By the geometry of the 3-fold quadric, we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every g ≥ 2.
{"title":"The Rationality of the Moduli Space of Two-pointed Ineffective Spin Hyperelliptic Curves","authors":"Francesco Zucconi","doi":"10.1093/qmath/haab006","DOIUrl":"https://doi.org/10.1093/qmath/haab006","url":null,"abstract":"By the geometry of the 3-fold quadric, we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every g ≥ 2.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1329-1356"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978792","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}
A Guesmia;J E MuÑoz Rivera;M A Sepúlveda CortÉs;O Vera VillagrÁn
We study in this paper the well-posedness and stability of a linear system of a thermoelastic Cosserat medium with infinite memory, where the Cosserat medium is a continuum in which each point has the degrees of freedom of a rigid body.
{"title":"Well-Posedness and Stability of a Generalized Micropolar Thermoelastic Body With Infinite Memory","authors":"A Guesmia;J E MuÑoz Rivera;M A Sepúlveda CortÉs;O Vera VillagrÁn","doi":"10.1093/qmath/haab014","DOIUrl":"https://doi.org/10.1093/qmath/haab014","url":null,"abstract":"We study in this paper the well-posedness and stability of a linear system of a thermoelastic Cosserat medium with infinite memory, where the Cosserat medium is a continuum in which each point has the degrees of freedom of a rigid body.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1495-1515"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978798","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}
We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma's restrictions on the size of d. This answers a question raised in Elma's paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1leq dleqfrac{log p}{3loglog p}$ , we obtain a significant improvement $h_{p,d}^- leq 2(frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- ll (frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $pequiv 1mod{2d}$ and degree $m=(p-1)/d$ , where d ≥ 1 is odd.
{"title":"Second Moment Of Dirichlet L-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers","authors":"Stéphane R Louboutin;Marc Munsch","doi":"10.1093/qmath/haab010","DOIUrl":"https://doi.org/10.1093/qmath/haab010","url":null,"abstract":"We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums \u0000<tex>${mathcal{A}}(p,d)$</tex>\u0000 recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for \u0000<tex>${mathcal{A}}(p,d),$</tex>\u0000 which holds true for any odd divisor d of p − 1, thus removing E. Elma's restrictions on the size of d. This answers a question raised in Elma's paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range \u0000<tex>$1leq dleqfrac{log p}{3loglog p}$</tex>\u0000, we obtain a significant improvement \u0000<tex>$h_{p,d}^- leq 2(frac{(1+o(1))p}{24})^{m/4}$</tex>\u0000 over the trivial bound \u0000<tex>$h_{p,d}^- ll (frac{dp}{24} )^{m/4}$</tex>\u0000 on the relative class numbers of the imaginary number fields of conductor \u0000<tex>$pequiv 1mod{2d}$</tex>\u0000 and degree \u0000<tex>$m=(p-1)/d$</tex>\u0000, where d ≥ 1 is odd.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1379-1399"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978794","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}
The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.
{"title":"Pullback of the Normal Module of Ideals with Low Codimension","authors":"Cleto B Miranda-Neto","doi":"10.1093/qmath/haaa065","DOIUrl":"10.1093/qmath/haaa065","url":null,"abstract":"The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1147-1166"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49664747","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}
We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K. Pak, “Transversal harmonic theory for transversally symplectic flows”, J. Aust. Math. Soc. 84 (2008), 233–245) to general transversely symplectic foliations.
{"title":"Transversal Hard Lefschetz Theorem on Transversely Symplectic Foliations","authors":"Jesús A Álvarez López;Seoung Dal Jung","doi":"10.1093/qmath/haaa071","DOIUrl":"https://doi.org/10.1093/qmath/haaa071","url":null,"abstract":"We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K. Pak, “Transversal harmonic theory for transversally symplectic flows”, J. Aust. Math. Soc. 84 (2008), 233–245) to general transversely symplectic foliations.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1235-1251"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947645","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}
We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Renyi entropies of information theory. We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X. Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.
{"title":"The Maximum Entropy of a Metric Space","authors":"Tom Leinster;Emily Roff","doi":"10.1093/qmath/haab003","DOIUrl":"https://doi.org/10.1093/qmath/haab003","url":null,"abstract":"We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Renyi entropies of information theory. We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X. Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1271-1309"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016816/9690900/09690907.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947669","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}
Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.
{"title":"A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated with Locally Compact Groups","authors":"Anthony to-Ming Lau;Ali Ülger","doi":"10.1093/qmath/haab005","DOIUrl":"https://doi.org/10.1093/qmath/haab005","url":null,"abstract":"Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1311-1327"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947670","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}
We generalize a Theorem of Ricci and count Gaussian primes $mathfrak{p}$ with short interval restrictions on both the norm and the argument of $mathfrak{p}$ . We follow Heath-Brown's method for counting rational primes in short intervals.
