— 1401 articles found.

Actions of $({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})$ on Lattice Ordered Dimension Groups

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (3) 2024, pp. 105–116
Vol.46 (3) 2024
Andrew J. Dean; Sarah K. Lucky Details
(Received: 2024-09-15 )
(Received: 2024-09-15 )

Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Sarah K. Lucky , Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: sklucky@lakeheadu.ca

Abstract/Résumé:

It is shown that if \(G\) is a lattice ordered countable group, then every action of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on \(G\) arises as an inductive limit of actions of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on simplicial groups. Some parts of the argument work in greater generality, and are proved for general finite abelian groups. A template is given for proving similar results for other such groups.

On montre que si \(G\) est un groupe dénombrable treillis-ordonné, alors toute action de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur \(G\) provient d’une limite inductive d’actions de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur des groupes simpliciaux. Des parties de cet argument fonctionnent dans une généralité plus grande et sont prouvées pour des groupes abéliens finis en général. Un modèle est donné pour prouver des résultats similaires pour d’autres groupes de ce type.

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Noncommutative dynamical systems 46L35, 46L55

PDF(click to download): Actions of $({Bbb Z}/2{Bbb Z})times ({Bbb Z}/2{Bbb Z})$ on Lattice Ordered Dimension Groups

Optimal Specification Testing for the Fixed Design Spatial Regression Model

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (3) 2024, pp. 79–104
Vol.46 (3) 2024
Stephane Bouka Details
(Received: 2024-06-24 , Revised: 2024-07-19 )
(Received: 2024-06-24 , Revised: 2024-07-19 )

Stephane Bouka, Laboratoire de Probabilites, Statistique et Informatiquer Unite de Recherche en Mathematiques et Informatique, Universite des Sciences et Techniques de Masuku, BP 813 Franceville, GABON; e-mail: stephane.bouka@univ-masuku.org

Abstract/Résumé:

In this paper, we address the problem of testing the specification of the regression function in a fixed N-dimensional setting when the errors are stationary isotropic mixing random fields. We propose a test statistic that takes into account the proximity between sites, establish its asymptotic normality and prove that it achieves the minimax rate.

Dans cet article, nous abordons le problème du test de spécification de la fonction de régression dans un cadre de design fixe de dimension N, lorsque les erreurs sont des champs aléatoires mélangeant isotropiques stationnaires. Nous proposons une statistique de test qui tient compte de la proximité entre les sites, établissons sa normalité asymptotique et prouvons qu’elle atteint la vitesse minimax du test.

Keywords: Optimal rate of testing, regression model, spatial data

AMS Subject Classification: Minimax procedures, Density estimation, Hypothesis testing, Spatial processes 62C20, 62G07, 62G10, 62M30

PDF(click to download): Optimal Specification Testing for the Fixed Design Spatial Regression Model

Homogeneity Tests for Several Distributions in Hilbert Space Based on Multiple Maximum Variance Discrepancy

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (2) 2024, pp. 46–78
Vol.46 (3) 2024
Armando Sosthène Kali Balogoun; Guy Martial Nkiet Details
(Received: 2024-04-29 , Revised: 2024-05-21 )
(Received: 2024-04-29 , Revised: 2024-05-21 )

Armando Sosthène Kali Balogoun, Département de Mathématiques, Faculté des Sciences et Techniques, Université Dan Dicko Dankoulodo de Maradi, BP 465 Maradi, Niger; e-mail: armando.balogoun@uddm.edu.ne

Guy Martial Nkiet, Département de Mathématiques et Informatique, Faculté des Sciences, Université des Sciences et Techniques de Masuku, BP 813 Franceville, Gabon; e-mail: guymartial.nkiet@univ-masuku.org

Abstract/Résumé:

This paper deals with the problem of testing for the equality of \(k\) probability distributions on Hilbert spaces, with \(k\geqslant 2\). We introduce a generalization of the maximum variance discrepancy called multiple maximum variance discrepancy. A consistent estimator of this measure is proposed as test statistic, and its asymptotic distribution under the null hypothesis is derived. Since this asymptotic distribution is that of an infinite sum of random variables, we then propose another test statistic obtained from an appropriate modification of the first one, and we get its asymptotic normality both under homogeneity hypothesis and under the alternative hypothesis, so introducing a faster test for homogeneity of distributions of random variables valued into a Hilbert space. A simulation study investigating the finite sample performances of the two introduced tests and comparing them to existing ones is provided.

