We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.

The work described in this paper grew out of an attempt to generalize some results obtained in an earlier paper [4] on the water entry problem of a thin wedge or cone into an incompressible fluid. The object of the generalization was to include the effect of gravity terms. In most papers on hydrodynamic impact it is considered permissible to neglect this effect since gravity terms might be expected to play a minor role in the initial stages of the motion. However, it seems desirable to investigate the effect of including gravity terms in order both to examine the later stages of the motion and to estimate to what extent their neglect is justified in the early stages. It will be seen that it is possible to develop a fairly complete solution for the normal entry of a thin symmetric body, both for two-dimensional and axially symmetric cases, on the basis of a linearized theory. The restriction to a linearized theory means that the whole field of analysis associated with the theory of surface waves of small amplitude becomes available. Most of the problems considered in this paper are initial value problems in which the whole fluid is at rest at t = 0.

In this paper we characterize boundedly laterally complete Riesz spaces, boundedly laterally complete Riesz spaces with the lateral boundedness property and Riesz spaces in which every principal ideal is finite dimensional. The characterizations are given in terms of extension properties of certain Riesz homomorphisms.

Contrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.

In this paper we shall use the integral operator method of Bergman, B[1–6], to investigate solutions of the partial differential equation where s > −1. In particular, information concerning the growth, and location of singularities, of solutions of (1.1) will be obtained. Equations of the form (1.1) with s = 1, 2, arise from the (n+k+1)-dimensional Laplace equation Δn+k+1u = 0 in the “axially symmetric” coordinates x1, …xn, p where the relationship between cartesian and “axially symmetric” coordinates is given by

Let r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

Let f(z)=z + … be regular in the unit disc |z| < 1 (hereafter called E). In a recent paper Trimble [7] has proved that if f(z) be convex in E, then F(z) = (1 − λ)z + λf(z) is starlike with respect to the origin in E for (2/3) ≦ λ ≦ 1. The purpose of this note is to show that if certain additional restrictions be imposed on f(z), then F(z) becomes starlike for all λ, 0 ≦ λ ≦ 1. Also we consider some related problems.

Recently, attention has been drawn to the problem of estimation of a k-variate probability density and its partial derivatives of various orders. Specifically, let X1, …, Xn be i.i.d. k-variate random variables with common density f wrt Lebesgue measure μ on the k-dimensional σ-field Bk. Parzen (1962) in the k = 1 case and Cacoullos (1966) in the k ≧ 1 case gave the asymptotic properties of a class of kernel estimates fn(x), x ∈ Rk, of f(x) based on X1, …, Xn. The asymptotic properties given in the above two papers concern consistency, asymp-totic unbiasedness, bounds for the mean squared error and asymptotic normality of fn. Also in the context of an empirical Bayes two-action problem, Johns and Van Ryzin (1972) introduced kernel estimates for f(x) and the derivative f'(x)for x∈R1 when f is a mixture of univariate exponential densities wrt Lebesgue measure on B1. They also investigated the asymptotic unbiasedness and themean squared error convergence properties of these estimates. Lin (1968) statedsome generalizations of the results of Johns and Van Ryzin, with applicationsto empirical Bayes decision problems.

In this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a mod p representation of the absolute Galois group.

Let n be a natural with largest component sd. We prove that if xσ(n) = yn + z (x, y, z given positive integers), n is not primitive (y/x)-abundant and n/s is not (y/x)-perfect, then n < 4(z + ½)3/27y (if z ≥ 175). All solutions are tabled for the equation xσ(n) - yn when x = 1, y ≥ 2, 1 ≤ z ≤ 210, and n is not primitive y-abundant. We also prove that if n is primitive (y/x)-abundant, then s3d < (yn/2)2. A number of results are proved concerning the range of σ(n)/n when n is primitive αabundant, for any real number α > 1. For example, then α(n/n < α + min {½ 3 α e-5a/9 /2) and σ(n)/n < α /1.6α/log n. All primitive abundant numbers n with α(n)/ ≥ 2.05 are listed.

Let ${\mathcal{A}}$ be a unital ring with involution. Assume that ${\mathcal{A}}$ contains a nontrivial symmetric idempotent and ${\it\phi}:{\mathcal{A}}\rightarrow {\mathcal{A}}$ is a nonlinear surjective map. We prove that if ${\it\phi}$ preserves strong skew commutativity, then ${\it\phi}(A)=ZA+f(A)$ for all $A\in {\mathcal{A}}$, where $Z\in {\mathcal{Z}}_{s}({\mathcal{A}})$ satisfies $Z^{2}=I$ and $f$ is a map from ${\mathcal{A}}$ into ${\mathcal{Z}}_{s}({\mathcal{A}})$. Related results concerning nonlinear strong skew commutativity preserving maps on von Neumann algebras are given.

By an ordered semigroup we mean a semigroup with a simple order which is compatible with the semigroup operation. Several authors, for example Alimov [1], Clifford [2], Conrad [4] and Hion [7], studied the archimedean property in some special kinds of ordered semigroups. For a general ordered semigroup, Fuchs [6] defined the archimedean equivalence as follows: a ~ b if and only if one of the four conditionsholds for some positive integer n.

We show that if a unital injective endomorphism of a $C^{\ast }$-algebra admits a transfer operator, then both of them are compressions of mutually inverse automorphisms of a bigger algebra. More generally, every interaction group – in the sense of Exel – extending an action of an Ore semigroup by injective unital endomorphisms of a $C^{\ast }$-algebra, admits a dilation to an action of the corresponding enveloping group on another unital $C^{\ast }$-algebra, of which the former is a $C^{\ast }$-subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed, and it is faithful if and only if the interaction group is also faithful.

Let G be a finite group and let δ(G) be the number of prime order subgroups of G. We determine the groups G with the property δ(G)≥∣G∣/2−1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order.

We establish a measure of algebraic independence for values of E-functions which is more nearly effectively computable than the previous one. When the system of equations meets either of two criteria, then the measure becomes entirely effectively computable.

The isoperimetric problem in the Euclidean plane is completely solved for bounded, convex sets which are symmetric about the origin, and which contain no non-zero point of the integral lattice.

We denote lattice join and meet by ∨ and ∧ respectively and the associated partial order by ≧. A lattice L with 0 and I is said to be orthocomplemented if it admits a dual automorphism x → x′, that is a one-one mapping of L onto itself such that which is involutive, so that for each x in L and, further, is such that for each x in L.