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

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.

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.

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.

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.

We give a brief survey of the results obtained by numerous authors in so-called almost-Hermite-Fejér-interpolation and deal mainly with new quantitative assertions.

These are based upon more general theorems for certain continuous linear operators which yield estimates involving different types of moduli of continuity.

Our paper shows that in the case of almost-Hermite-Fejér-interpolation the underlying general technique can be used to treat three essentially different cases: sequences of positive operators, which converge uniformly for every continuous function on [−1, 1], sequences of non-positive operators doing the same, and sequences of operators which converge on proper subspaces of C[−1, 1] only.

The main purpose of this paper is to show that there exists a Souslin line if and only if there exists a countable chain condition space which is not weak-separable but has a generic π-base. If I is the closure of the isolated points in a space X, then X is said to be weak-separable if a first category set is dense in X – I. A π-base is said to be generic if, whenever a member of is included in the disjoint union of members of it is included in one of them.

It is shovn that simple algebras of characteristic not equal to 2, which contain non-trivial elements that satisfy xn = x for some fixed, minimal integer n > 1, are generated by these elements.

Let G1, G2, be locally compact groups and let S1, S2, be Segal algebras on G1, G2, respectively. Under certain conditions on G1, G2, and S1, S2, we prove that if there is a bipositive or isometric isomorphism between S1, S2, or between their multiplier algebras then G1, and G2, are topologically isomorphic.

Let a function be regular in the disk |z| < 1. The radius of univalence 0.164 … of the family of f with |an| ≤ n (n ≥ 2) is, actually, the radius of star-likeness. The radius of univalence 1 - [k/(l+K)]½ of the family of f with |an| ≤ k (n ≥ 2), where K > 0 is a constant, is, actually, the radius of starlikeness. The radii of convexity of the two families are estimated from below.

The statement used in proving Theorem 2 of [1] needs explanation. This was pointed out to us by Professor S. Raghwan of Tata Institute of Fundamental Research, Bombay, and we gave the explanation of this in [2]. For the sake of completeness we give here the full proof of the theorem; filling the gap in the proof. We use the same notations and definitions as those of [1]. Also for simplicity of notation we write kmn, gmn and amn to mean km,n, gm,n and a respectively. We need the following lemma.