Showing papers in "Canadian Journal of Mathematics in 1957"
TL;DR: In this paper, a matrix A of m rows and n columns, all of whose entries are 0's and 1's, is considered, and the sum of row i of A is denoted by r i (i = 1,..., m) and sum of column n of A are denoted as s i (1, n, n).
Abstract: This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0’s and 1’s. Let the sum of row i of A be denoted by r i (i = 1, ... , m) and let the sum of column i of A be denoted by S i (i = 1, ... ,n).
563 citations
TL;DR: A very simple algorithm for finding a maximal flow and minimal cut in a transportation network is described and then applied to obtain an efficient computational routine for the Hitchcock distribution problem.
Abstract: The network-flow problem, originally posed by T. Harris of the Rand Corporation, has been discussed from various viewpoints in (1; 2; 7; 16). The problem arises naturally in the study of transportation networks; it may be stated in the following way. One is given a network of directed arcs and nodes with two distinguished nodes, called source and sink, respectively. All other nodes are called intermediate. Each directed arc in the network has associated with it a nonnegative integer, its flow capacity. Source arcs may be assumed to be directed away from the source, sink arcs into the sink. Subject to the conditions that the flow in an arc is in the direction of the arc and does not exceed its capacity, and that the total flow into any intermediate node is equal to the flow out of it, it is desired to find a maximal flow from source to sink in the network, i.e., a flow which maximizes the sum of the flows in source (or sink) arcs. Thus, if we let P 1 be the source, P n the sink, we are required to find x ij (i,j =1, . . . , w) which maximize
228 citations
TL;DR: In this article, it was shown that given any finite group G there exist infinitely many nonisomorphic connected graphs X whose automorphism group is isomorphic to G. Theorem 1.
Abstract: n 1938 Frucht (2) proved the following theorem: (1.1). Theorem. Given any finite group G there exist infinitely many nonisomorphic connected graphs X whose automorphism group is isomorphic to G.
213 citations
TL;DR: In this paper, it was shown that B is imbeddable in A if there exists a unitary matrix U of order n such that U*AU contains B a s a principal submatrix.
Abstract: Let A, B be two square matrices with complex coefficients, of respective orders n and m, where n ≥ m. We shall say that B is imbeddable in A if there exists a unitary matrix U of order n such that U*AU contains B a s a principal submatrix. In other words, B is said to be imbeddable in A if there exists a matrix V of type n × m such that V*V = IW (= the identity matrix of order m) and V*AV = B.
134 citations
TL;DR: In this article, the authors present concavity results for symmetric functions and apply these to obtain inequalities connecting the characteristic roots of the non-negative Hermitian (n.n.h.) matrices A, B and A + B.
Abstract: The purpose of this paper is to present two concavity results for symmetric functions and apply these to obtain inequalities connecting the characteristic roots of the non-negative Hermitian (n.n.h.) matrices A, B and A + B.
83 citations
TL;DR: In this paper, the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ) is defined, and K and C are the closure and boundary of the open unit disc U in the complex plane.
Abstract: Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).
78 citations
TL;DR: In this paper, a non-Desarguesian projective plane of order p 2n for every positive integer n and every odd prime p was constructed, which can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets.
Abstract: In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p 2n for every positive integer n and every odd prime p.
52 citations
51 citations
TL;DR: In this paper, the Cauchy principal value is associated with an expression which is called its finite part and which possesses many important properties, e.g., it possesses the property that the integrand at x = u has a finite part.
Abstract: If a 0 then (1) is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely (2) . Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.
49 citations
TL;DR: In this paper, it was shown that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions, and that the analytical properties of the generalized Walsh function can be obtained from the character of the direct product of cyclic groups of order α = 2, 3, 4.
Abstract: It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α
(α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.
47 citations
TL;DR: In this article, it was shown that there is no loss in generality in assuming that the greatest common divisor of the values of k, for which k > 0, is one.
Abstract: The above-mentioned paper contained the restriction that ak > 0. In our present paper we remove this restriction and allow the coefficients ak to be positive, negative or zero. However we do retain am ^ 0. In (1) it was shown that there is no loss in generality in assuming that the greatest common divisor of the values of k, for which ak ^ 0, is one. This assumption we shall also retain throughout the present paper. Since the degree m of the polynomial Pm(x) is fixed we shall, wherever possible, suppress m from our notation. Iterated exponential functions occur throughout the paper. For this reason we shall use alternative notations e or exp(x) to denote the exponential function. In this notation we write (1.1) and (1.2) as
TL;DR: In this paper, a brief elaboration on the axioms for a bicategory introduced in (3) is presented, where the main aim is the development of the structure of certain systems of topological and uniform spaces.
