A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shape

where (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric series

for rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such as

and

where Γ(x) denotes the gamma function.

The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

For unexplained terms in semigroup theory see [1] or [4].

Let C, D be classes of semigroups such that every finite semigroup in the class C is embeddable in a finite semigroup in the class D. If n ≥ 2 then k is said to be a C – Dcover of n if every semigroup of order n in the class C is embeddable in a semigroup in the class D of order not greater than k. Let be the least C – D cover of n.

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

The original motivation for the introduction by Frobenius of group characters for non-abelian groups was the problem of the factorization of the group determinant corresponding to a finite group G. The original papers are [5] and [6] and a good historical survey of the work is given in [7] and [8]. If G is of order n, the group matrix XG is defined to be the n×n matrix {xg, h} where xg, h = xgh∈G. Here the xg, g∈G, represent variables. The group determinant ΘG is defined to be det(XG), and is thus a polynomial of degree n in the xg. This determinant is the same, up to sign, as that of the matrix obtained from the unbordered multiplication table of G by replacing each element g by xg. If there is no ambiguity ΘG will be written as Θ.

In this paper, the Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included are the two-relator groups introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology computations and a classification of finite subgroups. The methods are geometric; technical tools include the original theorems of Magnus and Lyndon, as well as an amalgamation technique due to J. H. C. Whitehead.

Based on ideas from semigroup theory, Fountain and Gould [2, 3, 4] introduced a notion of order in a ring which need not have an identity. In some important cases of rings with identity, e.g. if the larger ring is a semisimple artinian ring, this notion coincides with the classical one. The most important result of Fountain and Gould (see [4]) is a Goldie-like characterization of two-sided orders in a regular ring with minimum condition on principal one-sided ideals. In addition, for the same class of rings, a generalization of the Faith–Utumi theorem has been proved by Gould and Petrich[7]. The methods of these papers seem not to work for one-sided orders.

Let R be a Cohen-Macaulay local ring of dimension d, multiplicity e and embedding dimension v. Abhyankar [1] showed that v − d + 1 ≤ e. When equality holds, R is said to have minimal multiplicity. The purpose of this paper is to study the preservation of this property under the formation of Rees algebras of several ideals in a 2-dimensional Cohen-Macaulay (CM for short) local ring. Our main tool is the theory of joint reductions and mixed multiplicities developed by Rees [9] and Teissier[12].

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by Banchoff[1] and Koschorke[6]; see also [15]. We consider the range of dimensions n ≤ 3k where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo-torsion, it can take any value from Thorm's oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.

In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.

We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.

Some general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equation

Here the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.