— 1403 articles found.
Polynomial Power Residue Symbols and $q$-resultants
Yoshinori Hamahata,Department of Applied Mathematics, Okayama University of Science, Ridai-cho 1-1, Okayama, 700{0005, Japan; e-mail: hamahata@xmath.ous.ac.jp
Abstract/Résumé:
We establish a relation between polynomial power residue symbols and \(q\)-resultants of \(\mathbb{F}_q\)-linear polynomials. We then establish the \(q-1\)-st power reciprocity law.
On établit une relation entre le symbole de résidu de puissances en caractéristique \(p\) et le \(q\)-résultant de deux \(\mathbb{F}_q\)-polynômes linéaire. Alors on démontre la loi de réciprocité des puissances \(q-1\)-èmes.
Keywords: Power residues, function fields., reciprocity law
AMS Subject Classification:
Power residues; reciprocity, Drinfeld modules; higher-dimensional motives; etc., Arithmetic theory of polynomial rings over finite fields
11A15, 11G09, 11T55
PDF(click to download): Polynomial Power Residue Symbols and $q$-resultants
Cauchy Problem on Two Characteristic Hypersurfaces for the Einstein-Vlasov Scalar Field Equations in Temporal Gauge
Marcel Dossa,University of Yaounde I, Faculty of Sciences, Department of Mathematics, P. O. Box. 812, Yaounde, Cameroon; e-mail: marceldossa@yahoo.fr
Jean Baptiste Patenou,University of Dschang, Faculty of Sciences, Department of Mathematics and Computer Science, P. O. Box. 67 Dschang, Cameroon
e-mail: jeanbaptiste.patenou@univ-dschang.org,jpatenou@yahoo.fr
Abstract/Résumé:
In this paper, we consider the initial value problem for the Einstein-Vlasov scalar field equations in temporal gauge, where the initial data are prescribed on two characteristic smooth intersecting hypersurfaces. From a suitable choice of some free data, the initial data constraints’s problem is solved globally, then the evolution problem relative to the deduced initial data is solved locally in time.
Dans cet article, on considère le problème de Cauchy pour les équations d’Einstein-Vlasov-Champ scalaire en jauge temporelle, dans le cas où les données initiales sont préscrites sur deux hypersurfaces caractéristiques régulières sécantes. A partir d’un choix judicieux de certaines données indépendantes, le problème des contraintes initiales est globalement résolu, et ensuite le problème de l’évolution relatif aux données initiales déduites est résolu localement dans le temps.
Keywords: Characteristic Cauchy problem, general relativity, geometric-transport scalar field equations, initial data constraints's problem, temporal gauge, well posedness
AMS Subject Classification:
PDE in relativity, Einstein's equations (general structure; canonical formalism; Cauchy problems)
35Q75, 83C05
PDF(click to download): Cauchy Problem on Two Characteristic Hypersurfaces for the Einstein-Vlasov Scalar Field Equations in Temporal Gauge
Absence of Non-commutative Matrix Observables for q-State Potts Models
James McVittie,The Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, Canada H3A 0B9; e-mail: james.mcvittie@mail.mcgill.ca
Abstract/Résumé:
This article is an expository work introducing the subject of lattice models in statistical physics and the types of observables that can be used to prove convergence, as well as a proof for the q-state Potts model showing that non-commutative matrix observables do not exist.
Cet article est une introduction au sujet des modèles sur réseau en physique statistique et les types d’observables qui peuvent être utilisées pour démontrer la convergence, et aussi une démonstration qu’il n’existe pas d’observable matricielle non-commutative pour le modèle “q-state Potts”.
Keywords: Ising model, Potts model, lattice, matrix, noncommutative, observable
AMS Subject Classification:
Dynamics of random walks; random surfaces; lattice animals; etc., Dynamics of disordered systems (random Ising systems; etc.)
82C41, 82C44
PDF(click to download): Absence of Non-commutative Matrix Observables for q-State Potts Models
Comments Related to Infinite Wedge Representations
Nathan Grieve,Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB,Canada E3B 5A3; e-mail: n.grieve@unb.ca
Abstract/Résumé:
We study the infinite wedge representation and show how it is related to the universal central extension of \(g[t,t^{-1}]\), the loop algebra of a complex semi-simple Lie algebra \(g\). We also give an elementary proof of the boson-fermion correspondence. Our approach to proving this result is based on a combinatorial construction combined with an application of the Murnaghan-Nakayama rule.
