bluuebunny/crossref_metadata_2025 huggingface dataset:
A Class Of Algebras Without Unity Element
Authors: Thrall R. M.
Published: December 2010

1. Introduction. In a study of the commuting algebra of tensor space representations of the orthogonal group W. P. Brown encountered a class of algebras for which the existence of a unity element was equivalent to semisimplicity, but which were of interest whether or not semisimple. He gave these
algebras the name generalized-total matrix algebras and proved (2) that each such algebra was characterized by three integers l, r, m and was isomorphic to the algebra of all square matrices of degree r + I + m which have zeros in the first l rows and in the last r columns.
Ovals, Dualities, and Desargues's Theorem
Authors: Ostrom T. G.
Published: December 2010

Consider a projective plane with n + 1 points on each line, An oval is a set of n + 1 points, no three of which are collinear.

DEFINITION. (1) A line which contains two points of will be called a
secant of .

(2) A line which contains exactly one point of will be called a tangent of
or an absolute line.
Une Caracterisation Vectorielle-Metrique Des Espaces L1
Authors: Grothendieck A.
Published: December 2010

Soit C un espace de Banach (nous ne parlerons que d'espaces sur le corps des nombres réels, pour simplifier). On dit que C possède la propriété de Nachbin ou est un espace (N), si pour toute application linéaire continue u dans C d'un sous-espace vectoriel F d'un espace de Banach E, il existe une application linéaire de même norme de E dans C qui prolonge u.
Maximal Determinants In Combinatorial Investigations
Authors: Ryser H. J.
Published: December 2010

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t, and let the absolute value of the determinant of Q be denoted by |det Q|. In this paper we study the problem of
determining the maximum of |det Q| for fixed t and v. It turns out that this problem is closely related to the v, k, λ problem, which has been extensively
studied of late.
The Covering Of Space By Spheres
Authors: Barnes E. S.
Published: December 2010

Bambah (1) has recently determined the most economical covering of three dimensional space by equal spheres whose centres form a lattice, the density of this covering being

1.1.

As is well known, this problem may be interpreted in terms of the inhomogeneous minimum of a positive definite quadratic form.
Diophantine Approximation and Horocyclic Groups
Authors: Rankin R. A.
Published: December 2010

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.
Products of a C-Measure and a Locally Integrable
Mapping
Authors: Morse Marston , Transue William
Published: December 2010

Let C be the field of complex numbers and E a locally
compact topological space. The authors' theory of C-bimeasures Λ and their
Λ-integrals in (1; 2) leads to integral representation of bounded operators
from A to B' where A and
B are MT-spaces as defined in (3).
These MT-spaces include the -spaces and Orlicz spaces as special cases.
A Generalized Tauberian Theorem
Authors: Keogh F. R. , Petersen G. M.
Published: December 2010

Let {s(n)} be a real sequence and let
x be any number in the interval 0 <
x ⩽ 1. Representing x by a non-terminating binary
decimal expansion we shall denote by
{s(n,x)} the subsequence
of {s(n)} obtained by omitting
s(k) if and only if there
is a 0 in the decimal place in the expansion of x. With
this correspondence it is then possible to speak of “a set of subsequences of the
first category,” “an everywhere dense set of subsequences,” and so on.
Further Identities and Congruences for the Coefficients of Modular
Forms
Authors: Newman Morris
Published: December 2010

If n is a non-negative integer, define p
r(n) by

otherwise define p
r(n) as 0. (Here and in what
follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless
otherwise stated.) p
r(n) is thus generated by the
powers of

,
Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions
Authors: Marcus Marvin , Purves Roger
Published: December 2010

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

No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.
On the Symmetry of Cubic Graphs
Authors: Tutte W. T.
Published: December 2010

Let G be a connected finite graph in which each edge has two distinct ends and no two distinct edges have the same pair of ends. We suppose further that G is cubic, that is, each vertex is incident with just three edges.

An s-path in G, where s is any positive integer, is a sequence S = (v0, v1… , vs) of s + 1 vertices of G, not necessarily all distinct, which satisfies the following two conditions:

(i) Any three consecutive terms of S are distinct.

