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The Topology of Projective Codes and the Distribution of Zeros of Odd Maps 投影码的拓扑与奇映射的零分布
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-06-28 DOI: 10.1307/mmj/20216170
Henry Adams, Johnathan Bush, F. Frick
{"title":"The Topology of Projective Codes and the Distribution of Zeros of Odd Maps","authors":"Henry Adams, Johnathan Bush, F. Frick","doi":"10.1307/mmj/20216170","DOIUrl":"https://doi.org/10.1307/mmj/20216170","url":null,"abstract":"We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some specific parameter. Our main result is a generalization of the Borsuk--Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for the circular chromatic number of a graph; (ii) we study generic embeddings of spheres into Euclidean space and show that projective codes give quantitative bounds for a measure of genericity of sphere embeddings; and we prove generalizations of (iii) the Ham Sandwich theorem and (iv) the Lyusternik--Shnirel'man--Borsuk covering theorem for the case where the number of measures or sets in a covering, respectively, may exceed the ambient dimension.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"46 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88789500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Do the Hodge Spectra Distinguish Orbifolds from Manifolds? Part 1 霍奇光谱能区分轨道和流形吗?第1部分
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-06-15 DOI: 10.1307/mmj/20216126
Katie Gittins, Carolyn Gordon, Magda Khalile, I. M. Solis, Mary R. Sandoval, E. Stanhope
{"title":"Do the Hodge Spectra Distinguish Orbifolds from Manifolds? Part 1","authors":"Katie Gittins, Carolyn Gordon, Magda Khalile, I. M. Solis, Mary R. Sandoval, E. Stanhope","doi":"10.1307/mmj/20216126","DOIUrl":"https://doi.org/10.1307/mmj/20216126","url":null,"abstract":"We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $le 3.$","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"182 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77374887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
An Analogue of the Strengthened Hanna Neumann Conjecture for Virtually Free Groups and Virtually Free Products 虚自由群和虚自由积的强化Hanna Neumann猜想的一个模拟
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-06-10 DOI: 10.1307/mmj/20216105
A. Klyachko, A. Zakharov
{"title":"An Analogue of the Strengthened Hanna Neumann Conjecture for Virtually Free Groups and Virtually Free Products","authors":"A. Klyachko, A. Zakharov","doi":"10.1307/mmj/20216105","DOIUrl":"https://doi.org/10.1307/mmj/20216105","url":null,"abstract":"The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"95 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73625014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Khovanov Homology for Links in Thickened Multipunctured Disks 加厚多穿孔圆盘中连杆的Khovanov同调性
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-06-07 DOI: 10.1307/mmj/20216166
Zachary Winkeler
{"title":"Khovanov Homology for Links in Thickened Multipunctured Disks","authors":"Zachary Winkeler","doi":"10.1307/mmj/20216166","DOIUrl":"https://doi.org/10.1307/mmj/20216166","url":null,"abstract":"We define a variant of Khovanov homology for links in thickened disks with multiple punctures. This theory is distinct from the one previously defined by Asaeda, Przytycki, and Sikora, but is related to it by a spectral sequence. Additionally, we show that there are spectral sequences induced by embeddings of thickened surfaces, which recover the spectral sequence from annular Khovanov homology to Khovanov homology as a special case.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"76 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76270457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Self-Similar Surfaces: Involutions and Perfection 自相似曲面:对合与完美
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-06-07 DOI: 10.1307/mmj/20216114
Justin Malestein, Jing Tao
{"title":"Self-Similar Surfaces: Involutions and Perfection","authors":"Justin Malestein, Jing Tao","doi":"10.1307/mmj/20216114","DOIUrl":"https://doi.org/10.1307/mmj/20216114","url":null,"abstract":"We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"14 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74466271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Distribution mod p of Euler’s Totient and the Sum of Proper Divisors 欧拉定理的分布模p与真除数和
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-05-26 DOI: 10.1307/mmj/20216082
