On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

About Kacʼs program in kinetic theory

In this Note we present the main results from the recent work of Mischler and Mouhot (2011) [15], which answers several conjectures raised fifty years ago by Kac (1956) [10]. There Kac introduced a many-particle stochastic process (now denoted as Kac?s master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in Kac (1956) [10]: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the H-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.     

Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory

In this paper we study the arithmetic of the Rikunas generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that and k where is a primitive n-th root of unity and =+^–1. In particular, this result with k implies the classical Kummer theory.Mathematics Subject Classification (2000): 11R20, 12E10, 12G05     

Order of Selbergʼs and Ruelleʼs zeta functions for compact even-dimensional locally symmetric spaces

We prove that Selberg?s and Ruelle?s zeta functions considered by U. Bunke and M. Olbrich can be represented as quotients of two entire functions of order not larger than the dimension of the underlaying compact, even-dimensional, locally symmetric space.     

Approximation formulas for Landauʼs constants

In this paper, we establish approximation formulas for evaluating Landau?s constants.     

A generalization of Ohnoʼs relation for multiple zeta values

In this paper, we prove that certain parametrized multiple series satisfy the same relation as Ohno?s relation for multiple zeta values. This result gives us a generalization of Ohno?s relation for multiple zeta values. By virtue of this generalization, we obtain a certain equivalence between the above relation among the parametrized multiple series and a subfamily of the relation. As applications of the above results, we obtain some results on multiple zeta values.     

A new formulation of Harnackʼs inequality for nonlocal operators

We provide a new formulation of Harnack?s inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnack?s inequality given here generalizes Harnack?s classical result from 1887 to nonlocal situations. As a consequence we derive Hölder regularity estimates by an extension of Moser?s method. The inequality that we propose is equivalent to Harnack?s original formulation but seems to be new even for the Laplace operator.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Moritaʼs duality for split reductive groups

In this paper, we extend the work in [Z. Qi, C. Yang, Morita?s theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 [7]] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations.     

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller?s seminal paper. In particular, this paper extends Feller?s results for continuous Q-functions to measurable Q-functions and provides additional results.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

A generalization of Menonʼs identity

In this note we give a generalization of the well-known Menon?s identity. This is based on applying the Burnside?s lemma to a certain group action.     

Ramanujanʼs Master Theorem for Riemannian symmetric spaces

Ramanujan?s Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujan?s Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one.