bluuebunny/crossref_metadata_2025 huggingface dataset:
On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, III: Rectangular Matrices
Authors: Majindar Kulendra N.
Published: December 2010

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).
Determination of a Subset from Certain Combinatorial Properties
Authors: Cantor David G. , Mills W. H.
Published: December 2010

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.

This problem can be expressed as a coin-weighing problem (1; 7).

In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).
Remarks on Complementation in the Lattice of all Topologies
Authors: Gaifman Haim
Published: December 2010

Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement.
Principal Loop-Isotopes of Quasigroups
Authors: Bryant B. F. , Schneider Hans
Published: December 2010

If a quasigroup (L, .) has finite order n, then there are n2 principal loop-isotopes. Some of these n2 loops may be isomorphic, and the main purpose of this paper is to obtain theorems that describe the isomorphism classes. Using these results and a computer, we have determined all the loops of order 6. These are listed (using the Fisher and Yates (2) designations) at the end of the paper.
The Number of Circular Patterns Compatible with a Pseudo-Symmetric Connected Graph
Authors: Baum L. E. , Eagon J. A.
Published: December 2010

In this paper we prove:

Theorem. Let be an ordered pseudo-symmetric connected graph with l lines and v vertices. Let there be aij lines directed from vertex i to vertex j, i, j = 1, 2, … , v. Let gcd(aij) = d, and define
Invariant Subspace Theorems for Finite Riemann Surfaces
Authors: Hasumi Morisuke
Published: December 2010

The purpose of this paper is to extend various invariant subspace theorems for the circle group to multiply connected domains. Such attempts are not new. Actually, Sarason (4) studied the invariant subspaces of annulus operators acting on L2 and showed certain parallelisms between the unit disk case and the annulus case. Voichick (8) observed analytic functions on a finite Riemann surface and generalized the Beurling theorem on the closed invariant subspaces of H2 as well as the Beurling–Rudin theorem on the closed ideals of the disk algebra. Here we shall consider LP(Γ) and C(Γ) defined on the boundary Γ of a finite orientable Riemann surface R.
The Laplace Transform of Generalized Functions
Authors: Benedetto John
Published: December 2010

In this paper we develop a theory of Laplace transforms for generalized functions. Some fundamental results in this field were given by Schwartz in (3) for the n-dimensional bilateral case from the point of view of topological vector spaces, and in (4) in a form amenable to operational use. Our presentation characterizes a one-dimensional theory of Laplace transforms with a half-plane of convergence (indicating an extension of the usual classical transform) and with the property that Laplace transforms are analytic functions satisfying the fundamental convolution-multiplication theorem. Section 1 is devoted to defining the Laplace transform of generalized functions and also to showing how the property of a half-plane of convergence is intrinsic to this definition.
Bisecants of Finite Collections of Sets in Linear Spaces
Authors: Edelstein M. , Kelly L. M.
Published: December 2010

The question posed by Sylvester (6) concerning the collinearity of a finite set of points in E2 having the property that each two together with some third be collinear has been the inspiration for numerous investigations. The original question was answered by the following theorem.
Szegö Polynomials on a Compact Group with Ordered Dual
Authors: Hirschman I. I.
Published: December 2010

The Szegö polynomials are defined on T, the real numbers modulo 1. In this paper and in its sequel we give a generalization of Szegö polynomials in which T is replaced by an arbitrary locally compact abelian group θ on whose dual there has been distinguished a measurable order relation compatible with the group structure. The present paper is devoted to the case where θ is compact and therefore discrete. The general case will be taken up in the sequel mentioned above. It is desirable to proceed in this way because the case θ compact is much simpler and much more like the classical situation than is the general case, in which various measure-theoretic difficulties obtrude. Moreover, as it happens, it is possible to develop the theory in this way with relatively little repetition.
Modular Annihilator Algebras
Authors: Barnes Bruce A.
Published: December 2010

In a recent paper (7) Yood developed the beginnings of a theory of modular annihilator algebras. In this paper we extend his work on these algebras.

The definition of modular annihilator algebra is algebraic in nature (see §4) ; in fact the algebra need not be assumed even topological. However, a significant number of important normed algebras are modular annihilator algebras. A list of examples is given in §8.

