This paper presents a recent result for the problem introduced eleven years ago by Fraenkel and McLeod [A diffusing vortex circle in a viscous fluid. In IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Kluwer (2003), 489–500], but described only briefly there. We shall prove the following, as far as space allows. The vorticity ${\it\omega}$ of a diffusing vortex circle in a viscous fluid has, for small values of a non-dimensional time, a second approximation ${\it\omega}_{A}+{\it\omega}_{1}$ that, although formulated for a fixed, finite Reynolds number ${\it\lambda}$ and exact for ${\it\lambda}=0$ (then ${\it\omega}={\it\omega}_{A}$), tends to a smooth limiting function as ${\it\lambda}\uparrow \infty$. In §§1 and 2 the necessary background and apparatus are described; §3 outlines the new result and its proof.

A natural number $n$ is called abundant if the sum of the proper divisors of $n$ exceeds $n$. For example, $12$ is abundant, since $1+ 2+ 3+ 4+ 6= 16$. In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if $A(x)$ denotes the count of abundant numbers belonging to the interval $[1, x] $, does $A(x)/ x$ tend to a limit? Four years later, Davenport answered Bessel-Hagen’s question in the affirmative. Calling this density $\Delta $, it is now known that $0. 24761\lt \Delta \lt 0. 24766$, so that just under one in four numbers are abundant. We show that $A(x)- \Delta x\lt x/ \mathrm{exp} (\mathop{(\log x)}\nolimits ^{1/ 3} )$ for all large $x$. We also study the behavior of the corresponding error term for the count of so-called $\alpha $-abundant numbers.

Let l be a prime and let v ≥ 1 be an integer (when l = 2 we assume v ≥ 2). Any ring, A, with unit, possesses mod lv algebraic K-groups [B] denoted by Ki(A; Z/v) (i ≥ 0). For i ≥ 2, Ki(A; Z/lv) = [Pi(lv), BGLA +], the group of based homotopy classes of maps from the Moore space , to BGLA+, the classifying space of algebraic K-theory [G–Q ; W].

In 1933, K. Borsuk [1] established the well-known result that if n closed sets cover Sn−1 then at least one set contains antipodal points, where Sn−1 is the surface of the ball Tn of centre O and unit diameter in Rn. This result prompted H. Hadwiger [2] to make a still unresolved conjecture which, in the spirit of B. Griinbaum's survey [3], we state as follows: Let r be the largest integer such that whenever r closed sets cover Sn−1 at least one set realizes all distances between 0 and 1. Then r = n.

I investigate what can be said about a set E in a probability space X when the “square” E x E can be covered by the squares of stochastically independent sets of given measure.

Gale transforms are constructed for certain infinite dimensional α-polytopes. In a manner analogous to the finite dimensional case the Gale transform can be used to determine all closed faces and Radon partitions of the α-polytope. A by-product is a characterization of closed faces using nets of functionals.

Let η(z) be the Dedekind eta function defined by

where q = exp(2πiz) and Im (z) > 0. Then it is famous that

We investigate the steady pressure driven flow of conducting fluid through an insulating circular pipe in the presence of a transverse applied magnetic field. It is well known that an exact solution is available, and the series expressions for the fluid velocity and induced magnetic field have often appeared in the literature, Uflyand (1960), Uhlenbusch and Fischer (1961), Gold (1962); however, no progress has been made in deriving asymptotic expansions for large Hartmann number directly from these series. Boundary layer methods have been used by Shercliff (1953. 1962) to determine the core structure, the Hartmann layer structure, the current distribution and the flow rate; however, these analyses do not describe the tangential layers (called obscure layers by Shercliff), which are located in the vicinity of the points at which the applied magnetic field is tangential to the pipe.

Let K be the finite field of pn elements, and Zp its prime subfield. It was proved by Davenport [3] that if p > p0(n) and θ is any given generating element of K, then there exists an integer m such that θ + m is a primitive root of K.

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1F) = m(F) for all F ∈ A and all surjective group endomorphisms T of G onto G.

This note contains characterizations of those sigma-fields for which sigma-finiteness is a necessary condition in the Radon-Nikodym Theorem.

Our purpose is to consider those σ-fields for which σ-finiteness is a necessary condition in the Radon–Nikodym Theorem. We first prove a measure theoretic equivalence in the general case, and then use this to obtain an algebraic characterization in the case when the σ-field is the Borel field of a locally compact separable metric space. For undefined terminology we refer the reader to [1] for measure theoretic and [2] for algebraic properties.

By a measure, we mean a countably additive function from σ-field of sets or a Boolean σ-algebra into the non-negative extended real numbers. We will say that a measure μ on a σ-field of sets Σ is RN provided each μ-continuous finite measure on Σ has a Radon–Nikodym derivative in L1(μ).

In the present paper a general solution of the equations of elasticity in complete aeolotropy is found under the assumption that the stresses and therefore the strains are linear in the third cartesian coordinate. This solution is applied to the elastic equilibrium of a completely aeolotropic cylinder, under a distribution of tractions on the lateral surface and resultant forces and couples on the end sections of the cylinder. The problem of extension of a completely aeolotropic cylinder by longitudinal lateral loading and end forces is solved with an application to the elliptic cylinder. The writer hopes to present in later communications applications of this general solution to the following problems: (i) Bending of a completely aeolotropic cylinder by longitudinal lateral loading and end bending couples, (ii) Torsion of a completely aeolotropic cylinder by transverse lateral loading and end twisting couples, (iii) Flexure with shear of a completely aeolotropic cylinder with free lateral surface. Particular cases of (ii) and (iii) were considered by Luxenberg [1] and Lechnitzky [2, 3].

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x(p, q) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula

and

The following problem was posed by Paul Gauthier: if f(z) and g(z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f(z) and g(z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D. It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

For (x, t) ∈ Rn+1, we put

If f(z) is a function of the complex variable z, regular in a neighbourhood of z = 0, with f (0) = 0 and f' (0) ≠ 0, then the equation w = f(z) admits a unique solution, regular in some neighbourhood of w = 0, given by

where C is an appropriate contour encircling z = 0. These formulae are well-known, being stages in the proof of the classical reversion † formula

due to Lagrange.

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

The object of this paper is to solve the Saint-Venant torsion problem for those cross-sections with inclusions, which are such that the z-plane boundaries involved can be mapped into concentric circles in a complex ζ-plane by the transformation

with z´(ζ) ≠ 0 or ∞ within the cross-section. We shall consider both solid and hollow inclusions having different elastic rigidities μ. In the case of the solid inclusion we have to restrict the coefficients as to be zero for all negative s, but it is an advantage to leave this restriction to the end of the analysis, since the forms of certain coefficients in the two cases differ only in this respect.

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.