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 (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.

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.

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.

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.

Problem 188, [3], stated: Apart from finitely many primes p show that if npis the smallest abundant number for which p is the smallest prime divisor of np then np is not squarefree.

Considérons une fonction entiére f(z) de type exponentiel T. Il est bien connu (voir [1]) que si le module de f(z) reste borné le long de Taxe réel, on a

pour tout x réel.

It is generally known [1] that the singular partial differential equation

may not have a unique solution because of the existence of nontrivial representations of zero.

1

Let N be an ideal of a ring A. We say that idempotents modulo N can be lifted provided that for every a of A such that a2-a ∈ N there exists an element e2=e ∈ A such that e-a ∈ N. The technique of lifting idempotents is considered to be a fundamental tool in the classical theory of nonsemiprimitive Artinian rings (refer [2; p. 72]).

By a homomorphic relation over an algebra A we mean a subalgebra of A × A. A variety [1[ of algebras will be called Mal′cevian [2[ if the identities of include two identities of the form f (x, y, y)=x, f(x, x, y)=y. In [3[ many examples and interesting properties of Mal′cevian varieties have been quoted or proved. In [4[ it is noted that every reflexive homomorphic relation over an algebra of a Mal′cevian variety is a congruence. The purpose of this short note is to observe that the property of Mal′cevian varieties noted in [1] is in fact characteristic of such varieties.

The role of binomial distribution in the probability theory and its applications is well known. Recently M. P. Singh [4] has been discussing the socalled inflated binomial distribution. This distribution was introduced for situations which are described by simple binomial except for zero cell which is inflated, that is, there more observations than can be expected on the basis of simple binomial. The investigation of this inflated binomial distribution seems to be useful.

The results obtained here must have been known and settled centuries ago. However, they have proved impossible to locate in the available literature. H. K. Farahat has asked for proofs of the linear independence over the rationals of certain infinite sequences of real numbers such as √2, √3, √5.... He also raised the general question of determining the degree of the field extension generated over the rationals by a family of positive irrational numbers of the form x=a1/mwhere a, m are positive integers.