In earlier papers of the same main title Jones and Lewis(1) (here-after referred to as I†) and Jones (2) studied, using ‘virtual body force’ methods, the axially symmetric flows of elastico-viscous liquids caused by the rotation of surfaces of revolution of arbitrary section. Particular attention was paid in I to the (oblate and prolate) ellipsoid geometry, but, for reasons of mathematical facility, the analysis was restricted to that of small eccentricity, the interesting and degenerate scheme of maximum eccentricity being outside its range of validity. It is the main purpose of the present note to broaden the scope of the analysis of I to cover a wider class of liquids and to present an exact analytical solution (in simple closed form) for the ellipsoid geometry.

In a recent paper (1), we derived the solutions to the equilibrium equations for linearized plane strain of an isotropic mixture of two elastic solids. The solutions were given in terms of four complex potential functions. Here we examine the properties of these solutions and evaluate the forces and couples across an arc in the mixture.

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.

We deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and let

be any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):

For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such that

for all integers x1, x2, …, xm not all zero.

In the present paper Bethe's approximation is applied to the statistical treatment of the adsorption of dipoles considering (a) the electrostatic forces alone, and (b) the electrostatic and van der Waals forces jointly. In each case formulae are obtained for the adsorption isotherm and for the variation of the heat of adsorption with the fraction of surface covered. In case (a) the heat curves are compared with those obtained by Wang who used a different approximation to take into account the electrostatic interactions due to particles adsorbed on outer sites. The curves are of the same general shape; sources of differences in them are discussed. In case (b) the results are compared with those obtained by Roberts using a different method. This treatment confirms his result that the electrostatic and van der Waals forces give contributions to the variation of the heat of adsorption which are of opposite sign and almost counterbalance one another, so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one type only. This comparison shows further that the distribution of particles on the surface, which is taken into account in the statistical method used in this paper, has a considerable effect on the variation of the heat of adsorption.

The familiar W.K.B.J. method expresses in terms of elementary functions approximate solutions of second order differential equations in normal form that otherwise have no analytical solutions expressible in terms of the standard elementary or transcendental functions. Proofs of the connexion formulae required to trace these approximate solutions across transition points are usually derived by comparison with known functions such as the Airy integral or Bessel functions, or by an appeal to the Stokes phenomenon in the complex plane. A new device is developed for the synthesis of refractive index profiles with transition points, and yet yielding solutions in terms of the elementary functions. These are then used to derive the W.K.B.J. connexion formula by means of a novel limiting process based on solutions considered only along the real z-axis. The method is obviously more complicated than the more usual approach, but contains special features throwing new light on the connexion formula across a single transition point.

In a fundamental paper in 1984, Vaughan Jones developed his new polynomialinvariant of knots using a Markov trace on the Temperley–Liebalgebra. Subsequently, Lickorish used the associated bilinear pairing toprovided an alternative proof for the existence of the 3-manifold invariants ofWitten, Reshetinkin and Turaev. A key property of this form is thenon-degeneracy of this form except at the parameter values±2cos(π/(n+1)) [7]. Koand Smolinsky derived a recursive formula for the determinants of specificminors of Markov's form, establishing the needed non-degeneracy[6]. In this paper,we define a triangular change of basis in which the form is diagonal andexplicitly compute the diagonal entries of this matrix as products of quotientsof Chebyshev polynomials, corroborating the determinant computation of Ko andSmolinsky. The method of proof employs a recursive method for defining therequired orthogonal basis elements in the Temperley–Lieb algebra,similar in spirit to Jones' and Wenzl's recursive formulafor a family of projectors in the Temperley–Lieb algebra. We define apartial order on the non-crossing chord diagram basis and give an explicitformula for a recursive construction of an orthogonal basis, via a recursionover this partial order. Finally we relate this orthogonal basis to basesconstructed using the calculus of trivalent graphs developed by Kauffman andLins [5].

This classification is found by analyzing the action of a normal subgroup of B3 as hyperbolic isometries. This paper gives an example of an unfaithful specialisation of the Burau representation on B4 that is faithful when restricted to B3, as well as examples of unfaithful specialisations of B3.

For a suitable choice of E*, the solution as t becomes large of the equation

where Y is given independent of t over the space boundaries, tends to the ground state solution of the wave equation

with the same boundary conditions on P as on Y. As a preliminary to using this relation to solve wave equations in more than one variable, the solution of the equation

has been studied. Methods of numerical solution are discussed, and the convergence of these is examined. Some practical experiments using an electronic computing machine are described.

We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

Considérons la fonction arithmétique $$D^*(n,t)\colonequal \sum_{\di{d|n,\,d'|n}{0<|\log (d'/d)|\le t}} 1\qquad (n\,{>}\,1,\,t\,{>}\,0),$$ qui généralise la fonction $T^*(n,\alpha)\colonequal D^*(n,(\log n)^{-\alpha})$, étudiée dans la première partie de cet article [7]. Les variations de $D^*(n,t)$ au voisinage de l'origine permettent de mesurer, en moyenne quadratique, la propension des diviseurs de l'entier $n$ à s'agglutiner autour de certaines valeurs. Nous nous proposons ici de donner une majoration essentiellement optimale de cette fonction de $t$ sur $[0,1]$ lorsque $n$ est astreint à parcourir un ensemble d'entiers non exceptionnels en un certain sens quantitativement précisé.

The equivariant Heegaard genus of a 3-manifold W with the action of a finite group G of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for W. Although a Heegaard splitting of a reducible manifold is reducible and although if W is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number. Along the way, we make use of a new invariant for W which is much better behaved under equivariant sums.

Let (λm), (µn) (m, n = 0, 1, 2,…) satisfy

respectively. Let vp (p = 0, 1, 2, …) be the sequence (λm+µn) arranged in ascending order, equal sums λm+µn being considered as giving just one vp Then for given formal series Σam, Σbn the formal series C = Σ cp where

is called the general Dirichlet product of Σamand Σbn (see Hardy (2), p. 239). When λn = µn = n we have the Cauchy product. In the case λn = logm, µn = logn (m, n = 1, 2,…) we have vp =log p(p = 1, 2, …)and it is natural to call C the ordinary Dirichlet product.

A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .

Let $\mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $\mathfrak{A}$-covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$. We give a realisability criterion for harmonic $\mathfrak{A}$-covers by patching local monodromy data in an extended homology group on $\Gamma$. As an explicit example, we work out the case $\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.

In a recent paper (1) it has been established that a spherically symmetric space-time of class one cannot describe an electromagnetic distribution of energy. In this investigation the inconsistency of class one conditions with Rainich algebraic conditions was shown in case of spherical symmetry. In the present paper it is shown that the result is true for space-time of class one, in general.

If G is a locally compact group such that G/G0 contains a uniform compactly generated nilpotent subgroup, then G has a maximal compact normal subgroup K such that G/G is a Lie group. A topological group G is an N-group if, for each neighbourhood U of the identity and each compact set C ⊂G, there is a neighbourhood V of the identity such that for each g∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relation

where S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.