Cet article considère le problème de test d’égalité de \(k\) lois sur un espace de Hilbert, avec \(k\geqslant 2\). Nous introduisons une généralisation de l’écart maximal de variance appelé écart maximal de variance multiple. Un estimateur convergent de cette mesure est proposé comme statistique de test, et sa loi asymptotique sous l’hypothèse nulle est déterminée. Puisque cette loi limite est celle d’une somme infinie de variables aléatoires, nous proposons ensuite une autre statistique de test obtenue à partir d’une modification appropriée de la première statistique, et nous obtenons sa normalité asymptotique aussi bien sous l’hypothèse nulle que sous l’hypothèse alternative, introduisant ainsi un test plus rapide d’homogénéité de lois de variables aléatoires à valeurs dans un espace de Hilbert. Une étude par simulation, permettant d’apprécier les performances des deux tests proposés et de les comparer à des tests existants, est fournie.

Keywords: Asymptotic normality, functional data analysis, kernel-based conditional dependence, reproducing kernel Hilbert space

AMS Subject Classification: Asymptotic distribution theory, Hilbert spaces with reproducing kernels (= proper functional Hilbert spaces; including de Branges-Rovnyak and other structured spaces) 62E20, 46E22

PDF(click to download): Homogeneity Tests for Several Distributions in Hilbert Space Based on Multiple Maximum Variance Discrepancy

Representation of a Zero Trace Matrix as a Commutator

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (2) 2024, pp. 41–45
Vol.46 (2) 2024
Ry Cyna; Irwin S. Pressman Details
(Received: 2024-05-01 , Revised: 2024-05-22 )
(Received: 2024-05-01 , Revised: 2024-05-22 )

Ry Cyna, Department of Physics, University of Toronto, McLennan Physical Laboratories, 60 St George St., Toronto, Ontario, CANADA M5S 1A7; e-mail: ry.cyna@mail.utoronto.ca

Irwin S. Pressman, Fields Institute for Research in Mathematical Sciences, 222 College St., Toronto, Ontario,
CANADA M5S 1A7 and School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, CANADA K1S 5B6; e-mail: irwinpressman@cunet.carleton.ca

Abstract/Résumé:

A new solution to the problem of representing a zero-trace matrix as a commutator of a pair of matrices is presented. For diagonalizable matrices, the solution first consists of a Toeplitz matrix \(H\) with \(1\) on the superdiagonal, and a second matrix with cumulative sums of eigenvalues on the subdiagonal. Defective matrices use the Jordan Normal Form to add cumulative sums of the ones and zeros in the Jordan superdiagonal to the diagonal of the second matrix. We show that every matrix is a polynomial in \(H\) and its transpose.

Une nouvelle solution au problème de la représentation d’une matrice sans trace comme commutateur d’une paire de matrices est présentée. Pour les matrices diagonalisables, la solution consiste d’abord en une matrice de Toeplitz \(H\) avec \(1\) sur le superdiagonale, et une deuxième matrice avec des sommes cumulées de valeurs propres sur la sous-diagonale. Les matrices défectueuses utilisent la forme normale de Jordan pour ajouter les sommes cumulèes des uns et des zéros de la superdiagonale de Jordan à la diagonale de la deuxième matrice. Nous montrons que toute matrice est un polynôme dans \(H\) et sa transposée.

Keywords: Toeplitz matrices, Zero trace matrix, commutator, matrix exponential

AMS Subject Classification: Matrix equations and identities, Identities; free Lie (super)algebras 15A24, 17B01

PDF(click to download): Representation of a Zero Trace Matrix as a Commutator

On the Stiefel–Whitney Classes of GKM Manifolds

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (2) 2024, pp. 16–40
Vol.46 (2) 2024
Oliver Goertsches; Panagiotis Konstantis; Leopold Zoller Details
(Received: 2024-01-18 , Revised: 2024-02-28 )
(Received: 2024-01-18 , Revised: 2024-02-28 )

Oliver Goertsches, Philipps-Universitat Marburg, Fachbereich 12 Mathematik und Informatik, Hans-Meerwein-Str. 6, 35043 Marburg; e-mail: goertsch@mathematik.uni-marburg.de

