Let X, Y be two Banach spaces. By L(X, Y) (resp. K(X, Y)) we denote the Banach space of all bounded, linear (resp. compact, bounded, linear) operators from X into Y. Several papers have been devoted to the question of when c0 embeds isomorphically into K(X, Y) (see 5, 8, 9 and their references) and its relationship with the following question:

(i) is K(X, Y) always uncomplemented in L(X, Y) when L(X, Y)K(X, Y)?

H*-algebras were introduced and studied by Ambrose [1] in the associative case, and the theory has been extended to such particular classes of non-associative algebras as Lie [18, 19], Jordan[20, 21, 7], alternative [11] and non-commutative Jordan [6] algebras. In all these cases the core of the matter is showing that every H*-algebra (in the given class) with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals (each of which is a topologically simple H*-algebra), and then listing all the topologically simple H*-algebras in the class. In fact every nonassociative H*-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals [6, theorem 2·7], so the problem of the classification of topologically simple non-associative H*-algebras becomes interesting. In relation with this problem the question arises whether, once an algebra A has been structured as a topologically simple H*-algebra, every H*-algebra structure on A is (up to a positive multiple of the inner product) totally isomorphic to the given one (see [3] and [11, section 4]). As a consequence of the results in this paper we give a general affirmative answer to this question.

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniform supercritical stream (i.e. the Froude number , where c is the speed of the stream, h is its depth, and g is the gravity constant).

We also deduce a number of inequalities relating F2 to a/h, where a is the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 < F2 < 2.

This paper deals with an initial value problem of a neutral functional-differential equation with variable time delay. The effect of the neutral term on the existence and uniqueness of solutions is investigated. A comparison theorem is given and is applied to a functional Riccati equation. The asymptotic behaviour of solutions of a linear perturbation of the generalized pantograph equation is studied.

We prove that the class of transition matrices for the finite state time-inhomogeneous birth and death processes coincides with the class of non-singular totally positive stochastic matrices. Thus we obtain a complete solution to the embedding problem for this class of Markov chains.

It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10300 (see Brent[1]). Moreover it has been proved by Hagis[4]and Chein[2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in priniciple be obtained solely by calculation, in view of the result of Pomerance[6] who showed that if N is an odd perfect number with at most k prime factors, then

Pomerance's work was preceded by a theorem of Dickson[3]showing that there can be only a finite number of such N. Clearly however the above bound is vastly too large to be of any practical use. The principal object of the present paper is to sharpen the estimate (1). Indeed we shall handle odd ‘multiply perfect’ numbers in general, as did Kanold[5], who extended Dickson's work, and Pomerance. Our result is the following.

It is a well-known result in Diophantine approximation over the reals that the Lebesgue measure of the elements $x \in {\mathbb{R}}$, such that $\Vert qx \Vert < {1}/{q^v}$ for infinitely many $q \in {\mathbb{R}}$, is full for $v \leq 1$ and null otherwise, where $\Vert\cdot\Vert$ denotes the distance to nearest integer. Also, the Hausdorff dimension of the set of these numbers is known to be ${2}({v\,{+}\,1})$ for $v > 1$. Similar results are known for matrices over the reals as well as the $p$-adics. In this paper, we prove the corresponding multi-dimensional result for matrices over the field of Laurent series with coefficients from a given finite field.

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphism

in group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.

The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

A class of f-driven algorithms due to Chvátal for finding the stability number of a graph are studied. It is shown that, for almost all graphs, the computation time grows subexponentially with n, the number of vertices. If, however, each edge exists with probability δ/n independently on other edges then asymptotically almost certainly the computation time is exponential. Still, for all large enough δ's, these algorithms perform noticeably better than a naive algorithm. The results are extended to random graphs with a fixed number of edges.

The bifurcation problem of an infinite elastoplastic medium surrounding a spherical cavity and subjected to uniform radial tension or compression at infinity is studied. The material is assumed to be incompressible, and its behaviour is modelled by both hypoelastic (flow theory) and hyperelastic (deformation theory) constitutive relations. No bifurcation was found with the flow theory. Surface bifurcation modes were discovered with the deformation theory in both tension and compression. An independent study is also presented of surface bifurcations of a semi-infinite elastoplastic material under equi-biaxial stress. The critical strain for the half-space coincides with the strain at the spherical cavity at the lowest bifurcation.

The theory of variational bicomplexes was established at the end of the seventies by several authors [2, 17, 23, 26, 29–32]. The idea is that the operations which take a Lagrangian into its Euler–Lagrange morphism [9, 10, 12, 24] and an Euler–Lagrange morphism into its Helmholtz' conditions of local variationality [1–3, 7, 11, 13, 18, 27] are morphisms of a (long) exact sheaf sequence. This viewpoint overcomes several problems of Lagrangian formulations in mechanics and field theories [21, 28]. To avoid technical difficulties variational bicomplexes were formulated over the space of infinite jets of a fibred manifold. But in this formalism the information relative to the order of the jet where objects are defined is lost.

We refer to the recent formulation of variational bicomplexes on finite order jet spaces [13]. Here, a finite order variational sequence is obtained by quotienting the de Rham sequence on a finite order jet space with an intrinsically defined sub-sequence, whose choice is inspired by the calculus of variations. It is important to find an isomorphism of the quotient sequence with a sequence of sheaves of ‘concrete’ sections of some vector bundle. This task has already been faced locally [22, 25] and intrinsically [33] in the case of one independent variable.

In this paper, we give an intrinsic isomorphism of the variational sequence (in the general case of n independent variables) with a sequence which is made by sheaves of forms on a jet space of minimal order. This yields new natural solutions to problems like the minimal order Lagrangian corresponding to a locally variational Euler–Lagrange morphism and the search of variationally trivial Lagrangians. Moreover, we give a new intrinsic formulation of Helmholtz' local variationality conditions, proving the existence of a new intrinsic geometric object which, for an Euler–Lagrange morphism, plays a role analogous to that of the momentum of a Lagrangian.

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys).

Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2→R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.)

Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps.

In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms).

We refer the reader to [16, 18] for background material concerning singularity theory.

Recently, Hitoshi Murakami has shown that the quantum SU(2)- and SO(3)-invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev–Viro invariant is the square of the modulus of the Reshetikhin–Turaev invariant.

In [19], R. Kadison proved that every surjective linear isometry $\Phi{:}\, {\C A} \to {\C B}$ between two unital C*-algebras has the form $$\Phi (x) = u T (x), \hbox{ $x\in {\C A}$,}$$ where $u$ is a unitary element in ${\C B}$ and $T$ is a Jordan *-isomorphism from${\C A}$ onto ${\C B}$. This result extends the classical Banach–Stone theorem [3, 32] obtained in the 1930s to non-abelian unital C*-algebras. A. L. Paterson and A. M. Sinclair extended Kadison's result to surjective isometries between C*-algebras by replacing the unitary element $u$ by a unitary element in the multiplier C*-algebra of the range algebra [28]. Thus, every surjective linear isometry between C*-algebras preserves the triple products as $$\J xyz \,{=}\, 2^{-1} ( x y^* z + z y^* x).$$

We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a $2\times 2$-matrix Riemann–Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the $2\times 2$ RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.

Given a set Γ of low-degree k-dimensional varieties in $\mathbb{R}$n, we prove that for any D ⩾ 1, there is a non-zero polynomial P of degree at most D so that each component of $\mathbb{R}$n\Z(P) intersects O(Dk−n|Γ|) varieties of Γ.