We show that if {ak}k is bounded then for almost every 0 < x < 1 where is the dyadic expansion of x. It is also shown that almost everywhere where p > 1 is any fixed integer.

We prove the existence of a solution for a semilinear boundary value problem at resonance in the first eigenvalue. The nonlinearity is assumed to be bounded below or above; no further growth restrictions are assumed.

We give the relation between the (rigid) graded Morita duality and the Morita duality on a finitely group-graded ring and the relation between a left Morita ring and some of its matrix rings.

The known effective bounds for the magnitude of integer points on elliptic curves are exponential in the “height” of curve. The bound given in this note is polynomial in the height and depends slightly on the discriminant.

Some extension properties of maps defined on dense subsets are studied for approximate polyhedra. The latter are characterised as approximate extensors for finite maps with small oscillation.

In this paper we show that given a σ-algebra Σ of subsets of a set Ω and a normed space X, then the normed space B(Σ, X), endowed with the usual supremum-norm, of the X-valued functions defined on Ω that are the uniform limit of a sequence of σ-simple X-valued functions on Ω is barrelled of class s if and only if X is barrelled of class s. This extends in the normed case the well known result obtained by Mendoza (1982) for barrelled spaces.

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets Sj (j = 1, 2) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2 must be identical?”, where Ef(Sj) stands for the inverse image of Sj under f. In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions Ef(S) = Eg(S) and Ef({∞}) = Eg({∞}) imply f ≡ g. As a special case this also answers the question posed by Gross.

A characterisation is given of separable Banach spaces with symmetric Schauder bases which admit equivalent symmetric norms that are Gâteaux differentiable or uniformly rotund in every direction. Some applications to questions on distortion of norms on l∞ are discussed.

The simple modules of , the coordinate ring of quantum affine space, are classified in the case when q is a root of unity.

We prove the existence of an order-preserving function on a class of preordered topological spaces that are inductive limits of preordered spaces. Some applications to economics are given.

New covering properties of simplexes are obtained which extend and complement some covering theorems for simplexes of Fan.

Our main theorem describes FC-soluble and linear repetitive groups. As a corollary, we characterise algebraic linear repetitive semigroups.

For every nonsingleton finite set A, we construct three families of partial clones on A that contain all permutations of A and are of continuum cardinality.

Using the concept of equi-integrability we introduce a measure of weak noncompactness in the Lebesgue space L1(0, l). We show that this measure is equal to the classical De Blasi measure of weak noncompactness.

We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same asymptotics.

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

We show that the Murasugi conditions for the Alexander polynomial of a cyclically periodic knot imply a modified form of the Burde-Trotter condition.

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

It is shown that an amenable algebra of operators on Hibert space which is generated by its normal elements is necessarily self-adjoint, so it is a C*-algebra.

Let L = Q[α] be a cyclic number field of odd prime degree q over the field Q of rationals. In this paper we give an algorithm to compute the discriminant of L/Q, which relies upon a fast method to find Eisenstein elements in L. The algorithm accepts as input the minimal polynomial of α over Q and a complete factorisation of the discriminant of α, and computes, in time polynomial in the size of the input, a list consisting of all the ramified primes with corresponding Eisenstein elements.