A necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

This paper deals with determining in a constructive manner those members of a linear space of functions which are of integrable-square. The space considered is the set of solutions to an ordinary differential equation, and the solutions of integrable-square are delineated by way of initial conditions. Numerical procedures for implementing the construction are discussed, and application is made to the deficiency index problem. Results from some specific computations are given.

We prove that sufficiently separated sequences are interpolating sequences for f′(z)(1−|z|2) where f is a Bloch function. If the sequence {zn} is an η net then the boundedness f′(z)(1−|z|2) on {zn} is a sufficient condition for f to be a Bloch function. The essential norm of a Hankel operator with a conjugate analytic symbol acting on the Bergman space is shown to be equivalent to .

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

Certain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space , for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.

This is contrasted with a known property of the corresponding expression domains. Whereas for an expression M, the operator domains contain only functions y for which both y and My lie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary for My to exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.

We identify the extreme points of the unit sphere of the Lorentz space Lw,1 This yields a characterization of the surjective isometries of Lw,1(0,1). Our main result is that every element in the unit sphere of Lw,1 is the barycenter of a unique Borel probability measure supported on the extreme points of the unit sphere of Lw,1.

In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

A version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.

Let U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.

In [4], Ramanujan stated the following beautiful identity which was later proved by Bailey [2] and [3];

We denote by S(n) the coefficient of qn on the right hand side of (1). The object of this paper is to prove the following two theorems.

We study the asymptotic behaviour of Dirichlet problems in domains of R2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function. Using a capacitary method together with ergodic theorems for additive and superadditive processes, we are able to characterise the limit problem precisely.

All polynomials considered in this paper belong to ℚ[x] and reducibility means reducibility over ℚ. It has been established by one of us [5] that every binomial in ℚ[x] has an irreducible factor which is either a binomial or a trinomial. He has further raised the question “Does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible over ℚ which has at most K terms (i.e. K non-zero coefficients)?”

Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.

As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

An associative ring R is called a left SI-ring if every singular left R-module is injective. In Goodearl [4] it is shown that these rings have a finite ring decomposition into a ring K with K/Soc K left semisimple, and simple rings which are Morita equivalent to left SI-domains.

We give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

The probability that a neutron leads to a divergent chain reaction in a nuclear reactor is governed by a nonlinear integro-partial-differential equation [1]. A model case of this equation was completely analysed by Pazy and Rabinowitz [2,3]. The purpose of this paper is to extend their results to the general case and to tackle some related topics.