Nous nous occupons dans ce travail de séries dont voici le type le plus simple

et de leurs analogues lorsqu'on remplace xn par n-s et yn par n-s. Faute d'une autre désignation, nous avons cru pouvoir appeler ces séries respectivement séries de Lambert et séries de Dirichlet. Une série plus générale que la précédente est

où les ak sont des constantes et où g(x) est un polynôme dont les coefficients sont des symboles de Jacobi (1, pp. 132-40; 4 pp 361-9).

possédant la période 2ω. Les expressions

ont des propriétés multiplicatives analogues à celles des sommes de Ramanujan (6) d'où l'on peut déduire deux propositions concernant les racines de g(x).

Let denote the set of functions of a complex variable z, regular at z = 0, and let I denote the set of non-negative integers. For f ∈ put

For a given subset 0 of there arises the problem of characterizing the admissible gap sets If of functions f in 0. When 0 is the set R of rational functions a complete solution in given by the following theorem:

(A) Let f ∈ R and let If be infinite. Then there exist integers L, L1, L2… , L3 such that 0 ≤ L1 < L2 … < Ls < L, and If = {n|n ∈ I, n ≡ Lj (mod L), j =1, … , s} U I\ where V is a Unite exceptional set.

Recently I (1) gave some new basic hypergeometric identities of the Cayley and Orr type with the help of a certain basic differential operator. The present paper deals with some bilateral generalizations of those identities together with certain new identities of the same type. In § 4 is indicated how the generalizations of Orr's identities given recently by Shukla (8, Theorems I, II) may be connected with each other. Later in § 5 certain general expansions of hypergeometric functions are deduced.

The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).

Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditions

and

are equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.

Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).

Throughout this paper R denotes the set of all real numbers, m(K) the Lebesgue measure of K ⊆ R, H a Hilbert space, L(H) the set of all linear continuous mappings of H into H, endowed with the usual structure of a Banach space.

We consider the mapping F of the set R into L(H) such that

holds for all x, y ∈ R. In (2) we have solved this equation under the assumption that H is of finite dimension. In this paper we prove that a weak measurability of F implies its weak continuity in the case of separable Hilbert space. In Theorem 2 we prove that every weakly continuous solution of (1) in the set of normal transformations has the form F(x) = cos (xN), where the normal transformation N does not depend on x.

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, if

where the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) no

is divisible by any p1 … , ps.

We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n where

and we seek necessary and sufficient conditions on n so that

Clearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.

In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

Le radical d'un anneau, dans le sens général de N. Jacobson (1; 2), peut être caractérisé de plusieurs manières. On peut, par exemple, le définir comme l'intersection de tous les idéaux primitifs de l'anneau envisagé. Dans ce travail, nous considérons une classe particulière d'idéaux primitifs, la classe des idéaux corpoïdaux: un idéal est dit corpoïdal si et seulement si l'anneauquotient correspondant est un corps. L'objet principal de ce travail est de donner quelques caractérisations de l'intersection C de tous les idéaux corpoïdaux d'un anneau A, intersection appelée le radical corpoïdal de l'anneau A. Si C est distinct de A, l'anneau-quotient A/C est isomorphe à une somme sous-directe de corps Dans le cas d'un anneau commutatif, tout idéal primitif est corpoïdal, et par conséquent son radical corpoïdal coïncide avec son radical.

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

The problem of arranging v elements into v sets in such a way that every set contains exactly k distinct elements and that every pair of sets has exactly λ = k(k — l)/(v — 1) elements in common, where 0 < » < k < v, is equivalent to finding a normal integral v by v matrix A such that AT A = B, where B is the v by v matrix having k in every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ, k, v design it is necessary that I (the v by v identity matrix) represent B rationally, (2) and (3) have proved the non-existence of certain λ, k, v designs. Neither of the proofs utilize the fact that it is necessary that A be normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rational A such that ATA = B then there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ, k, v which were not previously excluded.

It is shown in (6) how to represent certain sets of orthogonal Latin squares as a group together with a set of permutations of the group elements. The correspondence between 3-nets and loops is well known; for example, see (8). We shall consider a loop G together with a certain set of permutations on the elements of G and shall interpret such a structure as an incidence system in which the 3-net of the loop is embedded. Specifically, the permutations or “adjoints” will give rise to lines which may be adjoined to the 3-net of G in the sense of (3). The group of autotopisms of the loop determines a group of automorphisms of its 3-net analogous to collineations in an affine plane. We shall study the problem of extending these incidence preserving mappings to the adjoined lines. By analogy with the study of loops with operators, we shall consider homomorphisms of loops with adjoints and examine geometric consequences. Particular attention will be paid to the case where G has the inverse property and the adjoints are “linear.” The special case in which G is an abelian group is of geometric interest in that the corresponding incidence systems include the Veblen-Wedderburn affine planes.

Given a set E of n elements we denote by S(l, m, n), (l ≤ m ≤ n) a system of subsets of E, having m elements each, such that every subset of E having l elements is contained in exactly one set of the system S (l, m, n).

It is clear (3), that a necessary condition for the existence of S (l, m, n) is that

1

is the number of elements of S(l, m, n) and

is the number of those elements of S (l, m, n) which contain h fixed elements of E.

It is known that condition (1) is not sufficient for S(l, m, n) to exist. It has been proved that no finite projective geometry exists with 7 points on every line. This implies non-existence of S(2, 7, 43).

Combinatorial configurations may generally be phrased in terms of arrangements of objects into sets subject to certain conditions. In view of this, the question arises as to whether given a set S and its power-set Us (the class of all subsets of S), it might be possible to structure Us in a combinatorially significant manner. This paper proposes and investigates one such structuring achieved by defining a distance function over US.

Given A, B in Us, define their distance by

where N(E) denotes the number of elements in E, + ∞ being an admissible value.

The object of this paper is to develop a method of constructing balanced incomplete block designs. It consists in utilizing the existence of two balanced incomplete block designs to obtain another such design by what may be called the method of composition.

1. Preliminary results on orthogonal arrays and balanced incomplete block designs. Consider a matrix A = (aij) of k rows and N columns, where each aij represents one of the integers 1, 2, … , s. Consider all t-rowed submatrices of N columns, which can be formed from this array, t ≤ k. Each column of any Crowed submatrix can be regarded as an ordered t-plet.

If

is the prime power decomposition of an integer v, and we define the arithmetic function n(v) by

then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated as

MacNeish conjectured that the actual value of N(v) is n(v).

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.

Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

One of the unsolved problems of plane topology is the following:

Question. What are the homogeneous bounded plane continua?

A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).

The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?

The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

On sait que la définition de la cohomologie d'un espace X à coefficients dans un faisceau de groupes non abéliens est liée à la classification de certains types d'espaces fibres principaux de base X (1; 8; 10). Cette cohomologie se définit assez facilement en dimensions zéro et un, mais pour certains problèmes, il serait utile de pouvoir la définir en dimension deux également.

J'ai abordé la question dans (3; 6) mais un certain nombre de difficultés subsistaient qu'il semble utile de chercher à éliminer, ce qui est l'objet de la présente note.

Remarquons d'abord que la définition de H°(X, ), groupe des sections de , est triviale, aucune difficulté ne résultant du caractère non abélien de . En ce qui concerne H1(X, ) une difficulté naît du fait que cet ensemble n'est plus un groupe, même non abélien; il possède toutefois un élément neutre privilégié.

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).