Nous étudions l’algèbre extérieure en dimension infinie et montrons comment elle est reliée à l’extension centrale universelle de \(g[t,\!t^{-1}]\), l’algèbre de lacets sur une algèbre de Lie \(g\) semi-simple complexe. De plus, nous donnons une preuve élémentaire de la correspondance boson-fermion. Pour ce faire, nous utilisons une construction combinatoire, ainsi que la règle de Murnaghan-Nakayama.
Keywords: Boson-fermion correspondence, Infinite wedge representation, Murnaghan-Nakayama rule
AMS Subject Classification:
Symmetric functions, Completely integrable systems; integrability tests; bi-Hamiltonian structures; hierarchies (KdV; KP; Toda; etc.)
05E05, 37K10
PDF(click to download): Comments Related to Infinite Wedge Representations
Composition Series for Degenerate Principal Series of ${GL}(n)$
Dmitry Gourevitch,The Incumbent of Dr. A. Edward Friedmann Career Development Chair in Mathematics, Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel; e-mail: dimagur@weizmann.ac.il
Abstract/Résumé:
In this note we consider representations of the group \(GL(n,F)\), where \(F\) is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field \(F\). We study reducibility and composition series of such representations.
Similar results were obtained already in [4,20,31], but we give a short uniform proof in the general case, using the tools from [7]. We also indicate some applications to cosine transforms in integral geometry.
Dans cette note on considère des représentations du groupe \(GL(n,F)\), où \(F\) est le corps des nombres réels ou complexes ou plus généralement, un corps local arbitraire, dans l’espace de fibres en droites équivariants sur des Grassmanniennes sur le même corps \(F\). On étudie la réductibilité et la suite de composition de telles représentations.
Des résultats similaires ont déjà été obtenus dans [4,20,31], mais nous présentons une courte preuve dans le cas général en utilisant les outils de [7]. On donne aussi quelques applications aux transformées en cosinus en géométrie intégrale.
Keywords: Bernstein-Zelevinsky derivative, Degenerate principal series, Line bundle over Grassmannian, alpha-cosine transform
AMS Subject Classification:
Analysis on real and complex Lie groups, Representations of Lie and linear algebraic groups over local fields, Homogeneous spaces, Integral transforms in distribution spaces, Integral geometry; differential forms; currents; etc.
22E30, 22E50, 22F30, 46F12, 53C65
PDF(click to download): Composition Series for Degenerate Principal Series of ${GL}(n)$
Group Actions on Filtered Modules and Finite Determinacy. Finding Large Submodules in the Orbit by Linearization
Genrich Belitskii,Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel; e-mail: genrich@math.bgu.ac.il
Dmitry Kerner,Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be'er Sheva 84105, Israel; e-mail dmitry.kerner@gmail.com
Abstract/Résumé:
Let \(M\) be a module over a local ring \(R\) and a group action \(G\circlearrowright M\), not necessarily \(R\)-linear. To understand how large is the \(G\)-orbit of an element \(z\in M\) one looks for the large submodules of \(M\) lying in \(Gz\). We provide the corresponding (necessary/sufficient) conditions in terms of the tangent space to the orbit, \(T_{(Gz,z)}\).
This question originates from the classical finite determinacy problem of Singularity Theory. Our treatment is rather general, in particular we extend the classical criteria of Mather (and many others) to a broad class of rings, modules and group actions.
When a particular ‘deformation space’ is prescribed, \(\Sigma\subseteq M\), the determinacy question is translated into the properties of the tangent spaces, \(T_{(Gz,z)}\), \(T_{(\Sigma,z)}\), and in particular to the annihilator of their quotient, \(ann\,{T_{(\Sigma,z)}}/{T_{(Gz,z)}}\).
Etant donné une action d’un groupe sur un module, \(G\circlearrowright M\), et un élément \(z\in M\), on étudie le plus grand sous-module de \(M\) contenu dans l’orbite \(Gz\). On donne des conditions nécessaires et suffisantes décrivant ce module en termes de l’espace tangent a l’orbite, \(T_{(Gz,z)}\). Cela prolonge les critères classiques de la théorie des singularités à une large classe d’anneaux, modules, et actions de groupes.
