Let b and c be bases of a matroid. Then for any integer r, there exists an injection σ from r−subsets I of b to r-subsets σ(I) of c such that b – I + σ(I) is a base for all I. This result has implications for the structure of matroid base graphs.

Using a slight generalisation of Brown's inequality, we show that for martingales the existence of a weak nonuniform bound on the rate of convergence in the central limit theorem yields the usual upper bound part of the law of the iterated logarithm.

In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalises a theorem of Mazhar [7] on |C, 1|k summability factors of infinite series, is proved. Also we apply the theorem to Fourier series.

We give a formula for the Euler-Poincare characteristic of the fixed point set of the Cartan involution on the set of integral equivalence classes of integral unimodular hermitian forms, in terms of a product of special values of Riemann zeta functions and Dirichlet L-functions. This is done via reduction by Galois cohomology to a volume computation using the Tamagawa measure on the unitary groups.

In this paper we examine a Lagrange optimal control problem driven by a nonlinear evolution equation involving a nonmonotone, state dependent perturbation term. For this problem we establish the existence of optimal admissible pairs. For the same system we also examine a time optimal control problem involving a moving target set. Finally we work out in detail an example of a strongly nonlinear parabolic distributed parameter system.

We show that any hyperbolic flow (X, π) on a metric space X is topologically stable by showing that it is expansive and has the chain-tracing property.

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

It is shown that a bounded linear operator T from Lρ(Rn) to itself which commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this we show that the ρ-norm of the Hilbert transform is the same as the ρ-norm of its truncation to any Lebesgue measurable subset of Rn with non-zero measure.

We consider C∞ manifolds endowed with torsionfree affine connections and C∞ projective submersions between them which, by definition, map geodesics into geodesics up to parametrisation. After giving a differential characterisation of these mappings, we deal with the case when one of the given connections is projectively flat or satisfies certain conditions concerning its Ricci tensor; under these hypotheses we prove that the projective submersion is actually a covering.

The obvious necessary conditions for a triple system of index two having a given automorphism group to underly a directed triple system having the same group are shown to be sufficient. An efficient algorithm for directing the triple system in a way that preserves its group is given. Applications to the existence of directed triple systems with various automorphisms are outlined.

For a ring R with identity we show that the existence of certain nilpotent elements forces R to be a matrix ring of size ≥ 2.

Let F be a free group on a1,…, ap (p ≥ 1), and X a finitely generated subgroup in F. Suppose either that X contains some non-trivial power of , or that p is even and X contains some nontrivial power of [a1, a2]…[ap−1, ap]. We discuss some properties of X which we can derive from this assumption.

Let {Sn, n ≥ 1} denote the partial sum of sequence (Xn) of identically distributed martingale differences. It is shown that E|X1|q (lg |X1|)r < ∞ implies E(sup((lg n)pr/q/npr/q)|Sn|p) < ∞, where 1 < p < 2, p < q, r ∈ R and lg x = max{1, log+x} For the independent identically distributed case, the converse of the above statement holds.

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.

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 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 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.

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.