area, and the quality of SGR is currently evaluated by chemical composition analysis with no regard for differences in the apparent color.
This aim of this study was to investigate the accuracy and feasibility of using color metrics and chemical fingerprint analysis to analyze the quality of SGR. In this study, we used image software to quantify the color information of the red and white SGR samples based on color data generated using an RGB model. We also used ultra-performance liquid chromatography/quadrupole time-of-flight mass spectrometry (UPLC/QTOF MS) to produce chemical chromatographic fingerprints for these different samples. Hierarchical analysis and dimensional reduction methods were used to generate color and chemical scores for the different SGR samples, and standard fingerprint chromatograms were established according to the clustering results of the main model. Pearson correlation coefficient was used to evaluate the relationship between the color and the chemical composition of the different samples, which revealed a statistically significant relationship between these two variables (Fig. 2a).
Flow chart of this work. a Flow chart of the experiment design process (b) Flow chart for establishing a standard fingerprint chromatogram \( R{\text{Intens}} .= \frac{{{\text{Intens}} .}}{{{\text{SPH}} \times {\text{W}}}},\quad R{\text{RT }} = \frac{\text{RT}}{\text{SRT}}, \) where RIntens is the relative Intensity, RRT is the relative retention time, SPH and SRT are the peak height and retention time, respectively, of the internal standard (IS), and W was sample weight (g)
Information pertaining to the different SGR samples is shown in Table 1. All of these samples were authenticated as the genuine rhizome of S. Glabra by Prof. Lisha Dong (School of Pharmacy, Gui Yang College of Traditional Chinese Medicine, China). These samples were authenticated based on a comparison of their flowers with those of Smilax Glabra Roxb., according to the Flora of China [10].
Table 1 Details of the different SGR samples
Analytical grade methanol (Labscan, Bangkok, Thailand) was used to prepare the reference standards, as well as the samples of the different extracts. Chromatography-grade formic acid (Fluka, Buchs, Switzerland), chromatography-grade acetonitrile (Labscan, Bangkok, Thailand) and Milli-Q water (Millipore, Bedford, MA, USA) were used to prepare the mobile phases for chromatography. Analytical standards of astilbin, engeletin, (+)-catechin, (–)-epicatechin and 7,4′-dihydroxyflavone were obtained from the National Institute for the Control of Pharmaceutical and Biological Products, China. All of these chemical standards were greater than 98 % in purity.
The SGR samples were pulverized, and the resulting powders were sieved through a 120-mesh screen. The sieved powders were then stored in the dark prior to being analyzed. Samples of the sieved powders were uniformly dispersed on the surface of a glass culture dish (diameter, 60 mm) which was placed in a Desktop Proeasy LTM-101 light source system (Nexor, Guangzhou, China) together with a standard color checker (Mennon, Beijing, China) and an 18 % Grey Card (Topimage, Taipei, Taiwan). Photographic images were recorded using a Nikon D5200 camera with a focal length of 24 mm (M grade, A:F8, S:1/40). The photographic images were visualized using the Adobe Photoshop CS5 software (Adobe, San Jose, CA, USA). The Grey Card was used to correct the white balance in the images using the gray-pipette tool from Photoshop, whereas the standard color checker was used to produce a file containing less than five color errors. Five different areas of each photograph were selected for sampling, with each sampling area set as 51 × 51 pixels. Color data were obtained by calculating the average values of the RGB data, which were measured using the color information derived from each sample with the eyedropper tool from Photoshop (Fig. 3).
Photos of the 43 SGR powders. a Red cross-sectional SGR. b White cross-sectional SGR
Color data were subjected to partial least squares (PLS) regression analysis using the MATLAB software (MathWorks, Natick, MA, USA) to analyze the relationship between the colors in R-SGR and W-SGR, as well as quantifying the boundary between R-SGR and W-SGR. The dependent variable used in this model was G, whereas R and B were used as independent variables.
An SGR power sample of 0.5 g was added to 20.0 mL of 60 % (v/v) methanol in water, and the resulting mixture was sonicated (CP1800 HT, Crest, Penang-, Malaysia) for 30 min at room temperature. The sonicated mixture was centrifuged (5810, Eppendorf, Hamburg, Germany) at 2100×g for 5 min at room temperature, and 400 µL of the supernatant was placed in a volumetric flask, followed by 100 µL of 7,4′-dihydroxyflavone (0.1249 mg/mL), which was added as an internal standard (IS). The total volume of the mixture in the volumetric flask was then adjusted to 10 mL by the addition of methanol, and the resulting sample solution was filtered through a 0.22-µm filter before being analyzed by LC-MS. Two replicates of each SGR sample were prepared and analyzed in the same way. All of the extracts were stored at 4 °C before use.
UPLC analysis was performed on a Waters Acquity system (Waters Corporation, Milford, MA, USA) equipped with a quaternary solvent manager and a sample manager. This system was also coupled to a Micromass QTOF premier mass spectrometer (Bruker Daltonics, Milford, MA, USA) equipped with an electrospray ionization (ESI) interface. Chromatographic separation was conducted over a Waters C18 T3 column (1.8 µm, 2.1 × 100 mm) eluting with a mobile phase consisting of 0.1 % formic acid in water (A) and 0.1 % formic acid in acetonitrile (B) with linear gradient elution at a flow rate of 0.35 mL/min. The gradient elution was conducted as follows: 5–14 % B (0–1.5 min), 14–16 % B (1.5–6.0 min), 16–20 % B (6.0–20.0 min), 20–100 % B (20.0–30.0 min). The column temperature was kept at 40 °C throughout the entire process. The sample size for injection into the UPLC system was set at 2 µL. The ESI source of the MS was connected to the UPLC system via a capillary through the UV cell outlet. MS data were collected with a capillary voltage of 3.5 kV in the negative ionization mode. Nitrogen was used as a desolvation gas at a flow rate was of 8 L/h. The scan range (m/z) was set at 100–1000 Da.
Extracted ion chromatograms (EIC) were used to obtain the peak areas of the different components, and the relative peak areas were then calculated based on the peak area of the IS. The relative peak areas were calculated using the following equation:
$$ RPA_{i} = \frac{{A_{i} }}{{A_{S} \times W}}, $$
where RPA is the relative peak area, A is the peak area of a specific component, A s is the peak area of the IS, W is the sample weight and i is the peak number.
The original data were saved as CDF files using version 4.0 of the DataAnalysis software (Bruker, USA). The MATlab R2012b software (MathWorks, USA) was then used to extract information from the CDF files, including retention time (RT) and intensity (Intens.) data (Fig. 2b). The scan range (m/z) was set at 100–800 Da. According to the result of chemical hierarchical cluster, chromatogram maps corrected by IS were combined to generate standard chromatogram by mean value method.
The correlation coefficient, r con , was used in the current study to assess the consistency of the chemical composition of SGR using the following equations [3, 11, 12]:
$$ r_{con} = \tfrac{{\sum\nolimits_{i = 1}^{num} {x_{i} y_{i} } }}{{\sqrt {\left( {\sum\nolimits_{i = 1}^{num} {\left( {x_{i} } \right)^{2} } } \right)\left( {\sum\nolimits_{i = 1}^{num} {\left( {y_{i} } \right)^{2} } } \right)} }}, $$
$$ \bar{x} = \left( {\sum\limits_{i = 1}^{num} {x_{i} } } \right)/n,\quad \bar{y} = \left( {\sum\limits_{i = 1}^{num} {y_{i} } } \right)/n, $$
where x i and y i are the ith elements in two different fingerprints (i.e., x and y), num is the number of the elements in the fingerprints and \( \bar{x} \) and \( \bar{y} \) are the mean values of the num elements in fingerprints x and y, respectively.
Correlation of color metrics and chemical fingerprint
Dimensional reduction was used to convert complex chemical information and color data into one or a few simple variables. Factor analysis was used to reduce the number of dimensions and extract the most important information from the original data. To verify that the resulting data sets were suitable for factor analysis, we checked that the Kaiser–Meyer–Olkin (KMO) values were greater than 0.6 and that the Bartlett's test of Sphericity value was significant (significant values should be less than 0.05). The color and chemical scores were subsequently calculated using factor analysis, and the results were used to evaluate the correlation between the color metrics and the chemical fingerprint by Pearson correlation coefficient.
Hierarchical clustering was conducted using version 20.0 of the IBM SPSS software (IBM, Armonk, NY, USA) to allow for the standardization of the color and chemical data, followed by hierarchical clustering by classification analysis.
Dimensional reduction
Factor analysis was used for the dimensional reduction of the color data obtained for the different SGR samples, and the resulting factor score was subsequently regarded as the color score. Factor analysis was used for dimensional reduction processing, and the principal components were considered to be independent based on the factor rotation method. Based on the different values of the principal component analysis, we calculated a cumulative score for the chemical data. P value less than 0.05 were considered statistically significant.
Correlation analysis
The relationship between the color and chemical scores was evaluated using Pearson correlation coefficient. A scatter plot of these data was then generated using version 20.0 of the SPSS software with the chemical and color scores on the x and y axes, respectively. P value less than 0.05 were considered statistically significant.
A color dendrogram of the 43 different SGR samples evaluated in this study is shown in Fig. 4a. This figure shows that the SGR samples were divided into two major clusters (i.e., R-SGR and W-SGR) based on their color. The R-SGR samples were grouped into one cluster defined as "CLA", whereas the W-SGR samples were grouped into another cluster defined as "CLB". The results of quantitative color analysis supported the result of this subjective classification, suggesting that SGR samples were composed of both white and red materials. However, it is noteworthy that all of these samples were defined as being the same based on their original botanical identification.
Division of the samples into two classes using hierarchical analysis; CLA and CLB, based on color; and CHA and CHB, based on chemical composition. A multiple regression model was established in color space based on our regression analysis of the color data. a Cluster dendrogram of the different colors. b Cluster dendrogram constructed from the EIC peak areas of the 43 SGR samples. c Regression model of SGR into the RGB color space and the standard tape of the SGR samples
The major colors of the different SGR samples were represented by one of 256 different values in the RGB color space. Based on the results of the regression analysis, we established a multiple regression model in color space using the following regression equation (Fig. 4c):
$$ G = -17.18 + 0.62R + 0.46B \,(R^2 = 0.998; P < 0.001).$$
All of the R-SGR and W-SGR samples were represented in this regression model. As shown in the expanded view, this model established a standard tape of SGR, with R-SGR and W-SGR on either side based on the color location, allowing for the two clusters to be differentiated into two different zones.
The 43 SGR samples were also subjected to chemical analysis and separated into two categories based on the results by hierarchical clustering (Fig. 4b). The R-SGR samples were grouped in one cluster, which was defined as "CHA", whereas the W-SGR samples were grouped in anther cluster, which was defined as "CHB", indicating that there were two different types of SGR based on differences in their chemical composition.
The chemical compositions in the extracts of R-SGR and W-SGR were analyzed by UPLC/QTOF MS. Twenty-eight characteristic peaks were found in the EICs and nine compounds were identified based on their fragmentation patterns and a comparison of their MS data with those available in the literature (Table 2) [13, 14, 15]. The R-SGR and W-SGR samples contained different numbers of peaks (compositions), as shown in Fig. 5a, b. The amounts of the different components also changed between the two different sample types.
Table 2 Information concerning the different compounds found in the chromatogram of SGR
Standard fingerprint chromatographs and mass spectra of the CHA and CHB samples collected by UPLC/QTOF MS (a) Standard fingerprint chromatographs for CHA showing 95 % confidence intervals. b Standard fingerprint chromatographs for CHB showing 95 % confidence intervals (the 95 % confidence intervals resulted in increased precision of the standard fingerprint chromatographs). c The correlation coefficients of CHA. d The correlation coefficients of CHB (the red and blue lines represent the correlation coefficients of CHA and CHB, respectively). * indicates that the peak was identified based on a comparison with the literature [13–15]
The precision of this analytical method was determined by injecting the same sample solution into the system on six consecutive occasions. The peak areas of the different components were then taken as measures of the precision and expressed as relative standard deviation (RSD) values, which resulted in a precision of 2.97 %. Six peaks with peak areas greater than 3 % and relative retention time (RRT) of 0.25, 0.35, 0.45, 0.49, 0.65 and 0.70 were chosen to calculate the repeatability and stability of the method. The RSD of RT repeatability was less than 0.24 %, whereas the relative peak area (RPA) repeatability was less than 4.72 %, which showed good stability over 8 h at less than 4.57 %.
Standard fingerprint chromatograms of samples belonging to the CHA and CHB groups were established based on the result of the main clustering model (Fig. 5a, b), and calculated with a 95 % confidence interval. A pronounced difference was observed between the two modes of examination used in the standard fingerprint chromatograms. The standard fingerprint chromatograms of CHA contained 28 peaks, whereas standard fingerprint chromatograms of CHB contained only 17 peaks without peak numbers 12, 17, 21, 25 and 26. Moreover, the amount of the different components varied considerably between CHA and CHB. For example, the peak corresponding to astilbin (peak 18), which was the biggest of all of the peaks detected in CHA and CHB, was much smaller in CHB than it was in CHA. Five other peaks, including peaks 2, 5, 7, 22 and 23, also appeared at much lower levels in the standard fingerprint in CHB.
The relationships between the fingerprint chromatograms could be analyzed by comparing the similarities between specific reference points. The correlation coefficient was used in this study to examine the similarities between the samples clustered into the CHA and CHB groups. The correlation coefficients of the CHA were more than 0.9 (the red line in Fig. 5c), indicating that the standard fingerprint chromatograms represented the chemical characteristics of R-SGR. All of the correlation coefficients of the CHB were greater than 0.93 (the blue line in Fig. 5d), and the similarities in this group were higher than those of the CHA group, indicating that the standard fingerprint chromatograms represented the chemical characteristics of W-SGR. The standard fingerprint chromatograms of CHA and CHB were stable and consistent.
Correlation of the color metrics and chemical fingerprint data
For this example, we obtained a KMO value of 0.656, and the results of a Bartlett's test revealed that this result was significant (P < 0.001), indicating that the use of factor analysis was appropriate [16]. The three variables (RGB) for the color data were transformed into one factor score, affording a total cumulative variance of 97.851 %. The color score was calculated as follows:
$$ Color \, score \, = \, ZR\, \times \,0. 3 3 4\, + \,\, \, ZG\,\, \times \,0. 3 40\, + \,ZB \, \times \,0. 3 3 7, $$
where ZR, ZG and ZB are standardized data for R, G and B, respectively. The constants were defined as the component score coefficients. A high color score was indicative of white SGR, whereas a low color score was indicative of red SGR.
The results of the chemical data led to the identification of six principal factors based on 28 peaks by factor analysis. Principal component analysis was then used to translate these six variables into a chemical score, which gave a total cumulative variance of 88.298 % together with a KMO value of 0.639. Furthermore, Bartlett's test was significant (P < 0.001). The formula used to compute the chemical score was as follows: Chemistry score = FAS1 × 20.947 % + FAS2 × 16.818 % + FAS3 × 16.702 % + FAS4 × 15.418 % + FAS5 × 13.717 % + FAS6 × 4.696 %.
FAS was used as a factor score, with constant squared loadings for each factor. This method allowed for the evaluation of increasingly complex factors, and also provided a novel strategy for evaluating quality.
The Pearson correlation coefficient was determined to be −0.769 (double side test P < 0.001), suggesting that there was a significant correlation between the color and chemistry scores (Fig. 6).
Scatter diagram of the color and chemistry scores
The result of the dimensional reduction and correlation analysis indicated that there was a statistically significant relationship between the colors and chemical components of the different SGR samples, as did the results of the hierarchical clustering. These results therefore suggest that it is feasible to use color (red and white) to distinguish between two different qualities or specifications of SGR.
SGR samples were divided into two major clusters and the variation in their color provided a good indication of the differences in their quality. Notably, we observed a statistically significant correlation between the color metrics and the quality of the different SGR samples (Pearson's r = −0.769, P < 0.001).
CM:
SGR:
Smilacis Glabrae Rhizoma
R-SGR:
red SGR
W-SGR:
white SGR
Red–Green–Blue
UPLC-Q-TOF-MS:
ultra-performance liquid chromatography-quadrupole time-of-flight mass spectrometry
CLA:
color A
CLB:
color B
CHA:
chemical A
CHB:
chemical B
ESI:
electrospray ionization
extracted ion chromatogram
RPA:
relative peak area
RT:
retention time
internal standard
KMO:
Kaiser–Meyer–Olkin
RSD:
relative standard deviation
RRT:
relative retention time
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HC, LD, XH and TY designed the study. XH, TY and YT performed the experiments, TY and XH analyzed the data. XH, TY, JX, JZ and YZ wrote the manuscript. All authors read and approved the final manuscript.
This study was partially supported by grants from the National Natural Science Foundation of China (Project codes: 30960503 and 81460588) and QianKeHe ZY Zi of the Development of Guizhou Science and Technology Agency, China (Project No. 30011). We would also like to thank Ho Hing Man and Yeung Wing Ping for providing technical support.
School of Chinese Medicine, Hong Kong Baptist Univesity, Hong Kong, China
Xicheng He
, Tao Yi
, Yina Tang
, Jun Xu
, Jianye Zhang
, Yazhou Zhang
& Hubiao Chen
School of Pharmacy, Gui Yang Collage of Traditional Chinese Medicine, Guiyang, 550002, China
& Lisha Dong
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Corresponding authors
Correspondence to Lisha Dong or Hubiao Chen.
Xicheng He and Tao Yi contributed equally to this work
He, X., Yi, T., Tang, Y. et al. Assessing the quality of Smilacis Glabrae Rhizoma (Tufuling) by colormetrics and UPLC-Q-TOF-MS. Chin Med 11, 33 (2016) doi:10.1186/s13020-016-0104-y
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\begin{document}
\title{A note on cusp forms as $p$-adic limits}
\author{Scott Ahlgren} \address{Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \email{[email protected]} \author{Detchat Samart} \address{Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \email{[email protected]}
\subjclass[2010]{11F33, 11F11, 11F03} \keywords{Modular forms, Cusp forms as $p$-adic limits} \date{\today} \thanks{The first author was supported by a grant from the Simons Foundation (\#208525 to Scott Ahlgren).} \maketitle \begin{abstract} Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms. \end{abstract} \section{Introduction}
In a recent paper \cite{EO}, El-Guindy and Ono study a cusp form and a modular function related to the elliptic curve $y^2=x^3-x$. Following their notation, define \begin{align} g(z)&=\eta^2(4z)\eta^2(8z)= \sum_{n\ge 1}a(n)q^n=q-2q^5-3q^9+6q^{13}+\cdots,\label{eq:gdef}\\ L(z)&=\frac{\eta^6(8z)}{\eta^2(4z)\eta^4(16z)}=\frac1q+2q^3-q^7-2q^{11}+\cdots,\label{eq:Ldef}\\ {F}(z) &= -g(z)L(2z)=\sum_{n\ge -1}C(n)q^n=-\frac{1}{q}+2 q^3+q^7-2 q^{11}+\cdots.\label{eq:Fdef} \end{align} The main result of \cite{EO} states that if $p\equiv 3\pmod 4$ is a prime for which $p\nmid C(p)$, then as a $p$-adic limit, we have \begin{equation}\label{eq:elg-ono}
\lim_{m\to\infty}\frac{ F\big | U(p^{2m+1})}{C(p^{2m+1})}=g. \end{equation} The proof involves the theory of harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono \cite{GKO} on the $p$-adic coupling of mock modular forms and their shadows. Similar results were proved in \cite{GKO} and \cite{BG}.
Our goal is to prove strengthened versions of these results. We use a direct method; it does not involve harmonic Maass forms but rather an investigation of the action of the Hecke operators on a family of weakly holomorphic modular forms. A similar approach was recently employed in the study of the congruences of Honda and Kaneko \cite{AA:heckegrids}.
For the modular forms described above, we prove the following, of which \eqref{eq:elg-ono} is an immediate corollary. Note in addition that the $m=0$ case of \eqref{eq:vpcp} gives $p\nmid C(p)$. Let $v_p(\cdot)$ denote the $p$-adic valuation on $\mathbb{Z}[\![q]\!]$. \begin{theorem}\label{T:main} Let $p\equiv 3 \pmod 4$ be prime. Then for all integers $m\ge 0$ we have \begin{align} v_p(C(p^{2m+1})) &= m,\label{eq:vpcp}\\
v_p\left(\frac{{F}| U(p^{2m+1})}{C(p^{2m+1})}-g\right)&\ge m+1.\label{eq:thmmain} \end{align} \end{theorem}
In Theorems~\ref{T:main4} and \ref{T:main3} below we obtain similar improvements of results given in \cite{GKO} and \cite{BG}. It is clear that the present approach would give similar results for a number of other spaces of modular forms.
\section{Background} If $k$ is an integer, $f$ is a function of the upper half-plane, and $\gamma=\pmatrix abcd \in \GL_2^+(\mathbb{Q})$, we define \[
f (z)\big|_k \gamma := (\det \gamma)^{k/2} (cz+d)^{-k} f\pfrac{az+b}{cz+d}. \]
If $N\ge 1$, $k\in \mathbb{Z}$, and $\chi$ is a Dirichlet character modulo $N$, let $M_k(N,\chi)$ be the space consisting of functions $f$ which satisfy $f|_k\pmatrix abcd=\chi(d)f$ for all $\pmatrix abcd\in \Gamma_0(N)$ and which are holomorphic on the upper half plane and at the cusps. Let $M_k^!(N,\chi)$ be the space of forms which are meromorphic at the cusps, and let $M_k^\infty (N,\chi)$ denote the subspace of forms which are holomorphic at all cusps of $\Gamma_0(N)$ other than $\infty$. We drop the character from this notation when it is trivial. Each $f\in M_k^!(N,\chi)$ can be identified with its $q$-expansion; with $q:=\exp(2\pi iz)$ we have $f(z)=\sum a(n)q^n$ for some coefficients $a(n)$.
For each positive integer $m$, the $U$ and $V$-operators are defined on $q$-expansions by \begin{align*}
\sum a(n)q^n \big | U(m) &:= \sum a(mn)q^n,\\
\sum a(n)q^n \big | V(m) &:= \sum a(n)q^{mn}. \end{align*} Let $T_{k,\chi}(m)$ be the usual Hecke operator on $M_k^!(N,\chi)$. If $p$ is prime, then for $n\geq 1$ and $f\in M_k^!(N,\chi)$ we have \begin{equation}\label{E:TUV}
f | T_{k,\chi}(p^n) = \sum_{j=0}^n \chi(p^j)p^{(k-1)j} f| U(p^{n-j}) | V(p^j). \end{equation} Define \begin{equation*} \Theta : =\frac1{2\pi i}\frac{d}{dz}=q \frac{d}{dq}. \end{equation*}
\begin{lemma}\label{P:HD} If $(m,N)=1$, then we have \begin{equation} T_{k,\chi}(m) : M_k^\infty (N,\chi) \rightarrow M_k^\infty (N,\chi). \label{E:T} \end{equation} If $k\geq 2$ then \begin{equation} \Theta^{k-1} : M_{2-k}^\infty (N,\chi) \rightarrow M_k^\infty (N,\chi). \label{E:Th} \end{equation} \end{lemma} \begin{proof} For the first statement, it suffices to show that for each prime $p\nmid N$ we have \begin{equation*} T_{k,\chi}(p) : M_k^\infty (N,\chi) \rightarrow M_k^\infty (N,\chi). \end{equation*} We have \begin{equation}\label{E:Hecke}
f|T_{k,\chi}(p) = p^{\frac k2-1}\left(\sum_{j=0}^{p-1} f\big |_k \pMatrix 1j0p
+ \chi(p) f\big |_k
\pMatrix p001 \right). \end{equation} Let $r\in\mathbb{Q}$ be a cusp of $\Gamma_0(N)$ inequivalent to $\infty$ and choose $\gamma= \pmatrix abcd \in \operatorname{SL}_2(\mathbb{Z}) \backslash \Gamma_0(N)$ with $\gamma\infty = r$. Given $j\in \{0,\dots,p-1\}$ set $\lambda:=(a+c j, p)$. By a standard argument
(see e.g. \cite[\S 6.2]{Iwaniec}) we find that \[ \pMatrix 1j0p \pMatrix abcd =\pMatrix
{\frac{a+cj}{\lambda}}{*}
{\frac{cp}{\lambda}}{*} \pMatrix \lambda *
0 {\frac{p}\lambda}
\]
where the first matrix on the right is in $\operatorname{SL}_2(\mathbb{Z}) \backslash \Gamma_0(N)$.
It follows that each term from the sum on $j$ in \eqref{E:Hecke} is holomorphic at cusps other than $\infty$. To see that the last summand is also holomorphic at these cusps, let $\lambda':=(p, c)$. Then \[ \pMatrix p001 \pMatrix abcd =\pMatrix
{\frac{ap}{\lambda'}}*
{\frac{c}{\lambda'}}* \pMatrix
{\lambda'}*0{\frac{p}{\lambda'}}
\]
where the first matrix on the right is in $\operatorname{SL}_2(\mathbb{Z}) \backslash \Gamma_0(N)$.
Let $R_k$ be the Maass raising operator in weight $k$, so that we have the basic relation
\[R_{k-2}\(f \big|_{k-2}\gamma\)=\(R_{k-2} f\)\big|_{k}\gamma.\] Bol's identity (see for example \cite[Lemma 2.1]{BOR}) states that for $k\geq 2$ we have \[\Theta^{k-1}=\frac1{(-4\pi)^{k-1}}R_{k-2}\circ R_{k-4}\circ\cdots\circ R_{4-k}\circ R_{2-k}.\] It follows that \[\Theta^{k-1} : M_{2-k}^!(N,\chi) \rightarrow M_k^! (N,\chi)\] and that
\[\(\Theta^{k-1}f\)\big|_k\gamma=\Theta^{k-1}\(f\big|_{2-k}\gamma\). \] The claim \eqref{E:Th} follows from these two facts. \end{proof}
If $p\nmid 6N$ and $k\geq 0$, let $M_k^{(p)}(N)$ denote the subset of $M_k(N)$ consisting of forms whose coefficients are $p$-integral rational numbers. If $f\in M_k^{(p)}(N)$, define the filtration \[ w_p(f):=\inf\{k' : f\equiv g \pmod p \text{ for some }g \in M_{k'}^{(p)}(N)\}. \] We require two facts, which can be found for example in \cite[\S1]{Joch}. First, if $f\in M_k^{(p)}(N)$ and $w_p(f)\neq -\infty$, then $w_p(f)\equiv k\pmod{p-1}$. Also, we have \begin{equation}\label{E:fil1}
w_p\(f|V(p)\) = p \,w_p(f). \end{equation}
\section{Proof of Theorem~\ref{T:main}}\label{sec:main}
Recall the definitions \eqref{eq:gdef}--\eqref{eq:Fdef}, and note that $F=F_1$ and $g=-F_{-1}$ in the notation of the next proposition. \begin{proposition}\label{T:T2} We have the following. \begin{enumerate} \item For every odd integer $m\geq -1$ there exists a unique $F_m\in M_2^{\infty}(32)\bigcap\mathbb{Z}[\![q]\!]$ of the form \[F_m= -q^{-m}+O(q^3).\] \item Suppose that $p$ is an odd prime and that $n\geq 0$. Then \begin{equation*}
{F} | T_2(p^n) = p^n F_{p^n}+C(p^n)g. \end{equation*} \end{enumerate} \end{proposition} \begin{proof}
For each integer $r\ge 0$, let \begin{equation*} E_r(z) = -g(z)L^r(2z)=-\frac{\eta^2(4z)\eta^{6r}(16z)}{\eta^{2r-2}(8z)\eta^{4r}(32z)}= -q^{-2r+1}+2q^{-2r+5}+O(q^{-2r+9}). \end{equation*} Using standard criteria (see, e.g. \cite[Thm. 1.64, Thm. 1.65]{O:web}) we find that
$E_r\in M_2^{\infty}(32)$. The forms $F_m$ can then be constructed as linear combinations of forms $E_r$ with $2r-1\equiv m \pmod 4$.
Uniqueness follows since the space $S_2(32)$ is one-dimensional. This gives the first assertion.
From \eqref{E:TUV} we have \[
F|T_2(p^n)=F|U(p^n)+\sum_{j=1}^{n-1}p^j F| U(p^{n-j}) | V(p^j)+p^nF|V(p^n).\] Observe that
\[
{F}| U(p^n)=C(p^n)q + O(q^3)= C(p^n)g +O(q^3)
\]
and that
\[
\sum_{j=1}^{n-1} p^j {F}| U(p^{n-j}) | V(p^j)+ p^n {F}| V(p^n) = -p^n q^{-p^n} +O(q^3).