(ii) Any two consecutive terms of S are the two ends of some edge of G.
A Metrization for Power-Sets with Applications to Combinatorial Analysis
Authors: Silverman Robert
Published: December 2010

Combinatorial configurations may generally be phrased in terms of arrangements of objects into sets subject to certain conditions. In view of this, the question arises as to whether given a set S and its power-set Us (the class of all subsets of S), it might be possible to structure Us in a combinatorially significant manner. This paper proposes and investigates one such structuring achieved by defining a distance function over US.

Given A, B in Us, define their distance by

where N(E) denotes the number of elements in E, + ∞ being an admissible value.
A Class of Function Algebras
Authors: Anderson F. W.
Published: December 2010

A problem which has generated considerable interest during the past couple of decades is that of characterizing abstractly systems of realvalued continuous functions with various algebraic or topological-algebraic structures. With few exceptions known characterizations are of systems of bounded continuous functions on compact or locally compact spaces. Only recently have characterizations been given of the systems C(X) of all realvalued continuous functions on an arbitrary completely regular space X (1). One of the main objects of this paper is to provide, by using certain special techniques, a characterization of C(X) for a particular class of (not necessarily compact) completely regular spaces.
The Number of k-Coloured Graphs on Labelled Nodes
Authors: Read R. C.
Published: December 2010

By a labelled graph we shall mean a set of “nodes,“ distinguishable from one another and denoted by A1, A2, …, and a collection of “edges” viz., pairs of nodes. We say that an edge “joins” the pair of nodes which specifies it. We further stipulate that at most one edge joins any two nodes, and that no edge joins a node to itself.

By a “colouring” of a graph in k colours we shall mean a mapping of the nodes of the graph onto a set of k colours C1, C2, …, Ck such that no two nodes which are joined by an edge are mapped onto the same colour. A graph so coloured in exactly k colours will be called a k-coloured graph. Since it is usually possible to colour a graph in more than one way, there will, in general, be many k-coloured graphs corresponding to a given graph.
Nodal Non-Commutative Jordan Algebras
Authors: Kokoris Louis. A.
Published: December 2010

A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰. An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰+ is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰+ is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.
Automorphisms of Finite Linear Groups
Authors: Steinberg Robert
Published: December 2010

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.
Invariants of Finite Reflection Groups
Authors: Steinberg Robert
Published: December 2010

Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a reflection group to be a group generated by reflections. Chevalley (1) (and also Coxeter (2) together with Shephard and Todd (4)) has shown that a reflection group G, acting on a space of n dimensions, possesses a set of n algebraically independent (polynomial) invariants which form a polynomial basis for the set of all invariants of G.
A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems
Authors: Abian Smbat , Brown Arthur B.
Published: December 2010

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.

Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.

Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.

Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).

The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).

The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.
Univalent Solutions of W″ + pW = 0
Authors: Brown R. K.
Published: December 2010

Consider the differential equation

1.1

where

1.2

is regular in|z| < R.

The indicial equation associated with (1.1) is of the form

We shall denote the two roots of this equation byαandβ, where. Corresponding to the rootαthere exists a unique solution of (1.1) of the following form

1.3

In those cases for whichthere exists a unique second solution of (1.1) of the form

1.4
Weakly Compact Sets in Orlicz Spaces
Authors: Andô Tsuyoshi
Published: December 2010

The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.

The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.
Uniform Boundedness for Groups
Authors: Glicksberg Irving
Published: December 2010

Let G and H be locally compact abelian groups with character groups G*, H*, and let < . , . > denote the pairing between a group and its dual.

In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments.

Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that < rg, h* > = < g, τ*h* > for all g in G, h* in H*). Then τ is continuous.

