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.

A standard example of a matrix for which the computation of eigenvectors and eigenvalues is very awkward is the matrix A with ars=1/(r+s—1), 1≤r≤n, 1 ≤s≤n. It is therefore of interest that A commutes with a tridiagonal matrix.

Let A and B be positive semidefinite Hermitian n-square matrices. If A—B is positive semidefinite, write A≥B. Haynsworth [1] has proved that if A≥B then det(A+B)≥det A+n det B.

Let G be a subgroup of the symmetric group, Sn, and let λ be a character on G. Let

where A = (aij) and Er is the rth elementary symmetric function.

The Hermite-Fejér interpolation polynomial Hn[f] of degree ≤2n—1 is defined by

(1)

Where

(2)

are the zeroes of Chebyshev polynomial of first kind Tn(x)=cos n(arc cos x).

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.

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].

Before we give the example, we state the following, which depends on [2] for terminology.

Purdy asked whether the following conjecture is true

Conjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if

(1)

then

The proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):

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 a left near ring and let M be a right N-module. We recall [1] that M is called injective iff every diagram can be embedded into a commutative diagram where A and B are right N-modules with exact.

The purpose of this note is to show that if N is a d.g. near-ring with identity, then M is injective iff for every right ideal u of N and every N-homomorphism f:u→N, there exists an element m in M such that f(a)=ma for all a in u.

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.

Let A be a uniform algebra on the compact set X and let ψ be a non-trivial linear functional on A. A finite non-negative measure μ on X is called a Jensen measure for ψ if

(1)

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.

Let T be a linear operator on a Banach space X and consider the sequence of ranges

where the inclusions are not necessarily proper. The linear subspaces Xn=TnX (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X0=X and denote by |x|0 (x∈X0) the norm of X0.

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.