Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

For all positive integers m, n, m ≤ n, let m,n denote the variety of algebras with m n-ary operations ω1, …, ωm and n m-ary operations φ1, …, φn satisfying the system of identities The varieties m,n are considered by Swierczkowski [1] and by Akataeb and Smirnov [2]. Jonsson and Tarski [3] consider m,n in the case m = 1, n = 2. In [2] it is shown that the lattice Lm,n of subvarieties of m,n, is uncountable except when m = 1 in which case Lm,n has a very simple description. In particular, for n > 1 the lattice L1,n is the two element chain which means that 1,n has no proper subvarieties.

The standard methods of constructing generalized free products of groups (with a single amalgamated subgroup) and permutational products of groups are to consider groups of permutations on sets. Although there is an apparent similarity between these two constructions, the exact nature of the relationship is not clear. The following addendum to [4] grew out of an attempt to determine this relationship. By noting that the original construction of permutational products (B. H. Neumann [7]) deals with a group of permutations on a group (although the group structure has been previously ignored; see [7], [8]) we here give an extension of the original permutational product-construction which yields both the generalized free product and the permutational products as groups of permutations on groups. A generalized free product is represented as a group of permutations on the ordinary free product of the constituents of the underlying group amalgam and a permutational product is a group of permutations on the direct product of the constituents of the amalgam.

Until recently there has been little systematic work on the second-order properties of queueing processes. The aim of this paper is to study systematically the second-order properties of the queue length processes embedded at departure epochs in the M/G/1 and bulk service M/G/1 queues, and at arrival epochs in the GI/M/1 queue. In the latter case our results extend those of Daley [7], while in the ordinary M/G/1 queue our work parallels Daley's [6] discussion of waiting times in the same system. In the final section we briefly discuss two discrete time queueing systems.

In this paper we suppose that the function s(x) is integrable in the Lebesgue sense for every finite interval of x ≥ 0. If exists, we say that the integral exists.

A large number of results are available on the lattice of subvarieties of the variety of metabelian groups. When considering metabelian p-groups (for odd p), the immediate division is between groups of nilpotency class less than p, and those of class at least p. The first case was dealt with in some detail in [1], and this paper extends the results to the next interesting cases, classes p and p + 1. The main results are stated in Theorems 2 and 4, which give the basis laws for certain varieties, and 3 and 5, which assert the existence of specific generating groups for these varieties, and hence their non-trivial existence. The notation, and the essential parts of the logic, are as in [1]; for the purposes of this paper, the following modification of Lemma 1.1 of [1], and also the ring-of-integer operations used in its proof, are together dubbed the ‘Stirling manipulation’:

Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

Let G be a commutative lattice ordered group. Theorem 1 gives necessary and sufficient conditions under which a⊥ with a∈G is a maximal l-ideal. A wide family of, l-groups G having the property that the orthogonal complement of each atom is a maximal l-ideal is described. Conditionally σ-complete and hence conditionally complete vector lattices belong to the family.It follows immediately that if a is an atom in a conditionally complete vector lattice then a⊥ is a maximal vector lattice ideal. This theorem has been proved in [7] by Yamamuro. Theorem 2 generalizes another result contained in [7]. Namely we prove that if M is a closed maximal l-ideal of an archimedean l-group G then there exists an atom a ∈ G such that M = a⊥.

Eshelby et al. [1] and Stroh [2] have developed the theory of anisotropic elasticity for a three dimensional state of stress in which the stress is independent of one of the Cartesian coordinates. Various problems involving dislocations in an infinite anisotropic medium are solved in the first paper, while Stroh considers dislocation problems as well as determining the stresses round a crack subjected to an arbitrary non-uniform applied stress. In this note, which follows their treatment, we consider the problem of determining the stresses produced by the indentation of the plane surface of an anisotropic half space by a rigid punch. Problems of this type have been solved by Green and Zerna [3], Lekhnitskii [4], Brilla [5], Gallin [6] and Milne-Thomson [7] but if we take Cartesian coordinates x1, x2, x3 and let the stress be independent of x3, then these authors all assume the x1x2 plane to be one of elastic symmetry. The solution presented in this paper does not require this assumption so that it has a more general application than has been the case with previous solutions to problems of this type. The first part of the analysis given here is for general anisotropy, but in order to obtain a solution to the punch problem by the method of this paper, it is necessary to consider only a particular class of anisotropic materials. The indentation of such materials by a circular block is discussed in section 4 and the results are used in section 5 to examine the case when the circular block is on a transversely isotropic half space.

