bluuebunny/crossref_metadata_2025 huggingface dataset:
Characterizations of Developable Topological Spaces
Authors: Worrell J. M. , Wicke H. H.
Published: December 2010

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).

This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.
Bipositive and Isometric Isomorphisms of Some Convolution Algebras
Authors: Edwards R. E.
Published: December 2010

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces
Upsets in Round Robin Tournaments
Authors: Fulkerson D. R.
Published: December 2010

Consider a round robin tournament in which each of n players is required to play precisely one game with each other player, and assume that each game ends in a win or a loss. The results of such a tournament can be conveniently recorded in a square (0, 1)-matrix T = (tij) of order n by setting tij = 1 if player i defeats player j , tij = 0 if player i loses to player j , and tii = 0. Thus T has 0's along the main diagonal, and in the off-diagonal positions T satisfies the "skew-symmetry" condition that tij = 1 if and only if tji = 0. We call such a (0, 1)-matrix T a tournament matrix.
The K-Product of Arithmetic Functions
Authors: Gioia A. A.
Published: December 2010

In this note we introduce a natural generalization of the ordinary convolution of arithmetic functions: If f and g are arithmetic functions,

defines the K-product of f and g. If the kernel K(n) ≡ E(n) = 1, the K-product is the ordinary convolution Σd|nf(d)g(n/d);, if K(n) ≡ ∊(n) = [1/n], then the K-product is the unitary product Σf(d)g(n/d), summed over d|n, (d, n/d) = 1 (1, 2). We give in Theorem 1 a characterization of all associative kernels, i.e., kernels for which the corresponding K-product is associative.
Sums of Rational Numbers
Authors: Webb W. A.
Published: December 2010

Previously, the problem of expressing rational numbers as finite sums of rational numbers of a given type has been concerned with the Egyptian, or unit, fractions. It has long been known that any rational number is the sum of distinct unit fractions. In response to a problem proposed by E. P. Starke (4), R. Breusch (1) and B. M. Stewart (5) showed that every rational number with an odd denominator is a sum of distinct odd unit fractions. P. J. Van Albada and J. H. Van Lint (6) extended this result to show that any integer is a sum of unit fractions with denominators from an arithmetic progression.
Prime Modules
Authors: Feller E. H. , Swokowski E. W.
Published: December 2010

Characterizations for prime and semi-prime rings satisfying the right quotient conditions (see § 1) have been determined by A. W. Goldie in (4 and 5). A ring R is prime if and only if the right annihilator of every non-zero right ideal is zero. A natural generalization leads one to consider right R-modules having the properties that the annihilator in R of every non-zero submodule is zero and regular elements in R annihilate no non-zero elements of the module. This is the motivation for the definition of prime module in § 1.
On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, I: Square Matrices
Authors: Majindar Kulendra N.
Published: December 2010

A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.

To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.
Some Finitely Generated Analogues of a Group of A. H. Clifford
Authors: Lewin Jacques
Published: December 2010

Let n1 and n2 be two elements of a commutative field of characteristic different from 2 that satisfy: (i) n1 ≠ 0; (ii) n2 ≠ 0; (iii) n1 + n2 ≠ 0. We define the “weighted average” α * β of two arbitrary elements α and β of as
An Existence Theorem for Generalized Direct Products with Amalgamated Subgroups
Authors: Tang C. Y.
Published: December 2010

Generalized direct products with amalgamated subgroups were introduced by B. H. Neumann and Hanna Neumann in their joint paper (4). In general, we call a given collection of groups with specified subgroups amalgamated an amalgam of groups; if all groups are abelian we speak of an abelian amalgam. The group freely generated by the amalgam is called the abelian free sum of the amalgam provided it contains the amalgam isomorphically. The free abelian sum need not exist. Hence one of the problems is to find necessary and sufficient conditions for its existence.
The Representation of a Graph by Set Intersections
Authors: Erdös Paul , Goodman A. W. , Pósa Louis
Published: December 2010

