Voronoĭ conjectured that every parallelotope is affinely equivalent to a Voronoĭ polytope. For some m, a parallelotope is defined by a set of m facet vectors pi, and defines a set of m lattice vectors ti, for 1≤i≤m. It is shown that Voronoĭ's conjecture is true for an n-dimensional parallelotope P if and only if there exist scalars γi, and a positive definite n × n matrix Q such that γipi = Qti for each i. In this case, the quadratic form f(x) = xTQx is the metric form of P.

A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.

There is an error in the proof of Theorem 2 of my paper [1]. It appears on; age 91, lines 10 and 11: the application of the affine transformation T changes he measure on the Grassmannian G(d, d-i) which is not taken into account. As a result, in the statement of Theorem 2 the coefficient is not correct.

Let L(f) denote the Legendre transform of a function f: ℝn → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f(x) = f(x−a) such that

This paper introduces a wide generalization of a family of integral lattices defined by Coxeter, which share with the Coxeter lattices the following properties: they are perfect, often with an odd minimum, and have no non-trivial perfect sections with the same minimum.

Let A⊆ℕ, let p be a prime and w a word over ℤ pℤ ending with a non-zero digit. The relationship is investigated between the density of A. the length of w and the density of the set of numbers n for which the base p expansion of ends with w0n for some a ∈ A. Also considered is the analogous problem on Pascal's triangle. This leads in particular to answering a question of Granville and Zhu [7] regarding the asymptotic frequency of sums of 3 squares in Pascal's triangle.

For the purpose of this paper, we call a set of positive reals ν a well-spaced set if there is a c > 0 such that

Let q be a natural number. When the multiplicative iroup (ℤ/qℤ)* is a cyclic group, its generators are called primitive roots. Note that the generators are also elements with the maximum order if (ℤ/qℤ)* is cyclic. Thus, when (ℤ–qℤ)* is not a cyclic goup, we then call an element with: he maximal possible order a primitive root, which was initially introduced by R. Carmichael [1].

A subsemigroup S of a semigroup Q is a left order in Q, and Q is a semigroup of left quotients of S, if every element of Q can be written as a−1b for some a, b∈S with a belonging to a group -class of Q. Necessary and sufficient conditions on a semigroup S are obtained in order that S be a left order in a completely 0-simple semigroup Q. The class of all completely 0-simple semigroups of left quotients of S is related to the set of certain left congruences on S. Axioms are provided for semigroups which occur in the discussion of left orders in completely 0-simple semigroups.

This article studies the non-homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

The set ℳ* of numbers which occur as Mahler measures of integer polynomials and the subset ℳ of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number α of degree d in ℳ* is the Mahler measure of a separable integer polynomial of degree at most with all its roots lying in the Galois closure F of ℚ(α), and every unit in ℳ is the Mahler measure of a unit in F of degree at most over ℚ This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.

Given a Hausdorff topological vector space with dimensiongreater than one, the barycentre of simple masses can be seen as the unique associative, internal and continuous mapping defined on these masses. Moreover, if the associated dual space separates points, by extending the continuity property, one can characterize also the barycentre of masses with compact convex support.

In this note we generalize the following result of Sawyer [5]:

Theorem 1. There is a function ψ on ℝ such that, whenever g is a real-valued Borel measurable function on (a subset of) ℝ. × ℝn-1 with the property that y ↦ g(y, t) is C1 for a.e. t, the set

has measure zero.

The present paper extends the idea of characterizing topological properties of a space X by means of continuous selections for its closed subsets (X) endowed with a “natural” hyperspace topology. In this particular case, it is proved that the property of X to be topologically well-orderable is equivalent to the existence of a selection for (X) which is continuous with respect to the Fell topology.

The problem of finding necessary and sufficient condi-tions for the existence of trapped modes in waveguides has been known since 1943. [10]. The problem is the following: consider an infinite strip M in ℝ2(or an infinite cylinder with the smooth boundary in ℝn). The spectrum of the(positive) Laplacian, with either Dirichlet or Neumann boundary conditions, acting on this strip is easily computable via the separation of variables; the spectrum is absolutely continuous and equals [v0,+∞). Here, v0 is the first threshold, i.e., eigenvalue of the cross-section of the cylinder (so v0 = 0 in the case of Neumann conditions). Let us now consider the domain (the waveguide) which is a smooth compact perturbation of M (for example, weinsert an obstacle inside M). The essential spectrum of the Laplacian acting on still equals [v0, +ℝ), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large, see examples in [11] and [8].

This paper studies the existence and multiplicity of positive solutions of the following problem:

where Ω⊂RN(N≥3) is a smooth bounded domain, , 1 < p < N, and 0 < α < 1, p - 1 < β < p* - 1 (p* = Np/(N - p)) and 0 < γ < N + ((β + 1)(p - N)/p) are three constants. Also δ(x) = dist(x, ∂Ω), a ∈ Lp and λ < 0 is a real parameter. By using the direct method of the calculus of variations, Ekeland's Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if λ is small enough.