Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation are given. Applications for the trapezoid and mid-point inequalities are also provided. Received: 19 May 2008     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.      

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that , for θ > 1/2, with the assumption of an heuristic hypothesis weaker than the Lindel?f hypothesis. Received: 8 October 2007, Revised: 14 April 2008     

On values of n 2 + 1 free of large prime factors

Let P ⁺(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Zeta functions of<Emphasis Type="Italic">q</Emphasis>-perfect numbers

We introduce a notion ofq-analogue of the perfect numbers. We also define a new zeta function which we call a zeta function ofq-perfect numbers. In this paper, the properties of theq-perfect numbers and the zeta functions are studied. Especially, we determine theq-perfect numbers whenq is a root of unity.     

Energy decay rate of the thermoelastic Bresse system

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions.      

Using null strain energy functions in compressible finite elasticity to generate exact solutions

In this paper we characterize those strain energy functions in unconstrained nonlinear elasticity that satisfy the equations of equilibrium identically. The idea is to construct a useful, physically reasonable strain–energy function containing one or more components which are null, in such a way that exact solutions may be obtained from the resulting equilibrium equations. We show that the dilatation is a universal null energy while there may be others that depend on the actual problem. To obtain the null energies for a given problem it is often convenient to formulate the variational problem and look at the Euler–Lagrange equations. Specific examples are used to illustrate some of the potential uses of the method in finding exact solutions for physically meaningful constitutive models.      

Localizable spectrum and bounded local resolvent functions

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Received: 15 November 2007     

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

Let D be a quaternion division algebra with an involution of the second kind. We show that the quotient group SU(1,D)/[U(1,D),U(1,D)] is nontrivial in general. For global fields, we completely determine this group. Received: 11 May 2007, Revised: 15 December 2007     

On the probability of having rational \ell-isogenies

We calculate the chance that an elliptic curve over a finite field has a specified number of -isogenies which emanate from it. We give a partial answer for abelian varieties of arbitrary dimension. Received: 20 September 2007     

Decay properties for the solutions of a partial differential equation with memory

Decay properties in energy norm for solutions of a class of partial differential equations with memory are studied by means of frequency domain methods. Our results are optimal for this class, as we are able to characterize polynomial as well as exponential decay rates. The results apply to models for viscoelastic materials. An extension to a semilinearly perturbed problem is also included. Received: 9 July 2008, Revised: 16 September 2008     

Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data

We obtain existence results for some strongly nonlinear Cauchy problems posed in and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on , they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.     

A Math Query Language with an Expanded Set of Wildcards

Math search is a new area of research with many enabling technologies but also many challenges. Some of the enabling technologies include XML, XPath, XQuery, and MathML. Some of the challenges involve enabling search systems to recognize mathematical symbols and structures. Several math search projects have made considerable progress in meeting those challenges. One of the remaining challenges is the creation and implementation of a math query language that enables the general users to express their information needs intuitively yet precisely. This paper will present such a language and detail its features. The new math query language offers an alternative way to describe mathematical expressions that is more consistent and less ambiguous than conventional mathematical notation. In addition, the language goes beyond the Boolean and proximity query syntax found in standard text search systems. It defines a powerful set of wildcards that are deemed important for math search. These wildcards provide for more precise structural search and multi-levels of abstractions. Three new sets of wildcards and their implementation details will also be discussed.