Geometrically, a graph is a collection of points (or vertices) together with a set of edges (or curves) each of which joins two distinct vertices of the graph, and no two of which have points in common except possibly end points. Two given vertices of the graph may be joined by no edge or one edge, but may not be joined by more than one edge. From an abstract point of view, a graph G is a collection of elements ﹛x1, x2, …﹜ called points or vertices, together with a second collection of certain pairs (xα, Xβ) of distinct points of G. It is helpful to retain the geometric language, and refer to any pair in as an edge (or a curve) of G that joins the points xα and Xβ.
Groups Associated with Certain Loci In [5]
Authors: Horadam A. F.
Published: December 2010

If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation

1.1

represents the well-known (11) Perazzo cubic primalP43 of order 3 and dimension 4. With it is associated the Segre cubic threefoldS33 (12); specifically, S33 is the section of P43 by a tangent [4].
Classes of Functions on Algebras
Authors: Cullen C. G. , Hall C. A.
Published: December 2010

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .

Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.
The Construction of Perfect and Extreme Forms
Authors: Scott P. R.
Published: December 2010

Let

be a positive quadratic form of determinantD,and letMbe the minimum off(x) for integral x ≠ 0. Thenf(x) assumes the valueMfor a finite number of integral vectors x =±mk(k= 1 , … ,s)called theminimal vectors.
A Weight Theory for Unitary Representations
Authors: Sherman Thomas
Published: December 2010

Over a field of characteristic 0 certain of the simple Lie algebras have a root theory, namely those called “split” in Jacobson's book (3). We shall assume some familiarity with the subject matter of this book. Then the finite-dimensional representations of these Lie algebras have a weight theory. Our purpose here is to present a kind of weight theory for the representations of these Lie algebras when their ground field is the real numbers, and when the representation comes from a unitary group representation.
On a Restricted Class of Block Design Games
Authors: Di Paola Jane W.
Published: December 2010

Block design games have been developed by Richardson (12) and by Hoffman and Richardson (8), who proved a number of theorems concerning such games by studying the number of elements in a blocking coalition. Hoffman and Richardson listed as unsolved (except for PG(2, 3)) the following problem: What is the minimum number of elements in a blocking coalition of a block design game?

This note considers blocking coalitions in those games that are dual to block designs having λ = 1 and r — k > 0. For such games certain blocking coalitions are shown to be related to sets of mutually disjoint blocks in the design to which the game is dual. In particular, for Steiner triple systems the largest odd-numbered set of mutually disjoint triples is shown to yield a minimum blocking coalition in the dual.
Normal Functions and Non-Tangential Boundary Arcs
Authors: Lappan P. , Rung D. C.
Published: December 2010

Let D and C denote respectively the open unit disk and the unit circle in the complex plane. Further, γ = z(t), 0 ⩽ t ⩽ 1, will denote a simple continuous arc lying in D except for Ƭ = z(l) ∈ C, and we shall say that γ is a boundary arc at Ƭ.

We use extensively the notions of non-Euclidean hyperbolic geometry in D and employ the usual metric

where a and b are elements of D. For a ∈ D and r > 0 let

For details we refer the reader to (4).
Generalization of the Fibonacci Sequence to n Dimensions
Authors: Raney George N.
Published: December 2010

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.
Gap Series on Groups and Spheres
Authors: Rider Daniel
Published: December 2010

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E where

dx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.
Invariant Subspaces on Riemann Surfaces
Authors: Voichick Michael
Published: December 2010

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.

Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.
Equivalent Formulations of the Borsuk-Ulam Theorem
Authors: Bacon Philip
Published: December 2010

Let R° denote a space consisting of just one point and for each positive integer n let Rn denote euclidean n-space. For each non-negative integer n let Sn denote the n-sphere

In 1933 K. Borsuk published proofs of the following two theorems (2, p. 178).
Local Properties of the Embedding of a Graph in a Three-Manifold
Authors: McMillan D. R.
Published: December 2010

LetGbe a finite graph topologically embedded in the interior of a 3-manifoldM.Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each pointp∈Gis necessary in order for the embedding ofGto be tame:

For each sufficiently small open set U containing p, there is an open set V such that p∈V⊂U and if W is any connected open set such that p∈W⊂V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n —1generators.
Continuous Families of Curves
Authors: Grünbaum Branko
Published: December 2010

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.
A Class of Loops with the Isotopy-Isomorphy Property
Authors: Wilson Eric L.
Published: December 2010

A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.

It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).
Note on the Descendent Theorem of Slepian, Moore, and Prange
Authors: Shatz Stephen S.
Published: December 2010

In this note we prove the Descendent Theorem (2) of Slepian, Moore, and Prange in an abstract form. Our proof shows that the theorem is valid in much more general settings than that of vector spaces over Z/2Z. Applications of the descendent theorem to coding theory may be found in (2), and a study of Prange's method of proof is carried out by Dade in (1).
Extrema in Space-Time
Authors: Quintas Louis V. , Supnick Fred
Published: December 2010

Consider an astronomer and his observation field, i. e., the set of observable (light or radio) signal-emitting loci of the universe. Let the observation field be ordered by attaching a date to each observable locus indicating the time in the history of the universe that the signal was emitted from its source. Whereas both the astronomer and his observation field age with time, the observations of the astronomer may trace a sequence of loci whose time labels proceed forward or backward in time (cf. Appendix).

Consider now a finite set S of events in Ln,n-dimensional space-time (Riemannian n-space having the fundamental form
Embeddings of Topological Products of Circularly Chainable Continua
Authors: Fearnley L.
Published: December 2010

In a recent paper (5), the author has established the Euclidean spaces of least dimension in which the topological products of finite collections of k-cell-like continua can be embedded. Specifically, it was shown that, for each pair of positive integers k and n, the topological product of any collection of nk-cell-like continua can be embedded in Euclidean space of dimension k(n + 1). This result includes a theorem of Bennett (1) that the topological product of any finite collection of n snakelike continua can be embedded in Euclidean space of dimension n + 1.
Measures and Tensors II
Authors: Gil de Lamadrid Jesús
Published: December 2010

The present work is a sequel to our previous article (10) with the same title. The major theme remains the study of the relationship between tensor products of spaces of functions and vector-valued measures on a space S. In (10) S was a compact Hausdorff space. Here we extend our considerations to locally compact Hausdorff spaces. B(S) still stands for the Borel class of S.

Three types of vector-valued measures m : B(5) → E, E a Banach space, are considered here (§3), namely, weak*, weak, and strong vector measures, but the concepts of weak and strong measures coincide. This result is due to Bartle, Dunford, and Schwartz (1) for abstract vector measures, i.e. defined on an abstract σ-algebra of sets.
On Products of Symmetries
Authors: Fillmore Peter A.
Published: December 2010

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.
Self-Centred Sets
Authors: Kestelman H.
Published: December 2010

A subset S of an abelian group G is said to have a centre at a if whenever x belongs to S so does 2a — x. This note is mainly concerned with self-centred sets, i.e. those S with the property that every element of S is a centre of S. Such sets occur in the study of space groups: the set of inversion centres of a space group is always self-centred. Every subgroup of G is self-centred, so is every coset in G: this is the reason why the set of points of absolute convergence of a trigonometric series is self-centred or empty (1). A self-centred set of real numbers that is either discrete or consists of rational numbers must in fact be a coset (see §3); this does not hold for an arbitrary enumerable self-centred set of real numbers (§3.3).
On Redfield's Range-Correspondences
Authors: Foulkes H. O.
Published: December 2010