Keywords: Group actions, finite determinancy, matrix families, matrix singularities, modules over local rings, open orbits, sufficiency of jets
AMS Subject Classification:
Deformations of singularities, Canonical forms; reductions; classification, Normal families of functions; mappings, Classification; finite determinacy of map germs, Normal forms
14B07, 15A21, 32A19, 58K40, 58K50
PDF(click to download): Group Actions on Filtered Modules and Finite Determinacy. Finding Large Submodules in the Orbit by Linearization
Cartan-Remez Type Inequalities for Analytic and Plurisubharmonic Functions
Alexander Brudnyi,Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4; e-mail: albru@math.ucalgary.ca
Abstract/Résumé:
Recently there has been a considerable interest in Cartan-Remez type inequalities in connection with various problems of analysis. In this paper we formulate and prove several basic results in this area and describe some of their applications. The text is based on the material of the minicourse given by the author at the workshop on Analytic Microlocal Analysis held at the Northwestern University in May 2013.
Récemment, il y a eu un intérêt considérable aux inégalités de types de Cartan-Remez dans le cadre de divers problèmes de l’analyse. Dans cet article, nous formulons et nous démontrons plusieurs résultats de base dans ce domaine et nous décrivons certaines de leurs applications. Le texte est basé sur le matériau de la mini-course donnée par l’auteur à l’atelier sur l’analyse analytique microlocale tenu à l’Université Northwestern en mai 2013.
Keywords: Cartan-Remez inequality, Hilbert 16th problem, Jensen inequality, holomorphic function, subharmonic function
AMS Subject Classification:
Inequalities in approximation (Bernstein; Jackson; Nikol\cprime ski\u\i-type inequalities), Multidimensional problems (should also be assigned at least one other classification number in this section)
41A17, 41A63
PDF(click to download): Cartan-Remez Type Inequalities for Analytic and Plurisubharmonic Functions
Divergent Series: Past, Present, Future …
Christiane RousseauDepartement de mathematiques et de statistique and CRM, Universite de Montreal, C.P. 6128, Succursale Centre-ville, Montreal, QC , H3C 3J7, Canada;
e-mail: rousseac@dms.umontreal.ca
Abstract/Résumé:
The paper presents some reflections of the author on divergent series and their role and place in mathematics over the centuries. The point of view presented here is limited to differential equations and dynamical systems.
L’article présente quelques réflexions de l’auteure sur les séries divergentes, et leur rôle et place en mathématiques au courant des siècles. Le point de vue présenté ici est limité aux équations différentielles et aux systèmes dynamiques.
Keywords: Borel summability, Divergent series, Euler dierential equation, Stokes phenomenon, resummation
AMS Subject Classification:
Asymptotics; summation methods, Stokes phenomena and connection problems (linear and nonlinear)
34M30, 34M40
PDF(click to download): Divergent Series: Past, Present, Future ...
On Dependence of Rational Points on Elliptic Curves
Mohammad Sadek,Department of Mathematics and Actuarial Science, American University in Cairo, Cairo, Egypt; e-mail: mmsadek@aucegypt.edu
Abstract/Résumé:
Let \(E\) be an elliptic curve defined over \(Q\). Let \(\Gamma\) be a subgroup of \(E(Q)\) and \(P\in E(Q)\). In \cite{Arithmetic}, it was proved that if \(E\) has no nontrivial rational torsion points, then \(P\in\Gamma\) if and only if \(P\in \Gamma\) mod \(p\) for finitely many primes \(p\). In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when \(E\) does not have complex multiplication and either \(E\) is a semistable curve or \(E\) has no exceptional prime.
Soit \(E\) une courbe elliptique définie sur \(Q\). Soit \( \Gamma\) un sous-groupe de \( E(Q) \) et \( P \in E (Q) \). Dans \cite{Arithmetic}, il on a prouvé que si \( E \) n’a pas de points de torsion rationels non trivials, alors \( P \in \Gamma \) si et seulement si \( P \in \Gamma \) mod \( p \) pour un nombre fini de nombres premiers \( p \). Dans cette note, supposant l’hypothèse général de Riemann, nous fournissons une borne-supérieure explicite sur ces nombres premiers quand \( E \) n’a pas de multiplication complexe et soit \( E \) est une courbe semi-stable soit \( E \) n’a aucun nombre premier exceptionnel.