Panagiotis Konstantis , University of Cologne, Department of Mathematics and Computer Science, Division of Mathematics, Weyertal 86-90, 50931 Koln; e-mail: pako@math.uni-koeln.de

Leopold Zoller , Ludwig-Maximillians-Universitat Munchen, Mathematisches Institut, Theresienstr. 39, 80333 Munchen; e-mail: zoller@math.lmu.de

Abstract/Résumé:

We show that under standard assumptions on the isotropy groups of an integer GKM manifold, the equivariant Stiefel–Whitney classes of the action are determined by the GKM graph. This is achieved via a GKM-style description of the equivariant cohomology with coefficients in a finite field \(\mathbb{Z}_{p}\) even though in this setting the restriction map to the fixed point set is not necessarily injective. This closes a gap in our argument why the GKM graph of a 6-dimensional integer GKM manifold determines its nonequivariant diffeomorphism type. We introduce combinatorial Stiefel–Whitney classes of GKM graphs and use them to derive a nontrivial obstruction to realizability of GKM graphs in dimension 8 and higher.

Nous montrons que sous des hypothèses standard surles groupes d’isotropie d’une variété GKM entière, les classes de Stiefel–Whitney équivariantes de l’action sont déterminées par le grapheGKM. Ceci est réalisé par une description de style GKM de la cohomologie équivariante avec coefficients dans un corps fini \(\mathbb{Z}_{p}\), même si dans ce cadre l’application de restriction à l’ensemble des points fixes n’est pas nécessairement injective. Cela répare une erreur dans notre preuve du fait que le graphe GKM d’une variété GKM entière de dimension 6 détermine son type de difféomorphisme non équivariant. Nous introduisons des classes combinatoires de Stiefel–Whitney pour les graphes GKM et nous les utilisons pour dériver une obstruction non triviale à la réalisabilité des graphes GKM en dimension 8 et plus.

Keywords: (equivariant) Stiefel-Whitney classes, GKM theory, equivariant cohomology, realizability of GKM graphs

AMS Subject Classification: Equivariant homology and cohomology, Equivariant algebraic topology of manifolds 55N91, 57R91

PDF(click to download): On the Stiefel--Whitney Classes of GKM Manifolds

Corrigendum to “A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup” [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112]

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (1) 2024, pp. 11–15
Vol.46 (1) 2024
Chris Bruce; Charles Starling Details
(Received: 2024-02-24 )
(Received: 2024-02-24 )

Chris Bruce, School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom; e-mail: Chris.Bruce@glasgow.ac.uk

Charles Starling, Carleton University, School of Mathematics and Statistics, 4302 Herzberg Laboratories, Ottawa, ON K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Nous corrigeons la démonstration du théorème 4.1 dans l’article [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): Corrigendum to ``A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup'' [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88--112]

Bounded Circles on a Complex Hyperbolic Space are Expressed by Trajectories on Geodesic Spheres

C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (1) 2024, pp. 1–10
Vol.46 (1) 2024
Yusei Aoki; Toshiaki Adachi Details
(Received: 2023-12-06 , Revised: 2023-12-27 )
(Received: 2023-12-06 , Revised: 2023-12-27 )

Yusei Aoki , Division of Mathematics and Mathematical Science, Nagoya Institute of Technology, Nagoya 466-8555, Japan; e-mail: yusei11291@outlook.jp

Toshiaki Adachi , Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan; e-mail: adachi@nitech.ac.jp

Abstract/Résumé:

We take a bounded circle on a complex hyperbolic space. We show that if it has complex torsion either \(\pm 1\) or \(0\) then it is expressed by a geodesic on some geodesic sphere, and show that if it has complex torsion \(\tau\) with \(0 < |\tau| < 1\) then it is uniquely expressed by a non-geodesic trajectory on a geodesic sphere up to congruency.

Nous prenous un cercle borné en l’espace hyperbolique complexe. Nous montrons que il est exprimé par une géodésique sur une sphère géodésique si sa torsion complexe est \(0\) ou \(\pm 1\), et montrons que il est uniquement exprimé par une trajectoire sur une sphère géodésique qui n’est pas une géodésique si sa torsion complexe est \(0 < |\tau| < 1\).

