Showing papers in "Canadian Mathematical Bulletin in 1971"
TL;DR: In this paper, the following result was proved: the Lipschitz constant k < 1 for a complete metric space with fixed points u and un respectively is a contraction mappings of X into itself.
Abstract: The following result is proved in [1, p. 6]. Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.
400 citations
TL;DR: In this paper, the authors consider associative rings with unity elements and compare various results on the representation of rings by rings of sections of special rings, and show that these results can all be deduced from one statement.
Abstract: Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.
185 citations
TL;DR: In this article, it was shown that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. Schutte [2] raised the following question: given k > 0, is there a tournament Tn (k) such that, given k = 0, there is an edge x dominating y which is directed from x to y?
Abstract: By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schutte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk .)
138 citations
TL;DR: In this paper, a more conceptual proof was devised by N. P. Dekker, and the one offered below may have some claim to be regarded as "the reason the theorem is true".
Abstract: Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".
54 citations
TL;DR: In this paper, Erdös and Szekeres proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein.
Abstract: P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein (who later became Mrs. G. Szekeres).
54 citations
TL;DR: In this paper, the authors provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]) and [4] and Theorem 1, which they state as the following: if p>1, r≠1, and ƒ(x) is nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1.
Abstract: The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state as Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by 1 then 2 unless f≡0. The constant is the best possible.
39 citations
TL;DR: In this article, it was shown that if R is a commutative ring with 1, then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0 P, for each prime ideal P, and.
Abstract: In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0 P , for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.
35 citations
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TL;DR: In this article, the authors define the (t, K)-permanent of A to be (1) the summation taken over all m-tuples (i 1, i 2, …, im ) (repetitions allowed) of 1, 2, etc.
Abstract: Let A = (aij ) be an m × n matrix and let K = {s 1, …, sk } be a k-subset from {1, 2, …, n}. For 0≤t≤k≤n define the (t, K)-permanent of A to be (1) the summation taken over all m-tuples (i 1, i 2, …, im ) (repetitions allowed) of 1, 2, …, n each containing exactly t distinct entries from K and any number of distinct entries from the remaining n-k integers. For example, (4, 4, 7, 1, 1, 2), (4, 4, 6, 6, 6, 5) are 6-tuples, each containing exactly two distinct entries from K={2, 4, 5} for n ≥ 7.
33 citations
TL;DR: A large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them.
Abstract: After the publication of the important paper by Rota [9] on Mobius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Mobius functions is now recognized as a valuable tool in combinatorial and arithmetical research.
23 citations
TL;DR: In this article, a recursive construction for quasigroups orthogonal to their transposes is given, which is based on the singular direct product (SDP) construction of A. Sade.
Abstract: In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.
22 citations
TL;DR: In this paper, the set of all column matrices equipped with componentwise addition and scalar multiplication and the scalar product is identified with the set K = H ⊕ H.
Abstract: Let H be a complex Hilbert space and let K = H ⊕ H. Then K can be identified with the set of all column matrices equipped with componentwise addition and scalar multiplication and the scalar product
TL;DR: In this paper, the lattice of right ideal lattices of a ring and a ring is defined as a prime ideal lattice and if R is commutative then an ideal is prime if and only if it is a prime right ideal.
Abstract: Let R be a ring and let Lγ(R) be the lattice of right ideals. We define that I ∊ Lγ(R) is a prime right ideal provided that if AB⊆I for some A, B in Lγ(R) such that AI⊆I then either A⊆I or B⊆I. Any prime ideal of a ring R is a prime right ideal and if R is commutative then an ideal is prime if and only if it is a prime right ideal. If R is a ring and a∊R, let aR={x ∊ R | x=ar for some r∊R} and aR 1={x ∊ R | x = na+ar for some integer n and r ∊ R}.
TL;DR: In this paper, the oscillatory behavior of the solutions of the linear differential equation (1.1) where all functions are assumed to be continuous on a bounded interval [a, b] was studied.
Abstract: In this paper, we study the oscillatory behavior of the solutions of the linear differential equation (1.1) where (1.2) and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.
TL;DR: In this paper, the authors give some characterizations of collectionwise normality by means of the normality of certain product topological spaces; they are a consequence of a theorem proved by M. Katětov, but they appear to be new.
Abstract: The purpose of this note is to give some characterizations of collectionwise normality by means of the normality of certain product topological spaces; they are a consequence of a theorem proved by M. Katětov, but they appear to be new.
TL;DR: In this paper, the authors give necessary conditions for unique quasi-uniformizability of a topological space (X, ), with finite topology, and show that X, with finite, has a unique compatible quasiuniformity.