Keywords: elliptic curves, linear dependence, rational points
AMS Subject Classification:
Elliptic curves over global fields, Rational points
11G05, 14G05
PDF(click to download): On Dependence of Rational Points on Elliptic Curves
Analytic Compactifications of $C^2$ Part I—Curvettes at Infinity
Pinaki Mondal,The School of Mathematics, Physics & Technology, College of The Bahamas, Nassau, Bahamas; e-mail: pinakio@gmail.com
Abstract/Résumé:
We study normal analytic compactifications of \(C^2\) and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of Brenton (1973). As a by product we give new proofs of Jung’s theorem on polynomial automorphisms of \(C^2\) and Remmert and Van de Ven’s result that \(P^2\) is the only smooth analytic compactification of \(C^2\) for which the curve at infinity is irreducible. We also give a complete answer to the question of existence of compactifications of \(C^2\) with prescribed divisorial valuations at infinity. In particular, we show that a valuation on \(C(x,y)\) centered at infinity determines a compactification of \(C^2\) iff it is positively skewed in the sense of Favre and Jonsson (2004).
Nous étudions les compactifications analytiques normales de \(C^2\) et décrivons leurs singularités et la configuration des courbes à l’infini, en particulier ameliorant et généralisant les résultats de Brenton (1973). Comme un sous-produit, nous donnons de nouvelles preuves du théorème de Jung sur les automorphismes polynomiaux de \(C^2 \) et le résultat de Remmert et Van de Ven que \(P^2\) est la seule compactification analytique lisse de \(C^2\) pour laquelle la courbe à l’infini est irréductible. Nous donnons aussi une réponse complète à la question de l’existence de compactifications de \(C^2 \) avec des valorisations divisorielles préscrites à l’infini. En particulier, nous montrons qu’une évaluation sur \(C(x,y) \) centrée à l’infini détermine une compactification de \(C^2\) ssi elle est positivement asymétrique dans le sens de Favre and Jonsson (2004).
Keywords: Compactifications of $C^2$, curvettes, discreet valuations., polynomial automorphisms
AMS Subject Classification:
Rational and ruled surfaces, , Normal analytic spaces
14J26, 14M27, 32C20
PDF(click to download): Analytic Compactifications of $C^2$ Part I---Curvettes at Infinity
Résolution du $\partial \bar{\partial}$ pour les courants prolongeables définis sur la boule euclidienne de $C^n$
Salomon Sambou,Université Assane SECK de Ziguinchor, Sénégal; email: ssambou@univ-zig.sn
Eramane Bodian,Université Assane SECK de Ziguinchor, Sénégal; email: eramane20era@yahoo.fr
Dian Diallo,Université Assane SECK de Ziguinchor, Sénégal; email: diandiallo1086@yahoo.fr
Abstract/Résumé:
We solve the \(\partial \bar{\partial}\)-problem for extendable currents defined on the euclidean ball of \({C}^n\).
On résout le \(\partial \bar{\partial}\) pour les courants prolongeables définis dans la boule euclidienne de \({C}^n\).
Keywords: $\partial \bar{\partial}$, Courant prolongeable, cohomologie de De Rham
AMS Subject Classification:
Analytical consequences of geometric convexity (vanishing theorems; etc.)
32F32
PDF(click to download): Résolution du $partial bar{partial}$ pour les courants prolongeables définis sur la boule euclidienne de $C^n$
Sharp Maximal Function Estimates and Boundedness for the Toeplitz Type Operator Associated to a Multiplier Operator
Dazhao Chen,Department of Science and Information Science, Shaoyang University, Hunan Shaoyang, 422000, P. R. of China; e-mail: chendazhao27@sina.com
Abstract/Résumé:
In this paper, we establish sharp maximal function estimates for the Toeplitz type operator associated to a certain multiplier operator. As an application, we obtain the boundedness of the operator on Lebesgue, Morrey and Triebel-Lizorkin spaces.
