bluuebunny/crossref_metadata_2025 huggingface dataset:
Multiplicities and Minimal Widths for (0, 1)-Matrices
Authors: Fulkerson D. R. , Ryser H. J.
Published: December 2010

In a previous paper (1) the notion of the α-width ∈A(α) of a (0, 1)-matrix A was introduced, and a formula for the minimal α-width taken over the class of all (0, 1)-matrices having the same row and column sums as A, was obtained. The main tool in arriving at this formula was a block decomposition theorem (1, Theorem 2.1; repeated below as Theorem 2.1) that established the existence, in the class generated by A, of certain matrices having a simple block structure. The block decomposition theorem does not itself directly involve the notion of minimal α-width, but rather centres around a related class concept, that of multiplicity. We review both of these notions in § 2, together with some other pertinent definitions and results.
Connectivity and Reducibility of Graphs
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S.
Published: December 2010

Corresponding to every graph, bipartite graph, or directed bipartite graph there exists a directed graph which is connected if and only if the original graph is connected.

In this paper, it is shown that for every directed graph there exists a certain bipartite graph such that the directed graph is connected if and only if the bipartite graph is irreducible. Other connections between reducibility and connectivity are established.
Trace Forms on Lie Algebras
Authors: Block Richard
Published: December 2010

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:

Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient
trace form if and only if there exists a Lie algebra L with a representation Δ such that
The General Static Spherically Symmetric Solution of the "Weak" Unified Field Equations
Authors: Vanstone J. R.
Published: December 2010

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where

(1)

Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:

(2a)

(2b)

In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notation

for the symmetric and skew-symmetric parts of geometric objects Q is employed.
A Theory of Integration
Authors: Hayes Charles A.
Published: December 2010

The usual development of the Lebesgue integral starts with a measure that may have been derived from some simpler set function, its associated class of measurable sets, the corresponding set of measurable functions, and operations which ultimately define the integral of any given function of this class, except for certain ones which are unbounded above and below. Here we propose to define a process of integration with respect to a set function more general than a measure. This process allows us to integrate virtually all functions real-valued on our space. The integrals thus obtained are all completely additive on a certain completely additive class of sets. Under rather mild hypotheses, we are able to delineate this class of sets explicitly.
Some New Difference Sets
Authors: Gordon Basil , Mills W. H. , Welch L. R.
Published: December 2010

A difference set is a set D = {d1, d2, … , dk] of k distinct residues modulo v such that each non-zero residue occurs the same number of times among the k(k — 1) differences di — dj, i ≠ j. If λ is the number of times each difference occurs, then

(1)

When we wish to emphasize the particular values of v, k, and λ involved we will call such a set a (v, k, λ) difference set. Another (v, k, λ) difference set E = {e1, e2, … ek} is said to be equivalent to the original one if there exist a and t such that (t, v) = 1 and E = {a + td1, … , a + tdk}. If t = 1 we will call the set E a slide of the set D. If D = E, then t is called a multiplier of D.
Asymptotic Behaviour of Solutions of Parabolic Differential Inequalities
Authors: Lees Milton
Published: December 2010

Let there be given a parabolic differential operator

where A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality

(1.1)

Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.
Osculating Spaces
Authors: Scherk Peter
Published: December 2010

In this paper an attempt is made to prove some of the basic theorems on the osculating spaces of a curve under minimum assumptions. The natural approach seems to be the projective one. A duality yields the corresponding results for the characteristic spaces of a family of hyperplanes. A duality theorem for such a family and its characteristic curve also is proved. Finally the results are applied to osculating hyperspheres of curves in a conformai space.

