Dans la perspective de construire une théorie de Galois infinie pour les extensions non extérieures, nous montrons dans ce texte que le monoïde des endomorphismes d'une $Z$-extension intérieure filtrée s'identifie à la complétion procentrale du groupe de ses automorphismes intérieurs. En particulier, ce monoïde a une structure topologique naturelle qui fait de lui un espace complet et totalement discontinu. In order to build an infinite Galois theory for non-outer extensions, we show that the monoïd of endomorphisms of a filtered inner $Z$-extension identifies with the procentral completion of the group of its inner automorphisms. In particular, this monoid has a natural topological structure for which it is a complete and totally discontinuous space.

Let $H$ be a Hilbert space and $(\Omega,\mathcal{F},\mu)$ a probability space. A Hilbert point in $L^p(\Omega; H)$ is a nontrivial function $\varphi$ such that $\|\varphi\|_p \leq \|\varphi+f\|_p$ whenever $\langle f, \varphi \rangle = 0$. We demonstrate that $\varphi$ is a Hilbert point in $L^p(\Omega; H)$ for some $p\neq2$ if and only if $\|\varphi(\omega)\|_H$ assumes only the two values $0$ and $C>0$. We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.

In this paper we consider sequences of nonlinear functionals of Gaussian random fields. We prove their convergence to multifractional processes which generalize Hermite processes.

In this work, we numerically investigate how a defect can affect the behavior of traveling explosive solutions of quintic NLS equation in the one-dimensional case. Our numerical method is based on a Crank-Nicolson scheme in the time, finite difference method in space including a Perfectly Matched Layer (PML) treatment for the boundary conditions. It is observed that the defect splits the incident wave in one reflected part and one transmitted part; hence the dynamics of the solution may be changed and the blow-up may be prevented depending on the values of the defect amplitude $Z$. Moreover, it is numerically found that the defect can be considered as a barrier for large $Z$.

Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals on horizontal lines in the complex plane. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane) that are independent of the size of the graph. We also deduce partial results indicating that Benjamini-Schramm convergence of open quantum graphs should imply convergence of the empirical spectral measures.

Following an idea of J. Shapiro, we give a simple proof of the fact that an element of the Gordon Hedenmalm class $\Phi$ such that $\Phi(\infty)=\infty$ defines a contractive composition operator $C_\Phi$ on the space $\mathcal{H}^2$ of Dirichlet series.

We revisit, in an elementary way, the \emph{classical statement} of various ``Main Conjectures'' for $p$-class groups ${\mathscr{H}}_{ K}$ and $p$-ramified torsion groups ${\mathscr{T}}_{ K}$ of abelian fields $K$, in the non semi-simple case $p \mid [K : \bbQ]$. The classical ``algebraic'' definition of the $p$-adic isotypic components, ${\mathscr{H}}^{\scriptscriptstyle\text{alg}}_{K,\varphi}$, used in the literature, is inappropriate with respect to analytical formulas. For that reason we have introduced, in the 1970's, an ``arithmetic'' definition, ${\mathscr{H}}^{\scriptscriptstyle \mathrm{ar}}_{K,\varphi}$, in perfect correspondence with all analytical formulas and giving a natural ``Main Conjecture'', still unproved for real fields in the non semi-simple case. The two notions coincide for relative class groups ${\mathscr{H}}_{ K}^-$ and groups ${\mathscr{T}}_{ K}$ since transfer maps are injective, in $p$-extensions for these groups, but not necessarily for real class groups. Numerical evidence of the gap between the two notions is given (Examples \vref{Ex1}, \vref{Ex2}) and PARI calculations corroborate that the true Real Abelian Main Conjecture writes $\# {\mathscr{H}}^{\scriptscriptstyle\mathrm{ar}}_{ K,\varphi} = \order ({\mathscr{E}}_{ K} / \widehat{\mathscr{E}}_{ K} \, {\mathscr{F}}_{ K})^{e_{\varphi_0}}$ ($\varphi = \varphi_0^{} \varphi_p$, $\varphi_0^{}$ of prime-to-$p$ order, $\varphi_p$ of $p$-power order, $e_{\varphi_0}$ being the corresponding idempotent), in terms of units ${\mathscr{E}}_{ K}$, $\widehat{\mathscr{E}}_{ K}$ (units of the strict subfields) and ${\mathscr{F}}_{ K}$ (Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation of ${\mathscr{H}}_{ K}$ in some auxiliary cyclotomic extensions $K(\mu_\ell^{})$, $\ell \equiv 1 \pmod {2p^N}$ prime, proves the difficult non semi-simple real case.