Keywords: Geodesic spheres, circles, complex torsions, congruent, extrinsic shapes

AMS Subject Classification: Hermitian and K_õhlerian structures, Sub-Riemannian geometry, Geodesics 53B35, 53C17, 53C22

PDF(click to download): Bounded Circles on a Complex Hyperbolic Space are Expressed by Trajectories on Geodesic Spheres

Weakly Purely Infinite C*-algebras with Topological Dimension Zero are Purely Infinite

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (4) 2023, pp. 87–91
Vol.45 (4) 2023
George A. Elliott, FRSC; Mohammad Rouzbehani Details
(Received: 2023-09-27 , Revised: 2023-10-16 )
(Received: 2023-09-27 , Revised: 2023-10-16 )

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Mohammad Rouzbehani, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran; e-mail: rouzbehani.m.math@gmail.com

Abstract/Résumé:

We show that a C*-algebra with topological dimension zero is purely infinite if it is weakly purely infinite (a question of Kirchberg and Rørdam). We give an application of this result.

On démontre qu’une C*-algèbre de dimension topologique égale à zéro est purement infinie si elle est faiblement purement infinie (une question de Kirchberg et de Rørdam). On donne une application de ce résultat.

Keywords: (weak) pure infiniteness, C*-algebra, topological dimension zero

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): Weakly Purely Infinite C*-algebras with Topological Dimension Zero are Purely Infinite

Hyperbolicity of Renormalization for Bi-cubic Circle Maps with Bounded Combinatorics

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (3) 2023, pp. 64–86
Vol.45 (3) 2023
Gabriela Estevez; Michael Yampolsky Details
(Received: 2023-05-20 , Revised: 2023-10-16 )
(Received: 2023-05-20 , Revised: 2023-10-16 )

Gabriela Estevez, Instituto de Matematica e Estatıstica, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, 24.210-201, Bloco H, Campus do Gragoata, Niteroi, Rio de Janeiro RJ, Brasil; e-mail: gestevez@id.uff.br

Michael Yampolsky, Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, Canada; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

We construct a hyperbolic attractor of renormalization of bi-cubic circle maps with bounded combinatorics, with a codimension-two stable foliation.

On construit un attracteur hyperbolique de renormalisation pour une application de cercle bicubique à combinatoire bornée, dont la foliation est de codimension deux.

Keywords: Bi-cubic circle maps, bounded type rotation number, hyperbolicity of renormalization, renormalization operator

AMS Subject Classification: Maps of the circle, Universality; renormalization 37E10, 37E20

PDF(click to download): Hyperbolicity of Renormalization for Bi-cubic Circle Maps with Bounded Combinatorics

Maximal regularity for the Neumann-Stokes problem in $H^{r/2,r}$ spaces

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (3) 2023, pp. 56–63
Vol.45 (3) 2023
Igor Kukavica; Linfeng Li; Amjad Tuffaha Details
(Received: 2022-11-05 , Revised: 2023-09-28 )
(Received: 2022-11-05 , Revised: 2023-09-28 )

Igor Kukavica, Department of Mathematics, University of Southern California, Los Angeles, CA 90089; e-mail: kukavica@usc.edu

Linfeng Li, Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095; e-mail: lli265@math.ucla.edu

Amjad Tuffaha, Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE; e-mail: atufaha@aus.edu

Abstract/Résumé:

We provide a maximal regularity theorem for the linear Stokes equation with a non-homogeneous divergence condition in a bounded domain \(\Omega \subseteq \mathbb{R}^3\) and with the Neumann boundary conditions. We prove the existence and uniqueness of solutions such that the velocity belongs to the space \(H^{(s+1)/2,s+1}((0,T) \times \Omega)\), where \(s\in [1, 1.5 )\cup (1.5, 2)\).

Nous fournissons un théorème de régularité maximale pour l’équation linéaire de Stokes avec une condition de divergence non homogène dans un domaine borné \(\Omega \subseteq \mathbb{R}^3\) et avec les conditions aux limites de Neumann. On prouve l’existence et l’unicité de solutions telles que la vitesse appartient à l’espace \(H^{(s+1)/2,s+1}((0,T) \times \Omega)\), où \(s\in [1, 1.5 )\cup (1.5, 2)\).