Dans cet article, on établit des estimations de la fonction maximale optimale pour l’opérateur de type Toeplitz associé à un certain opérateur multiplicateur. Comme application, nous obtenons le caractère borné de l’opérateur sur les espaces de Lebesgue, de Morrey et de Triebel-Lizorkin.
Keywords: BMO, Lipschitz function, Morrey space, Toeplitz type operator, Triebel-Lizorkin space, multiplier operator, sharp maximal function
AMS Subject Classification:
Singular integrals (Calder__n-Zygmund; etc.), Maximal functions; Littlewood-Paley theory
42B20, 42B25
PDF(click to download): Sharp Maximal Function Estimates and Boundedness for the Toeplitz Type Operator Associated to a Multiplier Operator
Fermionic Realization of Two-Parameter Quantum Affine Algebra $U_{r;s}(C_l^{(1)})$
Naihuan Jing,School of Mathematical Sciences, South China University of Technology, Guangzhou 510640,
China and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; e-mail: jing@math.ncsu.eduHonglian Zhang,Department of Mathematics, Shanghai University, Shanghai 200444, China; e-mail: hlzhangmath@shu.edu.cn
Abstract/Résumé:
We construct a Fock space representation and the action of the two-parameter quantum algebra \(U_{r,s}(\frak{gl}_{\infty})\) using extended Young diagrams. In particular, we obtain an integrable representation of the two-parameter quantum affine algebra of type \(C_n^{(1)}\) which is a two-parameter generalization of Kang-Misra-Miwa’s realization.
Nous construisons une représentation sur un espace de Fock de l’algèbre quantique à deux paramètres \(U_{r,s}(\frak{gl}_{\infty})\) en utilisant les diagrammes de Young prolongés. En particulier, on obtient une représentation intégrable de l’algèbre quantique affine à deux paramètres de type \(C_n^{(1)}\) qui est une généralization à deux paramètres de la réalization de Kang-Misra-Miwa.
Keywords: Fock space, Two-parameter quantum ane algebra, Young diagram, fermionic realization
AMS Subject Classification:
Quantum groups (quantized enveloping algebras) and related deformations
17B37
PDF(click to download): Fermionic Realization of Two-Parameter Quantum Affine Algebra $U_{r;s}(C_l^{(1)})$
Laplacians for Derived Graphs of Regular Kähler Graphs
Yaermaimaiti Tuerxunmaimaiti,Division of Mathematics and Mathematical Science, Nagoya Institute of
Technology, Nagoya 466-8555, Japan; e-mail: yarimamat@gmail.comToshiaki Adachi,Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan; e-mail: adachi@nitech.ac.jp
Abstract/Résumé:
We consider \((p,q)\)-step adjacency on a Kähler graph which is compounded from a principal graph and an auxiliary graph. We attach probabilistic weights to \((p,q)\)-step paths so that these paths show trajectories under the influence of a magnetic field of strength \(q/p\) on this graph. We study eigenvalues of Laplacians corresponding to \((p,q)\)-step paths on some regular Kähler graphs and give examples of pairs of regular Kähler graphs whose Laplacians for arbitrary pairs \((p,q)\) of positive integers have the same eigenvalues.
On considère la relation de contiguité à un pas \((p,q)\) près sur un graphe kählerien qui se constitue d’un graphe principal et un graphe auxiliaire. On attache des poids probabilistiques aux sentiers dans un tel graphe composés de pas \((p,q)\) de sorte que ces sentiers montrent des trajectoires sous l’influence d’un champs magnétique de force \(q/p\) sur le graphe. On étudie le spectre de l’opérateur laplacien correspondant aux sentiers aux pas \((p,q)\) sur certains graphes kähleriens regulières. On donne un example de deux graphes différents dont les opérateurs laplaciens ont le même spectre pour toute paire \((p,q)\).