The analytical tools are collected in the first three sections. Some of them may be of independent interest.
The Construction of Certain Graphs
Authors: Erdös P. , Rogers C. A.
Published: December 2010

A graph G is called complete if any two of its vertices are connected by an edge; a set of vertices of G are said to be independent if no two of them are connected by an edge. It follows from a well-known theorem of Ramsay (1) that for each pair of positive integers k, l there is an integer f(k, l), which we take to be minimal, such that every graph with f(k, l) vertices either contains a complete graph of k vertices or a set of l independent points.
Cluster Sets of Functions on an N-Ball
Authors: Church P. T.
Published: December 2010

Let Dn be the open unit ball in En, and let Sn be the n-sphere. The cluster set C(f, p) of a function f : Dn —> Sn at a point p on the boundary of Dn is the set of points y in Sn such that there exists a sequence of points xm —> p, xm in Dn, with f(xm) —> y. Given an arc γ in , meeting the boundary bdy(Dn) only in p, the arc cluster set C(f, γ) is the set of points y in Sn such that there exists a sequence xm —> p, xm in γ, with f(xm) —> y.
Characters With Preassigned Values
Authors: Mills W. H.
Published: December 2010

Let k and t be positive integers; let q1, q2, . . . , qt be distinct prime numbers; and let ζ1, ζ2, ... , ζt be kth roots of unity, not necessarily primitive. Recent investigations on consecutive kth power residues have led to the following question: Under what conditions do there exist primes p that have a kth power character X such that
Space Forms of Grassmann Manifolds
Authors: Wolf Joseph A.
Published: December 2010

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.
On Redfield's Group Reduction Functions
Authors: Foulkes H. O.
Published: December 2010

In 1927, J. H. Redfield (12), discussed some of the links between combinatorial analysis and permutation groups, including such topics as group transitivity, the enumeration of certain geometrical configurations, and the construction of various permutation isomorphs of a given group. Except for a revision of Redfield's treatment of transitivity by D. E. Littlewood (6), this 1927 paper appears to have been overlooked. However, it has recently been described (5) as a remarkable pioneering paper which appears to contain or anticipate virtually all of the enumeration results for graphs which have been discovered and developed during the last thirty years.
Cayley Symmetries in Associative Algebras
Authors: Taft Earl J.
Published: December 2010

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.
The Number of Coloured Graphs
Authors: Carlitz L.
Published: December 2010

Let Fn(k) denote the total number of k-coloured graphs on n labelled nodes and let Mn(k) denote the number of graphs on n nodes that are coloured in at most kcolours ; also let fn(k) denote the number of connected k coloured graphs on n nodes. Read (3) has proved the following formulas:
Integral Bases for Quadratic Forms
Authors: Chalk J. H. H.
Published: December 2010

Let

be an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :
On the Sum of the Reciprocals of the Fermat Numbers and Related Irrationalities
Authors: Golomb Solomon W.
Published: December 2010

In (1), P. Erdös showed that the function

takes on irrational values whenever z = 1/t, t = 2 , 3 , 4 , 5 , . . . . The method of proof uses Lambert's identity,

where d(n) is the number of divisors of n; and it is shown that
Simultaneous Automorphisms in Spaces of Analytic Functions
Authors: Alexiewicz A. , Arsove M. G.
Published: December 2010

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .
Enumeration of Non-Separable Planar Maps
Authors: Brown William G.
Published: December 2010

In (2), Tutte has shown that the number, Bn, of rooted non-separable planar maps having n edges is [2(3n — 3)!]/[n! (2n — 1)!]. Rooting was accomplished by designating one edge as the root, orienting it, and distinguishing between its sides as left and right. We shall here compute the number, Bn,m, of rooted non-separable planar maps having n edges and such that the face to the left of the root is incident with exactly m edges, which maps will be said to be of type [n, m].
A Decision Problem for Transformations of Trees
Authors: Evans Trevor
Published: December 2010

For a given class of transformations on graphs there arises naturally the problem of deciding when one graph can be transformed into another by a finite sequence of such transformations. We consider, in this paper, the special case of this problem when the graphs are finite trees and the transformations consist of rearranging the order of the segments from a point and replacing subtrees by other trees according to a given set of pairs of interchangeable trees. This decision problem is, in fact, equivalent to the word problem for one-generator algebras in a certain variety of algebraic systems and we exhibit a procedure for solving the word problem for these algebras.
On The Number of Faces of a Convex Polytope
Authors: Gale David
Published: December 2010

