This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

Let $G$ be a finite group and $\chi $ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi H$ has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi $ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi H$ has a linear constituent with multiplicity 1.

We show that certain ${{C}^{*}}$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

In this paper we study nonlinear elliptic problems of Neumann type driven by the $p$-Laplacian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a ${{C}^{1}}$-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-$p$-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition.

We define a motivic analogue of the Haar measure for groups of the form $G(k((t)))$, where $k$ is an algebraically closed field of characteristic zero, and $G$ is a reductive algebraic group defined over $k$. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a certain natural Boolean algebra of subsets of $G(k((t)))$. This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of varieties over the base field. It is invariant under translations by all elements of $G(k((t)))$, and therefore we call it a motivic analogue of Haar measure. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process.

Let $p$ be a prime number ≥ 5 and $a,\,b,\,c$ be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation $a{{x}^{p}}\,+\,b{{y}^{p}}\,=\,c{{z}^{2}}$, in case the product of the prime divisors of $abc$ divides $2\ell $, with $\ell $ an odd prime number. For instance, under some conditions on $a,\,b,\,c$, we provide a constant $f(a,\,b,\,c)$ such that there are no such solutions if $p\,>\,f(a,\,b,\,c)$. In application, we obtain information concerning the $\mathbb{Q}$-rational points of hyperelliptic curves given by the equation ${{y}^{2}}\,=\,{{x}^{p}}\,+\,d$ with $d\,\in \,\mathbb{Z}$.

We prove ${{L}^{p}}({{\mathbb{T}}^{2}})$ boundedness, $1\,<\,p\,\le \,2$, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.

The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.