The theory of modular annihilator algebras is related to the theory of certain important topological algebras. In §5 we consider the relationships between dual and annihilator algebras and modular annihilator algebras, and in §7, the relationship between completely continuous normed algebras and modular annihilator algebras.
Engel Congruences in Groups of Prime-Power Exponent
Authors: Glauberman George , Krause Eugene F. , Struik Ruth Rebekka
Published: December 2010

In this paper a simplified proof of a theorem of Sanov (4) is given. No mention is required of Lie elements or of the Baker-Hausdorff formula, both of which played central roles in Sanov's proof.

Let G be a group. Define, for all x, y ∈ G,
On the Representation of Integers as Sums of Distinct Terms from a Fixed Sequence
Authors: Folkman Jon
Published: December 2010

Let A = (a1, a2, a3, …) be a sequence of positive integers. We let

denote the set of integers that are sums of distinct terms of A. If P(A) contains all sufficiently large integers, we say that A is complete. We shall show that certain classes of sequences that are characterized by their rate of growth are complete.
Trees in Polyhedral Graphs
Authors: Barnette David
Published: December 2010

A graph is said to be d-polyhedral provided it is isomorphic with the graph formed by the vertices and edges of a d-dimensional bounded (convex) polyhedron (d-polyhedron). A k-tree is a connected acyclic graph in which each vertex is of valence ⩽k.
On the Permanent of a Doubly Stochastic Matrix
Authors: Wilf Herbert S.
Published: December 2010

If is an n X n matrix, the permanent of A, Per A, is defined by

1

where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.
P and D in P-1XP = dg(λ, … , λn) = D As Matrix Functions of X
Authors: Rinehart R. F.
Published: December 2010

Let be the algebra of n X n matrices over the complex field C, X = ‖xrs‖ a matrix of , and f(X) = ‖(frs(x11, x12, … , xnn)‖ a function with domain and range in . If the frs are differentiable with respect to each of the xij on some open set R of , then the differential df(X) = ‖dfrs(xij)‖ exists, and, moreover, f(X) is Hausdorff-differentiable (HD) (1, 3, 7) i.e. df(X)is expressible in the form

where dX = ‖dxrs‖, and the matrices Ai Bi are independent of dX. The Hausdorff derivative fI(X) is defined to be

i.e. the value of df(X) for dX = I, the identity matrix (2).
Singular Perturbations of Non-Linear Elliptic and Parabolic Variational Boundary-Value Problems
Authors: Ton Bui An
Published: December 2010

Singular perturbations of linear elliptic and parabolic boundary-value problems have been studied extensively by Visik and Lyusternik (7), Huet (5), and others. It is the purpose of this paper to extend the results of (5) to the non-linear elliptic and parabolic variational boundary-value problems considered during the last few years by Browder (2, 4).

In §1, we give the notations and state the main assumptions on the nonlinearity of the elliptic operators. In §2 we study the singular perturbations of non-linear elliptic variational boundary problems. In §3, we consider the case of non-linear parabolic variational boundary problems with a small parameter.
A Mountain-Climbing Problem
Authors: Whittaker James V.
Published: December 2010

Suppose that two men stand at the same elevation on opposite sides of a mountain range and begin to climb in such a way that their elevations remain equal at all times. Will they ever meet along the way? It is this question, restated in mathematical terms, that we shall consider. We replace the mountain range by the graph of a continuous, real-valued function f(x) defined for x ∈ [0, 1], where f(0) = f(1) = 0, and we ask whether there exist continuous mappings ϕ(t), ψ(t) from [0, 1] into [0, 1] such that

1

2
Topologies of Lattice Products
Authors: Alo Richard A. , Frink Orrin
Published: December 2010

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely theidealtopology, theintervaltopology, and thenew intervaltopology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.

Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.
A Condition of Halo Type for the Differentiation of Classes of Integrals
Authors: Hayes C. A.
Published: December 2010

We shall consider a space S, a σ-algebra M of subsets of S, a measure μ defined on M, and the μ-integrals of certain μ-integrable functions f. To each point x of a certain set E of S we associate certain ones of the sets V ∈ M and form the quotients ∫ vf(x)dμ(x)/μ(V) for each such set V. In case these quotients tend to f(x) as the sets V converge to x in accordance with a definition we adopt in §2, then we say that the integral of f is differentiable or derivable at x. It is of interest to assert conditions that ensure the differentiability of a given integral or class of integrals at μ-almost all points of E.
Minimal Regular Graphs of Girths Eight and Twelve
Authors: Benson Clark T.
Published: December 2010

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal to

Here the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.
The L(r, t) Summability Transform
Authors: Powell Robert E.
Published: December 2010