We obtained a moment estimate for the sum of Rademacher random variables under condition that they are dependent in the way that their sum is zero.

Let $A$ be a nonempty finite subset of $\mathbb{Z}^d$ which is not contained in a hyperplane, $q\in\mathbb{Z}$ with $|q|>1$ and $m\in \mathbb{Z}$ such that $|q|+2d-1\leq m\leq (|q|+2d-1)^2$. In this paper it is shown that \begin{equation*} |A+q\cdot A|\geq \left(\frac{m}{|q|+2d-1}\right)|A|-c \end{equation*} where $c$ depends only on $q,d$ and $m$. In particular, taking $m=(|q|+2d-1)^2$, this results confirms a conjecture of A. Balog and G. Shakan.

In this paper, we study a class of convolution operators on the space of distributions that enlarge the well-studied class of passive operators. In this larger class, we are able to associate, to each operator, a holomorphic function in the right half-plane with a specific constraint on its range, determined by the operator. Afterwards, we investigate whether the properties of causality and slow growth hold automatically in our larger class of convolution operators. Finally, an alternative class of convolution operators is also considered.

In this paper, we consider the space of entire functions of minimal type growth for a proximate order. We study the surjectivity of corresponding in- finite order differential equations in the cases of regular singular type and of Korobeı̆nik type.

We obtain expected number of arrivals, absorption probabilities and expected time until absorption for an asymmetric discrete random walk on a graph in the presence of multiple function barriers. On each edge of the graph and in each vertex (barrier) specific probabilities are defined.

Les E- et G-fonctions de Siegel ont été définies en deux sens, strict et large, conjecturalement équivalents. En reprenant et complétant une esquisse d’André, nous énonçons et démontrons l’analogue au sens large du théorème d’André-Chudnovsky-Katz, qui est un théorème de structure sur les G-opérateurs au sens strict (il s’agit d’opérateurs différentiels annulant les G-fonctions au sens strict). Nous en déduisons un théorème de structure sur les E-opérateurs au sens large, qui sont des opérateurs différentiels annulant les E-fonctions au sens large. En application de ce dernier théorème, nous donnons une nouvelle preuve d’une généralisation par André du théorème de Siegel-Shidlovskii sur l’indépendance algébrique des valeurs des E-fonctions au sens large.

In Coulombel (2015) a multiplier technique, going back to Leray and Gård- ing for scalar hyperbolic partial differential equations, has been extended to the context of finite difference schemes for evolutionary problems. The key point of the analysis in Coulombel (2015) was to obtain a discrete energy-dissipation balance law when the initial difference operator is multiplied by a suitable quan- tity (the so-called multiplier). The construction of the energy and dissipation functionals was achieved in Coulombel (2015) under the assumption that all modes were separated. We relax this assumption here and construct, for the same multiplier as in Coulombel (2015), the energy and dissipation functionals when some modes cross. Semigroup estimates for fully discrete hyperbolic initial boundary value problems are deduced in this broader context.

We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if $G$ is a simply connected simple linear algebraic group and $\phi:G\to {\mathrm{GL}}(V)$ is a non-trivial irreducible representation for which there exists a non-regular non-central semisimple element $s\in G$ such that $\phi(s)$ has almost simple spectrum, then, with few exceptions, $G$ is of classical type and $\dim V$ is minimal possible. Here the spectrum of a diagonalizable matrix is called \emph{simple} if all eigenvalues are of multiplicity 1, and \emph{almost simple} if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.

