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.

For Empedocles, a little over 2400 years ago, there were four chemical elements, fire, water, earth, and air, and they were continually brought together and torn apart by two opposing forces, harmony and discord. For Aristotle, a hundred years later, two binary classification schemes took the place of one: instead of harmony-discord, he had wet-dry and hot-cold. The serious alchemists of the middle ages found that nature was even more complicated than that; they classified matter by its luster, heaviness, combustilibity, solubility, and so on. Boyle, in the 1600's, offered a definition similar to that of a prime number – an element is a substance from which other substances can be made but which cannot be separated into different substances – and analytical chemistry was off and running. A hundred years later Lavoisier (inspired in part by Newton's insight into the paramount importance of weight) could formulate a modified definition that led for the first time to quantitative tables of chemical elements similar to our modern ones.

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.

Jonathan A. Hillman asked “Must a 2-knot whose group has non-trivial center be fibered?” We will answer this question negatively.

The familiar Fuglede-Putnam theorem asserts that AX = XB implies A*X = XB* when A and B are normal. We prove that A and B* be hyponormal operators and let C be a hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert-Schmidt operator X, the Hilbert-Schmidt norm of AXD + CXB is greater than or equal to the Hilbert-Schmidt norm of A*XD* + C*XB*. In particular, AXD = CXB implies A*XD* = C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert-Schmidt operator of the hypothesis on X to be every operator and still retain the inequality under some suitable hypotheses.

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

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.

The purpose of this note is to establish results of a technical nature concerning a stochastic process that appears to be useful in the study of certain problems in statistical inference. These problems concern a test for symmetry, a method for obtaining an adaptive estimator of the centre of symmetry, and the detection of outliers with respect to the normal distribution. Details of the applications will be presented elsewhere.

A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

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.

It is shown that for a solution of a Riccati equation with polynomial coefficients an expansion can be constructed as a Stieltjes continued fraction, with coefficients given by a recurrence relation, which is in general non-linear. Particular expansions associated with hypergeometric and confluent hypergeometric equations are given, and are shown to have a uniquely simple form.

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 the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.

It is shown that a transformation group with a locally compact Hausdorff phase space and an abelian phase group is locally weakly almost periodic if and only if it is P-locally weakly almost periodic for some replete semigroup P in the phase group.

Let D be a domain bounded by a Jordan curve. For 1 ≤ p ≤ ∞, let Lp(D) be the class of all functions f holomorphic in D such that where A is the area of D. For f ∈Lp(D), set

πn consists of all polynomials of degree at most n. Recently, Andre Giroux (J. Approx. Theory 28 (1980), 45–53) has obtained necessary and sufficient conditions, in terms of the rate of decrease of the approximation error , such that has an analytic continuation as an entire function having finite order and finite type. In the present paper we have considered the approximation error (*) on a Carathéodory domain and have extended the results of Giroux for the case 1 ≤ p < 2.

Bernardi has proved that if f if starlike univalent in the uni disc E, then so is the function g given by

In the first part of the paper, we extend Bernardi's theorem to a certain class of p-valent starlike functions in E. We prove that if then g, defined by

also belongs to . In the second part of the paper we examine the converse problem for functions with negative coefficients, satisfying certain conditions.

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.

We consider finite affine planes having a collineation group acting as a rank 3 permutation group on the affine lines. By a classical result of A. Wagner, such affine planes are translation planes. We show that if, in addition, the plane has odd dimension or dimension 2 over its kernel, then the plane is Desarguesian.

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.