Keywords: Local existence, Navier-Stokes equations, maximal regularity, trace regularity

AMS Subject Classification: , , Smoothness and regularity of solutions of PDE 35-11, 35A01, 35B65

PDF(click to download): Maximal regularity for the Neumann-Stokes problem in $H^{r/2,r}$ spaces

Lebesgue Decomposition for Positive Operators Revisited

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (2) 2023, pp. 37–55
Vol.45 (2) 2023
Yoshiki Aibara; Yoshimichi Ueda Details
(Received: 2023-06-29 )
(Received: 2023-06-29 )

Yoshiki Aibara, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: y.aibara.math95@gmail.com

Yoshimichi Ueda, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: ueda@math.nagoya-u.ac.jp

Abstract/Résumé:

We explain how Pusz–Woronowicz’s notion of functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.

On montre comment la notion de calcul fonctionnel de Pusz–Woronomicz s’adapte à la théorie de décomposition de Lebesgue pour les opérateurs positifs sur un espace de Hilbert, initialement dévelopée par Ando. De cette façon on reconstruit la partie essentielle et fondamentale de cette théorie.

Keywords: Binary operation, Functional calculus, Lebesgue decomposition, Positive operator

AMS Subject Classification: None of the above; but in this section, Functional calculus, Operator means; shorted operators; etc., 28E99, 47A60, 47A64, 47B02

PDF(click to download): Lebesgue Decomposition for Positive Operators Revisited

Generalized Tracially Approximated C*-algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (2) 2023, pp. 13–36
Vol.45 (2) 2023
George A. Elliott, FRSC; Qingzhai Fan; Xiaochun Fang Details
(Received: 2023-06-12 , Revised: 2023-07-03 )
(Received: 2023-06-12 , Revised: 2023-07-03 )

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Qingzhai Fan , Department of Mathematics, Shanghai Maritime University, Shangha, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai, China 200092; e-mail: xfang@tongji.edu.cn

Abstract/Résumé:

In this paper, we introduce some classes of generalized tracial approximation C*-algebras. Consider the class of unital C*-algebras which are tracially 𝒵-absorbing (or have tracial nuclear dimension at most n, or have the property SP, or are m-almost divisible). Then A is tracially 𝒵-absorbing (respectively, has tracial nuclear dimension at most n, has the property SP, is weakly (n, m)-almost divisible) for any simple unital C*-algebra A in the corresponding class of generalized tracial approximation C*-algebras. As an application, let A be an infinite-dimensional unital simple C*-algebra, and let B be a centrally large subalgebra of A. If B is tracially 𝒵-absorbing, then A is tracially 𝒵-absorbing. This result was obtained by Archey, Buck, and Phillips in Archey et al. (2018).

On introduit la notion d’approximation traciale généralisée d’une C*-algèbre par des C*-algèbres dans une class donnée. Cette notion généralise la notion de Lin d’approximation triviale simple, et aussi la notion d’Archey et de Phillips de centralement grande sousalgèbre, deux notions qui se sont démontrées très importantes.

Keywords: Cuntz Semigroup, C∗-algebras, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Generalized Tracially Approximated C*-algebras

$D$-module Approach to Special Functions and Generating Functions

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (1) 2023, pp. 1–12
Vol.45 (1) 2023
Kam Hang Cheng; Yik Man Chiang; Avery Ching Details
(Received: 2022-08-31 , Revised: 2023-03-31 )
(Received: 2022-08-31 , Revised: 2023-03-31 )

Kam Hang Cheng, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: henry.cheng@family.ust.hk

Yik Man Chiang, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: machiang@ust.hk

Avery Ching, Department of Statistics, The University of Warwick, Coventry, CV4 7AL, United Kingdom. (Current address: Department of Mathematics, University of Dundee, Nethergate, Dundee DD1 4HN, Scotland, United Kingdom. e-mail: AChing001@dundee.ac.uk); e-mail: avery.ching@warwick.ac.uk

Abstract/Résumé:

This is a research announcement on a unifying study of generating functions of various sequences of special functions, using Bernstein’s theory of holonomic \(D\)-modules. Both new and well-known generating functions have been obtained in a systematic and algebraic way. New difference analogues of some special functions are also discovered. This announcement focuses on particular results about Hermite functions, Bessel functions and polynomials, Laguerre polynomials, and Gegenbauer polynomials.

