Let {ank} be a matrix defined by

1

and n taking only non-negative integer values.

Let f(x) ∈ L [0, 2π] and be periodic with period 2π outside this interval. Let the Fourier series associated with the function f(x) be given by

and let

where s is a constant.

A purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

Let P stand for a polynomial set (p.s.), i.e., a sequence {P0(x), P1(x), P2(x),...} such that for each n P0(x) is a polynomial in x of exact degree n and P0(x)≠0. We refer to Pn(x) as the nth component of P.

Let S denote the semigroup of all rectifiable, piecewise continuously difïerentiable paths in ℝn under concatenation. We prove a theorem to the effect that every finite collection of paths is contained in a subsemigroup of S which has the unique factorization property with respect to certain primes and straight lines. We also determine an abstract necessary sufficient condition for a subsemigroup of S to have this unique factorization property.

A surface of order three F in the real projective three-space P3 is met by every line, not in F, in at most three points.

In the present paper, we determine the existence and examine the distribution of elliptic, parabolic and hyperbolic points; that is, the differentiable points of F which do not lie on any line contained in F.

J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

Almost from the day that B. L. van der Waerden [10] proved his now famous theorem on arithmetic progressions, mathematicians have been working to find a new or an improved constructive proof of that result, but without much success.

L'objet de cette note est de montrer que le dérivé d'un cuboïde eulérien n'est jamais parfait.

(1) Appelons cuboïde entier un cuboïde dont les arêtes et les diagonales des faces sont des entiers; appelons cuboïde parfait un cuboïde entier dont la diagonale intérieure est un entier.

Let A1, A2 be commutative semi-simple Banach algebras and M(A1), M(A2) their multiplier algebras. Birtel in [2] has proved that every isomorphism of A1 onto A2 induces an isomorphism of M(A1) onto M(A2). In this note, we extend this result to the noncommutative case. We also show that if A is a dual A*-algebra which is a dense two-sided ideal of a B*-algebra B, then M(A) is isomorphic to M(B). Thus the converse of the previous result cannot hold. All algebras under consideration are over the complex field.

Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M2 be the maximum of the absolute values of the r × r minors of the augmented matrix (A | B). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M2. (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then .

All rings R considered here are commutative with identity and all the modules are unital right modules. As defined by Mehdi [6] a module MR is said to be a multiplication module if for every pair of submodules K and N of M, K ⊂ N implies K=NA for some ideal A of R. This concept generalizes the well known concept of a multiplication ring.

In this paper, we are concerned with the functional inequality

1

where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.

Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case

2

and this provided a characterization of Rényi's entropy.

This note answers a question raised by Lee Mohler in 1970, by exhibiting a finite topological space X which is the union of closed subspaces Y, Z, such that Y, Z, and Y ⋂ Z, but not X, have the fixed point property. The example is a triangulation △ of S3, the points of X being the simplices of Δ and the closed sets the subcomplexes of △.

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximation

and the extremal transformations U whenever they exist.

This paper contains several characterizations of regular-closed and minimal regular topological spaces along with some related properties.

For a bounded linear operator T on a Hilbert space , denote by Lat0T the lattice of all linear submanifolds of such that , and by (resp. Lat T) the sublattice consisting of operator ranges (resp. closed subspaces).

Without appealing to the Cauchy theorem or its corollaries, it is proved that the real and imaginary parts of a non-constant complex-valued analytic function of several complex variables are functionally independent. This unifies and generalizes some results sporadically treated in standard treatises on function theory.

Certain operators essentially defined by convolution are considered. Their possible domain and range spaces are determined; then conditions are given under which the construction of the optimal continuous partner may be carried out for a suitable domain or range. Special cases of operators of convolution-type are useful in studying the boundedness properties of conjugate function operators and, more generally, classes of operators satisfying restricted weak-type conditions.

In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d(x) d(y) = d(y) d(x) for all x, y∈R. Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.

We show that under mild conditions the Central Limit Theorem holds for general stochastic processes.