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Variance of squarefull numbers in short intervals 平方数在短间隔内的方差
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10972707
Tsz Ho Chan
{"title":"Variance of squarefull numbers in short intervals","authors":"Tsz Ho Chan","doi":"10.1215/00192082-10972707","DOIUrl":"https://doi.org/10.1215/00192082-10972707","url":null,"abstract":"In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0<theta<1/2$, almost all short intervals $(x, x + x^{1/2 + theta}]$ contain about $frac{zeta(3/2)}{2 zeta(3)} x^theta$ squarefull numbers.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"273 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Lefschetz numbers of Verdier monodromy and the motivic monodromy conjecture for toric varieties 环型变异的Verdier单态的Lefschetz数和动机单态猜想
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10950709
Jen-Chieh Hsiao
{"title":"Lefschetz numbers of Verdier monodromy and the motivic monodromy conjecture for toric varieties","authors":"Jen-Chieh Hsiao","doi":"10.1215/00192082-10950709","DOIUrl":"https://doi.org/10.1215/00192082-10950709","url":null,"abstract":"We give an expression of the Lefschetz number of iterates of the Verdier monodormy associated to an ideal on a complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. This extends the result of Denef and Loeser in the case of principal ideal on a smooth variety, and motivates a definition of motivic Milnor fiber for ideals. A discussion of the monodromy conjecture for motivic zeta functions of torus-invariant ideals on toric varieties is also included.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135604122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ore localization of amenable monoid actions and applications toward entropy—addition formulas and the bridge theorem 可服从单形动作的局部化及其在熵加公式和桥定理中的应用
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10950854
Dikran Dikranjan, Anna Giordano Bruno, Simone Virili
{"title":"Ore localization of amenable monoid actions and applications toward entropy—addition formulas and the bridge theorem","authors":"Dikran Dikranjan, Anna Giordano Bruno, Simone Virili","doi":"10.1215/00192082-10950854","DOIUrl":"https://doi.org/10.1215/00192082-10950854","url":null,"abstract":"For a left action $Soverset{lambda}{curvearrowright}X$ of a cancellative right amenable monoid $S$ on a discrete Abelian group $X$, we construct its Ore localization $Goverset{lambda^*}{curvearrowright}X^*$, where $G$ is the group of left fractions of $S$; analogously, for a right action $Koverset{rho}curvearrowleft S$ on a compact space $K$, we construct its Ore colocalization $K^*overset{rho^*}{curvearrowleft} G$. Both constructions preserve entropy, i.e., for the algebraic entropy $h_{mathrm{alg}}$ and for the topological entropy $h_{mathrm{top}}$ one has $h_{mathrm{alg}}(lambda)=h_{mathrm{alg}}(lambda^*)$ and $h_{mathrm{top}}(rho)=h_{mathrm{top}}(rho^*)$, respectively. Exploiting these constructions and the theory of quasi-tilings, we extend the Addition Theorem for $h_{mathrm{top}}$, known for right actions of countable amenable groups on compact metrizable groups, to right actions $Koverset{rho}{curvearrowleft} S$ of cancellative right amenable monoids $S$ (with no restrictions on the cardinality) on arbitrary compact groups $K$. When the compact group $K$ is Abelian, we prove that $h_{mathrm{top}}(rho)$ coincides with $h_{mathrm{alg}}(hat{rho})$, where $Soverset{hat{rho}}curvearrowright X$ is the dual left action on the discrete Pontryagin dual $X=hat{K}$, that is, a so-called Bridge Theorem. From the Addition Theorem for $h_{mathrm{top}}$ and the Bridge Theorem, we obtain an Addition Theorem for $h_{mathrm{alg}}$ for left actions $Soverset{lambda}curvearrowright X$ on discrete Abelian groups, so far known only under the hypotheses that either $X$ is torsion or $S$ is locally monotileable. The proofs substantially use the unified approach towards entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"418 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multiparameter ergodic theorems of Abelian type for power-bounded operators 幂有界算子的Abel型多参数遍历定理
IF 0.6
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10817368
T. Yoshimoto
{"title":"Multiparameter ergodic theorems of Abelian type for power-bounded operators","authors":"T. Yoshimoto","doi":"10.1215/00192082-10817368","DOIUrl":"https://doi.org/10.1215/00192082-10817368","url":null,"abstract":"","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43748757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Direct hyperfinite representations of finitely additive probabilities 有限可加概率的直接超有限表示
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10908708
Maxwell B. Stinchcombe
{"title":"Direct hyperfinite representations of finitely additive probabilities","authors":"Maxwell B. Stinchcombe","doi":"10.1215/00192082-10908708","DOIUrl":"https://doi.org/10.1215/00192082-10908708","url":null,"abstract":"Fix a standard measurable space (X,X) and an internal, hyperfinite H⊂∗X that contains each x∈X. All finitely additive probabilities on (X,X) can be represented by setting p(B) equal to the standard part of Q(∗B∩H) for some internal probability Q supported on H. From this starting point, we have the following: a decomposition of two ways in which a probability can fail to be countably additive, a nonstandard characterization of countable additivity for probabilities on complete separable metic spaces, and results on the multiplicity of hyperfinitely supported probabilities that represent a given finitely additive p, from which we have set-valued integrals with respect to products of finitely additive probabilities that respect statistical independence, and for subfields or sub-σ-fields of X, a proper ∗X-based disintegration of finitely additive probabilities as a countably additive integral over the set of finitely additive probabilities. We end with several applications for which the alternative Stone space approach to representing finitely additive probabilities is an impediment to the analyses.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135609918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A pointwise inequality for derivatives of solutions of the heat equation in bounded domains 有界区域内热方程解的导数的一个逐点不等式
