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.

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.

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 An unpublished proof due to Peter Freyd is extended slightly to yield that a lex-total category, which has a set of objects of cardinality at most that of the universe such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.

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.

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.

Finite groups G possessing a proper subgroup U such that for each element g of G there exists an automorphism of G mapping g into U are considered. The question of how the structure of U determines the structure of G is examined. For example, if G is soluble and U is nilpotent then G is nilpotent.

Let Un be the open unit polydisc, and let T be a mapping from Un into itself. Then the composition transformation CT is a mapping on the Hardy space H2(Un) into the space of complex functions on Un defined as CTf = f ∘ T for every f ∈ H2(Un). An attempt is made to study some properties of CT in this note. A partial generalization of a result of Schwartz, and a relation between intertwining analytic Toeplitz operators and composition operators are reported.

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.

In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

It Is known that a ring R with Krull dimension is an order in an Artinian ring if R is K-homogeneous and the prime radical N of R is weakly ideal invariant. The notion of weak ideal invariance can be interpreted in torsion theoretic terms, yielding a shorter and more conceptual proof of this result. In addition, it is shown that the orders in Artinian rings which arise in this fashion are precisely those for which R/N is K-homogeneous.

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

Given a ŕeal-valued function f on [−1, 1], n ∈ N, and the following partition of [−1, 1[:

there exists a unique polynomial R4n−1(f; x) of degree not exceeding 4n − 1 such that

and, for j = 1, 2 and 3,

In this paper we study the exponential dichotomy property for linear systems, the evolution of which can be described by a semigroup of class C0 on a Banach space. We define the class of (p, q) dichotomic semigroups and establish the connections between the dichotomy concepts and admissibility property of the pair (Lp, Lq) for linear control systems. The obtained results are generalizations of well-known results of W.A. Coppel, J.L. Massera and J.J. Schäffer, K.J. Palmer.

Let R be a ring in which for each x, y in R there exists a positive integer n = n(x, y) such that (xy)n − (yx)n is in the center of R. Then R has a nil commutator ideal.

It is noted that the translation planes of Rao and Rao may be constructed from a Desarguesian plane by the replacement of a set of disjoint derivable nets. Their plane of order 25 which admits a collineation group splitting the infinite points into orbits of lengths 18 and 8 may be obtained by replacing exactly three disjoint derivable nets and may be viewed as being derived from the Andre nearfield plane of order 25.