In a recent article Cheney and Sharma (1) studied the linear operator Pn defined by

where

here Lj(n)(t) denotes the Laguerre polynomial of degree j. Cheney and Sharma proved that if f is continuous on [0, 1], then Pn(f, x) converges uniformly to f(x) on [0, a] where 0 < a < 1.
The Steiner Point of a Convex Polytope
Authors: Shephard G. C.
Published: December 2010

Associated with each bounded convex set K in n-dimensional euclidean space En is a point s(K) known as its Steiner point. First considered by Steiner in 1840 (6, p. 99) in connection with an extremal problem for convex regions, this point has been found useful in some recent investigations (for example, 4) because of the linearity property

1

Addition on the left is vector addition of convex sets.
Interpolation by Polynomials in z and z–1 in the Roots of Unity
Authors: Sharma A.
Published: December 2010

Given a function ƒ(z), continuous on C: |z| = 1 in the complex plane, there is a close analogy between approximation in the sense of least squares by polynomials on the unit circle and interpolation by polynomials in the nth roots of unity to the same function. For detailed discussion of the problem and its generalization for a suitable Jordan curve one can refer to Walsh (3) or to a recent paper by Curtiss (2). More recently, Curtiss (1) has considered the problem of interpolation by polynomials in non-equally spaced points on the unit circle and has pointed out the limitations inherent in the problem.
Un Théorème Sur L'Arithmétique des Groupes Ordonnés
Authors: Aubert K. E.
Published: December 2010

La caractérisation fondamentale des sous-groupes du groupe additif R des nombres réels en tant que groupes ordonnés est la suivante:

A. Un groupe abélien totalement ordonné G est isomorphe à un sous-groupe de R si et seulement si G est archimédien.

Nous nous proposons ici de donner une nouvelle caractérisation des sousgroupes de R qui nous semble présenter un certain intérêt pour plusieurs raisons. D'abord elle est très différente de la caractérisation A ci-dessus, bien que notre démonstration utilise A. Puis elle ajoute un nouveau théorème de r-groupe à ceux déjà connus. Finalement, en employant les caractérisations des anneaux de valuation généraux et discrets données dans (1), le théorème en question possède quelques corollaires parmi lesquels se trouve le théorème de s-groupe pour un domaine d'intégrité (“s-Gruppensatz” dans la terminologie de Krull (5)).
Oscillation Criteria for Matrix Differential Equations
Authors: Howard H. C.
Published: December 2010

We shall be concerned at first with some properties of the solutions of the matrix differential equation

1.1

where

is an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).
Splitting and Decomposition by Regressive Sets, II
Authors: McLaughlin T. G.
Published: December 2010

In (3), Dekker drew attention to an analogy between (a) the relationship of the recursive sets to the recursively enumerable sets, and (b) the relationship of the retraceable sets to the regressive sets. As was to be expected, this analogy limps in some respects. For example, if a number set α is split by a recursive set, then it is decomposed by a pair of recursively enumerable sets; whereas, as we showed in (6, Theorem 2), α may be split by a retraceable set and yet not decomposable (in a liberal sense of the latter term) by a pair of regressive sets. The result for recursive and recursively enumerable sets, of course, follows from the trivial fact that the complement of a recursive set is recursive.
Permutability of Semilattice Congruences on Lattices
Authors: Fujiwara Tsuyoshi
Published: December 2010

Many authors have studied lattice congruences on lattices, but it seems that there are few studies concerning semilattice congruences on lattices. However, it seems that the semilattice congruences on lattices are closely connected with their structure. In this paper, we shall study the characterizations of modular, distributive, and relatively complemented lattices by the permutability of semilattice congruences.

We can obtain the dual statements of the following discussion, but we shall not write them as a rule to avoid double descriptions.
A General Perron Integral, II
Authors: Bullen P. S.
Published: December 2010

This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.