The following problem is as yet unsolved: Given a convex polytope with N vertices in n-space, what is the maximum number of (n — 1)-faces which it can have? Aside from its geometric interest this question arises in connection with solving systems of linear inequalities and linear equations in non-negative variables. The problem is equivalent to asking for the best bound on the number of basic solutions for such problems and hence a bound (though a weak one) for the number of iterations needed in the simplex method for solving linear programmes.
Completions of Quadrangles in Projective Planes
Authors: Killgrove R. B.
Published: December 2010

This paper discusses projective planes from the viewpoint of their classification into singly-generated and non-singly-generated planes. (Singly-generatedness, a property explicit in Hall (6) and Wagner (19), and implicit in Hughes (10), is defined in Section 5.) The elements (points and lines) of a singly-generated plane are expressible in four basic points called a quadrangle. A t'completion procedure" enables us to obtain expressions for the elements of a plane.
Traces of a Class Of (0, 1)-Matrices
Authors: Mesner Dale M.
Published: December 2010

If A = (aij) is a matrix whose elements are 0's and l's, more briefly a (0, l)-matrix, the trace of A, defined as the sum of the diagonal elements aii and denoted by Tr A, is the number of l's on the main diagonal. Matrices which can be obtained from A by permutation of its rows or of its columns, or both, may be expressed as PA, AQ, or PAQ, respectively, where P and Q are permutation matrices.
Conical Differentiation
Authors: Lane N. D. , Singh K. D.
Published: December 2010

This paper follows naturally a note on parabolic differentiation (2) in which the parabolically differentiable points in the real affine plane were discussed. In the parabolic case, the four-parameter family of parabolas in the affine plane led to three differentiability conditions. In the present paper, the five-parameter family of conies in the real projective plane gives rise to four differentiability conditions and a point of an arc in the projective plane is called conically differentiable if these four conditions are satisfied. The differentiable points are classified by the nature of their families of osculating conies, superosculating conies, and their ultraosculating conies.
Modular Representations of Sn
Authors: Robinson G. de B.
Published: December 2010

The purpose of this paper is to clarify and sharpen the argument in the last two chapters of the author's Representation theory of the symmetric group(3). When these chapters were written the peculiar properties of the case p = 2 were not fully appreciated. No difficulty arises in the definition of the block in terms of the p-core, or in the application of the general modular theory based on the formula
Weak Compactness and Separation
Authors: James Robert C.
Published: December 2010

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.
The Multiplier Theorem for Difference Sets
Authors: Turyn Richard J.
Published: December 2010

In a recent paper (5) Newman proved the following theorem: if D is a difference set in a cyclic group G and n = q is prime, then q is a multiplier of D. If n = 2q and (v, 7) = 1, then q is a multiplier of D. The purpose of this note is to point out that a stronger statement than the first part was proved in (1), to remove the restriction (v, 7) = 1 in the second part, and to give again and make some comments about the proof of the theorem which asserts that a prime divisor of n is a multiplier of D if prime to v.
The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes
Authors: Foulser David A.
Published: December 2010

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.

The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).
A Combinatorial Analogue of Poincaré's Duality Theorem
Authors: Klee Victor
Published: December 2010

For a non-negative integer s and a finite simplicial complexK, letβS(K) denote thes-dimensional Betti number ofKand letfs(K) denote the number ofs-simplices ofK. Our theorem, like Poincaré's, applies to combinatorial manifoldsM, but it concerns the numbersfs(M) instead of the numbersβS(M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices ofn-dimensional convex poly topes which have a given numberiof (n— 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system ofilinear inequalities innvariables, and was the original motivation for our study.
Connections Satisfying a Generalized Ricci Lemma
Authors: Vanstone J. R.
Published: December 2010

In this paper we shall consider a generalization of a very old problem in differential geometry; namely, given a second-order covariant tensor field aij(x) on an n-dimensional manifold, when does there exist a connection such that the covariant derivative, defined by

vanishes?

