In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.

We adopt the following definition of a completely regular semigroup S: for every element a of S, there exists a unique element a-1 of S such that

Soit A un anneau intègre de corps des fractions K, on s'interesse ici aux polynômes de K[X] qui prennent sur A leurs valeurs dans A. Ils forment bien évidemment un sous anneau de K[X] qui contient A[X], nous le notons As.

Cet article reprend, dans un langage moderne, les idées de deux publications assez anciennes, dues à Polya [3] et Ostrowsky [4], en 1919. Les auteurs montraient que si A est principal, As est un A-module libre qui admet sur A une base formée de polynômes de degré croissant ; si A est plus généralement l'anneau des entiers d'un corps de nombres ils mettaient en évidence une suite d'idéaux fractionnaires de A. En exposant le cas de l'anneau A principal, pour un anneau de valisation discrète dans le premier paragraphe, puis en généralisant aux anneaux de Dedekind dans notre second paragraphe, nous pouvons énoncer de façon plus explicite, que As est un A-module projectif, que l'on peut décomposer canoniquement en somme d'idéaux fractionnaires de A.

Kothe spaces, in the terminology of Diendonné [2], are certain spaces X of real valued integrable functions. In this paper we consider the problem of representation of continuous linear functional on vector valued Kothe spaces. The elements of a Kôthe space X(B) are functions with values in a Banach space B (see §2).

We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue–Orlicz spaces of a discrete group $\Gamma$ and relative Toeplitz-Schur multipliers on Schatten–von-Neumann–Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\Lambda \,\subseteq \,\Gamma$ , the norm of the Hilbert transformand the Riesz projection on Schatten–von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten–von-Neumann classes with exponent less than 1.

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.

Let $K/\mathbb{Q}$ be Galois and let $\eta \,\in \,{{K}^{\times }}$ be such that $\text{Re}{{\text{g}}_{\infty }}\left( \eta \right)\,\ne \,0$ . We define the local $\theta $ -regulator $\Delta _{p}^{\theta }\left( \eta \right)\,\in \,{{\mathbb{F}}_{p}}$ for the ${{\mathbb{Q}}_{p}}$ -irreducible characters $\theta $ of $G=\,\text{Gal}\left( K/\mathbb{Q} \right)$ . Let ${{V}_{\theta }}$ be the $\theta $ -irreducible representation. A linear representation ${{\mathfrak{L}}^{\theta }}\,\simeq \,\delta \,{{V}_{\theta }}$ is associated with $\Delta _{p}^{\theta }\left( \eta \right)$ whose nullity is equivalent to $\delta \,\ge \,1$ . Each $\Delta _{p}^{\theta }\left( \eta \right)$ yields $\text{R}eg_{p}^{\theta }\left( \eta \right)$ modulo $p$ in the factorization ${{\Pi }_{\theta }}{{\left( \text{Reg}_{p}^{\theta }\left( \eta \right) \right)}^{\phi \left( 1 \right)}}$ of $\text{Reg}_{p}^{G}\,\left( \eta \right)\,:=\frac{\text{Re}{{\text{g}}_{p}}\left( \eta \right)}{_{p}[K\,:\,\mathbb{Q}]}$ (normalized $p$ -adic regulator). From Prob $\left( \Delta _{p}^{\theta }\left( \eta\right)=0\,\text{and}\,{{\mathfrak{L}}^{\theta }}\simeq \delta {{V}_{\theta }} \right)\,\le {{p}^{-f{{\delta }^{2}}}}$ ( $f\,\ge \,1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that for $p$ large enough, $\text{Reg}_{p}^{G}\left( \eta \right)$ is a $p$ -adic unit or ${{p}^{\phi \left( 1 \right)}}\,||\,\text{Reg}_{p}^{G}\left( \eta\right)$ (a single $\theta $ with $f\,=\,\delta \,=\,1$ ); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups ${{C}_{3,}}\,{{C}_{5}},\,{{D}_{6}}$ ) is conjecture would imply that for all $p$ large enough, Fermat quotients, normalized $p$ -adic regulators are $p$ -adic units and that number fields are $p$ -rational.We recall some deep cohomological results that may strengthen such conjectures.

