We showed in [1] that the Eulerian family of cuboids with integral edges and face diagonals did not have integral inner diagonals. We now show that the derived family does not have integral inner diagonals except possibly when the generators are divisible by 705180. In this case there appears to be no inherent reason why the diagonals cannot be integral.

Let T be a bounded linear operator defined on a Hilbert space H. An element z∈H is called a least squares solution of the equation

if . It is easily shown that z is a least squares solution of (1) if and only if z satisfies the normal equation

It is well-known that Wilks’ Λ criterion is distributed as the product of p independent beta variables in the p-variable null-case [3]. In the collinear case, Λ is still distributed as the product of p independent beta variables, one of them following a non-central beta density. Thus when p=2, the exact non-null distribution of Λ in the collinear case is given by the product of two independent beta variables, one central and the other having non-centrality parameter λ.

If the samples of sizes n1, n2,…, nq+1 are available from (q+1) normal populations with different mean vectors μα, α=l, 2,…, q+1 and the same covariance matrix ∑ and if x’=(x1, x2, …, xp) denotes the vector of p variates on which the measurements are made, then we obtain the following multivariate analysis of variance table.

J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

Recently, in the Research Problems of Canadian Mathematical Bulletin, Vol. 14, No. 4, 1971, there appeared a problem which asks “Is a prime Von Neumann regular ring pimitive?” While we are not able to settle this question one way or the other, we prove that in a Von Neumann regular ring, there is a maximal annihilator right ideal if and only if there is a minimal right ideal.

Let (X, d) be a metric space. Let Xc be the family of all non-empty closed subsets of X endowed with the Hausdorff metric D induced by d. Let f be a non-expansive mapping of X into Xc (D(f(x), f(y))≤d(x, y) for all x, y in X). Some fixed point theorems are obtained by imposing certain conditions on f and X.

The problem of finding bounds for the number h(n) of Hamiltonian circuits on the n-cube has been studied by several authors, (1), (2), (3). The best upper bound known is due to Larman (5) who proved that .

In this paper we use a result of Nijenhuis and Wilf (4) on permanents of (0, 1)- matrices to show that for n≥5

where τ, a and c are constants.

Let C(n) be the number of partitions of n such that all even multiplicities of the parts are less than 2m, m>1; and all odd multiplicities are at least (2r+1) and at most 2(m+r)—1, r≥0. Let D(n) be the number of partitions ofn into parts which are either odd multiples of (2r+1) or are even and not divisible by 2m.

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

This is a supplement to a previous paper [4] in which integration was developed for arbitrary extended-real functions over arbitrary sets in an outer measure space S, ℳ*, m* where m*; also written m, is a regular outer measure on all subsets of an arbitrary set S≠ø, and ℳ* is thsetse family of all m*-measurable sets.

Let A be a commutative algebra contained in Mn(F), F a field. Then A is nilpotent if there exists v such that Av=(0), and is said to have nilpotency class k (denoted Cl(A)=k) if Ak=(0), but Ak-1≠(0). A well known result asserts that matrix algebras are nilpotent if and only if every element is nilpotent. Let N = {A | A is a nilpotent commutative subalgebra of Mn(F)}.

In [2: p. 415], P. Gabriel proves that if R is a ring with 1 and S is a non-empty multiplicative set such that 0∉S, then S-1R exists if and only if for every pair (a, s)∈R×S, there is a pair (b, t)∈R×S such that at=sb and if s1a=0 for some s1 in S then as2=0 for some s2 in S. The purpose of this note is to give a self contained elementary proof of Gabriel’s result.

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].

Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

Let p be a rational prime and n a positive integer ≥2. We denote by an(p) the least positive integral value of a for which the polynomial xn+x+a is irreducible (mod p), and set

(1)

Ulam in [7] has conjectured that any graph G with p≥3 nodes is uniquely reconstructable from its collection of subgraphs Gi=G-vi, i=1,2, … p. This conjecture has been proved for various finite graphs including regular, Eulerian, unicyclic, separable, trees and cacti. Since Ulam′s conjecture seems difficult to prove or disprove, some authors have tried to strengthen the conjecture [3]. One of these stronger conjectures is Harary′s conjecture [2].

It was shown in [1] that certain classes of groups which are closed under quotients, and extensions contain a unique maximal inclosed subclass. These results prompted the question whether there exists a class of groups which is closed under quotients and extensions and yet does not have a unique maximal R0-closed subclass. This note provides an example of such a class.

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will prove

Let G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.

In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined by

W(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.