The earliest question of this type arose in the case when is symmetric and positive definite. A solution connection of the problem is then given by the Christoffel symbols
On the Enumeration of Rooted Non-Separable Planar Maps
Authors: Brown W. G. , Tutte W. T.
Published: December 2010

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges is

In the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).

Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.
On the Number of Vertices of a Convex Polytope
Authors: Klee Victor
Published: December 2010

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.
Summability- (Z, p) and Sequences of Periodic Type
Authors: Hill J. D. , Sledd W. T.
Published: December 2010

We shall say that the sequence x = {sk} is summable- (Z, p) to the value s if

where p is a positive integer and

Ignoring the values of n, 1 — p ≤ n < 0, which are clearly irrelevant, the transformation (Z, p) coincides with the regular Nörlund transformation defined by the sequence ( 1 , 1 , . . . , 1, 0, 0 , . . .) containing p initial 1's. This class of methods was first studied systematically by Silverman and Szász (8).
Screenability, Pointwise Paracompactness, and Metrization of Moore Spaces
Authors: Heath R. W.
Published: December 2010

F. B. Jones (6) has shown that, if , then every separable normal Moore space is metrizable. It is not known whether this assumption is necessary, though perhaps some progress is made in (5). However, it is easily seen from R. H. Bing's Example E in (3) that a certain condition (see (1) below) implied by is necessary. Also in (3), Bing showed that every screenable normal Moore space is metrizable. In this paper we establish that: (1) every separable normal Moore space is metrizable if and only if every uncountable subspace M of E1 contains a subset which is not an Fσ (in M); (2) if every pointwise paracompact normal Moore space is metrizable, then so is every separable normal Moore space; (3) every screenable Moore space is pointwise paracompact but not conversely; (4) a T3-space is a pointwise paracompact Moore space if and only if it has a uniform base (in the sense of (1, p. 40), not a uniformity).
A Property of a Triangle Inscribed in a Convex Curve
Authors: Zirakzadeh A.
Published: December 2010

The purpose of this paper is to prove the following theorem:

Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.
The Inverse Multiplier for Abelian Group Difference Sets
Authors: Johnsen E. C.
Published: December 2010

In(1)Bruck introduced the notion of a difference set in a finite group. LetGbe a finite group ofvelements and let D = {di},i= 1, . . . ,kbe ak-subset ofGsuch that in the set of differences {di-1dj} each element ≠ 1 inGappears exactly λ times, where 0 < λ <k<v— 1. When this occurs we say that (G,D) is av,k,λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that av,k, λ group difference set is equivalent to a transitivev,k,λconfiguration, i.e., av,k,λconfiguration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parametersv,kandλ, then, we have the relation shown by Ryser(5)
Sur Une Famille de Groupes de Permutations Doublement Transitifs
Authors: Ree Rimhak
Published: December 2010

Dans (6), l'auteur a défini, pour chaque entier n ≥ 0, un groupe fini Gn d'ordre q3(q —1)(q3 + 1), où q = 32n+1. Les groupes G1, G2, . . . sont simples, tandis que est simple.

En réalisant G = Gn comme un groupe de permutations, opérant à droite, de l'ensemble de q3 + 1 classes à droite modulo le normalisateur N(P) d'un 3-sous-groupe de Sylow de G, on voit aisément que G satisfait aux conditions (0.1)-(0.4) suivantes :

(0.1)

G est un groupe de permutations doublement transitif d'un ensemble E de m + 1 lettres, où m ≥ 3;

(0.2)

si a et b sont deux lettres distinctes quelconques de E, le sous-groupe formé de la totalité des permutations dans G qui laissent invariantes a et b contient une, et une seule, permutation ≠ 1 qui laisse au moins 3 lettres invariantes;