In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.
Nearly Regular p-Groups
Authors: Hobby C. R.
Published: December 2010

A finite p-group G is said to be regular if for every pair of elements a, b in G,

1

where c is an element of the derived group of the subgroup generated by a and b. We shall say that G is nearly regular if (i) there is a central element z of order p such that G/⧼z⧽ is regular, and (ii) the subgroup generated by x and y is regular whenever x ∈ G and y ∈ G′, the derived group of G.
A New Proof of Some Results of Rényi and the Asymptotic Distribution of the Range of his Kolmogorov-Smirnov Type Random Variables
Authors: Csörgö Miklós
Published: December 2010

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.
Sieve-Generated Sequences with Translated Intervals
Authors: Buschman R. G. , Wunderlich M. C.
Published: December 2010

Consider the following sieve process. Let A(1) be the sequence of integers greater than 1. Let A(n+1) be obtained from A(n) = {a1(n), a2(n), …} by eliminating one element from each of the intervals Ik(n), where

We let an = an(n) and A = {an} be the sequence of integers that survive the sieve. M. C. Wunderlich (8) has found a necessary and sufficient condition for an ∼ n log n and, in a more recent paper, M. Wunderlich and W. E. Briggs (9) have studied a subclass of the sequences defined above for which an ∼ n log n.
Green's Functions for Singular Ordinary Differential Operators
Authors: Brauer Fred
Published: December 2010

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.
n-ANR's for Certain Normal Spaces
Authors: Mancuso Vincent J.
Published: December 2010

For various classesQof metric spaces, there are several well-known results characterizing the localn-connectivity of a metric space in terms ofn-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classesQ.In the last part of the paper we specifyQto be the class of totally normal spaces and characterize the localn-connectivity of ann-dimensional separable metric space in terms of ANR(Q)'s.
The Distribution of Sequences Modulo 1
Authors: Zame Alan
Published: December 2010

In a recent paper (2), Helson and Kahane consider the problem of the existence of real numbers x such that the sequence (λn x) (when reduced modulo 1) is not summable by a given regular Toeplitz method, where (λn) is a lacunary sequence of positive real numbers. Thus, as an example, they show the existence of uncountably many x such that the sequence (θn x) does not have a distribution function modulo 1, where θ is some fixed number > 1.
A Packing Problem for Measurable Sets
Authors: Sankoff D. , Dawson D. A.
Published: December 2010

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?

In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?
Unique Extension and Product Measures
Authors: Luther Norman Y.
Published: December 2010

Following (2) we say that a measure μ on a ring is semifinite if

Clearly every σ-finite measure is semifinite, but the converse fails.

In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.
On Pólya's Theorem
Authors: Sheehan J.
Published: December 2010

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.
Symmetric Coordinate Spaces and Symmetric Bases
Authors: Ruckle William
Published: December 2010

In this paper properties of symmetric coordinate spaces and symmetric bases are investigated. Since a space which possesses a basis is essentially a space of sequences (12, p. 207), the interrelation of these two concepts naturally suggests itself.

Section 2 is a summary of the terminology and methods employed, which fall into four categories: (1) set theoretical properties of coordinate spaces such as symmetry and dual spaces; (2) the notion of FK and BK space (12, p. 202; 13); (3) the theory of the Schauder basis in F-space applied to the case when (see § 2) is a basis for a coordinate space; (4) the concept of a sequential norm, which the author introduced in (7) to illustrate the underlying unity of the first three ideas.
A Class of Operators on the Lorentz Space M(ϕ)
Authors: Boyd David W.
Published: December 2010

In order to deal with certain problems in the theory of interpolation spaces, it is convenient to consider operators of the following form:

Let k be a non-negative measurable function on the half-line R+, and let ƒ be a measurable function on R+ with

1

Then the operator T is defined by

2

with the domain of T, D(T), consisting of all ƒ which satisfy (1).
On the Number of Associative Triples in an Algebra of n Elements
Authors: Brockwell P. J.
Published: December 2010

Consider a set of n elements α1, … , αn (denoted by S) and the set of all possible multiplication tables on these elements. The total number of such tables is clearly and each table can be represented by a square matrix [μij] where μij is the product αiαj (μij ∈ S, i = 1, … , n; j = 1, … , n). The triple (αi, αj, αk) is said to be associative if the following equation is satisfied:

1.1
Collineation Groups Containing Perspectivities
Authors: Dembowski Peter
Published: December 2010

Let P be a projective plane of finite order n and Γ a group of collineations of P. Gleason (6) and Wagner (10) have shown that if every point of P is the centre, and every line the axis, of a non-trivial perspectivity in Γ, then Γ contains a subgroup of order n2 which consists entirely of elations. It then follows that either P or its dual is a translation plane with respect to at least one line; in fact if Γ has no fixed elements, then P is desarguesian and Γ contains all elations of P. It was shown by Piper (7) and Cofman (4) that the hypotheses of Gleason and Wagner can be relaxed in certain cases, while the same conclusions hold.
A New Look at the Kummer Surface
Authors: Edge W. L.
Published: December 2010