Il s’agit d’une annonce de recherche sur une étude unificatrice des fonctions génératrices de diverses séquences de fonctions spéciales, en utilisant la théorie de Bernstein des \(D\)-modules holonomes. Des fonctions génératrices nouvelles et bien connues ont été obtenues de manière systématique et algébrique. De nouveaux analogues discrets de certaines fonctions spéciales sont également découverts. Cette annonce se concentre sur des résultats particuliers concernant les fonctions d’Hermite, les fonctions et polynômes de Bessel, les polynômes de Laguerre et les polynômes de Gegenbauer.

Keywords: D-modules, generating functions, holonomic systems of PDEs, special functions, transmutation formulae

AMS Subject Classification: Representations of entire functions by series and integrals, Monodromy; relations with differential equations and $D$-modules, Orthogonal polynomials and functions of hypergeometric type (Jacobi; Laguerre; Hermite; Askey scheme; etc.), None of the above; but in this section, Weyl theory and its generalizations, Algebraic aspects (differential-algebraic; hypertranscendence; group-theoretical), Commutators; derivations; elementary operators; etc. 30D10, 32S40, 33C45, 33E99, 34B20, 34M15, 47B47

PDF(click to download): $D$-module Approach to Special Functions and Generating Functions

A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (4) 2022, pp. 88–112
Vol.44 (4) 2022
Charles Starling Details
(Received: 2022-11-30 )
(Received: 2022-11-30 )

Charles Starling, Carleton University, School of Mathematics and Statistics. 4302 Herzberg Laboratories, Ottawa ON, K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We prove a new uniqueness theorem for the tight C*-algebras of an inverse semigroup by generalising the uniqueness theorem given for étale groupoid C*-algebras by Brown, Nagy, Reznikoff, Sims, and Williams. We use this to show that in the nuclear and Hausdorff case, a *-homomorphism from the boundary quotient C*-algebra of a right LCM monoid is injective if and only if it is injective on the subalgebra generated by the core submonoid. We also use our result to clarify the identity of the tight C*-algebra of an inverse semigroup we previously associated to a subshift and erroneously identified as the Carlsen-Matsumoto algebra.

Nous prouvons un nouveau thèoréme d’unicité pour les C*-algèbres serrées d’un semi-groupe inverse en généralisant le théorème d’unicité donné pour les C*-algèbres groupoides étales par Brown, Nagy, Reznikoff, Sims et Williams. Nous utilisons ceci pour montrer que dans le cas nucléaire et de Hausdorff, un *-homomorphisme de l’algèbre C* du quotient aux limites d’un monoïde LCM droit est injectif si et seulement s’il est injectif sur la sous-algèbre générée par le sous-monoide de noyau. Nous utilisons également notre résultat pour clarifier l’identité de l’algèbre C* serrée d’un semi-groupe inverse que nous avons précédemment associé à un sous-décalage et identifié à tort comme l’algèbre de Carlsen-Matsumoto.

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup

On the Parabolic Gluing Method and Singularity Formation

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (4) 2022, pp. 69–87
Vol.44 (4) 2022
Juncheng Wei, FRSC; Qidi Zhang; Yifu Zhou Details
(Received: 2022-11-22 )
(Received: 2022-11-22 )

Juncheng Wei, FRSC, Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada; e-mail: jcwei@math.ubc.ca

Qidi Zhang, Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada; e-mail: qidi@math.ubc.ca

Yifu Zhou, Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA; e-mail: yzhou173@jhu.edu

Abstract/Résumé:

Singularity formation for evolution equations has attracted much
attention in recent years. In this survey article, we will introduce
some recent progress on the parabolic gluing
method
and its applications in investigating the mechanism
of singularity formation for parabolic flows. Two model problems will be
revisited to illustrate the ideas, and recent developments and
techniques will be presented.

La formation de singularités pour les équations d’évolution a attiré
beaucoup d’attention ces dernières années. Dans cet article d’enquête,
nous présenterons quelques progrès récents sur la méthode de
collage parabolique
et ses applications dans l’étude du
mécanisme de formation de singularités pour les écoulements
paraboliques. Deux problèmes modèles seront revisités pour illustrer les
idées, et les développements et techniques récents seront présentés.

