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).

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

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

with the boundary condition

§1. Introduction. The spectral function ρα(μ) (−∞<μ<∞) associated with the Sturm–Liouville equation

and a boundary condition

is a non-decreasing function of μ which is defined in terms of the Titchmarsh–Weyl function mα(λ) for (1.1) and (1.2).

The Newhouse gap lemma is generalized by finding a geometric condition which ensures that N-fold sums of compact sets, which might even have thickness zero, are intervals. A new proof is also obtained of a lower bound on the thickness of the sum of two Cantor sets.

Let ϑ be an integer of multiplicative order t≥1 modulo a prime p. Sums of the form

are introduced and estimated, with a sequence such that kz1, …, kzT is a permutation of z1, …, zT, both sequences taken modulo t, for sufficiently many distinct modulo t values of k. Such sequences include

xn for x = 1 ,…,t with an integer n≥1;

xn for x = 1 ,…,t and gcd (x, t) = 1 with an integer n≥1;

ex for x = 1 ,…,T with an integer e, where T is the period of the sequence ex modulo t.

Some of the results can be extended to composite moduli and to sums of multiplicative characters as well. Character sums with the above sequences have some cryptographic motivation and applications and have been considered in several papers by J. B. Friedlander, D. Lieman and I. E. Shparlinski. In particular several previous bounds are generalized and improved.

Let q be a prime number and let a = (a1, …, as) be an s-tuple of distinct integers modulo q. For any x coprime with q, let be such that . For fixed s and q→∞ an asymptotic formula is given for the number of residue classes x modulo q for which

The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q], is also treated.

One of the converse statements to Lagrange's theorem is that, for each subgroup H of G and any prime factor p of |G: H|, there exists a subgroup K such that H≤K≤G with |K: H | = p. This paper treats integers n such that all groups of order n have this property.

For a completely regular space X, denote by Cp(X) the space of continuous real valued functions on X, endowed with the pointwise convergence topology. The spaces X and Y are t-equivalent if Cp(X) and Cp(Y) are homeomorphic. It is proved that, for metrizable spaces X, the countable dimensionality is preserved by t-equivalence. It is also shown that this relation preserves absolute Borel classes greater than 2 and all projective classes.

The key result of this paper proves the existence of functions ρn(h) for which, whenever H is a (Lebesgue) measurable subset of the n-dimensional unit cube In with measure |H| > h and ℛ is a class of subintervals (n-dimensional axis-parallel rectangles) of In that covers H, then there exists an interval R∈ℛ in which the density of H is greater than ρn(h); that is, |H∩R|/|R|>ρn (h) (=(h/2n)2). It is shown how to use this result to find 4 points of a measurable subset of the unit square which are the vertices of an axis-parallel rectangle that has quite large intersection with the original set. Density and covering properties of classes of subsets of ℝn are introduced and investigated. As a consequence, a covering property of the class of intervals of ℝn is obtained: if ℛ is a family of n-dimensional intervals with , then there is a finite sequence R1, …, Rm∈ℛ such that and .