This article deals with the currents set up in thin-walled cylinders by two-dimensional field variation in the plane of the section, and is applied particularly to the problem of the distribution of alternating currents in strip conductors, a matter of some technical importance.

Based on recent work by Futer, Kalfagianni and Purcell, we prove that the volume of sufficiently complicated positive braid links is proportional to the signature defect Δσ = 2g−σ.

In a recent paper (5) the author gave an alternative proof of the result due to Hille(2) for the representation of the infinitesimal generator of the Poisson operator. In this paper we obtain the n-dimensional analogue of the result by using arguments similar to those employed in (5).

1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way that

for x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.

The Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

A method is derived for converting a pair of coupled singular integral equations of a certain form into a single equation of the same (Cauchy-separable) type. Reduction methods for systems of singular integral equations are generally directed towards the construction of equivalent Fredholm equations. Preservation of the singular nature of the kernel in the reduction process permits the powerful techniques associated with Cauchy kernels to be used in seeking closed solutions of the original pair.

The example given, derived previously from a problem in wave diffraction theory, illustrates many aspects of the method.

We investigate the functional quantization problem for stochastic processes with respect to Lp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the space Lp(IRd, μ).

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given by

where the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form as

where P(t) is the vector with components pi(t) and the constant matrix A has elements

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

1. Introduction. The problems associated with low-frequency diffraction by a circular disc have attracted the attention of many research workers. A review article by Bouwkamp(1) gives an excellent account of the research that was done prior to 1953 and a report by Eggimann(3) provides a summary of the papers that were published between 1953 and 1960. References to work published since 1960 may be found in papers of Noble (10) and Williams (16), (17).

1. In the equation in the title (referred to hereafter as the equation (E)) a, b are positive constants and h, p are continuous functions depending only on the arguments shown, with p ω-periodic in t, that is p(t, x, y, z) = p(t + ω, x, y, z) for some constant ω > 0 and for arbitrary t, x, y and z. We shall be concerned specifically with the existence of ω-periodic solutions of equations (E), in which the function h is such that |h(x)| → ∞ as |x| → ∞.

This paper follows (4) in a quantitative study of derivations. The object is to investigate the properties of the spectrum of an inner derivation. In particular, we shall be interested in obtaining a general description of the spectrum of a derivation on a W*-algebra.

1. Ganea has shown (Theorem 2·2 of (2)) that h-cogroup structures on a CW space coincide, in the homotopy category, with coalgebra structures defined with respect to the cotriple determined by SΩ. The construction used by Ganea to prove this does not dualize. It seems possible moreover that the dual statement is false. In this note I show, using the results of (4), that, for suitable spaces, there is a surjection from the set of algebra structures defined with respect to the triple determined by ΩS to the set of h-group structures, and I construct a right inverse for this surjection.

Let k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.