This paper deals with a general bulk service queueing system with one server, for which customers arrive in a Poisson stream and the service is in bulk. The maximum number of customers to be served in one lot is B (capacity), but the server does not start service until A (quorum, less than B) customers have accumulated; and the service time follows a general probability distribution. Probability generating functions of the distributions Pn and , under equilibrium, have been derived by using the supplementary variable technique. An expression has been derived for the expected value of the queue size and its relation with the expected value of the waiting time of a customer has been explored. A numerical case has been worked out on the assumption that bulk service follows a specified Erlangian distribution.

In this paper the anti-normal property is studied. A space is anti-normal if its only normal subspaces are those whose cardinalities require them to be normal. It is shown that every topological space of at least four elements contains a normal three point subspace from which it follows that there is only one non-trivial anti-normal space.

Nonlinear hydromagnetic connection is investigated using the modal equations for cellular convection. The boundary layer method is used assuming large Rayleigh number R, moderate Prandtl number σ and for different ranges of the Chandrasekhar number Q. The heat flux F is determined for the value of the horizontal wave number which maximizes F. For a weak field, the inertial force dominates over the Lorentz force. F is independent of Q, but it increases with R and σ. For a moderate field, the Lorentz force is significant. F increases with R and σ and decreases as Q increases. For a strong field, the Lorentz force dominates over the inertial force. F is independent of σ, but it increases with R and decreases as Q increases.

Since the early 1950's, when Kodaira “discovered” positive line bundles, the notion of positivity has undergone a continuous evolution. This paper is intended as an introduction to the study of positivity notions. More specifically, I consider the simplest case - line bundles over compact Riemann surfaces - and compare five positivity notions for such bundles. The results obtained are certainly not new; they are, in fact, known in much greater generality. However, by restricting to the dimension one case, I am able to make use of Riemann surface techniques to significantly simplify the proofs. In fact, this article should be easily understood by anyone familiar with the contents of Gunning's Lectures on Riemann surfaces.

Completely simple semigroups form a variety if we consider them both with the multiplication and the operation of inversion. Denote the lattice of all varieties of completely simple semi-groups by L(CS) and that of varieties of groups by L(G). We prove that the mappings V → V ∩ G and V → V v G are homomorphisms of L(CS) onto L(G) and the interval [G, CS], respectively. The homomorphism V → (V ∩ G, V v G) is an isomorphism of L(CS) onto a subdirect product. We explore different properties of the congruences on L(CS) induced by these homomorphisms.

A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

A classical theorem of Brudno, dealing with the consistency of summability with regular matrices is shown by example not to hold over a non-archimedian field.

In this paper we calculate the number of congugacy classes in the following finite classical groups: GLn(Fq); PGLn(Fq), SLn(Fq), and more generally G(Fq), where G is any algebraic group isogenous to SLn; PSLn(Fq); ; , , and more generally where G is any group isogenous to SUn over Fq; and .

In a recent paper, Cusack has given a criterion, in terms of the Fitting class “join” operation, for a normal Fitting class to be closed under the taking of Hall π-subgroups. Here we show that Cusack's result can be slightly modified so as to give a criterion for any Fitting class of finite soluble groups to be closed under taking Hall π-subgroups.

The measure of a polynomial is defined as the exponential of a certain intractable-looking integral. However, it is shown how the measures of certain polynomials can be evaluated explicitly: when all their irreducible factors are linear, and belong to one of two special classes. Asymptotic values for the measures of two sequences of polynomials in large numbers of variables are also found. The proof of this result uses a quantitative form of the central limit theorem.

The lattice of all tolerance relations (that is, reflexive, symmetric compatible relations) on a lattice is investigated. For modular lattices some examples are given which show that such relations do naturally occur.

A counter example is constructed to show that an integral inequality established by Sree Hari Rao (Theorem 3.1, J. Math. Anal. Appl. 72 (1979), 545–550) is erroneous.

A Grothendieck topos has the property that its Yoneda embedding has a left-exact left adjoint. A category with the latter property is called lex-total. It is proved here that every lex-total category is equivalent to its category of canonical sheaves. An unpublished proof due to Peter Freyd is extended slightly to yield that a lex-total category, which has a set of objects of cardinality at most that of the universe such that each object in the category is a quotient of an object from that set, is necessarily a Grothendieck topos.

This paper presents sufficient conditions, involving accretive operators, for the existence, uniqueness and continuous dependence on a control parameter of the solutions of some initial and boundary value problems for neutral functional differential equations.

The existence of metrizable (LF)-spaces was announced by Stephen A. Saxon (“Metrizable generalized (LF)-spaces”, 701–46–14), in Notices Amer. Math. Soc. 20 (1973), A–143. Elsewhere, the authors have discovered an abundant existence of metrizable and normable (generalized) (LF)-spaces, while observing that an (LF)-space is metrizable if and only if it is Baire-like. Recently, W. Robertson, I. Tweddle and F.E. Yeomans introduced the class of locally convex spaces E having the property

(db) if E is the union of an increasing sequence (En) of vector subspaces, then some En is dense and barrelled.

It has long been known that, in terms of right division, groups can be defined by a single law. In this paper a single law defining groups in terms of multiplication and inversion is proposed. This law is in 4 variables, and it is conjectured that no fewer than 4 variables will do, and that the proposed law is of minimal length as well. Some extensions of the result, and an alternative single law with the same length and number of variables, are also discussed. By contrast, groups in terms of multiplication, inversion, and a unit element can not be defined by a single law. Most of these results were stated by Tarski at the Logic Colloquium at Hannover in 1966, but apparently no proof has yet been published.

It is shown that, for a monoidal category V, not every commutation is a symmetry and also that a commutation does not suffice to define the tensor product A ⊗ B of V-categorles A and B. Moreover, it is shown that every symmetry can be transported along a monoidal equivalence.

Two theorems of T.M. Flett [Quart. J. Math. Oxford Ser. (2) 7 (1956), 81–95] on the degree of approximation to a function by the Cesàro means of its Fourier series are extended to Nörlund means. Their conjugate analogues are also proved.

Let G denote a locally compact metrisable zero dimensional group with left translation invariant metric d. The Lipschitz spaces are defined by

where af: x → f(ax) and α > 0; when r = ∞ the members of Lip(α; r) are taken to be continuous. For a suitable choice of metric it is shown that , where 1 ≤ p ≤ 2, α > q−1, p, q are conjugate indices and . It is also shown that for G infinite the range of values of α cannot be extended.

For r-partite and for r-uniform hypergraphs bounds are given for the minimum degree which ensures d independent edges.