\] Assertion (2) follows from assertion (1) together with Lemma~\ref{P:HD}. \end{proof}
Before proving Theorem~\ref{T:main} we require two lemmas. \begin{lemma}\label{L:Cp} For each prime $p\equiv 3 \pmod 4$ and each integer $m\ge 0$ we have \begin{equation*}\label{E:Cp} C(p^{2m+1}) \equiv (-1)^m p^m \, C(p) \pmod {p^{m+1}}. \end{equation*} \end{lemma} \begin{proof} Lemma 2.3 and Corollary 2.4 of \cite{EO} show that for each $p\equiv 3\pmod 4$, there is a modular function $\phi_p\in M_0^\infty(32)$ of the form \begin{equation}\label{eq:phi_p} \phi_p(z) = q^{-p} +C(p)q+O(q^3) \end{equation}
(we have corrected a sign error in the proof of the corollary).
From Lemma~\ref{P:HD} we have
\[ \Theta(\phi_p) = -pq^{-p}+C(p)q+O(q^3)\in M_2^\infty(32). \] On the other hand, Proposition~\ref{T:T2} gives \[
{F}| T_2(p) = -pq^{-p}+C(p)q +O(q^3). \] Therefore \begin{equation}\label{E:T2T}
{F}| T_2(p) = \Theta(\phi_p), \end{equation} or equivalently \begin{equation}\label{E:Up1}
{F}| U(p) = \Theta(\phi_p)-p\,{F}| V(p). \end{equation} Applying $U(p^2)$ to both sides of \eqref{E:Up1} and arguing inductively, we obtain the following for each $m\geq 0$: \begin{equation}\label{E:Up2}
{F}| U(p^{2m+1}) = \sum_{k=0}^m (-1)^{m-k} p^{m-k} \Theta(\phi_p)| U(p^{2k})+(-1)^{m+1}p^{m+1}{F}| V(p). \end{equation}
For any $k\ge 0$ we have $\Theta(\phi_p)| U(p^{2k}) \equiv 0 \pmod {p^{2k}}$. Therefore for each each $m\ge 0$ we have \begin{equation}\label{E:PT}
{F} | U(p^{2m+1}) \equiv (-1)^m p^m \Theta(\phi_p) \pmod {p^{m+1}}. \end{equation} The lemma follows by comparing coefficients of $q$ in \eqref{E:PT}. \end{proof}
The authors of \cite{EO} speculated that $v_p(C(p))=0$ for every prime $p\equiv 3 \pmod 4$. We prove that this is the case. \begin{lemma}\label{L:Cp2} For each prime $p\equiv 3 \pmod 4$ we have $p\nmid C(p)$. \end{lemma} \begin{proof} Assume to the contrary that $p \mid C(p).$ From \eqref{E:T2T} and Proposition~\ref{T:T2} it follows that \[
\Theta(\phi_p) = {F}| T_2(p) = pF_p +C(p)g \equiv 0 \pmod p, \] from which it follows that for some integral coefficients $A_p$ we have \[ \phi_p \equiv q^{-p}+\sum_{n=1}^\infty A_p(np)q^{np} \pmod p. \]
Let \[f(z)= \frac{\eta^8(32z)}{\eta^4(16z)} = q^8+4q^{24}+O(q^{40})\in M_2(32).\] Then $f^p \in M_{2p}(32)$ has the form \[f^p \equiv \sum_{n= 8}^\infty B_p(np)q^{np}\equiv q^{8p}+\cdots\pmod p.\] Since $\phi_p\in M_0^\infty(32)$, we find that $h_p:=\phi_p f^p\in M_{2p}(32)$ has the form \[h_p \equiv \sum_{n =7}^\infty D_p(pn) q^{pn}\equiv q^{7p}+\cdots\pmod p.\] so that \begin{equation}\label{E:hp}
h_p \equiv h_p| U(p)|V(p) \pmod p. \end{equation} Using \eqref{E:fil1} we obtain \[
w_p(h_p)=p\,w_p(h_p| U(p)). \] Since $w_p(h_p)\equiv 2p\pmod{p-1}$ and $p\mid w_p(h_p)$ we must have
$w_p(h_p)=2p$, so that $w_p(h_p| U(p))=2$. Thus there exists $h_0\in M_2^{(p)}(32)$ such that \[
h_0 \equiv h_p | U(p) = q^7+O(q^8) \pmod p. \] However, by examining a basis for the eight-dimensional space $M_2(32)$
we find that there is no such form $h_0$. This provides the desired contradiction. \end{proof}
\begin{proof}[Proof of Theorem~\ref{T:main}] Assertion \eqref{eq:vpcp} follows from Lemmas~\ref{L:Cp} and \ref{L:Cp2}. To prove \eqref{eq:thmmain}, we use Proposition~\ref{T:T2} and \eqref{E:TUV} to find that \begin{equation}\label{E:FU}
\frac{{F}| U(p^{2m+1})}{C(p^{2m+1})}- g =\frac{1}{C(p^{2m+1})}\left(p^{2m+1}F_{p^{2m+1}} - \sum_{j=1}^{2m+1}p^j {F} | U(p^{2m+1-j})| V(p^j)\right). \end{equation} Using \eqref{E:TUV} we obtain \[
{F}| T_2(p^{2m})
= \sum_{j=1}^{2m+1}p^{j-1} {F} | U(p^{2m+1-j})| V(p^{j-1}). \] Since $C(n)=0$ for $n\not\equiv 3 \pmod 4$, we see from Proposition~\ref{T:T2} that
${F}| T_2(p^{2m})=p^{2m}F_{p^{2m}}$. It follows that \begin{equation*}
\sum_{j=1}^{2m+1}p^{j} {F} | U(p^{2m+1-j})| V(p^{j}) = p^{2m+1}F_{p^{2m}}| V(p) \equiv 0 \pmod {p^{2m+1}}. \end{equation*} Assertion \eqref{eq:thmmain} now follows from
\eqref{E:FU} and \eqref{eq:vpcp}. \end{proof}
\section{An example in weight $4$ and level $9$}\label{sec:GKO} In \cite{GKO}, the authors study the $p$-adic coupling of mock modular forms and their shadows. As an application of their general result, they prove two $p$-adic limit formulas involving the hypergeometric functions $_2F_1\left(\frac{1}{3},\frac{1}{3};1;z\right)$ and $_2F_1\left(\frac{1}{3},\frac{2}{3};1;z\right)$ evaluated at certain modular functions. We will use the following notation: \begin{align*} g_1(z)&=\eta^8(3z)= \sum_{n\ge 1}a(n)q^n=q-8 q^4+20 q^7-70 q^{13}+\cdots\in S_4(9),\\ L_1(z)&=\frac{\eta^3(z)}{\eta^3(9z)}+3=\frac{1}{q}+5 q^2-7 q^5+3 q^8+15 q^{11}+\cdots,\\ G(z) &= g_1(z)L_1^2(z)=\sum_{n\ge -1}C(n)q^n=\frac{1}{q}+2 q^2-49 q^5+48 q^8+771 q^{11}+\cdots. \end{align*} After rewriting using (3.3) and (3.4) of \cite{GKO}, we find that each of the two formulas in Theorem~1.3 of \cite{GKO} is equivalent to the assertion that for every prime $p\equiv 2 \pmod 3$ with $p^3\nmid C(p)$ we have \begin{equation}\label{E:g4}
\lim_{m\rightarrow \infty} \frac{G | U(p^{2m+1})}{C(p^{2m+1})}=g_1(z). \end{equation} Here we prove a strengthened version of this result. \begin{theorem}\label{T:main4} Let $p\equiv 2 \pmod 3$ be a prime. Then for each integer $m\ge 0$ we have \begin{align} v_p(C(p^{2m+1}))&=\begin{cases}
3m+1, & \text{if } p=2,\\
3m, & \text{if } p\neq2.
\end{cases} \label{E:vp41}\\
v_p\left(\frac{G| U(p^{2m+1})}{C(p^{2m+1})}-g_1\right)&\ge \begin{cases}
3m+2, & \text{if } p=2,\\
3m+3, & \text{if } p\neq2.
\end{cases}
\label{E:vp42} \end{align} \end{theorem} The proof follows the argument in Section \ref{sec:main}, so we give fewer details here. \begin{proposition}\label{P:Fm4} We have the following. \begin{enumerate} \item For every integer $m\ge -1$ with $3\nmid m,$ there exists a unique $G_m\in M_4^\infty(9)\bigcap\mathbb{Z}[\![q]\!]$ of the form \[G_m=q^{-m}+O(q^2).\] \item Let $p\neq 3$ be prime and let $n$ be a nonnegative integer. Then we have \begin{equation*}
G| T_4(p^n) = p^{3n}G_{p^n}+C(p^n)g_1. \end{equation*} \end{enumerate} \end{proposition}
\begin{proof} For each integer $r\ge 0,$ let \[E_r(z) = g_1(z)L_1(z)^r = q^{1-r}+(5r-8)q^{4-r}+O(q^{7-r}).\] Then $E_r(z)\in M_4^\infty(9)$. We construct each form $G_m$ by taking a linear combination of $E_r$ with $r-1\equiv m \pmod 3.$ Uniqueness
follows since $S_4(9)$ is spanned by the form $g_1=G_{-1}$.
We deduce assertion (2) as in the last section using \eqref{E:T}, \eqref{E:TUV}, and assertion (1).
\end{proof} \begin{lemma}\label{L:Cp4} If $p\equiv 2 \pmod 3$ is prime, then \begin{equation*} C(p^{2m+1}) \equiv (-1)^{m}p^{3m}C(p) \pmod {p^{3m+3}}. \end{equation*} \end{lemma} \begin{proof} Define
\[
\phi_2(z)= \frac{\eta^2(3z)}{\eta^6(9z)} = \sum_{n\ge -2}A_2(n)q^n=\frac{1}{q^2}-2 q-q^4+O\left(q^5\right).
\] It is seen from the expression of $\phi_2$ as an infinite product that $A_2(n)=0$ if $n\not\equiv 1 \pmod 3$. Similarly, if \[ L_1(z) = \frac{\eta^3(z)}{\eta^3(9z)}+3 = \sum_{n\ge -1}b(n)q^n, \] then $b(n)=0$ for all $n\not\equiv 2 \pmod 3.$ Therefore, for each positive integer $l\equiv 2 \pmod 3$ there exist $c_0,c_1,\ldots, c_{\frac{l-2}{3}}\in\mathbb{Z}$ such that \begin{equation*} \phi_l(z) = \phi_2(z)\sum_{j=0}^{\frac{l-2}{3}}c_j L_1^{l-2-3j}(z) = q^{-l}+\sum_{n\ge 1} A_l(n)q^n \in M_{-2}^\infty(9), \end{equation*} with $A_l(n)\in\mathbb{Z}$ and $A_l(n)=0$ if $n\not\equiv 1 \pmod 3$ (these coincide with the forms $w_l$ in \cite[Prop. 3.1]{GKO}). Since the constant term in the weight two modular form $\phi_l L_1$ must be zero, we find as in the last section that $A_l(1)=-C(l)$. In particular, for any prime $p\equiv 2 \pmod 3$ we have \[\phi_p = q^{-p}-C(p)q +O(q^2).\] By Lemma \ref{P:HD}, we have \[\Theta^3(\phi_p)= -p^3q^{-p}-C(p)q+O(q^2)\in M_4^\infty(9).\] Hence it follows from Proposition \ref{P:Fm4} that \begin{equation}\label{E:ph4}
\Theta^3(\phi_p) = -p^3 G_p -C(p)g_1 = -G| T_4(p) =-G| U(p) -p^3 G| V(p), \end{equation} so that \begin{equation}\label{E:Up4}
G| U(p) = -\Theta^3(\phi_p)-p^3 G| V(p). \end{equation} Applying $U(p^2)$ iteratively leads to \begin{equation}\label{E:Up4m}
G| U(p^{2m+1}) = \sum_{l=0}^m (-1)^{m+1-l}p^{3(m-l)}\Theta^3(\phi_p)| U(p^{2l})+(-1)^{m+1}p^{3(m+1)}G| V(p) \end{equation}
for any non-negative integer $m$. Since $\Theta^3(\phi_p)| U(p^{2l}) \equiv 0 \pmod {p^{6l}}$, we have from \eqref{E:Up4m} that \begin{equation}\label{E:Up4mc}
G| U(p^{2m+1}) \equiv (-1)^{m+1}p^{3m}\Theta^3(\phi_p) \pmod {p^{3m+3}}. \end{equation} Comparing coefficients of $q$ in \eqref{E:Up4mc} gives the result. \end{proof}
The authors of \cite{GKO} verified that $p^3\nmid C(p)$ for every prime $p\equiv 2 \pmod 3$ less than $32,500$. Here we prove \begin{lemma}\label{L:Cp4d} For every odd prime $p\equiv 2 \pmod 3$, we have $p\nmid C(p)$. \end{lemma} \begin{proof} Suppose by way of contradiction that $p\equiv 2\pmod 3$ is an odd prime with $p\mid C(p).$ Then \eqref{E:ph4} gives \[\Theta^3(\phi_p) \equiv 0 \pmod p,\] which implies that for some coefficients $A_p$ we have \[ \phi_p\equiv q^{-p}+\sum_{n\ge 1}A_p(np)q^{np} \pmod p. \] Since $\phi_2$ has no zeros on the upper half plane (and does not vanish at any cusp), we have $h_p:=\phi_p \phi_2^{-p} \in M_{2p-2}(9)$. Moreover, \[h_p\equiv \sum_{n\ge p}D_p(pn)q^{pn}\equiv q^p+\cdots\pmod p.\] Therefore
$h_p| U(p)|V(p) \equiv h_p \pmod p$ so that
$w_p(h_p)=p w_p(h_p| U(p))$. Since $w_p(h_p)\equiv 2p-2\pmod {p-1}$ and $w_p(h_p)\equiv 0\pmod p$, we must have $w_p(h_p)=0$, but this is impossible since $M_0(9)$ contains no non-constant elements. \end{proof}
\begin{proof}[Proof of Theorem \ref{T:main4}] Assertion \eqref{E:vp41} follows from Lemma~\ref{L:Cp4}, Lemma~\ref{L:Cp4d}, and the fact that $C(2)=2$. Next, we use Proposition \ref{P:Fm4} and \eqref{E:TUV} to write \begin{equation}\label{E:FU4}
\frac{G| U(p^{2m+1})}{C(p^{2m+1})}- g_1 =\frac{1}{C(p^{2m+1})}\left(p^{6m+3}G_{p^{2m+1}} - \sum_{j=1}^{2m+1}p^{3j} G | U(p^{2m+1-j})| V(p^j)\right). \end{equation} Since $C(n)=0$ for any $n\not\equiv 2 \pmod 3$, Proposition~\ref{P:Fm4} and \eqref{E:TUV} give
\[\sum_{j=1}^{2m+1}p^{3j} G | U(p^{2m+1-j})| V(p^{j}) = p^3G| T_4(p^{2m})| V(p) = p^{6m+3}G_{p^{2m}}| V(p) \equiv 0 \pmod {p^{6m+3}}.\] The result follows from \eqref{E:FU4} and \eqref{E:vp41}. \end{proof}
\section{An example in weight $3$ and level $16$}\label{sec:BG} In \cite{BG} the authors establish an analogous representation of a weight $3$ cusp form as a $p$-adic limit. Let $\chi$ denote the non-trivial Dirichlet character modulo $4$, and define \begin{align*} g_2(z)&:= \eta^6(4z) = \sum_{n\ge 1}a(n)q^n=q-6 q^5+9 q^9+\cdots \in S_3(16,\chi),\\ L_2(z) &:= \frac{\eta^6(8z)}{\eta^2(4z)\eta^4(16z)}=\frac{1}{q}+2 q^3-q^7-2 q^{11}+\cdots,\\ H(z) &:= g_2(z)L_2^2(z)=\sum_{n\ge -1}C(n)q^n=\frac{1}{q}-2 q^3-13 q^7+26 q^{11}+\cdots. \end{align*} The two formulas stated in the main theorem of \cite{BG} involve the hypergeometric function $_2F_1(\frac{1}{2},\frac{1}{2};1;z);$ after rewriting they are equivalent to the following statement: for every prime $p\equiv 3 \pmod 4$ with $p^2\nmid C(p)$ we have
\[\lim_{m\rightarrow \infty} \frac{H| U(p^{2m+1})}{C(p^{2m+1})}=g_2(z).\] Here we prove \begin{theorem}\label{T:main3} For every prime $p\equiv 3 \pmod 4$ and every integer $m\ge 0$ we have \begin{align} v_p(C(p^{2m+1}))&=2m, \label{E:vp31}\\
v_p\left(\frac{H| U(p^{2m+1})}{C(p^{2m+1})}-g_2\right)&\ge 2m+2.\label{E:vp32} \end{align} \end{theorem} We give only a sketch of the proof. \begin{proposition}\label{L:Fm3} We have the following. \begin{enumerate} \item For every odd integer $m\ge -1,$ there exists a unique $H_m\in M_3^\infty(16,\chi)\bigcap\mathbb{Z}[\![q]\!]$ of the form \[H_m = q^{-m}+O(q^3).\] \item Let $p$ be an odd prime and let $n$ be a nonnegative integer. Then we have
\[ H | T_{3,\chi}(p^n) = \chi(p^n)p^{2n} H_{p^n}+C(p^n)g_2.\] \end{enumerate} \end{proposition} \begin{proof} For each integer $r\ge 0$ define \[ E_r(z) := g_2(z)L_2^r(z)= \frac{\eta^{6r}(8z)}{\eta^{2r-6}(4z)\eta^{4r}(16z)}\in M_3^\infty(16,\chi).\] We construct the form $H_m$ with the desired properties by taking an appropriate linear combination of $E_r$, and uniqueness follows since $S_3(16,\chi)$ is one-dimensional. Assertion (2) is proved as before. \end{proof}
\begin{lemma}\label{L:Cp3} If $p\equiv 3 \pmod 4$ is prime and $m\geq 0$ then \[C(p^{2m+1}) \equiv p^{2m}C(p) \pmod {p^{2m+2}}.\] \end{lemma} \begin{proof} For each $l\ge 2$, let $\phi_l\in M_{-1}^\infty(16,\chi)$ be the form given in \cite[Lem. 3.3]{BG}. We have $\phi_2(z)=\frac{\eta^2(8z)}{\eta^4(16z)}$. For $l\ge 3$ we have \[\phi_l(z) = \phi_2(z)P_l(L_2(z)), \] where $P_l(x)\in\mathbb{Z}[x]$ has $\deg P_l = l-2.$ Let $p\equiv 3 \pmod 4$ be prime. As above we find that \[\phi_p(z)=q^{-p}-C(p)q+\sum_{n\ge 5}A_p(n)q^n.\] It follows from Proposition~\ref{P:HD} that \begin{equation}\label{E:ph3} \Theta^2(\phi_p) = p^2q^{-p}-C(p)q+O(q^5) \in M_3^\infty(16,\chi), \end{equation} and we deduce using Proposition \ref{L:Fm3} that
\[H| U(p) = H| T_{3,\chi}(p)+p^2 H| V(p) = -\Theta^2(\phi_p)+p^2 H| V(p).\] Iteratively applying $U(p^2)$ results in \begin{equation*}
H | U(p^{2m+1}) =-\sum_{l=0}^m p^{2(m-l)}\Theta^2(\phi_p)| U(p^{2l})+p^{2(m+1)}H| V(p), \end{equation*} so we have \begin{equation}\label{E:Up3}
H | U(p^{2m+1}) \equiv -p^{2m}\Theta^2(\phi_p) \pmod {p^{2m+2}}. \end{equation} Comparing coefficients gives the result. \end{proof}
\begin{lemma}\label{L:Cp3d} For every prime $p\equiv 3 \pmod 4$ we have $p\nmid C(p).$ \end{lemma} \begin{proof} Suppose by way of contradiction that $p\mid C(p).$ Then \eqref{E:ph3} and Lemma~\ref{L:Fm3} show that $\Theta^2(\phi_p) \equiv 0 \pmod p,$ whence \[\phi_p \equiv q^{-p}+\sum_{n\ge 1}A_p(np)q^{np} \pmod p.\] Let $f(z)=\frac{\eta^{12}(16z)}{\eta^6(8z)}= q^6+6q^{14}+O(q^{22})\in M_3(16,\chi).$ Then $h_p := \phi_p f^p \in M_{3p-1}(16)$ has the form \[h_p\equiv \sum_{n \ge 5p}D_p(pn)q^{pn}\equiv q^{5p}+\cdots\pmod p,\] so that
\[h_p \equiv h_p| U(p)|V(p) \pmod p.\]
Analyzing the filtration yields $w_p(h_p) = 2p$ and $w_p(h_p | U(p)) =2$. However, we find by examining a basis that there is no form $h_0\in M_2^{(p)}(16)$ with $h_0\equiv q^5+\cdots\pmod p$. This provides the desired contradiction. \end{proof} The proof of Theorem~\ref{T:main3} follows as before.
\end{document} | arXiv |
Abstract: We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves in the same connected component can be joined by a minimizing geodesic. These results then imply that the shape space of unparametrized curves has the structure of a complete length space. | CommonCrawl |
Towards a multi-tracer timeline of star formation in the LMC -- I.\ Deriving the lifetimes of H\,{\sc i} clouds
Ward, JL, Chevance, M, Kruijssen, JMD, Hygate, APS, Schruba, A and Longmore, SN Towards a multi-tracer timeline of star formation in the LMC -- I.\ Deriving the lifetimes of H\,{\sc i} clouds. Monthly Notices of the Royal Astronomical Society. ISSN 0035-8711 (Accepted)
2007.03691v1.pdf - Accepted Version
Publisher URL: https://dx.doi.org/10.1093/mnras/staa1977
The time-scales associated with the various stages of the star formation process remain poorly constrained. This includes the earliest phases of star formation, during which molecular clouds condense out of the atomic interstellar medium. We present the first in a series of papers with the ultimate goal of compiling the first multi-tracer timeline of star formation, through a comprehensive set of evolutionary phases from atomic gas clouds to unembedded young stellar populations. In this paper, we present an empirical determination of the lifetime of atomic clouds using the Uncertainty Principle for Star Formation formalism, based on the de-correlation of H$\alpha$ and H\,{\sc i} emission as a function of spatial scale. We find an atomic gas cloud lifetime of 48$\substack{+13\\-8}$\,Myr. This timescale is consistent with the predicted average atomic cloud lifetime in the LMC (based on galactic dynamics) that is dominated by the gravitational collapse of the mid-plane ISM. We also determine the overlap time-scale for which both H\,{\sc i} and H$\alpha$ emission are present to be very short ($t_{over}<1.7$\,Myr), consistent with zero, indicating that there is a near-to-complete phase change of the gas to a molecular form in an intermediary stage between H\,{\sc i} clouds and H\,{\sc ii} regions. We utilise the time-scales derived in this work to place empirically determined limits on the time-scale of molecular cloud formation. By performing the same analysis with and without the 30 Doradus region included, we find that the most extreme star forming environment in the LMC has little effect on the measured average atomic gas cloud lifetime. By measuring the lifetime of the atomic gas clouds, we place strong constraints on the physics that drives the formation of molecular clouds and establish a solid foundation for the development of a multi-tracer timeline of star formation in the LMC.
This is a pre-copyedited, author-produced PDF of an article accepted for publication in Monthly Notices of the Royal Astronomical Society following peer review. The version of record Jacob L Ward, Mélanie Chevance, J M Diederik Kruijssen, Alexander P S Hygate, Andreas Schruba, Steven N Longmore, Towards a multi-tracer timeline of star formation in the LMC – I. Deriving the lifetimes of H I clouds, Monthly Notices of the Royal Astronomical Society, , staa1977, https://doi.org/10.1093/mnras/staa1977 is available online at:https://dx.doi.org/10.1093/mnras/staa1977
Q Science > QC Physics
10.1093/mnras/staa1977 | CommonCrawl |
for Algebraic Geometry Seminar events the year of Friday, September 14, 2018.
4:00 pm in Illini Hall 1,Tuesday, January 16, 2018
Graduate Student Algebraic Geometry Seminar
Abstract: We'll have a cookies party while deciding what'll be in the seminar this semester.
Submitted by lzhao35
Moduli space of compact Riemann surfaces
Jin Hyung To (UIUC)
Abstract: We will overview the moduli space of compact Riemann surfaces.
Riemann-Roch Formula and The Dimension of Our Universe
Lutian Zhao [email] (UIUC)
Abstract: In this talk we'll introduce the classical Riemann-Roch formula, which appears as a vast generalization of the Euler-Maclaurin formula for the integrals. As an interesting application, the critical dimension for the bosonic string theory can be calculated by these formula to be d=26, which matches with the physical prediction using light-cone quantization. No basic knowledge on string theory and Riemann-Roch will be assumed.
4:00 pmTuesday, February 6, 2018
Abstract: Cancelled
3:00 pm in 243 Altgeld Hall,Tuesday, February 13, 2018
A noncommutative McKay correspondence
Chelsea Walton (UIUC)
Abstract: The aim of this talk is two-fold-- (1) to recall the classic McKay correspondence in the commutative/ classic setting for group actions on polynomial rings, and (2) to present a generalization of the McKay correspondence in the noncommutative/ quantum setting of Hopf algebra actions on noncommutative analogues of polynomial rings. Many notions will be defined from scratch during the talk and pre-talk, and useful examples will be provided during the pre-talk.
Submitted by ecliff
4:00 pm in Illini Hall 1,Tuesday, February 13, 2018
Introduction to Cohomological Field Theory
Sungwoo Nam (UIUC)
Abstract: Cohomological field theory(CohFT) was first introduced by Kontsevich and Manin to organize the data of Gromov-Witten theory and quantum cohomology into a list of axioms. Although its main model is Gromov-Witten theory, it has been also successful dealing with problems outside of Gromov-Witten theory. In this talk, I will introduce the notion of CohFT, Givental-Teleman's classification of semisimple CohFTs and some concrete examples. Basic knowledge of Gromov-Witten theory will be helpful, but it is not assumed in this talk.
A Bird's-Eye View of Seiberg Witten Integrable Systems
Matej Penciak (UIUC)
Abstract: In this talk I will give a rudimentary description of supersymmetric gauge theories, and focus on the particular case of $N=2$ supersymmetry in dimension $4$ with gauge group $SU(2)$. In this setting, originally noticed and explained by Seiberg and Witten in 1994, the moduli of vacua exhibits the structure of an algebraic integrable system. I will explain how this structure manifests itself, and the give a sketch of the calculation that Seiberg and Witten made in their original paper. If time permits, I will explain the generalization of this story to more general gauge groups, and with possible additional matter fields included in the theory.
4:00 pm in 1 Illini Hall,Tuesday, February 27, 2018
Itziar Ochoa de Alaiza Gracia (UIUC)
Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.
3:00 pm in Illini Hall 2,Tuesday, March 6, 2018
(normally ordered) Tensor product of Tate objects and decomposition of higher adeles
Aron Heleodoro (Northwestern University)
Abstract: In this talk I will introduce the different tensor products that exist on Tate objects over vector spaces (or more generally coherent sheaves on a given scheme). As an application, I will explain how these can be used to describe higher adeles on an n-dimensional smooth scheme. Both Tate objects and higher adeles would be introduced in the talk. (This is based on joint work with Braunling, Groechenig and Wolfson.)
4:00 pm in 1 Illini Hall,Tuesday, March 6, 2018
Introduction to GIT, II
Itziar Ochoa de Alaiza Gracia
Constructible 1-motives
Simon Pepin Lehalleur (Freie Universität Berlin)
Abstract: Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise, we have at our disposal a mature theory of triangulated categories of mixed motivic sheaves with rational coefficients over general base schemes, with a "six operations" formalism and the expected relationship with algebraic cycles and algebraic K-theory. A parallel development has taken place in the study of Voevodsky's category of mixed motives over a perfect field, where the subcategory of 1-motives (i.e., generated by motives of curves) has been completely described by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to combine these two sets of ideas to study the triangulated category of 1-motivic sheaves over a base. Our main results are the definition of the motivic t-structure for constructible 1-motivic sheaves, a precise relation with Deligne 1-motives, and the extraction of the "1-motivic part" of a general motivic sheaves via a "motivic Picard functor".
4:00 pm in 1 Illini Hall,Tuesday, March 13, 2018
Algebraic Morse theory from GIT
Jesse Huang (UIUC)
Abstract: Birational geometry is closely tied to GIT quotients and variations. In this episode of GIT series, I will apply the machinery to a countable set of basic examples, through which we shall see how the change of linearization produces elementary birational transformations.
