Pub Date : 2022-12-20DOI: 10.4153/s0008439523000504
Maximilian Hauck, I. Shparlinski
We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.
我们使用字符和的边界和一些组合参数来显示非常光滑数的丰度，这些数也有很少的非零二进制数。
{"title":"Smooth numbers with few non-zero binary digits","authors":"Maximilian Hauck, I. Shparlinski","doi":"10.4153/s0008439523000504","DOIUrl":"https://doi.org/10.4153/s0008439523000504","url":null,"abstract":"We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48991747","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 : 2022-11-22DOI: 10.4153/S0008439522000698
O. Bogopolski, A. Ivanov
Abstract We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $mathbb {Z}$ . We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.
{"title":"Decidability problem for exponential equations in finitely presented groups","authors":"O. Bogopolski, A. Ivanov","doi":"10.4153/S0008439522000698","DOIUrl":"https://doi.org/10.4153/S0008439522000698","url":null,"abstract":"Abstract We study the following decision problem: given an exponential equation \u0000$a_1g_1^{x_1}a_2g_2^{x_2}dots a_ng_n^{x_n}=1$\u0000 over a recursively presented group G, decide if it has a solution with all \u0000$x_i$\u0000 in \u0000$mathbb {Z}$\u0000 . We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"731 - 748"},"PeriodicalIF":0.6,"publicationDate":"2022-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44524963","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 : 2022-11-15DOI: 10.4153/S0008439522000741
Ángel Chávez, S. Garcia, Jackson Hurley
Abstract We introduce a family of norms on the $n times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.
{"title":"Norms on complex matrices induced by random vectors","authors":"Ángel Chávez, S. Garcia, Jackson Hurley","doi":"10.4153/S0008439522000741","DOIUrl":"https://doi.org/10.4153/S0008439522000741","url":null,"abstract":"Abstract We introduce a family of norms on the \u0000$n times n$\u0000 complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"808 - 826"},"PeriodicalIF":0.6,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49456108","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 : 2022-11-14DOI: 10.4153/S0008439522000686
Adam H. Fuller
Abstract Słociński gave sufficient conditions for commuting isometries to have a nice Wold-like decomposition. In this note, we provide analogous results for row isometries satisfying certain commutation relations. Other than known results for doubly commuting row isometries, we provide sufficient conditions for a Wold decomposition based on the Lebesgue decomposition of the row isometries.
{"title":"Słociński–Wold decompositions for row isometries","authors":"Adam H. Fuller","doi":"10.4153/S0008439522000686","DOIUrl":"https://doi.org/10.4153/S0008439522000686","url":null,"abstract":"Abstract Słociński gave sufficient conditions for commuting isometries to have a nice Wold-like decomposition. In this note, we provide analogous results for row isometries satisfying certain commutation relations. Other than known results for doubly commuting row isometries, we provide sufficient conditions for a Wold decomposition based on the Lebesgue decomposition of the row isometries.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"780 - 790"},"PeriodicalIF":0.6,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41475756","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 : 2022-11-02DOI: 10.4153/S0008439522000662
C. Touré, R. Brits, Geethika Sebastian
Abstract We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^star $ -algebra, and if $phi :Ato mathbb C $ is a continuous function satisfying $ phi (x)phi (y) in sigma (xy) $ for all $x,yin A$ (where $sigma $ denotes the spectrum), then either $phi $ is a character of A or $-phi $ is a character of A.
