A quick distributional way to the prime number theorem

We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results are employed.     

Multivariate versions of Bartlett’s formula

This paper quantifies the form of the asymptotic covariance matrix of the sample autocovariances in a multivariate stationary time series—the classic Bartlett formula. Such quantification is useful in many statistical inferences involving autocovariances. While joint asymptotic normality of the sample autocovariances is well-known in univariate settings, explicit forms of the asymptotic covariances have not been investigated in the general multivariate non-Gaussian case. We fill this gap by providing such an analysis, bookkeeping all skewness terms. Additionally, following a recent univariate paper by Francq and Zakoian, we consider linear processes driven by non-independent errors, a feature that permits consideration of multivariate GARCH processes.     

Extension of Caristi’s fixed point theorem to vector valued metric spaces

The paper deals with the classical Caristi fixed point theorem in vector valued metric spaces. The results obtained seem to be new in this setting.     

Distributional Taylor formula

We derive a distributional Taylor formula with precise integral remainder. We give applications of it and estimates for the remainder.     

Distributions vectorielles homogènes sur une algèbre de Jordan

We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions exist if and only if the representation is spherical, and that then the dimension of the space of these distributions is r+1 (where r is the rank of V). We give also construction of these distributions and of those that are invariant under the semi-simple part of G.     

Itô’s stochastic calculus and Heisenberg commutation relations

Stochastic calculus and stochastic differential equations for Brownian motion were introduced by K. Itô in order to give a pathwise construction of diffusion processes. This calculus has deep connections with objects such as the Fock space and the Heisenberg canonical commutation relations, which have a central role in quantum physics. We review these connections, and give a brief introduction to the noncommutative extension of Itô’s stochastic integration due to Hudson and Parthasarathy. Then we apply this scheme to show how finite Markov chains can be constructed by solving stochastic differential equations, similar to diffusion equations, on the Fock space.     

System of nonlinear Volterra’s integral equations with polar kernel and singularities

We study an existence-uniqueness theorem for a system of nonlinear Volterra-type integral equations with polar kernel and singularities. To do so, we use a method of regularization of fractional derivatives with a delta sequence to avoid singularities over diagonal {y|x−y=0}$\left\{y\right|x-y=0\}$. The other singularities we regularize with delta sequences with different growth. As a framework, we use Colombeau special algebra.     

Factor maps and invariant distributional chaos

The main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic.     

Remarks on Caristi’s fixed point theorem and Kirk’s problem

Without assumptions on the continuity and the subadditivity of η, by means of Caristi’s fixed point theorem, we investigated the existence of fixed points for a Caristi type mapping which partially answered Kirk’s problem and improved Caristi’s fixed point theorem, Jachymski’s fixed point theorem and Khamsi’s fixed point theorem since φ is not necessarily assumed to be bounded below on X.     

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n  -Massey Conjecture for n≥3$n\ge 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.     

On convolution in weighted ‐spaces

A distributional generalization of Young's inequality was stated by L. Schwartz. It asserts that convolution yields a continuous bilinear map if A generalization to the weighted ‐spaces in the form was given by N. Ortner and the author in 1989. By means of interpolation theory, we improve this result with respect to the image space under certain restrictions on μ and ν. This implies limit relations in for the Poisson kernel and yields a solution of the Dirichlet problem for the half‐space with boundary values in the space By this we generalize a former result of J. Alvarez, M. Guzmán‐Partida and S. Pérez‐Esteva referring to the special case of      

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M   in the context of an inclusion M⊆M_?αG$M\subseteq M{?}_{\alpha }G$, where G is a discrete group acting on a factor M by outer ?-automorphisms. We characterize the M  -bimodules X⊆M_?αG$X\subseteq M{?}_{\alpha }G$ that are closed in the Bures topology in terms of the subsets of G  . We show that this characterization also holds for w^?$w^{?}$-closed bimodules when G has the approximation property (AP  ), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w^?$w^{?}$-continuous surjective isometric maps on X.     

A characterization of the minimal invariant sets of Alspach’s mapping

In 1981, Dale Alspach modified the baker’s transform to produce the first example of a nonexpansive mapping T on a weakly compact convex subset C of a Banach space that is fixed point free. By Zorn’s lemma, there exist minimal weakly compact, convex subsets of C which are invariant under T and are fixed point free.In this paper we produce an explicit formula for the nth power of T, Tⁿ, and prove that the sequence (Tⁿf)_n∈N converges weakly to , for all f∈C. From this we derive a characterization of the minimal invariant sets of T.     

Banach operator pairs and common fixed points in hyperconvex metric spaces

The purpose of this paper is to establish DeMarr’s well-known theorem for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without the compactness assumption. We also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings. As an application, several results on the invariant best approximation are proved.     

Abel?s summation formula in partially ordered groups

Abel?s partial summation formula has been used classically to obtain convergence tests for certain types of series of real or complex numbers. We generalize the formula and convergence tests to the setting of directed partially ordered topological groups, where the notion of a convergent series is replaced by that of a Cauchy multipliable sequence.     

A coercive James’s weak compactness theorem and nonlinear variational problems

We discuss a perturbed version of James’s sup theorem for weak compactness that not only properly generalizes that classical statement, but also some recent extensions of this central result: the sublevel sets of an extended real valued and coercive function whose subdifferential is surjective are relatively weakly compact. Furthermore, we apply it to generalize and unify some facts in mathematical finance and to prove that the unique possible framework in the development of an existence theory for a wide class of nonlinear variational problems is the reflexive one.     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

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we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

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we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

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we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.