{"title":"Gaussian Primes in Narrow Sectors","authors":"Joshua Stucky","doi":"10.1093/qmath/haab009","DOIUrl":"https://doi.org/10.1093/qmath/haab009","url":null,"abstract":"We generalize a Theorem of Ricci and count Gaussian primes \u0000<tex>$mathfrak{p}$</tex>\u0000 with short interval restrictions on both the norm and the argument of \u0000<tex>$mathfrak{p}$</tex>\u0000. We follow Heath-Brown's method for counting rational primes in short intervals.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1357-1377"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978793","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}
We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,mathtt{C}}:hom_mathtt{C}(x, x) to hom_mathtt{C}(unicode{x1D7D9},unicode{x1D7D9})$ for all dualizable objects x in any symmetric monoidal $infty$ -category $mathtt{C}$ . This generalizes the trace from linear algebra that assigns a scalar $operatorname{Tr}(,f,) in k$ to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace $operatorname{Tr}$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the $infty$ -categorical trace.By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f mapsto operatorname{Tr}(,f^{,n})$ . Unlike for $operatorname{Tr}$ , the relevant connected component of the moduli space is not contractible, but rather equivalent to $Bmathbb{Z}/nmathbb{Z}$ or BS 1 for n = 0. As a result, we obtain a $mathbb{Z}/nmathbb{Z}$ -action on $operatorname{Tr}(,f^{,n})$ as well as a circle action on $operatorname{Tr}(operatorname{id}_x)$ .
我们将仿迹变换定义为任意对称单轴$infty$ -范畴$mathtt{C}$中所有可对偶对象x的共轭不变映射的自然族$T_{x,mathtt{C}}:hom_mathtt{C}(x, x) to hom_mathtt{C}(unicode{x1D7D9},unicode{x1D7D9})$。这推广了将标量$operatorname{Tr}(,f,) in k$赋给有限维k向量空间的任意自同态f: V→V的线性代数迹。我们的主要定理是利用带奇异点的一维协方差假设来计算类迹变换的模空间。因此，我们证明了迹$operatorname{Tr}$可以唯一地扩展到一个类迹变换，直至一个可收缩的选择空间。这允许我们给出$infty$ -分类跟踪的几个与模型无关的特征。通过将前面提到的从自同态到自同态的类迹变换的概念限制在一定范围内，我们可以特别地恢复Toën和Vezzosi的定理。其他类似跟踪的转换的例子例如由$f mapsto operatorname{Tr}(,f^{,n})$给出。与$operatorname{Tr}$不同，模空间的相关连通分量是不可收缩的，而是等价于$Bmathbb{Z}/nmathbb{Z}$或n = 0时的BS1。结果，我们在$operatorname{Tr}(,f^{,n})$上得到一个$mathbb{Z}/nmathbb{Z}$ -作用，在$operatorname{Tr}(operatorname{id}_x)$上得到一个圆作用。
{"title":"The Space of Traces in Symmetric Monoidal Infinity Categories","authors":"Jan Steinebrunner","doi":"10.1093/qmath/haab013","DOIUrl":"https://doi.org/10.1093/qmath/haab013","url":null,"abstract":"We define a tracelike transformation to be a natural family of conjugation invariant maps \u0000<tex>$T_{x,mathtt{C}}:hom_mathtt{C}(x, x) to hom_mathtt{C}(unicode{x1D7D9},unicode{x1D7D9})$</tex>\u0000 for all dualizable objects x in any symmetric monoidal \u0000<tex>$infty$</tex>\u0000-category \u0000<tex>$mathtt{C}$</tex>\u0000. This generalizes the trace from linear algebra that assigns a scalar \u0000<tex>$operatorname{Tr}(,f,) in k$</tex>\u0000 to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace \u0000<tex>$operatorname{Tr}$</tex>\u0000 can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the \u0000<tex>$infty$</tex>\u0000-categorical trace.By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by \u0000<tex>$f mapsto operatorname{Tr}(,f^{,n})$</tex>\u0000. Unlike for \u0000<tex>$operatorname{Tr}$</tex>\u0000, the relevant connected component of the moduli space is not contractible, but rather equivalent to \u0000<tex>$Bmathbb{Z}/nmathbb{Z}$</tex>\u0000 or BS\u0000<sup>1</sup>\u0000 for n = 0. As a result, we obtain a \u0000<tex>$mathbb{Z}/nmathbb{Z}$</tex>\u0000-action on \u0000<tex>$operatorname{Tr}(,f^{,n})$</tex>\u0000 as well as a circle action on \u0000<tex>$operatorname{Tr}(operatorname{id}_x)$</tex>\u0000.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1461-1493"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978797","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}
We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results of Alon-Fischer-Newman and Lovász-Szegedy for graphs of bounded VC-dimension. We also consider the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the corresponding results for graphs in the literature. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples, in particular for graphs definable in the p-adics.
{"title":"Definable Regularity Lemmas for Nip Hypergraphs","authors":"Artem Chernikov;Sergei Starchenko","doi":"10.1093/qmath/haab011","DOIUrl":"https://doi.org/10.1093/qmath/haab011","url":null,"abstract":"We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results of Alon-Fischer-Newman and Lovász-Szegedy for graphs of bounded VC-dimension. We also consider the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the corresponding results for graphs in the literature. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples, in particular for graphs definable in the p-adics.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1401-1433"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978795","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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