§1. Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equation

with integers k≥2 and m≥2, has only the trivial solutions. x = −j(j = i, …, m), y = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].

By introducing the concept of polarity in convex sets, it is possible, in a natural way, to generalize several classic characterizations of ellipsoids, showing that all of them depend upon and are related to the concept of projective centre of symmetry. Using these ideas, it is also possible to develop new characterizations of ellipsoids and to propose new problems.

Let L/K be an extension of number fields and let be the subgroup of the unit group consisting of the elements that are roots of units of . Denote by (L/K, B) the number of points in with relative height in the sense of Bergé-Martinet at most B. Here ℙ1(L) stands for the one-dimensional projective space over L. In this paper is proved the formula (L/K, B) = CB2 + O(B2−1/[L:Q]), where C is a constant given in terms of invariants of L/K such as the regulators, class number and discriminant.

A function is called strongly unbounded on a domain D if there exists a sequence in D on which f and all its derivatives tend to infinity. A result of Gordon is generalized to show that an unbounded analytic function on a quasidisk is always strongly unbounded there.

An extension of Asplund's theorem concerning the n-extreme and the n-exposed points of a convex body in ℝn and an extension of Liberman's characterization of convexity are given for closed convex bounded sets with the RNP.

Consider a real valued Morse function f on a C2 closed connected n-dimensional manifold M. It is proved that a suitable Riemannian metric exists on M, such that f is harmonic outside the set of critical points of f of index 0 and n. The proof is based on a result of Calabi [1], providing a criterion for a closed one-form on a closed connected manifold to be harmonic with respect to some Riemannian metric.

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

§1. Introduction. In this paper, we consider the spectral functions ρα(μ) for μ ∈ R associated with a Dirac equation on [0, ∞) given by

together with the initial condition

Let X and Y be separable metrizable spaces, and f: X→Y a function. It is wished to recover f from its values on a small set via a simple algorithm. It is shown that this is possible if f is Baire class one, and in fact a characterization is obtained. This leads to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

This paper gives a partial answer to a problem posed by Volčič and shows, in particular, that a three-dimensional convex body K is uniquely determined if p′ and p″ are two points interior to K and the lengths of all the chords of K through p′ and the areas of all sections of K with planes through p″ are known, provided that a specific condition on the positions of p′ and p″ with respect to K is satisfied. The problem will be studied in the more general framework of i-chord functions, and the results will also cover cases where the points p′ and p″ are not interior to K, possibly with one of them at infinity.