Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

In studying the geometry of normed linear space it is useful to draw attention to the following mapping.

In the theory of multivalent functions there are several different levels of postulates for p-valency. Perhaps the most well-known is the class of mean p-valent functions in the sense of Spencer [8] (we shall refer to them as areally mean p-valent functions), whose basic properties are found, e.g., in Hayman [4]. Recently Eke [1, 2] extended to these functions a number of results which had been known for circumferentially mean p-valent functions.

The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the front- and end-adjunctions are closed transformations. (The terms ‘Closed Category’ etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a ‘closed monad’; this fact was also proved in [4].

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

This is a sequel to the author's (Phatarfod [9]) paper in which an analogue of Wald's Fundamental Identity (F.I.) for random variables defined on a Markov chain with a finite number of states was derived. From it the sampling properties of sequential tests of simple hypotheses about the parameters occurring in the transition probabilities were obtained. In this paper we consider the case of continuous Markovian variables. We restrict our attention to the practically important case of a Normal Markov sequence X0,X1,X2,… such that the Yr being independent normal variables with mean zero and variance σ2.

This paper is a continuation of [1]1 We shall use the same notations as those in [1]. Let F(X) ∈ R[X], X = (X1, …, Xn), be a polynomial of degree d > 0 and h(x) ∈ SP(Rn), i.e. h(x) is the sum of a polynomial and a Schwartz funtion. We shall consider Dirichlet series of the type where NF = {x∈Rn: F(x) = 0}. We proved, in [1], that Z(h, F, s) is regular for σ > (n+p)/d and possesses the analytic continuation to the whole s-plane when Fd(x) (the highest homogeneous part of F(X)) ≠ 0 for x ≠ 0. In this paper, we shall say the following.

In this paper first we prove the uniqueness theorem for an integral transform whose kernel is H-function. Later on we establish a new and interesting theorem concerning this transform and a generalized Laplace transform whose kernel is Meijer's G-function.

Let S be a pseudo-metric space with pseudo-metric d. Then for each nonempty A ⊆ S, x ∈ cl (A) if and only if d(x, A) = 0, and d(x, y) = d(y, x) for all x, y ∈S. Thus, if d(x, y) = 0, then x∈cl({Y}) by the first requirement and Y∈cl({x}) by the second requirement. It is natural then to expect that if a topological space S is to be pseudo-metrizable, it should at least satisfy the requirement:

Lattices of congruences are studied in section II.6 of Cohn [2]. Papers [5] and [6] by Munn deal with lattices of congruences on semigroups and conditions under which these lattices are modular. In [4] Lallement shows that the lattice of congruences on a completely 0-simple semigroup is semimodular, giving an alternative proof of the result, due to Preston [7], that the lattice of congruences on a completely 0-simple semigroup satisfies a certain chain condition which is a natural extension to arbitrary lattices of the Jordan-Dedekind chain condition for finite lattices.

It is well known that the projections of a pair of points from the vertices of a triangle onto the opposite sides lie on a conic and that when the points are the centroid and orthocentre of the triangle, this conic is a circle. Analogously the projections of the centroid and orthocentre of a simplex from its vertices onto the opposite (n—1)-dimensional faces, if the simplex is orthocentric, lie on a hypersphere [2, 5]. Further the projections of two points onto the edges of a general simplex from the opposite faces lie on quadric [1]; and when the points are the centroid and orthocentre respectively and the simplex is orthocentric, this quadric is a hypersphere [2]. The results as regards projections onto (n—l)-dimensional and 1-dimensional faces being thus known, it remains to see what results hold in the case of intermediary faces. And in this note we prove that a similar result holds for projections onto intermediary faces as well.

The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind:

If S1, S2,… are the times of successive births in a pure birth-process with linear birthrate and i > 0 individuals initially, then given that there are k > 0 births in (0, t), the random variables eλS1—1, eλS2−1,λS1−1 are distributed (conditionally) like the order statistics of k independent random variables, all uniformly distributed on the interval (0, eλt–1), regardless of the initial population size. A similar property holds for the pure death process.

Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, l, …) of giving rise to j progeny in the following generation and in which there is an independent immigration component where bj (j = 0, l, …) is the probability that j individuals enter the population at each generation. Then letting Xn (n = 0, l, …) be the population size of the n-th generation, it is known (Heathcote [4], [51]) that {Xn} defines a Markov chain on the non-negative integers. Unless otherwise stated, we shall consider only those offspring and immigration distributions that make the Markov chain {Xn} irreducible and aperiodic.

In 1921 von Karman [1] showed that the Navier-Stokes equations for steady viscous axisymmetric flow can be reduced to a set of ordinary differential equations if it is assumed that the axial velocity component is independent of the radial distance from the axis of symmetry. He used these similarity equations to obtain a solution for the flow near an infinite rotating disk. Later Batchelor [2] and Stewartson [3] applied these equations to the problem of steady flow between two infinite disks rotating in parallel planes a finite distance apart.

For any set X denote by m(X) the Banach space of all bounded real-valued functions on X, equipped with the supremum norm, and denote by (X) the semigroup (under functional composition) of all transformations of X, i.e. mappings with domain X and range contained in X. A pair (X, S), where S is a subsernigroup of (X), will be called a transformation semigroup. Important examples are obtained by letting X be the underlying set in an abstract semigroup and considering the pairs (X, S1) and (X, S2), where S1 [Sn] denotes the set of left [right] multiplication mappings of X. We shall call transformation semigroups in these classes of examples l-[r-] semigroups.