On démontre un théorème de Voronoï (caractérisation des maxima locaux de l’invariant d’Hermite) pour les familles de réseaux paramétrées par les espaces symétriques irréductibles non exceptionnels de type non compact.

This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.

We study the local $L$-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for $\text{G}{{\text{L}}_{2}}\,\times \,\text{G}{{\text{L}}_{3}}$.

We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\varepsilon \left( K \right)$ can be described for Cantor-type compact sets.

By means of the Pucci operator, we construct a function ${{u}_{0}}$, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

A quasi-Poisson manifold is a $G$-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ${{\Lambda }^{3}}\mathfrak{g}$ associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

We study the symplectic geometry of the moduli spaces ${{M}_{r}}={{M}_{r}}\left( {{\mathbb{S}}^{3}} \right)$ of closed $n$-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in $\text{SU}\left( 2 \right)$ by the diagonal conjugation action of $\text{SU}\left( 2 \right)$. Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\text{SU}\left( 2 \right)$-manifold in the sense of $\left[ \text{AKSM} \right]$. An integrable Hamiltonian system is constructed on ${{M}_{r}}$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on ${{M}_{r}}$ relates to the symplectic structure obtained from gauge-theoretic description of ${{M}_{r}}$. The results of this paper are analogues for the 3-sphere of results obtained for ${{M}_{r}}\left( {{\mathbb{H}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in hyperbolic 3-space $\left[ \text{KMT} \right]$, and for ${{M}_{r}}\left( {{\mathbb{E}}^{3}} \right)$, the moduli space of $n$-gons with fixed side-lengths in ${{\mathbb{E}}^{3}}\left[ \text{KM}1 \right]$.

Form domains are characterized for regular $2n$-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first $n\,-\,1$ derivatives.

We investigate representations of the Cuntz algebra ${{\mathcal{O}}_{2}}$ on antisymmetric Fock space ${{F}_{a}}({{\mathcal{K}}_{1}})$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi {{P}_{1}}$. We pay special attention to the vector states on ${{\mathcal{O}}_{2}}$ corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra ${{\mathcal{F}}_{2}}$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi {{P}_{1}}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to ${{\mathcal{F}}_{2}}$.

The endomorphisms of $B\left( {{F}_{a}}({{\mathcal{K}}_{1}}) \right)$ associated with these representations of ${{\mathcal{O}}_{2}}$ are also considered.

We describe the set of numbers ${{\sigma }_{k}}\left( {{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}} \right)$, where ${{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}}$ are complex numbers of modulus 1 for which ${{z}_{1}}{{z}_{2}}\cdot \cdot \cdot {{z}_{n+1}}=1$, and ${{\sigma }_{k}}$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${{\overset{\sim }{\mathop{\text{A}}}\,}_{n}}$.

Nous prouvons un encadrement optimal pour la quantité $H\left( x,\,s \right)\,=\,\sum{_{n\ge 1}\,\frac{1}{{{\left( x+n \right)}^{s}}}}$ pour $x\,\ge \,0$ et $s\,>\,1$, qui améliore l'encadrement standard par des intégrales. Cet encadrement entraîne des inégalités sur la fonction $\zeta $ de Riemann, et amène à conjecturer la monotonie de la fonction $s\,\mapsto \,{{[(s\,-\,1)\text{ }\!\!\zeta\!\!\text{ (}s\text{)}]}^{\frac{1}{S-1}}}$. On donne des applications à l'étude de la convexité de fonctions liées à la fonction $\Gamma $ d'Euler et à la majoration optimale des fonctions élémentaires intervenant dans les opérateurs de Baskakov. Puis, nous étendons aux fonctions complètement monotones sur ]$0,\,+\infty $[ les résultats établis pour la fonction $x\,\mapsto \,{{x}^{-s}}$, et nous en déduisons des preuves élémentaires du comportement, quand $z$ tend vers 1, des séries génératrices de certaines fonctions arithmétiques. Enfin, nous prouvons qu'une partie du résultat se généralise à une classe de fonctions convexes positives décroissantes.

An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,\,y)\,=\,P(x)y-q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value ${{\text{ }\!\!\zeta\!\!\text{ }}_{F}}(2)$ for certain quadratic and quartic fields.

Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\mathfrak{g}$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\mathfrak{g}\left( F \right)$ sont des distributions ${{J}_{M}}\left( X,\,f \right)-f$ est une fonction test—indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X\,\in \,\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d’Arthur.

Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont borńees sur $\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$ : c’est un résultat bien connu de Harish-Chandra. Si $M\subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_{M}^{b}\left( X,f \right)$, égales à ${{J}_{M}}\left( X,f \right)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, etc.) restent borńees quand $X$ parcourt $\mathfrak{m}\left( F \right)\,\cap \,{{\mathfrak{g}}_{\text{reg}}}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

We compute the category of perverse sheaves on Hermitian symmetric spaces in types $\text{A}$ and $\text{D}$, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety $\Lambda$.

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\text{ }{{\varepsilon }_{q}}$.

Let $X$ be a complex Banach space and let ${{B}_{p}}\left( X \right)$ denote the vector-valued Bergman space on the unit disc for $1\,\le \,p\,<\,\infty $. A sequence ${{\left( {{T}_{n}} \right)}_{n}}$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between ${{B}_{p}}\left( X \right)$ and ${{B}_{q}}\left( Y \right)$ (resp. ${{B}_{p}}\left( X \right)$ and ${{l}_{q}}\left( Y \right)$) if for any function $f\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{x}_{n}}{{z}^{n}}$ in ${{B}_{p}}\left( X \right)$ we have that $g\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{T}_{n}}\left( {{x}_{n}} \right){{z}^{n}}$ belongs to ${{B}_{q}}\left( Y \right)$ (resp. ${{\left( {{T}_{n}}\left( {{x}_{n}} \right) \right)}_{n}}\,\in \,{{\ell }_{q}}\left( Y \right)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in ${{B}_{p}}\left( X \right)$ are introduced.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in ${{\mathbf{R}}^{n}}$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by “analytic” and “meromorphic” solutions of the equation $Lu\,=\,0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

We verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

Let $\mathbf{A}$ be a $k$-element algebra whose chief factor size is $c$. We show that if $\mathbf{B}$ is in the variety generated by $\mathbf{A}$, then any abelian chief factor of $\mathbf{B}$ that is not strongly abelian has size at most ${{c}^{k-1}}$. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to $c$ in the situation where the variety generated by $\mathbf{A}$ omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

Let $X:\,{{\mathbb{R}}^{2}}\to \,{{\mathbb{R}}^{2}}$ be a ${{C}^{1}}$ map. Denote by $\text{Spec}(X)$ the set of (complex) eigenvalues of $\text{D}{{\text{X}}_{p}}$ when $p$ varies in ${{\mathbb{R}}^{2}}$. If there exists $\in \,>\,0$ such that $\text{Spec(}X)\,\bigcap \,(-\in ,\,\in )\,=\,\varnothing $, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.