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10908733
Stefan Steinerberger
{"title":"A pointwise inequality for derivatives of solutions of the heat equation in bounded domains","authors":"Stefan Steinerberger","doi":"10.1215/00192082-10908733","DOIUrl":"https://doi.org/10.1215/00192082-10908733","url":null,"abstract":"Let $u(t,x)$ be a solution of the heat equation in $mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $Omega subset mathbb{R}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-Delta phi_k = lambda_k phi_k$ with Dirichlet conditions on smooth domains $Omega subset mathbb{R}^n$.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135180997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
A note on domain monotonicity for the Neumann eigenvalues of the Laplacian 拉普拉斯算子的诺伊曼特征值的域单调性的一个注记
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10972651
Kei Funano
{"title":"A note on domain monotonicity for the Neumann eigenvalues of the Laplacian","authors":"Kei Funano","doi":"10.1215/00192082-10972651","DOIUrl":"https://doi.org/10.1215/00192082-10972651","url":null,"abstract":"Given a convex domain and its convex sub-domain we prove a variant of domain monotonicity for the Neumann eigenvalues of the Laplacian. As an application of our method we also obtain an upper bound for Neumann eigenvalues of the Laplacian of a convex domain.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"182 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Critical metrics of the volume functional with pinched curvature 具有压缩曲率的体积函数的临界度量
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10972626
H. Baltazar, C. Queiroz
{"title":"Critical metrics of the volume functional with pinched curvature","authors":"H. Baltazar, C. Queiroz","doi":"10.1215/00192082-10972626","DOIUrl":"https://doi.org/10.1215/00192082-10972626","url":null,"abstract":"","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fundamental heaps for surface ribbons and cocycle invariants 表面带状和循环不变量的基本堆
Illinois Journal of Mathematics Pub Date : 2023-01-01 DOI: 10.1215/00192082-10972597
Masahico Saito, Emanuele Zappala
{"title":"Fundamental heaps for surface ribbons and cocycle invariants","authors":"Masahico Saito, Emanuele Zappala","doi":"10.1215/00192082-10972597","DOIUrl":"https://doi.org/10.1215/00192082-10972597","url":null,"abstract":"We introduce the notion of fundamental heap for compact orientable surfaces with boundary embedded in $3$-space, which is an isotopy invariant of the embedding. It is a group, endowed with a ternary heap operation, defined using diagrams of surfaces in a form of thickened trivalent graphs called surface ribbons. We prove that the fundamental heap has a free part whose rank is given by the number of connected components of the surface. We study the behavior of the invariant under boundary connected sum, as well as addition/deletion of twisted bands, and provide formulas relating the number of generators of the fundamental heap to the Euler characteristics. We describe in detail the effect of stabilization on the fundamental heap, and determine that for each given finitely presented group there exists a surface ribbon whose fundamental heap is isomorphic to it, up to extra free factors. A relation between the fundamental heap and the Wirtinger presentation is also described. Moreover, we introduce cocycle invariants for surface ribbons using the notion of mutually distributive cohomology and heap colorings. Explicit computations of fundamental heap and cocycle invariants are presented.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135509004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
The Disney Animation Renaissance 迪士尼动画复兴
Illinois Journal of Mathematics Pub Date : 2022-11-22 DOI: 10.5622/illinois/9780252044779.001.0001
Mary E. Lescher
{"title":"The Disney Animation Renaissance","authors":"Mary E. Lescher","doi":"10.5622/illinois/9780252044779.001.0001","DOIUrl":"https://doi.org/10.5622/illinois/9780252044779.001.0001","url":null,"abstract":"The Disney Animation Renaissance examines the unique situation of an animation studio, known as Walt Disney Feature Animation Florida (WDFAF), existing within a theme-park attraction, the Magic of Disney Animation. This dual-identity facility debuted in May 1989 at the Disney-MGM Studios at the Walt Disney World Resort in Orlando, Florida. Operating during the period now known as the Disney Renaissance (1989-1999), this studio/attraction represents a microcosm of what was occurring in the animation due to the introduction of computer technology. This book utilizes contemporary journalism, subsequent academic investigation, and oral histories from the artists, administrators, and support personnel to illustrate what the dramatic shift from traditional hand-drawn animation methods to a purely digital format—the movement from mechanization to computerization. Moving through a chronological format, this book demonstrates how the history of the Florida studio is emblematic of how the computer revolutionized animation as an art form, growing from a supporting technology to a dominant position in the industry.","PeriodicalId":56298,"journal":{"name":"Illinois Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147919981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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