The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.
Sets of Equally Inclined Spheres
Authors: Mauldon J. G.
Published: December 2010

In n-dimensional Euclidean space En, where we shall throughout assume that n ≥ 2, the maximum number of (n — 1)-dimensional spheres which can be mutually orthogonal is n + 2, and it is well known that the sum of the squares of the reciprocals of their radii is zero, so that the spheres cannot be all real. The maximum number of such spheres which can be mutually tangent is also n + 2, and in 1936 Soddy (7) indicated the beautiful relation connecting their radii. These two formulae are the particular cases γ = 0, γ - 1 of Theorem 1 below, which gives the relation connecting the radii of a set of n + 2 such spheres when every pair is inclined at a given non-zero angle 0, where γ is written for cos θ.
An Extremum Result
Authors: Kiefer J.
Published: December 2010

The main object of this paper is to prove the following:

Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that

(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;

(b)

The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).
A Note on Normal Matrices
Authors: Thompson R. C.
Published: December 2010

Let Un be an n-dimensional unitary space with inner product (u, v). For vectors u1, . . . , ur ∊ Un, r ≤ n, let denote the Grassmann exterior product (4) of the ut; it is a vector in Um where m = nCr. If also , then is the determinant of the r X r matrix If A is a linear transformation of Un to itself, the rth compound of A is defined by
On Utumi's Ring of Quotients
Authors: Lambek Joachim
Published: December 2010

The purpose of this note is to establish and exploit the fact that Utumi's maximal ring of right quotients (6) of an associative ring R (let us say with 1) is the bicommutator of the minimal injective extension of R regarded as a right R-module. Nothing new will be said about Johnson's ring of quotients (4), which is still the most important case.
Some Applications of Spaces of Entire Functions
Authors: De Branges Louis
Published: December 2010

A Hilbert space, whose elements are entire functions, is of particular interest if it has these properties:

(H1) Whenever F(z) is in the space and has a non-real zero w, the function is in the space and has the same norm as F(z).

(H2) For each non-real number w, the linear functional defined on the space by F(z) —> F(w) is continuous.

(H3) Whenever F(z) is in the space, is in the space and has the same norm as F(z). If E(z) is an entire function satisfying
Non-Existence of Some Unsymmetrical Partially Balanced Incomplete Block Designs
Authors: Shrikhande S. S. , Raghavarao D. , Tharthare S. K.
Published: December 2010

A partially balanced incomplete block (PBIB) design with m-associate classes is defined by Bose and Shimamoto (4) as follows:

(i) The experimental material is divided into b blocks of k units each, different treatments being applied to the units in the same block.

(ii) There are v treatments each of which occurs in r blocks.

(iii) There can be established a relation of association between any two treatments satisfying the following requirements.
A Characterization of Comparability Graphs and of Interval Graphs
Authors: Gilmore P. C. , Hoffman A. J.
Published: December 2010

Let < be a non-reflexive partial ordering defined on a set P. Let G(P, <) be the undirected graph whose vertices are the elements of P, and whose edges (a, b) connect vertices for which either a < b or b < a. A graph G with vertices P for which there exists a partial ordering < such that G = G(P, <) is called a comparability graph.

In §2 we state and prove a characterization of those graphs, finite or infinite, which are comparability graphs. Another proof of the same characterization has been given in (2), and a related question examined in (6). Our proof of the sufficiency of the characterization yields a very simple algorithm for directing all the edges of a comparability graph in such a way that the resulting graph partially orders its vertices.
Anneaux Métaprimitifs
Authors: Thierrin G.
Published: December 2010

Un anneau R est dit primitif (3) s'il existe un R-module irréductible et fidèle M. L'objet de cet article est d'étudier une généralisation naturelle de la notion d'anneau primitif en remplaçant "irréductible" par "sous-directement irréductible" dans la définition ci-dessus. D'une façon plus précise, un anneau R sera dit métaprimitif s'il existe un R-module sous-directement irréductible et fidèle M, avec SR ≠ {0}, S étant le sous-module minimal de M. Nous établissons pour les anneaux métaprimitifs un théorème de structure (théorème 2.3) faisant apparaître certaines analogies avec le théorème de structure correspondant pour les anneaux primitifs.
On Matrix Commutators of Higher Order
Authors: Robinson D. W.
Published: December 2010

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the following

Proposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.
The Spin Representation of the Symmetric Group
Authors: Morris A. O.
Published: December 2010

Let Γnbe therepresentation group or spin group(9; 4) of the symmetric groupSn. Then the irreducible representations of Γncan be allocated into two classes which we shall call (i) ordinary representations, which are the irreducible representations of the symmetric group, and (ii)spinorprojectiverepresentations.

