Pub Date : 2022-08-20DOI: 10.4153/S0008414X22000402
Sean D. Cox, Matthew Elpers
Abstract Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of $mathrm{Homeo}^+(mathbb {R})$ ) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).
{"title":"How robustly can you predict the future?","authors":"Sean D. Cox, Matthew Elpers","doi":"10.4153/S0008414X22000402","DOIUrl":"https://doi.org/10.4153/S0008414X22000402","url":null,"abstract":"Abstract Hardin and Taylor proved that any function on the reals—even a nowhere continuous one—can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman, who provided upper and lower frontiers (in the subgroup lattice of \u0000$mathrm{Homeo}^+(mathbb {R})$\u0000 ) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder’s Theorem (that every Archimedean group is abelian).","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"22 1","pages":"1493 - 1515"},"PeriodicalIF":0.7,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79180227","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}
Pub Date : 2022-08-02DOI: 10.4153/s0008414x23000226
D. Lenz, Timo Spindeler, Nicolae Strungaru
We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.
{"title":"ABSTRACT ALMOST PERIODICITY FOR GROUP ACTIONS ON UNIFORM TOPOLOGICAL SPACES","authors":"D. Lenz, Timo Spindeler, Nicolae Strungaru","doi":"10.4153/s0008414x23000226","DOIUrl":"https://doi.org/10.4153/s0008414x23000226","url":null,"abstract":"We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"69 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76362270","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}
Pub Date : 2022-08-01DOI: 10.4153/s0008414x22000335
Antonio Lei, J. Mashreghi, K. Bringmann, Huaxin Lin, Stefan Friedl, Takahiro Kitayama, Lukas Lewark, Matthias Nagel
{"title":"CJM volume 74 issue 4 Cover and Front matter","authors":"Antonio Lei, J. Mashreghi, K. Bringmann, Huaxin Lin, Stefan Friedl, Takahiro Kitayama, Lukas Lewark, Matthias Nagel","doi":"10.4153/s0008414x22000335","DOIUrl":"https://doi.org/10.4153/s0008414x22000335","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"17 1","pages":"f1 - f3"},"PeriodicalIF":0.7,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84384237","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}
Pub Date : 2022-08-01DOI: 10.4153/s0008414x22000347
{"title":"CJM volume 74 issue 4 Cover and Back matter","authors":"","doi":"10.4153/s0008414x22000347","DOIUrl":"https://doi.org/10.4153/s0008414x22000347","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"49 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77320132","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}
Pub Date : 2022-07-19DOI: 10.4153/s0008414x22000591
Brendan Owens, Savso Strle
This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball. To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon-Litherland to the general setting. Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the $n$-ball, which generalizes previous results of Akbulut-Kirby and others.
{"title":"Disoriented homology and double branched covers","authors":"Brendan Owens, Savso Strle","doi":"10.4153/s0008414x22000591","DOIUrl":"https://doi.org/10.4153/s0008414x22000591","url":null,"abstract":"This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball. To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon-Litherland to the general setting. Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the $n$-ball, which generalizes previous results of Akbulut-Kirby and others.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82305400","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}
Pub Date : 2022-07-08DOI: 10.4153/S0008414X22000372
X. Fang, P. Tien
Abstract Let $f(z)=sum _{n=0}^infty a_n z^n$ be an entire function on the complex plane, and let ${mathcal R} f(z) = sum _{n=0}^infty a_n X_n z^n$ be its randomization induced by a standard sequence $(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions $f(z)$ such that ${mathcal R} f(z)$ is almost surely in the Fock space ${mathcal F}_{alpha }^p$ for any $p, alpha in (0,infty )$ . Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space ${mathcal F}(infty , q, alpha )$ , an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.
设$f(z)=sum _{n=0}^infty a_n z^n$为复平面上的一个完整函数，设${mathcal R} f(z) = sum _{n=0}^infty a_n X_n z^n$为由独立伯努利、斯坦豪斯或高斯随机变量的标准序列$(X_n)_n$引起的随机化。在本文中，我们描述了这些函数$f(z)$，使得${mathcal R} f(z)$对于任何$p, alpha in (0,infty )$几乎肯定在Fock空间${mathcal F}_{alpha }^p$中。然后，利用这样的表征和独立的嵌入定理，得到了一个littlewood型定理，也称为Fock空间尺度下随机化下的正则性改进。本文得到的其他结果包括:(a)混合范数空间${mathcal F}(infty , q, alpha )$ (Fock空间的端点版本)中随机解析函数的熵积分表征;(b)对Fock空间中随机空缺元素的完整描述;(c)不同Fock空间间随机乘法器的完整描述。
{"title":"Two problems on random analytic functions in Fock spaces","authors":"X. Fang, P. Tien","doi":"10.4153/S0008414X22000372","DOIUrl":"https://doi.org/10.4153/S0008414X22000372","url":null,"abstract":"Abstract Let \u0000$f(z)=sum _{n=0}^infty a_n z^n$\u0000 be an entire function on the complex plane, and let \u0000${mathcal R} f(z) = sum _{n=0}^infty a_n X_n z^n$\u0000 be its randomization induced by a standard sequence \u0000$(X_n)_n$\u0000 of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions \u0000$f(z)$\u0000 such that \u0000${mathcal R} f(z)$\u0000 is almost surely in the Fock space \u0000${mathcal F}_{alpha }^p$\u0000 for any \u0000$p, alpha in (0,infty )$\u0000 . Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space \u0000${mathcal F}(infty , q, alpha )$\u0000 , an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"97 1","pages":"1176 - 1198"},"PeriodicalIF":0.7,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77224498","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}
Pub Date : 2022-07-08DOI: 10.4153/S0008414X22000360
Theresa C. Anderson, Bingyang Hu
Abstract Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$ , not $2^d$ , grids is the optimal number in an adjacent dyadic system in $mathbb {R}^d$ . As a byproduct, we show that a collection of $d+1$ dyadic systems in $mathbb {R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $mathbb {R}$ . The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n-adic, for any n) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa.
