In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G. More precisely, let D be a closed subgroup of G and let X denote the coset space G/D. Let S be a subgroup of G and let Z denote (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, the set of fixed points of S on X. We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G; again we consider the orbits of (GS)0 on XS.

We construct handle-bundle decompositions of compact G-manifolds, G a compact Lie group, that are particularly well adapted to the orbit structure of the group action.

We apply methods of Mahler to obtain explicit lower bounds for certain combinations of E-functions satisfying systems of linear differential equations as studied by Makarov. Our results sharpen and generalise earlier results of Mahler, Shidlovskii, and Väänänen.

A nonlinear nondifferentiable program with linear constraints is considered and a converse duality theorem is discussed. First we weaken an assumption previously made by Bhatia, and then give a simple proof under this weaker hypothesis, using the Fritz John conditions. Finally, defining a generalized Slater constraint qualification which implies Abadie's constraint qualification, we give a simple condition for the dual problem to satisfy this constraint qualification.

Let G be a p-group of order n and embed G into Sn by the Cayley representation. If X is a group such that G≨X≤Sn and CX(G) = G, then it is proved that G is properly contained in NX(G).

In this paper we introduce two classes, namely Γp(m, M) and Σp(m, M), of functions

regular and p-valent in D − {0} where D = {z : |z| < 1}. We show that, for suitable choices of real constants α, β and γ, the integral operators of the form

map into , where is the class of p-valent meromorphically starlike functions of order ρ, 0 ≤ ρ < 1. For the classes Γp(m, M) and Σp(m, M), we obtain class preserving integral operators of the form

with suitable restrictions on real constants α and γ.

Our results generalize almost all known results obtained so far in this direction.

In this paper we prove two results relating the cardinalities of discrete subsets to those of dense subsets of a topological space. In particular, we show that

(1) closed discrete subsets of separable normal spaces are countable, and

(2) discrete subsets of separable regular spaces have cardinalities at most that of the continuum.

We prove first that a logical fraction functor from a topos to a topos must be a filter-power functor, then we prove that such functors can have adjoints only when the filter is principal. Finally we refine this so that we are able to prove that the filter-power of a Grothendieck topos is Grothendieck if and only if the filter is principal.

The full collineation group of the flag transitive plane of order 27 constructed by Hering is determined. It is shown that the stabilizer of the origin of this plane is of order 2184.

We apply recent results of Vaaler that simultaneously bound linear forms so as to obtain a ‘Siegel lemma’, sharper than those that have appeared in the literature, and in a shape convenient for application in transcendence theory.

In a metric space (X, d) a ball B(x, ε) is separated if d(B(x, ε), X\B(x, ε)] > 0. If the separated balls form a sub-base for the d-topology then Ind X = 0. The metric is gap-like at x if dx(X) is not dense in any neighbourhood of 0 in [0, ∞). The usual metric on the irrational numbers, P, is the uniform limit of compatible metrics (dn), each dn being gap-like on P. In a completely metrizable space X if each dense Gδ is an Fσ then Ind X = 0.

If μ and γ are ordinal numbers such that for every α ≤ γ·2, then γ < ω. This localizes a result due to Patai.

There are two Hardy and Littlewood theorems which describe the rate of growth of functions in Hp on the unit circle T. In this paper we first establish their analogues on Euclidean space Rn and then apply them to solve multiplier and factorization problems on Hp(Rn).