Abstract: This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.
TL;DR: In this article, the author established the following congruence: if n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p n r as 0.
Abstract: If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p 2 − l)/24.
TL;DR: In this paper, a generalized limit is a linear functional ϕ on a normed linear space whose general element, x, is a bounded sequence of real numbers and x = l.u.b. |ξ n |.
Abstract: Let M be the normed linear space whose general element, x, is a bounded sequence of real numbers, and ‖x‖ = l.u.b. |ξ n |. Let T denote the linear operation (of norm 1) defined by Tx = (ξ2, ξ3, … , ξ n+1,…). A generalized limit is a linear functional ϕ on M which satisfies the conditions .
TL;DR: The condition used in Jacobson's theorem is a sufficient condition for commutativity as discussed by the authors, but it is not a necessary condition, as it is satisfied by a very restricted class of commutative rings.
Abstract: A well-known theorem of Jacobson (1) asserts that if every element a of a ring A satisfies a relation a n(a) = a where n(a) > 1 is an integer, then A is a commutative ring. Thus the condition used in Jacobson's theorem is a sufficient condition for commutativity. However the condition is by no means a necessary one, as it is satisfied by a very restricted class of commutative rings.
TL;DR: In this article, it was shown that if r has any of the values 2, 4, 6, 8, 10, 14, 26 and p is a prime > 3 such that r(p + 1) ≡ 0 (mod 24), then 1,, where n is an arbitrary integer.
Abstract: If n is a non-negative integer, define pr (n) as the coefficient of xn in ; otherwise define pr (n) as 0. In a recent paper (1) the author has proved that if r has any of the values 2, 4, 6, 8, 10, 14, 26 and p is a prime > 3 such that r(p + 1) ≡ 0 (mod 24), then 1 , , where n is an arbitrary integer.
TL;DR: In this article, it was shown that for h > 1, the countries of any map on a surface of connectivity h = 3 − x can be coloured using at most colours.
Abstract: Heawood (3) showed that for h > 1 the countries of any map on a surface of connectivity h = 3 − x can be coloured using at most colours. In a previous paper (1)1 proved the following
TL;DR: In this article, Touchard constructed a set of polynomials Ω n (z) such that after expansion of the left member B m is replaced by B m,,, and (2).
Abstract: In a recent paper (4) Touchard has constructed a set of polynomials Ω n (z)
such that (1) , where after expansion of the left member B m is replaced by B m , , and (2) .
TL;DR: In this article, it is shown that there exists a positive real number h such that the inequality (1) has infinitely many solutions in coprime integers a and c, and that the set of all such numbers h is a closed set with supremum √ 5.
Abstract: 1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality (1) has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.
TL;DR: In this article, the authors show that the pyramid behaves polyhedral under projection precisely because of its polyhedral nature, and this principle can be reinterpreted to give a criterion for the positive extendibility of positive functions defined on a subspace of a partially ordered vector space.
Abstract: 1. I n t r o d u c t i o n , A pyramid clearly has all its projections closed, even when the line segments from vertex to base are extended to infinite half-lines. N o t so a circular cone. For if the cone is on its side and supported by the (x, y) plane in such a way tha t its infinite half-line of support coincides with the positive x axis, then its horizontal projection on the (y, z) plane is the open upper half-plane y > 0, together with the single point (0, 0). I t is our purpose to show tha t the pyramid behaves bet ter under projection precisely because of its polyhedral nature . And this principle can be reinterpreted to give a criterion for the positive extendibility of positive func t iona l defined on a subspace of a partially ordered vector space. Throughout our discussion E will be a real finite-dimensional vector space and E' its dual space. A subset P of E stable under vector addition and multiplication by non-negative scalars is called a convex cone. In particular, every linear subspace is a convex cone. The smallest subspace containing P is ( — P) + P, and the largest subspace contained in P is ( — P)r\\ P. Omitting parentheses, we shall write —P + P and —Pr\\P. I t is customary to call d im( — P + P ) the dimension of P , and d im( — P C\\ P) the lineality of P. We define the polar P° of P to be the set of all func t iona l f £ E' such t ha t f(x) > 0 for all x Ç P. P° is a closed convex cone, and in fact is the most general such cone, since the double polar P°° coincides with the closure of P. This fact authorizes us to use the notation P°° for the closure of P (provided t h a t P is a convex cone). The elementary duali ty theory of closed convex cones can be summed up as follows: (1) Galois connection: ( P + Q)° = P ° H Q° and ( P P\\ Q)° = (P° + Ç) 0 . (2) d i m ( P + P ) + d i m ( P ° + P°) = dim E. (3) If P ° = £ ' , then P = E.