Abstract: In [ 2 ] P. Fletcher proved that a finite topological space has a unique compatible quasi-uniformity; C. Barnhill and P. Fletcher showed in [1] that a topological space (X, ), with finite, has a unique compatible quasiuniformity. In this note we give some necessary conditions for unique quasiuniformizability.
TL;DR: In this paper, a Bernoullian symmetric random walk with positive and negative steps was studied, denoting respectively the number of positive steps and number of times the particle crossed the origin, given that it returned there on the last step.
Abstract: In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution of g and k, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.
TL;DR: In this paper, the singular direct product is used to construct quasigroups satisfying the identity x(xy)=yx, and it is shown that singular direct products preserve the identity X(xy) =yx.
Abstract: In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.
TL;DR: In this article, the cardinality of a maximal sum-free set in additive groups is defined as λ(G) where λ is the number of subsets in the group.
Abstract: Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.
TL;DR: In this paper, the authors considered the problem of finding an n × n complex matrix A with a 11, …, ann as principal elements and λ 1, λ n as singular values.
Abstract: We shall be concerned with the following problem. Let a11, …, ann be complex numbers and λ1, …, λ n nonnegative real numbers. Under what conditions does there exist an n × n complex matrix A with a 11, …, ann as principal elements and λ1, …, λ n as singular values? This problem has been suggested in [3] but, to our knowledge, has not yet been solved.
TL;DR: The interesting conjecture of Conway [1] states that p < m unless A is symmetric as mentioned in this paper, and the interesting conjecture holds even when A is a finite set of integers and A + A denote {ai + aj } with p different members.
Abstract: Let A be a finite set of integers {ai } and A + A denote {ai + aj } with p different members and A — A denote {ai —aj ) with m different members, the interesting conjecture of Conway [1] states that p < m, unless A is symmetric.
TL;DR: In this article, it was shown that if a chainable continuum has exactly two arc components, one of them is an arc and the other is a half-ray, then it is shown that one of the components is a double arc.
Abstract: Abstract It is shown that if a chainable continuum has exactly two arc components, then one of them is an arc and the other is a half-ray.
TL;DR: In this paper, an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff is given.
Abstract: In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.
TL;DR: In this article, it was shown that min A ∊ Λn (per A) ≥ 3(n-1) for the class of n × n (0, 1)-matrices with exactly three l's in each row and column.
Abstract: In [3, p. 77] Ryser notes the importance of the minimum of the permanent function on the class of (0, 1)-matrices having exactly k ones in each row and column. In [4] a lower bound was found for the minimum of the permanent on the class Λn of n × n (0, 1)-matrices with exactly three l's in each row and column. The purpose of our work is to improve this result, in particular we show that min A ∊ Λn (per A) ≥ 3(n-1).
TL;DR: It is known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover, for any finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero, there exists a decomposition of L having at least one weight space.
Abstract: Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.
TL;DR: A function f(x) is said to be completely monotonic on (0, ∞) if 1 Familiar examples of such functions are given by f (x) = exp(exp(αx) and f(X) = (x+β)-α, where α ≥ 0, β ≥ 0.
Abstract: A function f(x) is said to be completely monotonic on (0, ∞) if 1 Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].
TL;DR: In this article, the authors considered the square functional equation, where ǫ is a real-valued function of two real variables x, y on the whole xy-plane and t is an arbitrary real variable.
Abstract: We consider the following functional equation 1 where ƒ = ƒ(x, y) is a real-valued function of two real variables x, y on the whole xy-plane and t is a real variable. With regard to the geometric meaning of (1), the equation is called the “square” functional equation.
TL;DR: The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1] as mentioned in this paper, and was proved again by Glimcher [3].
Abstract: Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa } the corresponding family of their Stone-Čech compactifications and ΠaXa the usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].
TL;DR: The concept of p-space was introduced by Arhangel'skii and Alexadroff [2] as discussed by the authors, and the problem of finding a direct intrinsic definition (without appeal to compactification) is studied in this paper.
Abstract: The concept of p-space is quite recent. It was introduced by Arhangel'skii [2]. The definition of p-space given in [2] involves compactification of the space. In view of the interesting properties of p-spaces obtained in [2], Alexadroff [1] suggested a problem of finding a direct intrinsic definition (without appeal to compactification). The main aim of this note is to answer the above problem. I am grateful to Dr. S. K. Kaul for his comments.
TL;DR: In this paper, the concepts of Σinjective module, essential extension, and hull extension were extended to modules over arbitrary rings, and the maximal Johnson and Utumi Σ-quotient rings of R were constructed.
Abstract: Abstract Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.
TL;DR: In this paper, the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort 1 to unbounded domains G has been extended to bounded domains G with suitably regular boundaries.
Abstract: The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort 1 to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings 2 are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.