Completions of categories were studied by Lambek in [3], using the contravariant Horn functor to embed a small category C into the functor category (C *, S), where C * is the opposite category of C, and S is the category of sets. Three completions of C were considered; the completion (C*, S), the full subcategory (C *, C)inf⊆(C *, S) whose objects consist of all inf-preserving functors, and the full sub-category B⊆(C*, S)inf consisting of all subobjects of products of representable functors of the form HomC (—, C), C an object of C.

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

Let X 1X 2,… be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.

Let S k =X 1+…+X k and v x =max{k:S k ≤x}, x≥0 and v x =0 if X 1>x. Billingsley [1] proved if X1≥0 then

converges weakly to the Wiener measure W.

Let τx (ω)=inf{k≥1|S k >x}. In §2 we prove that

converges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.

Since Cauchy's time the theory of analytic functions of a complex variable has depended on complex integration theory, and in particular on the fundamental integral theorem (1825) and integral formulas bearing his name. Cauchy defined an analytic function to be one which had a continuous first derivative in a region D, and showed that an analytic function had derivatives of all orders in D. It was not until 1900, with E. Goursat's famous proof of Cauchy's integral theorem, that the continuity of the first derivative could be inferred from its mere existence at all points of D.

In this paper diagonals of various orders in a (strict) convex polygon P n 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 P m, k denote the set of pyramidal numbers

1.1

The question has been asked whether there exist elements p, q, r in Pm, k such that p+q = r or, as the problem is usually posed,

1.2

The case k=2 has been studied by Sierpinski [6] and Khatri [3]; the case k=3 by Oppenheim [4] and Segal [5]; recently Fraenkel [2] has generalized (1.1) to the larger set

1.3

The purpose of this note is to obtain an extension of the classical Sturm separation theorem for the second order, linear selfadjoint differential equation

1

to the case of a noncompact interval. The classical theorem (cf. [3, p. 209], [4, p. 224]) assumes that r and s are continuous with r positive on a compact interval I, and concludes that between each pair of zeros (on I) of one (nontrivial) solution of (1) there lies precisely one zero of any other linearly independent solution of (1). If I is not compact, a function y which is a solution of (1) on /may often be extended (continuously) to an endpoint, say a, of I.

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={D n}, 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.

We shall call a semigroup S a σ-reflexive semigroup if any subsemigroup H in S is reflexive (i.e. for all a, b ∈ S, ab ∈ H implies ba∊H ([2], [5]). It can be verified that any group G is a σ-reflexive semigroup if and only if any subgroup of G is normal. In this paper, we characterize subdirectly irreducible o-reflexive semigroups. We derive the following commutativity result: any generalized commutative ring R ([1]) in which the integers n=n(x, y) in the equation (xy)n = (yx)m can be taken equal to 1 for all x, y ∈ R must be a commutative ring.

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, F h (b;x) is the generating function of p h (m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:

Luxemburg and Zaanen [5] call an element φ of the topological dual of a normed or seminormed vector space V an integral if

We denote the space of integrals by V I , For the L λ function spaces introduced by Ellis and Halperin [2] another Banach subspace of the dual emerges, namely the conjugate space Lλ* which is the Lλ space determined by the conjugate length function λ*-L λ* is contained in (Lλ)I but need not coincide with it.

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.

Let M(n) be the set of all the points (x 1, x 2,…, x n)∈E n such that x i∈{0,1,2} for each i= 1, 2,…, n and let f(n) be the cardinality of a largest subset of M(n) containing no three distinct collinear points. L. moser [4] asked for a proof of the inequality

Let us consider the set Sn of those points (x 1x 2,…, x n)∈M(n) which satisfy |{i:X i= 1}| = [(n +1)/3]. As S n is a subset of the sphere with center at (1, 1,…, 1) and radius (n-[(n+1)/3])1/2, no three distinct points of S n are collinear. Thus we have

1

By "multiplicative inverse eigenvalue problem" (m.i.e.p., for short) we mean the following. Let A be an n×n matrix and let s1,…, sn be n given numbers. Under what conditions does there exist an n×n diagonal matrix V such that VA has eigenvalues s1,…,s n ?

In the "additive inverse eigenvalue problem" (a.i.e.p., for short) we seek the diagonal matrix V so that A + V has eigenvalues s1,…, s n ?.

In the present paper we extend to the m.i.e.p. the ideas used in [7] for the a.i.e.p.

By per X we denote the permanent of the square matrix X.

If S is an ordered set we write tp S to denote the order type of S and |5| for the cardinal of S. We also write [S]k for the set {X:X ⊂ S, |X|=k}. The partition symbol

(1)

J. Ch. Boland suggested, and Mrs. Sheehan named, the idea of the slimming number of a graph G, i.e. the minimum number, s(G), of edges, e 1e 2,…, e s, which must be removed from G in order that G—∪ e i be planar.

For the complete graph, K n (n≥3), it may be seen by Euler's formula that a planar subgraph contains at most 3n—6 edges; moreover one may construct such a subgraph inductively, starting from K 3, and adding points successively, joining them to the three vertices of the region in which they lie, so

1

P n will denote a set of n points in the plane. A well known theorem of Gallai- Sylvester (see e.g. [4]) states that if the points of P n do not all lie on a line then they always determine an ordinary line, i.e. a line which goes through precisely two of the points of P n .

Using this theorem I proved that if the points do not all lie on a line, they determine at least n lines. I conjectured that if n>n0 and no n—1 points of Pn are on a line, they determine at least 2n-4 lines. This conjecture was proved by Kelly and Moser [3], who, in fact, proved the following more general result: Let P n be such that at most n—k of its points are collinear.

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.