1. In a recent paper [6], the spectrum of the Cesàro operator C on c0 (the space of null sequences of complex numbers with the sup norm) was obtained by finding the eigenvalues of the adjoint operator on and showing that the operator (C–λI)−1 lies in B(c0) for all λ outside the closure of this set of eigenvalues. In this paper we apply a similar method to find the spectrum of the two-dimensional Cesàro operator on a space of double sequences c0(c0) (defined in §2). We shall introduce a simplification to the proof in [6] by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of [11] for the single variable case on the space of convergent sequences c), and the crux of our proof is to show that the moment constant associated with the method (C – λI)−1 is regular for the space c0(c0) and the set of λ under consideration. It turns out that c0(c0) ≅ c0c0 (see page 237 of [7]) and that the two-dimensional Cesàro operator on c0(c0) is the tensor product C ⊗ C of the Cesàro operator C on c0. Thus our result gives a direct proof that the spectrum σ(C ⊗ C) equals σ(C)σ(C), which is a special case of the result of Schechter in [8].

Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.

Introduction. For applications in differential topology it is desirable to be able to find restrictions on the possible real and complex vector bundles over a given manifold. These restrictions usually state that the top component of some specific rational multinomial in the various characteristic classes is an integral multiple of the fundamental cocycle. For example, in proving non-embedding or non-immersion theorems (compare (1); (10)) one could test whether the normal bundle is consistent with these integrality conditions.

Taylor and Fisk have discussed the internal conversion of magnetic multipole radiation by K electrons and have found internal conversion coefficients which are much larger than those for electric multipoles, and which increase with the order of the multipole. The object of the present note is to determine whether this state of affairs holds also for pair production by internal conversion of magnetic multipole radiation.

We study permanence results for almost quasi-isometries, the maps arising from the Gromov construction of finitely generated random groups that contain expanders (and hence that are not C*-exact). We show that the image of a sequence of finite graphs of large girth and controlled vertex degrees under an almost quasi-isometry does not have Guoliang Yu's property A. We use this result to broaden the application of Gromov's techniques to sequences of graphs which are not necessarily expanders. Thus, we obtain more examples of finitely generated non-C*-exact groups.

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups.

The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

Let H be a complex Hilbert space. Recall that a bounded linear operator A, on H, is positive if (Ax, x) ≥ 0 (x ∈ H) (so that A = A* necessarily) and positive definite if A is positive and invertible.

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

The object of this paper is to find the characteristic functions and the characteristic numbers of the partial differential equation

valid in a domain G, and where on Γ, the boundary of G. The method employed is to transform two quadratic forms to their common self-conjugate “triangle“ of reference. The solution to the problem is given by this method in a simple manner, without the use of the integral equation theory, or the use of minimal sequences.

In the above paper it is pointed out that before we may use the usual wave equation for the propagation of current along a straight wire in free space it is necessary that (a) the inductance L and capacity C per unit length of the wire are independent of the current distribution and (b) that the product LC = 1/c2. This has been shown to be the case.

The inductance and capacity of an element of a conductor are defined generally in the way suggested by Moullin and it is shown that the radiation resistance term, which is a function of the current distribution, is small compared with the corresponding terms involving inductance and capacity. The usual wave equation is derived from Maxwell's equations for the case of a tubular conductor in which the current is a function of the distance z along the conductor only. If the curl of the current is not equal to zero this will not represent the true state of affairs.

It is pointed out in the earlier part of the paper that a complete solution of the problem, taking into account the radiation resistance, and the skin effect, involves the solution of an integral equation which is given.

The field about a tubular current, with a current distribution J0 sin kz along it, is examined. It is found that, for a long wire, the electric force is perpendicular to the surface at every point since we neglect the contribution from the ends. The magnetic force is in circles about the axis as would be expected.

A small component parallel to the current, and in phase with it, is introduced by the charges upon the ends of the tubular conductor, and this component determines the radiation resistance. From this it follows that the radiation resistance could be increased by placing a capacity on the ends—thus increasing the charges there.

Processes related to the empirical distribution are shown to converge to a time changed Brownian motion. In particular, the time change of the Brownian motion differs from the time change obtained by using the distribution function: this gives rise to two different representations of a related Brownian bridge.

If A = (am, n) is a regular matrix, then for any sequence x = {xn}, Am(x) will denote the transform and A-lim x denotes if that limit exists. We shall denote by the set of bounded sequences which are summed by A. If B is another regular matrix with then we say that B is b-stronger than A. In that case B must be b-consistent with A (see (4) and (6)), i.e. if then

If {μn} is a sequence of positive real numbers with we say that A and B are (μn)-consistent if every sequencer x = {xn} satisfying xn = 0(μn) which is summed by both A and B is summed to the same value by both matrices. A finite set of matrices A1, A2, …, AN is said to be simultaneously (μn)-inconsistent (b-consistent) if whenever is summed by Ai with (i = l, 2, …, N) then implies that The set of sequences, is denoted by

We measure the distance of a general shape from the set of shapes having a certain sequence of their vertices collinear and also describe the behaviour of shapes as they move along a geodesic towards this set. This models problems which arise in computer vision and other applications of shape theory.

The convexity of the internal energy, regarded as a function of any tensor strain measure, determines the stability of a material in a particular loading environment. Such convexity criteria reflect properties of the material alone and therefore have the alternative interpretation of constitutive inequalities. An assessment of the relative strengths of these criteria is therefore of interest because it provides an ordering of the constitutive inequalities. Existing comparison theorems, allowing such an assessment, suppose that work-conjugate measures of stress and strain are coaxial. The main content of this paper is a theorem which makes no such supposition and so enables a comparison of criteria of convexity, and an ordering of constitutive inequalities, for arbitrarily anisotropic materials.

In the preceding paper, Blackett and Williams described an automatic curve follower for use with the differential analyser. The device would only operate if the x-coordinate was either continuously increasing or continuously decreasing; it could not handle input functions where the x value pulsated, as may often happen when the x-coordinate is not the independent variable.

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.

For each diagram D of a 2-knot, we provide a way to construct a new diagram D′ of the same knot such that any sequence of Roseman moves between D and D′ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in 4-space.

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.

The goodness of fit of hypothetical discriminant functions has been discussed by Bartlett(1), Kshirsagar(2, 3), Radcliffe(4) and Williams (5–8). The associated problem of obtaining confidence intervals for the coefficients of a single discriminant function or canonical variable has been dealt with by Bartlett (1).