Consider the following neutral delay differential equation

where p ∈ R, τ ∈ (0, ∞), δ ∈ R+ = (0, ∞) and Q ∈ (C([t0, ∞), R). We show that if

then Equation (*)has a nonoscillatory solution when p ≠ –1. We also deal in detail with a conjecture of Chuanxi, Kulenovic and Ladas, and Györi and Ladas.

Let n be an odd integer greater than 9. It is proved that the alternating group An has spread 3 in the sense that for any non-trivial elements x1, x2, x3 of An, there is an element y in An such that 〈xi, y〉 = An for i = 1, 2, 3.

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the $k$th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each $k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.

In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions $f$ are uniquely determined by their poles and the zeros of $f-e_{j}$ (counting multiplicities) for two finite complex numbers $e_{1}\neq e_{2}$.

The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such that

where S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S.

Let Cvp(G) be the space of convolution operators on the Lebesgue space LP(G), for an arbitrary locally compact group G. We describe Cvp(G) as a dual space, whose predual, is a Banach algebra of functions on G, under pointwise operations, with maximal ideal space G. This involves a variation of Herz's definition of AP(G); the benefit of this new definition is that all of Cvp(G) is obtained as the dual in the nonamenable setting. We also discuss further developments of this idea.

Let for t ∈ [a, b] ⊂ [0, ∞)

where Ws is an n-dimensional Wiener process, f(s) an n-vector process and G(s) an n × m matrix process. f and G are nonanticipating and sample continuous. Then the set of limit points of the net

in Rn is equal, almost surely, to the random ellipsoid Et = G(t)Sm, Sm = {x ∈ Rm: |x| ≤ 1}. The analogue of Lévy's law is also given. The results apply to n-dimensional diffusion processes which are solutions of stochastic differential equations, thus extending the versions of Hinčin's and Lévy's laws proved by H.P. McKean, Jr, and W.J. Anderson.

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.

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

Let k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

A local theory of holomorphic curves in the complex hyperquadric is worked out using the method of moving frames. As a consequence a complete global characterization of totally isotropic curves is obtained.

We prove that every bounded representation of the tensor product of two C*-algebras, one of which is nuclear and contains matrices of any order, is similar to a *-representation.

We introduce a new class of generalized arcwise connected functions and discuss their basic properties. Our generalization is illustrated by an example and an application is given for a mathematical programming problem involving this new class of functions.

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.

Dedicated to George Szekeres on his 90th birthday

We discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

In this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.