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

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

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

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

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

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

Tatsuyoshi Hamada,Department of Applied Mathematics, Fukuoka University, Fukuoka, 814-0180, Japan; e-mail: hamada@fukuoka-u.ac.jp

Katsufumi Yamashita,Department of Mathematics, Saga University, Saga, 840-8502, Japan; e-mail: ky-karatsucity@sgr.bbiq.jp

Abstract/Résumé:

In Theorem 1, we show a new condition for a real hypersurface \(M\) isometrically immersed into a nonflat complex space form to be a hypersurface of type (A). This condition is expressed by the parallelism of a certain tensor of type (1, 1) on \(M\) . Furthermore, using the discussion in the proof of Theorem 1, we can give a condition for a Kähler manifold to be a complex space form (see Theorem 2).

Dans le théorème 1, nous donnons une nouvelle condition pour qu’une hypersurface réelle \(M\) immergée dans une “space form” complexe non plate soit une hypersurface de type (A). Cette condition est exprimée par le parallélisme d’un certain tenseur de type (1, 1) sur \(M\) . De plus, en utilisant la discussion dans la démonstration du théorème 1, nous donnons une condition pour qu’une variété de Kähler soit une “space form” complexe (voir le théorème 2).

Keywords: Kahler manifold, complex space form, real hypersurfaces of type (A), structure tensor

AMS Subject Classification: Local submanifolds, Geodesics, Global submanifolds 53B25, 53C22, 53C40

PDF(click to download): The Parallelism of a Certain Tensor of Real Hypersurfaces in a Nonflat Complex Space Form

Atsushi Kanazawa,Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada; e-mail: kanazawa@10.alumni.u-tokyo.ac.jp

Abstract/Résumé:

We provide a simple sufficient condition for the \(L^p\)-convergence of the Laplace–Beltrami eigenfunction expansions of functions on a compact Riemannian manifold with a Dirichlet boundary condition.

Nous fournissons une condition suffisante simple pour la convergence \(L ^ p\) de Laplace–Beltrami expansions de fonctions sur une variété riemannienne compacte avec une condition aux limites de Dirichlet.

Keywords: Dirichlet boundary condition, Green's function, Laplace-Beltrami operator, Lp-convergence, eigenfunction expansion

AMS Subject Classification: Elliptic equations on manifolds; general theory 58J05

PDF(click to download): $L^p$-Convergence of the Laplace--Beltrami Eigenfunction Expansions

Zhuang Niu,Department of Mathematics, University of Wyoming, Laramie, Wyoming, 82071 USA; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras. III. Uniqueness Theorem and Isomorphism Theorem

Y. Karshon,Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3; e-mail: karshon@math.toronto.edu

L. Kessler,Department of Mathematics, Physics, and Computer Science, University of Haifa, at Oranim, Tivon 36006, Israel; e-mail: liatke.math@gmail.com

M. Pinsonnault,Department of Mathematics, Middlesex College The University of Western Ontario London, Ontario N6A 5B7 Canada; e-mail: mpinson@uwo.ca

Abstract/Résumé:

Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not tell whether the underlying symplectic manifolds are (non-equivariantly) symplectomorphic. For two dimensional torus actions on closed symplectic four-manifolds, we reduce the counting question to combinatorics, by expressing the manifold as a symplectic blowup in a way that is compatible with all the torus actions simultaneously.

Nous nous intéressons aux différentes actions d’un tore sur une variété symplectique donnée. En géométrie symplectique équivariante, les théorèmes de classification permettent parfois de distinguer des actions hamiltoniennes de tores géométriquement inéquivalentes. Par contre, ces théorèmes ne permettent habituellement pas de déterminer si les variétés symplectiques sous-jaçentes sont symplectomorphes. Dans le cas des variétés symplectiques de dimension \(4\), nous réduisons le problème d’énumération des actions toriques inéquivalentes à un problème combinatoire en exprimant la variété considérée comme un éclatement symplectique qui est compatible simultanément avec toutes les actions toriques. Ce résultat est obtenu en employant des techniques pseudo-holomorphes.

Keywords:

AMS Subject Classification: Momentum maps; symplectic reduction 53D20

PDF(click to download): Counting Toric Actions on Symplectic Four-Manifolds

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, Wyoming, 82071 USA; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu ("splitting interval"), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras. II. Existence Theorem

Gilles G. de Castro, Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil; e-mail: gilles.castro@ufsc.br

Fernando de L. Mortari, Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil; e-mail: fernando.mortari@ufsc.br

Abstract/Résumé:

Given a positive function on the set of edges of an arbitrary directed graph \(E=(E^0,E^1)\), we define a one-parameter group of automorphisms on the C*-algebra of the graph \(C^*(E)\), and study the problem of finding KMS states for this action. We prove that there are bijective correspondences between KMS states on \(C^*(E)\), a certain class of states on its core, and a certain class of tracial states on \(C_0(E^0)\). We also find the ground states for this action and give some examples.