Keywords: (p; q)-Laplacians, Kähler graphs, complement graphs, derived graphs, isospectral, products of graphs
AMS Subject Classification:
Graphs and matrices, Hermitian and K_õhlerian manifolds
05C50, 53C55
PDF(click to download): Laplacians for Derived Graphs of Regular Kähler Graphs
The Pólya-Schur Problem on the Unit Circle
Peter C. Gibson,Dept. of Mathematics & Statistics, York University, 4700 Keele St., Toronto, Ontario,
Canada, M3J 1P3; e-mail: pcgibson@yorku.ca
Abstract/Résumé:
The Pólya-Schur problem for a region \(Z\) in the complex plane is to characterize the semigroup of linear operators \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) that map polynomials whose zeros are confined to \(Z\) to polynomials of the same type, or to 0. We give a constructive solution to the Pólya-Schur problem in the case where \(Z\) is the unit circle. This shows that the associated semigroup is qualitatively simpler than in the classical case where \(Z\) is the real line, whereas recent results have not clearly distinguished the two cases.
Le problème Pólya-Schur pour une région \(Z\) dans le plan complexe est de charactériser le semigroupe des opérateurs linéaires \(A:\mathbb{C}[z]\rightarrow \mathbb{C}[z]\) envoyant chaque polynôme dont les racines appartiennent à \(Z\) vers un polynôme du même type, ou vers 0. Nous présentons une solution constructive au problème Pólya-Schur dans le cas où \(Z\) est le cercle unité. Cela démontre que le semigroupe associé est qualitativement plus simple que dans le cas classique de la ligne réelle, tandis que les résultats récents n’ont pas distingué les deux cas.
Keywords: Polya-Schur type theorems, composition operators, stable polynomials
AMS Subject Classification:
Polynomials, Zeros of polynomials; rational functions; and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), Operators on function spaces (general)
30C10, 30C15, 47B38
PDF(click to download): The Pólya-Schur Problem on the Unit Circle
Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups
Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail: walters@unbc.ca
Abstract/Résumé:
We prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)
Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.
Keywords: C*-algebra, Fourier transform, Gaussian sums, Hilbert space, L2 spaces, Locally compact Abelian groups, characters, cyclic groups, integral transforms, projective char- acter, self-dual groups, unitary operators
AMS Subject Classification:
Harmonic analysis and almost periodicity, General properties and structure of LCA groups, Compact groups, General properties and structure of locally compact groups, $C^*$-algebras and $W$*-algebras in relation to group representations, General properties and structure of real Lie groups, Integral representations; integral operators; integral equations methods, Integral operators, Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory
11K70, 22B05, 22C05, 22D05, 22D25, 22E15, 31A10, 45P05, 46L35, 46L40, 46L80
PDF(click to download): Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups
Periodic Integral Transforms and Associated Noncommutative Orbifold Projections
Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC V2N 4Z9, Canada; e-mail: walters@unbc.ca
Abstract/Résumé:
We report on recent results on the existence of Cubic and Hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \(\mathcal E(t)\) of the continuous field of rotation C*-algebras \(\{A_t\}_{0 \le t \le 1}\) that is invariant under the noncommutative Hexic transform automorphism. This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical Theta function theory.
On décrit des résultats récents sur l’existence d’une transformation intégrale d’ordre trois (ou d’ordre six) sur un groupe localement compact abélien self-dual. On étudie l’application possible à la construction d’un champs continu de projecteurs invariants sous l’automorphisme associé du champs de C*-algèbres de rotation. On calcule certains invariants topologiques de ces projecteurs.
Keywords: C*-algebra, Fourier transform, automorphisms, inductive limits, noncommutative tori, orbifold, rotation algebra, symmetries, topological invariants, unbounded traces
AMS Subject Classification:
Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory, $K$-theory, String and superstring theories; other extended objects (e.g.; branes), Topological field theories, String and superstring theories
46L35, 46L40, 46L80, 55N15, 81T30, 81T45, 83E30
PDF(click to download): Periodic Integral Transforms and Associated Noncommutative Orbifold Projections
Constructive Geometrization of Thurston Maps
Nikita Selinger,Department of Mathematics, Stony Brook University, Stony Brook NY, USA; e-mail: nikita@math.sunysb.edu
Michael Yampolsky,Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4; e-mail: yampol@math.toronto.edu
Abstract/Résumé:
We prove that every Thurston map can be constructively geometrized in a canonical fashion. According to Thurston’s theorem, a map with hyperbolic orbifold has a canonical geometrization – a combinatorially equivalent postcritically finite rational map of the Riemann sphere – if and only if there is no Thurston obstruction. We follow Pilgrim’s idea of a canonical decomposition of a Thurston map to handle the obstructed case. A key ingredient of our proof is a geometrization result for marked Thurston maps with parabolic orbifolds – an analogue of Thurston’s theorem for the exceptional case not covered by it.