To a real reductive group G there is attached a family of (real) groups, each of lower dimension but sharing Cartan subgroups with G (cf. [8]). The purpose of these groups is to provide “building blocks” (in a specific sense (cf. [11])) for analysis on G. Their définition is via an L-group construction; the connected component of the identity, LH 0, in the L-group of such a group H is naturally a subgroup of LG 0 but the requirement that H “share” Cartan subgroups with G precludes defining LH the full L-group of H, as a subgroup of LG. Nevertheless, the principle of functoriality in the L-group suggests that the embeddings of LH in LG will play a role in analysis. In this paper, we study the embeddings of LH in LG in order to resolve a problem about the normalization of orbital integrals.

In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right),\,\left| q \right|\,<\,1$ , which he called mock $\theta $ -functions. He observed that as $q$ radially approaches any root of unity $\zeta $ at which $F\left( q \right)$ has an exponential singularity, there is a $\theta $ -function ${{T}_{\zeta }}\left( q \right)$ with $F\left( q \right)\,-\,{{T}_{\zeta }}\left( q \right)\,=\,O\left( 1 \right)$ . Since then, other functions have been found that possess this property. These functions are related to a function $H\left( x,\,q \right)$ , where $x$ is usually ${{q}^{r}}$ or ${{e}^{2\pi ir}}$ for some rational number $r$ . For this reason we refer to $H$ as a “universal” mock $\theta $ -function. Modular transformations of $H$ give rise to the functions $K,\,{{K}_{1}},\,{{K}_{2}}$ . The functions $K$ and ${{K}_{1}}$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mock $\theta $ -functions of even order and $H$ are listed.

The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\,\mapsto \,\text{Det}\,{{\text{D}}^{2}}u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function $u$ on a bounded set $\Omega \,\subset \,{{\mathbb{R}}^{2}}$ satisfies the equation $\text{Det}\,{{D}^{2}}u\,=\,0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

This paper is a continuation of $[6]$ . We consider the model subspaces ${{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}$ of the Hardy space ${{H}^{2}}$ generated by an inner function $\Theta $ in the upper half plane. Our main object is the class of admissible majorants for ${{K}_{\Theta }}$ , denoted by Adm $\Theta $ and consisting of all functions $\omega $ defined on $\mathbb{R}$ such that there exists an $f\ne 0,f\in {{K}_{\Theta }}$ satisfying $|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$ . Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any ${{K}_{\Theta }}$ generated by a meromorphic inner function. In contrast with $[6]$ , we consider the generating functions $\Theta $ such that the unit vector $\Theta \left( x \right)$ winds up fast as $x$ grows from $-\infty \,\text{to}\,\infty $ . In particular, we consider $\Theta \,=\,B$ where $B$ is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$ . It is shown, among other things, that for any such $B$ , any even $\omega $ decreasing on $\left( 0,\,\infty\right)$ with a finite logarithmic integral is in Adm $B$ (unlike the “vertical” case treated in $[6]$ ), thus generalizing (with a new proof) a classical result related to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ . Some oscillating $\omega $ 's in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ , and to de Branges’ space $H\left( E \right)$ .

In this article we refine the method of Bertolini and Darmon $\left[ \text{BD}1 \right],\,\left[ \text{BD2} \right]$ and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

If a is a degree of unsolvability, a is called high if a ≦ 0’ and a’ = 0”. In [1], S. B. Cooper showed that if a is high, then (i) a is not a minimal degree, and (ii) there is a minimal degree b < a. We give new proofs of these results which avoid the intricate priority and recursive approximation arguments of [1] in favor of “oracle” constructions using the recursion theorem. Also our constructions apply to degrees a which are not below 0'. Call a degree a generalized high if a’ = (a U 0')' . Among the degrees ≦ 0', the generalized high degrees obviously coincide with the high degrees.

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.

On sait que la méthode classique de Schneider (en une variable) permet de minorer des combinaisons linéaires de deux logarithmes de nombres algébriques avec des coefficients algébriques. Nous généralisons cette méthode en plusieurs variables pour minorer des combinaisons linéaires de plusieurs logarithmes.

If X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X * is not a w *-G δ subset of X *. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w *-G δ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.

We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

The fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals.