Mathematical literature contains a number of theorems which assert the existence of a structure ‘universal’ in relation to some given system of objects. Two examples will suffice to indicate the nature of these theorems.

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself (1) and his principal followers and expositors, Tait(2) and Joly(3), only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.

This paper presents an algebraic model for a general class of two-person perfect-information partizan games which may contain cycles, i.e. sequences of positions that may be assumed more than once. An algorithm is given for determining the winning, losing and tie-positions of disjunctive compounds of partizan games.

Let G be a group with the lower central series

Let

where Σ runs over all non-negative integers a1, a2,…, an such that and is the aith symmetric power of the abelian group Gi/Gi+1 where

Let I (G) be the augmentation ideal of G in , the group ring of G over . Define the additive group Qn (G) = In (G) / In+1 (G) for any n ≥ 1. Then it is well known that Q1(G) ≅ W1(G) for any group G. Losey (4,5) proved that Q2(G) ≅ W2(G) for any finitely generated group G. Furthermore recently Tahara(12) proved that Q3(G) is a certain precisely defined quotient of W3(G) for any finite group G.

The purpose of the present note is to investigate the closure operations common to both the operations of forming classes and formations. Furthermore, new descriptions of formations and saturated formations in terms of closure operations are given.

1. Let G be a group, ZG its integral group ring and Δ = ΔG the augmentation ideal ZG By an augmentation quotient of G we mean any one of the ZG-modules

where n, r ≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotients Qn(G) = Qn,1(G) and

(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determine Qn(G) and Pn(G) for finite G it is sufficient to assume that G is a p-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelian p-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: if G is a finite group then there exist natural numbers n0 and π such that Qn(G) ≅ Qn+π (G) for all n ≥ n0; if Gω is the nilpotent residual of G and G/Gω has class c then π divides l.c.m. {1, 2, …, c}. There do not appear to be any examples in the literature of this periodic behaviour for c > 1. One of goals here is to present such examples. These examples will be from the class of finite p-groups in which the lower central series is an Np-series.

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .

This paper is concerned with the notion of independence relating sets of elements in a ring A to a proper ideal a of A. A set of elements a1, …, an ∈A is called a-independent if every form in A[X1, …, Xn] vanishing at a1,…, an has all its coefficients in a. This notion leads to many questions (cf. (2) and (12)), which are of some interest in their own right, several of which are considered here. On the other hand, this independence is related to the structure of the graded ring associated to the ideal generated by the set of elements, hence is often relevant to some problems concerning regular sequences and complex.

Let AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) if

for every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.

Let X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.

Let S be a semigroup with a compact topology in which multiplication is continuous on the left (i.e. xi→x implies xiy→xy for each y in S). Then S has a minimal left ideal L which is compact; each idempotent e in L is a right identity for L (xe = xfor each x ∈ L)and L = Se; Ge = eL is a group and L is the union of all such groups; and if f is a second idempotent in L, the canonical map x ↦ fx of Ge to Gf is an algebraic isomorphism (see Ruppert(2) for these facts). Baker and Milnes(1), §4(A), have observed that, in the case in which S is the Stone–Cech compactification of a discrete abelian group, the canonical map from Ge to Gf may not be a homeomorphism. (This contrasts with the situation in compact semigroups with separately continuous multiplication.) We present a simple proof of a more definitive result.

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions that

and

for all elements a and b in A. It follows from (1.1) and (l.2) that

for all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition that

for all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

Let T be a linear operator on a complex normed space X. Its spatial numerical range V(T) is denned as

We give a ‘picture’ proof to a theorem of M. Freedman (2) which shows the failure of the 4-dimensional homology surgery theory and the homology splitting theorem. Our proof employs the language of the framed links (3) and it involves calculating Casson-Gordon invariant of a certain algebraically slice knot. We use framed links to represent 1-connected 4-manifolds with boundary by attaching 2-handles along them onto B4 via the framings. We adapt the notation ≈ for diffeomorphisms, and for diffeomorphisms between the boundaries of manifolds. Here manifolds refer to smooth manifolds.

The concept of strong almost convergence was introduced in (2), where the matrices summing every strongly almost convergent sequence, leaving the limit invariant, were characterized.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ and perpendicular distance t from the origin, that is H(t, θ) = {x ε Rn: x.θ = t} (Note that H(t, θ) and H(−t, −θ) are the same hyperplane.) If f(x)ε L1(Rn) we will denote by F(t, θ) the projection of f perpendicular to θ, that is the integral of f(x) over H(t, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ε L1 (Rn), F(t, θ) exists for almost all t for every θ.

The main result of the present paper is the construction of a very oscillatory inner function F in the unit disc. The existence of such an F leads to a certain extension of Fatou's classical theorem concerning the boundary behaviour of H∞-functions.

The potential barrier governed by a second-order ordinary differential equation has been studied for decades, both exactly and approximately. These concepts are generalized so as to be applicable to differential equations of arbitrary even order. The barrier is defined, and the approximate solutions both inside and outside are derived. Transmission through this barrier is investigated by deriving connexion formulae through the two transition points of arbitrary odd order. The transmitted solution is linked through the barrier to the incident and reflected solutions in a unique manner, enabling an approximate transmission coefficient to be calculated and its structure to be ascertained. Every concept in the investigation is a suitable generalization of the elementary case, the generalization possible being a product of the approximate processes adopted and how they limit the nature of the generalization. An inspection of the elementary case and its transmission coefficient gives no hint as to the nature of the generalization, the limitations necessarily imposed, and the structure of the final transmission coefficient, since the elementary case has special features not present in the generalization.

Theoretical and numerical calculations of the mechanical properties of single crystals usually presuppose pairwise interactions between the atoms of the lattice. It follows from this assumption that the Cauchy relations hold in respect of the Green measure of strain. Here we account for many-body interactions between the atoms of the lattice. The main result is that a continuum model of n-body interactions must possess completely symmetric (n − 1)th order Green strain moduli. Existing strain energy functions, used to model crystal elasticity, are thereby given some physical significance.