As is well known (3; 5), there is an ordinary irreducible representation [λ] corresponding to every partition (λ) = (λ1, λ2, . . . , λm) ofnwith

λ1≥ λ2≥ . . . ≥ λm> 0.
Minimal Interchanges of (0, 1)-Matrices and Disjoint Circuits in a Graph
Authors: Walkup David W.
Published: December 2010

In this paper we obtain a partial answer in graph-theoretic form to a question raised by Ryser (2, p. 68) concerning the minimal number of interchanges required to transform equivalent (0, l)-matrices into each other.

For given positive integersmandnwe consider the collection ofm×n(0, 1)-matricesA= {aij}, i.e.aij= 0 or 1 for 1 ≤i≤m, 1 ≤j≤n. We say the (0, 1)-matricesA= {aij} andB= {bij} areequivalentand writeA~Bif and only if they have the same row and column sums, that is, if and only if.
The Thickness of the Complete Graph
Authors: Beineke Lowell W. , Harary Frank
Published: December 2010

A graph G consists of a finite set of p points and q lines joining pairs of these points. Each line joins two distinct points and no pair of points is joined by more than one line. A subgraph of G is a graph whose points and lines are also in G. If every pair of points of a graph is joined by a line, the graph is called complete and is denoted by Kp. A planar graph can be embedded in the plane, that is, drawn in the plane in such a way that none of its lines intersect.
A Periodic Jacobi-Perron Algorithm
Authors: Bernstein Leon
Published: December 2010

In the first part of this paper I shall demonstrate that one irrational root of the algebraic equation

creates an algebraic number field, out of whichn— 1 irrationals can be chosen so that they yield a periodic Jacobi-Perron algorithm. The coefficients in (1) are subject to certain restrictions which will be elaborated below.
The Lototsky Transform and Bernstein Polynomials
Authors: King J. P.
Published: December 2010

The Bernstein polynomials

1

associated with a function f denned on [0, 1] have been the subject of much recent research and have been generalized in several directions (1 ; 2 ; 5). The generalized Lototsky or [F, dn] matrix (3) has also been the subject of extensive research.
Alternative Metrization Proofs
Authors: Rolfsen Dale
Published: December 2010

Alternative methods of proving several classical metrization theorems are offered in this paper, showing that they follow by elementary methods from an early theorem of Alexandroff and Urysohn. A simplified proof of the latter theorem is also given. Theorem 5 and a corollary to Theorem 3 state the main results.
Note on Primary Ideal Decompositions
Authors: McCarthy P. J.
Published: December 2010

Let R be a ring with a unity element. An ideal Q of R is called (right) primary if for ideals A and B of R, AB ⊂ Q and A ⊄ Q imply that Bn ⊂ Q for some positive integer n. If R satisfies the ascending chain condition for ideals (ACC), then R is said to have a Noetherian ideal theory if every ideal of R is an intersection of a finite number of primary ideals. If R is a commutative ring that satisfies the ACC, then R has a Noetherian ideal theory. However, it is known that in general R may satisfy the ACC without having a Noetherian ideal theory (an example of such a ring is given in (2)). Thus there is some interest in conditions that imply that a ring R satisfying the ACC will have a Noetherian ideal theory.
The Maximal Co-Rational Extension by a Module
Authors: Courter R. C.
Published: December 2010

Modules are S-modules where S is an arbitrary ring with or without a unit element. We consider a projective module P having a submodule K such that K + Y = P implies that the submodule Y is P (P, then, is a projective cover of P/K (Definition 4 in this section)) and we define the submodule X of P by

Our main result states that up to isomorphism P/X is the maximal co-rational extension over P/K (by P/K, in the more precise wording of the title).
A Characterization of 2-Betweenness in 2-Metric Spaces
Authors: Freese Raymond W. , Andalafte Edward Z.
Published: December 2010

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.
Magic Graphs
Authors: Stewart B. M.
Published: December 2010

In this paper we use number-theoretic properties to classify ordinary graphs that are finite and have no isolated vertices. The classification depends on whether there is an assignment of real values, usually rational integer values, to the edges of the graph, such that the set of assigned values and the set of vertex sums of these values, summed at each vertex over all the edges incident to the vertex, will be a pair of sets with prescribed properties. Then we seek corresponding graph-theoretic properties.