In an important paper (7), long overlooked, J. H. Redfield dealt with several aspects of enumerative combinatorial analysis. In a previous paper (1) I showed the relation between a certain repeated scalar product of a set of permutation characters of a symmetric group and Redfield's composition of his group reduction functions. Here I consider, from a group representational point of view, Redfield's idea of a range-correspondence and its application to enumeration of linear graphs. The details of the application of these ideas to more general enumerations are also given.
A Permanental Inequality—The Case of Equality
Authors: Marcus Marvin , Minc Henryk
Published: December 2010

In (3) we proved that if A is a complex n-square normal matrix with characteristic roots α1 … , αn,then

1

If A is positive semi-definite hermitian, the inequality (1) becomes

2

This inequality partially answers the problem of determining the maximum permanent of a positive semi-definite hermitian matrix with prescribed characteristic roots (6). In (1), Brualdi and Newman proved that (2) also holds when A is an n-square circulant with non-negative entries. In a recent conversation Dr. Newman raised the question of determining the cases of equality in (1). In the present note we answer this question.
Generalizations of the Converse of the Contraction Mapping Principle
Authors: Wong James S. W.
Published: December 2010

This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).
Commutators in Factors of Type III
Authors: Brown Arlen , Pearcy Carl
Published: December 2010

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.
A Semigroup Approach to Lattices
Authors: Janowitz M. F.
Published: December 2010

In (3, p. 85) we denned a Baer semigroup to be a multiplicative semigroup with 0 having the property that the left (right) annihilator of every element is a principal left (right) ideal generated by an idempotent. We showed (3, Lemma 1(vi) and Theorem 5, p. 86) that with set inclusion as the partial order, the set of left annihilators and the set of right annihilators of elements of a Baer semigroup form dual isomorphic lattices with 0 and 1.
Group Algebra Modules. I
Authors: Gulick S. L. , Liu T. S. , Van Rooij A. C. M.
Published: December 2010

Some time ago, J. G. Wendel proved that the operators on the group algebra L1(G) which commute with convolution correspond in a natural way to the measure algebra M(G) (13). One might ask if Wendel's theorem can be restated in a more general setting. It is this question that is the point of departure for our present paper. Let K be a Banach module over L1(G). Our interest is in operators from L1(G) into K, and from K into L∞(G), which commute with the module composition (where L∞(G) is thought of as a module over L1(G) also). Such operators we call (L1(G), K)- and (K, L∞(G))-homomorphisms, respectively. Investigations of various other kinds of module homomorphisms occur in A. Figà-Talamanca (6) and B. E. Johnson (9; 10).
Conjugates and Legendre Transforms of Convex Functions
Authors: Rockafellar R. T.
Published: December 2010

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.
Notes on Sphere Packings
Authors: Leech John
Published: December 2010

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.

The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.
A Note on Weierstrass Points
Authors: McQuillan Donald L.
Published: December 2010

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).
Gocommutative Hopf Algebras
Authors: Larson Richard G.
Published: December 2010

A coalgebra over the field F is a vector space A over F, with maps δ: A → A ⊗ A and ∊: A → F such that

1

and

2

The notion of coalgebra is dual to the notion of algebra with unit, with δ as coproduct (equation (1) says that δ is associative) and ∊ as the unit map (equation (2) is just the statement that ∊ is a unit for the coproduct δ). If A is also an algebra with unit and δ and ∊ are algebra homomorphisms, A is a Hopf algebra.
On Arcs in a Finite Projective Plane
Authors: Martin G. E.
Published: December 2010

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments.