In a fully ordered (f.o.) ring with identity, the set of all bounded elements as defined below might be an Archimedian subring. Most of the examples of f.o. rings constructed in literature having the bounded set as Archimedian subring are polynomial rings. For example I[x], R[x] etc., where I is the ring of integers and R is the field of rationals, with lexicographic ordering. Now we ask whether a f.o. ring with identity, with the set of bounded elements as Archimedian subring can be a polynomial ring over an Archimedian subring. This is answered affirmatively in Theorem 1. It is proved in Theorem 3 that f.o. rings with identity and with every positive element a large element, belong to the above class. The problem then arises as to when the set of all bounded elements, called a weak Archimedian sub- ring in [2], becomes an Archimedian subring. This problem is completely solved in Theorem 2. The concept of weak Archimedian rings is found to be useful by the author in characterizing some f.o. rings as algebraic algebras in [3].

Let σan be an infinite series, with sequence of partial sums {un}. Let {pn} be a sequence of constants, real or complex, and write Pn = po+p1+ … +pn The sequence-to-sequence transformation defines the sequence {tn} of Nörlund means of the sequence {u}, generated by the sequence {pn}. The series σan is said to be surnmable (N, pn), to sum s, if limn→∞ tn = s. It is said to be absolutely sum.mable (N, pn), or summable |N, pn|, if {tn} ∈BV.

Until recently, very little work has been done on the second order properties of Markov chains. Craven [1] has studied the joint distributions of Markov chains having a Borel subset of n-dimensional Euclidean space as state space. His idea was to consider the process as a time series.

Suppose A and B are continuous linear operators mapping a complex Banach space X into itself. For any polynomial pC, it is obvious that when A commutes with B, then p(A) commutes with B. To see that the reverse implication is false, let A be nilpotent of order n. Then An commutes with all B but A cannot do so. Sufficient conditions for the implication: p(A) commutes with B implies A commutes with B: were given by Embry [2] for the case p(λ) = λn and Finkelstein and Lebow [3] in the general case. The latter authors proved in fact that if f is a function holomorphic on σ(A) and if f is univalent with non-vanishing derivative on σ(A), then A can be expressed as a function of f(A).

Some concepts in measure theory can be generalized by means of classes of null sets. When measures are considered then the classes of all sets of measure zero play the role of classes of null sets. The purpose of this paper is to give an abstract formulation and proof of the Lebesgue decomposition theorem.

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

A map f: A→B in category is called monic if fg = fh implies that g = h for all maps g, h: C → A; it is called epic if gf = hf implies that g = h for all maps g, h: B → C. An object A ∈ is called an S-object if every monic map f: A → A is also epic; it is called a Q-object if every epic map f: A → A is also monic. If A is both an S-object and a Q-object then A is called an SQ-object. In the category of sets the SQ-sets are the finite sets. In the category of vector spaces over a field F the SQ-spaces are precisely the finite dimensional spaces. In the light of these simple examples, it seems reasonable to view the SQ-objects of a category as being of ‘finite type’. We shall be chiefly concerned with investigating the SQ-objects in certain subcategories of the category of locally compact abelian groups.

The symbol D will be written for the space of indefinitely differentiable functions on the n-dimensional Euclidean space Rn which have compact support and Dapos; will denote the space of Schwartz distribution on Rn, the topological dual of D. Except where contrary is explicitly stated, it will be assumed that D′ is equipped with the strong topology β (D′, D) induced by D.