It is possible to describe the problem in terms of a symmetric matrix having specified properties for its row sums, but in this paper we make no use of this interpretation; however, see (3).
The Isomorphism of Certain Continuous Rings
Authors: Dawkins Brian P. , Halperin Israel
Published: December 2010

In this paper we shall prove the following two theorems (the terminology is explained in § 2 below; all rings are assumed to be associative).

THEOREM 1. Suppose that is a division ring of finite order m over its centre Z and let μ(m) denote the factor sequence 1, m, m2, … , mn, … . Then the rings μ(w) and Zμ(m) are isomorphic.
Subordinate and Pseudo-Subordinate Semi-Algebras
Authors: Barbeau Edward J.
Published: December 2010

Throughout this paper E denotes a compact Hausdorff space, which, to avoid trivial complications, is assumed to contain at least two points. C(E), with the uniform norm, is the Banach algebra of all continuous real-valued functions defined on E; C+(E) is the set of those functions in C(E) which take only non-negative values. A subset of C(E) is a wedge if and only if it is closed under addition and multiplication by nonnegative scalars; a semi-algebra is a wedge closed under (pointwise) multiplication. The set C+(E) is a semi-algebra, and all semi-algebras considered in this paper are contained in C+(E). For a subset K of C+(E), the closed wedge (semi-algebra) generated by K is the least closed wedge (semi-algebra) containing K.
Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions
Authors: Botta E. P.
Published: December 2010

Let Mm(F) be the vector space of m-square matrices

where F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).
Equilibrium Points for Open Acyclic Relations
Authors: Peleg Bezalel
Published: December 2010

A formulation of a fixed point theorem, which can be applied conveniently to non-cooperative games and cooperative games, is suggested in this note.

Let N1, … , Nm be m non-empty, finite disjoint sets. For k = 1, … , m we denote by Sk the simplex the coordinates of whose points are indexed by the members of Nk; thus Sk is the collection of all real functions xk defined on Nk which satisfy:

1.1

1.2
Supermagic Complete Graphs
Authors: Stewart B. M.
Published: December 2010

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums

1

have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.
Finiteness of Semigroups of Operators in Universal Algebra
Authors: Nelson Evelyn
Published: December 2010

This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.

This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.
Expansion of Continuous Differentiable Functions in Fourier Legendre Series
Authors: Saxena R. B.
Published: December 2010

Let

1.1

denote the nth partial sum of the Fourier Legendre series of a function ƒ(x). The references available to us, except (5), prove only that Sn(ƒ, x) converges uniformly to ƒ(x) in [— 1, 1] if ƒ(x) has a continuous second derivative on [—1, 1]. Very recently Suetin (5) has shown by employing a theorem of A. F. Timan (7) (which is a stronger form of Jackson's theorem) that Sn(ƒ, x) converges uniformly to ƒ(x) ƒ(x) belongs to a Lipschitz class of order greater than 1/2 in [—1, 1]. More generally he has proved the following theorem.
Matrix Links, An Extremization Problem, and the Reduction of a Non-Negative Matrix to One With Prescribed Row and Column Sums
Authors: Menon M. V.
Published: December 2010

The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum” and “jth column-sum” of the matrix under consideration. r1 … , rm; c1, … , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = ri and σj = cj for all i and j.
Finite Groups with a given Number of Conjugate Classes
Authors: Poland John
Published: December 2010

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation

*

has only finitely many solutions over the positive integers.
Von Neumann's Manuscript on Inductive Limits of Regular Rings
Authors: Halperin Israel
Published: December 2010