A gentle approach to the de Rham-Witt complex
Akhil Mathew (The University of Chicago)
Abstract: The de Rham-Witt complex of a smooth algebra over a perfect field provides a chain complex representative of its crystalline cohomology, a canonical characteristic zero lift of its algebraic de Rham cohomology. We describe a simple approach to the construction of the de Rham-Witt complex. This relates to a homological operation L\eta_p on the derived category, introduced by Berthelot and Ogus, and can be viewed as a toy analog of a cyclotomic structure. This is joint work with Bhargav Bhatt and Jacob Lurie.
3:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2018
Equal sums of higher powers of binary quadratic forms, I
Bruce Reznick (UIUC)
Abstract: We will describe all non-trivial solutions to the equation $f_1^d(x,y) + f_2^d(x,y) = f_3^d(x,y) + f_4^d(x,y)$ for quadratic forms $f_j \in \mathbb C[x,y]$. No particular prerequisites are needed and tools will be derived during the talk. Lots of fun stuff. The content of the second talk, next week, will be shaped by the reaction to this one.
3:00 pm in 243 Altgeld Hall,Tuesday, April 10, 2018
Equal sums of higher powers of binary quadratic forms, II
Abstract: A continuation of last week, but newcomers are welcome and will be brought up to speed. I promise to completely satisfy any curiosity you might have about the representation of binary sextic forms as a sum of two cubes of binary forms and as a sum of three cubes of binary forms. A theorem about universal representations as a sum of three cubes resolves the first non-trivial case of a conjecture of Boris Shapiro.
4:00 pm in 1 Illini Hall,Tuesday, April 10, 2018
GIT quotients of flag varieties
Joshua Wen (UIUC)
Abstract: As we've seen, GIT quotients depend on a choice of line equivariant line bundle, and varying this choice can lead to drastic or subtle changes between quotients. After introducing a framework for 'variation of GIT' by Dolgachev and Hu, I want to consider a case of the flag variety and its action either by a torus or semisimple group. Here, one already knows many equivariant line bundles, and studying dimensions of invariant sections leads to results of representation-theoretic significance.
Moduli of Twisted Curve
Hao Sun (UIUC)
Abstract: I'll give an introductory talk about the twisted curves. Twisted curves are related to the study of r-spin Witten classes and r-spin geometry of the moduli space of curves.
3:00 pm in 243 Altgeld Hall,Tuesday, August 28, 2018
The ghost of Arthur Coble (?)
Submitted by nevins
4:00 pm in 2 Illini Hall,Wednesday, August 29, 2018
3:00 pm in 243 Altgeld Hall,Tuesday, September 4, 2018
Tamely ramified geometric Langlands correspondence in positive characteristic
Shiyu Shen (UIUC Math)
Abstract: I will describe a generic version of tamely ramified geometric Langlands correspondence (GLC) in positive characteristic for $GL_n$, generalizing the work of Bezrukavnikov-Braverman on the unramified case. Let $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic $p>n$. I will give a spectral description of the parabolic Hitchin fiber over an open subset of the Hitchin base, and describe a correspondence between flat connections with regular singularities on $X$ and twisted Higgs bundles on the Frobenius twist $X^{(1)}$. Then I will explain how to use a twisted version of Fourier-Mukai transform to establish the GLC.
4:00 pm in 2 Illini Hall,Wednesday, September 5, 2018
(Crystalline) differential operators in positive characteristic
Abstract: I will talk about several features of (Crystalline) differential operators in characteristic $p$, including the Azumaya property and two theorems by Cartier.
3:00 pm in 243 Altgeld Hall,Tuesday, September 11, 2018
A geometric model for complex analytic equivariant elliptic cohomology
Arnav Tripathy (Harvard)
Abstract: Elliptic cohomology has always been a natural big brother to ordinary cohomology and K-theory and is often implicated in the trichotomies of integrable systems or geometric representation theory. However, computations with elliptic cohomology are often made difficult by the fact that we do not know geometric representatives for elliptic cohomology classes. I will explain in this talk a step forward, in joint work with D. Berwick-Evans, for the case of equivariant elliptic cohomology over the complex numbers by using geometric constructions inspired by supersymmetric field theory. This talk will need no prior knowledge of either elliptic cohomology or field theories.
4:00 pm in 2 Illini Hall,Wednesday, September 12, 2018
Differential Equations from Hodge Theory
Lutian Zhao [email] (UIUC Math)
Abstract: The classical theory of elliptic integrals is the milestone in the history of various fields in math: algebraic geometry, differential equations, number theory,.. etc. In this talk, I'll use this as the motivating example for the theory of periods. I'll talk about how we get some equations for the periods and interpret these equations in terms of Hodge theory. As an interesting application, I'll calculate the number of rational curves on quintic threefold by these differential equations. Only complex analysis is assumed.
Window equivalences and spherical functors
Jesse Huang [email] (UIUC Math)
Abstract: We will appreciate some recent results on the derived categories of GIT quotients through the basic example of a flip/flop.
Kasteleyn Operators from Mirror Symmetry
Eric Zaslow (Northwestern University)
Abstract: Kenyon-Okounkov-Sheffied showed that the statistical properties of dimer configurations on bipartite graphs on a two-torus are determined by a spectral curve. Goncharov-Kenyon showed how edge weights of the graph define a cluster integrable system. I will show how both of these results follow from sheaf quantization in the context of homological mirror symmetry (HMS): the spaces involved are moduli spaces of objects in two categories related by HMS. This talk is based on joint work with David Treumann and Harold Williams.
Line bundles on abelian varieties
Matej Penciak [email] (UIUC Math)
Abstract: In this talk I want to give classical results on line bundles on abelian varieties. I'll begin with the Appel-Humbert theorem. Then I will give an introduction to theta functions and show that they can be identified with sections of line bundles. This will connect the older approach with the more well-known description in terms of divisors. Finally, I'll consider more general problem of classifying vector bundles on elliptic curves, and describe Atiyah's solution to the classification problem.
3:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018
S(ymplectic) duality
Justin Hilburn
Abstract: In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G. By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands. This is based on joint work with PhilsangYoo, Tudor Dimofte, and Davide Gaiotto.
4:00 pm in 2 Illini Hall,Wednesday, October 3, 2018
Flops and derived categories of threefolds, Part 1
Sungwoo Nam (UIUC Math)
Abstract: In his paper, Bridgeland showed that derived categories of threefolds, which are related by flopping operations, are equivalent. Besides its own interest, this result can be used to study behavior of invariants of threefolds under birational morphisms. In this talk, we will present Bridgeland's work for two weeks. As the main idea involves constructing flop as a moduli space of perverse point sheaves, I'll introduce some notions such as derived categories and t-structures and their properties relevant to the proof. After that, I will give application of the theorem on birational Calabi-Yau threefolds and curve counting invariants.
On the Hitchin fibration for algebraic surfaces
Tsao-Hsien Chen (University of Chicago)
Abstract: In his work on non-abelian Hodge theory, Simpson constructs the Hitchin map from the moduli spaces of Higgs bundles over an arbitrary smooth algebraic variety X to an affine space, generalizing Hitchin's construction in the case of when X is a Riemann surface. Very little is known about the geometry of Simpson's Hitchin map except in the case when X is one-dimensional. In the talk I will report on some recent developments on the structure of Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces. Joint work with B.C. Ngo.
4:00 pm in 2 Illini Hall,Wednesday, October 10, 2018
Ciaran O'Neill (UIUC Math)
Abstract: We will give the proof of Bridgelands theorem, stated last time. We will introduce Fourier-Mukai transforms, an important part of the proof.
A Spectral Description of the Ruijsenaars-Schneider System
Matej Penciak (UIUC Math)
Abstract: The Ruijsenaars-Schneider (RS) integrable hierarchy is a many-particle system which can be viewed as a relativistic analogue of the Calogero-Moser system. The integrability and Lax form of the system has been known since it was introduced by Ruijsenaars and Schneider. In this talk I will give background on the RS system, and some classical results on elliptic functions. Then I will explain work in preparation that identifies the RS system and its Lax matrix in terms of spectral sheaves living in the total space of projective bundles on cubic curves. This work provides input to a larger project (some of it joint with David Ben-Zvi and Tom Nevins), and I will give an outline for why this spectral description will be useful in the larger project.
Derived categories of abelian categories and applications
Jin Hyung To
Abstract: TBD
Derived categories of abelian categories, II
Abstract: TBA
Orientation data for coherent sheaves on local $\mathbb{P}^2$
Yun Shi (UIUC Math)
Abstract: Orientation data is an ingredient in the definition of Motivic Donaldson-Thomas (DT) invariant. Roughly speaking, it is a square root of the virtual canonical bundle on a moduli space. It has been shown that there is a canonical orientation data for the stack of quiver representations for a quiver with potential. In this talk, I will briefly introduce Motivic DT invariant, and the role of orientation data in its definition. I will then give a construction of orientation data for the stack of coherent sheaves on local $\mathbb{P}^2$ based on the canonical orientation data from quiver representations.
Congruences of modular forms from an algebro-geometric perspective
Ningchuan Zhang (UIUC Math)
Abstract: In this talk, I'll give an algebro-geometric explanation of congruences of normalized Eisenstein series following chapter 4 of Nicholas Katz's paper "$p$-adic properties of modular schemes and modular forms". The key idea in Katz's paper is to establish a $p$-adic Riemann-Hilbert correspondence that can translate congruences of normalized Eisenstein series to that of continuous $\mathbb{Z}_p^\times$-representations in rank $1$ free $\mathbb{Z}_p$-modules. The latter is very easy to compute given that $\mathbb{Z}_p^\times$ is topologically cyclic when $p\neq 2$.
4:00 pm in 2 Illini Hall,Wednesday, November 7, 2018
Model theory and ideas from algebraic geometry
Chieu Minh Tran (UIUC Math)
Abstract: In this (very soft) talk, I will give a model theory crash course and explain how ideas from classical (or ancient?) algebraic geometry have played an important role in shaping the field as we know it today.
3:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2018
Severi degrees via representation theory
Yaim Cooper (IAS)
Abstract: The Severi degrees of $\mathbb{P}^1 \times \mathbb{P}^1$ can be computed in terms of an explicit operator on the Fock space $F[\mathbb{P}^1]$. We will discuss this and variations on this theme. We will explain how to use this approach to compute the relative Gromov-Witten theory of other surfaces, such as Hirzebruch surfaces and $E \times\mathbb{P}^1$. We will also discuss operators for calculating descendants. Joint with R. Pandharipande.
4:00 pm in 243 Altgeld Hall,Friday, November 16, 2018
De Rham epsilon lines and the epsilon connection
Michael Groechenig (University of Toronto)
Abstract: De Rham epsilon lines for holonomic D-modules on curves were introduced by Deligne and Beilinson-Bloch-Esnault. This formalism includes a product formula, expressing the determinant of cohomology of a holonomic D-module as a tensor product of the epsilon lines computed with respect to a non-zero rational 1-form. Patel generalised the theory of de Rham epsilon factors to arbitrary dimensions. A curious feature of BBE's 1-dimensional theory, is the epsilon connection which appears when studying the variation of the epsilon lines on the space of non-zero 1-forms. In this talk I will explain how properties of algebraic K-theory yield a conjectural candidate for the epsilon connection in arbitrary dimensions.
Effective generation problems for families of varieties
Yajnaseni Dutta (Northwestern University)
Abstract: In birational geometry, a great deal of interest lies in the study of space of global sections (i.e. linear system) associated to a line bundle and when it produces morphisms to projective spaces (i.e. are globally generated). For instance, Takao Fujita, in 1988, conjectured that there is an effective bound on the twists of ample line bundles to obtain global generation of canonical bundles. Even though the conjecture remains unsolved as of today, partial progress was made by Angehrn-Siu, Ein-Lazarsfeld, Heier, Helmke, Kawamata, Reider, Ye-Zhu et al. In this talk I will focus on similar global generation problems for pushforwards of canonical and pluricanonical bundles under certain morphisms f: Y --> X. The canonical bundle case was first approached by Kawamata and the proof combines the Hodge theoretic properties of the morphism and the birational geometric techniques from the known cases of the Fujita conjecture. I will show how birational techniques and invariants of positivity of line bundles allow us to avoid Hodge theory to make a global generation statement on an open subset (of X) over which f behaves particularly nicely, partially proving a conjecture proposed by Popa and Schnell. This, in particular, ensures an effective non-vanishing of global sections of the pushforward. As an upshot I will discuss positivity properties of such pushforwards and an effective vanishing theorem. Finally, If time permits I will give a crash course on Kawamata's method and explain why Hodge theory is indispensable to hope for further improvements in this direction.
4:00 pm in 2 Illini Hall,Wednesday, December 5, 2018
Tiered UFD
Justin Kelm (UIUC Math)
Abstract: We proved the proporty that a strongly tiered UFD will satisfy. | CommonCrawl |
Ultrafilter on a set
In the mathematical field of set theory, an ultrafilter on a set $X$ is a maximal filter on the set $X.$ In other words, it is a collection of subsets of $X$ that satisfies the definition of a filter on $X$ and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of $X$ that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set $X$ can also be characterized as a filter on $X$ with the property that for every subset $A$ of $X$ either $A$ or its complement $X\setminus A$ belongs to the ultrafilter.
This article is about specific collections of subsets of a given set. For more general ultrafilters on partially ordered sets, see Ultrafilter. For the physical device, see ultrafiltration.
Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set $\wp (X)$ and the partial order is subset inclusion $\,\subseteq .$ This article deals specifically with ultrafilters on a set and does not cover the more general notion.
There are two types of ultrafilter on a set. A principal ultrafilter on $X$ is the collection of all subsets of $X$ that contain a fixed element $x\in X$. The ultrafilters that are not principal are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exists models of ZF where every ultrafilter on a set is principal.
Ultrafilters have many applications in set theory, model theory, and topology.[1]: 186 Usually, only free ultrafilters lead to non-trivial constructions. For example, an ultraproduct modulo a principal ultrafilter is always isomorphic to one of the factors, while an ultraproduct modulo a free ultrafilter usually has more complex structures.
Definitions
See also: Filter (mathematics) and Ultrafilter
Given an arbitrary set $X,$ an ultrafilter on $X$ is a non-empty family $U$ of subsets of $X$ such that:
1. Proper or non-degenerate: The empty set is not an element of $U.$
2. Upward closed in $X$: If $A\in U$ and if $B\subseteq X$ is any superset of $A$ (that is, if $A\subseteq B\subseteq X$) then $B\in U.$
3. π−system: If $A$ and $B$ are elements of $U$ then so is their intersection $A\cap B.$
4. If $A\subseteq X$ then either $A$ or its complement $X\setminus A$ is an element of $U.$[note 1]
Properties (1), (2), and (3) are the defining properties of a filter on $X.$ Some authors do not include non-degeneracy (which is property (1) above) in their definition of "filter". However, the definition of "ultrafilter" (and also of "prefilter" and "filter subbase") always includes non-degeneracy as a defining condition. This article requires that all filters be proper although a filter might be described as "proper" for emphasis.
A filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter subbase is a non-empty family of sets that is contained in some (proper) filter. The smallest (relative to $\subseteq $) filter containing a given filter subbase is said to be generated by the filter subbase.
The upward closure in $X$ of a family of sets $P$ is the set
$P^{\uparrow X}:=\{S:A\subseteq S\subseteq X{\text{ for some }}A\in P\}.$
A prefilter or filter base is a non-empty and proper (i.e. $\varnothing \not \in P$) family of sets $P$ that is downward directed, which means that if $B,C\in P$ then there exists some $A\in P$ such that $A\subseteq B\cap C.$ Equivalently, a prefilter is any family of sets $P$ whose upward closure $P^{\uparrow X}$ is a filter, in which case this filter is called the filter generated by $P$ and $P$ is said to be a filter base for $P^{\uparrow X}.$
The dual in $X$[2] of a family of sets $P$ is the set $X\setminus P:=\{X\setminus B:B\in P\}.$ For example, the dual of the power set $\wp (X)$ is itself: $X\setminus \wp (X)=\wp (X).$ A family of sets is a proper filter on $X$ if and only if its dual is a proper ideal on $X$ ("proper" means not equal to the power set).
Generalization to ultra prefilters
A family $U\neq \varnothing $ of subsets of $X$ is called ultra if $\varnothing \not \in U$ and any of the following equivalent conditions are satisfied:[2][3]
1. For every set $S\subseteq X$ there exists some set $B\in U$ such that $B\subseteq S$ or $B\subseteq X\setminus S$ (or equivalently, such that $B\cap S$ equals $B$ or $\varnothing $).
2. For every set $S\subseteq \bigcup \limits _{B\in U}}B$ there exists some set $B\in U$ such that $B\cap S$ equals $B$ or $\varnothing .$
• Here, $ \bigcup \limits _{B\in U}}B$ is defined to be the union of all sets in $U.$
• This characterization of "$U$ is ultra" does not depend on the set $X,$ so mentioning the set $X$ is optional when using the term "ultra."
3. For every set $S$ (not necessarily even a subset of $X$) there exists some set $B\in U$ such that $B\cap S$ equals $B$ or $\varnothing .$
• If $U$ satisfies this condition then so does every superset $V\supseteq U.$ In particular, a set $V$ is ultra if and only if $\varnothing \not \in V$ and $V$ contains as a subset some ultra family of sets.
A filter subbase that is ultra is necessarily a prefilter.[proof 1]
The ultra property can now be used to define both ultrafilters and ultra prefilters:
An ultra prefilter[2][3] is a prefilter that is ultra. Equivalently, it is a filter subbase that is ultra.
An ultrafilter[2][3] on $X$ is a (proper) filter on $X$ that is ultra. Equivalently, it is any filter on $X$ that is generated by an ultra prefilter.
Ultra prefilters as maximal prefilters
To characterize ultra prefilters in terms of "maximality," the following relation is needed.
Given two families of sets $M$ and $N,$ the family $M$ is said to be coarser[4][5] than $N,$ and $N$ is finer than and subordinate to $M,$ written $M\leq N$ or N ⊢ M, if for every $C\in M,$ there is some $F\in N$ such that $F\subseteq C.$ The families $M$ and $N$ are called equivalent if $M\leq N$ and $N\leq M.$ The families $M$ and $N$ are comparable if one of these sets is finer than the other.[4]
The subordination relationship, i.e. $\,\geq ,\,$ is a preorder so the above definition of "equivalent" does form an equivalence relation. If $M\subseteq N$ then $M\leq N$ but the converse does not hold in general. However, if $N$ is upward closed, such as a filter, then $M\leq N$ if and only if $M\subseteq N.$ Every prefilter is equivalent to the filter that it generates. This shows that it is possible for filters to be equivalent to sets that are not filters.
If two families of sets $M$ and $N$ are equivalent then either both $M$ and $N$ are ultra (resp. prefilters, filter subbases) or otherwise neither one of them is ultra (resp. a prefilter, a filter subbase). In particular, if a filter subbase is not also a prefilter, then it is not equivalent to the filter or prefilter that it generates. If $M$ and $N$ are both filters on $X$ then $M$ and $N$ are equivalent if and only if $M=N.$ If a proper filter (resp. ultrafilter) is equivalent to a family of sets $M$ then $M$ is necessarily a prefilter (resp. ultra prefilter). Using the following characterization, it is possible to define prefilters (resp. ultra prefilters) using only the concept of filters (resp. ultrafilters) and subordination:
An arbitrary family of sets is a prefilter if and only it is equivalent to a (proper) filter.
An arbitrary family of sets is an ultra prefilter if and only it is equivalent to an ultrafilter.
A maximal prefilter on $X$[2][3] is a prefilter $U\subseteq \wp (X)$ that satisfies any of the following equivalent conditions:
1. $U$ is ultra.
2. $U$ is maximal on $\operatorname {Prefilters} (X)$ with respect to $\,\leq ,$ meaning that if $P\in \operatorname {Prefilters} (X)$ satisfies $U\leq P$ then $P\leq U.$[3]
3. There is no prefilter properly subordinate to $U.$[3]
4. If a (proper) filter $F$ on $X$ satisfies $U\leq F$ then $F\leq U.$
5. The filter on $X$ generated by $U$ is ultra.
Characterizations
There are no ultrafilters on the empty set, so it is henceforth assumed that $X$ is nonempty.
A filter subbase $U$ on $X$ is an ultrafilter on $X$ if and only if any of the following equivalent conditions hold:[2][3]
1. for any $S\subseteq X,$ either $S\in U$ or $X\setminus S\in U.$
2. $U$ is a maximal filter subbase on $X,$ meaning that if $F$ is any filter subbase on $X$ then $U\subseteq F$ implies $U=F.$[6]
A (proper) filter $U$ on $X$ is an ultrafilter on $X$ if and only if any of the following equivalent conditions hold:
1. $U$ is ultra;
2. $U$ is generated by an ultra prefilter;
3. For any subset $S\subseteq X,$ $S\in U$ or $X\setminus S\in U.$[6]
• So an ultrafilter $U$ decides for every $S\subseteq X$ whether $S$ is "large" (i.e. $S\in U$) or "small" (i.e. $X\setminus S\in U$).[7]
4. For each subset $A\subseteq X,$ either[note 1] $A$ is in $U$ or ($X\setminus A$) is.
5. $U\cup (X\setminus U)=\wp (X).$ This condition can be restated as: $\wp (X)$ is partitioned by $U$ and its dual $X\setminus U.$
• The sets $P$ and $X\setminus P$ are disjoint for all prefilters $P$ on $X.$
6. $\wp (X)\setminus U=\left\{S\in \wp (X):S\not \in U\right\}$ is an ideal on $X.$[6]
7. For any finite family $S_{1},\ldots ,S_{n}$ of subsets of $X$ (where $n\geq 1$), if $S_{1}\cup \cdots \cup S_{n}\in U$ then $S_{i}\in U$ for some index $i.$
• In words, a "large" set cannot be a finite union of sets none of which is large.[8]
8. For any $R,S\subseteq X,$ if $R\cup S=X$ then $R\in U$ or $S\in U.$
9. For any $R,S\subseteq X,$ if $R\cup S\in U$ then $R\in U$ or $S\in U$ (a filter with this property is called a prime filter).
10. For any $R,S\subseteq X,$ if $R\cup S\in U$ and $R\cap S=\varnothing $ then either $R\in U$ or $S\in U.$
11. $U$ is a maximal filter; that is, if $F$ is a filter on $X$ such that $U\subseteq F$ then $U=F.$ Equivalently, $U$ is a maximal filter if there is no filter $F$ on $X$ that contains $U$ as a proper subset (that is, no filter is strictly finer than $U$).[6]
Grills and filter-grills
If ${\mathcal {B}}\subseteq \wp (X)$ then its grill on $X$ is the family
${\mathcal {B}}^{\#X}:=\{S\subseteq X~:~S\cap B\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}\}$
where ${\mathcal {B}}^{\#}$ may be written if $X$ is clear from context. For example, $\varnothing ^{\#}=\wp (X)$ and if $\varnothing \in {\mathcal {B}}$ then ${\mathcal {B}}^{\#}=\varnothing .$ If ${\mathcal {A}}\subseteq {\mathcal {B}}$ then ${\mathcal {B}}^{\#}\subseteq {\mathcal {A}}^{\#}$ and moreover, if ${\mathcal {B}}$ is a filter subbase then ${\mathcal {B}}\subseteq {\mathcal {B}}^{\#}.$[9] The grill ${\mathcal {B}}^{\#X}$ is upward closed in $X$ if and only if $\varnothing \not \in {\mathcal {B}},$ which will henceforth be assumed. Moreover, ${\mathcal {B}}^{\#\#}={\mathcal {B}}^{\uparrow X}$ so that ${\mathcal {B}}$ is upward closed in $X$ if and only if ${\mathcal {B}}^{\#\#}={\mathcal {B}}.$
The grill of a filter on $X$ is called a filter-grill on $X.$[9] For any $\varnothing \neq {\mathcal {B}}\subseteq \wp (X),$ ${\mathcal {B}}$ is a filter-grill on $X$ if and only if (1) ${\mathcal {B}}$ is upward closed in $X$ and (2) for all sets $R$ and $S,$ if $R\cup S\in {\mathcal {B}}$ then $R\in {\mathcal {B}}$ or $S\in {\mathcal {B}}.$ The grill operation ${\mathcal {F}}\mapsto {\mathcal {F}}^{\#X}$ induces a bijection
${\bullet }^{\#X}~:~\operatorname {Filters} (X)\to \operatorname {FilterGrills} (X)$
whose inverse is also given by ${\mathcal {F}}\mapsto {\mathcal {F}}^{\#X}.$[9] If ${\mathcal {F}}\in \operatorname {Filters} (X)$ then ${\mathcal {F}}$ is a filter-grill on $X$ if and only if ${\mathcal {F}}={\mathcal {F}}^{\#X},$[9] or equivalently, if and only if ${\mathcal {F}}$ is an ultrafilter on $X.$[9] That is, a filter on $X$ is a filter-grill if and only if it is ultra. For any non-empty ${\mathcal {F}}\subseteq \wp (X),$ ${\mathcal {F}}$ is both a filter on $X$ and a filter-grill on $X$ if and only if (1) $\varnothing \not \in {\mathcal {F}}$ and (2) for all $R,S\subseteq X,$ the following equivalences hold:
$R\cup S\in {\mathcal {F}}$ if and only if $R,S\in {\mathcal {F}}$ if and only if $R\cap S\in {\mathcal {F}}.$[9]
Free or principal
If $P$ is any non-empty family of sets then the Kernel of $P$ is the intersection of all sets in $P:$[10]
$\operatorname {ker} P:=\bigcap _{B\in P}B.$
A non-empty family of sets $P$ is called:
• free if $\operatorname {ker} P=\varnothing $ and fixed otherwise (that is, if $\operatorname {ker} P\neq \varnothing $).
• principal if $\operatorname {ker} P\in P.$
• principal at a point if $\operatorname {ker} P\in P$ and $\operatorname {ker} P$ is a singleton set; in this case, if $\operatorname {ker} P=\{x\}$ then $P$ is said to be principal at $x.$
If a family of sets $P$ is fixed then $P$ is ultra if and only if some element of $P$ is a singleton set, in which case $P$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter $P$ is ultra if and only if $\operatorname {ker} P$ is a singleton set. A singleton set is ultra if and only if its sole element is also a singleton set.
The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.
Proposition — If $U$ is an ultrafilter on $X$ then the following are equivalent:
1. $U$ is fixed, or equivalently, not free.
2. $U$ is principal.
3. Some element of $U$ is a finite set.
4. Some element of $U$ is a singleton set.
5. $U$ is principal at some point of $X,$ which means $\operatorname {ker} U=\{x\}\in U$ for some $x\in X.$
6. $U$ does not contain the Fréchet filter on $X$ as a subset.
7. $U$ is sequential.[9]
Every filter on $X$ that is principal at a single point is an ultrafilter, and if in addition $X$ is finite, then there are no ultrafilters on $X$ other than these.[10] In particular, if a set $X$ has finite cardinality $n<\infty ,$ then there are exactly $n$ ultrafilters on $X$ and those are the ultrafilters generated by each singleton subset of $X.$ Consequently, free ultrafilters can only exist on an infinite set.
Examples, properties, and sufficient conditions
If $X$ is an infinite set then there are as many ultrafilters over $X$ as there are families of subsets of $X;$ explicitly, if $X$ has infinite cardinality $\kappa $ then the set of ultrafilters over $X$ has the same cardinality as $\wp (\wp (X));$ that cardinality being $2^{2^{\kappa }}.$[11]
If $U$ and $S$ are families of sets such that $U$ is ultra, $\varnothing \not \in S,$ and $U\leq S,$ then $S$ is necessarily ultra. A filter subbase $U$ that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by $U$ to be ultra.
Suppose $U\subseteq \wp (X)$ is ultra and $Y$ is a set. The trace $U\vert _{Y}:=\{B\cap Y:B\in U\}$ is ultra if and only if it does not contain the empty set. Furthermore, at least one of the sets $U\vert _{Y}\setminus \{\varnothing \}$ and $U\vert _{X\setminus Y}\setminus \{\varnothing \}$ will be ultra (this result extends to any finite partition of $X$). If $F_{1},\ldots ,F_{n}$ are filters on $X,$ $U$ is an ultrafilter on $X,$ and $F_{1}\cap \cdots \cap F_{n}\leq U,$ then there is some $F_{i}$ that satisfies $F_{i}\leq U.$[12] This result is not necessarily true for an infinite family of filters.[12]
The image under a map $f:X\to Y$ of an ultra set $U\subseteq \wp (X)$ is again ultra and if $U$ is an ultra prefilter then so is $f(U).$ The property of being ultra is preserved under bijections. However, the preimage of an ultrafilter is not necessarily ultra, not even if the map is surjective. For example, if $X$ has more than one point and if the range of $f:X\to Y$ consists of a single point $\{y\}$ then $\{y\}$ is an ultra prefilter on $Y$ but its preimage is not ultra. Alternatively, if $U$ is a principal filter generated by a point in $Y\setminus f(X)$ then the preimage of $U$ contains the empty set and so is not ultra.
