Using a variation on the concept of a CS module, we describe exactly when a simple ring is isomorphic to a ring of matrices over a Bézout domain. Our techniques are then applied to characterise simple rings which are right and left Goldie, right and left semihereditary.

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

Let Γ be a totally ordered abelian group and I an order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebra T(Γ) to the structure of the Toeplitz algebras T(I) and T(Γ/I). We then describe the primitive ideal space of the Toeplitz algebra T(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T(Γ) when 0→ I→ Γ→ Γ/I→ 0 is a non-trivial group extension.

Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. A further important point is that new examples of marginally trapped biharmonic Lagrangian surfaces in an indefinite complex Euclidean plane are obtained. This fact suggests that Chen and Ishikawa's classification of marginally trapped biharmonic surfaces [6] is not complete.

We give an asymptotic bound for the size of the n-generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.

In this paper we study linear fractional relations defined in the following way. Let i, 'i, i = 1,2, be Banach spaces. We denote the space of bounded linear operators by . Let T ε ( 1 ⊕ 2, '1 ⊕ '2). To each such operator there corresponds a 2 × 2 operator matrix of the form(*) where T ij ε ( j , 'i . For each such T we define a set-valued map G T from ( 1, 2) into the set of closed affine subspaces of ('1, '2) by

The map G T is called a linear fractional relation.

The paper is devoted to the following two problems.

• • Characterization of operator matrices of the form (*) for which the set G T(K) is non-empty for each K in some open ball of the space (1,2).

• • Characterizations of quadruples (1, 2, '1, '2) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem “a bounded entire function is constant”.

In this paper we obtain a Hermite-Hadamard type inequality for a class of subharmonic functions. Our proofs rely essentially on the properties of elliptic partial differential equations of second order. Our study extends some recent results from [1], [2] and [6].

Starting from results of Dubé and Mingarelli, Wahlén, and Ehrström, who give conditions that ensure the existence and uniqueness of nonnegative nondecreasing solutions asymptotically constant of the equation we have been able to reduce their hypotheses in order to obtain the same existence results, at the expense of losing the uniqueness part. The main tool they used is the Banach Fixed Point Theorem, while ours has been the Schauder Fixed Point Theorem together with one version of the Arzelà-Ascoli Theorem.

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of with ramification index e. The orders are given explicitly when G is a p-group of order p or p 2. An example is given when G is not abelian.

We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.

Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.

We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

The weighted energy theory for Navier-Stokes equations in 2D strips is developed. Based on this theory, the existence of a solution in the uniformly local phase space (without any spatial decaying assumptions), its uniqueness and the existence of a global attractor are verified. In particular, this phase space contains the 2D Poiseuille flows.

Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value. In a previous paper, we had found URSCM of 7 points for the whole set of unbounded analytic functions inside an open disk. Here we show the existence of URSCM of 5 points for the same set of functions. We notice a characterization of BI-URSCM of 4 points (and infinity) for meromorphic functions in K and can find BI-URSCM for unbounded meromorphic functions with 9 points (and infinity). The method is based on the p-Adic Nevanlinna Second Main Theorem on 3 Small Functions applied to unbounded analytic and meromorphic functions inside an open disk and we show a more general result based upon the hypothesis of a finite symmetric difference on sets of zeros, counting multiplicities.

There is a commutative algebra of differential-difference operators, acting on polynomials on , associated with the reflection group B 2. This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4 F 3-type. These properties are in the form of recurrence and contiguity relations and are proved herein.

Daniel Martin B.Sc., M.A., Ph.D., F.R.S.E. was born in Carluke on 16 April 1915, the only child of William and Rose Martin (née Macpherson). The family home in which he was born, Cygnetbank in Clyde Street, had been remodelled and extended by his father, and it was to be Dan's home all his life. His father, who was a carpenter and joiner, had a business based in School Lane, but died as a result of a tragic accident when Dan was only six. Thereafter Dan was brought up single handedly by his mother.

After attending primary school in Carluke from 1920 to 1927, Dan entered the High School of Glasgow. It was during his third year there that he started studying calculus on his own. He became so enthused by the subject that he set his sights on a career teaching mathematics, at university if at all possible. On leaving school in 1932, he embarked on the M.A. honours course in Mathematics and Natural Philosophy at the University of Glasgow. At that time the Mathematics Department was under the leadership of Professor Thomas MacRobert; the honours course in Mathematics consisted mainly of geometry, calculus and analysis, and the combined honours M.A. with Natural Philosophy was the standard course for mathematicians. A highlight of his first session at university was attending a lecture on the origins of the general theory of relativity, given on 20th June 1933 by Albert Einstein. This was the first of a series of occasional lectures on the history of mathematics funded by the George A. Gibson Foundation which had been set up inmemory of the previous head of the Mathematics Department. From then on, relativity was to be one of Dan's great interests, lasting a lifetime; indeed, on holiday in Iona the year before he died, Dan's choice of holiday reading included three of Einstein's papers.

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.

In this paper we prove that if R is a Prüfer domain, then the R-module R⊕ R satisfies the radical formula.

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then d = 4k or d = 64k 5−48k 3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.

In this paper we shall first show that if T is a class A(k) operator then its operator transform is hyponormal. Secondly we prove some spectral properties of T via . Finally we show that T has property (β).