If A is a finite matrix with complex elements, and if A = AT (where AT denotes the transpose of A ), it is known (see [8] ) that there exists a unitary matrix U such that UA UT = D is a real diagonal matrix with non-negative elements which is a canonical form for A relative to the given U, UT transformation.

On note D le disque unité ouvert du plan complexe, le disque fermé et T l'ensemble des réels modulo 2π.

Soit une suite de réels positifs; on désigne par {Nn}+ la classe des fonctions f, analytiques dans D, continues ainsi que toutes leurs dérivées dans , qui vérifient la propriété suivante: il existe des constantes Af et Mf telles que pour tout z dans et tout entier n positif ou nul .

Let {λn} be an increasing sequence of positive numbers. The question of the closure in Lp(0, ∞ ) (1 ≦ p ≦ ∞ ) of the linear span of the sequence has been considered by several authors, notably by Boas (1) and Fuchs [3;4].

In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.

In [4] it was conjectured that a light map p : E → B for which paths can be lifted and lifting of arcs is unique is a Serre fibration. As is well-known this implies that paths have unique liftings. In this paper we shall prove several special cases of this conjecture.

The two main theorems are: (3.5) Let p be a light compact map of a metric space E onto a connected semi-locally contractible along arcs metric space B. If arcs can be lifted uniquely then p is locally trivial.

Meier [8, Hauptsatz] proved a remarkable theorem concerning the boundary behavior of functions meromorphic in the upper half plane; but his techniques are very complicated. So Noshiro [10, p. 72-73] proved an analogous (but somewhat weaker) result to Meier's by a simple method using the theorem of Gross and Iversen.

In this paper, we sharpen and generalize Noshiro's theorem in some directions by making use of the notion “porosity”, and we state some applications.

Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.

For graph theoretic terms, see Tutte [1], A rank 2 tactical configuration of girth 2g and order (s, t) may be regarded as a (1 + s, 1 + t)-regular bipartite graph of girth 2g. We assume s ≦ t. Using a technique of Friedman [2] we show (i) and (ii).

If A is a real normal matrix At (the transpose of A) is a real polynomial in A. We study here those normal (0, 1)-matrices A with constant row sums which have At a polynomial of degree two in A.

An unsolved problem in metrization theory is whether every Hausdorff M-space with a Gg-diagonal is metrizable. There are several recent results which have a bearing upon this question. In [9], P. Zenor showed that an M-space is metrizable if and only if it has a regular Gδ-diagonal; in [1], Borges showed that a regular meta-Lindelöf M-space is metrizable if and only if it has a -diagonal.

In the development of any mathematical theory it is often advisable to test the known theoretical results in particular cases, for in this way we are able to judge the state of the subject, its strengths and its weaknesses. It is the purpose of this paper to put forward three remarkable graphs which can be used to indicate the progress being made in research in graph theory. We shall undertake three tasks - the construction of the graphs themselves, the investigation of their properties (insofar as the theory allows), and the indication of possible lines for further enquiry.

The purpose of this note is to give a new proof of a theorem of L. Carlitz [2] and R. McConnel [5]. The theorem is as follows:

THEOREM 1. Let F = GF(pn) be the finite field of order q = pn and let K — {x ∈ F|xd = 1} for some proper divisor d of q — 1.

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R

(i) (d1q)+ Λ (d2q)- = 0, and

(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.

We remark that this characterization holds even when R does not have an identity element.

The purpose of this paper is two-fold : first, to show that Dedekind domains satisfy a generalization of the Wedderburn-Mal'cev Theorem and, secondly, to classify certain types of finitely generated projective algebras over a Dedekind domain.

With respect to the first problem, E. C. Ingraham has shown that a Dedekind domain R is an inertial coefficient ring (IC-ring) if and only if R has zero radical or R is a local Hensel ring.

Let ωn(x) denote the Jacobi polynomials with the weight Function

If we denote the corresponding normalized Jacobi polynomials by we Have

(1.1)

Now let

be the nth partial sum of the Fourier series of Jacobi polynomials of a function f(x).

It is well-known that if is a quotient group of a group G, then (i) is a partition of G, and (ii) the usual complex product (in the usual sense of multiplication of complexes) of every pair of members of is a member of . There arises the question whether, conversely, (i) and (ii), perhaps in a weaker form, suffice for to be a quotient group of G.

Let denote the class of functions f(z) = z + that are analytic and univalent in and will denote the collection of f ∈ that map U onto a domain that is respectively starlike with respect to the origin and convex.

In [4, p. 85] Hayman used Steiner symmetrization to solve a problem, a special case of which is the following.

An integral domain D is called an essential ring if D = ∩αVα where the Vα are valuation rings which are quotient rings of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin [2, pp. 717-718] has observed that every v-multiplication ring is essential and that an essential ring having a defining family of valuation rings {Vα} which is of finite character (i.e. every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring; but he conjectures that, in general, there exists an essential ring which is not a v-multiplication ring.

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation

(1)

where

(i) denotes the order of differentiation with respect to t. The delay terms τi σi are assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constant M on the half line (t0, + ∞ ) for some t0 ≧ 0.

Let J be a simple closed curve in a 3-manifold M3. We say M — J is 1-ALG at p ∈ J (or has locally abelian fundamental group at p) if and only if for each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and each loop in V — J which bounds in U — J is contractible to a point in U — J.