To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.
A Note on d-Groups
Authors: Wehrfritz B. A. F.
Published: December 2010

This note concerns subgroups of the general linear group GL(n, F), where n is finite and F algebraically closed, g ∈ GL(n, F) is called a d-element if there exists an x ∈ GL(w, F) such that x–1gx is diagonal, and a u-element if (g — 1)n = 0. A subgroup G of GL(n, F) is called a d-group (or a u-group) if every element of G is a d-element (or a u-element). In view of the Jordan decomposition of the elements of GL(n, F) into products of d-elements and u-elements it is important to know the structure of d-groups and u-groups, u-groups present very little difficulty and their structure is well known (1, 19.4), but d-groups seem to have a more complicated structure.
A Note on Hypernilpotent Radical Properties for Associative Rings
Authors: Dickson S. E.
Published: December 2010

We work entirely in the category of associative rings. We show that if P1 is a homomorphically closed class which contains the zero rings, then the lower Kurosh radical P of P1 is the class P2 of all rings R such that every non-zero homomorphic image of R has non-zero ideals in P1, provided that P1 is closed under extensions by zero rings (i.e., if I is a P1-ideal of R and (R/I)2 = 0, then R ∈ P1). The latter assumption replaces the hypothesis that P1 be hereditary for ideals in a similar result of Anderson-Divinsky-Sulinsky in (2). This leads to a brief proof that the lower radical construction of Kurosh terminates at Pω0 (where ω0 is the first infinite ordinal) when P1 is a homomorphically closed class of associative rings containing the zero rings. This was proved for arbitrary homomorphically closed classes P1 of associative rings in (2).
On Extensions of Topologies
Authors: Borges Carlos J. R.
Published: December 2010

If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:

(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.

(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).

(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.
Covering Classes of Residues
Authors: Jordan James H.
Published: December 2010

A set of ordered pairs of integers {(ai, mi)} is said to cover the integers if each integer x satisfies the congruence x ≡ ai (mod mi) for some i. We may assume that the mi are positive. Trivially {(0, 1)} covers, as does {(0, m), (1, m), (2, m), … , (m — 1, m)}. In order to arrive at some non-trivial problems concerning covers, the following definition is given: A finite set of ordered pairs of integers with mi > 1 and mi ≠ mj if i ≠ j, is called a covering class of residues if every integer x satisfies the congruence x ≡ ai (mod mi) for some i.
On the Non-Existence of a Type of Regular Graphs of Girth 5
Authors: Brown William G.
Published: December 2010

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that

1.1

Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.
Pairs of Consecutive Residues of Polynomials
Authors: Williams Kenneth S.
Published: December 2010

Let p be a large prime and let ƒ(x) be a polynomial of fixed degree d ⩾ 4 with integral coefficients, say,

1.1

Recently Mordell (8) has considered the problem of estimating the least positive residue of ƒ(x) (mod p), that is, the unique integer l (0 ⩽ l ⩽ p — 1) such that the congruence

1.2

is soluble for r = l but not for r = 0, 1, … , l — 1.
Some Results on v-Multiplication Rings
Authors: Griffin Malcolm
Published: December 2010

A family Ω of valuations of the field K is said to be of finite character if only a finite number of valuations are non-zero at any non-zero element of K. If w ∈ Ω has ring and maximal ideal , then A = ∩w∈Ω is said to be defined by Ω and ∩ A is a prime ideal called the centre of w on A and denoted by Z(w). If = Az(w), then w is said to be an essential valuation for A. A domain defined by a family of finite character in which every valuation is essential is called a ring of Krull type.
Cell Growth Problems
Authors: Klarner David A.
Published: December 2010

Thesquare latticeis the set of all points of the plane whose Cartesian coordinates are integers. Acellof the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. Ann-ominois a union ofncells which is connected and has no finite cut set.