摘要本文给出了经典Kowalski-Słodkowski定理的一个乘法版，该定理识别了复一元Banach代数a上的所有泛函集合中的特征。特别地，我们证明了，如果a是一个$C^star $ -代数，如果$phi :Ato mathbb C $是一个连续函数，对于所有$x,yin A$ ($sigma $表示谱)都满足$ phi (x)phi (y) in sigma (xy) $，那么要么$phi $是a的一个字符要么$-phi $是a的一个字符。
{"title":"A multiplicative Kowalski–Słodkowski theorem for \u0000$C^star $\u0000 -algebras","authors":"C. Touré, R. Brits, Geethika Sebastian","doi":"10.4153/S0008439522000662","DOIUrl":"https://doi.org/10.4153/S0008439522000662","url":null,"abstract":"Abstract We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a \u0000$C^star $\u0000 -algebra, and if \u0000$phi :Ato mathbb C $\u0000 is a continuous function satisfying \u0000$ phi (x)phi (y) in sigma (xy) $\u0000 for all \u0000$x,yin A$\u0000 (where \u0000$sigma $\u0000 denotes the spectrum), then either \u0000$phi $\u0000 is a character of A or \u0000$-phi $\u0000 is a character of A.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"951 - 958"},"PeriodicalIF":0.6,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42053749","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 : 2022-10-25DOI: 10.4153/S0008439522000650
Yuchen Ding
Abstract Let $omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Recently, M. R. Murty and V. K. Murty proved that $$ begin{align*}x(loglog x)^3llsum_{nle x}omega^*(n)^2ll xlog x.end{align*} $$ They further conjectured that there is some positive constant C such that $$ begin{align*}sum_{nle x}omega^*(n)^2sim Cxlog x,end{align*} $$ as $xrightarrow infty $ . In this short note, we give the correct order of the sum by showing that $$ begin{align*}sum_{nle x}omega^*(n)^2asymp xlog x.end{align*} $$
摘要设$omega^*（n）$是素数p的个数，使得$p-1$除n。最近，M.R.Murty和V.K.Murty证明了$$boot｛align*｝x（loglogx）^3lllsum_｛nle x｝omega^*（n xrightarrowinfty$。在这个简短的注释中，我们给出了和的正确顺序，通过显示$$ begin｛align*｝sum_｛nle x｝omega^*（n）^2 symp x log x end｛align*｝$$
{"title":"On a conjecture of M. R. Murty and V. K. Murty","authors":"Yuchen Ding","doi":"10.4153/S0008439522000650","DOIUrl":"https://doi.org/10.4153/S0008439522000650","url":null,"abstract":"Abstract Let \u0000$omega ^*(n)$\u0000 be the number of primes p such that \u0000$p-1$\u0000 divides n. Recently, M. R. Murty and V. K. Murty proved that \u0000$$ begin{align*}x(loglog x)^3llsum_{nle x}omega^*(n)^2ll xlog x.end{align*} $$\u0000 They further conjectured that there is some positive constant C such that \u0000$$ begin{align*}sum_{nle x}omega^*(n)^2sim Cxlog x,end{align*} $$\u0000 as \u0000$xrightarrow infty $\u0000 . In this short note, we give the correct order of the sum by showing that \u0000$$ begin{align*}sum_{nle x}omega^*(n)^2asymp xlog x.end{align*} $$","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"679 - 681"},"PeriodicalIF":0.6,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42280774","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 : 2022-10-19DOI: 10.4153/S000843952300005X
A. Brudnyi
Abstract The paper investigates the algebraic properties of weakly inverse-closed complex Banach function algebras generated by functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank 1 and are projective-free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure described by the classical H. Alexander theorem.
{"title":"Projective freeness and stable rank of algebras of complex-valued BV functions","authors":"A. Brudnyi","doi":"10.4153/S000843952300005X","DOIUrl":"https://doi.org/10.4153/S000843952300005X","url":null,"abstract":"Abstract The paper investigates the algebraic properties of weakly inverse-closed complex Banach function algebras generated by functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank 1 and are projective-free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure described by the classical H. Alexander theorem.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"844 - 853"},"PeriodicalIF":0.6,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43189482","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 : 2022-10-18DOI: 10.4153/S0008439522000625
Kaloyan Slavov
Abstract We improve to nearly optimal the known asymptotic and explicit bounds for the number of $mathbb {F}_q$ -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.
{"title":"Nearly sharp Lang–Weil bounds for a hypersurface","authors":"Kaloyan Slavov","doi":"10.4153/S0008439522000625","DOIUrl":"https://doi.org/10.4153/S0008439522000625","url":null,"abstract":"Abstract We improve to nearly optimal the known asymptotic and explicit bounds for the number of \u0000$mathbb {F}_q$\u0000 -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"654 - 664"},"PeriodicalIF":0.6,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42649683","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 : 2022-10-17DOI: 10.4153/S0008439522000613
J. Falcó, P. Gauthier, Myrto Manolaki, V. Nestoridis
Abstract In Gauthier, Manolaki, and Nestoridis (2021, Advances in Mathematics 381, 107649), in order to correct a false Mergelyan-type statement given in Gamelin and Garnett (1969, Transactions of the American Mathematical Society 143, 187–200) on uniform approximation on compact sets K in $mathbb C^d$ , the authors introduced a natural function algebra $A_D(K)$ which is smaller than the classical one $A(K)$ . In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set K. Finally, we introduce a notion of O-hull of K and strengthen known results.
在Gauthier, Manolaki, and Nestoridis (2021, Advances In Mathematics 381, 107649)中，为了纠正Gamelin and Garnett (1969, Transactions of American Mathematical Society 143, 187-200)关于紧集K在$mathbb C^d$上一致逼近的一个错误的Mergelyan-type陈述，作者引入了一个比经典函数代数$ a (K)$小的自然函数代数$A_D(K)$。在本文中，我们研究了这两个代数重合的情况，并将它们与包含紧集K的开集上的多项式、有理函数或全纯函数的所有似似逼近函数的类进行了比较。最后，我们引入了K的o壳的概念，并加强了已知的结果。
{"title":"Some remarks on approximation in several complex variables","authors":"J. Falcó, P. Gauthier, Myrto Manolaki, V. Nestoridis","doi":"10.4153/S0008439522000613","DOIUrl":"https://doi.org/10.4153/S0008439522000613","url":null,"abstract":"Abstract In Gauthier, Manolaki, and Nestoridis (2021, Advances in Mathematics 381, 107649), in order to correct a false Mergelyan-type statement given in Gamelin and Garnett (1969, Transactions of the American Mathematical Society 143, 187–200) on uniform approximation on compact sets K in \u0000$mathbb C^d$\u0000 , the authors introduced a natural function algebra \u0000$A_D(K)$\u0000 which is smaller than the classical one \u0000$A(K)$\u0000 . In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set K. Finally, we introduce a notion of O-hull of K and strengthen known results.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"643 - 653"},"PeriodicalIF":0.6,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44229267","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 : 2022-10-13DOI: 10.4153/s0008439523000486
Luca Gennaioli, N. Gigli
This note is motivated by recent studies by Eriksson-Bique and Soultanis about the construction of charts in general metric measure spaces. We analyze their construction and provide an alternative and simpler proof of the fact that these charts exist on sets of finite Hausdorff dimension. The observation made here offers also some simplification about the study of the relation between the reference measure and the charts in the setting of $text {RCD}$ spaces.
{"title":"A note about charts built by Eriksson-Bique and Soultanis on metric measure spaces","authors":"Luca Gennaioli, N. Gigli","doi":"10.4153/s0008439523000486","DOIUrl":"https://doi.org/10.4153/s0008439523000486","url":null,"abstract":"\u0000 This note is motivated by recent studies by Eriksson-Bique and Soultanis about the construction of charts in general metric measure spaces. We analyze their construction and provide an alternative and simpler proof of the fact that these charts exist on sets of finite Hausdorff dimension. The observation made here offers also some simplification about the study of the relation between the reference measure and the charts in the setting of \u0000 \u0000 \u0000 \u0000$text {RCD}$\u0000\u0000 \u0000 spaces.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46462070","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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