Keywords: Blow-up, Fujita equation, Sobolev critical exponent, parabolic gluing method, slow decay

AMS Subject Classification: Asymptotic behavior of solutions, 35B40, 35K58

PDF(click to download): On the Parabolic Gluing Method and Singularity Formation

Cops and Robber Game in Higher-dimensional Manifolds with Spherical and Euclidean Metric

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (3) 2022, pp. 50–68
Vol.44 (3) 2022
Vesna Iršič; Bojan Mohar, FRSC; Alexandra Wesolek Details
(Received: 2022-11-04 )
(Received: 2022-11-04 )

Vesna Iršič, Faculty of Mathematics and Physics, University of Ljubljana and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia; e-mail: vesna.irsic@fmf.uni-lj.si

Bojan Mohar, FRSC, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada; e-mail: mohar@sfu.ca

Alexandra Wesolek, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada; e-mail: agwesole@sfu.ca

Abstract/Résumé:

A recently introduced variation of the game of cops and robber is played on geodesic spaces. In this paper we establish some general strategies for the players, in particular the generalized radial strategy and the covering space strategy. Those strategies are then applied to the game on the \(n\)-dimensional ball, the sphere, and the torus.

Le jeu aux gendarmes et aux voleurs sur les espaces géodésiques est analysé. On établit quelques stratégies générales pour les joueurs, en particulier la stratégie radiale généralisée et la stratégie d’espaces de couverture universels. Ces stratégies sont ensuite appliquées aux jeux sur la boule de dimension \(n\), sur la sphère, et sur le tore.

Keywords: Game of cops and robber, geodesic space, radial strategy

AMS Subject Classification: Metric spaces; metrizability, Differential games, Discrete-time games 54E35, 91A23, 91A50

PDF(click to download): Cops and Robber Game in Higher-dimensional Manifolds with Spherical and Euclidean Metric

Interpolation Polynomials and Linear Algebra

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (2) 2022, pp. 33-49
Vol.44 (2) 2022
Askold Khovanskii, FRSC; Sushil Singla; Aaron Tronsgard Details
(Received: 2022-03-11 , Revised: 2022-04-05 )
(Received: 2022-03-11 , Revised: 2022-04-05 )

Askold Khovanskii, FRSC, University of Toronto, Toronto, Canada; e-mail: askold@math.toronto.edu

Sushil Singla, Department of Mathematics, Shiv Nadar University, Greater Noida, India 201314; e-mail: ss774@snu.edu.in

Aaron Tronsgard, University of Toronto, Toronto, Canada; e-mail: tronsgar@math.utoronto.ca

Abstract/Résumé:

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. In particular, we show that one can evaluate a meromorphic function at a matrix, using only an interpolation polynomial.

On reconsidère la thèorie des polynômes d’interpolation de Lagrange et l’applique à l’algèbre linéaire. En particulier, on peut évaluer une fonction méromorphe à une matrice seulement avec un polynôme d’interpolation.

Keywords: Cayley Hamilton theorem, Interpolation polynomials, meromorphic function at a matrix

AMS Subject Classification: Instructional exposition (textbooks; tutorial papers; etc.), , Canonical forms; reductions; classification, Interpolation 15-01, 15A16, 15A21, 41A05

PDF(click to download): Interpolation Polynomials and Linear Algebra

On Duistermaat-Heckman Measure for Filtered Linear Series

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (1) 2022, pp. 16-32
Vol.44 (1) 2022
Nathan Grieve Details
(Received: 2021-09-07 )
(Received: 2021-09-07 )

Nathan Grieve, Department of Mathematics & Computer Science, Royal Military College of Canada, P.O. Box 17000, Station Forces, Kingston, ON, K7K 7B4, Canada; School of Mathematics and Statistics, 4302 Herzberg Laboratories, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada; Departement de mathematiques, Universite du Quebec a Montreal, Local PK-5151, 201 Avenue du President-Kennedy, Montreal, QC, H2X 3Y7, Canada; e-mail: nathan.m.grieve@gmail.com

Abstract/Résumé:

We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. 18 (2009), no. 2, 279–308); Boucksom and H. Chen (Compos. Math. 147 (2011), no. 4, 1205–1229); and S. Boucksom, A. Küronya, C. Maclean, and T. Szemberg (Math. Ann. 361 (2015), no. 3–4, 811–834). The key point is to associate a Duistermaat-Heckman measure to a filtered big linear series on a given projective variety. The expectation of the measure admits a description via the theory of Newton-Okounkov bodies. Such considerations have origins in symplectic geometry. They have applications for \(\mathrm{K}\)-stability and Diophantine arithmetic geometry of projective varieties.

