In this paper we are concerned with functional equations of the type

1 .

in which x, y do not appear explicitly. J. Aczel [1] has given a method forfinding real solutions of some of these equations. We prove a theorem whichcan sometimes be used to solve problems concerning the uniqueness ofsolutions of such equations.

Some years ago, G. Pall observed that the invariant factors of the incidence matrices of a certain pair of non-isomorphic projective planes of order 9 were different. With the aim of investigating such phenomena experimentally, we have constructed a code to calculate the invariant factors of rational integral matrices (actually, we compute the Smith's normal form of these matrices, as described, e.g., in MacDuffee [2; p. 41]), and this note is in the nature of a report on some preliminary experiments in the use of this code. In particular, we have computed the invariant factors of all (0, 1) matrices of order ≤ 8, with constant row and column sums, and these data are presented in the Appendix.

In the study of algorithms for the iterative solution of an arbitrary analytic equation f(z) = 0, acceleration procedures are of importance in practice and of considerable interest in the theory of the subject. Let

.

be an iteration formula which has a zero ξ of f(z) as attractive fixed point. An algorithm of this type is said to converge towards a root ξ of f(z) = 0 for all initial approximations z = z0 in a vicinity of ξ, of order k > 0, when

1 .

In a recent paper, [l], Dixmier has proved Heinz' inequality by deducing it from a lemma due to Thorin. In this note it is proved directly from a convexity theorem.

Let(M(k), ℳ(k), μ(k)), k = 0, …, n, be measure spaces and Lq(k) (M(k), ℳ(k), μ(k)) be all the functions on M(k) such that

A. P. Guinand [1] has shown that for appropriate infinite sequences an, αn,(an, αn real andpositive for all n) the relation

1 .

implies that

2 .

The following two theorems are due to Capelli.

THEOREM 1. Let g(x) and h(x) be polynomials over a field R of characteristic 0; let f(x) = g(h(x)). Then f(x) is irreducible over R if and only if

1. (i) g(x) is irreducible over R

and

1. (ii) h(x) - β is irreducible over R(β), where β is a root of g(x).

This note is based on Hu's Homotopy Theory (New York, 1959), pp. 115 f.

The theorem that the homotopy sequence is exact splits into six statements. The fifth in Hu's exposition can be stated as follows: Let f:In, In-1, Jn-1→X, A, x0,

1

Then there exists a map

2

such that f~j g: In, In-1, Jn-1→X, A, x0.

A special case of a well known theorem of Ramsay [3] states that an infinite graph either contains an infinite complete subgraph or it contains an infinite independent set; in other words there exists an infinite subset of its vertices so that either every two of them are joined by an edge or no two of them are joined by an edge. Thus if we have a graph whose vertices are the integers, and which has no infinite complete sub-graph, it certainly has an infinite independent set. The question can now be asked if there exists an independent set whose vertices n1 < n2 < … do not tend to infinity too fast.

Let

1

be n+2 distinct points on the real line and let us denote the corresponding real numbers, which are at the moment arbitrary, by

2

The problem of Hermite-Fejér interpolation is to construct the polynomials which take the values (2) at the abscissas (1) and have preassigned derivatives at these points. This idea has recently been exploited in a very interesting manner by P. Szasz [1] who has termed qua si-Hermite-Fejér interpolation to be that process wherein the derivatives are only prescribed at the points x1, x2, …, xn and the points -1, +1 are left out, while the values are prescribed at all the abscissas (1).

In this note we do not intend to establish new results but only to suggest avery simple proof of Lusin's theorem, direct for σ-finite regular measures,a proof that bypasses the usual procedure of first establishing this theoremfor sets of finite measure only. The proposed proof utilizes the notion ofsubuniform convergence, a method which seems not yet to have been used,despite its simplicity and adaptability. Simultaneously, a useful supplementto Egoroff's theorem will be obtained.

Let G be a region in Eućlidean n-space En and consider the eigenvalue problem Δ2u = λu on G, with boundary conditions u = 0 on Γ, the boundary of G. (To be precise, we are considering the eigenvalue problem for the self-adjoint 2 realization L associated with the Laplacian -Δ2and zero boundary condition, acting in L2(G), cf Browder [2]). If G is bounded, the spectrum of this problem is discrete, but Rellich showed in 1952 [6] that the spectrum could also be discrete for certain unbounded regions which he introduced and called "infinitely narrow tubes".

Let p(z) be a polynomial of degree n, i. e. a finite sum of the form where cν are any given numbers and z=x+iy is a complex variable. To answer a question raised by the chemist Mendelieff, A. Markoff [3] proved the following theorem.

In this note, we present two matrix lemmas (one without proof) which haveinteresting applications in statistical estimation theory.

LEMMA 1. Let A be a k X k positive definite matrix. Then for any k X 1vector c, we have that

1 .

The identity of Abel [1] [2] we deal with here can be stated in thefollowing form: If n is a positive integer,

.

(In order that all terms be defined we require a ≠ 0, b ≠ n.)

This identity and deductions from it have been very useful in many problems,for instance in mathematical statistics [3]. Usually this identity isestablished by means of the Lagrange - Bűrman theorem [4]. Here we willderive it very simply.

We consider round-robin tournaments of n players in which, at each encounter, the winner is awarded 1 point and the loser 0 (ties are excluded).

Let

1

be the n scores, ordered in a non-decreasing sequence. Clearly

2