The elementary filter induced by an infinite sequence, all of whose points are distinct, is not an ultrafilter.[12] If $n=2,$ then $U_{n}$ denotes the set consisting all subsets of $X$ having cardinality $n,$ and if $X$ contains at least $2n-1$ ($=3$) distinct points, then $U_{n}$ is ultra but it is not contained in any prefilter. This example generalizes to any integer $n>1$ and also to $n=1$ if $X$ contains more than one element. Ultra sets that are not also prefilters are rarely used.
For every $S\subseteq X\times X$ and every $a\in X,$ let $S{\big \vert }_{\{a\}\times X}:=\{y\in X~:~(a,y)\in S\}.$ If ${\mathcal {U}}$ is an ultrafilter on $X$ then the set of all $S\subseteq X\times X$ such that $\left\{a\in X~:~S{\big \vert }_{\{a\}\times X}\in {\mathcal {U}}\right\}\in {\mathcal {U}}$ is an ultrafilter on $X\times X.$[13]
Monad structure
The functor associating to any set $X$ the set of $U(X)$ of all ultrafilters on $X$ forms a monad called the ultrafilter monad. The unit map
$X\to U(X)$
sends any element $x\in X$ to the principal ultrafilter given by $x.$
This ultrafilter monad is the codensity monad of the inclusion of the category of finite sets into the category of all sets,[14] which gives a conceptual explanation of this monad.
Similarly, the ultraproduct monad is the codensity monad of the inclusion of the category of finite families of sets into the category of all families of set. So in this sense, ultraproducts are categorically inevitable.[14]
The ultrafilter lemma
The ultrafilter lemma was first proved by Alfred Tarski in 1930.[13]
The ultrafilter lemma/principle/theorem[4] — Every proper filter on a set $X$ is contained in some ultrafilter on $X.$
The ultrafilter lemma is equivalent to each of the following statements:
1. For every prefilter on a set $X,$ there exists a maximal prefilter on $X$ subordinate to it.[2]
2. Every proper filter subbase on a set $X$ is contained in some ultrafilter on $X.$
A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.[15][note 2]
The following results can be proven using the ultrafilter lemma. A free ultrafilter exists on a set $X$ if and only if $X$ is infinite. Every proper filter is equal to the intersection of all ultrafilters containing it.[4] Since there are filters that are not ultra, this shows that the intersection of a family of ultrafilters need not be ultra. A family of sets $\mathbb {F} \neq \varnothing $ can be extended to a free ultrafilter if and only if the intersection of any finite family of elements of $\mathbb {F} $ is infinite.
Relationships to other statements under ZF
See also: Boolean prime ideal theorem and Set-theoretic topology
Throughout this section, ZF refers to Zermelo–Fraenkel set theory and ZFC refers to ZF with the Axiom of Choice (AC). The ultrafilter lemma is independent of ZF. That is, there exist models in which the axioms of ZF hold but the ultrafilter lemma does not. There also exist models of ZF in which every ultrafilter is necessarily principal.
Every filter that contains a singleton set is necessarily an ultrafilter and given $x\in X,$ the definition of the discrete ultrafilter $\{S\subseteq X:x\in S\}$ does not require more than ZF. If $X$ is finite then every ultrafilter is a discrete filter at a point; consequently, free ultrafilters can only exist on infinite sets. In particular, if $X$ is finite then the ultrafilter lemma can be proven from the axioms ZF. The existence of free ultrafilter on infinite sets can be proven if the axiom of choice is assumed. More generally, the ultrafilter lemma can be proven by using the axiom of choice, which in brief states that any Cartesian product of non-empty sets is non-empty. Under ZF, the axiom of choice is, in particular, equivalent to (a) Zorn's lemma, (b) Tychonoff's theorem, (c) the weak form of the vector basis theorem (which states that every vector space has a basis), (d) the strong form of the vector basis theorem, and other statements. However, the ultrafilter lemma is strictly weaker than the axiom of choice. While free ultrafilters can be proven to exist, it is not possible to construct an explicit example of a free ultrafilter (using only ZF and the ultrafilter lemma); that is, free ultrafilters are intangible.[16] Alfred Tarski proved that under ZFC, the cardinality of the set of all free ultrafilters on an infinite set $X$ is equal to the cardinality of $\wp (\wp (X)),$ where $\wp (X)$ denotes the power set of $X.$[17] Other authors attribute this discovery to Bedřich Pospíšil (following a combinatorial argument from Fichtenholz, and Kantorovitch, improved by Hausdorff).[18][19]
Under ZF, the axiom of choice can be used to prove both the ultrafilter lemma and the Krein–Milman theorem; conversely, under ZF, the ultrafilter lemma together with the Krein–Milman theorem can prove the axiom of choice.[20]
Statements that cannot be deduced
The ultrafilter lemma is a relatively weak axiom. For example, each of the statements in the following list can not be deduced from ZF together with only the ultrafilter lemma:
1. A countable union of countable sets is a countable set.
2. The axiom of countable choice (ACC).
3. The axiom of dependent choice (ADC).
Equivalent statements
Under ZF, the ultrafilter lemma is equivalent to each of the following statements:[21]
1. The Boolean prime ideal theorem (BPIT).
2. Stone's representation theorem for Boolean algebras.
3. Any product of Boolean spaces is a Boolean space.[22]
4. Boolean Prime Ideal Existence Theorem: Every nondegenerate Boolean algebra has a prime ideal.[23]
5. Tychonoff's theorem for Hausdorff spaces: Any product of compact Hausdorff spaces is compact.[22]
6. If $\{0,1\}$ is endowed with the discrete topology then for any set $I,$ the product space $\{0,1\}^{I}$ is compact.[22]
7. Each of the following versions of the Banach-Alaoglu theorem is equivalent to the ultrafilter lemma:
1. Any equicontinuous set of scalar-valued maps on a topological vector space (TVS) is relatively compact in the weak-* topology (that is, it is contained in some weak-* compact set).[24]
2. The polar of any neighborhood of the origin in a TVS $X$ is a weak-* compact subset of its continuous dual space.[24]
3. The closed unit ball in the continuous dual space of any normed space is weak-* compact.[24]
• If the normed space is separable then the ultrafilter lemma is sufficient but not necessary to prove this statement.
8. A topological space $X$ is compact if every ultrafilter on $X$ converges to some limit.[25]
9. A topological space $X$ is compact if and only if every ultrafilter on $X$ converges to some limit.[25]
• The addition of the words "and only if" is the only difference between this statement and the one immediately above it.
10. The Alexander subbase theorem.[26][27]
11. The Ultranet lemma: Every net has a universal subnet.[27]
• By definition, a net in $X$ is called an ultranet or an universal net if for every subset $S\subseteq X,$ the net is eventually in $S$ or in $X\setminus S.$
12. A topological space $X$ is compact if and only if every ultranet on $X$ converges to some limit.[25]
• If the words "and only if" are removed then the resulting statement remains equivalent to the ultrafilter lemma.[25]
13. A convergence space $X$ is compact if every ultrafilter on $X$ converges.[25]
14. A uniform space is compact if it is complete and totally bounded.[25]
15. The Stone–Čech compactification Theorem.[22]
16. Each of the following versions of the compactness theorem is equivalent to the ultrafilter lemma:
1. If $\Sigma $ is a set of first-order sentences such that every finite subset of $\Sigma $ has a model, then $\Sigma $ has a model.[28]
2. If $\Sigma $ is a set of zero-order sentences such that every finite subset of $\Sigma $ has a model, then $\Sigma $ has a model.[28]
17. The completeness theorem: If $\Sigma $ is a set of zero-order sentences that is syntactically consistent, then it has a model (that is, it is semantically consistent).
Weaker statements
Any statement that can be deduced from the ultrafilter lemma (together with ZF) is said to be weaker than the ultrafilter lemma. A weaker statement is said to be strictly weaker if under ZF, it is not equivalent to the ultrafilter lemma. Under ZF, the ultrafilter lemma implies each of the following statements:
1. The Axiom of Choice for Finite sets (ACF): Given $I\neq \varnothing $ and a family $\left(X_{i}\right)_{i\in I}$ of non-empty finite sets, their product $ \prod \limits _{i\in I}}X_{i}$ is not empty.[27]
2. A countable union of finite sets is a countable set.
• However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set.
3. The Hahn–Banach theorem.[27]
• In ZF, the Hahn–Banach theorem is strictly weaker than the ultrafilter lemma.
4. The Banach–Tarski paradox.
• In fact, under ZF, the Banach–Tarski paradox can be deduced from the Hahn–Banach theorem,[29][30] which is strictly weaker than the Ultrafilter Lemma.
5. Every set can be linearly ordered.
6. Every field has a unique algebraic closure.
7. Non-trivial ultraproducts exist.
8. The weak ultrafilter theorem: A free ultrafilter exists on $\mathbb {N} .$
• Under ZF, the weak ultrafilter theorem does not imply the ultrafilter lemma; that is, it is strictly weaker than the ultrafilter lemma.
9. There exists a free ultrafilter on every infinite set;
• This statement is actually strictly weaker than the ultrafilter lemma.
• ZF alone does not even imply that there exists a non-principal ultrafilter on some set.
Completeness
The completeness of an ultrafilter $U$ on a powerset is the smallest cardinal κ such that there are κ elements of $U$ whose intersection is not in $U.$ The definition of an ultrafilter implies that the completeness of any powerset ultrafilter is at least $\aleph _{0}$. An ultrafilter whose completeness is greater than $\aleph _{0}$—that is, the intersection of any countable collection of elements of $U$ is still in $U$—is called countably complete or σ-complete.
The completeness of a countably complete nonprincipal ultrafilter on a powerset is always a measurable cardinal.
Ordering on ultrafilters
The Rudin–Keisler ordering (named after Mary Ellen Rudin and Howard Jerome Keisler) is a preorder on the class of powerset ultrafilters defined as follows: if $U$ is an ultrafilter on $\wp (X),$ and $V$ an ultrafilter on $\wp (Y),$ then $V\leq {}_{RK}U$ if there exists a function $f:X\to Y$ such that
$C\in V$ if and only if $f^{-1}[C]\in U$
for every subset $C\subseteq Y.$
Ultrafilters $U$ and $V$ are called Rudin–Keisler equivalent, denoted U ≡RK V, if there exist sets $A\in U$ and $B\in V$ and a bijection $f:A\to B$ that satisfies the condition above. (If $X$ and $Y$ have the same cardinality, the definition can be simplified by fixing $A=X,$ $B=Y.$)
It is known that ≡RK is the kernel of ≤RK, i.e., that U ≡RK V if and only if $U\leq {}_{RK}V$ and $V\leq {}_{RK}U.$[31]
Ultrafilters on ℘(ω)
There are several special properties that an ultrafilter on $\wp (\omega ),$ where $\omega $ extends the natural numbers, may possess, which prove useful in various areas of set theory and topology.
• A non-principal ultrafilter $U$ is called a P-point (or weakly selective) if for every partition $\left\{C_{n}:n<\omega \right\}$ of $\omega $ such that for all $n<\omega ,$ $C_{n}\not \in U,$ there exists some $A\in U$ such that $A\cap C_{n}$ is a finite set for each $n.$
• A non-principal ultrafilter $U$ is called Ramsey (or selective) if for every partition $\left\{C_{n}:n<\omega \right\}$ of $\omega $ such that for all $n<\omega ,$ $C_{n}\not \in U,$ there exists some $A\in U$ such that $A\cap C_{n}$ is a singleton set for each $n.$
It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters.[32] In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.[33] Therefore, the existence of these types of ultrafilters is independent of ZFC.
P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of $[\omega ]^{2}$ there exists an element of the ultrafilter that has a homogeneous color.
An ultrafilter on $\wp (\omega )$ is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters.[34]
See also
• Extender (set theory) – in set theory, a system of ultrafilters representing an elementary embedding witnessing large cardinal propertiesPages displaying wikidata descriptions as a fallback
• Filter (mathematics) – In mathematics, a special subset of a partially ordered set
• Filter (set theory) – Family of sets representing "large" sets
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
• Łoś's theorem – Mathematical constructionPages displaying short descriptions of redirect targets
• Ultrafilter – Maximal proper filter
• Universal net – A generalization of a sequence of pointsPages displaying short descriptions of redirect targets
Notes
1. Properties 1 and 3 imply that $A$ and $X\setminus A$ cannot both be elements of $U.$
2. Let ${\mathcal {F}}$ be a filter on $X$ that is not an ultrafilter. If $S\subseteq X$ is such that $S\not \in {\mathcal {F}}$ then $\{X\setminus S\}\cup {\mathcal {F}}$ has the finite intersection property (because if $F\in {\mathcal {F}}$ then $F\cap (X\setminus S)=\varnothing $ if and only if $F\subseteq S$) so that by the ultrafilter lemma, there exists some ultrafilter ${\mathcal {U}}_{S}$ on $X$ such that $\{X\setminus S\}\cup {\mathcal {F}}\subseteq {\mathcal {U}}_{S}$ (so in particular $S\not \in {\mathcal {U}}_{S}$). It follows that ${\mathcal {F}}=\bigcap _{S\subseteq X,S\not \in {\mathcal {F}}}{\mathcal {U}}_{S}.$ $\blacksquare $
Proofs
1. Suppose ${\mathcal {B}}$ is filter subbase that is ultra. Let $C,D\in {\mathcal {B}}$ and define $S=C\cap D.$ Because ${\mathcal {B}}$ is ultra, there exists some $B\in {\mathcal {B}}$ such that $B\cap S$ equals $B$ or $\varnothing .$ The finite intersection property implies that $B\cap S\neq \varnothing $ so necessarily $B\cap S=B,$ which is equivalent to $B\subseteq C\cap D.$ $\blacksquare $
References
1. Davey, B. A.; Priestley, H. A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
2. Narici & Beckenstein 2011, pp. 2–7.
3. Dugundji 1966, pp. 219–221.
4. Bourbaki 1989, pp. 57–68.
5. Schubert 1968, pp. 48–71.
6. Schechter 1996, pp. 100–130.
7. Higgins, Cecelia (2018). "Ultrafilters in set theory" (PDF). math.uchicago.edu. Retrieved August 16, 2020.
8. Kruckman, Alex (November 7, 2012). "Notes on Ultrafilters" (PDF). math.berkeley.edu. Retrieved August 16, 2020.
9. Dolecki & Mynard 2016, pp. 27–54.
10. Dolecki & Mynard 2016, pp. 33–35.
11. Pospíšil, Bedřich (1937). "Remark on Bicompact Spaces". The Annals of Mathematics. 38 (4): 845-846. doi:10.2307/1968840. JSTOR 1968840.
12. Bourbaki 1989, pp. 129–133.
13. Jech 2006, pp. 73–89.
14. Leinster, Tom (2013). "Codensity and the ultrafilter monad" (PDF). Theory and Applications of Categories. 28: 332–370. arXiv:1209.3606. Bibcode:2012arXiv1209.3606L.
15. Bourbaki 1987, pp. 57–68. sfn error: no target: CITEREFBourbaki1987 (help)
16. Schechter 1996, p. 105.
17. Schechter 1996, pp. 150–152.
18. Jech 2006, pp. 75–76.
19. Comfort 1977, p. 420.
20. Bell, J.; Fremlin, David (1972). "A geometric form of the axiom of choice" (PDF). Fundamenta Mathematicae. 77 (2): 167–170. doi:10.4064/fm-77-2-167-170. Retrieved 11 June 2018. Theorem 1.2. BPI [the Boolean Prime Ideal Theorem] & KM [Krein-Milman] $\implies $ (*) [the unit ball of the dual of a normed vector space has an extreme point].... Theorem 2.1. (*) $\implies $ AC [the Axiom of Choice].
21. Schechter 1996, pp. 105, 150–160, 166, 237, 317–315, 338–340, 344–346, 386–393, 401–402, 455–456, 463, 474, 506, 766–767.
22. Schechter 1996, p. 463.
23. Schechter 1996, p. 339.
24. Schechter 1996, pp. 766–767.
25. Schechter 1996, p. 455.
26. Hodel, R.E. (2005). "Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem". Archive for Mathematical Logic. 44 (4): 459–472. doi:10.1007/s00153-004-0264-9. S2CID 6507722.
27. Muger, Michael (2020). Topology for the Working Mathematician.
28. Schechter 1996, pp. 391–392.
29. Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (PDF). Fundamenta Mathematicae. 138: 13–19. doi:10.4064/fm-138-1-13-19.
30. Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox" (PDF). Fundamenta Mathematicae. 138: 21–22. doi:10.4064/fm-138-1-21-22.
31. Comfort, W. W.; Negrepontis, S. (1974). The theory of ultrafilters. Berlin, New York: Springer-Verlag. MR 0396267. Corollary 9.3.
32. Rudin, Walter (1956), "Homogeneity problems in the theory of Čech compactifications", Duke Mathematical Journal, 23 (3): 409–419, doi:10.1215/S0012-7094-56-02337-7, hdl:10338.dmlcz/101493
33. Wimmers, Edward (March 1982), "The Shelah P-point independence theorem", Israel Journal of Mathematics, 43 (1): 28–48, doi:10.1007/BF02761683, S2CID 122393776
34. Jech 2006, p. 91(Left as exercise 7.12)
Bibliography
• Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications. Vol. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489.
• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
• Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
• Jech, Thomas (2006). Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-44085-7. OCLC 50422939.
• Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753.
Further reading
• Comfort, W. W. (1977). "Ultrafilters: some old and some new results". Bulletin of the American Mathematical Society. 83 (4): 417–455. doi:10.1090/S0002-9904-1977-14316-4. ISSN 0002-9904. MR 0454893.
• Comfort, W. W.; Negrepontis, S. (1974), The theory of ultrafilters, Berlin, New York: Springer-Verlag, MR 0396267
• Ultrafilter at the nLab
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KFU > Academics > Institute of Computer Science and Information Technology
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Department of Mathematical Statistics
Avkhadiev Farit Gabidinovich
[email protected]
Doctor of Science (D. Sc., Habilitate) in Mathematics, Sobolev Institute of Math. in Siberian branch of Academy of Science, Novosibirsk, USSR (1990).
D. Sc. Thesis: Geometric Properties of Conformal and Locally Quasiconformal Maps with given Boundary Behavior
PhD in Mathematics, Kazan State University, USSR (1972). Advisor: Prof. L. A. Aksent'ev.
PhD Thesis: Some Sufficient Conditions of Univalence and their Applications to Inverse Boundary-value Problems
M. Sc. (cum laudi) in Mathematics, Kazan State University, USSR (1969)
Undergraduate at the Kazan State University, Faculty of Mechanics and Mathematics, USSR (1964-1969)
1969 - present: Chebotarev Inst. of Math. and Mech. of the Kazan State University / Kazan Federal University: Researcher (1969 - 1974),
Senior Researcher (1974 - 1990),
Leading Researcher (1990 - 1999),
Principal Researcher (1999 - present),
Head of the Department of Mathematical Analysis (1993 - present),
Head of the Department of Mathematics (1999 - present)
September 1977 - June 1980: University of Annaba, Algeria, Associate Professor
1994 - present: Department of Mathematical Statistics of the Faculty of Computer Science and Cybernetics of the Kazan University, Professor
Developed and taught new graduate and undergraduate courses
at the Kazan University: Mathematical Analysis, Geometric Function Theory-Selected Topics, Inverse Boundary-value Problems, Functional Analysis, Convex Analysis, Methods for the Dirichlet Boundary-value Problem, Isoperimetric Inequalities in Mathematical Physics, Complex Analysis
at the University of Annaba, Algeria (1977 - 1980): Mathematics, Theory of Functions of a Complex Variable, Algebra
at the Kazan State Pedagogic University (2 Semesters): Algebra
at the Kazan State Academy of Architecture and Constructions (3 Semesters): Mathematics
Advisor for PhD Thesis of 4 Students: F. Kh. Arslanov(1991), I. R. Kayumov (1997), R. G. Salahudinov (1998), Mohamed Sabri Salem Ali (2000)
Grants of the Russian Foundation of Basic Research: No 96 - 01 - 00110 (1996 - 1998), No 99 - 01 - 00366 (1999 - 2001), No 02 - 01 - 00168 (2002 - 2004), No 05 - 01 - 00523 (2005 - 2007)
Grants of the Deutsche Forschungsgemeinschaft in 1998, 1999, 2001, 2002, 2004, 2005, 2007
Mushtary Prize in the field of Mathematics, Mechanics and Technique (2004, Tatarstan Academy of Science competition)
Internal Prizes of the Kazan State University for Best Research Paper of the Year (1979, 1983, 1993)
CONFERENCE PRESENTATIONS (incomplete list)
Int. Conf. on Function Spaces, Approximation Theory, Nonlinear Analysis, dedicated to the centennial of S. M. Nikolskii, Moscow, Russia, May 23-29, 2005
Int. Conf. on Algebra and Analysis, Kazan, Russia, July 2 - 9, 2004
Int. Conf. on Geometric Analysis and Appl., Volgograd, Russia, May 24-30, 2004
Int.Workshop on Potential Theory and Appl., Kiev, Ukraine, August, 2003
Int. Conf. on Geometric Function Theory, dedicated to Herbert Groetzsch, Halle, Germany, May 27-30, 2002
Int. Conf., dedicated to D. F. Egorov, Kazan, Russia, September 13-18, 1999
Int. Conf., dedicated to P. L. Chebyshev, Moscow, Russia, May, 1996
Int. Workshop on Planar Harmonic Mappings, Technion, Haifa, Israel, May 8-15, 1995
IX Int. Conf. on Finite Elements in Fluids, Venice, Italy, October 15-21, 1995
Int. Congress of Mathematics, Zurich, July 2-10, 1994
Int. Conf. and the VII-th Romanian - Finnish Seminar on Complex Analysis, Timishoara, Rumania, August 23-27, 1993
I European Congress of Mathematics, Paris, France, July 6 - 10, 1992
LIST of MAIN PUBLICATIONS
63. Avkhadiev, F.G.; Wirths, K.-J. Unified Poincar\'e and Hardy inequalities with sharp constants for convex domains. ZAMM, 14, No. 8-9, 532-632 (2007).
62. Avkhadiev, F.G.; Chuprunov, A.N. The probability of a successful allocation of ball groups by boxes. Lobachevskii J. Math., 25, 3-7 (2007).
61. Avkhadiev, F.G.; Wirths, K.-J. The punishing factors for convex pairs are 2^n-1. Revista Mat. Iberoamericana, 23, No. 3, 847-860 (2007).
60. Avkhadiev, F.G.; Wirths, K.-J. Punishing factors and Chua's conjecture. Bull. Belgian Math. Soc., Simon Stevin, 14, 333-340 (2007).
59. Avkhadiev, F.G.; Wirths, K.-J. Sharp Bounds for Sums of Coefficients of Inverses of Convex Functions. Comp. Methods and Function Theory, 7, No. 1, 105-109 (2007).
58. Avkhadiev, F.G.; Wirths, K.-J. A proof of the Livingston conjecture. Forum Mathematicum, 19, 149-157 (2007).
57. Avkhadiev, F.G.; Wirths, K.-J. Subordination under concave univalent functions. Complex Variables and Elliptic Equations, 52, No. 4, 299-305 (2007).
56. Avkhadiev, F.G. Hardy-Type Inequalities on Planar and Spatial Open Sets. (Russian, English) Proc. Steklov Inst. Math., 255, 1-11 (2006); translation from Trudy Mat. Inst. V. A. Steklova, 255, 8-18 (2006).
55. Avkhadiev, F.G.; Pommerenke, Ch.; Wirths, K.-J. Sharp inequalities for the coefficients of concave univalent functions. Commentarii Mathematicae Helvetici, 81, 801-807 (2006).
54. Avkhadiev, F.G. Hardy type inequalities in higher dimensions with explicit estimate of constants. Lobachevskii J. Math., 21, 3-31 (2006).
53. Avkhadiev, F.G.; Wirths, K.-J. Punishing Factors for Finitely Connected Domains. Monatshefte Math., 147, 103-115 (2005).
52. Avkhadiev, F.G. A Simple Proof of the Gauss-Winckler Inequality. Amer. Math. Monthly, 112, No.5, 459-462 (2005).
51. Avkhadiev, F.G.; Kacimov, A.R. Analytic solutions and estimations for microlevel flows in fracturs. J. Porous Media, 8, No.2, 125-148 (2005).
50. Avkhadiev, F.G.; Wirths, K.-J. The Conformal Radius as a Function and its Gradient Image. Israel J. Math., 149, 349-374 (2005).
49. Avkhadiev, F.G. New isoperimetric inequalities for moments of domains and torsional rigidity. (Russian, English) Russ. Math. 48, No.7, 1-9 (2004); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2004, 48 No.7, 3-11 (2004).
48. Avkhadiev, F.G. Conformally Invariant Inequalities in Mathematical Physics . (Russian) Naukoemkie Tekhnol., 5(4), 47-51 (2004).
47. Avkhadiev, F.G. Mobius transformations and multiplicative representations for spherical potentials. Publications de l'Institut Mathematique. Nouvelle serie (Beograd), Tome 75(89), 253-260 (2004).
46. Avkhadiev, F.G.; Kayumov, I.R. Comparison theorems of isoperimetric type for moments of compact sets. Collectanea Math. 55, no. 1, 1-9 (2004).
45. Avkhadiev, F.G.; Pommerenke, Ch.; Wirths, K.-J. On the coefficients of concave univalent functions. Math. Nachr. 271, 3-9 (2004).
44. Avkhadiev, F.G.; Wirths, K.-J. Schwarz-Pick Inequality for Hyperbolic Domains in the Extended Plane. Geometriae Dedicata, 106, 1-10 (2004).
43. Avkhadiev, F.G.; Wirths, K.-J. Poles Near the Origin Produce Lower Bounds for Coefficients of Meromorphic Univalent Functions. Michigan Math. J. 52(1), 119-130 (2004).
42. Avkhadiev, F.G.; Wirths, K.-J. On a conjecture of Livingston. Mathematika, 46(69), No.1, 19-23 (2004).
41. Avkhadiev, F.G.; Wirths, K.-J. Punishing Factors for Angles. Computational Methods and Function Theory, 3, No.1, 127-141 (2003).
40. Avkhadiev, F.G.; Wirths, K.-J. Schwarz-Pick Inequality for Derivatives of Arbitrary Order. Constr. Approx. 19, 265-277 (2003).
39. Avkhadiev, F.G.; Salahudinov, R.G. Isoperimetric inequalities for conformal moments of plane domains. J. Inequal. Appl. 7, No.4, 593-601 (2002).
38. Avkhadiev, F.G.; Wirths, K.-J. Convex holes produce lower bounds for coefficients. Complex Variables,Theory Appl. 47, No.7, 553-563 (2002).
37. Avkhadiev, F.G.; Maklakov, D.V. New equations of convolution type obtained by replacing the integral by the maximum. (Russian, English) Math. Notes 71, No.1, 17-24 (2002); translation from Mat. Zametki 71, No.1, 18-26 (2002).
36. Avkhadiev, F.G.; Wirths, K.-J. On the coefficient multipliers theorem of Hardy and Littlewood. Lobachevskii J. Math. 11, 7-12, electronic only (2002).
35. Avkhadiev, F.G.; Elizarov, A.M. Bilateral estimates of the critical Mach number for some classes of carrying wing profiles. ANZIAM J. 42, No.4, 494-503 (2001).
34. Avhadiev, F.G.; Salahutdinov, R.G. Bilateral isoperimetric inequalities for boundary moments of plane domains. Lobachevskii J. Math. 9, 3-5, electronic only (2001).
33. Avkhadiev, F.G.; Wirths, K.-J. Asymptotically sharp bounds in the Hardy-Littlewood inequalities on mean values of analytic functions. Bull. Lond. Math. Soc. 33, No.6, 695-700 (2001).
32. Avhadiev, F.G.; Schulte, N.; Wirths, K.-J. On the growth of nonvanishing analytic functions. Arch. Math. 74, No.5, 356-364 (2000).
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28. Avkhadiev, F.G. Solution of the generalized Saint Venant problem. (Russian, English) Sb. Math. 189, No.12, 1739-1748 (1998); translation from Mat. Sb. 189, No.12, 3-12 (1998).
27. Avkhadiev, F.G.; Aksent'ev, L.A.; Bil'chenko, G.G. Classes of univalent and multivalent Schwarz-Christoffel integrals and their applications. (Russian, English) Russ. Math. 41, No.3, 63-66 (1997); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1997, No.3(418), 64-67 (1997).
