Showing papers in "Canadian Mathematical Bulletin in 1974"
TL;DR: In this article, the authors developed the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kovari [10, 11] which describes the local behaviour of f(z), near a point where | fz | is large, in terms of the power seriesf of f
Abstract: Suppose that 1.1 is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kovari [10, 11], which describes the local behaviour of f(z), near a point where | f(z) | is large, in terms of the power seriesf of f(z).
269 citations
110 citations
TL;DR: In this article, the order of a maximum-sized chain in a lattice L is defined as the length of the longest chain in L minus one, where L(L) denotes the set of all join-irreducible elements in L.
Abstract: An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and l(L) the length of L, that is, the order of a maximum-sized chain in L minus one.
50 citations
TL;DR: In this paper, a module quasi-continu is defined as a module stable par tout projecteur de son enveloppe injective, i.e., a module that is not unstable.
Abstract: Le but de ce travail est de prolonger les travaux de Y. Utumi sur les anneaux continus. Nous appelons module quasi-continu un module stable par tout projecteur de son enveloppe injective. Un anneau unitaire A est dit quasi-continu a gauche (resp. a droite) si le A-module a gauche (resp. a droite ) est quasi-continu. Cette definition est plus generale que celle d'anneau continu a gauche donnee par Yuzo Utumi, et quand l'anneau est regulier, elles coincident (§111).
49 citations
TL;DR: In this paper, it was shown that α(G) is the best possible upper bound for d, where d is the minimum number of generators of G. G. Theorem 1.
Abstract: For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G. G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).
46 citations
TL;DR: In this paper, a structure theorem for modules satisfying the conditions of the dualization of Goldie dimension was proved for all rings considered in this paper and all modules are unital.
Abstract: The purpose of this note is to offer a equalization of the concept of Goldie dimension and to prove a structure theorem (Theorem 3.1) for modules satisfying the conditions of this dualization. In this paper, all rings considered are associative with unit and all modules are unital.
41 citations
TL;DR: In this article, it was shown that the derived Eulerian cuboid does not have integral inner diagonals except possibly when the generators are divisible by 705180.
Abstract: We showed in [1] that the Eulerian family of cuboids with integral edges and face diagonals did not have integral inner diagonals. We now show that the derived family does not have integral inner diagonals except possibly when the generators are divisible by 705180. In this case there appears to be no inherent reason why the diagonals cannot be integral.
27 citations
TL;DR: In this paper, the inverted incidence relation between the flats of a geometry has been interpreted as a geometric lattice, and the inverted lattice G' is not necessarily geometric, but it is a geometrical lattice.
Abstract: To give a geometric interpretation to the inverted incidence relation between the flats of a geometry has for years been a tempting idea in combinatorial geometries [1]. If G is a geometric lattice, the inverted lattice G' is not necessarily geometric.
24 citations
TL;DR: In this paper, the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G is discussed. But the authors focus on the stability groups of a G-set.
Abstract: This paper deals with the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G. It provides characterizations of pure extensions, pure-essential extensions, and equational compactness in terms of the stability groups of a G-set, proves the general existence of equationally compact hulls, and gives an explicit description of these. Further, it establishes, among other results, that all G-sets are equationally compact iff all subgroups of the group G are finitely generated, that every equationally compact G-set is a retract of a topologically compact one, and that for free groups G with infinite basis there are homogeneous G-sets which are not equationally compact.
24 citations
TL;DR: In this paper, a non-empty family of real valued continuous functions on [a, b] is considered, and the problem is to find an element * ∈ S if it exists, for which it exists.
Abstract: Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which
20 citations
TL;DR: The Hermite-Fejer interpolation polynomial Hn [f] of degree ≤ 2n-1 is defined by as mentioned in this paper, where (1) are the zeroes of Chebyshev polynomials of first kind Tn(x)=cos n (arc cos x).
Abstract: The Hermite-Fejer interpolation polynomial Hn [f] of degree ≤2n—1 is defined by (1) Where (2) are the zeroes of Chebyshev polynomial of first kind Tn(x)=cos n (arc cos x).
TL;DR: In this article, the authors considered the problem of continuous real valued functions on a non-trivial interval J of reals under the conditions (H 1 and H 2) and (H 2).
Abstract: In this paper we consider the equation (1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0 under the conditions ((H 0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J; and (H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation (1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0 under the conditions (H 1) and (H 2): the real valued functions m and n are continuous on a non-trivial interval J of reals.
TL;DR: The proof of the conjecture was obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337): as discussed by the authors. But the proof of this theorem was not proved.
Abstract: Purdy asked whether the following conjecture is true Conjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if (1) then The proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):
TL;DR: In this paper, the fixed and common fixed points of self mappings of a metric space were investigated when the pair T i (i=l, 2) satisfies a condition of the following type: (1)
Abstract: Let (X, d) be a metric space and Ti (i=l, 2) be self mappings of X. The purpose of this paper is to investigate the fixed and common fixed points of Ti , when the pair T i (i=l, 2) satisfies a condition of the following type: (1)
TL;DR: In this article, it was shown that a morphism A→B of a first-order theory T is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner.