It is now known (3) that if is a regular rank ring, then the rank function can be extended to the matrix ring in such a way that R(a) = R(a ꕕ n) ; here, a is an arbitrary element of is the n × n diagonal matrix with a for each entry on the diagonal, and R denotes rank in and also in . It is also known (2) that every regular rank ring has a rankmetric completion which is again a regular rank ring.
Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II
Authors: Botta Peter
Published: December 2010

Let Mm(F) be the vector space of m-square matrices X — (Xij), i,j= 1, … , m over a field ƒ;ƒ a function on Mm(F) to some set R. It is of interest to determine the structure of the linear maps T: Mm(F) → Mm(F) that preserve the values of the function ƒ (i.e., ƒ(T(x)) — ƒ(x) for all X). For example, if we take ƒ(x) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues.
A Note Concerning Simultaneous Integral Equations
Authors: Knopp Paul , Sinkhorn Richard
Published: December 2010

In (2) Sinkhorn showed that corresponding to each positive n × n matrix A (i.e., every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where each Dk is a diagonal matrix with a positive main diagonal. The Dk themselves are unique up to a scalar multiple. In (3) the result was extended to show that D1AD2 could be made to have arbitrarypositive row and column sums (with the reservation, of course, that the sum of the row sums equal the sum of the column sums) where A need no longer be square.
An Integral Representation for the Product of Spectral Measures
Authors: Derzko N. A.
Published: December 2010

Let be a Hilbert space with inner product (•, •) and let E(•) and E0(•) be spectral measures in corresponding to self-adjoint operators and . In this paper we consider the set function ƒ(I × J) = E(I)E0(J) defined on the semiring of bounded rectangles, and obtain an integral representation for this set function for disjoint I, J under the hypotheses that H — H0 is a type of Carleman operator.
On a Condition of J. Ohm for Integral Domains
Authors: Gilmer Robert
Published: December 2010

This paper originated mainly from results presented in a paper by J. Ohm (13), and, to a lesser degree, from results of Gilmer in (3). Ohm's paper is concerned with the validity of the equation (x, y)n = (xn, yn) for each pair of elements x, y of an integral domain D with identity. If D is a Prüfer domain, the above equation is valid for all x, y ϵ D (7, p. 244). Butts and Smith have shown (2) that if (x, y)2 = (x2, y2) for all x, y of the integrally closed domain D, then D is a Prüfer domain.
On the Distribution of nθ Modulo 1
Authors: Graham R. L. , Van Lint J. H.
Published: December 2010

In recent work of E. Arthurs and L. A. Shepp on a problem of H. Dym concerning the existence of an ergodic stationary stochastic process with zero entropy (cf. 1), the function dθ(n) was introduced as follows:

For an irrational number θ, let

be the sequence of points {lθ}, 1 ≦ l ≦ n, (where {x} denotes x — [x], the fractional part of x) and define*

dθ(n) = max(ai — ai-1), 1 ≦ in + 1.
The Bracket Function and Complementary Sets of Integers
Authors: Fraenkel Aviezri S.
Published: December 2010

The following result is well known (as usual, [x]denotes the integral part of x):

(A) Let α and β be positive irrational numbers satisfying

1

Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.

The theorem has a converse, and the following holds:

(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.
Some Selection Theorems for Measurable Functions
Authors: Himmelberg C. J. , Van Vleck F. S.
Published: December 2010

Let F: X → Y be a multifunction from X to Y. Then, given measure-theoretic or topological structures on X and Y, it is possible in various ways to define the measurability of F. The selection problem is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector. This problem has been studied recently in papers by Castaing (2) and Kuratowski and Ryll-Nardzewski (6). In the latter paper, the problem is studied for its own interest. The former uses solutions of the problem to obtain general Filippov-type theorems. (See, for example, the corollaries following Theorems 2 and 3 of Castaing's paper.) For other material on Filippov's results see, among others, (3; 4; 5; 7; 9).
A Fixed-Point Theorem for Commuting Monotone Functions
Authors: Gray William J.
Published: December 2010

Hamilton (1) proved that a hereditarily unicoherent, hereditarily decomposable metric continuum has the fixed-point property for homeomorphisms. In this paper we shall generalize this result by showing that if X is a hereditarily unicoherent, hereditarily decomposable Hausdorff continuum and 5 is an abelian semigroup of continuous monotone functions from X into X, then S leaves a point of X fixed.