(0.3)

toute involution dans G laisse au moins 3 lettres invariantes (une involution dans un group est par définition un élément d'ordre 2);

(0.4)

m est impair.
Matrix Rational Completions Satisfying Generalized Incidence Equations
Authors: Johnsen E. C.
Published: December 2010

Let us consider the following problem. Let there be v elements x1 , . . . , xv and v sets S1, . . . , Sv such that every set contains exactly k distinct elements and every pair of sets has exactly λ distinct elements in common. To avoid trivial situations we shall in general assume that 0 < λ < k < v — 1. This is known as a v, k, λconfiguration or design. We can give an equivalent characterization of a configuration in terms of a matrix A = [aij], called its incidence matrix, by writing the elements x1 , . . . , xv row and the sets S1, . . . , Sv in a column and setting aij = 1 if xj is in Si and aij = 0 if xj is not.
A Property of Two Chords which Divide a Convex Curve into Four Arcs of Equal Length
Authors: Eggleston H. G. , Zirakzadeh A.
Published: December 2010

It will be shown in this paper that if two chords of a closed plane convex curve θ divide θ into four arcs of equal length and intersect inside the domain bounded by θ, then the sum of the lengths of the two chords is at least equal to (√5 — 2)½ times the length of θ. We shall show firstly that we need only consider the case when θ is a convex (possibly degenerate) quadrilateral and then prove the result in this case.

This result is related to a conjecture of P. Ungar in which the chords are assumed to be perpendicular and the factor (√5 — 2)½ is replaced by §. But Ungar's conjecture is neither proved nor disproved by this result.
On the Total Area of the Faces Of a Four-Dimensional Polytope
Authors: Fejes Tóth L.
Published: December 2010

Let L be the total length of the edges of a convex polyhedron containing a unit sphere. If the number of edges is small, the edges must be, on the average, comparatively long. If, on the other hand, the edges are short, their number must be great. So the problem arises to find a polyhedron with a possibly small value of L.

By a simple argument the author (5; 6) proved that L > 20 and announced the conjecture that L ≥ 24 with equality only for the cube (of in-radius 1). The same argument shows that for trigonal-faced polyhedra L > 28. This supports the conjecture that for such polyhedra L ≥ 12√6 = 29.4 . . . , with equality only for the tetrahedron and octahedron.
An Extension of the Class of Alternative Rings
Authors: Outcalt D. L.
Published: December 2010

Define R to be a ring of characteristic not 2 or 3 satisfying the identity

(1)       (x,y,z) = (y,z,x)

for all x, y, z ∊ R, where by characteristic not p is meant x —> px is a one-toone mapping of R upon R. The associator (a, b, c) of R is defined by (a, b, c) = ab·c — a·bc. lf R also satisfies the identity

(2)      (x, x, x) = 0
Some Properties of Graphs with Multiple Edges
Authors: Fulkerson D. R. , Hoffman A. J. , McAndrew M. H.
Published: December 2010

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).
Ideal Decompositions in Noetherian Rings
Authors: Barnes Wilfred E. , Cunnea William M.
Published: December 2010

An interesting identity is obtained for ideals A and B in a Noetherian ring :

A = (A + Bn) ∩ (A : Bn)

for sufficiently large n. This identity is applied to obtain Fuchs' quasi-primary decomposition of A in an improved form, and to obtain Krull's theorem on the intersection of the powers of A, both developments making no use of the Noetherian primary decomposition of A. Finally, the identity is used to obtain the primary decomposition without reference to irreducible ideals, in a largely constructive manner which yields the decomposition in an illuminating, automatically normal form, and which, subject to certain simple conditions, is unique.
Vector Fields and Infinitesimal Transformations on Almost-Hermitian Manifolds with Boundary
Authors: Hilt Arthur L. , Hsiung Chuan-Chih
Published: December 2010

Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.
Extremal Problems for Schlicht Functions in the Exterior of the Unit Circle
Authors: Netanyahu E.
Published: December 2010

