An extension is proposed to the theory of Verwey and Overbeek on the force between two colloidal particles in a dilute sol. The effect of the Coulomb interaction of the ions in the dispersion medium is considered. It is shown that this contributes an additional repulsion term to the mutual free energy of the two particles. This correction term behaves as an inverse power and is the predominant effect at large separations, although it is a second-order effect at small separations. For parallel plates, this power is the fourth, and for two spheres, the sixth. A tentative calculation shows that, at least for thin plates and low electrolyte concentrations, our new repulsion will outweigh the attractive London-van der Waals force considered by Verwey and Overbeek, and more recently by Casimir and Polder, when the plates are far apart.

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.

For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.

Here we show the following results:

(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;

(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;

(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

In a recent paper Collins (l) has shown that Copson's method of solution of the problem of determining the electrostatic field due to an electrified disk (2) can be extended to determine the surface charge density induced on an electrified disk when it is situated inside an earthed coaxial infinitely long hollow cylinder. In Collins's solution the problem is reduced to that of solving a Fredholm integral equation of the second kind. The object of the present note is to show that Collins's solution can be obtained more simply by making use of a technique in the theory of dual integral equations developed recently by the author (3) and Copson(4).

We give an explicit description of the construction of inverse limits in the category of locales and show a localic version of Steenrod's theorem without the Axiom of Choice; as an application we generalize the classic result by Steenrod.

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

An ordinary differential equation for Hermite-Bell polynomials is derived. Bell's error concerning orthogonality of his polynomials is corrected.

We study a divisor computing the minimal log discrepancy on a smooth surface. Such a divisor is obtained by a weighted blow-up. There exists an example of a pair such that any divisor computing the minimal log discrepancy computes no log canonical thresholds.

This note is concerned with the relations between the invariants Ωi (i = 0,1,…, d – 1) of the canonical system on an algebraic Vd. Here Ω0 is the grade of the canonical system, and Ωi is the virtual arithmetic genus of the Vi of intersection of d – i canonical Vd–1's.

This paper continues the study initiated in [9] of the Markov branching process (MBP) with a density-independent catastrophe component. The population size process (Xt) was investigated in the subcritical and critical cases in [11], and the present paper is concerned with a parallel investigation of the supercritical case. We refer the reader to [11] for definitions and notation, and we will not repeat them here.

In an earlier paper (2) the author and Radhakrishna have, under certain conditions, obtained the following result.

1. A locus Vn, of dimension n, in [2n + 1], for example a curve in [3] or a surface in [5], has ∞ 2n chords, of which a finite number pass through a general point of the space.

The geometrical configuration consisting of five straight lines tangent to the stream line (ψ = const.), isobar (p = const.), isocline (θ = const.), isovel (q = const.) and isopycnic (ρ = const.) at a point P of the physical plane in the plane flow of a fluid has a number of interesting properties. The study of this geometric configuration for a wide class of ideal fluids (which includes polytropic gases and incompressible fluids), in particular its properties at sonic points, is the purpose of this paper.

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

1. Introduction. The open subsets of a topological space X form a complete Brouwerian, i.e. distributive and pseudo-complemented lattice L(X). Many topological properties of X can be formulated as properties of L(X), although, in general, X is not determined by L(X). Obvious examples are quasi-compactness and connectivity: X is quasi-compact if any set of elements of L(X) whose join is I has a finite subset whose join is I; X is connected if no element of L(X) has a complement. These properties which we shall call ‘paratopological’, can be defined for lattices that are not of the form L(X), and many topological theorems can be proved in this more general context. The purpose of this paper is to develop sections of general topology as part of the theory of complete Brouwerian lattices (CBL).

The purpose of this note is to improve Theorem 1·1 of (1) to the following result:

Theorem 1. Let H be the ordinary free product of a sequence of groups Gn, and let m be any integer > 0. If

where each gi є Gi and is not the identity, then h is not equal to any product of fewer than m commutators from H.

The purpose of the present note is to prove the following two theorems:

Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.

Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.

Let V be a smooth, 1-connected 4-manifold. It has long been known that there may be 2-dimensional homology classes in V which are not represented by smoothly embedded 2-spheres, owing to the failure of the Whitney lemma in this dimension. One of the simplest examples of this failure may be described as follows. Let K be a smooth knot in S3 and r an integer. By V(K, r) we mean the 4-manifold obtained by attaching a 2-handle to B4 along an r-framed tubular neighbourhood of K. Evidently each V(K, r) is homotopy equivalent to S2 and we say that K is an r-shake-slice knot if there is a 2-sphere, smoothly embedded in V(K, r), which represents a generator of the second homology group. For example, if K is a slice knot then it is r-shake-slice for each integer r. On the other hand, it is known that arf (K) = 0 whenever K is r-shake-slice and so in particular a trefoil is never such a knot.