Maximal isotropic varieties of the singular symplectic structure on R2n are characterised in terms of generating families. The normal forms of the simplest singularities (of codimension 1) are obtained with the help of the theory of boundary singularities.

The rôle of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

The purpose of this paper is to get some necessary conditions for a Poisson stable flow to be recurrent and to analyse the bilateral versions of positive and negative Lipschitz stability. Moreover, a characterisation of recurrent orbits is obtained in a certain flow.

Let Mn and N2n−2 be smooth, connected manifolds of dimension n and 2n − 2 respectively with n ≡ 2 mod 4 and 6 ≤ n ≤ 26. Let f: Mn → N2n−2 be a continuous map. Under certain suitable conditions on the stable normal bundle of f, we give a direct and simpler proof that f is homotopic to an immersion. For the case 6 ≤ n ≤ 26 and n ≠ 18, the result was proved by Li-Banghe and Peterson by using non-stable obstruction theory and their earlier result.

The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

A theorem is given on the interchange of integrals for the product of a Radon polymeasure and a measure. Examples show that if the conditions of the theorem are not satisfied, then the conclusions may not hold.

In this note the recent result of Corduneanu on almost periodicity of L2(G)-bounded solutions of nonlinear parabolic equations is extended to the case when the nonlinear growth rate is beyond the first eigenvalue of the associated elliptic boundary value problem.

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

In this paper the oscillatory and asymptotic properties of solutions of the equation

are investigated where δ = ±1, τ > 0, σ > 0, the functions r(s) and are non- decreasing and .

We obtain a topological zero-one law for sets with the Baire property which are invariant under a semigroup of open continuous maps acting on a topological space.

In this paper we prove that if R is a ring with unity satisfying [xy − xn, ym = 0, for all x, y ∈ R and fixed integers m ≥ 1, n ≥ 1, then R is commutative.

Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

In this paper, we give a new characterisation of pseudo-compact spaces, namely a space X is pseudocompact if and only if each σ-point finite open cover of X has a finite subfamily whose union is dense. As a corollary, we show that every pseudocompact σ-metacompact (or screenable) space is compact, which sharpens some known results.

This paper shows that amongst the p-groups of order p5, where p denotes an odd prime, there is only one group whose automorphism group is again a p-group. This automorphism group has order p6 and it is shown that this is the smallest order a p-group may have when it occurs as an automorphism group. The paper also shows that all groups of order p5 have an automorphism of order 2 apart from the group above and three other related groups.

Inserting new knots into B-spline curves is a well-known technique in CAGD to gain extra flexibility for design purposes. However, from a user's point of view, the insertion of knots is somewhat unsatisfactory since the newly generated control points sometimes show up in unexpected locations. The aim of this note is to show that these problems can be circumvented by inserting the control vertices directly, thus also providing a more natural user interface.