Let Σ represent the class of functions

(1)

which are schlicht and regular, except for the pole at infinity, in |z| > 1. Further let Σ-1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2)
A Characterization of Certain Ptolemaic Graphs
Authors: Kay David C. , Chartrand Gary
Published: December 2010

With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.
On Counting Rooted Triangular Maps
Authors: Mullin R. C.
Published: December 2010

Let R be a simply connected closed region in the Euclidean plane E2 whose boundary is a simple closed curve C. A triangular map, or simply "map," is a representation of R as the union of a finite number of disjoint point sets called cells, where the cells are of three kinds, vertices, edges, and faces (said to be of dimension 0, 1, and 2, respectively), where each vertex is a single point, each edge is an open arc whose ends are distinct vertices, and each face is a simply connected open region whose boundary consists of the closure of the union of three edges. Two cells of different dimension are incident if one is contained in the boundary of the other.
On Commutative Continuation of Partial Endomorphisms of Groups
Authors: Chehata C. G.
Published: December 2010

Given a homomorphic mapping θ of a subgroup A of a group G onto another subgroup B of G, necessary and sufficient conditions for the existence of a supergroup G* of G and an endomorphism θ* of G* such that θ* coincides with θ on A were derived by B. H. Neumann and Hanna Neumann (3). The homomorphism θ is called a partial endomorphism of G and θ* is said to continue, or extend, θ. Necessary and sufficient conditions for the simultaneous continuation of two partial endomorphisms of a group G to total endomorphisms of one supergroup G* ⊇ G were derived by the author (2).
The Genus of the n-Cube
Authors: Beineke Lowell W. , Harary Frank
Published: December 2010

The definition of the genus γ(G) of a graph G is very well known (König
2): it is the minimum genus among all orientable surfaces in which G can be
drawn without intersections of its edges. But there are very few graphs whose
genus is known. The purpose of this note is to answer this question for one
family of graphs by determining the genus of the n-cube.

The graph Qn called the n-cube has 2n vertices each of which is a binary
sequence a1a2. . . an of length n, where ai = 0 or 1.
On Measures of Symmetry of Convex Bodies
Authors: Chakerian G. D. , Stein S. K.
Published: December 2010

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and define

c(K) = V(K′)/V(K)

Let

c(n) = inf{c(K) : K ⊂ En}.
Hereditary Radicals in Associative and Alternative Rings
Authors: Anderson T. , Divinsky N. , Suliński A.
Published: December 2010

In the first part of this paper we shall consider associative rings, pointing out where associativity is required. In the second part we shall consider not necessarily associative rings and in particular alternative rings.

A property S of rings is said to be a radical property, in the sense of Kurosh (4), if it satisfies the following three conditions:

(a) Every homomorphic image of an S-ring (i.e. a ring with property S) is again an S-ring.
On Tchebycheff Quadrature
Authors: Erdös Paul , Sharma A.
Published: December 2010

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula

(1)

is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).
Colouring of Trivalent Polyhedra
Authors: Kotzig Anton
Published: December 2010

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.

In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?
The Construction of Fields with Infinite Cyclic Automorphism Group
Authors: Kuyk Willem
Published: December 2010

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?

The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.
Commutators of Operators on Hilbert Space
Authors: Brown Arlen , Halmos P. R. , Pearcy Carl
Published: December 2010

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.
Regular Rank Rings
Authors: Halperin Israel
Published: December 2010

1.1. Throughout this note, will denote an associative ring but we shall not require to possess a unit.

If A and B are subsets of , then A + B will denote the set {x + y| x ∊ A, y ∊ B}. Aτ will denote the set {u ∊ | au = 0 for all a ∊ A} .

