Pub Date : 2022-06-14DOI: 10.4153/S000843952200039X
Nathan Grieve
Abstract We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $mathrm {K}$ -semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.
{"title":"Vertices of the Harder and Narasimhan polygons and the laws of large numbers","authors":"Nathan Grieve","doi":"10.4153/S000843952200039X","DOIUrl":"https://doi.org/10.4153/S000843952200039X","url":null,"abstract":"Abstract We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of \u0000$mathrm {K}$\u0000 -semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"340 - 357"},"PeriodicalIF":0.6,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45132044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-13DOI: 10.4153/S0008439522000388
Masanari Kida
Abstract Let $F_{2^n}$ be the Frobenius group of degree $2^n$ and of order $2^n ( 2^n-1)$ with $n ge 4$ . We show that if $K/mathbb {Q} $ is a Galois extension whose Galois group is isomorphic to $F_{2^n}$ , then there are $dfrac {2^{n-1} +(-1)^n }{3}$ intermediate fields of $K/mathbb {Q} $ of degree $4 (2^n-1)$ such that they are not conjugate over $mathbb {Q}$ but arithmetically equivalent over $mathbb {Q}$ . We also give an explicit method to construct these arithmetically equivalent fields.
{"title":"Arithmetically equivalent fields in a Galois extension with Frobenius Galois group of 2-power degree","authors":"Masanari Kida","doi":"10.4153/S0008439522000388","DOIUrl":"https://doi.org/10.4153/S0008439522000388","url":null,"abstract":"Abstract Let \u0000$F_{2^n}$\u0000 be the Frobenius group of degree \u0000$2^n$\u0000 and of order \u0000$2^n ( 2^n-1)$\u0000 with \u0000$n ge 4$\u0000 . We show that if \u0000$K/mathbb {Q} $\u0000 is a Galois extension whose Galois group is isomorphic to \u0000$F_{2^n}$\u0000 , then there are \u0000$dfrac {2^{n-1} +(-1)^n }{3}$\u0000 intermediate fields of \u0000$K/mathbb {Q} $\u0000 of degree \u0000$4 (2^n-1)$\u0000 such that they are not conjugate over \u0000$mathbb {Q}$\u0000 but arithmetically equivalent over \u0000$mathbb {Q}$\u0000 . We also give an explicit method to construct these arithmetically equivalent fields.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"380 - 394"},"PeriodicalIF":0.6,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44014000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-06DOI: 10.4153/S0008439522000728
M. Meckes
Abstract Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.
{"title":"Magnitude and Holmes–Thompson intrinsic volumes of convex bodies","authors":"M. Meckes","doi":"10.4153/S0008439522000728","DOIUrl":"https://doi.org/10.4153/S0008439522000728","url":null,"abstract":"Abstract Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in \u0000$ell _1^n$\u0000 and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"854 - 867"},"PeriodicalIF":0.6,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44958529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-04DOI: 10.4153/S0008439523000024
N. Bergeron, K. Chan, F. Soltani, M. Zabrocki
Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ . (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ begin{align*}text{Hilb}_{R_n/I_n}(q) = sum_{k=0}^{lfloor{n/2}rfloor} f^{(n-k,k)} q^k,,end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ . (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.
{"title":"Quasisymmetric harmonics of the exterior algebra","authors":"N. Bergeron, K. Chan, F. Soltani, M. Zabrocki","doi":"10.4153/S0008439523000024","DOIUrl":"https://doi.org/10.4153/S0008439523000024","url":null,"abstract":"Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let \u0000$R_n$\u0000 denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in \u0000$R_n$\u0000 form a commutative subalgebra of \u0000$R_n$\u0000 . (2) There is a basis of the quotient of \u0000$R_n$\u0000 by the ideal \u0000$I_n$\u0000 generated by the quasisymmetric polynomials in \u0000$R_n$\u0000 that is indexed by ballot sequences. The Hilbert series of the quotient is given by \u0000$$ begin{align*}text{Hilb}_{R_n/I_n}(q) = sum_{k=0}^{lfloor{n/2}rfloor} f^{(n-k,k)} q^k,,end{align*} $$\u0000 where \u0000$f^{(n-k,k)}$\u0000 is the number of standard tableaux of shape \u0000$(n-k,k)$\u0000 . (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"997 - 1013"},"PeriodicalIF":0.6,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44232527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-03DOI: 10.4153/s0008439523000218
S. Garcia, J. Lagarias, Ethan S. Lee
We improve upon the traditional error term in the truncated Perron formula for the logarithm of an $L$-function. All our constants are explicit.