Noah Lebowitz-Lockard, P. Pollack, A. Roy
{"title":"Distribution mod p of Euler’s Totient and the Sum of Proper Divisors","authors":"Noah Lebowitz-Lockard, P. Pollack, A. Roy","doi":"10.1307/mmj/20216082","DOIUrl":"https://doi.org/10.1307/mmj/20216082","url":null,"abstract":"Abstract. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (with A fixed but arbitrary). We also show that the values of s(n), for n composite, are uniformly distributed among all p residue classes modulo every p ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73345891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Generic Stabilizers for Simple Algebraic Groups 简单代数群的泛型稳定器
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-05-20 DOI: 10.1307/mmj/20217216
S. Garibaldi, R. Guralnick
{"title":"Generic Stabilizers for Simple Algebraic Groups","authors":"S. Garibaldi, R. Guralnick","doi":"10.1307/mmj/20217216","DOIUrl":"https://doi.org/10.1307/mmj/20217216","url":null,"abstract":"We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of $G$ and the group $G(k)$ of $k$-points. For $G$ simple and $V$ faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those $G$ and $V$ for which the stabilizer in general position is smooth, or $dim V/G<dim G$, or there is a $v in V$ whose stabilizer in $G$ is trivial.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"74 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86354346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Weyl Sums over Integers with Digital Restrictions 带数字限制的整数上的Weyl和
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-05-11 DOI: 10.1307/mmj/20216094
I. Shparlinski, J. Thuswaldner
{"title":"Weyl Sums over Integers with Digital Restrictions","authors":"I. Shparlinski, J. Thuswaldner","doi":"10.1307/mmj/20216094","DOIUrl":"https://doi.org/10.1307/mmj/20216094","url":null,"abstract":"We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"85 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84339093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups 非圆柱形双曲群及其拟等距嵌入子群
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-05-05 DOI: 10.1307/mmj/20216112
Carolyn R. Abbott, J. Manning
{"title":"Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups","authors":"Carolyn R. Abbott, J. Manning","doi":"10.1307/mmj/20216112","DOIUrl":"https://doi.org/10.1307/mmj/20216112","url":null,"abstract":"We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"105 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80868968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Multiplicity Along Points of a Radicial Covering of a Regular Variety 沿正则变种的根覆盖点的多重性
IF 0.9 3区 数学
Michigan Mathematical Journal Pub Date : 2021-04-29 DOI: 10.1307/MMJ/20195775
Diego Sulca, O. Villamayor
{"title":"Multiplicity Along Points of a Radicial Covering of a Regular Variety","authors":"Diego Sulca, O. Villamayor","doi":"10.1307/MMJ/20195775","DOIUrl":"https://doi.org/10.1307/MMJ/20195775","url":null,"abstract":"We study the maximal multiplicity locus of a variety X over a field of characteristic p > 0 that is provided with a finite surjective radicial morphism δ : X → V , where V is regular, for example, when X ⊂ A is a hypersurface defined by an equation of the form T −f(x1, . . . , xn) = 0 and δ is the projection onto V := Spec(k[x1, . . . , xn]). The multiplicity along points of X is bounded by the degree, say d, of the field extension K(V ) ⊂ K(X). We denote by Fd(X) ⊂ X the set of points of multiplicity d. Our guiding line is the search for invariants of singularities x ∈ Fd(X) with a good behavior property under blowups X → X along regular centers included in Fd(X), which we call invariants with the pointwise inequality property. A finite radicial morphism δ : X → V as above will be expressed in terms of an O V -submodule M ⊆ OV . A blowup X → X along a regular equimultiple center included in Fd(X) induces a blowup V ′ → V along a regular center and a finite morphism δ : X → V . A notion of transform of the O V -module M ⊂ OV to an O V ′ -module M ′ ⊂ OV ′ will be defined in such a way that δ ′ : X → V ′ is the radicial morphism defined by M . Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on O V -submodules M ⊂ OV .","PeriodicalId":49820,"journal":{"name":"Michigan Mathematical Journal","volume":"39 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82553534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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