Abstract: Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .
TL;DR: In this paper, the authors improved Dixmier's result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are not smooth points of the third convolutional space.
Abstract: J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.
TL;DR: Theorem 1.1: If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0,b), then 1 where the constant b/4 is the best possible.
Abstract: Z. Opial [11] proved in 1960 the following theorem: Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then 1 where the constant b/4 is the best possible.
TL;DR: In this article, the authors consider continuous linear functionals on the convergence domain of a matrix and the extent to which their representation is unique; this turns out to be connected with the behaviour of the subsets.
Abstract: The purpose of this paper is to continue the study of certain “distinguished” subsets of the convergence domain of a matrix, as developed by A. Wilansky [6] and G. Bennett [1], We also consider continuous linear functionals on the domain, and the extent to which their representation is unique; this turns out to be connected with the behaviour of the subsets.
TL;DR: In this paper, it was shown that there is a strictly increasing map G-B and sets A(n+1) (v c B) s ch that {x}
Abstract: In order to make (19) correct we define X., S(n) (n<(o) a little more caref lly and replace lines 21-31 on Page 504 by the following : Let n < co and s ppose that we have already chosen elements xi c S (ia+a, then choose sets A,A'cA(' ) s ch that A
TL;DR: Chow as discussed by the authors showed that any one-toone adjacency preserving transformation of the Grassmann space of all the [r] of Sn (0 < r < n − 1) onto itself is a transformation of a basic group of the space.
Abstract: In his paper [1] on homogeneous spaces W. L. Chow states that “ Any one-toone adjacency preserving transformation of the Grassmann space of all the [r] of Sn (0 < r < n— 1) onto itself is a transformation of the basic group of the space.” In the proof both the transformation and its inverse are assumed to be adjacency preserving. See also Dieudonne [2] p. 81.
TL;DR: In this article, it was shown that every non-empty convex bounded subset of the plane is a convex subset of a bounded linear operator T on a Hilbert space H for some T.
Abstract: The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined by W(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.
TL;DR: In this paper, it was shown that the conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime and I is minimal prime.
Abstract: Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.
TL;DR: In this article, the analysis of normal arcs of cyclic order four in the conformai plane is discussed, i.e., arcs in which every circle is met by every circle at not more than three points.
Abstract: In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.
TL;DR: In this article, it was shown that the subgroup generated by a finite subset of a group G is locally finite, and that the fact that this series converges to an element of G (a finite sum) forces the subgroups generated by g 1,…,gn to be finite, proving the theorem.
Abstract: Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite. This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let g l,…,gn be a finite subset of G, and let 1—x is not a zero divisor so it is invertible and its inverse is 1+x+x 3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g 1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].
TL;DR: In this article, the concept of Rayleigh quotient was generalized to a complex Banach space, where the quotient is considered as a function of the components of a characteristic vector of a symmetric matrix pencil.
Abstract: In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient (1) considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that (2) where q i is a characteristic vector corresponding to the characteristic value λ i
TL;DR: The Levitzki radical has been shown to exist in the alternative and Jordan algebras as mentioned in this paper, which is fundamental in the study of polynomial identity algesbras.
Abstract: The Levitzki radical, which is fundamental in the study of algebras satisfying a polynomial identity, has been shown to exist in the varieties of alternative and Jordan algebras (see Zhevlakov [8], Zwier [9], and Tsai [7]— for an important application of this radical to alternative algebras satisfying a polynomial identity, see Slater [6]). In fact, Hartley [4] even investigated local nilpotence for Lie algebras, though this property can not be radical in the sense of Kurosh-Amitsur [3] for these algebras.
TL;DR: In this paper, the rth order moduli of continuity of R. de Vore stated the following conjectures: (a) Let ηv→0 decreasingly, (b)
Abstract: Let f∈Lp (1≤p<∞) 2π-periodic, (1) and let us consider the rth order moduli of continuity of f (2) R. de Vore stated the following conjectures: (a) Let ηv→0 decreasingly, (3)
TL;DR: In this paper, Harary's conjecture has been proved for various finite graphs including regular, Eulerian, unicyclic, separable, trees and cacti.
Abstract: Ulam in [7] has conjectured that any graph G with p≥3 nodes is uniquely reconstructable from its collection of subgraphs Gi=G-vi, i=1,2, … p. This conjecture has been proved for various finite graphs including regular, Eulerian, unicyclic, separable, trees and cacti. Since Ulam′s conjecture seems difficult to prove or disprove, some authors have tried to strengthen the conjecture [3]. One of these stronger conjectures is Harary′s conjecture [2].
TL;DR: In this paper, Nieto et al. showed that a compact linear operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q ∈ B(Y, X) such that QT=I+A, where I is the identity on X and A∈ B[X, X] is a compact operator, if and only if dim N(T) <∞ and R(T] has a closed complement.
Abstract: Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.
TL;DR: In this paper, the authors show that self-complementary graphs, digraphs, and relations provide special classes of self complementary generalized orbits, where a self complementary k-orbit is one in which for every k-subset S in it, X-S is also in it.
Abstract: Abstract A permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.