Let q be a prime power, Fq a field with q elements, f ∈ Fq[x] a polynomial of degree n ≥ 1, V(f) = #f(Fq) the number of different values f(α) of f, with α ∈ Fq, and p = q – V(f). It is shown that either ρ = 0 or 4n4 > q or 2pn > q. Hence, if q is “large” and f is not a permutation polynomial, then either n or ρ is “large”.

Weak spectral synthesis fails in the group algebra and the generalised group algebra of any non compact locally compact abelian group and also in the Fourier algebra of any infinite compact Lie group.

In the present paper a simple proof of a theorem of Sargent on CλP-integral of Burkill is given.

In this paper, sufficient conditions for oscillations of the first order neutral differential equation with variable coefficients

are obtained, where c, τ, σ and µ are positive constants, p, q ∈ C ([t0, ∞), R+).

Starting from the well-known classical theorem of Archimedes about the volumes of spheres and circumscribing cylinders in three-dimensional Euclidean space, one considers circumscribing tubes of small geodesic spheres in general Riemannian manifolds and one derives new characterisations of two-point homogeneous spaces from it.

We investigate the existence of best approximation of an element α in a function module from a subfunction module whose fibers satisfy the intersection property of balls. Also we investigate the lower semicontinuity of the metric projection associated with such a subfunction module.

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

Let R be a commutative ring with unit, T an R-module, and n a positive integer. It is proved that T is n-flat over R if B⊗RT is B-torsionfree for each n–generated commutative R-algebra B. The converse holds if T is n–generated, in which case T is actually flat over R. Several other instances of the converse are established, but it is shown that the converse fails in general, even for R an integral domain, T an ideal of R, and n = 1.

Let X be a compactification of the ray with the arc as remainder. The following characterisation of the open images of X is obtained: Let h: X → Y be an open onto map. If Y is not homeomorphic to [0, 1] or the one-point space, then h is a homeomorphism. In 1977 open images of the usual sin (1/x) continuum were characterised by Professor Sam B. Nadler.

We give a description of the structure of the semigroups for which each principal ideal is a retract. The globally idempotent case is solved quickly using a—suitably modified—construction which has been developed by Tully for the study of semigroups in which each ideal is a retract. The general case can be treated by a naturally obtained semigroup of subsets of the semigroup constructed for the globally idempotent case.

A new deformation lemma for functions satisfying the Palais-Smale condition on a real Banach space is obtained. This is used to deduce some critical point theorems which are extensions of some well known results.

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

The purpose of this paper is to study the chain recurrent sets under persistent dynamical systems, and give a necessary condition for a persistent dynamical system to be topologically stable. Moreover we show that the various recurrent sets depend continuously on persistent dynamical systems.

We prove an identity involving Nörlund polynomials, the proof of which is elementary and involves the enumeration of lattice points. The identity is slightly stronger than an identity of Carlitz which he obtained by using Apostol's transformation formula for Lambert series.

The object of the present paper is to derive some interesting coefficient estimates for quasi-subordinate functions. Furthermore, a conjecture for quasi-subordinate functions is shown.

The resolvent operator and the moment generating function of a reflected Ornstein-Uhlenbeck process are obtained. These results are then applied to determine the long-run average cost and the total expected discounted cost of operating a finite storage system with content-dependent release rate.

Given a finitely generated multiplicative subgroup Us in a number field, we employ a simple argument from the geometry of numbers and an inequality on multiplicative dependence in number fields to obtain a minimal set of generators consisting of elements of relatively small height.