我们改进了传统的截断Perron公式中的误差项，用于计算L函数的对数。所有的常数都是显式的。
{"title":"The error term in the truncated Perron formula for the logarithm of an -function","authors":"S. Garcia, J. Lagarias, Ethan S. Lee","doi":"10.4153/s0008439523000218","DOIUrl":"https://doi.org/10.4153/s0008439523000218","url":null,"abstract":"We improve upon the traditional error term in the truncated Perron formula for the logarithm of an $L$-function. All our constants are explicit.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45387011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Justis P Ehlers, Nikhil Patel, Peter K Kaiser, Jeffrey S Heier, David M Brown, Xiangyi Meng, Jamie Reese, Leina Lunasco, Thuy K Le, Ming Hu, Sunil K Srivastava
Purpose: To evaluate the association of fluid volatility with ellipsoid zone (EZ) integrity and subretinal hyperreflective material (SHRM) volume during anti-vascular endothelial growth factor (VEGF) therapy in neovascular age-related macular degeneration (nAMD).
Methods: This study was a post hoc analysis of the OSPREY study. Retinal volatility was quantified as the standard deviation across weeks 12 to 56 for six optical coherence tomography (OCT) metrics: central subfield thickness (CST), total fluid (TF) volume, subretinal fluid (SRF) volume, intraretinal fluid (IRF) volume, macular total retinal fluid index (TRFI), and central macular TRFI. Eyes with volatility ≤ 25th or ≥ 75th percentile values were compared.
Results: Eyes with low volatility in several exudative metrics showed greater change from baseline in SHRM volume at week 12 than eyes with high volatility. During the maintenance phase (weeks 12-56), eyes exhibiting high SRF volatility demonstrated increased SHRM volume compared to eyes with low SRF volatility (P = 0.027). Eyes exhibiting high volatility in CST, TF, and SRF demonstrated less improvement in EZ total attenuation (P < 0.001, P = 0.033, and P = 0.043, respectively) than eyes with low volatility. Early exudative instability (i.e., between weeks 4-8 or weeks 8-12) in multiple parameters (i.e., CST, TF, IRF, macular TRFI, or central macular TRFI) was associated with greater volatility during the maintenance phase (P < 0.05).
Conclusions: Greater volatility in exudative OCT metrics, particularly SRF volatility, was associated with a greater increase in SHRM and less improvement in EZ integrity, suggesting that volatility is detrimental to multiple anatomic features in nAMD. Early exudative instability during the loading phase of treatment was associated with longer-term volatility in exudation.
