Let 1 < a1 < a2 < … be a sequence of integers and let N(x) denote the number of a's not exceeding x. If N(x)/x tends to a limit as x tends to infinity we say that the a's have a density. Often one calls it the asymptotic density to distinguish it from the Schnirelmann or arithmetical density. The statement that almost all integers have a certain property will mean that the integers which do not have this property have density 0. Throughout this paper p, q, r will denote primes.

I conjectured for a long time that, if e > 0 is any given number, then almost all integers n have two divisors d1 and d2 satisfying

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If f ∈ Lp (— ∞, ∞), 1 < p ≤ 2, then f has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞) where p-1 + q-1 = 1. Also if ∣x∣1-2/qf(x) ∈ Lq (— ∞, ∞), q ≥ 2, then / has a Fourier-Plancherel transform F ∈ Lq (— ∞, ∞). These results can be found in (2, Theorems 74 and 79). In neither case, however, does the collection of transforms cover Lq, except when p = q = 2, and in neither case, with the same exception, has the collection of transforms been characterized.

Further, if f ∈ Lp, (— ∞, ∞), 1 < p ≤ 2, then its transform F has the property |x|1-2/pF(x) ∈ Lp (— ∞, ∞) (see 2, Theorem 80) but, except when p = 2, the collection of transforms does not cover the set of functions with this property, and again, except when p = 2, the collection of transforms has not been characterized.

This paper develops a case for adopting as the standard elliptic integrals of the third kind the function IIs(u, a) defined by

and the three functions lIs (u, a + Kc), Πs(u, a + Kn), Πss(u, a+ Kn) where KC, Kn, Kn are the three quarter-periods of the Jacobian system. The function Πs(u, a) is the same function whether qs u is cs u, ns u, or ds u.

The origin of the paper was a wish to understand how it has come about that the integrals commonly accepted as standard are not related symmetrically to the theta functions in terms of which they are expressed.

The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.

For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.

Let be a difference field of characteristic 0, m an irreducible manifold of effective order n over {y}, and F an algebraically irreducible difference polynomial in {y} of effective order n + k, k > 0, which vanishes on 3 m. In an earlier paper (2, p. 447) I gave necessary conditions, restated below as (a), (b), and (c) of the main theorem, for m to be an essential singular manifold of F. These conditions are analogous to the low power criterion of Ritt (1, p. 65) for the corresponding problem of differential algebra. Like that criterion they depend, in the special case that m is the manifold of y, only on which power products appear effectively in F. Unlike the low power criterion, however, conditions (a), (b), and (c) are only necessary, not sufficient.

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.

The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.

We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.

By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.

A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

A proper subgraph of a connected linear graph is said to disconnect the graph if removing it leaves a disconnected graph. In this paper we characterize, in the following sense, the disconnecting subgraphs of a fixed connected graph. We define two distinct types of disconnecting subgraphs (isthmuses and articulators) which are minimal in the sense that no proper subgraph of either type can disconnect the graph. We then show that any disconnecting subgraph must contain either an isthmus or an articulator. We also define a set of subgraphs (called dense) which form a lattice. We show that the union of the minimal dense subgraphs contains all isthmuses and articulators. In terms of these subgraphs we investigate some of the consequences of assuming that a disconnecting subgraph must contain at least m points.

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.

The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

Let A1, A2,… , An be a finite collection of subsets (not necessarily distinct) of a set A. By a transversal of A1, A2,… , An we shall mean a set of n distinct elements a1, a2,… , an of A such that, for some permutation i1 i2, … , in of the integers 1, 2, … , n,

More generally, we shall say that the set {a1, a2, … , ar}, (r ≤ n) is a partial transversal oi A1, A2, … An of length r if (i) a1, a2, … , ar are distinct elements of A and (ii) there exists a set of distinct integers i1, i2… , irsuch that

A well-known theorem of P. Hall (2) states that the sets A1, A2… , An have a transversal (of length n) if, and only if, every k of them contain collectively at least k distinct elements (k = 1, 2, … , n).

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.

The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.

More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.

Soit n un entier ≥ 1. On notera Mn l'ensemble des matrices carrées d'ordre n à éléments réels, et Gn le groupe des x = (εJk) ∈ Mn tels que εJk = 0 pour 1 ≤ j ≤ k ≤ n, εJk; = 1 pour 1 εJkj εJkn. Le groupe Gn est un groupe de Lie nilpotent simplement connexe, dont l'algèbre de Lie s'identifie à l'ensemble Qn des x = (εJk) ∈ Mn tels que εJk = 0 pour 1 ≤ j ≤ k ≤ n. Nous allons déterminer:

1. (1°) le centre de l'algèbre enveloppante de gn;

2. (2°) la série “principale” de représentations unitaires irréductibles de Gn; (la recherche de toutes les représentations unitaires irréductibles de Gn ne semble pas facile) ;

3. (3°) la formule de Plancherel pour Gn ;

4. (4°) les caractères globaux (au sens de (5)) des représentations de la série principale.

Throughout this note K will denote a field of characteristic p > 0. Let I be the set {1,2, … , m}, and a finite additive group of functions on I with values in K. We assume that is total in the sense that, for any λ1, … , λm in K Σi=imλiσ(i) = 0 for all a in G implies all σi = 0. It is clear that is an elementary p-group. Let pn be the order of . A generalized Witt algebra is defined as an algebra over K with basis elements {e(σ, i)∣ σ ∈ , i ∈ I} and the multiplication table

is a simple Lie algebra except when p = 2, m = 1.

Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:

If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).

Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:

1. (i) K is supersolubly immersed in G.

2. (ii) K/ϕK is supersolubly immersed in G/ϕK.

3. (iii) If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.

Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.

The purpose of this paper is to investigate non-linear recursive sequences of maximum length with elements from GF(2). In particular, the question of whether or not a recursive sequence of maximum length can be equal to its dual is settled. This question, as far as the author knows, was originally asked by Rosser. Part I contains the necessary background for Part II, and in the main is a condensation of some unpublished work (1955) of W. A. Blankinship and R. P. Dilworth.

Let H be an n-square Hermitian matrix with eigenvalues h1 ≥ h2 ≥ . . . ≥ hn. Fan (2) showed that

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k — 1,2, … , n, where the max and min are taken over all sets of k orthonormal (o.n.) vectors in unitary w-space Vn. Marcus and McGregor (3) have generalized this result in the case that H is non-negative Hermitian. For vectors X1, … , xr r ≤ n, in Vn, let X1 Λ X2 Λ … Λ xT denote the Grassmann exterior product of the xt; it is a vector in Vm,, where

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:

No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

Let denote an arbitrary Boolean algebra. Let Latin letters a, b, … denote general elements of while the symbols 0, 1 denote the special smallest and largest elements. Let Greek letters α, β, … denote various prime dual ideals of elements of . It is recalled that a prime dual ideal of is a proper subset of closed under finite intersections of its elements and maximal with respect to those properties. Every prime dual ideal includes the element 1 and for each element a of includes either a or ā (complement of ain ) but not both. Occasional reference will be made to principal dual ideals of . These are subsets of composed of all elements of majorizing some fixed non-zero element of . Finally, let X() denote the collection of all prime dual ideals of .