The set of alln-ominoes,Rnis an infinite set for eachn; however, we are interested in the elements of two finite sets of equivalence classes,SnandTn, which are defined on the elements ofRnas follows: Two elements ofRnbelong to the same equivalence class (i) inSn, or (ii) inTn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.
A Characterization of the Finite Simple Group PSp4(3)
Authors: Janko Zvonimir
Published: December 2010

The aim of this paper is to characterize the finite simple group PSp4(3) by the structure of the centralizer of an involution contained in the centre of its Sylow 2-subgroup. More precisely, we shall prove the following result.
Der Höhensatz in der Geometrie Involutorischer Gruppenelemente
Authors: Bachmann F.
Published: December 2010

In der ebenen absoluten Geometrie, die hier stets in der Allgemeinheit meines Buches Aufbau der Geometrie aus dem Spiegelungsbegriff (AGS) verstanden sei, hat man guten Grund, die Geraden vor den Punkten zu bevorzugen. Grundlegende Relationen in der Menge der Geraden sind das Senkrechtstehen von zwei Geraden und das Im-Büschel-Liegen—die Abhängigkeit—von drei Geraden, das man nach Hjelmslev wie folgt definiert: Drei Geraden liegen im Büschel, wenn das Produkt der Spiegelungen an ihnen eine Geraden-Spiegelung ist.
Generalized Convex Bodies of Revolution
Authors: Firey WM. J.
Published: December 2010

The figures studied in this paper are special convex bodies in Euclidean three-dimensional space which we shall call generalized convex bodies of revolution (GCBR). Such a set is obtained by the following procedure. Let K1 be a convex body of revolution and let x, y, z denote Cartesian coordinates in a system for which the z-axis is the axis of K1.
The Number of Very Reduced 4 × n Latin Rectangles
Authors: Moser W. O. J.
Published: December 2010

Two permutations (displayed in the two rows)

of the integers 1, 2, … , n are called discordant if ai ≠ bi, i = 1, 2, …, n. Let v(4, n), n ⩾ 4, be the number of permutations discordant with the three Permutations
Two Isomorphs of the Four-Colour Problem
Authors: Synge J. L.
Published: December 2010

It is known that the four-colour problem for the faces of a map on a sphere is isomorphic with the four-colour problem for the vertices of its dual, and the problem is here discussed in the latter form. The isomorphs described below are concerned with codes for the four colours and for a change in colour as we pass along an edge from vertex to vertex. In the first (algebraic) isomorph, the coding involves the four fourth roots of unity, and leads to a graphical representation in the complex plane. In the second (arithmetical) isomorph, the coding involves the integers mod 4, and also the face-edge incidence matrix of the dual.
Differences, Derivatives, and Decreasing Rearrangements
Authors: Duff G. F. D.
Published: December 2010

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and

(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.
An Elementary Proof of Gram's Theorem for Convex Polytopes
Authors: Shephard G. C.
Published: December 2010

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .
Bisimple Inverse Semigroups as Semigroups of Ordered Triples
Authors: Reilly N. R. , Clifford A. H.
Published: December 2010

In (8) and (13) it has been shown that certain bisimple inverse semigroups, called bisimple ω-semigroups and bisimple Z-semigroups, can be represented as semigroups of ordered triples. In these cases, two of the components of each triple are integers, and the third is drawn from a fixed group. This representation is analogous to that given by the theorem of Rees (1, p. 94) concerning completely simple semigroups, and shares the same advantages.
Enumeration of Graphs with Given Partition
Authors: Parthasarathy K. R.
Published: December 2010

In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.
On Independent Complete Subgraphs in a Graph
Authors: Moon J. W.
Published: December 2010

A graph G = G(n, e) consists of a set of n nodes e pairs of which are joined by a single edge; we assume that no edge joins a node to itself. A graph with modes is called a complete -graph if each pair of its nodes is joined by an edge. The graphs belonging to some collection of graphs are independent if no two of them have a node in common. The maximum number of independent complete -graphs contained in a given graph G will be denoted by Ik(G).
Generating Functions for Ultraspherical Functions
Authors: Viswanathan B.
Published: December 2010