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24. Avkhadiev, F.G. Conformal mappings that satisfy the boundary condition of equality of metrics. (Russian, English) Dokl. Math. 53, No.2, 194-196 (1996); translation from Dokl. Akad. Nauk 347, No.3, 295-297 (1996).
23. Avkhadiev, F.G.; Elizarov, A.M. Exact bounds of solution to a variation inverse boundary value problem in a countable-connected domain. (Russian, English) Russ. Math. 40, No.3, 1-11 (1996); translation from Isv. Vyssh. Zaved., Mat. 1996, No.3(406), 3-13 (1996).
22.Avhadiev, F.G.; Kayumov, I.R. Estimates for Bloch functions and their generalization. Complex Variables, Theory Appl. 29, No.3, 193-201 (1996).
21. Avkhadiev, F.G.; Maklakov, D.V. The analytic method for construction of the hydrodynamic profile according to a given cavitation diagram. (Russian, English) Phys.-Dokl. 40, No.7, 356-358 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 343, No.2, 195-197 (1995).
20. Avkhadiev, F.G.; Elizarov, A.M.; Fokin, D.A. Estimates for critical Mach number under isoperimetric constraints. Eur. J. Appl. Math. 6, No.5, 385-398 (1995).
19. Avkhadiev, F.G.; Shokleva, L.I. Generalizations of certain Beurling's theorem with applications for inverse boundary value problem. (Russian, English) Russ. Math. 38, No.5, 78 81 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No.5(384), 80-83 (1994).
18. Avkhadiev, F.G.; Shabalina, S.B. Zeros of coefficients of transforms and conditions for the solvability of inverse boundary value problems. (Russian, English) Russ. Math. 38, No.8, 1-9 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No.8(387), 3-10 (1994).
17. Avkhadiev, F.G.; Maklakov, D.V. A solvability criterion for a problem of design of profiles by cavitation diagram. (Russian, English) Russ. Math. 38, No.7, 1-10 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No.7(386), 3-12 (1994).
16. Avkhadiev, F.G.; Elizarov, A.M.; Fokin, D.A. Maximization of the critical Mach number for lifting airfoils. (Russian, English) Fluid Dyn. 27, No.3, 423-428 (1992); translation from Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza 1992, No.3, 155-162 (1992).
15. Avkhadiev, F.G. Estimates in the Zygmund class and their application to boundary value problems. (Russian, English) Sov. Math., Dokl. 40, No.1, 217-220 (1990); translation from Dokl. Akad. Nauk SSSR 307, No.6, 1289-1292 (1989).
14. Avkhadiev, F.G. Admissible functionals under injectivity conditions for differentiable mappings of n-dimensional domains. (Russian, English) Sov. Math. 33, No.4, 1-12 (1989); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No.4(323), 3-12 (1989).
13. Avkhadiev, F.G. On injectivity in the domain of open isolated mappings with given boundary properties. (Russian, English) Sov. Math., Dokl. 35, 117-120 (1987); translation from Dokl. Akad. Nauk SSSR 292, 780-783 (1987).
12. Avkhadiev, F.G.; Aksent'ev, L.A.; Elizarov, A.M. Sufficient conditions for the finite-valence of analytic functions and their applications. (Russian, English) J. Sov. Math. 49, No.1, 715-799 (1990); translation from Itogi Nauki Tekh., Ser. Mat. Anal. 25, 3-121 (1987).
11. Avkhadiev, F.G.; Aksent'ev, L.A. Progress and problems in sufficient conditions for finite-valence of analytic functions. (Russian, English) Sov. Math. 30, No.10, 1-20 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No.10(293), 3-16 (1986).
10. Avkhadiev, F.G.; Nasyrov, S.R. Construction of a Riemann surface with respect to its boundary. (Russian, English) Sov. Math. 30, No.5, 1-12 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No.5(288), 3-11 (1986).
9. Avkhadiev, F.G. Some geometric inequalities and sufficient conditions for p-valence. (Russian, English) Sov. Math. 27, No.10, 1-13 (1983); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1983, No.10(257), 3-12 (1983).
8. Avhadiev, F.G. Sufficient conditions for the univalence of quasiconformal mappings. (Russian, English) Math. Notes 18, 1063-1067 (1975); translation from Mat. Zametki 18, 793- 802 (1975).
7. Avhadiev, F.G.; Aksent'ev, L.A. The main results on sufficient conditions for an analytic function to be schlicht. (Russian, English) Russ. Math. Surv. 30, No.4, 1-63 (1975); translation from Usp. Mat. Nauk 30, No.4(184), 3-60 (1975).
6. Avkhadiev, F.G. Sufficient conditions for one-sheetedness in nonconvex domains. (Russian, English) Sib. Math. J. 15(1974), 679-685 (1975); translation from Sib. Mat. Zh. 15, 963-971 (1974).
5. Avhadiev, F.G.; Aksent'ev, L.A. Functions of the Bazilevic class in the disc and in an annulus. (Russian, English) Sov. Math., Dokl. 15, 78-82 (1974); translation from Dokl. Akad. Nauk SSSR 214, 241-244 (1974).
4. Avhadiev, F.G.; Aksent'ev, L.A. The subordination principle in sufficient conditions for univalence. (Russian, English) Sov. Math., Dokl. 14, 934-939 (1973); translation from Dokl. Akad. Nauk SSSR 211, 19-22 (1973).
3. Avkhadiev, F.G.; Aksent'ev, L.A. Sufficient conditions for univalence of analytic functions. (Russian, English) Sov. Math., Dokl. 12, 859-863 (1971); translation from Dokl. Akad. Nauk SSSR 198, 743-746 (1971).
2. Avkhadiev, F.G. Radii of convexity and close-to-convexity of certain integral representations. (Russian, English) Math. Notes 7, 350-357 (1970); translation from Mat. Zametki 7, 581-592 (1970).
1. Avkhadiev, F.G. On sufficient conditions for univalence of solutions of inverse boundary value problems. (Russian, English) Sov. Math., Dokl. 11, 109-112 (1970); translation from Dokl. Akad. Nauk SSSR 190, 495-498 (1970).
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\begin{document}
\title{Oblique and checkerboard patterns in the quenched Cahn-Hilliard model}
\begin{abstract} We consider transversely modulated fronts in a directionally quenched Cahn-Hilliard equation, posed on a two-dimensional infinite channel, with both parameter and source-term type heterogeneities. Such quenching heterogeneities travel through the domain, excite instabilities, and can select the pattern formed in their wake. We in particular study striped patterns which are oblique to the quenching direction and checkerboard type patterns. Under generic spectral assumptions, these patterns arise via an $O(2)$-Hopf bifurcation as the quenching speed is varied, with symmetries arising from translations and reflections in the transverse variable. We employ an abstract functional analytic approach to establish such patterns near the bifurcation point. Exponential weights are used to address neutral continuous spectrum, and a co-domain restriction is used to address neutral mass-flux. We also give a method to determine the direction of bifurcation of fronts. We then give an explicit example for which our hypotheses are satisfied and for which bifurcating fronts can be investigated numerically. \end{abstract}
\paragraph{Statements and Declarations}\textit{Competing Interests:} The authors have no competing interests to declare.
\paragraph{Keywords:} Cahn-Hilliard Equation; Front Solutions; Transverse Patterns; $O(2)$-Hopf; Heterogeneity; Directional Quenching
\paragraph{MSC Classification:} 35B36, 35B32, 37C81, 37L20
\section{Introduction} \subsection{Motivation} The Cahn-Hilliard equation \begin{equation} \partial_tu=-\Delta(\Delta u+f(u)), \quad f(u) = u - u^3, \quad u(\mathbf{x},t)\in \mathbb{R},\quad (\mathbf{x},t)\in\mathbb{R}^d\times \mathbb{R}, \end{equation} is a prototypical and well-studied model for phase separation processes in two-phase systems in a variety of contexts; see for example \cite{Miranville} for a mathematical review with many references. Through different initial conditions and boundary conditions, this equation can exhibit many different types of patterns. In particular, small random initial data, say on the unbounded domain, tends to lead to the formation of a random assortment of layers, stripes, spots, and defects, most of which are unstable via local coarsening. We remark that this equation is a $H^{-1}$ gradient-flow with respect to the following free-energy
\begin{equation}\mathcal{E}[u]=\int_{\Omega}\frac{1}{2}|\nabla u|^2+F(u)d\mathbf{x},\nonumber\end{equation} defined on a generic domain $\Omega$, with symmetric double-well potential $ F(u) = \frac{1}{4}(1-u^2)^2$ which favors the states $u = \pm1$.
In several experimental and phenomenological settings we can observe phase separation in a binary phase alloy, or a model thereof \cite{Foard12,Guo,Thiele21,Krekhov,Wilczek_2015}. In many of these experiments, a process known as directional quenching has been used to induce phase separation in a controlled manner and select the pattern formed in the wake. Indeed, depending on the initial data, and the shape and speed of the quench, a variety of patterns can be formed including regular spot arrangements, stripes of different orientations and wavenumbers, layers between pure $\pm 1$ states, as well as square and rhomb patterns. Here a quench travels through the spatial domain inciting instability in a given homogeneous equilibrium state, typically by spatiotemporally mediating the potential $F$ between single-well and double-well configuration, the latter of which is given above. In our work, and in parallel with several of the above mentioned references, we consider quenched patterns on a two-dimensional spatial domain.
Recent work in this area includes \cite{Goh}, which studied quenched stripe formation in one spatial dimension for a variety of quenching and source type heterogeneities with spinodally unstable regions which are bounded in $x$. The work \cite{Monteiro} studies two-dimensional directional quenched fronts with unbounded quenching domain in both the Allen-Cahn and Cahn-Hilliard equations showing that with zero quenching speed pure phase selection, vertical stripes, horizontal interfaces, and horizontal stripes can be formed, though oblique stripes cannot be formed. Existence results can also be obtained for non-zero quenching speeds for many of the aforementioned structures, but interestingly, these works did not address the formation of stripes which are oblique to the quenching interface in the moving interface case. Such an ambiguity motivates and is one the main focuses of our work. Results from \cite{GohScheel}, which studied the Swift-Hohenberg equation, showed that weakly oblique stripes exist as perturbations of parallel stripes. Because of this, and results of numerical simulations (discussed below), we expect oblique stripes to exist in the quenched Cahn-Hilliard equation. Our work also reveals cellular, or ``checkerboard" type patterns which biurcate with oblique stripes.
Following the functional analytic methods of \cite{Goh}, we study the bifurcations of transversely modulated patterns in the presence of quenching terms which have localized or bounded spinodal unstable regimes. That is we look for bifurcating fronts which are spatially patterned but are still asymptotically constant, with the pattern state lying in a potentially moving localized spatial region. This modeling assumption allows us to focus on the pattern forming dynamics and behavior just behind the quenching line. One hopes to build upon these results to establish large amplitude patterns, as well as fronts which converge to these patterns asymptotically in the far-field.
We seek to understand how a spatial heterogeneity can select patterns in the Cahn-Hilliard equation in two spatial dimensions under directional quenching. Our assumptions are roughly as follows. We consider nonlinearities of the form $f(x-ct,u)$ with a given front solution $u_*(x-ct)$, both of which converge exponentially fast in the co-moving frame $\tilde x:= x - ct$ with quenching speed $c$, to states $f_\pm(u)$ and $u_\pm$. Further we assume for $\Tilde{u}_\pm$ close to $u_\pm$ there exist a family of smooth asymptotically constant front solutions asymptotic to $\Tilde{u}_\pm$. We further assume a generic transverse Hopf instability of the associated linearization $L$ about $u_*$. In particular, we assume that an isolated, semi-simple pair of complex conjugate eigenvalues with transversely modulated eigenfunctions cross the imaginary axis as the quench speed $c$ varies while there are no other resonant spectrum at integer multiples of the Hopf frequency. As the quenched equation, with heterogeneity varying only in the $x$ direction, possesses a reflection symmetry $y\mapsto -y$ in the vertical direction along the quenching line, we generically assume that the Hopf eigenvalues have algebraic and geometric multiplicity two. As there is also a translation symmetry in $y$, we hence study Hopf-instabilities in the presence of a transverse $O(2)$ symmetry. We mention the works \cite{barker2021transverse,pogan20152}, which study $O(2)$-Hopf bifurcations in viscous slow magnetohydrodynamic shocks and in viscous shock waves in a channel, respectively, and handle similar problems to ours in different ways.
Under these assumptions, we establish the existence a pair of one-parameter families of time-periodic solutions which bifurcate from the front solution $u_*$. These branches, which take the form of oblique stripe and checkerboard patterns, respectively correspond to ``rotating" and ``standing" waves under the $O(2)$ symmetry group. Our results also give computable bifurcation coefficients which can be used determine whether these bifurcations are subcritical or supercritical.
The proof is done through an abstract functional analytic approach. We use exponentially weighted spaces to push neutral continuous spectrum away from the imaginary axis and, along with co-domain restriction which takes into account mass-conservation, we obtain a linearization of \eqref{comoving_inhom} which is a Fredholm operator of index 0. This allows us to perform a Lyapunov-Schmidt reduction to produce a set of finite-dimensional bifurcation equations, which we can then use to establish bifurcating solutions, obtain their leading order expansions, and determine their bifurcation direction.
The rest of the introduction is devoted to developing our setting, stating our hypotheses, and finally stating the theorem we wish to prove.
\subsection{Our Setting} We consider the following modified Cahn-Hilliard equation with spatiotemporal heterogeneities in both the nonlinearity $f$ (corresponding to changes in the potential well) as well as a moving source term $\chi$ which adds mass to the system as it travels. We consider heterogeneities which rigidly propagate in the horizontal direction with fixed speed $c$, leaving a front solution its wake. Our equation takes the form, \begin{equation}\label{e:ch-1} \partial_tu=-\Delta(\Delta u+f(x-ct,u))+c\chi(x-ct;c), \quad \mathbf{x} = (x,y)\in\mathbb{R}^2, \,\, t\in \mathbb{R}, \,\,\, \Delta = \partial_x^2+\partial_y^2. \end{equation}
Before continuing with our general hypotheses on the heterogeneities, front solutions, and their spectra, we give a specific example which will motivate and guide our work. \begin{Example} \label{ex:top} {\it Tophat quench} \end{Example} Consider a cubic-quintic nonlinearity, \begin{equation}\label{e:nlcq}
f(x-ct,u)=h(x-ct)u+\gamma u^3-u^5,\qquad \gamma \in \mathbb{R}, \end{equation} with a parameter heterogeneity $h(x-ct)$ on the linear term. Here $h$ is 1 in the interval $[-K+\delta,K-\delta]$, -1 outside the interval $[-K,K]$, and smoothly and monotonically transitions between the two states in between the intervals; see \eqref{e:hex} for an approximate example.
When $h\equiv1$, the sign of the parameter $\gamma$, roughly speaking, mediates between supercritical ($\gamma<0$) and subcritical ($\gamma>0$) pattern-forming dynamics. Indeed in this situation, local perturbations of the trivial state $u\equiv0$ will grow, invade, and form patterned states in different manners for these two cases, with the former corresponding to \emph{pulled} front invasion where the linear dynamics about $u\equiv0$ govern front dynamics, and \emph{pushed} front invasion where the nonlinear dynamics behind the interface accelerate invasion. For both parameter domains, one finds patterned states in the quenched system for quenching speeds $c$ approximately below the free invasion speed. We remark that in the pushed case such quenches have been observed to form patterns for quenching speeds faster than the free invasion speed; see \cite{goh2016pattern}.
Figure \ref{fig:ch-ex1} gives snapshots of numerical simulations of such a quenched nonlinearity inducing oblique stripes and checkerboard patterns in a periodic channel in the pulled case, $\gamma = -1$. We note these solutions are time-periodic so that a $y$ cross-section of $u$ resembles a traveling, or rotating, wave in the former and a standing wave in the latter. \begin{figure}
\caption{Numerically, one can find these checkerboards (right) and oblique stripes (left) arising from Example \ref{ex:top}. Here $\gamma = -1$, and patterns arise from the trivial front $u_*=0$. The quench speed was chosen to be $c=1$. }
\label{fig:ch-ex1}
\end{figure} This example is investigated in more detail in Section \ref{s:ex}.
\paragraph{General Setting} In the general case, we consider solutions $u(x,y,t)$ of \eqref{e:ch-1} which are periodic in both $t$ and $y$. In particular we have $t\in[0,2\pi/\omega)$ and $y\in[0,2\pi/k)$, where $\omega$ is the frequency in time and $k$ is the frequency in the second spatial variable. We then rescale time to the new variable $\tau=\omega t$, we rescale the vertical spatial variable to $\Tilde{y}=ky$, and put the system into the co-moving frame $\Tilde{x}=x-ct$. Simplifying our notation by removing the tildes and writing $\Delta_k:=\partial_x^2+k^2\partial_y^2$, we can then write the Cahn-Hilliard equation as \begin{equation}\label{comoving_inhom} \begin{split}
\omega\partial_\tau u&=-\Delta_k(\Delta_k u+f(x,u))+c\partial_xu+c\chi(x;c)\\
u(x,y,\tau)&=u(x,y,\tau+2\pi)\\
u(x,y,\tau)&=u(x,y+2\pi,\tau). \end{split} \end{equation} Note that there are symmetries in the $y$ variable. In particular \eqref{comoving_inhom} is invariant under translations $y\mapsto y+\theta$ and reflections $y\mapsto-y$. Hence any bifurcating fronts will occur in the presence of spatial symmetries. In particular, the symmetry group will be $O(2)$, being the semidirect product of $SO(2)$ (which contains rotations) with $\mathbb{Z}_2$ (which represents the reflections). With these preliminaries set, we are now ready to present our hypotheses.
\subsection{Hypotheses and Main Result} We begin our hypotheses by specifying restrictions on the form of the nonlinearity $f$ and an associated traveling front solution, which propagates with fixed speed, and is asymptotically constant in space, $u_*.$ We remark that these hypotheses are an extension of the one-dimensional setting of \cite{Goh} to the two-dimensional case. The quenching speed $c$ will serve as the main bifurcation parameter of our study.
\begin{Hypothesis}\label{hyp1}
The nonlinearity $f$ is smooth in both $x$ and $u$, and converges with an exponential rate to smooth functions $f_\pm:=f_\pm(u)$ as $x\to\pm\infty$. This convergence is uniform for $u$ in bounded sets. \end{Hypothesis}
\begin{Hypothesis}\label{hyp2}
There exists a front solution $u_*(x;c_*)$ of \eqref{comoving_inhom} for some $c_*>0$ with \[\lim_{x\to\pm\infty}u_*(x;c_*)=u_\pm.\]
Moreover, $u_*\in\mathcal{C}^4(\mathbb{R})$ and \begin{equation}|u_*(x)-u_\pm|+\sum_{j=1}^3|\partial_x^ju_*(x)|\leq Ce^{-\beta|x|}\nonumber\end{equation}
for some $C,\beta>0$. We refer to this front solution as the primary front. \end{Hypothesis}
This primary front will be the solution to the Cahn-Hilliard equation from which our patterns bifurcate. In our previous example, the trivial solution $u_*\equiv0$ plays this role.
Next, we assume that there are no additional neutral modes of the spatial linearization at the origin, and thus that the Hopf instability is the sole neutral mode at the bifurcation speed $c_*$
\begin{Hypothesis}\label{hyp3}
The point $0\in\mathbb{C}$ is not contained in the extended point spectrum of the linearization $L:H^4(\mathbb{R}\times\mathbb{T})\subset L^2(\mathbb{R}\times\mathbb{T})\to L^2(\mathbb{R}\times\mathbb{T})$ defined as \[Lv=-\Delta_k(\Delta_k v+\partial_uf(x,u_*(x))v)+c_*\partial_xv.\] \end{Hypothesis} Furthermore, we assume that fronts persist for perturbations of asymptotic states of the front which preserve the difference between values at $x = \pm\infty.$ \begin{Hypothesis}\label{hyp4}
Assuming the above hypotheses, for $\Tilde{u}_\pm$ in a small neighborhood of $u_\pm$ with $\Tilde{u}_+-\Tilde{u}_-=u_+-u_-$, and $c$ close to $c_*$, there exists a family of smooth front solutions $u_*(x;c)$ asymptotic to $\Tilde{u}_\pm$ satisfying Hypothesis \ref{hyp2}. \end{Hypothesis}
This implies that our assumptions are open; that we can vary $u_\pm$ and still be able to find a front solution. The next hypothesis ensures that there exists a generic Hopf-instability, with transversely modulated eigenfunctions, in the presence of a $y$-reflection symmetry. \begin{Hypothesis}\label{hyp5}
The operator $L$ defined on $L^2(\mathbb{R}\times\mathbb{T})$ as above has an isolated pair of eigenvalues, $\lambda_\pm(c)=\mu(c)\pm i\kappa(c)$, with algebraic and geometric multiplicity two and $L^2(\mathbb{R}\times\mathbb{T})$-eigenfunctions $e^{iy}p(x), e^{-iy}\overline{p(x)}$ along with their images under the reflection symmetry $y\mapsto-y$, such that for some $\omega_*\neq0$ and $c_*>0$
\begin{equation}
\mu(c_*)=0,\qquad \mu'(c_*)>0\qquad \kappa(c_*)=\omega_*.
\end{equation} \end{Hypothesis}
Here we note that the geometrically double eigenvalues are induced by the $y$-reflection symmetry, and the above assumption guarantees that they are semi-simple. The higher multiplicity of the Hopf modes precludes application of the standard Hopf Theorem and, due to the presence of symmetry, requires the application an equivariant Hopf theorem. For more information see Appendix \ref{app2}.
Recall our symmetries are the $SO(2)$ action $y\mapsto y+\theta$, and $\mathbb{Z}_2$ reflection action $y\mapsto-y$. The action of $SO(2)$ leads to rotating, or traveling, waves, which appear in our system as oblique striped patterns. The $\mathbb{Z}_2$-action produces standing waves, which appear as checkerboard patterns.
We also define the $L^2$-adjoint eigenfunctions to $p$ and $\overline{p}$, as $\psi_+$ and $\psi_-$ respectively, whose corresponding eigenvalues must have the same algebraic and geometric multiplicity as the eigenvalues $\lambda_\pm(c)$. We further normalize $\psi_+$ and $\psi_-$ such that $\langle p,\psi_+\rangle_{L^2}=\langle\overline{p},\psi_-\rangle_{L^2}=1$.
Finally, we make a non-resonance assumption which guarantees that there are no point or essential spectrum touching the imaginary axis at frequencies which are non-zero multiples of the Hopf frequency $\omega_*.$ Here $\lambda = 0$ is not included to take into account the neutral continuous spectrum which touches the origin in a quadratic tangency induced by the neutral mass flux/conservation structure of \eqref{e:ch-1}. \begin{Hypothesis}\label{hyp6}
For all $\lambda\in i\omega_*(\mathbb{Z}\setminus\{0,\pm1\})$, the operator $L-\lambda$ is invertible when considered on the unweighted space $L^2(\mathbb{R}\times\mathbb{T})$. \end{Hypothesis}
With these hypotheses, we are now ready to state our result.
\begin{Theorem}\label{thm1}
Assume Hypotheses \ref{hyp1} through \ref{hyp6}. Then there exist a pair of one-parameter families of time-periodic solutions which bifurcate from the front solution $u_*(x;c)$ as the speed $c$ varies through $c_*$.
The bifurcation equation has, to leading order, the form\begin{equation}
\Theta(a,b;\Tilde{c},\Tilde{\omega})=\left(\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega}+\frac{\theta_1+\theta_2}{2}N\right)\begin{pmatrix}
a\\b
\end{pmatrix}+\frac{\theta_2-\theta_1}{2}\delta\begin{pmatrix}
a\\-b
\end{pmatrix},
\end{equation}
with amplitudes $a,b\in\mathbb{C}$, $N=|a|^2+|b|^2, \delta=|b|^2-|a|^2,$ and
\begin{align}
\theta_1&=\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+\overline{p}\phi_{aa}]\right),\psi_+\right\rangle_{L^2},\\
\theta_2&=\left\langle-\Delta_k(\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{b\overline{b}}+p\phi_{a\overline{b}}+\overline{p}\phi_{ab}]),\psi_+\right\rangle_{L^2}.
\end{align}
Here, the $\phi_{ij}$'s are particular functions of $x$ which can be obtained using the linear operator and evaluations of derivatives of $f$ on the front $u_*$ (see equations \eqref{phi_a_a} to \eqref{phi_barb_barb}), and $a$ and $b$ are the coordinates of the kernel of \eqref{linearized}, the time-dependent linearization of the Cahn-Hilliard equation. The direction of the bifurcation can be determined by examining the relationship between $\frac{\theta_1+\theta_2}{2}$ and $\frac{\theta_2-\theta_1}{2}$.
\begin{itemize}
\item If $\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)>0$, then standing waves (checkerboard patterns) will bifurcate as $c$ increases through $c_*$
\item If $\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)<0$, then standing waves will bifurcate as $c$ decreases through $c_*$
\item If $\mathrm{Re}\,\left(\frac{\theta_2-\theta_1}{2}\right)<\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)$, then rotating waves (oblique stripe patterns) will bifurcate as $c$ increases through $c_*$
\item If $\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)<\mathrm{Re}\,\left(\frac{\theta_2-\theta_1}{2}\right)$, then rotating waves bifurcate as $c$ decreases through $c_*$
\end{itemize} The family of oblique stripe and checkerboard solutions can be parameterized in terms of the amplitude $a$ as
\begin{align}
u_{os} &= u_*+2a\mathrm{Re}\,(e^{i(\tau+y)}p(x))+O(a^2),\\
u_{cb}&=u_*+4a\cos(y)\mathrm{Re}\,(e^{i\tau}p(x))+O(a^2),
\end{align} and the bifurcation parameter $c$ can also be parameterized in terms of the amplitude $a$ for both oblique stripes and checkerboards as \begin{align}
c_{os}&=c_*-\frac{\mathrm{Re}\,(\theta_1)}{\mu'(c_*)}a^2+O(|a|^3),\label{c_os}\\
c_{cb}&=c_*-\frac{\mathrm{Re}\,(\theta_1+\theta_2)}{\mu'(c_*)}a^2+O(|a|^3).\label{c_cb}
\end{align} \end{Theorem}
This theorem tells us when patterns bifurcate from our front solution $u_*$, as well as giving computable coefficients which determine the direction of bifurcation of patterns.
We prove this theorem in Section \ref{s:ab}. In Section \ref{s:ex} we provide an example of a specific nonlinearity and heterogeneity for which we observe this behavior. In Section \ref{s:disc} we discuss our work and mention a few open areas of research stemming from it. Finally, in the appendices we provide proofs establishing Fredholm properties of the linearization we consider, and give a short summary of the abstract theory of Hopf bifurcation with $O(2)$ symmetry.
\section{Abstract Results}\label{s:ab} In this section, we seek to prove our theorem. We approach \eqref{comoving_inhom} as an abstract nonlinear equation, for which the primary front $u_*$ is a zero. We first establish Fredholm properties of the linearization at $u_*$ in a space of $y,t$ periodic functions. The independence of the front $u_*$ in $y$ and $t$ allows for a Fourier decomposition of the operator and its Fredholm index. We use exponentially weighted spaces to push neutral continuous spectrum away from the imaginary axis in the $t,y$-independent component. Then, using a co-domain restriction which projects off of the constants to address neutral mass flux in $x$, we obtain an operator which has Fredholm index 0, with kernel spanned by time-modulated forms of the transverse eigenfunctions. We then perform a Lyapunov-Schmidt decomposition to reduce the infinite-dimensional equation to a finite dimensional bifurcation equation in terms of the kernel variables, which can be put into $O(2)$-Hopf normal form, allowing us to establish bifurcating transversely patterned solutions and give expressions for the direction of bifurcation. To begin, we introduce some useful notation.
