A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

It is proved that there are infinitely many integers n such that n and n + 1 have the same number of distinct prime divisors.

We show that, if f : M2×2 → R is rank-one convex on the hyperboloid is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on M2×2 uniformly on compact subsets of . Equivalently, every gradient Young measure supported on a compact subset of is a laminate.

Let R be an integral domain with quotient field K and let X be an indeterminate. A result of W. C. Waterhouse states that, if each quadratic polynomial f ∈ R[X] which factors into linear polynomials in K[X] also factors into linear polynomials in R[X], then every irreducible element in R is prime. In this note the rings which satisfy the hypothesis of this theorem are characterized, and compared to the rings for which each polynomial f ∈ R[X] which factors into two polynomials of positive degree in K[X] also factors into two polynomials of positive degree in R[X]. Relevant examples are furnished via the pullback construction.

A generalization of the following two facts is proved: the class of generalized zonoids is dense in the class of convex compact bodies with a centre of symmetry, and the class of generalized triangle bodies is dense in the class of all convex compact bodies (proved by Schneider). The main result is deduced from a more general representation theoretical statement.

A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

The structures of the module Q/R over certain domains R are investigated, where Q denotes the field of quotients of R.

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form If(u) = ∫Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.

Generalizing an earlier notion of secondary polytopes, Billera and Sturmfels introduced the important concept of fibre polytopes, and showed how they were related to certain kinds of subdivision induced by the projection of one polytope onto another. There are two obvious ways in which this concept can be extended: first, to possibly unbounded polyhedra, and second, by making the definition a categorical one. In the course of these investigations, it became clear that the whole subject fitted even more naturally into the context of finite tilings which admit strong duals. In turn, this new approach provides more unified and perspicuous explanations of many previously known but apparently quite disparate results.

§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 applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

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.

For positive rational integers λ, we study the Hecke L-series attached to elliptic curves y2 = x3 − 2433Dλ over the quadratic field Q(√−3) and obtain various bounds of p(= 2, 3)-adic valuations of their values at s = 1 according to the cases of D and λ. In particular, for the case of even λ, we obtain a criterion of reaching the bounds of 3-adic valuations. From this, combining with the work of Coates and Wiles and Rubin, we obtain some results about the conjecture of Birch and Swinnerton-Dyer of these curves.