Goodman [1] has pointed out the applications of the distributional results of the complex multivariate normal statistical analysis. Khatri [4], has suggested the maximum latent root statistic for testing the reality of a covariance matrix. The joint distribution of the latent roots under certain null hypotheses can be written as, [2], [3],

1

where

and

In this paper diagonals of various orders in a (strict) convex polygon Pn are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.

Let U be an open set in the Euclidean plane which has finite area. A complete (or solid) packing of U is a sequence of pairwise disjoint open disks C={Dn}, each contained in U and whose total area equals that of U. A simple osculatory packing of U is one in which the disk Dn has, for each n, the largest radius of disks contained in (S- denotes the closure of the set U.) If rn is the radius of Dn, then the exponent of the packing, e(C, U) is the infimum of real numbers t for which In the sequel we refer to a complete packing simply as a packing.

The Prefrattini subgroups of a finite soluble group were introduced by Gaschutz [3]. These are a conjugacy class of subgroups which avoid complemented chief factors and cover Frattini chief factors. Gaschutz [3, Satz 7.1] showed that if G has p-length 1 for each prime p, and if U≤G avoids all complemented chief factors and covers all Frattini factors, then U is a Prefrattini subgroup of G. We begin by proving the analogous result for the f-Prefrattini subgroups introduced by Hawkes [5], If f is a saturated formation, then the f-Prefrattini subgroups of G are a conjugacy class of subgroups which avoid f-eccentric complemented chief factors of G and cover all other chief factors of G.

By an r-graph G we mean a finite set V(G) of elements called vertices and a set E(G) of some of the r-subsets of V(G) called edges. This paper defines certain colour classes of r-graphs which connect the material of a variety of recent graph theoretic literature in that many existing results may be reformulated as structural properties of the classes for some special cases of r-graphs. It is shown that the concepts of Ramsey Numbers, chromatic number and index may be defined in terms of these classes. These concepts and some of their properties are generalized. The final subsection compares two existing bounds for the chromatic number of a graph.

The purpose of this article is to present a new generalization of the classical Stone-Weierstrass theorem for commutative C*-algebras.

Under the assumption that B is a sub-C*-algebra of A separating the pure states of A and zero, Kaplansky has conjectured that B=A [4, p. 246]. He gave a proof for the case that A is postliminary ([4, Theorem 7.2]; see also [2, 11.1.8]). Glimm, Akemann, and Sakai have established the conjecture in the presence of various other additional hypotheses, most of which hold in the commutative case ([3], [1], [7]).

In certain boundary value problems associated with two-parameter ordinary differential equations defined and having 2p, (p>1), turning points in a given interval, there arises certain equations in integers whose solutions determine the coefficients in the asymptotic expansions for the eigenvalues [1,2, 3, pp. 134-139].

In 1962, V. Ju. Sandberg gave a definition of a Lipschitz structure more general than that of a metric space. In [1], the author gave alternate definitions of a Lipschitz structure, one in terms of a family of pseudometrics and the other in terms of sequences of entourages. These definitions were proved equivalent to each other. In 1965, Sandberg gave conditions under which his Lipschitz structure was generated by a metric.

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that

1.1

convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by

1.2

where then fi(xj) = δij and fi∊X* for each positive integer i.

A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).

In [2] Volger proved that the underlying functor of a category of set-valued models of an r-ary theory has a left adjoint. We want to show that his proof remains valid if instead of set valued models of an r-ary theory models of a finitary theory with values in an arbitrary cocomplete cartesian closed category are considered. As Volger for sets we show for any cocomplete cartesian closed category T C that for every finitary theory (S being a skeleton of the full subcategory of finite sets) the restriction of the left adjoint of on C(s) is a functor in ; here brackets around the exponent indicate as usual a restriction to functors which preserve finite products. We are very much indebted to the referee for pointing out that our proof of the last statement is only based on the properties of C mentioned above and the fact that S has and T preserves finite products. With this in mind and retaining only that part of the cartesian closedness which is relevant for the following considerations we can state the following.

Nous présentons dans cette note une caractérisation catégorique de la catégorie Gr des groupes et homomorphismes. Pour cela nous introduisons la notion de "catégorie projective de groupes" et montrons qu'elle correspond exactement à celle de "demi-variété" de Gr; cette démonstration dépend essentiellement du théorème de caractérisation des catégories algébriques Lawvere [4]. De même la notion de "monomorphisme prénormal" est inspirée de ce théorème et permet de formuler un critère un peu plus faible que celui de Lawvere sur les précongruences et distinguant les variétés parmi les demi-variétés de Gr. Finalement, la caractérisation de Gr que nous donnons fait appel à la classification des variétés de Shreier de Gr de Neumann et Wiegold [7] et au théorème de Scott [8] permettant de plonger tout groupe dans un groupe simple.

Let (X1, d1) and (X2, d2) be metric spaces. A mapping/: X1→X2 is said to be a Lipschitz mapping (with respect to d1 and d2) if and only if (*)d2(f(x), f(y))≤λ⋅ d1(x, y) for all x, y∈X1, where λ is a fixed real number. The constant λ is called a Lipschitz constant for f. If (*) is satisfied for λ=l, then f is called nonexpansive (see, for example, [21]) and if (*), again with λ=1, is replaced by a strict inequality for all x≠y, then f is called contractive [1]. If x∊X1 and X1=X2, then the sequence where f1(x)=f(x) and fn(x)=f(fn-1(x)) for each n>1, is called the sequence of iterates of f at x.

A. Granas [4] has studied single-valued compact vector fields in Banach spaces. In [3], he extended the fixed point theorems of Roth, Boknenblust and Karlin to the case of multi-valued functions. Closely following [4], we give here some general theorems in a class of multi-valued functions in Banach spaces.

Distributions often arise in practice where one or more values of the variate are unobserved. Various practical problems have been referred by Finney [1], David and Johnson [4], Bliss and Fisher [3]. It is of interest to know the exact distribution of the sum of variâtes from such truncated discrete population. In this paper, utilising the property of characteristic function certain general results are shown. The distribution of the sum of independent variables from a discrete population, truncated by any set of s distinct values, follows from them immediately. Using these results the exact distributions of the sum, from binomial, poisson, negative binomial and geometric population, truncated from anywhere, are derived.

Let f be a complex-valued function defined on the real line R with the property that for every x∊R. If k is a nonnegative integer,f is said to have k negative squares, or to be indefinite of order k, if the Hermitian form

The purpose of this paper is to continue the work of [7]. Throughout the paper all notations shall have the same meanings as those in [7]; that is, the ring R is commutative with identity, B(R) is the set of idempotents in R, Spec B(R) is the set of prime ideals in B(R), and Ue for e in B(R) denotes the set {x/x in Spec B(R) with 1—e in x}. Then from [6] we know that {Ue/e in B(R)} forms a basic open set for a topology imposed in Spec B(R) and this topological space is totally disconnected, compact and Hausdorff.

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].

The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

We find several equivalent conditions for an A*-algebra with dense socle to be completely continuous. Such an A*-algebra is modular annihilator [10]. We also study modular annihilator A*-algebras with the weak (βk)-property and obtain a necessary and sufficient condition for such algebras to be dual.

In [2] and [3] the homology of reduced product spaces of certain type of polyhedra was studied. Let Xn=X×X×…×X be the Cartesian product of n copies of a topological space X.

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:

A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that

• (i) exits for every real t but no kind of uniformity of convergence is stipulated;

• (ii) exits for every t;

• (iii) for every t.