Étant donné une fonction positive sur l’ensemble des arcs d’un graphe orienté arbitraire \(E=(E^0,E^1)\), nous définissons un groupe à un paramètre d’automorphismes de la \(C^*\)-algèbre du graphe \(C^*(E)\), et nous étudions le problème de trouver les états KMS pour cette action. Nous prouvons qu’il existe des bijections entre les états KMS sur \(C^*(E)\), une certaine classe d’états sur le core, et une certaine classe détats traciaux sur \(C_0(E^0)\). Nous trouvons également les états fondamentaux pour cette action et nous donnons quelques exemples.

Keywords: C*-algebra, Graph, Ground state, KMS state

AMS Subject Classification: Noncommutative dynamical systems 46L55

PDF(click to download): KMS States for the Generalized Gauge Action on Graph Algebras

Leonid Monin, Department of Mathematics, University of Toronto, Toronto, ON Canada M5S 2E4 e-mail: lmonin@math.toronto.edu

Abstract/Résumé:

In this paper the geometry of the lattice is used to prove basic theorems about subgroups and factor groups of \(\Bbb Z^n\). We suggest a geometric algorithm which reduces a finitely generated abelian group to its normal form.

Dans ce papier, un argument de géométrie sur les réseaux est utilisé pour prouver des théorèmes fondamentaux sur les sous-groupes ou les groupes quotients de \(\Bbb Z^n\). Nous proposons un algorithme géométrique qui réduit un groupe abélien finement engendré à sa forme normale.

Keywords: Integer lattice, Integer volume, Normal Smith form

AMS Subject Classification: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 52B20

PDF(click to download): Lattice Geometry and Reduction of Finitely Generated Abelian Groups to a Normal Form

Dan Z. Kučerovský, Department of Mathematics and Statistics, University of New Brunswick, Federicton, NB, Canada E3B 5A3; e-mail: dkucerov@unb.ca

Abstract/Résumé:

We give a complete invariant for finite-dimensional Hopf C*-algebras. Algebras that are equal under the invariant are the same up to a Hopf *-(co-anti)isomorphism.

On donne un invariant complet pour les C*-algèbres de Hopf de dimension finie.

Keywords: C*-algebra, Hopf algebras

AMS Subject Classification: , Algebras of specific types of operators (Toeplitz; integral; pseudodifferential; etc.) 16T05, 47L80

PDF(click to download): On an Abstract Classification of Finite-dimensional Hopf C*-algebras

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

Abstract/Résumé:

It is shown that the index map in Banach algebra K-theory, as a natural map from the K\(_1\)-group of a quotient of a Banach algebra to the K\(_0\)-group of the corresponding ideal, is unique (up to an integral multiple).

Il est démontré que l’application index dans la K-théorie des algèbres de Banach est unique, dans un sens très naturel.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory 19K56

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-Theory

Uffe Haagerup, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark; e-mail: haagerup@math.ku.dk

Abstract/Résumé:

It is shown that all 2-quasitraces on a unital exact \(C^*\)-algebra are traces. As consequences one gets: (1) Every stably finite exact unital \(C^*\)-algebra has a tracial state, and (2) if an \(AW^*\)-factor of type \(II_1\) is generated (as an \(AW^*\)-algebra) by an exact \(C^*\)-subalgebra, then it is a von Neumann \(II_1\)-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that \(RR(A)=0\) for every simple non-commutative torus of any dimension.

On démontre que toute 2-quasitrace sur une C*-algèbre exacte à élément unité est une trace. On en déduit les deux conséquences suivantes: (1) Toute C*-algèbre stablement finie et exacte possède un état tracial, et (2) si un AW*-facteur de type \(II_1\) est engendré (comme AW*-algèbre) par une sous-C*-algèbre exacte, il est une algèbre de von Neumann. Ceci est une solution partielle à un problème bien connu de Kaplansky. Le résultat principal a été utilisé par Blackadar, Kumjian, et Rørdam pour démontrer que \(RR(A) = 0\) pour tout tore non-commutatif simple de dimension quelconque.

Keywords: Quasitraces, classification of C*-algebras, exact C*-algebras

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

PDF(click to download): Quasitraces on Exact C*-algebras are Traces

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, Wyoming, USA 82071; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras~~I. The Building Blocks and the Limit Algebras

Calvin Tadmon, Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, P. O. Box 67, Dschang, Cameroon; e-mail: tadmonc@yahoo.fr and Department of Mathematics and Applied Mathematics, University of Pretoria, 0002, Pretoria, South Africa; e-mail: calvin.tadmon@up.ac.za

Abstract/Résumé:

We show how to assign, on two intersecting null hypersurfaces, initial data for the Einstein-Vlasov system in harmonic coordinates. As all the components of the metric appear in each component of the stress-energy tensor, the hierarchical method of Rendall cannot apply strictly speaking. To overcome this difficulty, additional assumptions have been imposed to the metric on the initial hypersurfaces. Consequently, the distribution function is constrained to satisfy some integral equations on the initial hyper surfaces.

