The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L Schrödinger data and wave data in H 1 2 ×H − 1 2 . In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces – previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-...

We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line s = 1/2.

We give the (Ahumada type) Selberg trace formula for a semiregular bipartite graph G: Furthermore, we discuss the distribution on arguments of poles of zeta functions of semiregular bipartite graphs. As an application, we present two analogs of the semicircle law for the distribution of eigenvalues of specified regular subgraphs of semiregular bipartite graphs. r 2003 Elsevier Science (USA). Al...

We study the twisted homology group attached to a Selberg type integral under some resonance condition, which naturally appears in the su2conformal field theory and the representation of the Iwahori-Hecke algebra. We determine the dimension of the space of the regularizable cycles. The dimensionformula is given in terms of the generalized hypergeometric series 3F2.

This paper is based on the introduction to the monograph ”Double affine Hecke algebras” to be published by Cambridge University Press. It is based on a series of lectures delivered by the author in Kyoto (1996–1997), at University Paris 7 (1997–1998), at Harvard University in 2001, and in several other places, including recent talks at the conferences ”Quantum Theory and Symmetries 3” (Cincinna...

It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succe...

Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The purpose of this work is to use Hadamard fractional integral to establish some new integral inequalities of Gruss type by using one or two parameters which ensues four main results . Furthermore, other integral inequalities of reverse ...

In this paper, we consider the evaluation of two kinds of oscillatory integrals with a Hankel function as kernel. We first rewrite these integrals as the integrals of Fourier-type. By analytic continuation, these Fourier-type integrals can be transformed into the integrals on [0, + ), the integrands of which are not oscillatory, and decay exponentially fast. Consequently, the transformed integr...

1. In [12] and [13] Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite vol...