Let $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that

$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$

for all $x\in X$ and $f\in\mathcal{A}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.

The aim of this paper is to investigate and study the possible spectral pair $(\mu_{M,D},\varLambda(M,S))$ associated with the iterated function systems $\{\phi_{d}(x)= M^{-1}(x+d)\}_{d\in D}$ and $\{\psi_{s}(x)=M^{\ast}x+s\}_{s\in S}$ in $\mathbb{R}^n$. For a large class of self-affine measures $\mu_{M,D}$, we obtain an easy check condition for $\varLambda(M,S)$ not to be a spectrum, and answer a question of whether we have such a spectral pair $(\mu_{M,D},\varLambda(M,S))$ in the Eiffel Tower or three-dimensional Sierpinski gasket. Further generalization of the given condition as well as some elementary properties of compatible pairs and spectral pairs are discussed. Finally, we give several interesting examples to illustrate the spectral pair conditions considered here.

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.

The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”

occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the type

where A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.

In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.

AMS 2000 Mathematics subject classification: Primary 16D20

A theory of positive definite kernels in the context of Hilbert C*-modules is presented. Applications are given, including a representation of a Hilbert C*-module as a concrete space of operators and a construction of the exterior tensor product of two Hilbert C*-modules.

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question posed by Vladimir L. Popov.

The relationship between certain non-associative algebras and the deterministic theory of population genetics was first investigated by Etherington (3)-(8), who defined the concepts of baric, train and special train algebras. Gonshor (10) dealt with, among other topics, algebras corresponding to autopolyploidy, on the assumption that chromosome segregation operated. In this paper [ discuss algebras corresponding to more general systems of inheritance among polyploids, which have been discussed without using algebras by Haldane (11), Geiringer (9), Moran (13) and Seyffert (16). These algebras are special cases of what I have defined as segregation algebras, and mixtures of them. All the algebras corresponding to a fixed ploidy have a relationship which I have called special isotopy. An example shows that algebras arise in other genetic systems which are not isotopic to segregation algebras.

Let A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation

with initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.

Throughout this paper any near-ring N will be left distributive, zero symmetric and have an identity. Furthermore, all N-groups will be unitary. All groups considered will be written additively. This does not imply commutativity.

We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

Let Vk denote the class of functions

which map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.

In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.

Etherington introduced certain algebraic methods into the study of population genetics (6). It was noted that algebras arising in genetic systems tend to have certain abstract properties and that these can be used to give elegant proofs of some classical stability theorems in population genetics (4, 5, 9, 10).

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))

and the generalised Mehler transform pair (7)

These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

In this paper, I introduce a class of arrangements called arrays of strength d and discuss methods of constructing them with the help of finite geometrical configurations and algebraic groups involving elements of a Galois field. The definitions of arrays of strength d and other configurations that are used are given below.

1. The cardinal function is the interpolation function

which takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that

(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.

This paper is mainly concerned with the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary-value problem of the fast diffusive polytropic filtration equation with reaction sources.

In some chemical reaction–diffusion processes, the reaction takes place only at some local sites, due to the presence of a catalyst. In this paper we study the well-posedness of a model problem of this type. Sufficient conditions are found to ensure global existence and finite time blowup. The blowup rate and the blowup set are also investigated in the case of special nonlinearity.

We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.