Elements a and b will be said to be orthogonal if ab = ba = 0.
A Note on Well-Distributed Sequences
Authors: Murdoch B. H.
Published: December 2010

A sequence is said to be well distributed (mod 1) (3, 4, 5) if the limit

(1.1)

exists, uniformly in p ≥ 0, for all intervals I in [0, 1], with length |I|, characteristic function where (x) is the fractional part of x. If (1.1) is true for p = 0 and all I in [0, 1] we say that is uniformly distributed (mod 1).
On The Kakeya Constant
Authors: Cunningham F. , Schoenberg I. J.
Published: December 2010

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.
Affine Subplanes of Finite Projective Planes
Authors: Rigby J. F.
Published: December 2010

Letπbe a finite projective plane of orderncontaining a finite projective subplane π* of orderu<n. Bruck has shown (1, p. 398) that ifπcontains a point that does not lie on any line of π*, thenn≥u2+u, while if every point ofπlies on a line of π* thenn=u2.

Letπbe a finite projective plane of orderncontaining a finiteaffinesubplane π0of orderm<n.
Two Theorems on Mosaics
Authors: Gordon B. , Robertson M. M.
Published: December 2010

The concept of a mosaic was recently introduced by A. A. Mullin (1). By the fundamental theorem of arithmetic, every integer n > 1 can be uniquely expressed in the form

where the pi are primes satisfying p1 < p2 < . . . < pr. We then express any exponents aj which are greater than unity in the same manner, and continue in this way until the process terminates. The resulting planar configuration of primes is called the mosaic of n.
A Normal form for Riemann Matrices
Authors: Albert A. A.
Published: December 2010

A matrix ω having p rows and 2p columns of complex number elements is called a Riemann matrix of genus p if there exists a rational 2p-rowed skew matrix C such that

(1)

is positive definite Hermitian. The matrix C is then called a principal matrix of ω. If ω and ω0 are two Riemann matrices of the same genus, we say that ω and ω0 are isomorphic if there exists a non-singular p-rowed complex matrix α, and a non-singular 2p-rowed rational matrix A, such that

(2)
On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, II: Rectangular Matrices
Authors: Majindar Kulendra N.
Published: December 2010

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.

Our theorem can be stated as follows:

THEOREM 2. Let A be a v X b matrix with integer elements such that

2.1
On Ingham's Summation Method
Authors: Segal S. L.
Published: December 2010

Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s if

where as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)
Convex Polyhedra with Regular Faces
Authors: Johnson Norman W.
Published: December 2010

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.

That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.
Isotopy of 2-Dimensional Cones
Authors: Copeland Arthur H.
Published: December 2010

Knowing the isotopy of cones is a crucial first step in knowing the isotopy of finitely triangulable spaces, for the cones are exactly the stars of vertices. Furthermore, they are the simplest examples of contractible spaces, and the non-triviality of the contractible spaces is one of the distinguishing characteristics of isotopy theory as contrasted with homotopy theory.

The present paper is concerned with the cones over 1-dimensional finitely triangulable spaces. It is clear that homeomorphic spaces have homeomorphic cones, hence cones of the same isotopy type. The surprising result of §2 is that there are very few exceptions to the converse statement. The exceptional isotopy classes of cones all contain cones over spaces that are themselves cones.
Simple Algebras that Generalize the Jordan Algebra M38
Authors: Sagle Arthur A.
Published: December 2010

In this paper we discuss a generalization of the split exceptional Jordan algebra M38() of the 3 X 3 hermitian matrices with elements in the split Cayley-Dickson algebra (1). The generalization consists of replacing by the non-commutative Jordan algebra ≡ (A, ƒ, s, t) discussed in (2; 3) and forming the set of 3 X 3 hermitian matrices M3m() ≡ M with elements in the m-dimensional algebra . With the usual definition of multiplication X · Y = ½(XY + YX), M becomes a commutative algebra and we have the following theorem, which shows how the structure of M is reflected by that of .
On the Continuity and Self-Injectivity of a Complete Regular Ring
Authors: Utumi Yuzo
Published: December 2010

Let S be a ring, and let (ei) be an orthogonal system of a finite number of idempotents. Then e = Σei has the following properties:

(i) Se Σ Sei and eS = Σ ei S.

