Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type x（t） （*）∫Rtα（s)x(s)ds=f(t),t∈R,（1）as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Bauach space-valued defined on a compact interval R of R^n Rt is a subinterval of R depending on t ∈ R and (*) f denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems, The Henstock integral setting enables us to consider highly oscillating functions.     

Existence of Solutions for Second Order Partial Neutral Functional Differential Equations

We establish existence of mild solutions for a class of abstract second-order partial neutral functional differential equations with unbounded delay in a Banach space. The work of the second author was supported by FONDECYT-CONICYT, Grants 1050314 and 7050034.     

A Construction of Subsets of the Reals which have a Similarity Decomposition

We construct a subset of the set R of real numbers of cardinality |R| which has a similarity decomposition, and which has an ordertype < that of R. Seymour Ginsburg had posed the question whether there exist sets with another ordertype than that of R which also have a similarity decomposition.      

On a conjecture of Schmutz

In this work we discuss Schmutz’s conjecture that in dimension 2 to 8 the distinct norms that occur in the lattices with the best known sphere packings are strictly greater than those in any other lattice of the same covolume. We see that the ternary conjecture is not true. However, it seems that there is but one exception: one lattice, where for one length the conjecture fails. Received: 11 February 2008, Revised: 20 May 2008     

Unique continuation of semi-conformality for a harmonic mapping onto a surface

We show that a harmonic mapping ϕ from either a three-manifold (with a condition on its Ricci curvature) or from a surface with values in a surface which has rank 2 somewhere, satisfies the following unique continuation property: if ϕ is semi-conformal on an open set, then it is semi-conformal everywhere.     

On a class of slowly changing functions

We study the slowly changing functions and use them in the study of entire functions. An erratum to this article is available at .     

Inverse-Definiteness of the Fourth-Order Symmetric Differential Operator （Ⅰ）

Abstract   We give a linear symmetric differential operator L defined by

A generalization of an integral of Ramanujan

This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral is itself a version of the dilogarithm,

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.     

On the diophantine equation x 2 + p 2k = y n

Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23]. Received: 25 April 2008     

A dynamic location problem with maximum decreasing capacities

In this paper a capacitated dynamic location problem with opening, closure and reopening of facilities is formulated and a primal-dual heuristic that can solve this problem is described. The problem formulated considers the situation where a facility is open (or reopens) with a certain maximum capacity that decreases as clients are assigned to that facility during its operating periods. This problem is NP-hard. Computational results are presented and discussed. This research was partially supported by research project POCTI/ISFL-1/152 and POCTI/MAT/139/2001.     

Intersection domains and the sum of two complex hyperbolic metrics

In this article, we prove that the intersection of the unit ball in ℂⁿ with an affine transformation of it is a negatively curved domain. The two-dimensional case is due to Cheung and Wu.     

The Commutators of Fractional Integrals on Besov Spaces

Abstract   In this paper we consider the commutators of fractional integrals

A p-adic analogue of a formula of Ramanujan

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures. Received: 11 April 2008     

Asymptotic behavior of geophysical fluids in highly rotating balls

In this note we study the convergence in the limit of small Ekman and Rossby numbers of the magnetohydrodynamics equations relevant to describe the flow in the Earth core. In particular, we prove the nonlinear stability of Ekman-Hartmann type boundary layers in a spherical geometry for some class of well-prepared initial data.     

On the birational rigidity of some pencils of Del Pezzo surfaces

In this note, we discuss birational properties of some three-dimensional Del Pezzo fibrations of degree two. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 62, Algebraic Geometry-10, 1999.     

On a Bernoulli-integrable magnetohydrodynamic system

A geometric procedure is used to isolate four Bernoulli integrals for a nonlinear, two-dimensional magnetohydrodynamic system. The general solution of the latter may thereby be constructed.     

On a discrete norm of the self-inversive unimodular polynomials

The quotient of divided by , whereP is a self-inversive and unimodular polynomial of any degree, dominates an absolute constantK>1. A 1989 paper gaveK=1.0252 on which its authors conjetured that the best constant is . We supply counter examples to their claim and provide a partial result for whenever theL _q norm is replaced by some “discrete” type norm. Research supported by the Shiraz University Grant 72-SC-784-432.     

Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros

We prove that an absolute constantc>0 exists such that

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered.An-alyticity in time for a class of solutions taking values in a Gevrey class of functions is proven.Exponentialdecay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by theexponential decay are also obtained.