Let C be a convex subset of a normed linear space E. The following properties of C were studied by Clark [2]:

1. (1) C is of finite width/ that is, C lies between two parallel closed hyperplanes;

2. (2) C is of finite width in some direction; that is, for some line L in E there is a finite upper bound for the lengths of C's intersections with lines parallel to L;

3. (3) C is partially bounded; that is, there is a finite upper bound for the radii of balls contained in C.

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 L be a positive definite symmetric matrix having a noncentral multivariate beta density of an arbitrary rank, see, e.g. Hayakawa ([2, p. 12, Equation 38]). Then an explicit procedure is given for decomposing the density of L in terms of densities of independent beta variates.

We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4].

We consider the initial-boundary value problem for the parabolic partial differential equation

1.1

in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

A group G is called a T3-group if it contains subgroups K and H, HΔK, with the property that if g and gb are members of G—K there is exactly one h∊H which satisfies the equation gh=gb. In these circumstances (G, K, H) is called a T3-triple.

As its name implies, this paper consists of observations on various topics in graph theory that stem from the concept of Hamiltonian cycle. We shall mainly adopt the notation and terminology of Harary [5]. However, we use vertices and edges for what are called "points" and "lines" in [5]. V(G), E(G) respectively will denote the sets of vertices and edges of graph G, and |X| will denote the cardinal of the set X.|V(G)| is the order of G, and |E(G)| the size of G. Throughout n is reserved for the order of G.

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.

It is shown in this paper that Rao's criterion of comparing two unbiased estimators on the basis of definiteness of the difference between their dispersion matrices is equivalent to Cramer's criterion based on their concentration ellipsoids. When the estimators have normal distributions it is shown that both the criteria have a desirable property in terms of the probabilities of the estimators lying in ellipsoids with the parameter point as the center.

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].

All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

Throughout this paper H, H1 etc. will denote positive constants which will not necessarily be the same at different occurrences. If is a series, we shall use the notation For a real, define

1

Let {pn} be a sequence with p0>0 and pn≥0 for n > 0. Define

2

The following identities are immediate:

3

4

where

5

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 A be an additive abelian group and Dn(A) the set of those n x n matrices over A all of whose row and column sums are equal. Such matrices can be regarded as a possible generalization of doubly stochastic real matrices; alternatively, if A is a commutative ring, it turns out that Dn(A) is exactly the image of the permutation representation of Sn, the symmetric group of degree n, over A.

Various generalizations of the Bernstein operator, defined on C[0, 1] by the relation

1.1

where

have been given. Note that bnk(x) is the well-known binomial distribution.

Almost tangent structures have been studied by Eliopoulos [1] and certain Riemannian structures subordinate to almost tangent structures have been studied by Closs [2]. In this paper we investigate those subordinate Riemannian structures for which the underlying almost tangent structure is without torsion and those for which the fundamental form is closed.

Similar studies have been carried out with respect to Riemannian structures subordinate to almost complex structures by Lichnerowicz [4] and with respect to Riemannian structures subordinate to almost product structures by Legrand [5].

The classical mean value theorem asserts that if f is a real, bounded, Riemann integrable function defined on a finite real interval a≤t≤b, then , where infa≤t≤bf(t)≤y0supa≤t≤bf(t). The extensions of Choquet [3], Price [15], and of this paper generalize the fact that y0 belongs to the closure of the convex hull of f([a, b]). The version of Choquet ([3, p. 38]) applies to a continuous function on a compact interval with values in a Banach space; that of Price ([15, p. 24]) applies to a bilinear integral of a special type containing the Birkhoff integral [2]. The m-integral of Dinculeanu [6] (specialization of Bartle's *- integral [1]) leaves intact the Lebesgue dominated convergence theorem and is strong enough to support an extended development. The paper is organized as follows: the object of §2 is to express the integral of a bounded m-integrable function as a limit of Riemann sums; §3 gives Price's generalization of "convex hull" [15]; the theorem of the paper is established in §4; §5 gives applications to vector differentiation which, for continuously differentiable functions, contain results of Dieudonné [5] and McLeod [13].

Let A be a ring and s2(A) the set of idempotents in A. It is a familiar fact that s2(A) becomes a ring if we define 0, 1 and multiplication as in A, and a new negative and new addition by

1

R. A. Melter [3] made the surprising observation that s3(A), where in general sq(A)={x∈A:xq=x}, is a ring if 2 is a unit in A and we define 0,1 minus and multiplication as in A, and a new addition by

2

The nonobvious facts are that s3(A) is closed under ⊕ and that ⊕ is associative when applied to the elements of s3(A). The ⊕ in (1) is actually a formal group over A, and so is associative when applied to any elements of A. However in (2) (and similarly in other cases we shall define) the ⊕ is not a formal group, and the associative law depends on the fact that the elements involved are in s3(A).

Let p and q be polynomial symbols of a type of algebras having operations ∨, ∧, and; (interpreted as the join, meet, and product of congruence relations). If is an algebra, L(), the local variety of , is the class of all algebras such that for each finite subset G of there is a finite subset F of such that every identity of F is also an identity of G.

THEOREM. There is an algorithm which, for each inequality

p≤q,

and pair of integers n, k≥2, determines a set Un, k of (Malcev) equations with the property:

For each algebra, p≤q is true in the congruence lattice offor each∊L() if and only if for each finite subset F ofand integer n≥2 there is a k=k(n, F) such that Un, kare identities of F.

This generalizes a corresponding result for varieties due to Wille (Kongruenzklassengeometrien, Lect. Notes in Math. Springer- Verlag, Berlin-Heidelberg, New York, 1970) and at the same time provides a more direct proof.

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