Let ℳ be the collection of all intersections of balls, considered as a subset of the hyperspace ℳ of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. It is proved that ℳ is uniformly very porous if and only if the space fails the Mazur intersection property.

§1. Introduction. In 1946, Davenport and Heilbronn [9] proved a result which opened up the study of Diophantine inequalities. Suppose that Q(x) is a diagonal quadratic form with non-zero real coefficients in s variables. We write

The integral representation for the solution of the 2-D Dirichlet problem for harmonic functions with boundary data on closed and open curves is obtained. The solution is expressed as a sum of potentials, the density of which obeys the uniquely solvable Fredholm integral equation of the second kind.

§0. Introduction. Low-dimensional topology is dominated by the fundamental group. However, since every finitely presented group is the fundamental group of some closed 4-manifold, it is often stated that the effective influence of π1 ends in dimension three. This is not quite true, however, and there are some interesting border disputes. In this paper, we show that, by imposing the extra condition of parallelizability on the tangent bundle, the dominion of π1 is extended by an extra dimension.

Consider a convex polytope X and a family of convex sets, satisfying a given property P. Moreover, assume that is closed under operations of cutting and convex pasting along hyperplanes. Necessary and sufficient conditions are given to have . As a consequence, it follows that, if all simplices or small enough simplices have the property in question, then X also has that property.

The problem of scattering of tidal waves by reefs and spits of arbitrary shape is reduced to a skew derivative problem for the two-dimensional Helmholtz equation in the exterior of open arcs in a plane. The resulting boundary-value problem is studied by potential theory and a boundary integral equation method. After some transformations, the skew derivative problem is reduced to a Fredholm integral equation of the second kind, which is uniquely solvable. In this way the solvability theorem is proved and an integral representation of the solution is obtained. A uniqueness theorem is also proved.

Let d≥2 and let K⊂ℝd be a convex body containing the origin 0 in its interior. For each direction ω, let the (d−l)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum when the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on Sd−1, whose null-space needs to be, and will be determined.

First, a special case of Knaster's problem is proved implying that each symmetric convex body in ℝ3 admits an inscribed cube. It is deduced from a theorem in equivariant topology, which says that there is no S4–equivariant map from SO(3) to S2, where S4 acts on SO(3) on the right as the rotation group of the cube, and on S2 on the right as the symmetry group of the regular tetrahedron. Some generalizations are also given.

An original linear algebraic approach to the basic notion of Freiman's isomorphism is developed and used in conjunction with a combinatorial argument to answer two questions, posed by Freiman about 35 years ago.

First, the order of growth is established of t(n), the number of classes isomorphic n-element sets of integers: t(n) = n(2 + σ(1))n. Second, it is proved linear Roth sets (sets of integers free of arithmetic progressions and having Freiman rank 1) exist and, moreover, the number of classes of such cardinality n is amazingly large; in fact, it is “the same as above”: .

§1. Introduction. We study in this paper some properties of the Lusternik-Schnirelmann category of isolated invariant sets of continuous dynamical systems. There are several different definitions of this coefficient, although most of them agree in the important case of ANR's (Absolute Neighbourhood Retracts). We refer to the review articles [10] by R. H. Fox and [15, 16] by I. M. James for general information about this topological invariant. We shall use in this paper the definition of the Lusternik-Schnirelmann category of a compactum introduced by K. Borsuk in [4].

Let φ be the Euler function, and consider the error term H in the asymptotic formula

Let P⊂ℝ2 be a polyhedron, that is, the intersection of a finite number of closed half-spaces, and suppose that its characteristic function lP can be expressed as a linear combination

where each Ai is a relatively open and convex set. Let n(P) be the number of all non-empty facets of P. One of the main objectives of this paper is to show that

Let E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation υ (we do not require any finiteness condition on the residue class field of E). Let f(X) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of f(X). This method consists of attaching to each side S of a Newton polygon of f(X) a factor (not necessarily irreducible) of f(X), the degree of which is the length of the horizontal projection of S.

A new criterion on Catalan's equation is proved by elementary means

This shows, without appealing either to the theory of linear forms in logarithms, or to any computation, that (C) has no solution (x, y, p, q) with min {p, q}≤41, except (3,2, 2, 3).

We construct non-trivial laminates distributed continuously along a smooth closed ‘evolute’ curve by approximating an auxiliary (‘involute’) curve by a polygon with rank-1 sides. We give an explicit example in which the evolute has no rank-1 connections. Using these techniques, given a finite set of matrices B1,…,Br and any other matrix A, we show how to construct a laminate with positive mass at each Bi, any proportion less than 1 of its support on the Bi, and centre of mass at A.

It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.

The Cauchy problem for the time-dependent Ginzburg–Landau equations of superconductivity in Rd (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L2. When d = 3, we establish the existence result of solutions for L3 initial data and the uniqueness result for L4 initial data.

§1. Introduction. We consider the spectral function ρα(λ) associated with the Sturm–Liouville equation

with the boundary condition

A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.

We establish the uniqueness of positive radial solutions of −Δu = up − u in B(R1, R2), u = 0 on ∂B(R1, R2), where B(R1, R2) is an annulus and 0 < R1 < R2 ≤ ∞, in the following cases.

(a) n ∈ {3, 4} and 1 < p ≤ n/(n − 2).

(b) n ∈ {5, 6, 7, 8} and 1 < p ≤ p0(n) for some p0(n) < n/(n − 2).

Earlier to this result, the uniqueness has been obtained by Coffman for n = 3 and 1 < p ≤ 3 and by Yadava for p ≥ (n + 2)/(n − 2) and n ≥ 3.