The ultraspherical function

1.1

for |1 — x| < 2 is a solution of the differential equation

1.2

This equation has two independent solutions; of the two, only Pn(λ)(x) is analytic at x = 1, aside for some special values of λ, which we shall not consider.
Products of Zero-One Matrices
Authors: Kelly John B.
Published: December 2010

Let P be a finite set with p objects oj, j = 1, 2, … , p, and let {Si}, i = 1, 2, … , n, be a family of n subsets of P. The incidence matrix A = (aij) for the family {Si} is defined by the rules: aij = 1 if 0j, ∈ Si and aij = 0 if 0j ∉ Si. Then, if AAT = B = (bij) (where AT denotes the transpose of A), it is easy to see that bij = |Si ⋂ Sj|, i = 1, … , n, j = 1, … , n, so that the elements of B are integers with bii ⩾ bij ⩾ 0.
Vector Lattices with Duals of Integral Type
Authors: Ellis H. W.
Published: December 2010

Marston Morse and William Transue (4; 5; 6) have developed a theory of seminormed Banach spaces of complex or real-valued functions called MT spaces. For E a locally compact space, the space of real-valued functions on E, and K the space of real-valued functions on E with compact supports, a real MT space A is a vector subspace of containing K and provided with a suitable seminorm NA.
Composition Operators
Authors: Nordgren Eric A.
Published: December 2010

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by
On Maximal Subsystems of Root Systems
Authors: Wallach Nolan R.
Published: December 2010

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0. Let h be a Cartan subalgebra of g and let Δ be the root system of g with respect to h.
Totally Variant Sets in Finite Groups and Vector Spaces
Authors: Hoffman Frederick , Welch Lloyd R.
Published: December 2010

We are concerned here with the question of which finite groups and vector spaces possess subsets which are moved by every non-identity automorphism (in the vector-space case—non-singular linear transformation). We find that this is the case for all but four finite-dimensional vector spaces (2-, 3-, and 4-dimensional space over Z2, 2-dimensional space over Z3), and for all finite groups except for those corresponding to the vector-space exceptions, and the quaternion group of order eight. The question was first posed to the authors, in the vector-space case, by Morris Marx.
Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set
Authors: Hallam T. G. , Komkov V.
Published: December 2010

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).
A Representation Theorem for Relatively Complemented Distributive Lattices
Authors: Nanzetta Philip
Published: December 2010

In this note, we are concerned with the following generalization of a wellknown theorem of M. H. Stone; see (2, 8.2).

Theorem 1. Let L be a relatively complemented distributive lattice.

(I) If L has no least element, then L is isomorphic to the lattice of non-empty compact-open subsets of an anti-Hausdorff, nearly-Hausdorff, T1-space with a base of open sets consisting of compact-open sets.

(II) (3, Theorem 1) If L has a least element, then L is isomorphic to the lattice of all compact-open subsets of a locally compact totally disconnected space. Moreover, the spaces of (I) and (II) are compact if and only if L has a greatest element.

The space in question is the space of prime ideals of L with the hull-kernel topology.

The author is indebted to M. G. Stanley for several conversations concerning this note.
Maps which Induce the Zero Map on Homotopy
Authors: Hoo C. S.
Published: December 2010

In this paper, all spaces will have the homotopy type of simply connected CW-complexes, and will have base points which are preserved by maps and homotopies. We denote by [X, Y] the set of homotopy classes of maps from X to Y, and by N[X, Y] the subset of those homotopy classes [ƒ] which induce the zero homomorphism on homotopy, that is, is the zero homomorphism for each i.
The Lattice of all Topologies is Complemented
Authors: Van Rooij A. C. M.
Published: December 2010

In (3), J. Hartmanis raised the question whether the lattice of all topologies in a given set is complemented and gave the affirmative answer for the case of a finite set. H. Gaifman (2), has extended this result to denumerable sets. Using Gaifman's paper, Anne K. Steiner (4) has proved that the lattice is always complemented. Our aim in this article is to give an alternative proof, independent of Gaifman's results. So far, Steiner's proof has not been available to the author.
Rings in which all Subrings are Ideals. I
Authors: Kruse Robert L.
Published: December 2010