Nous montrons comment construire, sur deux hypersurfaces caractéristiques sécantes, les données initiales pour le problème de Cauchy associé aux équations d’Einstein-Vlasov en jauge harmonique. Comme toutes les composantes de la métrique apparaissent dans chaque composante du tenseur d’impulsion-énergie, la méthode de construction hierarchisée de Rendall ne peut pas s’appliquer stricto sensu. Pour surmonter cette difficulté, une condition supplémentaire est imposée à la métrique sur les hypersurfaces initiales. Par conséquent la fonction de distribution des particules est contrainte à vérifier des équations intégrales sur les hypersurfaces initiales.

Keywords: Goursat problem

AMS Subject Classification: Gases, Einstein's equations (general structure; canonical formalism; Cauchy problems) 82D05, 83C05

PDF(click to download): The Goursat problem for the Einstein-Vlasov system: (I) The initial data constraints

S. Subburam, Department of Mathematics, SASTRA University, Thanjavur - 613401, Tamil Nadu, India; e-mail: ssubburam@maths.sastra.edu

R. Thangadurai, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211019, India; e-mail: thanga@hri.res.in

Abstract/Résumé:

In this paper, we shall prove that all positive integral solutions \((x, y, z)\) of the diophantine equation \(x^{3} + by + 1 – xyz = 0\) satisfy \(x \le b\left((2b^{3} + b)^{3} + 1\right) + 1,\) \(y \le (2b^{3} + b)^{3} + 1,\) and \(z \le \left(b\left((2b^{3} + b)^{3} + 1\right) + 1\right)^{2} + 2b^{3} + b\) for a given positive integer \(b\). As an application of this result, we investigate the divisors of the sequence \(\{n^3+1\}\) in residue classes. More precisely, we study the following sums: \[\displaystyle\sum_{b \le X }\displaystyle\sum_{\tiny\begin{array}{c}d \mid n^{3} + 1 \\ d \equiv – b \pmod{n} \end{array}} 1 \hspace{0.1in}\text{and} \hspace{0.1in} \displaystyle\sum_{n \le X}\displaystyle\sum_{\tiny\begin{array}{c}d \mid n^{3} + 1 \\ d \equiv – b \pmod{n} \end{array}} 1\] for a given positive real number \(X\) and a positive integer \(b\).

Keywords: Diophantine equations, divisors, residue classes

AMS Subject Classification: Cubic and quartic equations 11D25

PDF(click to download): On the Diophantine Equation $x^{3} + by + 1 - xyz = 0$

Alexander Brudnyi, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada; e-mail: albru@math.ucalgary.ca

Abstract/Résumé:

The concept of a weak Markov set takes its origin from Whitney problems for differentiable functions on \(\mathbb R^n\). These are the only sets for which differential calculus similar to that for open subsets of \(\mathbb R^n\) can be developed to some extent. This paper surveys some recent results in this direction obtained by the author. In particular, we show that some classical results for smooth functions and differential forms (such as the Poincaré Lemma, the de Rham and Hartogs theorems, the Künneth formulas, etc.) are valid also on certain weak Markov sets and more generally on certain topological spaces with weak Markov structures. The class of such spaces includes \(C^\infty\) manifolds with boundaries and some Lipschitz and fractal topological manifolds.

Le concept d’un ensemble faible de Markov provient des problèmes de Whitney pour des fonctions différentiables sur \(\mathbb R^n\). Ce sont les seuls ensembles pour lesquels le calcul différentiel semblable à celui pour les sous-ensembles ouverts de \(\mathbb R^n\) peut être développé dans une certaine mesure. Cet article examine quelques résultats récents dans cette direction obtenus par l’auteur. En particulier, on prouve que quelques résultats classiques pour les fonctions lisses et les formes différentielles (comme le lemme de Poincaré, les théorèmes de de Rham et de Hartogs, les formules de Künneth, etc.) sont également valides sur quelques ensembles faibles de Markov et plus généralement sur quelques espaces topologiques munis de structures faibles de Markov. La classe de tels espaces inclut les variétés lisses à bord et quelques variétés topologiques lipschitziennes et fractales.

Keywords: Ck function, Weak Markov set, Whitney problems, de Rham cohomology, extension, trace

AMS Subject Classification: Continuity and differentiation questions 26B05

PDF(click to download): Differential Relations on Weak Markov Sets