{"title":"On the general dyadic grids on \u0000${mathbb {R}}^d$","authors":"Theresa C. Anderson, Bingyang Hu","doi":"10.4153/S0008414X22000360","DOIUrl":"https://doi.org/10.4153/S0008414X22000360","url":null,"abstract":"Abstract Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that \u0000$d+1$\u0000 , not \u0000$2^d$\u0000 , grids is the optimal number in an adjacent dyadic system in \u0000$mathbb {R}^d$\u0000 . As a byproduct, we show that a collection of \u0000$d+1$\u0000 dyadic systems in \u0000$mathbb {R}^d$\u0000 is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on \u0000$mathbb {R}$\u0000 . The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n-adic, for any n) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"22 1","pages":"1147 - 1175"},"PeriodicalIF":0.7,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82499486","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}
Pub Date : 2022-06-30DOI: 10.4153/s0008414x22000323
Sílvia Anjos, M. Barata, Martin Pinsonnault, Ana Alexandra Reis
{"title":"Loops in the fundamental group of which are not represented by circle actions","authors":"Sílvia Anjos, M. Barata, Martin Pinsonnault, Ana Alexandra Reis","doi":"10.4153/s0008414x22000323","DOIUrl":"https://doi.org/10.4153/s0008414x22000323","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"53 25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86936063","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}
Pub Date : 2022-06-27DOI: 10.4153/s0008414x22000670
Elia Fusi, Luigi Vezzoni
. We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize left- invariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution ω t which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of 11+ t ω t to the universal covering of the manifold converges in the Cheeger-Gromov sense to ( H s × C s , ˜ ω ∞ ) where ˜ ω ∞ is an algebraic soliton.
． 研究了Oeljeklaus-Toma流形上的多闭流。我们对Oeljeklaus-Toma流形上的左不变多闭度量进行了参数化，并对在泛覆盖上提升为多闭流的代数孤子的测度进行了分类。我们进一步证明了从左不变多闭度量开始的多闭流具有一个长时间解ω t，该解一旦归一化坍缩为Gromov-Hausdorff意义上的环面。此外，11+ t ω t对流形的泛覆盖的提升在Cheeger-Gromov意义下收敛于(H s × C s，≈ω∞)，其中≈ω∞是一个代数孤子。
{"title":"ON THE PLURICLOSED FLOW ON OELJEKLAUS-TOMA MANIFOLDS","authors":"Elia Fusi, Luigi Vezzoni","doi":"10.4153/s0008414x22000670","DOIUrl":"https://doi.org/10.4153/s0008414x22000670","url":null,"abstract":". We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize left- invariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution ω t which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of 11+ t ω t to the universal covering of the manifold converges in the Cheeger-Gromov sense to ( H s × C s , ˜ ω ∞ ) where ˜ ω ∞ is an algebraic soliton.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"193 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74196886","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}
Pub Date : 2022-06-24DOI: 10.4153/S0008414X22000311
Erman Isik, Yasemin Kara, Ekin Ozman
Abstract In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=mathbb {Q}(sqrt {-d})$ , where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.
{"title":"On ternary Diophantine equations of signature \u0000$(p,p,text{3})$\u0000 over number fields","authors":"Erman Isik, Yasemin Kara, Ekin Ozman","doi":"10.4153/S0008414X22000311","DOIUrl":"https://doi.org/10.4153/S0008414X22000311","url":null,"abstract":"Abstract In this paper, we prove results about solutions of the Diophantine equation \u0000$x^p+y^p=z^3$\u0000 over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation \u0000$x^p+y^p=z^3$\u0000 does not have a particular type of solution over \u0000$K=mathbb {Q}(sqrt {-d})$\u0000 , where \u0000$d=1,7,19,43,67$\u0000 whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"11 1","pages":"1293 - 1313"},"PeriodicalIF":0.7,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76436594","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}
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