We give a new proof of the existence of a surjective symbol whose associated composition operator on $H^2 (\mathbb{D})$ is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the $1$-dimensional case, for $N \geq 2$, the behavior of the approximation numbers $a_n = a_n (C_\phi)$, or rather of {$\beta^-_N = \liminf_{n \to \infty} [a_n]^{1/ n^{1/ N}}$ or $\beta^+_N = \limsup_{n \to \infty} [a_n]^{1/ n^{1/ N}}$}, of composition operators on $H^2 (\mathbb{D}^N)$ cannot be determined by the image of the symbol.

If $\mathcal{D}$ is a Reedy category and $\mathcal{M}$ is a model category, the category $\mathcal{M}^{\mathcal{D}}$ of $\mathcal{D}$-diagrams in $\mathcal{M}$ is a model category under the Reedy model category structure. If $\mathcal{C} \to \mathcal{D}$ is a Reedy functor between Reedy categories, then there is an induced functor of diagram categories $\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}$. Our main result is a characterization of the Reedy functors $\mathcal{C} \to \mathcal{D}$ that induce right or left Quillen functors $\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}$ for every model category $\mathcal{M}$. We apply these results to various situations, and in particular show that certain important subdiagrams of a fibrant multicosimplicial object are fibrant.

In this paper, we consider the space of entire functions with normal type growth for a given proximate order and a continuous linear operator from such space into itself which is defined by a partial differential operator of infinite order. We will study corresponding partial differential equations with variable coefficients in the cases of regular singular type and of Korobeı̆nik type.

An increasing sequence of positive integers $(n_k)$ is said to be Jamison if whenever $T$ is a linear bounded operator on a complex separable Banach space, the following holds : \[\sup_{k}||T^{n_k}||<\infty \Rightarrow \sigma_{p}(T)\cap S^1 \text{ is countable}\] In this paper, we study certain perturbations on the set of Jamison sequences and prove a stability result.

We discuss the evaluation of convolution sums involving the divisor function, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, for the class of levels $\alpha\beta$ belonging to all natural numbers. The evaluation of convolution sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. We illustrate our method with the explicit examples for the levels $\alpha\beta=33$, $40$, $45$, $50$, $54$, and $56$. As a corollary, the known convolution sums for the levels $\alpha\beta=10$, $11$, $12$, $15$, $16$, $18$, $24$, $25$, $27$, $32$ and $36$ are improved when we revisit their evaluations. If the level $\alpha\beta\equiv 0 \pmod{4}$, we determine natural numbers $a,b$ and use the evaluated convolution sums together with other known convolution sums to carry out the number of representations of $n$ by the octonary quadratic forms $a\,\underset{i=1}{\overset{4}{\sum}}x_{i}^{2}+ b\,\underset{i=5}{\overset{8}{\sum}}x_{i}^{2}$. Similarly, if the level $\alpha\beta\equiv 0 \pmod{3}$, we compute natural numbers $c,d$ and make use of the evaluated convolution sums together with other known convolution sums to determine the number of representations of $n$ by the octonary quadratic forms $c\,\underset{i=1}{\overset{2}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,) + d\,\underset{i=3}{\overset{4}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$. In addition, we determine formulae for the number of representations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(1,6)$, $(2,3)$.

Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. We study consequences of $K$ having a large or undefined Szlenk index and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the Szlenk indices of $X$. We also show that for a given ordinal $\xi$ the class of operators whose Szlenk index is not an ordinal less than or equal to $\xi$ admits a universal element if and only if $\xi<\omega_1$; W.B. Johnson's theorem that the formal identity map from $\ell_1$ to $\ell_\infty$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.

We consider a multidimensional SDE with a Gaussian noise and a drift vector being a vector function of bounded variation. We prove the existence of generalized derivative of the solution with respect to the initial conditions and represent the derivative as a solution of a linear SDE with coefficients depending on the initial process. The obtained representation is a natural generalization of the expression for the derivative in the smooth case. In the proof we use the results on continuous additive functionals.

We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences.