Nous revisitons les travaux de S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. 18 (2009), no. 2, 279–308); Boucksom et H. Chen (Compos. Math. 147 (2011), no. 4, 1205–1229); et S. Boucksom, A. Küronya, C. Maclean, et T. Szemberg (Math. Ann. 361 (2015), no. 3–4, 811–834). Nous étudions deux résultats, qui sont à l’intersection de la \(\mathrm{K}\)-stabilité, de la géométrie arithmétique Diophantienne, et de la théorie des corps convexe de Newton-Okounkov. Ils se rapportent à des filtrations de grands systèmes linéaires sur des variétés projectives et à l’existence de une mesure de Duistermaat-Heckman. La mesure de Duistermaat-Heckman vient de la géométrie symplectique. L’espérance de la mesure peut être calculée par la théorie de Newton-Okounkov.

Keywords: Duistermaat-Heckman measure, Newton-Okounkov body, restricted volume functions

AMS Subject Classification: Valuations and their generalizations, Divisors; linear systems; invertible sheaves 13A18, 14C20

PDF(click to download): On Duistermaat-Heckman Measure for Filtered Linear Series

K-Theory and Traces

C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (1) 2022, pp. 1-15
Vol.44 (1) 2022
George A. Elliott Details
(Received: 2021-09-16 , Revised: 2021-12-19 )
(Received: 2021-09-16 , Revised: 2021-12-19 )

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that for a unital C*-algebra, what is sometimes referred to as the Elliott invariant—loosely speaking, K-theory and traces— i.e., the order-unit K\(_0\)-group, the K\(_1\)-group, and the trace simplex, paired in the natural way with K\(_0\), can be expressed purely in terms of K-theory, with the trace simplex and its pairing with K\(_0\) recoverable in a simple way (using polar decomposition) from algebraic K\(_1\), defined as in the purely algebraic context using invertible elements rather than just unitaries.

L’invariant naïf d’Elliott, qui est à la base de la classification complète récente d’une énorme classe de C*-algèbres simples (celles qui sont de dimension nucléaire finie, qui sont séparables, et qui satisfont à l’UCT), peut s’exprimer entièrement dans le cadre de K-théorie algébrique.

Keywords: Algebraic K1-group of a C*-algebra encodes bounded traces

AMS Subject Classification: None of the above; but in this section, $K_0$ as an ordered group; traces, General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 19B99, 19K14, 46L05, 46L35, 46L80

PDF(click to download): K-Theory and Traces

The Bundle of KMS State Spaces for Flows on a Unital AF C*-algebra

C. R. Math. Rep. Acad. Sci. Canada Vol. 43 (4) 2021, pp. 103-121
Vol.43 (4) 2021
George A. Elliott; Klaus Thomsen Details
(Received: 2021-09-21 )
(Received: 2021-09-21 )

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Klaus Thomsen, Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark; e-mail: matkt@math.au.dk

Abstract/Résumé:

It is shown that, for any unital simple infinite-dimensional AF algebra, the KMS-state bundle for a one-parameter automorphism group is isomorphic to an arbitrary proper simplex bundle over the real line with (as is necessary) fibre at (inverse temperature) zero isomorphic to the trace simplex.

On démontre que, pour toute C*-algèbre AF simple à élément unité et à dimension infinie, le faisceau d’états KMS pour un grouped’automorphismes à un paramètre est isomorphe à un faisceau de simplices propre arbitraire sur la ligne réelle tel que (nécessairement) le fibre sur la température inverse zéro est isomorphe au simplex tracial.

Keywords: One-parameter automorphism group of an AF algebra, bundle of equilibrium (KMS) states arbitrary

AMS Subject Classification: General theory of $C^*$-algebras, Noncommutative dynamical systems 46L05, 46L55

PDF(click to download): The Bundle of KMS State Spaces for Flows on a Unital AF C*-algebra

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