(ii) The mappings v: Se → Π Sei and w: eS → Π ei S defined by v(x) = [xei] and w(x) = [ei x] respectively are isomorphisms.

Next assume that (ei)i∈I is a set of idempotents indexed by a totally ordered set I such that ei ej = 0 for every i < j. If I is finite, it is evident that

has the above two properties.
Simplicial and Homotopical Cohomology of Polyhedra
Authors: Stamm Emil
Published: December 2010

It is well known that, on the category of finite polyhedra, any two cohomology theories, satisfying the Eilenberg-Steenrod axioms, are isomorphic. Examples of such theories are simplicial cohomology and homotopical cohomology (the latter is defined by means of homotopy classes of maps into Eilenberg-MacLane spaces). In the case of polyhedra, using triple sequences and spectral sequences, one obtains a deep insight into the relationship between general cohomology theories (without the dimension axiom) and ordinary simplicial cohomology (1, p. 66). As a corollary the abovementioned uniqueness of cohomology theories satisfying the dimension axiom is obtained.
Weakly Semi-Simple Finite-Dimensional Algebras
Authors: Edwin Clark W.
Published: December 2010

Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group S is a translate of A if there exist an algebra B over F and an epimorphism ϕ: B → F such that A = 0→-1 and S = 1→-1. It is shown in (2) that any such semi-group S has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I of S such that some power In of I lies in the kernel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate S such that R (S) = S.
Pairs of Bilinear Equations in a Finite Field
Authors: Duane Porter A.
Published: December 2010

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations

1.1

where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.
Inequalities for the Permanental Minors of Non-Negative Matrices
Authors: Brualdi R. A. , Newman M.
Published: December 2010

Let A be an n X n non-negative matrix, that is, a matrix whose entries are non-negative numbers. The permanent of A is the scalarvalued function of A defined by

where the summation extends over all permutations i1 … , in of the integers 1, … , n. The purpose of this paper is to prove several inequalities involving the permanent of A and the permanent of submatrices of A when suitable restrictions are placed on the row sums.
Countable Compagtifications
Authors: Magill Kenneth D.
Published: December 2010

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.
A Note on Convergence Fields
Authors: Berg I. D.
Published: December 2010

The purpose of this note is to show that the (bounded) convergence field of a conservative matrix is closed under a certain diagonalization procedure.

As an application of the above result we establish a conjecture of Hill and Sledd in (1) and obtain a result of Lorentz originally proved in (2).

First we introduce some notation and definitions, most of which are standard. Let l∞ denote the Banach space of bounded sequences with the supremum norm and let c denote the closed subspace of l∞ consisting of convergent sequences.
Some Results on Small-Deviation Probability Convergence Rates for Sums of Independent Random Variables
Authors: Heyde C. C.
Published: December 2010

Let﹛Xj,j =1, 2, 3, …﹜ be a sequence of independent, non-degenerate random variables and write

Under quite a diverse variety of conditions we may obtain

asn→ ∞ for allx,— ∞ <x< ∞, and some realp⩾ 0. For example, suppose the ﹛Xj﹜ happen to be distributed identically and belong to the domain of normal attraction of a symmetric stable law with characteristic exponentα,0 <α⩽ < 2,α≠ 1.
The Kernel of the General-Sum Four-Person Game
Authors: Peleg B.
Published: December 2010

In this paper we apply various results and methods of previous papers on the kernel to four-person games.

Section 2 contains the basic definitions needed. In §3 we prove that the kernel of the general-sum four-person game consists of a line segment (which may shrink to a point). A method for classifying games according to their kernels is suggested in §4 and is used there to characterize all four-person games whose kernel consists of a non-degenerate interval. In the last section, §5, we offer a bargaining procedure, based on principles established in (1), which leads to the kernel in the case of a non-degenerate interval.
On the Periodicity of Compositions of Entire Functions
Authors: Gross Fred
Published: December 2010

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).

The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.