\paragraph{Function spaces}
Recall, we consider the domain $(x,y,\tau)\in\mathbb{R}\times\mathbb{T}_y\times\mathbb{T}_\tau$, where $\mathbb{T}_{y,\tau}=[0,2\pi)$. We let \[X:=L^2(\mathbb{T}_\tau),\qquad Y:=H^1(\mathbb{T}_\tau).\] We then define the exponentially weighted $L^2$-space \begin{equation}
\mathcal{X}:=L_\eta^2(\mathbb{R}\times\mathbb{T}_y,X)=\{v:\mathbb{R}\times\mathbb{T}_y\rightarrow X\,\,|\,\,\|v\|^2_{2,\eta}<\infty\}\nonumber \end{equation} with weighted norm \begin{equation}
\|v\|^2_{2,\eta}:=\int_{\mathbb{R}\times \mathbb{T}_y}\| e^{\eta\langle x\rangle}v(x,y,\cdot)\|_{X}^2dxdy,\qquad \langle x\rangle=\sqrt{1+x^2}. \end{equation} Given the following inner product, $\mathcal{X}$ becomes a Hilbert space: \[\langle u,v\rangle_\mathcal{X}:=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\int_{-\infty}^\infty u(x,y,\tau)\overline{v(x,y,\tau)}e^{2\eta\langle x\rangle}dxdy d\tau.\]
We can similarly define Sobolev spaces $H_\eta^k$ as \begin{equation}
H_\eta^k(\mathbb{R}\times\mathbb{T},X)=\{v:\mathbb{R}\times\mathbb{T}\rightarrow X\,\,|\,\,\|D^\alpha v\|^2_{2,\eta}<\infty,\,\, |\alpha|\leq k\}.\nonumber \end{equation} Finally, we define the Banach space \[ \mathcal{Y}:=L^2_\eta(\mathbb{R}\times\mathbb{T}_y,Y)\cap H_\eta^4(\mathbb{R}\times\mathbb{T}_y,X), \]
with norm \begin{equation}\|u\|^2_\mathcal{Y}:=\int_0^{2\pi}\int_{-\infty}^\infty \|u(x,y,\cdot)\|_Y^2+\sum_{|\alpha|\leq4}\|D^\alpha u(x,y,\cdot)\|^2_Xdxdy.\nonumber\end{equation}
\paragraph{Abstract nonlinear equation}
We consider perturbations $u=u_*+v$ of the front solution $u_*(x)$ in \eqref{comoving_inhom} at the parameters $(\omega,c) = (\omega_*,c_*)$. Defining $\Tilde{\omega}=\omega-\omega_*,\, \, \Tilde{c}=c-c_*$, and $\Omega=(\Tilde{\omega},\Tilde{c})$, and subtracting off $v$-independent parts, we obtain: \begin{equation} 0 = (\tilde \omega + \omega_*)\partial_\tau v+\Delta_k(\Delta_k v+g(x,v;u_*))-(\tilde c + c_*)\partial_xv =: \mathcal{F}(v; \Omega) \end{equation} where $g(x,v;u_*):=f(x,u_*+v)-f(x,u_*)$. This defines a locally smooth mapping $\mathcal{F}:\mathcal{Y}\times\mathbb{R}^2\to\mathcal{X}$ with $(0;0,0)$ corresponding to the base front solution $u_*$ and general zeros $(v;\Omega)$ of $\mathcal{F}$ corresponding to $y,\tau$-periodic solutions of \eqref{comoving_inhom}. The smoothness of $\mathcal{F}$ in the $v$ variable is dependent on the smoothness of $f$ in the same variable. By Hypothesis \ref{hyp1} we have that $f$ is smooth in $v$, and hence so is $\mathcal{F}$.
\subsection{Linear Properties} Linearizing $\mathcal{F}$ at the front solution $(v;\tilde{\omega},\tilde{c})=(0;0,0)$, we obtain the following closed and densely defined linear operator \begin{align} \mathcal{L}:\mathcal{Y}\subset\mathcal{X}&\to\mathcal{X},\nonumber\\ v&\mapsto\omega_*\partial_\tau v+\Delta_k(\Delta_k v+\partial_uf(x,u_*)v)-c_*\partial_xv.\label{linearized} \end{align} We can further define the (formal) $L^2$-adjoint to be \[\mathcal{L}^*=-\omega_*\partial_\tau+(\Delta_k+\partial_uf(x,u_*))\Delta_k+c_*\partial_x.\] By restricting its co-domain, the operator $\mathcal{L}$ has the following Fredholm properties. \begin{Proposition}\label{pfred}
Let $\mathring{\mathcal{X}}:=\{u\in\mathcal{X}|\langle u,e^{-2\eta\langle x\rangle}\rangle_\mathcal{X}=0\}$. Then for $\eta>0$ small, $\mathcal{L}:\mathcal{Y}\to\mathring{\mathcal{X}}$ is Fredholm of index 0, with four dimensional kernel. \end{Proposition} We leave the proof of this proposition to Appendix \ref{app1}. The general scheme is to decompose the space $\mathcal{X}$ into various Fourier subspaces, study the Fredholm properties on each, and then use Fredholm algebra to determine the index of the full operator. We find on all but one subspace that $\mathcal{L}$ is a Fredholm operator with Fredholm index 0, while on the $t,y$-independent subspace $\mathcal{L}$ has index $-1$ when considered as a mapping into $\mathcal{X}$. Restricting the codomain to $\mathring{\mathcal{X}}$, which projects off constants, then yields an index 0 operator. We find the kernel of $\mathcal{L}$ lies in the subspaces spanned by the base modes of the form $e^{\pm iy}e^{\pm i \tau}$.
As we wish to perform a Lyapunov-Schmidt reduction on $\mathcal{F}$, the first step is to decompose the domain and codomain of $\mathcal{L}$. Recall that $e^{iy}p(x), e^{-iy}\overline{p}(x)$, along with their $y$-reflections, give Hopf eigenfunctions of the linear operator $Lv=-\Delta_k(\Delta_k v+\partial_uf(x,u_*)v)+c_*\partial_xv$. We then define \begin{align} &P_+=e^{i(\tau+y)}p(x),\quad P_-=\overline{P_+},\nonumber\\ &Q_+=e^{i(\tau-y)}p(x),\quad Q_-=\overline{Q_+}.\nonumber \end{align} Then, by Hypotheses \ref{hyp3} and \ref{hyp5}, $\ker\mathcal{L}=span\{P_+,P_-,Q_+,Q_-\}$, and hence any $u_0\in\ker\mathcal{L}$ is given by \[u_0=aP_++\overline{a}P_-+bQ_++\overline{b}Q_-,\] with $a,b\in\mathbb{C}$. We similarly have elements of the $L^2$-adjoint kernel: $$ \Psi_+=e^{i(\tau+y)}\psi_+,\quad \Psi_- = \overline{\Psi_+},\qquad \Phi_+=e^{i(\tau-y)}\psi_+,\quad \Phi_- = \overline{\Phi_+}. $$
Then we can decompose the domain and codomain of $\mathcal{L}$ as
\[
\mathcal{Y}=\ker\mathcal{L}\oplus M \qquad \mathring{\mathcal{X}}=N\oplus\ker\mathcal{L}^*,
\] where $N=(\ker\mathcal{L}^*)^\perp$ and $M=(\ker\mathcal{L})^\perp$. We then define projections $E:\mathring{\mathcal{X}}\to\ker\mathcal{L}^*$ and $1-E$ with \[ E\mathcal{F}=\sum_{i=+,-}\langle\mathcal{F},\Psi_i\rangle_\mathcal{X}\cdot\Psi_i+\langle\mathcal{F},\Phi_i\rangle_\mathcal{X}\cdot\Phi_i. \] Finally, for $u_0\in\ker\mathcal{L}$ and $u_h\in M$, we obtain the decomposed system of equations \begin{align} E\mathcal{F}(u_0+u_h;\Omega)&=0\label{e:bif}\\ (1-E)\mathcal{F}(u_0+u_h;\Omega)&=0. \end{align}
Since $(1-E)\mathcal{F}$ has invertible linearization in $u_h$ at $u_0=0$, the implicit function theorem gives that there exists a smooth mapping $w:\ker\mathcal{L}\times\mathbb{R}^2\to M$ satisfying \[(1-E)\mathcal{F}(u_0+w(u_0;\Omega);\Omega)=0.\] Furthermore, this solution satisfies $w(0;\Omega)=0.$ We also have \begin{align}
0 = \frac{\partial\mathcal{F}}{\partial u_0}\Big|_{(0;0,0)}&=\omega_*\partial_\tau\frac{\partial w}{\partial u_0}+\Delta_k\left(\Delta_k\frac{\partial w}{\partial u_0}+\partial_uf(x,u_*)\frac{\partial w}{\partial u_0}\right)-c_*\partial_x\frac{\partial w}{\partial u_0}=\mathcal{L}\frac{\partial w}{\partial u_0}\nonumber \end{align}
and so we have that $\frac{\partial w}{\partial u_0}\in\ker\mathcal{L}$, while $w\in(\ker\mathcal{L})^\perp$. Hence $\frac{\partial w}{\partial u_0}(0;0,0)$ is both tangent and normal to $w$, and so $\frac{\partial w}{\partial u_0}(0;0,0)=0$. So we can expand $w$ in terms of the kernel coordinates $a$ and $b$ as \[w(a,\overline{a},b,\overline{b};\Omega)=a^2e^{2i(\tau+y)}\phi_{aa}(x)+|a|^2\phi_{a\overline{a}}(x)+abe^{2i\tau}\phi_{ab}(x)+a\overline{b}e^{2iy}\phi_{a\overline{b}}(x)+\]
\[\overline{a}^2e^{-2i(\tau+y)}\phi_{\overline{aa}}(x)+\overline{a}be^{-2iy}\phi_{\overline{a}b}(x)+\overline{a}\overline{b}e^{-2i\tau}\phi_{\overline{a}\overline{b}}(x)+b^2e^{2i(\tau-y)}\phi_{bb}(x)+|b|^2\phi_{b\overline{b}}(x)+\] \[\overline{b}^2e^{-2i(\tau-y)}\phi_{\overline{bb}}(x)+\mathcal{O}(u_0^3)\]
Here $\phi_{ij}\in M\subset\mathcal{Y}$ for all $i,j\in\{a,\overline{a},b,\overline{b}\}$. Rearranging $(1-E)\mathcal{F}=0$, we obtain \begin{align} &\mathcal{L}(u_0+w(u_0;\Omega))=-\Delta_k(f(x,u_*+u_0+w(u_0;\Omega))-f(x,u_*)-\partial_uf(x,u_*)(u_0+w(u_0;\Omega)))\nonumber\\
&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)(u_0+w(u_0;\Omega))^2+\frac{1}{6}\partial_u^3f(x,u_*)(u_0+w(u_0;\Omega))^3+\mathcal{O}(\|u_0\|_\mathcal{Y}^4)\right). \end{align} Substituting in the above expansion and projecting onto the different Fourier modes $e^{2i(j\tau+\ell y)}$ with $j,\ell\in\{0,\pm1\}$, we find the following system of equations at quadratic order in $u_0$, after dividing out constants: \begin{align}
\mathcal{L}(e^{2i(\tau+y)}\phi_{aa})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{2i(\tau+y)}p^2(x)\right),\label{phi_a_a}\\
\mathcal{L}(\phi_{a\overline{a}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)|p(x)|^2\right),\label{phi_a_bara}\\
\mathcal{L}(e^{2i\tau}\phi_{ab})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{2i\tau}p^2(x)\right),\label{phi_a_b}\\
\mathcal{L}(e^{2iy}\phi_{a\overline{b}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{2iy}|p(x)|^2\right),\label{phi_a_barb}\\
\mathcal{L}(e^{-2i(\tau+y)}\phi_{\overline{a}\overline{a}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{-2i(\tau+y)}\overline{p}^2(x)\right),\label{phi_bara_bara}\\
\mathcal{L}(e^{-2iy}\phi_{\overline{a}b})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{-2iy}|p(x)|^2\right),\label{phi_bara_b}\\
\mathcal{L}(e^{-2i\tau}\phi_{\overline{a}\overline{b}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{-2i\tau}\overline{p}^2(x)\right),\label{phi_bara_barb}\\
\mathcal{L}(e^{2i(\tau-y)}\phi_{bb})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{2i(\tau-y)}p^2(x)\right),\label{phi_b_b}\\
\mathcal{L}(\phi_{b\overline{b}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)|p(x)|^2\right),\label{phi_b_barb}\\
\mathcal{L}(e^{-2i(\tau-y)}\phi_{\overline{b}\overline{b}})&=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{-2i(\tau-y)}\overline{p}^2(x)\right).\label{phi_barb_barb} \end{align}
\begin{Lemma}
Equations \eqref{phi_a_a} to \eqref{phi_barb_barb} can be uniquely solved for the functions $\phi_{ij}\in M$, $i,j\in\{a,\overline{a},b,\overline{b}\}$. \end{Lemma} \begin{proof} First, the right hand side in each of \eqref{phi_a_a} - \eqref{phi_barb_barb} is contained in $\mathring{\mathcal{X}}$. We note each right-hand side takes the form $\Delta_k(H)$ for some function $H$. As $p(x)$ is exponentially localized, so is $H$. Hence, integration by parts gives that $\langle \Delta_k H, e^{-2\eta\langle x \rangle} \rangle_{L^2_\eta} = \langle \Delta_k H, 1\rangle_{L^2} = 0.$
Next, we claim that each right hand side of \eqref{phi_a_a} - \eqref{phi_barb_barb} is contained in the range of $\mathcal{L}$. This is seen by observing that $\ker \mathcal{L}^*$ is spanned by the functions $\Psi_+,\Psi_-,\Phi_+,$ and $\Phi_-$, all of which have $\tau$-dependence of the form $e^{\pm i\tau}$, while the right hand sides of \eqref{phi_a_a} - \eqref{phi_barb_barb} contain $\tau$ dependence of the form $e^{0i\tau}$ or $e^{\pm 2i\tau}$ and hence must lie in $(\ker\mathcal{L}^*)^\perp$. Uniqueness follows from the fact that each $\phi_{ij}\in M=(\ker\mathcal{L})^\perp$.
\end{proof}
Thus we can solve equations \eqref{phi_a_a}-\eqref{phi_barb_barb} for $\phi_{ij}$, which in turn will give the leading-order expansion of $w(u_0;\Omega)$. Plugging this into the bifurcation equation \eqref{e:bif} we obtain \[0=E\mathcal{F}(u_0+w(u_0;\Omega);\Omega) = \sum_{i=+,-}\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Psi_i\rangle_\mathcal{X}\cdot\Psi_i+\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Phi_i\rangle_\mathcal{X}\cdot\Phi_i,\] which, since $\Psi_+,\Psi_-,\Phi_+,$ and $\Phi_-$ are linearly independent, is equivalent to the finite dimensional system of equations
\begin{align}
0=\Theta_1(u_0;\Omega):=\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Psi_+\rangle_\mathcal{X}&\\ 0=\Theta_2(u_0;\Omega):=\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Psi_-\rangle_\mathcal{X}&\\ 0=\Theta_3(u_0;\Omega):=\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Phi_+\rangle_\mathcal{X}&\\ 0=\Theta_4(u_0;\Omega):=\langle\mathcal{F}(u_0+w(u_0;\Omega);\Omega),\Phi_-\rangle_\mathcal{X}.& \end{align}
Then by the definitions of $\Psi_i$ and $\Phi_i$ we have that the only terms which are nonzero in the inner product will be terms of the form $e^{i\ell_\tau\tau}e^{i\ell_yy}$ for $\ell_\tau,\ell_y=\pm1$. Each of the functions $\Theta_i$ is zero trivially when $a=b=0$, and so we can decompose each equation as $\Theta_i = r_i(a,\overline{a},b,\overline{b};\Omega)\cdot h_i(a,\overline{a},b,\overline{b})$, where $h(0,0,0,0) = 0$. In the coordinates $(a,\bar a, b, \bar{b})\in \mathbb{C}^4$, the reduced equations are equivariant under the following actions induced by the symmetries of the original nonlinear equation: time translation induces the action $(a,b)\mapsto (e^{i \theta} a, e^{i \theta} b),\quad \theta\in[0,2\pi)$, $y$-translation induces $(a,b)\mapsto (e^{i \phi} a, e^{-i \phi} b), \phi\in[0,2\pi)$, while $y$-reflection induces $(a,b)\mapsto (b,a)$. Under these symmetries, we readily conclude that $r_i$ can be written as a function of $|a|^2, |b|^2$ and $\Omega$. Because of the inner product with the adjoint kernel elements $\Psi_+,\Psi_-,\Phi_+$, and $\Phi_-$, each $h_i$ will be linear in exactly one of the kernel coordinates, and will not depend on any of the others. Thus, possibly after redefinition by a constant, we can write
\[\Theta_1=r_1(|a|^2,|b|^2;\Tilde{\omega},\Tilde{c})a\]
\[\Theta_2=r_2(|a|^2,|b|^2;\Tilde{\omega},\Tilde{c})\overline{a}\]
\[\Theta_3=r_3(|a|^2,|b|^2;\Tilde{\omega},\Tilde{c})b\]
\[\Theta_4=r_4(|a|^2,|b|^2;\Tilde{\omega},\Tilde{c})\overline{b}.\] We note that expansions of each $r_i$ are determined by taking derivatives of $\mathcal{F}$ and taking inner products with the appropriate $\Psi_i$ or $\Phi_i$. We consider these in cases by the signs of $\ell_\tau$ and $\ell_y$, in the following way:
In the case $\ell_\tau = \ell_y = 1$, we must consider $\langle\mathcal{F},e^{i(\tau+y)}\psi_+\rangle_\mathcal{X}$, and so the only nonzero terms in the inner product will be terms with the mode $e^{i(\tau+y)}$ by the orthogonality of the exponential.
\textit{To find $\partial_{\Tilde{c}}r_1$:} We consider $\mathcal{F}_{a\Tilde{c}}$, and we will then take the appropriate inner product, finding \begin{align}
\mathcal{F}_{a\Tilde{c}}|_0&=\Bigg[(\Tilde{c}+c_*)\partial_x\Bigg(ae^{i(\tau+y)}p(x)+\overline{a}e^{-i(\tau+y)}\overline{p}(x)\nonumber\\ &\qquad\qquad +be^{i(\tau-y)}p(x)+\overline{b}e^{-i(\tau-y)}\overline{p}(x)+w(a,\overline{a},b,\overline{b};\Omega)\Bigg)\Bigg]_{a\Tilde{c}} = e^{i(\tau+y)}p'(x). \end{align} So $\partial_{\Tilde{c}}r_1(0)=\langle e^{i(\tau+y)}p'(x),e^{i(\tau+y)}\psi_+\rangle_\mathcal{X}=\lambda_{\Tilde{c}}(0).$
\textit{To find $\partial_{\Tilde{\omega}}r_1$:} We consider $\mathcal{F}_{a\Tilde{\omega}}$ and take the appropriate inner product, finding \begin{align}
\mathcal{F}_{a\Tilde{\omega}}|_0&=\Bigg[(\Tilde{\omega}+\omega_*)\Bigg(ae^{i(\tau+y)}p(x)+\overline{a}e^{-i(\tau+y)}\overline{p}(x)+be^{i(\tau-y)}p(x)+\nonumber\\ &\qquad\qquad+\overline{b}e^{-i(\tau-y)}\overline{p}(x)+w(a,\overline{a},b,\overline{b};\Omega)\Bigg)_\tau\Bigg]_{a\Tilde{\omega}}=ie^{i(\tau+y)}p(x), \end{align} and so $\partial_{\Tilde{\omega}}r_1(0)=\langle ie^{i(\tau+y)}p(x),e^{i(\tau+y)}\psi_+\rangle_\mathcal{X}=i$.
\textit{To find $\partial_{|a|^2}r_1$:} We must consider all the different ways one can produce nonzero terms of the correct mode, $e^{i(\tau+y)}$. From the right-hand sides of the equations defining the $\phi_{ij}$'s, this can only occur from terms of the form $u_0^3$ or $u_0\cdot w(u_0;\Omega)$. By considering these terms and taking the appropriate inner products, we will find that
\[\partial_{|a|^2}r_1=\left\langle-\Delta_k\left({3\choose 2,1}\frac{1}{6}\partial_u^3f(x,u_*)e^{i(\tau+y)}p^2(x)\overline{p}(x)+\right.\right.\]\[\left.\left.+{2\choose 1,1}\frac{1}{2}\partial_u^2f(x,u_*)e^{i(\tau+y)}p(x)\phi_{a\overline{a}}+{2\choose 1,1}\frac{1}{2}\partial_u^2f(x,u_*)e^{i(\tau+y)}\overline{p}(x)\phi_{aa}\right),e^{i(\tau+y)}\psi_+\right\rangle_\mathcal{X}=\] \[=\left\langle-\Delta_k\left(e^{i(\tau+y)}\frac{1}{2}\partial_u^3f(x,u_*)p^2\overline{p}+e^{i(\tau+y)}\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+\overline{p}\phi_{aa}]\right),e^{i(\tau+y)}\psi_+\right\rangle_\mathcal{X}+h.o.t.\]
where ${N\choose n_1...n_k}=\frac{N!}{n_1!\cdots n_k!}$.
\textit{To find $\partial_{|b|^2}r_1$:} Similar to how we found $\partial_{|a|^2}r_1$, we look for nonzero terms with the correct mode, which will only occur in the right-hand sides from $u_0^3$ or $u_0\cdot w(u_0;\Omega)$. By taking the appropriate inner products of these terms and rewriting the Laplacian as $\Delta_k=\partial_x^2-k^2$ so that the term $e^{i(\tau+y)}$ commutes with the Laplacian, we find
\[\partial_{|b|^2}r_1=\left\langle-\Delta_k\left(\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{b\overline{b}}+p\phi_{a\overline{b}}+\overline{p}\phi_{ab}]\right),\psi_+\right\rangle_{L^2}+h.o.t.\]
Thus in total we have (with $\Delta_k=\partial_x^2-k^2$) \begin{align}
\Theta_1&=(\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega})a+a|a|^2\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+\overline{p}\phi_{aa}]\right),\psi_+\right\rangle_{L^2}\nonumber\\
&+a|b|^2\left\langle-\Delta_k\left(\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{b\overline{b}}+p\phi_{a\overline{b}}+\overline{p}\phi_{ab}]\right),\psi_+\right\rangle_{L^2}+h.o.t. \end{align}
Employing similar computations for the subspaces $\ell_\tau=\ell_y=-1$, $\ell_\tau=-\ell_y=1$, and $-\ell_\tau=\ell_y=1$ we respectively find \begin{align} \Theta_2&=\overline{\Theta_1} = (\overline{\lambda}_{\Tilde{c}}(0)\Tilde{c}-i\Tilde{\omega})\overline{a}
+\overline{a}|a|^2\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)\overline{p}^2p+\partial_u^2f(x,u_*)[\overline{p}\phi_{a\overline{a}}+p\phi_{\overline{a}\overline{a}}]\right),\psi_-\right\rangle_{L^2}\nonumber\\
&+\overline{a}|b|^2\left\langle-\Delta_k\left(\partial_u^3f(x,u_*)\overline{p}^2p+\partial_u^2f(x,u_*)[\overline{p}\phi_{b\overline{b}}+p\phi_{\overline{a}\overline{b}}+\overline{p}\phi_{\overline{a}b}]\right),\psi_-\right\rangle_{L^2}+h.o.t.\\ \Theta_3&=(\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega})b
+b|a|^2\left\langle-\Delta_k\left(\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+p\phi_{\overline{a}b}+\overline{p}\phi_{ab}]\right),\psi_+\right\rangle_{L^2}\nonumber\\
&+b|b|^2\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{b\overline{b}}+\overline{p}\phi_{bb}]\right),\psi_+\right\rangle_{L^2}+h.o.t.\label{e:TH3}\\ \Theta_4&=\overline{\Theta_3} = (\overline{\lambda}_{\Tilde{c}}(0)\Tilde{c}-i\Tilde{\omega})\overline{b}
+\overline{b}|a|^2\left\langle-\Delta_k\left(\partial_u^3f(x,u_*)\overline{p}^2p+\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+p\phi_{\overline{a}\overline{b}}+\overline{p}\phi_{a\overline{b}}]\right),\psi_+\right\rangle_{L^2}\nonumber\\
&+\overline{b}|b|^2\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)\overline{p}^2p+\partial_u^2f(x,u_*)[p\phi_{\overline{b}\overline{b}}+\overline{p}\phi_{b\overline{b}}]\right),\psi_+\right\rangle_{L^2}+h.o.t. \end{align}
\subsection{Reduced Equations} Following \cite[Ch. 17, Prop. 2.1]{Golubitsky}, we suppress the complex conjugate equations and put our bifurcation equations in the form\begin{equation}
\begin{pmatrix}
\Theta_1(a,b)\\\Theta_3(a,b)
\end{pmatrix}=(p+iq)\begin{pmatrix}
a\\b
\end{pmatrix}+(r+is)\delta\begin{pmatrix}
a\\-b
\end{pmatrix},\label{eq:normal_form} \end{equation}
where $p,q,r,s$ are polynomials in the variables $N=|a|^2+|b|^2$ and $D=\delta^2=(|b|^2-|a|^2)^2$. This will allow us to determine the direction of the bifurcation for both the standing and rotating waves, whether supercritical or subcritical. We have
\[\Theta_1(a,b)=(\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega})a+\theta_1 a|a|^2+\theta_2 a |b|^2+\mathcal{O}(|u_0|^4+|u_0|^2\cdot |w|+|w|^2),\] with \begin{align}
\theta_1=\left\langle-\Delta_k\left(\frac{1}{2}\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{a\overline{a}}+\overline{p}\phi_{aa}]\right),\psi_+\right\rangle_{L^2},\\
\theta_2=\langle-\Delta_k(\partial_u^3f(x,u_*)p^2\overline{p}+\partial_u^2f(x,u_*)[p\phi_{b\overline{b}}+p\phi_{a\overline{b}}+\overline{p}\phi_{ab}]),\psi_+\rangle_{L^2}. \end{align}
Since we can write $|a|^2=\frac{1}{2}N-\frac{1}{2}\delta$ and $|b|^2=\frac{1}{2}N+\frac{1}{2}\delta$, this gives us the leading order form \[\Theta_1\approx a\left[\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega}+\frac{\theta_1+\theta_2}{2}N+\frac{\theta_2-\theta_1}{2}\delta\right].\]
Using this, we can deduce the leading order forms of the polynomials $p,q,r,$ and $s$ as \begin{align}
p\approx\mu_{\Tilde{c}}(0)\Tilde{c}+\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)N,\\
q\approx\Tilde{\omega}+\kappa_{\Tilde{c}}(0)\Tilde{c}+\mathrm{Im}\left(\frac{\theta_1+\theta_2}{2}\right)N,\\
r\approx\mathrm{Re}\,\left(\frac{\theta_2-\theta_1}{2}\right),\\
s\approx\mathrm{Im}\left(\frac{\theta_2-\theta_1}{2}\right). \end{align}
However this selection of $p,q,r,$ and $s$ must also hold for $\Theta_3$. $\Theta_3$ can be written similarly to $\Theta_1$ as \[\Theta_3\approx b\left[\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega}+\eta_1|a|^2+\eta_2|b|^2\right]=\] \[=b\left[\lambda_{\Tilde{c}}(0)\Tilde{c}+i\Tilde{\omega}+\frac{\eta_1+\eta_2}{2}N+\frac{\eta_1-\eta_2}{2}\delta\right],\] with $\eta_1$ and $\eta_2$ being the appropriate inner products from $\Theta_3$ in \eqref{e:TH3}. We can perform similar assignments as above to get forms for $p,q,r,$ and $s$ in terms of $\eta_1$ and $\eta_2$.
We claim that $\phi_{a\overline{a}}=\phi_{b\overline{b}}$, $\phi_{a\overline{b}}=\phi_{\overline{a}b}$, and $\phi_{aa}=\phi_{bb}$. Then it follows that $\eta_2=\theta_1$ and $\eta_1 = \theta_2$ and hence that $\theta_1+\theta_2=\eta_1+\eta_2$ and $\theta_2-\theta_1=\eta_1-\eta_2$, which in turn shows that $\begin{pmatrix}\Theta_1\\\Theta_3\end{pmatrix}$ can be written in the form \eqref{eq:normal_form}. We show the claimed equalities next.
\subsubsection{$\phi_{a\overline{a}}=\phi_{b\overline{b}}$}
Returning to equations \eqref{phi_a_bara} and \eqref{phi_b_barb}, we note that $\phi_{a\overline{a}}$ and $\phi_{b\overline{b}}$ solve the same equation and hence are equal since the equations can be solved uniquely.
\subsubsection{$\phi_{a\overline{b}}=\phi_{\overline{a}b}$}
Here we compare \eqref{phi_a_barb} and \eqref{phi_bara_b}. Recall we have $\mathcal{L}v=\omega_*\partial_\tau v+\Delta_k(\Delta_k v+\partial_uf(x,u_*)v)-c_*\partial_xv$. From \eqref{phi_a_barb} we have
\[\Delta_k(\Delta_ke^{2iy}\phi_{a\overline{b}}+\partial_uf(x,u_*)e^{2iy}\phi_{a\overline{b}})-c_*\partial_xe^{2iy}\phi_{a\overline{b}}=\]\[=-\Delta_k\left(\frac{1}{2}\partial_u^2f(x,u_*)e^{2iy}|p(x)|^2\right).\]
We then note that because we have functions $e^{2iy}\phi_{a\overline{b}}$ and $e^{2iy}|p(x)|^2$, the Laplacian becomes $\Delta_k=\partial_x^2+k^2\partial_y^2=\partial_x^2-4k^2$. Dividing by $e^{2iy}$, this leaves us with \[(\partial_x^2-4k^2)((\partial_x^2-4k^2)\phi_{a\overline{b}}+\partial_uf(x,u_*)\phi_{a\overline{b}})-c_*\partial_x\phi_{a\overline{b}}=\]
\[=\frac{1}{2}(4k^2-\partial_x^2)(\partial_u^2f(x,u_*)|p(x)|^2).\]
For \eqref{phi_bara_b}, note that again the Laplacian becomes $\Delta_k=\partial_x^2-4k^2$ because of the form of the functions. Evaluating the $y$ dependence and dividing by $e^{-2iy}$, we find: \[(\partial_x^2-4k^2)((\partial_x^2-4k^2)\phi_{\overline{a}b}+\partial_uf(x,u_*)\phi_{\overline{a}b})-c_*\partial_x\phi_{\overline{a}b}=\]
\[=-\frac{1}{2}(\partial_x^2-4k^2)(\partial_u^2f(x,u_*)|p(x)|^2)=\frac{1}{2}(4k^2-\partial_x^2)(\partial_u^2f(x,u_*)|p(x)|^2).\]
then we can see that $\phi_{a\overline{b}}$ and $\phi_{\overline{a}b}$ solve the same equation. Thus they are equal by the same reasoning as before.
\subsubsection{$\phi_{aa}=\phi_{bb}$}
This follows a similar procedure as the previous case, though now there is dependence on $\tau$. We can show that $\phi_{aa}$ and $\phi_{bb}$ solve the same equation, and thus must be equal. Thus we have shown that $\eta_2=\theta_1$ and $\theta_2=\eta_1$, and so we can assign the polynomials $p,q,r,$ and $s$ as desired.
\subsection{Bifurcations}
With our bifurcation equation in the desired form, we can now conclude the existence of the desired solutions and determine the bifurcation directions from the front solution $u_*$. The results of \cite[Ch. 17]{Golubitsky} readily give the existence of the pair of bifurcating solution branches, solving for $\Omega$ as a function of the solution amplitude. Furthermore, all that is required to determine the direction of bifurcation is $\frac{\partial p}{\partial N}(0)$ and $r(0)$, where $p$ and $r$ are the polynomials from the normal form. Straightforward calculation gives
\[
p_N(0)=\mathrm{Re}\,\left(\frac{\theta_1+\theta_2}{2}\right)=:\alpha,\qquad
r(0)=\mathrm{Re}\,\left(\frac{\theta_2-\theta_1}{2}\right)=:\beta.
\]
Using these, we can then determine the direction of bifurcation of the two branches of solutions. If $\alpha<0$ and $\beta>0$, then both solution branches bifurcate as $c$ moves below $c_*$, which we refer to as a \textbf{type 1} bifurcation. In contrast, if $\alpha>0$ and $\beta<0$, then both branches bifurcate to the right, which we will call a \textbf{type 3} bifurcation.
If $\alpha,\beta>0$, then we must consider two cases: $\alpha<\beta$ and $\alpha>\beta$. If $\alpha>\beta$, then both branches bifurcate to the right, a \textbf{type 3} bifurcation. If $\alpha<\beta$, then rotating waves, corresponding to oblique stripes, bifurcate to the left, and standing waves, corresponding to checkerboard patterns, to the right, a \textbf{type 2} bifurcation.
If $\alpha,\beta<0$, then we again must consider two cases: $\alpha<\beta$ and $\alpha>\beta$. If $\alpha<\beta$, then both branches bifurcate to the left, a \textbf{type 1} bifurcation. If $\alpha>\beta$, the standing waves bifurcate to the left, and rotating waves to the right, a \textbf{type 4} bifurcation.
\begin{figure}
\caption{Here we see examples of all the different bifurcations possible: Type 1 (left), Type 2 (second), Type 3 (third), and Type 4 (right). S represents the branch of standing waves (checkerboard patterns), and R denotes the rotating waves (oblique stripe patterns). Here $c$ is our bifurcation parameter, the quench speed, and $\|u\|$ is an $L^2$ norm.}
\label{fig:bif}
\end{figure}
\subsection{Leading Order Forms of the Solutions} Using the decomposition $v = u_0 + w$, and the fact that $w$ is higher order, we obtain the following expansion for the full patterned front solution
\[u= u_*+(aP_++\overline{a}P_-+bQ_++\overline{b}Q_-)+O(|a|^2+|b|^2).\]
Following \cite[Ch. 17, Sec. 3]{Golubitsky}, in order to find zeros of the normal form \eqref{eq:normal_form} it suffices to take the real representatives from the $O(2)\times S^1$-orbit of solutions and thus restrict to $a,b\in \mathbb{R}$.
Here rotating waves correspond to $a>b=0$, arising from the isotropy subgroup $\widetilde{SO}(2)=\{(\theta,\theta)\,\,|\,\,\theta\in S^1\}$, while standing waves correspond to $a=b>0$, arising from the isotropy subgroup $\mathbb{Z}_2\oplus\mathbb{Z}_2^c$. Using this information, we can determine the particular leading order form for both the rotating and standing waves, again which correspond to oblique and checkerboard patterns. This gives the solution forms \begin{align}
u_{os}&=u_*+a(P_++P_-)+O(|a|^2)\nonumber\\
&=u_*+2a\mathrm{Re}\,\left(e^{i(\tau+y)}p(x)\right)+O(|a|^2),\\
u_{cb}&=u_*+\left(aP_+ + aP_-+aQ_+ +aQ_-\right)+O(|a|^2)\nonumber\\
&=u_*+4a\cos(y)\mathrm{Re}\,\left(e^{i\tau}p(x)\right)+O(|a|^2). \end{align} The expansions for the bifurcation parameter in terms of the amplitude, Equations \eqref{c_os} and \eqref{c_cb}, come from \cite[Ch. 17, Sec. 3]{Golubitsky}. This concludes the proof of Theorem \ref{thm1}.
\section{Example}\label{s:ex} In this section, we lay out an explicit example of a nonlinearity and heterogeneity which give rise to such bifurcations. We will show using both rigorous and numerical evidence that these satisfy our hypotheses in Section \ref{sec3.1}, and we numerically determine the direction of the bifurcation in Section \ref{sec3.2}. In Section \ref{sec3.3} we examine what happens as we vary the transverse wavenumber $k$.
\subsection{Nonlinearity and Heterogeneity}\label{sec3.1} We fix $k=\frac{1}{2}$ and define the nonlinearity \begin{equation}\label{e:fex}
f(x-ct,u)=h(x-ct)u+\gamma u^3-u^5, \end{equation} with top-hat heterogeneity \begin{equation}\label{e:hex} h(\Tilde{x})=\tanh(\delta(\Tilde{x}-K))\tanh(-\delta(\Tilde{x}+K)), \end{equation} and $\delta\gg1$ and $K$ large. In our numerics, described below, we set $\delta=5$ and $K=10\pi$. To understand this nonlinearity, we briefly discuss free invasion fronts in the unquenched, homogeneous coefficient nonlinearity \begin{equation}\Tilde{f}(u)=u+\gamma u^3-u^5.\end{equation} In the regime $\gamma<0$, invasion fronts into the unstable state $u\equiv0$ are determined by linear information of the state ahead of the front, and are known as \emph{pulled} fronts. In the regime $\gamma>1$, fronts are governed by the strong nonlinear growth and travel faster than the linear information ahead of the front predicts, commonly referred to as \emph{pushed} fronts. An example of such behavior can be found in \cite{goh2016pattern}, and more detail can be found in \cite[Sec. 2.6]{van2003front}.
We now verify that equation \eqref{comoving_inhom} with nonlinearity \eqref{e:fex} satisfies our hypotheses. Clearly $f$ is smooth in both $x$ and $u$ and, as $x\to\pm\infty$, $f_\pm(u)=-u+\gamma u^3-u^5$ with the appropriate convergence rate. Thus Hypothesis \ref{hyp1} is satisfied.
For Hypothesis \ref{hyp2}, our primary front will be the trivial solution $u_*\equiv0$.
For the spectral assumptions, Hypotheses \ref{hyp3} - \ref{hyp6}, we first note that essential spectrum of the linearized operator \[Lv:=-\Delta_k(\Delta_kv+\partial_uf(x,u_*)v)+c_*\partial_xv\] can be calculated explicitly. We insert the ansatz $u(x,y,\tau)=e^{\lambda\tau+\nu x+i\ell y}$ into the linearized equation, $v_t = L v$ with $h \equiv -1$, to obtain the dispersion relation \begin{equation} d(\lambda,\nu;k,\ell,c)=-(\nu^2-k^2\ell^2)[(\nu^2-k^2\ell^2)-1]+c\nu-\lambda.\label{dispersion} \end{equation}
To determine the essential spectrum (on a space which is not exponentially weighted), we solve $d(\lambda,im)=0$ for $\lambda$ in terms of $m$ and all parameters, obtaining
\begin{equation}
\Sigma_{ess}=\{-(-m^2-k^2\ell^2)[(-m^2-k^2\ell^2)-1]+icm \,\,|\,\, m\in\mathbb{R}\}.\label{ess_spec}
\end{equation} Computation of such curves readily gives that the essential spectrum is contained in the left half-plane, and only touches the imaginary axis in a quadratic tangency at the origin. Furthermore, one can readily calculate that the real part of the numerical point spectrum corresponding to $\Sigma_{ess}$ also does not vary with $c$. Hence there is no resonant essential spectrum away from $\lambda = 0$ for $c$ near the Hopf point.
We study point spectrum numerically. We truncate the linear operator \[ Lv:=-\Delta_k(\Delta_kv+\partial_uf(x,u_*)v)+c_*\partial_xv, \] to a bounded computational domain $(x,y)\in [-M,M]\times [0,2\pi)$ with periodic boundary conditions, discretize it spectrally, and evaluate it using the Fast Fourier Transform. Approximate eigenvalues and eigenfunctions are then calculated using the MATLAB command `eigs'. See Figure \ref{fig:spec_at_bif} for a depiction of the numerical spectrum near the bifurcation speed. Here, as the system possesses periodic boundary conditions, the work \cite{sandstede2000absolute} implies that the point spectrum of the truncated operator accumulate onto the essential and point spectrum of the unbounded domain operator as $M\rightarrow+\infty$. We thus conjugate the operator with an exponential weight which pushes the essential spectrum $\Sigma_{ess}$ into the left half-plane, leaving only numerical eigenvalues which approximate the point spectrum of the unbounded domain operator. In the exponentially weighted case, there are no eigenvalues at 0 due to the lack of translation symmetry in $x$ and that the front $u_*$ is trivial. This gives Hypothesis \ref{hyp3}.
\begin{figure}
\caption{100 eigenvalues (dots) of the discretized and exponentially conjugated operator nearest the origin at approximately the time of bifurcation with $k=1/2$ and exponential weight $\eta=0.2$. Overlaid are the essential spectrum of the weighted operator on an unbounded domain (solid curve), and the absolute spectrum (dashed curves).}
\label{fig:spec_at_bif}
\end{figure}
In numerically solving for the spectrum of the operator $L$, we are able to also find discretized approximations of eigenfunctions. Three such eigenfunctions are shown in Figure \ref{fig:efuncs} left. The first eigenfunction corresponds to the patterns found in one dimension and comes from the most unstable branch, which reaches furthest into the right half-plane. The other two eigenfunctions, which are transversely modulated, correspond to the rightmost eigenvalues of the next most unstable branch. Varying $c$, we find the corresponding eigenvalues cross the imaginary axis generically for some unique $c_*$. Thus we can see numerically that there is an isolated pair of generic Hopf eigenvalues with algebraic and geometric multiplicity two, crossing at some non-zero speed $c_*$, indicating that Hypothesis \ref{hyp5} is satisfied; see Figure \ref{fig:efuncs} center.
\begin{figure}
\caption{(Left) Eigenfunctions: one depicting vertical stripes, and two others depicting transverse patterns. (Center) The real parts of the first 10 unstable eigenvalues varying with $c$, indicating leading order Hopf bifurcation locations. (Right) The real part of the branch points of the absolute spectrum varying with $c$. The $\ell=1$ (lower) branch is neutral at $c_*\approx 1.35$, and the $\ell=0$ (upper) branch is neutral at $c_*\approx 1.6$.}
\label{fig:efuncs}
\end{figure}
Moving back to the unbounded domain problem, with $x\in \mathbb{R}$, we note that for $K$ large, point spectrum can be located by computing the absolute spectrum \cite{rademacher2007computing} of the plateau state, where $h \equiv 1$. Indeed, the work of \cite{sandstede2000gluing} implies that all but finitely many of the point spectrum of $L$, posed on the unbounded domain, accumulate onto the absolute spectrum of the trivial state with $h\equiv1$ with rate $O(1/K^2)$ as $K\rightarrow+\infty$. Hence branch points of the absolute spectrum give leading-order predictions for the onset of instabilities.
To compute the absolute spectrum, we once again use the linear dispersion relation. For each $\ell\in\mathbb{Z}$, we seek curves $(\lambda(\gamma),\nu(\gamma)),\, \gamma\in\mathbb{R}$, which solve $d(\lambda,\nu;k,\ell,c)=d(\lambda,\nu+i\gamma;k,\ell,c)=0$; see Figure \ref{fig:spec_at_bif} for computations using Mathematica. Branch points of the absolute spectrum can then be located by evaluating solutions at $\gamma = 0$. As $c$ is decreased, we find that the branch points destabilize. In Figure \ref{fig:spec_at_bif}, the most unstable dashed line corresponds to $\ell=0$, and the next most unstable dashed line to $\ell=\pm1$. We can see from the figures that as the quench speed $c$ is decreased, the $\ell=0$ mode, corresponding to the $y$-independent mode, bifurcates first, followed by the $\ell=\pm1$ transverse modes. Plotting $c$ versus $\mathrm{Re}(\lambda(0))$ we find good agreement with the numerical eigenvalues.
Further, we can compute the speeds at which these branch points will cross the imaginary axis using the linear spreading speed calculations of \cite[Sec. 2.11]{van2003front}. Linear spreading speeds for the homogeneous system with $h\equiv 1$ can be obtained in the 2D channel by Fourier decomposing in $y$ and studying the spreading speed for each transverse modulation $e^{i\ell y},\, \ell\in\mathbb{Z}$. We obtain the following family of linearized equations \begin{equation}
\partial_{\tau}v=-(\partial_x^2-k^2\ell^2)[(\partial_x^2-k^2\ell^2)v+v], \quad \ell\in \mathbb{Z}, \end{equation}
Expanding this with $k=1/2$, \cite{van2003front} gives the $\ell = 0$ and $\ell = \pm 1$ linear spreading speeds respectively as \[ c_{*,0}=\frac{2}{3\sqrt{6}}(2+\sqrt{7})(\sqrt{7}-1)^{1/2} = 1.622..., \qquad c_{*,1}=\frac{7}{3\sqrt{3}}= 1.347... \]
We find branch points for $|\ell|>1$ lie in the open left half-plane and are bounded away from the imaginary axis for speeds $c$ near the transverse Hopf speed $c_{*,1} = 1.347...$. Since we have control of all other branch points near the transverse Hopf-speed $c_{*,1}=1.347$, we can conclude strong numerical evidence that no other resonant point spectra bifurcate at the same $c$ as the Hopf-instability. This, along with our numerical computations and the discussion on the essential spectrum above, indicates Hypothesis \ref{hyp6} also holds.
\subsection{Bifurcations}\label{sec3.2} Calculation of the bifurcation coefficients $\theta_1$ and $\theta_2$, and thus the direction of bifurcation, requires evaluation of the eigenfunctions, the corresponding adjoint eigenfunctions, as well as the evaluation of $u_*$ in the derivatives of $f$ (which is trivial in this case). Instead of expanding these coefficients theoretically, we instead investigate the direction of bifurcation numerically.
Using the transverse eigenfunctions described in Figure \ref{fig:efuncs} above as initial conditions, we use direct numerical simulation, with spectral discretization in space and a Crank-Nicholson method in time, to simulate the bifurcated nonlinear states. We then continue these solutions adiabatically, varying $c$ and letting the end-time solution of the previous $c$ value relax to a steady state for the new $c$ value before incrementing again. This is done for both $\gamma=-1$ and $\gamma=2$, the difference of which we find as being mediation between super- and subcritical bifurcations of standing and rotating waves. In doing so, we produce Figure \ref{fig:bif_diag}. As these are direct numerical solutions, only locally stable states are observed.
\begin{figure}
\caption{Numerical bifurcation diagram depicting the branching for both the pushed and pulled nonlinearities.}
\label{fig:bif_diag}
\end{figure}
For the $\gamma=-1$, or pulled, case, we note that there is bifurcation around $c=1.4$ where the $L^2$ norm of the solution $u$ begins branching off from 0. We would expect that the pushed case, here $\gamma=2$, would bifurcate from the same point, around $c=1.4$, with unstable subcritical branch bifurcating in $c>1.4$. While not observed, we expect this branch to continue up to some $c$ where it hits a fold point connecting with the large-amplitude nonlinear state observed in our numerics. We also remark that the adiabatic continuations given by Figure \ref{fig:bif_diag} indicate that the folds for checkerboard and oblique stripes occur at different speeds.
\subsection{Varying $k$}\label{sec3.3} We now wish to explore what happens as we vary the parameter $k$, which controls the vertical wavenumber of patterns. To do this, we vary $k$ between 0 and 0.9 and find the 100 eigenvalues closest to 0 for a fixed speed, $c=1.2$. We then find the most unstable (or least stable) transverse mode and plot its real part against $k$. In doing so, we obtain Figure \ref{fig:k} left.
\begin{figure}
\caption{(Left) The real parts of the transverse eigenvalues as $k$ varies in $[0,0.9]$. (Right) The spectrum with transverse wavenumber $k=94/99$ at speed $c=1.2$ with exponential weight $\eta=0.2$.}
\label{fig:k}
\end{figure}
For $c$ fixed positive, we find that as $k$ increases, and thus the vertical period decreases, the transverse mode stabilizes, indicating that no transverse patterns arise and that the transverse Hopf location happens for smaller speeds $c$. In Figure \ref{fig:k} right, we depict the numerical spectrum for $k= 94/99$, observing that the transverse branches of spectrum have moved to the left of the essential spectrum, which itself is shifted due to the exponential weight.
\section{Discussion}\label{s:disc} We have shown in Section \ref{s:ab} that, in the wake of a quench, the 2-dimensional Cahn-Hilliard equation can produce a pair of one-parameter families of time- and $y$-periodic solutions bifurcating from a given front solution under Hypotheses \ref{hyp1}-\ref{hyp6}. Our results give leading-order forms for these solution families as well as computable formulas for the bifurcation coefficients, allowing the determination of the bifurcation direction. In Section \ref{s:ex}, we have given an explicit example of this behavior, and shown numerically what happens as the vertical wavenumber $k$ increases.
There are several avenues of subsequent inquiry which could follow from our work. First of all, one naturally would wish to study how the local bifurcating branches established here continue globally in the quench speed $c$ and the vertical wavenumber $k$. Indeed in the subcritical pushed case, $\gamma >1$, considered in Section \ref{s:ex}, one would seek to locate the secondary fold bifurcation to the large amplitude nonlinear states. We expect such a location to be mediated by the interaction of the oscillatory tail of the patterned state with the quenching interface \cite{goh2016pattern}. Next it would be of interest to study pattern selection in a domain with large $y$ period (i.e., $k\rightarrow0^+$) where possibly several transverse modes can be excited. We expect the mode with the largest period to bifurcate first, but it would be interesting to see if subsequent bifurcations of higher harmonics lead to multi-mode interactions or defect nucleation.
In another direction, one could seek to establish transverse patterns where the spinodally unstable region is unbounded and bifurcating solutions are asymptotically periodic as $x\rightarrow-\infty$, for example taking a step-function like quench $h(\tilde x) = -\tanh(\delta\tilde x)$ in \eqref{e:fex}. One possible approach would be to first consider a heterogeneity of the form $h_K(\tilde x) = \tanh(\delta \tilde x) \tanh(-\delta(\tilde x + K))$, covered by our hypotheses, and take the large plateau limit $K\rightarrow+\infty$ to establish a full pattern forming front.
Next, there are several unanswered questions on stability of such fronts. Indeed, the stability of the parallel striped fronts, posed in either 1- or 2-dimensional spatial domains, has not been established. We expect a reduced stability principle \cite{kielhofer2011bifurcation} to provide a relatively straightforward approach to establishing stability in one dimension. Moving to the transversely modulated patterns studied in this work, one cannot use such reduced stability principles as the trivial state from which they bifurcate is already unstable due to the parallel striped Hopf instability. Hence we expect the transverse patterns to be unstable or metastable near the bifurcation point. Indeed, since we do observe these patterns numerically, it would be of interest to understand how initial conditions starting near the transverse patterns dynamically evolve, and how or whether they converge to another state such as the parallel striped fronts for long times. It would also be of interest to periodically extend both the parallel and transverse patterns in $y\in \mathbb{R}$ and consider stability to localized $L^2(\mathbb{R}^2)$ perturbations.
\appendix \section{Fredholm Properties}\label{app1}
In this appendix, we provide the proof of Proposition \ref{pfred}. The general approach will be to apply an abstract closed range lemma to the linear operator and its $\mathcal{X}$-adjoint to obtain that the linearization $\mathcal{L}$ is Fredholm. We then compute its index via a Fourier decomposition in $\tau$ and $y$. To begin, for $J>0$, let $\mathcal{X}(J)$ and $\mathcal{Y}(J)$ denote the spaces of functions, in $\mathcal{X}$ and $\mathcal{Y}$ respectively, which have $x$-support in the interval $[-J,J]$. Since the embedding $\mathcal{Y}(J)\hookrightarrow\mathcal{X}(J)$ is compact, we have the following:
\begin{Lemma}\label{l:a1}
There exist constants $C>0$ and $J>0$ such that the operator $\mathcal{L}$ as defined in Section 2 satisfies
\begin{equation}
\|\xi\|_\mathcal{Y}\leq C(\|\xi\|_{\mathcal{X}(J)}+\|\mathcal{L}\xi\|_\mathcal{X}),\quad \xi\in \mathcal{Y}.
\end{equation} \end{Lemma}
\begin{proof} We remark that the proof of this result follows a similar approach as \cite[Lem. 2.3]{Goh} but we include it for completeness. Throughout $C>0$ will be a changing constant, possibly dependent on the weight $\eta$, the front $u_*$, parameters $c_*,\omega_*$ and the nonlinearity $f$, but not $\xi$.
\textbf{Step 1:} We begin by proving that the estimate holds for $J=\infty$. Assume for the moment that we have the exponential weight $\eta=0$. Then
\begin{equation}
\|\mathcal{L}\xi\|_{\mathcal{X}}\geq\|(\partial_\tau+\Delta_k^2)\xi\|_{\mathcal{X}}-\|\Delta_k(\partial_uf(x,u_*)\xi)-c_*\partial_x\xi\|_{\mathcal{X}}.
\end{equation}
Since $f$ and $u_*$ are smooth, we have for all $\epsilon>0$\begin{align}
\|\Delta_k(\partial_uf(x,u_*)\xi)-c_*\partial_x\xi\|_{\mathcal{X}}&\leq C\|\xi\|_{H^2(\mathbb{R}\times\mathbb{T},X)}\nonumber\\
&\leq C\|\xi\|^{1/2}_{\mathcal{X}}\cdot\|\xi\|^{1/2}_{H^4(\mathbb{R}\times\mathbb{T},X)}\nonumber\\
&\leq C(\epsilon\|\xi\|_{H^4(\mathbb{R}\times\mathbb{T},X)}+\frac{1}{4\epsilon}\|\xi\|_{\mathcal{X}}).
\end{align}
By combining the two inequalities, we see that for $\epsilon$ sufficiently small\begin{align}
\|\mathcal{L}\xi\|_{\mathcal{X}}+\frac{C}{4\epsilon}\|\xi\|_{\mathcal{X}}&\geq\|(\partial_\tau+\Delta_k^2)\xi\|_{\mathcal{X}}-C\epsilon\|\xi\|_{H^4(\mathbb{R}\times\mathbb{T},X)}\nonumber\\
&\geq C_1\|\xi\|_{\mathcal{Y}}.
\end{align}
For $\eta>0$, one follows a similar procedure with the conjugated operator $\mathcal{L}_\eta:=e^{\eta\langle x\rangle}\mathcal{L}e^{-\eta\langle x\rangle}$. Additional terms which arise from the conjugation are small because the weight $\eta$ is small.
\textbf{Step 2:} Next, we wish to show that the estimate holds for the constant coefficient operators \begin{equation}
\mathcal{L}_{\pm}\xi=\omega_*\partial_\tau \xi+\Delta_k(\Delta_k \xi+f_\pm'(u_\pm)\xi)-c_*\partial_x \xi.
\end{equation}
Again, we must work with the conjugated operators $\mathcal{L}_{\pm,\eta}:=e^{\eta\langle x\rangle}\mathcal{L}_\pm e^{-\eta\langle x\rangle}$. If $\mathcal{L}_{\pm,\eta}\xi=h$, then by taking the Fourier transform in $x,y,$ and $\tau$ we see that
\begin{align}
&\hat{h}(i\zeta,i\chi,i\rho)=\nonumber\\
& \left[i\rho\omega_*-i(\zeta-\eta)c_*+\left((\zeta-\eta)^2+k^2\chi^2\right)^2-\left((\zeta-\eta)^2+k^2\chi^2\right)f'_\pm(u_\pm)\right]\hat{\xi}(i\zeta,i\chi,i\rho),\\
&\zeta\in\mathbb{R}, \chi,\rho\in\mathbb{Z}.\nonumber
\end{align}
By Hypothesis \ref{hyp6}, for $\eta>0$ the essential spectrum of the time-independent portion of the operator will not intersect $i\omega_*\mathbb{Z}$. Thus both equations $\mathcal{L}_{\pm,\eta}\xi=h$ are invertible, and thus so are their Fourier transforms. Using this, we get\begin{equation}
\hat{\xi}=[i\rho\omega_*-i(\zeta-\eta)c_*+((\zeta-\eta)^2+k^2\chi^2)^2-((\zeta-\eta)^2+k^2\chi^2)f'_\pm(u_\pm)]^{-1}\hat{h}.
\end{equation}
The coefficient on the right-hand side must be bounded by our assumptions, and so we have
\begin{align}
\|\hat{\xi}\|_{\mathcal{Y}}&\leq\sup_{\zeta,\chi,\rho}\left|[i\rho\omega_*-i(\zeta-\eta)c_*+((\zeta-\eta)^2+k^2\chi^2)^2-((\zeta-\eta)^2+k^2\chi^2)f'_\pm(u_\pm)]^{-1}\right|\|\hat{h}\|_{\mathcal{X}}\nonumber\\
&=C\|\widehat{\mathcal{L}_{\pm,\eta}\xi}\|_\mathcal{X}.
\end{align}
By Plancherel's Theorem, this gives us $\|\xi\|_{\mathcal{Y}}\leq C\|\mathcal{L}_\pm\xi\|_{\mathcal{X}}$.
\textbf{Step 3:} Finally, we seek to complete the proof by using the estimates previously established to perform a patching argument. For $J>1$, let $\xi^{\pm}\in\mathcal{Y}$ be such that $\xi^+(x,y)=0$ for all $x\leq J-1$ and $\xi^-(x,y)=0$ for all $x\geq 1-J$. The exponential convergence rates from Hypotheses \ref{hyp1} and \ref{hyp2} as $x\to\pm\infty$ give the following: for every $\epsilon>0$ there is some $J>0$ sufficiently large such that\begin{equation}
\|(\mathcal{L}_\pm-\mathcal{L})\xi^\pm\|_\mathcal{X}\leq\epsilon\|\xi^\pm\|_{H^2_\eta(\mathbb{R}\times\mathbb{T}_y,X)}.
\end{equation}
From this and the estimate from Step 2, we get that \begin{align}
\|\xi^\pm\|_\mathcal{Y}&\leq C\|\mathcal{L}_\pm\xi^\pm\|_\mathcal{X}\nonumber\\
&\leq C(\|(\mathcal{L}_\pm-\mathcal{L})\xi^\pm\|_\mathcal{X}+\|\mathcal{L}\xi^\pm\|_\mathcal{X})\nonumber\\
&\leq C(\epsilon\|\xi^\pm\|_{H^2_\eta(\mathbb{R}\times\mathbb{T}_y,X)}+\|\mathcal{L}\xi^\pm\|_{\mathcal{X}}). \end{align}
Choosing $\epsilon<\frac{1}{C}$, we see that
\begin{equation}\label{e:a11}
\|\xi^\pm\|_\mathcal{Y}\leq C\|\mathcal{L}\xi^\pm\|_\mathcal{X}.
