The Schur Theorem says that if G is a group whose center Z(G) has finite index n, then the order of the derived group G′ is finite and bounded by a number depending only on n. In this paper we show that if G is a finite group such that G/Z(G) has rank r, then the rank of G′ is r-bounded. We also show that a similar result holds for a large class of infinite groups.

It is shown that the problem of the diffraction of an arbitrary electromagnetic wave by a circular disk can be reduced to the solution of independent scalar problems. These latter problems are of a type which can be reduced to the solution of a Fredholm integral equation of the second kind and are particularly suitable for solution by approximate means at low frequencies.

A simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.

In this paper we study the existence and the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. An example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central configuration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice of x0, there should be at least one free time minimizer x(t) defined for all t ≥ 0 and satisfying x(0)=x0. We prove that such motions are completely parabolic. Using Marchal's theorem we deduce as a corollary that there are no entire free time minimizers, i.e. defined on $\mathbb{R}$. This means that the Mañé set of the Newtonian N-body problem is empty.

The Bruck-Slaby theorem on nilpotence of commutative Moufang loops may be combined with a theorem of Belousov to give a quick elementary proof of a stronger version of Fischer's Theorem on the solubility of right multiplication groups of finite distributive quasigroups.

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.

In the limit of small gap width (b − a)/b, the equation takes the form

which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression

for the capacity of the disc; for the opposite limit the expression

is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.

Let gn(z) be a polynomial of degree n in z. Then there exist constants Πk, nsuch that

.

W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurston's conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurston's work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincaré – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

It is shown that for an axisymmetric stationary system there exists, in vacuo, a canonical coordinate system, analogous to Weyl's canonical coordinates for the static case. Some approximate solutions of the vacuum field equations are then obtained in these canonical coordinates.

In the second part of this paper, the vacuum field equations for an isolated axisymmetric stationary system are set up and approximate solutions obtained by means of a multipole expansion. The physical components of the Riemann tensor are then examined and it is found that the ‘mass term’ predominates, the ‘rotation term’ being of the next order of magnitude.

Some applications of an alternating field method to various problems of superconductivity are described. It is shown how the method can be used to measure the resistance and self inductance of a metal ring, and this was applied to show how the resistance of a superconducting tin ring is restored by a magnetic field. The method is also convenient for a rapid determination of the order of magnitude of the specific resistance of a substance, and to see whether or not it becomes superconducting; several elements were examined in this way, but no new superconductors were found.

A method based upon the Lagrange identity has been used by Brun [2] in the linear theories of thermoelasticity and viscoelasticity to establish uniqueness of the solution to the initial boundary value problem on bounded three-dimensional regions. A major feature of Brun's analysis is that it does not require any sign-definiteness assumptions on, for instance, the elasticities. The technique was extended by Knops and Payne [14] to derive certain continuous dependence results in linear elastodynamics, again for a bounded region. These authors had earlier recovered Brun's uniqueness result for linear elasticity [11] and derived other continuous dependence results based upon logarithmic convexity arguments [12, 14] (see also [13] for a similar treatment of thermoelasticity). Levine [18] later treated an abstract version of the Brun approach and applied it to a family of abstract linear operator equations. Among his results is a simplified proof that equipartition of the kinetic and potential energies occurs. Other applications of the Lagrange identity in proofs of uniqueness for bounded regions include those by Naghdi and Trapp [19] for a Cosserat surface, and by Green [9] for a theory of linear thermoelasticity that allows second sound.

In this paper the major concern is in investigating various divisibility properties of the random variable W, the limit of the Heyde martingale in supercritical branching processes. Throughout the paper ø(s) will be used to denote the Laplace-Stieltjes transform of W.

If M is an arbitrary loopless matroid with ground set E(M) and rank function ρ, then let α(M) be the minimum size of a set of cocircuits of M, whose union is E(M), and let β(M) be the maximum size of a set of pairwise disjoint cocircuits of M. The following conjecture is based on Gallai's theorem that the vertex-stability and vertex-covering numbers of a graph G sum to the number of vertices of G (1).

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.

It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually prove

This may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

According to recent experiments by Cosyns(1), in which the probability that a cosmic-ray electron actuates a Geiger-Müller Counter is observed, the rate of production of primary ions is practically the same in helium as in hydrogen. The value obtained for hydrogen (which agrees very closely with that previously obtained by Danforth and Ramsey (2) using the same method) is moreover little different from that found for fast β-particles in expansion-chamber experiments.

Let K be an oriented knot in the 3-sphere S3. An exterior of K is the closure of the complement of a regular neighbourhood of K, and is denoted by E(K). A Seifert surface for K is an oriented surface S( ⊂ S3) without closed components such that ∂S = K. We denote S ∩ E(K) by Ŝ, and we regard S as obtained from Ŝ by a radial extension. S is incompressible if Ŝ is incompressible in E(K). A tunnel for K is an embedded arc τ in S3 such that τ ∪ K = ∂τ. We denote τ ∪ E(K) by τ, and we regard τ as obtained from τ by a radial extension. Let τ1, τ2 be tunnels for K. We say that τ1 and τ2 are homeomorphic if there is a self-homeomorphism f of E(K) such that f(τ1) = τ2. The tunnels τ1 and τ2 are isotopic if τ1 is ambient isotopic to τ2 in E(K). Then the main result of this paper is as follows: Theorem. Let K be a knot in S3, and let τ1, τ2 be tunnels for K. Suppose that there are incompressible Seifert surfaces S1 S2 for K such that S1 ∪ S2 = K, and τi ⊂ Si (i = 1, 2). If τ1 and τ2 are isotopic, then there is an ambient isotopyhτ (0 ≤ t ≤ 1) of S3 such that ht(K) = K, and h1(τ1) = τ2.

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

The origin of this paper lies in a study of the unfolding space of the stratum N16 of singularity theory, and the question, at which points of the stratum the versal deformation space ceases to be topologically trivial over the stratum. This question turns out to be closely related to the study of how a plane section (= binary quintic) of a quintic curve varies as we deform the curve, either rigidly (under GL3) or equisingularly.