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous continuous operator $T: Y\longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \mathbb{R}$ and every point $y\in Y$, $f\circ T$ is Fréchet differentiable at $y \in Y$ whenever $f$ is Gâteaux differentiable at $T(y)\in X$.

We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particular that either the $\partial\overline{\partial}$-Lemma or the property that the Hodge and Frölicher spectral sequence degenerates at the first level are not closed under deformations.

It is known for quite some time that the extension theorems play an important role in the homogenization of the periodic (heterogeneous) mediums. However, the construction of such extension operators depends on a reflection technique but for the functions in $H^{l,r}(\Omega_p^\varepsilon)$ $(l>2)$ this reflection technique is not so straightforward, and would lead to a rather cumbersome anaylsis. In this work, we will give a short overview of some extension operators mapping from $L^{r}(S;H^{l,r}(\Omega_p^\varepsilon))\cap H^{1,r}(S; H^{l,s}(\Omega_p^\varepsilon)^*) \to L^{r}(S;H^{l,r}(\Omega))\cap H^{1,r}(S; H^{l,s}(\Omega)^*)$ using a much simpler approach. This note also generalizes the previously known results to Lipschitz domains and for any $r\in \mathbb{N}$ such that (s.t.) $\frac{1}{r}+\frac{1}{s}=1$.

An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many postivie integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.

In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $\Delta/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category $\mathfrak{C}[B]$. We study the relationship with Lurie's straightening theorem. Along the way we also prove some results on localizations of simplicial categories and quasi-categories.

A metric compact space $M$ is seen as the closure of the union of a sequence $(M_n)$ of finite $\epsilon_n$-dense subsets. Extending to $M$ (up to a vanishing uniform distance) Banach-space valued Lipschitz functions defined on $M_n$, or defining linear continuous near-extension operators for real-valued Lipschitz functions on $M_n$, uniformly on $n$ is shown to be equivalent to the bounded approximation property for the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Several consequences are spelled out.

This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory.Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation.

This subordination principle states roughly: if a property is true for Hardy spaces in some kind of domains in ${\mathbb{C}}^{n}$ then it is also true for the Bergman spaces of the same kind of domains in ${\mathbb{C}}^{n-1}.$ We give applications of this principle to Bergman-Carleson measures, interpolating sequences for Bergman spaces, $A^{p}$ Corona theorem and characterization of the zeros set of Bergman-Nevanlinna class. These applications give precise results for bounded strictly-pseudo convex domains and bounded convex domains of finite type in ${\mathbb{C}}^{n}.$

Let $d\geq 2$, $A \subset \mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \in \mathbb{Z}$ such that $|q| \geq 2$. We show $$|A+ q \cdot A| \geq (|q|+d+1)|A| - O(1).$$

We characterise smooth curve in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two yield Sarkisov links that are birational selfmaps of the cubic threefold, namely $(g,d) = (0,5)$ and $(2,6)$. Using the link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold with Picard number $3$. We also prove that the group of birational selfmaps of any smooth cubic threefold contains elements contracting surfaces birational to any given ruled surface.

Cet article est consacré à l'étude détaillée du vaste projet éditorial de Borel dans les années 1920 et 1930 autour des probabilités et de leurs applications. Après avoir rappelé quelques éléments sur la biographie de Borel et décrit la mise en place du projet, nous examinons les acteurs qui y ont participé pour mieux cerner le réseau que Borel a mis en place pour arriver à ses fins. Enfin, dans une troisième partie les fascicules du Traité sont aussi examinés individuellement afin de dessiner le contour du domaine probabiliste tel que Borel le concevait, dont nous montrons qu'il est en fait déjà obsolète au moment où la publication s'achève.

We are interested here in a birth-and-growth process where germs are born according to a Poisson point process with intensity measure invariant under space translations. The germs can be born in free space and then start growing until occupying the available space. In order to consider various ways of growing, we describe the crystals at each time through their geometrical properties. In this general framework, the crystallization process can be characterized by the random field giving for a point in the state space the first time this point is reached by a crystal. We prove under general conditions that this random field is mixing in the sense of ergodic theory and obtain estimates for the coefficient of absolute regularity.