目的：评估新生血管性年龄相关性黄斑变性（nAMD）患者在接受抗血管内皮生长因子（VEGF）治疗期间，液体挥发性与椭圆形区（EZ）完整性和视网膜下超反光物质（SHRM）体积之间的关联：本研究是对 OSPREY 研究的一项事后分析。视网膜波动性被量化为第12周到第56周的六项光学相干断层扫描（OCT）指标的标准偏差：中央子场厚度（CST）、总液（TF）体积、视网膜下液（SRF）体积、视网膜内液（IRF）体积、黄斑总视网膜液指数（TRFI）和黄斑中央TRFI。对波动率≤第25百分位值或≥第75百分位值的眼进行比较：结果：与波动率高的眼睛相比，几项渗出指标波动率低的眼睛在第12周时的SHRM体积与基线相比变化更大。在维持阶段（第 12-56 周），与 SRF 波动率低的眼睛相比，SRF 波动率高的眼睛的 SHRM 容量有所增加（P = 0.027）。与波动率低的眼睛相比，CST、TF 和 SRF 波动率高的眼睛在 EZ 总衰减方面的改善较少（P < 0.001、P = 0.033 和 P = 0.043）。多个参数（即CST、TF、IRF、黄斑TRFI或黄斑中心TRFI）的早期渗出不稳定性（即第4-8周或第8-12周之间）与维持阶段的较大波动性有关（P < 0.05）：结论：渗出性 OCT 指标的波动性越大，尤其是 SRF 的波动性越大，与 SHRM 的增加和 EZ 完整性的改善程度越低有关，这表明波动性对 nAMD 的多种解剖特征不利。在治疗的负荷阶段，早期渗出的不稳定性与渗出的长期波动性有关。
{"title":"The Association of Fluid Volatility With Subretinal Hyperreflective Material and Ellipsoid Zone Integrity in Neovascular AMD.","authors":"Justis P Ehlers, Nikhil Patel, Peter K Kaiser, Jeffrey S Heier, David M Brown, Xiangyi Meng, Jamie Reese, Leina Lunasco, Thuy K Le, Ming Hu, Sunil K Srivastava","doi":"10.1167/iovs.63.6.17","DOIUrl":"10.1167/iovs.63.6.17","url":null,"abstract":"<p><strong>Purpose: </strong>To evaluate the association of fluid volatility with ellipsoid zone (EZ) integrity and subretinal hyperreflective material (SHRM) volume during anti-vascular endothelial growth factor (VEGF) therapy in neovascular age-related macular degeneration (nAMD).</p><p><strong>Methods: </strong>This study was a post hoc analysis of the OSPREY study. Retinal volatility was quantified as the standard deviation across weeks 12 to 56 for six optical coherence tomography (OCT) metrics: central subfield thickness (CST), total fluid (TF) volume, subretinal fluid (SRF) volume, intraretinal fluid (IRF) volume, macular total retinal fluid index (TRFI), and central macular TRFI. Eyes with volatility ≤ 25th or ≥ 75th percentile values were compared.</p><p><strong>Results: </strong>Eyes with low volatility in several exudative metrics showed greater change from baseline in SHRM volume at week 12 than eyes with high volatility. During the maintenance phase (weeks 12-56), eyes exhibiting high SRF volatility demonstrated increased SHRM volume compared to eyes with low SRF volatility (P = 0.027). Eyes exhibiting high volatility in CST, TF, and SRF demonstrated less improvement in EZ total attenuation (P < 0.001, P = 0.033, and P = 0.043, respectively) than eyes with low volatility. Early exudative instability (i.e., between weeks 4-8 or weeks 8-12) in multiple parameters (i.e., CST, TF, IRF, macular TRFI, or central macular TRFI) was associated with greater volatility during the maintenance phase (P < 0.05).</p><p><strong>Conclusions: </strong>Greater volatility in exudative OCT metrics, particularly SRF volatility, was associated with a greater increase in SHRM and less improvement in EZ integrity, suggesting that volatility is detrimental to multiple anatomic features in nAMD. Early exudative instability during the loading phase of treatment was associated with longer-term volatility in exudation.</p>","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"65 1","pages":"17"},"PeriodicalIF":5.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9206498/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80029691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-23DOI: 10.4153/S000843952300019X
Egor Lappo
We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated to the graph. This generalizes the result of Taniyama for $theta$-curves.
{"title":"CONCORDANCE OF SPATIAL GRAPHS","authors":"Egor Lappo","doi":"10.4153/S000843952300019X","DOIUrl":"https://doi.org/10.4153/S000843952300019X","url":null,"abstract":"We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated to the graph. This generalizes the result of Taniyama for $theta$-curves.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48785384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-19DOI: 10.4153/S0008439522000443
Eleanor Mcspirit, K. Ono
Abstract Let $mathcal {C}_n =left [chi _{lambda }(mu )right ]_{lambda , mu }$ be the character table for $S_n,$ where the indices $lambda $ and $mu $ run over the $p(n)$ many integer partitions of $n.$ In this note, we study $Z_{ell }(n),$ the number of zero entries $chi _{lambda }(mu )$ in $mathcal {C}_n,$ where $lambda $ is an $ell $ -core partition of $n.$ For every prime $ell geq 5,$ we prove an asymptotic formula of the form $$ begin{align*}Z_{ell}(n)sim alpha_{ell}cdot sigma_{ell}(n+delta_{ell})p(n)gg_{ell} n^{frac{ell-5}{2}}e^{pisqrt{2n/3}}, end{align*} $$ where $sigma _{ell }(n)$ is a twisted Legendre symbol divisor function, $delta _{ell }:=(ell ^2-1)/24,$ and $1/alpha _{ell }>0$ is a normalization of the Dirichlet L-value $Lleft (left ( frac {cdot }{ell } right ),frac {ell -1}{2}right ).$ For primes $ell $ and $n>ell ^6/24,$ we show that $chi _{lambda }(mu )=0$ whenever $lambda $ and $mu $ are both $ell $ -cores. Furthermore, if $Z^*_{ell }(n)$ is the number of zero entries indexed by two $ell $ -cores, then, for $ell geq 5$ , we obtain the asymptotic $$ begin{align*}Z^*_{ell}(n)sim alpha_{ell}^2 cdot sigma_{ell}( n+delta_{ell})^2 gg_{ell} n^{ell-3}. end{align*} $$
{"title":"Zeros in the character tables of symmetric groups with an \u0000$ell $\u0000 -core index","authors":"Eleanor Mcspirit, K. Ono","doi":"10.4153/S0008439522000443","DOIUrl":"https://doi.org/10.4153/S0008439522000443","url":null,"abstract":"Abstract Let \u0000$mathcal {C}_n =left [chi _{lambda }(mu )right ]_{lambda , mu }$\u0000 be the character table for \u0000$S_n,$\u0000 where the indices \u0000$lambda $\u0000 and \u0000$mu $\u0000 run over the \u0000$p(n)$\u0000 many integer partitions of \u0000$n.$\u0000 In this note, we study \u0000$Z_{ell }(n),$\u0000 the number of zero entries \u0000$chi _{lambda }(mu )$\u0000 in \u0000$mathcal {C}_n,$\u0000 where \u0000$lambda $\u0000 is an \u0000$ell $\u0000 -core partition of \u0000$n.$\u0000 For every prime \u0000$ell geq 5,$\u0000 we prove an asymptotic formula of the form \u0000$$ begin{align*}Z_{ell}(n)sim alpha_{ell}cdot sigma_{ell}(n+delta_{ell})p(n)gg_{ell} n^{frac{ell-5}{2}}e^{pisqrt{2n/3}}, end{align*} $$\u0000 where \u0000$sigma _{ell }(n)$\u0000 is a twisted Legendre symbol divisor function, \u0000$delta _{ell }:=(ell ^2-1)/24,$\u0000 and \u0000$1/alpha _{ell }>0$\u0000 is a normalization of the Dirichlet L-value \u0000$Lleft (left ( frac {cdot }{ell } right ),frac {ell -1}{2}right ).$\u0000 For primes \u0000$ell $\u0000 and \u0000$n>ell ^6/24,$\u0000 we show that \u0000$chi _{lambda }(mu )=0$\u0000 whenever \u0000$lambda $\u0000 and \u0000$mu $\u0000 are both \u0000$ell $\u0000 -cores. Furthermore, if \u0000$Z^*_{ell }(n)$\u0000 is the number of zero entries indexed by two \u0000$ell $\u0000 -cores, then, for \u0000$ell geq 5$\u0000 , we obtain the asymptotic \u0000$$ begin{align*}Z^*_{ell}(n)sim alpha_{ell}^2 cdot sigma_{ell}( n+delta_{ell})^2 gg_{ell} n^{ell-3}. end{align*} $$","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"467 - 476"},"PeriodicalIF":0.6,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48127336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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