\end{equation}
Next we consider an element $\xi\in\mathcal{Y}$ with $\xi=0$ for all $|x|\leq J-1$. Then, we can decompose $\xi=\xi^++\xi^-$, where
\begin{equation}
\xi^+=\left\{\begin{array}{cc}
\xi(x),&x\geq 0 \\
0,&x<0
\end{array}\right.,\xi^-=\left\{\begin{array}{cc}
0,& x\geq0 \\
\xi(x),&x<0
\end{array}\right..\nonumber
\end{equation}
Applying estimate \eqref{e:a11} and the triangle inequality, we obtain
\begin{align}
\|\xi\|_\mathcal{Y}^2&\leq\|\xi^+\|^2_\mathcal{Y}+\|\xi^-\|^2_\mathcal{Y}\nonumber\\
&\leq C(\|\mathcal{L}\xi^+\|^2_\mathcal{X}+\|\mathcal{L}\xi^-\|^2_\mathcal{X})\nonumber\\
&\leq C(\|\mathcal{L}\xi\|^2_\mathcal{X}+\|\mathcal{L}\xi\|^2_\mathcal{X})=C\|\mathcal{L}\xi\|^2_\mathcal{X}.\label{e:a12}
\end{align}
Lastly, for a general $\xi\in\mathcal{Y}$, we choose a smooth bump function $\beta$ such that $\beta=1$ when $|x|\leq J-1$ and $\beta=0$ for $|x|\geq J$. From the triangle inequality (first line), the results of Step 1 (second line), and estimate \eqref{e:a12} (second line), we see that\begin{align}
\|\xi\|_\mathcal{Y}&\leq\|\beta\xi\|_\mathcal{Y}+\|(1-\beta)\xi\|_{\mathcal{Y}}\nonumber\\
&\leq C(\|\beta\xi\|_\mathcal{X}+\|\mathcal{L}(\beta\xi)\|_\mathcal{X}+\|\mathcal{L}((1-\beta)\xi)\|_\mathcal{X})\nonumber\\
&\leq C(\|\xi\|_{\mathcal{X}(J)}+\|\mathcal{L}\xi\|_\mathcal{X}).\nonumber
\end{align} \end{proof}
We then have the following corollary: \begin{Corollary}
$\mathcal{L}$ has closed range and finite dimensional kernel. \end{Corollary}
\begin{proof}
Since the embedding $\mathcal{Y}(J)\hookrightarrow\mathcal{X}(J)$ is compact, we have that the identity operator is compact. Then, the proof follows by applying an abstract closed range lemma, such as in \cite[Ch. 6, Prop 6.7]{Taylor}. \end{proof}
We can define the $L^2$-adjoint by integration by parts, finding $\mathcal{L}^*=-\omega_*\partial_\tau+\Delta_k^2+\partial_uf(x,u_*)\Delta_k+c_*\partial_x$. Since we wish to work with exponentially weighted spaces, we must define the $L^2_\eta$-adjoint. This is done by once again working with conjugated operators posed on $L^2$. Recall the conjugated operator $\mathcal{L}_\eta$, posed on $L^2$, is given by $\mathcal{L}_\eta:=e^{\eta\langle x\rangle}\mathcal{L}e^{-\eta\langle x\rangle}$. Because of this, we have \begin{align}
\langle \mathcal{L}_\eta u,v\rangle_{L^2}&=\langle e^{\eta\langle x\rangle}\mathcal{L}e^{-\eta\langle x\rangle}u,v\rangle_{L^2}=\nonumber\\
&=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\int_{-\infty}^\infty e^{\eta\langle x\rangle}\mathcal{L}(e^{-\eta\langle x\rangle}u)\overline{v}dxdy d\tau=\nonumber\\
&=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\int_{-\infty}^\infty e^{-\eta\langle x\rangle}u\mathcal{L}^*(\bar v e^{\eta\langle x\rangle})dxdyd\tau=\nonumber\\
&=\langle u,e^{-\eta\langle x\rangle}\mathcal{L}^*e^{\eta\langle x\rangle}v\rangle_\mathcal{X}=\langle u,\mathcal{L}^*_\eta v\rangle_{\mathcal{X}} \end{align} and hence \begin{equation}\mathcal{L}^*_\eta:=e^{-\eta\langle x\rangle}\mathcal{L}^*e^{\eta\langle x\rangle}.\nonumber\end{equation}
Hence $\mathcal{L}^*$ is defined on a weighted $L^2$ space with weight $e^{-\eta\langle x\rangle}$. This conjugated operator can be run through the same estimates as in Lemma \ref{l:a1} for $\eta$ sufficiently small, and we reach the conclusion of the corollary. Thus $\mathcal{L}:\mathcal{Y}\to\mathcal{X}$ is a Fredholm operator.
To find the Fredholm index, we first form the Fourier series in $\tau$ and $y$ to get \begin{equation}
u(x,y,\tau)=\sum_{\ell_\tau,\ell_y}e^{i\ell_\tau\tau}e^{i\ell_yy}\hat{u}_{\ell_\tau,\ell_y}(x), \end{equation} where $\hat{u}_{\ell_\tau,\ell_y}\in L^2_\eta(\mathbb{R})$,
as well as the decomposition of $\mathcal{X}$ as $\bigoplus_{\ell_\tau,\ell_y}\mathcal{X}_{\ell_\tau,\ell_y}$, where $\mathcal{X}_{\ell_\tau,\ell_y}:=\{e^{i\ell_\tau\tau}e^{i\ell_yy}\hat{u}_{\ell_\tau,\ell_y}(x),\,\,|\,\hat{u}_{\ell_\tau,\ell_y}\in L^2_\eta(\mathbb{R})\}$; see \cite{AnB} for more detail. This induces a decomposition of $\mathcal{Y}$ as $\mathcal{Y}=\bigoplus_{\ell_\tau,\ell_y}\mathcal{Y}\cap\mathcal{X}_{\ell_\tau,\ell_y}=\bigoplus_{\ell_\tau,\ell_y}\mathcal{Y}_{\ell_\tau,\ell_y}$. We then define \begin{align} \mathcal{L}_{\ell_\tau,\ell_y}:\,\, \mathcal{Y}_{\ell_\tau,\ell_y}\subset\mathcal{X}_{\ell_\tau,\ell_y}&\to\mathcal{X}_{\ell_\tau,\ell_y}\nonumber\\
\hat u&\mapsto (\partial_x^2-k^2\ell_y^2)[(\partial_x^2-k^2\ell_y^2)\hat u+\partial_uf(x,u_*)\hat u]-\left(c_*\partial_x+i\omega_*\ell_\tau\right) \hat u.\nonumber \end{align}
We first organize this decomposition into three subspaces, $ \mathcal{X}_{0,0},
\bigoplus_{|\ell_\tau|=1, |\ell_y| = 1} \mathcal{X}_{\ell_\tau,\ell_y},
$
and their complement, $\mathcal{X}_h = X_{1,0}\oplus X_{0,1}\oplus X_{2,1}\oplus X_{1,2}\oplus \left( \bigoplus_{|\ell_y|,|\ell_\tau|\geq2} X_{\ell_y,\ell_\tau}\right)$, in $\mathcal{X}$. \ \begin{Lemma}\label{l:00}
$\mathrm{ind}\,\mathcal{L}_{0,0}=-1$. \end{Lemma} \begin{proof}
Recall, $\mathcal{L}_{0,0}=\partial_x^2[\partial_x^2+\partial_uf(x,u_*)]-c_*\partial_x=-L = -\partial_x\circ\Tilde{L}$, where $\Tilde{L}:=-\partial_x(\partial_x^2+\partial_uf(x,u_*))+c_*$ converges to the constant coefficient operators $\Tilde{L}_{\pm}$ as $x\rightarrow\pm\infty$ by our hypotheses. For $c>0$, each of the polynomials $\nu^3+f'_\pm(u_\pm)\nu-c=0$ has two positive roots and one negative root. Thus the difference between the number of unstable eigenvalues is zero and so $\Tilde{L}$ has Fredholm index 0. Then since $\partial_x$ has Fredholm index -1, we have that $\mathrm{ind}\,\mathcal{L}_{0,0}=-1$. \end{proof}
\begin{Lemma}\label{L:11}
For $|\ell_\tau|=|\ell_y|=1$, $\mathrm{ind}\,\mathcal{L}_{\ell_\tau,\ell_y}=0$. \end{Lemma} \begin{proof} There are four index pairs considered here, depending on the signs of $\ell_\tau$ and $\ell_y$. We only show the case $\ell_\tau=\ell_y=1$, as the other three cases follow the same reasoning. Here we have $$ \mathcal{L}_{1,1}=(\partial_x^2-k^2)[(\partial_x^2-k^2)+\partial_uf(x,u_*)]-c_*\partial_x+i\omega_*\partial_\tau = -L_{1,1} + i\omega_*\partial_\tau $$ with the spatial operator $L_{1,1}:=-(\partial_x^2-k^2)[(\partial_x^2-k^2)+\partial_uf(x,u_*)]+c_*\partial_x$. Hypothesis \ref{hyp5} implies that $L_{1,1}$ has a eigenvalue $\lambda = i \omega$ with one-dimensional eigenspace spanned by $e^{i(y+\tau)}p(x)$, and similarly for the adjoint $L_{1,1}^*$ for the eigenvalue $-i\omega_*$; see \cite[section 1.5.5]{Kato}. Thus the kernel of both $\mathcal{L}_{1,1}$ and its adjoint operator $\mathcal{L}_{1,1}^*$ is one-dimensional and hence $\mathrm{ind}\,\mathcal{L}_{1,1}=0$. \end{proof}
\begin{Lemma}
Defining $\mathcal{L}_h := \mathcal{L}\big|_{\mathcal{X}_h}$, we have $\mathrm{ind}\, \mathcal{L}_h = 0$. \end{Lemma} \begin{proof} First, we recall that for $\lambda\in i\mathbb{Z}\diagdown \{\pm i\omega_*,0\}$, $L - \lambda$ and thus $L^* - \lambda$ is invertible. This implies that for any $\ell_\tau\in \mathbb{Z}\diagdown \{0,\pm1\}$ and $\ell_y\in \mathbb{Z}$, that $\mathrm{ind}\, \mathcal{L}_{\ell_t,\ell_y} = 0$.
Next, operators $\mathcal{L}_{\ell_t,\ell_y}$ with $\ell_\tau = 0$ also have index 0 due to Hypothesis \ref{hyp3} which gives that $\lambda = 0$ is not in the extended point spectrum. Finally, operators with $\ell_\tau = \pm1$ but $|\ell_y|\neq1$ have index 0 due to Hypothesis \ref{hyp5} which gives that the Hopf eigenfunctions lie in the $\ell_y = \pm 1$ subspaces. \end{proof}
Combining the above Lemmas, we use standard Fredholm algebra results \cite{Taylor} to conclude \begin{Proposition}
$\mathrm{ind}\,\mathcal{L}=-1$ for $\mathcal{L}:\mathcal{Y}\to\mathcal{X}$. \end{Proposition}
Now we consider a closed subset defined $\mathring{\mathcal{X}}:=\{u\in\mathcal{X}|\langle u,e^{-2\eta\langle x\rangle}\rangle_\mathcal{X}=0\}$. We note that for any exponential weight $\eta>0$ and for any $u\in\mathcal{X}$ \begin{align}
\langle u,e^{-2\eta\langle x\rangle}\rangle_\mathcal{X}&=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\int_{-\infty}^\infty u(x,y,\tau)e^{-2\eta\langle x\rangle}e^{2\eta\langle x\rangle}dxdy d\tau=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\int_{-\infty}^\infty u(x,y,\tau)dxdy d\tau\nonumber \end{align}
For any $v=\mathcal{L}u, u\in\mathcal{Y}$ we have $\langle v,e^{-2\eta\langle x\rangle}\rangle_\mathcal{X}$=0 by integration by parts, and so $v\in\mathring{\mathcal{X}}$. Thus we have that $\mathcal{L}$ maps $\mathcal{Y}$ into $\mathring{\mathcal{X}}$. Finally, by composing $\mathcal{L}:\mathcal{Y}\rightarrow \mathcal{X}$ with the index 1 orthogonal projection $\mathcal{P}:\mathcal{X}\rightarrow\mathring{\mathcal{X}}$, Fredholm algebra gives that $\mathcal{L}:\mathcal{Y}\rightarrow \mathring{\mathcal{X}}$ has Fredholm index 0.
This concludes the proof of Proposition \ref{pfred}.
\section{Hopf Bifurcation with $O(2)$ symmetry}\label{app2}
The contents of this section can be found in detail in \cite[Ch. 16]{Golubitsky}. As was stated in the hypotheses, the presence of symmetry forces generic Hopf eigenvalues to have algebraic and geometric multiplicity 2. Hence an equivariant Hopf theorem is needed. In this appendix, we lay out an introduction to this theorem.
Suppose we have a bifurcation problem with $\Gamma$-symmetry, i.e., we have \[\partial_{\tau}u+G(u;c)=0\] where $G:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n$, and $G(\gamma\cdot u;c)=\gamma\cdot G(u;c)$ for $\gamma\in\Gamma$, with $c$ being our bifurcation parameter. We say that the space $\mathbb{R}^n$ is \textbf{$\Gamma$-simple} if either of the following conditions holds: (1) we can write $\mathbb{R}^n=V\oplus V$ for some subspace $V$ where linear mappings $F:V\to V$ such that $$F(\gamma\cdot v)=\gamma\cdot F(v), \forall\,\, v\in V, \gamma\in\Gamma$$ are multiples of the identity (called absolutely irreducible), or (2) $\mathbb{R}^n$ is such that the only $\Gamma$-invariant subspaces are $\{0\}$ and $\mathbb{R}^n$, but it does not meet condition (1) (called non-absolutely irreducible).
\begin{Lemma}\label{lem3}
Suppose that $\mathbb{R}^n$ is $\Gamma$-simple, that $G$ commutes with the action of $\Gamma$, and that $(dG)_{0;0}$ has $i$ as an eigenvalue. Then
(a) The eigenvalues of $(dG)_{0;c}$ consist of a complex conjugate pair $\mu(c)\pm i\kappa(c)$, each of multiplicity $m=n/2$. Moreover, $\mu$ and $\kappa$ are smooth functions of $c$.
(b) There is an invertible linear map $S:\mathbb{R}^n\to\mathbb{R}^n$, commuting with $\Gamma$, such that \[(dG)_{0;0}=SJS^{-1}\qquad \text{where}\qquad J=\begin{pmatrix}
0&-I_m\\I_m&0
\end{pmatrix}.\] \end{Lemma}
For a proof, see \cite[Ch. 16]{Golubitsky}. Lemma \ref{lem3}(a) gives that eigenvalues of multiplicity $m$ cross with nonzero speed. For $m>1$, an extension of the standard Hopf theorem is needed.
The amount of symmetry present in a solution $u$ to the system is measured by the isotropy subgroup
\[\Sigma_u=\{\sigma\in \Gamma\,|\,\sigma\cdot u=u\}.\] We also have the space of solutions fixed by the isotropy subgroups, \[\mathrm{Fix}(\Sigma_u)=\{v\in\mathbb{R}^n\,|\,\sigma\cdot v=v\,\,\forall\sigma\in\Sigma_u\}.\] Using these, we then have the following theorem:
\begin{Theorem}
Let $\Sigma_u$ be an isotropy subgroup of a group $\Gamma$ such that $\dim \mathrm{Fix}(\Sigma_u)=2$. Assume that \[(dG)_{0;0}=J\] meaning that $\Gamma$ acts absolutely irreducibly. Further assume that the eigenvalue crossing condition $\mu'(0)\neq0$ holds. Then there is a unique branch of small-amplitude periodic solutions to the bifurcation problem of period near $2\pi$ whose spatial symmetries are given by $\Sigma_u$. \end{Theorem}
For a proof, see \cite[Ch. 16]{Golubitsky}. Because this theorem has found solutions which are periodic in time, the symmetry group of the solutions is not just $\Sigma_u$, but $\Sigma_u\times S^1$, taking into account the time translation symmetry. This tells us that the full group of symmetries of the problem will be $\Gamma\times S^1$, which now not only accounts for the spatial symmetries, but also the temporal periodicity.
In our problem of the Cahn-Hilliard equation in two dimensions we note that from the reduced equations achieved by the Lyapunov-Schmidt reduction, our bifurcation equation becomes real four dimensional (considering $\Tilde{\omega}$ to be constant and taking $\Tilde{c}$ as our bifurcation parameter), mapping $\mathbb{R}^4\times\mathbb{R}\to\mathbb{R}^4$. Thus we have $n=4$, and hence $m=2$. Additionally, there are two different symmetries in the transverse spatial variable: the translation symmetry which corresponds to rotations, and the $y$-reflection symmetry, which corresponds to the standing waves. Thus we have the symmetry group $\Gamma=O(2)$ in our case, and we must consider the isotropy subgroups of $O(2)\times S^1$.
Notably, there are two maximal isotropy subgroups of $O(2)\times S^1$: one corresponding to rotations denoted $\widetilde{SO}(2)=\{(\theta,\theta)\,\,|\,\,\theta\in S^1\}$, and one corresponding to reflections denoted $\mathbb{Z}_2\oplus\mathbb{Z}_2^c$. The rotation corresponds to rotating waves, which appear as oblique stripes. The reflection corresponds to standing waves, which appear as checkerboard patterns.
\end{document} | arXiv |
# Data preprocessing: handling missing values and scaling data
Before you can perform nearest neighbor search, you need to preprocess your data. This involves handling missing values and scaling your data.
Handling missing values is crucial because missing data can lead to biased results. There are several methods to handle missing values, including:
- Deleting rows with missing values
- Imputing missing values using the mean, median, or mode
- Using regression models to predict the missing values
- Using machine learning algorithms to estimate the missing values
Scaling data is important because different features may have different scales, which can lead to one feature dominating the others in the search process. There are two common methods for scaling data:
- Min-max scaling: scales the data to a fixed range, usually [0, 1]. This is done by subtracting the minimum value and dividing by the range (max - min).
- Standardization: scales the data to have a mean of 0 and a standard deviation of 1. This is done by subtracting the mean and dividing by the standard deviation.
Here's an example of how to handle missing values and scale data using pandas and scikit-learn:
```python
import pandas as pd
from sklearn.preprocessing import MinMaxScaler, StandardScaler
# Load your data
data = pd.read_csv("your_data.csv")
# Handle missing values
data.fillna(data.mean(), inplace=True)
# Scale the data using MinMaxScaler
scaler = MinMaxScaler()
scaled_data = scaler.fit_transform(data)
# Scale the data using StandardScaler
scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)
```
## Exercise
1. Load a dataset with missing values.
2. Handle the missing values using the mean imputation method.
3. Scale the data using MinMaxScaler.
4. Scale the data using StandardScaler.
5. Compare the results of the two scaling methods.
# Distance metrics: understanding L1 and L2 norms
To perform nearest neighbor search, you need to understand distance metrics. Two common distance metrics are the L1 norm (also known as the Manhattan distance) and the L2 norm (also known as the Euclidean distance).
The L1 norm is defined as the sum of the absolute differences between the corresponding elements of two vectors:
$$\text{L1}(x, y) = \sum_{i=1}^{n} |x_i - y_i|$$
The L2 norm is defined as the square root of the sum of the squared differences between the corresponding elements of two vectors:
$$\text{L2}(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}$$
In scikit-learn, you can compute the L1 and L2 distances using the `linalg.norm` function:
```python
import numpy as np
from scipy.linalg import norm
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
# Compute the L1 distance
l1_distance = norm(x - y, ord=1)
# Compute the L2 distance
l2_distance = norm(x - y, ord=2)
```
## Exercise
1. Compute the L1 and L2 distances between two vectors.
2. Explain why L1 distance is also known as Manhattan distance.
3. Explain why L2 distance is also known as Euclidean distance.
# Feature extraction: selecting the right features for your data
Feature extraction is the process of selecting the most relevant features from your data. This is important because selecting too many features can lead to overfitting, while selecting too few features can result in underfitting.
There are several techniques for feature extraction, including:
- Filter methods: filter out irrelevant features based on their statistical properties.
- Wrapper methods: evaluate all possible subsets of features to find the best combination.
- Embedded methods: use machine learning algorithms to learn which features are important.
In scikit-learn, you can use the `SelectKBest` and `RFE` classes for feature extraction:
```python
from sklearn.feature_selection import SelectKBest, RFE
from sklearn.datasets import load_iris
from sklearn.linear_model import LinearRegression
# Load your data
data = load_iris()
# Use SelectKBest to select the top 2 features
selector = SelectKBest(k=2)
selected_data = selector.fit_transform(data.data, data.target)
# Use RFE to select the most important features
model = LinearRegression()
rfe = RFE(estimator=model, n_features_to_select=2)
selected_data = rfe.fit_transform(data.data, data.target)
```
## Exercise
1. Load a dataset with multiple features.
2. Use SelectKBest to select the top 2 features.
3. Use RFE to select the most important features.
4. Compare the results of the two feature extraction methods.
# K-Nearest Neighbors: the algorithm and its implementation in scikit-learn
K-Nearest Neighbors (KNN) is a popular algorithm for nearest neighbor search. It works by finding the k nearest neighbors of a query point and then making a prediction based on the majority class of these neighbors.
In scikit-learn, you can implement the KNN algorithm using the `KNeighborsClassifier` class:
```python
from sklearn.neighbors import KNeighborsClassifier
from sklearn.datasets import load_iris
# Load your data
data = load_iris()
# Create a KNN classifier with k=3
knn = KNeighborsClassifier(n_neighbors=3)
# Fit the classifier to the data
knn.fit(data.data, data.target)
# Make predictions on new data
new_data = np.array([[4, 5, 6, 7]])
predictions = knn.predict(new_data)
```
## Exercise
1. Load a dataset with multiple classes.
2. Create a KNN classifier with k=3.
3. Fit the classifier to the data.
4. Make predictions on new data.
5. Evaluate the accuracy of the classifier.
# Model evaluation: performance metrics and cross-validation
To evaluate the performance of your model, you need to use performance metrics and cross-validation.
Performance metrics are used to quantify the quality of your model's predictions. Some common metrics include:
- Accuracy: the proportion of correct predictions.
- Precision: the proportion of true positive predictions.
- Recall: the proportion of true positive predictions among all actual positives.
- F1 score: the harmonic mean of precision and recall.
- Area under the ROC curve: a measure of the model's ability to distinguish between classes.
Cross-validation is used to estimate the performance of your model on unseen data. There are several types of cross-validation, including:
- K-fold cross-validation: splits the data into k subsets and iteratively uses each subset as a validation set.
- Leave-one-out cross-validation: uses each data point as a validation point.
- Stratified k-fold cross-validation: ensures that each fold has a representative sample of each class.
In scikit-learn, you can use the `cross_val_score` and `cross_val_predict` functions for cross-validation:
```python
from sklearn.model_selection import cross_val_score, cross_val_predict
from sklearn.neighbors import KNeighborsClassifier
from sklearn.datasets import load_iris
# Load your data
data = load_iris()
# Create a KNN classifier with k=3
knn = KNeighborsClassifier(n_neighbors=3)
# Perform k-fold cross-validation with k=5
scores = cross_val_score(knn, data.data, data.target, cv=5)
# Perform leave-one-out cross-validation
loo_predictions = cross_val_predict(knn, data.data, data.target, cv=-1)
```
## Exercise
1. Load a dataset with multiple classes.
2. Create a KNN classifier with k=3.
3. Perform k-fold cross-validation with k=5.
4. Perform leave-one-out cross-validation.
5. Evaluate the performance of the classifier using accuracy, precision, recall, F1 score, and area under the ROC curve.
# Handling classification and regression problems
Nearest neighbor search can be used for both classification and regression problems.
For classification problems, you can use the majority class or the class with the highest probability as the prediction. For regression problems, you can use the average or the median of the k nearest neighbors as the prediction.
In scikit-learn, you can use the `KNeighborsClassifier` and `KNeighborsRegressor` classes for classification and regression problems, respectively:
```python
from sklearn.neighbors import KNeighborsClassifier, KNeighborsRegressor
from sklearn.datasets import load_iris, load_boston
# Load a classification dataset
classification_data = load_iris()
# Create a KNN classifier with k=3
knn_classifier = KNeighborsClassifier(n_neighbors=3)
# Fit the classifier to the data
knn_classifier.fit(classification_data.data, classification_data.target)
# Load a regression dataset
regression_data = load_boston()
# Create a KNN regressor with k=3
knn_regressor = KNeighborsRegressor(n_neighbors=3)
# Fit the regressor to the data
knn_regressor.fit(regression_data.data, regression_data.target)
```
## Exercise
1. Load a classification dataset.
2. Create a KNN classifier with k=3.
3. Fit the classifier to the data.
4. Load a regression dataset.
5. Create a KNN regressor with k=3.
6. Fit the regressor to the data.
# Tuning the K-Nearest Neighbors algorithm
To improve the performance of your KNN model, you can tune its hyperparameters, such as the number of neighbors k.
You can use techniques such as grid search and random search to find the best hyperparameter values. In scikit-learn, you can use the `GridSearchCV` and `RandomizedSearchCV` classes for hyperparameter tuning:
```python
from sklearn.model_selection import GridSearchCV, RandomizedSearchCV
from sklearn.neighbors import KNeighborsClassifier
from sklearn.datasets import load_iris
# Load your data
data = load_iris()
# Define the parameter grid
param_grid = {'n_neighbors': [2, 3, 4, 5]}
# Perform grid search
grid_search = GridSearchCV(KNeighborsClassifier(), param_grid, cv=5)
grid_search.fit(data.data, data.target)
# Perform random search
random_search = RandomizedSearchCV(KNeighborsClassifier(), param_grid, cv=5)
random_search.fit(data.data, data.target)
```
## Exercise
1. Load a dataset with multiple classes.
2. Define a parameter grid for the KNN classifier.
3. Perform grid search to find the best hyperparameter values.
4. Perform random search to find the best hyperparameter values.
5. Evaluate the performance of the tuned classifier.
# Applications of Nearest Neighbor search in various domains
Nearest neighbor search has been applied to various domains, including:
- Image recognition: finding similar images based on their pixel values.
- Recommender systems: finding similar users or items based on their preferences.
- Natural language processing: finding similar words or sentences based on their embeddings.
- Genomics: finding similar DNA sequences based on their nucleotide frequencies.
In each domain, the nearest neighbor search algorithm is used to find similar instances in the data, which can then be used for various tasks, such as image recognition, recommendation, or classification.
## Exercise
1. Research an application of nearest neighbor search in a domain of your choice.
2. Describe how the algorithm is used in that domain.
3. Explain the benefits and challenges of using nearest neighbor search in that domain.
# Comparing Nearest Neighbor search to other machine learning algorithms
Nearest neighbor search is a simple and intuitive algorithm, but it has some limitations, such as its sensitivity to the distance metric and its scalability to large datasets.
Other machine learning algorithms, such as decision trees, random forests, and neural networks, can often outperform nearest neighbor search in terms of accuracy and computational efficiency. However, these algorithms can be more complex to implement and may require more data preprocessing and feature engineering.
In practice, it is common to combine multiple algorithms to create ensemble models, which can improve the overall performance of the model. For example, you can use nearest neighbor search to find a small set of similar instances and then use a more complex algorithm, such as a decision tree, to make predictions based on these similar instances.
## Exercise
1. Compare the performance of nearest neighbor search to another machine learning algorithm of your choice.
2. Discuss the advantages and disadvantages of using nearest neighbor search in comparison to the other algorithm.
3. Describe a scenario where combining nearest neighbor search with another algorithm could improve the overall performance of the model.
# Real-world case studies
Nearest neighbor search has been successfully applied to various real-world problems, such as:
- Image recognition: finding similar images in a large dataset based on their pixel values.
- Recommender systems: finding similar users or items based on their preferences.
- Natural language processing: finding similar words or sentences based on their embeddings.
- Genomics: finding similar DNA sequences based on their nucleotide frequencies.
In each case, the nearest neighbor search algorithm has been used to find similar instances in the data, which can then be used for various tasks, such as image recognition, recommendation, or classification.
## Exercise
1. Research a real-world case study where nearest neighbor search was applied.
2. Describe the problem and the solution implemented using nearest neighbor search.
3. Discuss the benefits and challenges of using nearest neighbor search in that case study.
# Note: This is a summary of the textbook outline and does not contain the detailed content of the textbook. The detailed content of the textbook is written in the text blocks in the markdown format. | Textbooks |
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