A quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.

Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:

First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.

The equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.

In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.

This paper investigates the effect on the structure of a band of imposing conditions on the congruence or right congruence lattice of the band. Bands whose congruence lattices are distributive or Boolean are characterized, as are bands whose right congruence lattices are Boolean.

If the conic α, a0x2 + a1y2 + a2z2 + 2a3yz + 2a4zx + 2a5xy=0, is represented by the point (a0, a1, a2, a3, a4, a5) of S5, the transforms of α by projectivities that fix a second conic ω will generally be represented by points of a threefold in S5 containing (a0, a1,…, a5). It is shown that this threefold is in general a rational sextic belonging to an ∞2 family, that is composed of ∞1 sets of projectively equivalent threefolds. Special, exceptional members of the family are discussed.

Equations for the threefold are found in terms of the mutual projection invariants of ω and α.

An attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.

Some results are given for Weyl's m-coefficient in the algebra B = C(K). Included are a discussion of a theorem of Hille, a generalisation of Weyl's L.P./L.C. classification, and an example.

A new class of irregular boundary value problems—non-regular in the sense of Birkhoff—is studied. This class of strongly irregular problems includes the class of boundary value problems with irregular decomposing boundary conditions. For each strongly irregular problem we can find a problem with irregular decomposing boundary conditions so that we have equiconvergence with respect to Riesz typical means of the eigenfunction expansions arising from these two problems of an arbitrary summable function.

We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

The paper is concerned with a Three-dimensional theory of non-linear magnetosonic waves in a turbulent plasma. A perturbation method is used that allows us to obtain a transport equation, like Burgers equation, but with a variable coefficient.

Let M be a finitely generated module over the finitely generated abelian group U. Denote the group of all semilinear maps of M by SautUM, a ℤ-automorphism g of M being semilinear if there exists an automorphism γ of U, called an auxiliary automorphism of g, such that mug = mguγ for all m ∊ M and u ∊ U.

An isomorphic factorisation of a digraph D is a partition of its arcs into mutually isomorphic subgraphs. If such a factorisation of D into exactly t parts exists, then t must divide the number of arcs in D. This is called the divisibility condition. It is shown conversely that the divisibility condition ensures the existence of an isomorphic factorisation into t parts in the case of any complete digraph. The sufficiency of the divisibility condition is also investigated for complete m-partite digraphs. It is shown to suffice when m = 2 and t is odd, but counterexamples are provided when m = 2 and t is even, and when m = 3 and either t = 2 or t is odd.

Let β be a cyclotomic integer. The question of the solvability of the diophantine equation xq = β in a cyclotomic field has been considered by many authors (see [4], [5], [12]). Some of the methods used in these investigations also work in J-fields. (As to the definition, see Section 2.) It is well known that J-fields share some important properties with cyclotomic fields. It is also easy to give interesting examples where the solution belongs to a. J-field but not to a cyclotomic field. It seems therefore to be of some importance to consider in general the solvability of xq = β in a. J-field, or in other words whether β1/q generates a. J-field.

The isometries of the space of convex bodies of Ed with respect to the symmetric-difference metric are precisely the mappings generated by measurepreserving affinities of Ed.

The 24 classes of even-valued, unimodular, positive quadratic forms in 24 variables have been determined by Niemeier [6]. One class is distinguished from the rest by the property that its forms have arithmetic minimum 4, rather than 2. The 24-dimensional lattice Λ corresponding to this form-class can therefore be identified with one found earlier by Leech [4], which has been studied extensively in connection with sporadic simple groups (e.g. [1]).

The Dedekind η-function is defined by

and is well-known to satisfy a functional equation relating η(aτ + b/cτ + d) to η (τ), where a, b, c, d are rational integers with ad – bc = 1—see for instance Iseki [2], and the further references cited there.

In this paper we shall consider the well known mean value theorem,

where γ is Euler's constant. The error term E(T) has been estimated by various writers; in particular the bounds E(T) = o(T log T), E(T) ≪ T1/2 log T, E(T) ≪ T5/12(log T)2 and

where ε is any positive quantity, have been given by Hardy and Littlewood [7], Ingham [8], Titchmarsh [10], and Balasubramanian [2], respectively.