Let X be a Hausdorff continuum. X is unicoherent if, whenever X = A ∪ B, where A and B are subcontinua of X, A ∩ B is a continuum. If each subcontinuum of X is unicoherent, X is hereditarily unicoherent. X is decomposable if X is the union of two of its proper subcontinua. If each subcontinuum of X which contains more than one point is decomposable, X is hereditarily decomposable.
Semi-Valuations and Groups of Divisibility
Authors: Ohm Jack
Published: December 2010

Associated with any integral domain R there is a partially ordered group A, called the group of divisibility of R. When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A, and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A, first solve the problem there, and then pull back the solution if possible to R. Lorenzen (15) originally applied this technique to solve a problem of Krull, and Nakayama (16) used it to produce a counterexample to another question of Krull. More recently, Heinzer (7;8) has used the method to construct other interesting examples of rings.
Stone Lattices. II. Structure Theorems
Authors: Chen C. C. , Grätzer G.
Published: December 2010

Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice.

The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple.

It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).
Pascal Ovals in Projective Planes
Authors: Rigby J. F.
Published: December 2010

A projective plane consists of a set of points and a set of lines, together with a relation of incidence between points and lines, such that

(i) any two distinct points P, Q are incident with a unique line PQ,

(ii) any two distinct lines p, q are incident with a unique point p ∩ q,

(iii) there exist four points, no three of which are incident with the same line.

We shall use the usual geometrical terminology.

An oval in a projective plane π is a set of points of π such that:

(i) no three points of are collinear; this means that a line of π is either a secant of , containing two points of , or a tangent of , containing just one point of , or a non-secant of , containing no point of ;
Coefficients of Functions with Bounded Boundary Rotation
Authors: Robertson M. S.
Published: December 2010

For fixed k ≧ 2, let Vk denote the class of normalized analytic functions

such that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and

1

Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjecture

holds for functions convex in one direction.
Counting Paths, Circuits, Chains, and Cycles in Graphs: a Unified Approach
Authors: Biondi E. , Divieti L. , Guardabassi G.
Published: December 2010

The problem of counting partial subgraphs (or patterns, for short) in a given graph has been approached by several mathematicians from various points of view (see, e.g., [1; 3; 5; 13-15; 17-23; 26-29]; applications may also be found in [2; 8; 9; 16]). Specific algorithms have been presented and almost all of them are essentially based upon a careful analysis of the graph under consideration. In these cases, we say that a direct approach has been followed. Unfortunately, when large graphs are considered, all direct counting methods require rather cumbersome computations. For this reason, during the last few years many efforts have been made in finding suitable indirect counting methods. First, Biondi [5] faced the problem of counting cycles in non-oriented graphs by inspection of the complementary graph. More recently, a number of papers [1; 3; 21; 22; 28] have been concerned with counting trees in classes of non-oriented graphs having complementary graphs with special structural properties. However, to the best of our knowledge, no general indirect counting method is available in the literature.
Initial Segments of Many-One Degrees
Authors: Lachlan A. H.
Published: December 2010

Our aim in this paper is to give a characterization of the order types of the countable initial segments of many-one degrees (m-degrees). The basic definitions and background information can be found in [2] from where we draw most of our notation and terminology. We expand the usual notion of m-reducibility by adopting the convention that R ≦m ∅ and R ≦mN for every recursive set R. This has the effect of giving all recursive sets the same m-degree; that m-degree will be denoted by 0. We shall denote by ≦ the partial ordering of m-degrees induced by ≦m, and shall denote by a ∪ b the least upper bound of the m-degrees a, b. We call a ∪ b the union of a and b.
Cell Complexes, Valuations, and the Euler Relation
Authors: Perles M. A. , Sallee G. T.
Published: December 2010

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.