In analogy with Hamiltonian groups, an associative ring in which every subring is a two-sided ideal is called a Hamiltonian ring, or, more concisely, an H-ring. Several attempts at classification of H-rings have been made. H-rings generated by a single element have been studied by M. Šperling (5), L. Rédei (4), and A. Jones and J. J. Schäffer (2). H-rings enjoying additional properties have been characterized by F. Szász (e.g., 6), and by S.-X. Liu (3). A class of closely related rings has been studied by P. A. Freĭdman (1). In the present paper and its sequel all H-rings are classified and completely described in terms of their generators and relations.
Another Enumeration of Trees
Authors: Knuth Donald E.
Published: December 2010

Given a set of vertices which have each been assigned one of the colours C1, C2, …, Cm, with nj vertices Cj, a formula is derived for the number of oriented trees on these vertices, having a designated root, and subject to any number of restrictions of the form “no arc goes from a vertex of colour Ci to a vertex of colour Cj”. The formula is based on a combinatorial construction which defines a correspondence between such trees and certain sequences.
Associated Regular Spaces
Authors: Pelham Thomas J.
Published: December 2010

Let be any topological space. In this paper, we show that there is a unique regular topology on X which is coarser than such that if Y is any regular space, the continuous maps are the same for and . We shall call the regular topology associated with and the regular space associated with .
Sur Les Espaces Localement Quasi-Compacts
Authors: Dixmier J.
Published: December 2010

Un espace topologique est dit quasi-compact s'il vérifie l'axiome de Borel-Lebesgue sans être nécessairement séparé. Un espace topologique est dit localement quasi-compact si chaque point admet un système fondamental de voisinages quasi-compacts.
Arcs, Semigroups, and Hyperspaces
Authors: McWaters M. M.
Published: December 2010

Several years ago Kelley (2) showed that if X is a metric continuum then S(X), the space of non-null, closed subsets of X, and C(X), the space of non-null, closed, connected subsets of X, with the Vietoris topology, are arcwise connected continua. He further showed that S(X) is acyclic. In this note we extend these results to non-metric continua.
Inductive Extension of a Vector Measure Under a Convergence Condition
Authors: Fox Geoffrey
Published: December 2010

Let μ be a vector measure (countably additive set function with values in a Banach space) on a field. If μ is of bounded variation, it extends to a vector measure on the generated σ-field (2; 5; 8). Arsene and Strătilă (1) have obtained a result, which when specialized somewhat in form and context, reads as follows: “A vector measure on a field, majorized in norm by a positive, finite, subadditive increasing set function defined on the generated σ-field, extends to a vector measure on the generated σ-field”.
Sets of Polynomials Orthogonal Simultaneously on Four Ellipses
Authors: Goodman Ruth
Published: December 2010

It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are five essentially different sets, two of which are orthogonal on concentric circles, and three of which are orthogonal on confocal ellipses. Merriman (1) has shown that the orthogonality of a set of polynomials on both of two concentric circles is sufficient to guarantee their orthogonality on the entire family of circles.
Operators of the form PAQ-QAP
Authors: Brown Arlen , Pearcy Carl
Published: December 2010

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .

It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .
Topological Homotheties on Compact Metrizable Spaces
Authors: Janos Ludvik
Published: December 2010

Notation and definitions.

Definition 1. Let (X, ρ) be a metric space and ϕ: X → X a continuous self-mapping of X. We shall call ϕ and α-contraction on (X, ρ) if and only if α ϵ [0,1) and . We shall call ϕ an α-homothety on (X, ρ) if and only if α > 0 and .
Unitary Representations of Generalized Symmetric Groups
Authors: Puttaswamaiah B. M.
Published: December 2010

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute with

Obviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) and

then {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).