On montre que toute application de Thurston peut être géométrisée de façon constructive et canonique. Selon le théoreme de Thurston, une telle application ayant un orbifold hyperbolique possède une géométrisation canonique, c’est-à-dire une fonction rationnelle combinatoriellement équivalente dont les orbites critiques sont finies, si et seulement s’il n’existe pas d’obstruction de Thurston. On traite le cas où il existe une obstruction en utilisant l’idée de Pilgrim d’une décomposition canonique d’une application de Thurston. L’ingrédient principal de la preuve est un résultat de géométrisation pour les applications de Thurston marquées ayant un orbifold parabolique – un analogue du théorème de Thurston pour le cas exceptionnel.
Keywords: Thurston equivalence, Thurston obstruction, decidability, geometrization
AMS Subject Classification:
Combinatorics and topology, Special coverings; e.g. branched
37F20, 57M12
PDF(click to download): Constructive Geometrization of Thurston Maps
Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras
Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail: walters@unbc.ca
Abstract/Résumé:
We demonstrate, in a rather quantitative manner, the existence of topological obstructions to approximating the irrational rotation C*-algebra \(A_\theta\) by Fourier invariant unital C*-subalgebras of either of the forms \[M \oplus B \oplus \sigma(B), \qquad M \oplus N \oplus D \oplus \sigma(D) \oplus \sigma^2(D) \oplus \sigma^3(D),\] where \(M, N\) are Fourier invariant matrix algebras (over \(\mathbb C\)), \(B\) is a C*-subalgebra whose unit projection is flip invariant and orthogonal to its Fourier transform, and \(D\) is a C*-subalgebra whose unit projection is orthogonal to its orbit under the Fourier transform. Here, \(\sigma\) is the noncommutative Fourier transform automorphism of \(A_\theta\) defined by \(\sigma(U) = V^{-1},\ \sigma(V)=U\) on the canonical unitary generators \(U,V\) obeying the unitary Heisenberg commutation relation \(VU = e^{2\pi i\theta}UV\).
On montre l’existence d’obstructions topologiques à l’approximation du tore non-commutatif par sous-algèbres de certains types qui sont invariantes sous l’automorphisme de Fourier.
Keywords: C*-algebra, Fourier transform, automorphisms, inductive limits, noncommutative tori, rotation algebra, topological invariants, topological obstructions, unbounded traces
AMS Subject Classification:
Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory
46L35, 46L40, 46L80
PDF(click to download): Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras
Commutativity Criteria in Banach Algebras
Cheikh O. Hamoud,Department of Mathematics and Science, Ajman University of Science and Technology,
Ajman Campus, United Arab Emirates; e-mail: c.hamoud@ajman.ac.ae
Abstract/Résumé:
We consider complex Banach algebras satisfying the condition \(\displaystyle (xy)^k=x^ky^k\) for all \(x\,,\,y\,\) in the algebra where \(k\) is an integer \((k\geq 2)\).
We show that for \(k=2\) and \(k=3\), this condition yields commutativity in unital Banach algebras. For higher values of \(k\), commutativity is obtained for semi-simple algebras and the conclusions are quite similar to the ones in Cheikh 1995.
The extension of the results to wider classes of algebras is also considered.
Nous considérons des algèbres de Banach complexes vérifiant la condition \(\displaystyle (xy)^k=x^ky^k\) pour tout \(x\,,\,y\,\) dans l’algèbre, \(k\) étant un entier \((k\geq 2)\).
Nous montrons que pour \(k=2\) et \(k=3\), cette condition entraine la commutativité dans les algèbres de Banach unitaires. Pour les valeurs plus elevées de \(k\), la commutativité est établie dans les algèbres semi-simples avec des résultats similaires à ceux obtenus dans Cheikh 1995.
L’extension des résultats à d’autres classes d’algèbres topologiques est également considérée.
Keywords: Banach algebra, commutativity, radical, spectral radius
AMS Subject Classification:
Representations of topological algebras, General theory of commutative topological algebras
46H15, 46J05
PDF(click to download): Commutativity Criteria in Banach Algebras