To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.

A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.
Singer Groups
Authors: Hestenes Marshall D.
Published: December 2010

Interest in the Singer groups has arisen in various places. The name itself results from the connection Singer [7] made between these groups and perfect difference sets, and this is closely associated with the geometric property that a Singer group is regular on the points of a projective space. Some information about these groups appears in Huppert's book [3, p. 187]. Singer groups are frequently useful in constructing examples and counterexamples. Our aim in this paper is to make a systematic study of the Singer subgroups of the linear groups, with a particular view to analyzing the examples they provide of Frobenius regular groups. Frobenius regular groups are a class of permutation groups generalizing the Zassenhaus groups, and Keller [5] has shown recently that they provide a new characterization of A6 and M11.
Products of Reflections in an Affine Moufang Plane
Authors: Götzky K. Martin
Published: December 2010

Let be a Moufang plane. By specializing one line ω, the line at infinity, weobtain an affine Moufang plane . The group generated by the shears of is called the equiaffine group. Veblen [9, § 52] asked whether every equiaffinity is a product of two affine reflections. He gave a proof which will work in an affine Pappian plane, using the following two properties.

Property 1. If an equiaffinity fixes two distinct proper points of , it fixes every point collinear with them.

Property 2. Let e be an equiaffinity and P a point such that ppe2pe3 is a triangle. Then the lines PePe2 and PPe3 are parallel.

Without using these properties, it will be proved that the answer to Veblen's question is “yes“ if and only if the Moufang plane is Pappian.
Systems of equations and generalized characters in groups
Authors: Isaacs I. M.
Published: December 2010

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.

Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.
Homology of Deleted Products of Contractible 2-Dimensional
Polyhedra. III
Authors: Patty C. W.
Published: December 2010

Our aim in this paper is to continue our investigation of the homology of
deleted products of finite, contractible, 2-dimensional polyhedra. In
[1], we observed that if X is such a
polyhedron, then a homeomorph of X can be constructed by
starting with a 2-simplex and appending n-simplexes
(n = 1, 2). In this paper, we are concerned with those
polyhedra which have the property that if they are constructed as above,
then at some stage we are forced to add to Xi–1 a 2-simplex τ at two of its 1-faces,
〈u3, u1〉 and 〈u3, u2〉, where there is a simple closed curve S in
∂(St(u3, Xi–1)) such that u1 and u2 are not in S but every sequence of 1-simplexes
in ∂(St(u3, Xi–1)) from u1 to u2 intersects S. The cone over the complete graph
on five vertices and the cone over the houses-and-wells figure are examples
of such spaces.
The Structure of the Algebra of Hankel Transforms and the Algebra of Hankel-Stieltjes Transforms
Authors: Schwartz Alan
Published: December 2010

Let M be the space of all bounded regular complex-valued Borel measures defined on I = [0, ∞). M is a Banach space with ‖μ‖ = ∫d|μ|(x) (μ ∈ M). (Integrals in this paper extend over all of I unless otherwise specified.) Let v be a fixed real number no smaller than and let if z ≠ 0 and , where Jv, is the Bessel function of the first kind of order v and cv =[2vΓ(v + 1)]–1; is an entire function, as can be seen from the power series definition of

The Hankel-Stieltjes transform of order v is given by .
The Maximal p-Extension of a Local Field
Authors: Marshall Murray A.
Published: December 2010

1. Let k denote a local field, that is, a complete discrete-valued field with perfect residue class field . Let G denote the Galois group of the maximal separable algebraic extension M of k, and let g denote the corresponding object over . For a given prime integer p, let G(p) denote the Galois group of the maximal p-extension of k. The dimensions of the cohomology groups

considered as vector spaces over the prime field Z/pZ, are equal, respectively, to the rank and the relation rank of the pro-p-group G(p); see [4; 9]. These dimensions are well known in many cases, especially when k is finite [6; 3; (Hoechsmann) 2, pp. 297-304], but also when k has characteristic p, or when k contains a primitive pth root of unity [4, p. 205].