This paper is concerned with homogenisation processes for parametrised families of transport equations in ℝn symmetric hyperbolic systems in several space dimensions and anisotropic wave equations. The main tools for carrying out the homogenisation are the Young measures and Radon transform combined with the integral representation of holomorphic functions of Nevanlinna-Pick's type.

An initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variables

The complex function H is measurable on ℂ ×I × ℂ5, with I being an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

We use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u =(u1, …, uN), . We also prove the boundedness of the minimiser of l1 by using the reverse Holder inequality.

Weakly coupled semilinear elliptic systems of the form

are considered in RN, N≧2, where k = 1, 2, …, M, u = (u1, …, uM) and λ is a real constant. The aim of this paper is to give sufficient conditions for (*) to have entire solutions whose components are positive in RN and converge to non-negative constants as |x| tends to ∞. For this purpose a new supersolution-subsolution method is developed for the system (*) without any hypotheses on the monotonicity of the non-linear terms fk with respect to u.

The question of “correctness” of cardinal interpolation with shifted three-directional box splines is solved for arbitrary orders of the directional vectors. It is shown that the corresponding symbol can be viewed as a collection of curves with certain properties (convexity, increasing argument, etc.) which are investigated in detail. The method of proof involves an induction argument which is based on properties of the exponential Euler splines (studied in [6]).

Components in the function space of maps from a space X to the classifying space BG of a topological group G can sometimes be distinguished up to homotopy type by a Samelson product method. When X is a closed Riemann surface and G is a unitary group, this method is nearly sufficient to classify the components up to homotopy type.

In an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This is used to estimate the exponential rank of certain algebras of operator-valued continuous functions.

In this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.

In this paper we characterize the universal pointed actions of a semigroup S on a compact space such that the orbit of the distinguished point is dense; such actions are called transitive. The characterization is given in terms of the universal right topological monoidal compactification of S. All transitive actions are shown to arise as quotients modulo left congruences on this universal compactification. Minimal actions are considered, and close connections between these and minimal left ideals of the compactification are derived.

On a convex surface S ⊂ Rd, two points x, y are conjugate if there are at least two shortest paths, called segments, from x to y. This paper is about the set of points conjugate to some fixed point xєS.

The problem of finding rational points on varieties defined by two additive cubic equations has attracted some interest. Davenport and Lewis [12], Cook [8] and Vaughan [16] showed that the pair of equations

with integer coefficients a,, bt always has a nontrivial solution when s = 18, s = 17, and 5 = 16 respectively. Vaughan's result in s = 16 variables is best possible since there are examples of pairs of equations (1) with s = 15 which fail to vanish simultaneously in the 7-adic field. However if the existence of a 7-adic solution is assured then Baker and Briidern [2], building on work of Cook [9], showed that s = 16 could be replaced by s = 15, and recently Briidern [5] has obtained the result with s = 14.

The flow induced by an oscillating circular cylinder which may perform transverse, torsional and axial vibrations is considered. The steady streaming associated with purely transverse vibrations of the cylinder may be significantly modified by the presence of, and interaction with, torsional oscillations. Similarly the interaction between the transverse and axial vibrations introduces a modification to the axial flow, which results in a steady streaming motion in the axial direction.

Let K0(x) be a simple transcendental extension of a field K0, υ0 be a valuation of K0 with value group G0 and residue field K0. Suppose is an inclusion of totally ordered abelian groups with [G1: G0] < ∞ such that G is the direct sum of G1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K0(x) extending υ0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k0. This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ0 to K0(x).

Let Q(x) = Q(x1,…, xn) є ęZ x1, …, xn] be a quadratic form. The primary purpose of this paper is to bound the smallest non-zero solution of the congruence Q(x) = 0 (mod q). The problem may be formulated as follows. We ask for the least bound Bn(q) such that, for any Ki > 0 satisfying

and any Q, the congruence has a non-zero solution satisfying