Kummer's surface has the base surface F of a certain net of quadrics in [5] for a non-singular model. All the quadrics of have a common self-polar simplex ∑, and can, in a double-infinity of ways, be based on a quadric Ω1 and two quadrics that Ω1 reciprocates into each other. F is invariant under harmonic inversions in the vertices and opposite bounding primes of ∑ and (§2) contains 32 lines. In §3 it is shown, conversely, that those quadrics for which a given simplex is self-polar and which contain a line of general position constitute a net of this kind.
The Inversive Distance Between Two Circles
Authors: Bottema O.
Published: December 2010

H. S. M. Coxeter (3) has recently studied the correspondence between two geometries the isomorphism of which was well known, but to which he was able to add some remarkable consequences. The two geometries are the inversive geometry of a plane E (the Euclidean plane completed with a single point at infinity or, what is the same thing, the plane of complex numbers to which ∞ is added) on the one hand, and the hyperbolic geometry of three-dimensional space S.
The Classification of Non-Euclidean Plane Crystallographic Groups
Authors: Macbeath A. M.
Published: December 2010

This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).
On The Pseudo-Euclidean Geometry Due to G. Hessenberg
Authors: Szász Paul
Published: December 2010

G. Hessenberg (2) showed that euclidean plane geometry can be realized on the surface of the sphere without assuming the parallel axiom. This geometry will be called the pseudo-euclidean geometry due to G. Hessenberg. In the present paper we give a slightly different treatment, which is perhaps simpler than that of Hessenberg and which has a greater transparency.
On The Commutativity of J-Rings
Authors: Luh Jiang
Published: December 2010

A ring R is called a J-ring if there exists an integer n > 1 such that xn = x for every x ∈ R. The following beautiful theorem due to Jacobson (4, 5) is a generalization of Wedderburn's theorem, which asserts that every finite division ring must be a field.
Pseudovaluations of Polynomials
Authors: Nelson R.
Published: December 2010

A valuation of a ring K is a function

where A is an archimedean ordered field and W has the properties of the absolute valuation; see (2, chap. X). The theory was extended in 1936 by K. Mahler (1), who introduced the concept of pseudovaluations. Whereas for a valuation we must always have

for a pseudovaluation it is sufficient that
On the Properties of an Entire Function of Two Complex Variables
Authors: Agarwal Arun Kumar
Published: December 2010

1. Let

be an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . Let

Following S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).
Generalized Radical Rings
Authors: Edwin Clark W.
Published: December 2010

Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.
On the Gotlieb-Csima Time-Tabling Algorithm
Authors: Dempster M. A. H.
Published: December 2010

This paper concerns an algorithm, proposed by C. C. Gotlieb (4) and modified by J. Csima (1;2), for a recent combinatorial problem whose application includes the construction of school time-tables. Theoretically, the problem is related to systems of distinct subset representatives, the construction of Latin arrays, the colouring of graphs, and flows in networks (1;2;3). I t was conjectured by Gotlieb and Csima that if solutions to a given time-table problem existed, i.e. if time-tables incorporating certain pre-assigned meetings existed, their algorithm would find one.
On Events Represented by Probabilistic Automata of Different Types
Authors: Salomaa Arto
Published: December 2010

The usual definition of a finite probabilistic automaton (4; 5) does not involve any outputs. In this paper, such automata are referred to as Rabin automata. Various other types of probabilistic automata are obtained by introducing output functions. Similarly, as in connection with finite deterministic automata, the distinction between Moore automata and Mealy automata can then be made. Rabin automata may be regarded as automata where the output function is deterministic. One can also consider the case where the transition function is deterministic.
Peripheral Covering Properties Imply Covering Properties
Authors: Grace E. E.
Published: December 2010

Recently several papers (11; 12; 13; 14) have been published in which it is shown that a Moore space (normal, in one case) is metrizable if it has the peripheral version (in the sense defined below) of a certain covering property that was known to imply metrizability of Moore spaces. Each of these metrization theorems can be proved more easily by using a slight variation of the appropriate standard proof to show that such a space is collectionwise normal and hence (2, Theorem 10) metrizable. But this approach, as well as that followed in (11 ; 12; 13 ; 14), obscures the point that, in Moore spaces and in more general settings, the peripheral versions of these covering properties imply the covering properties.
Holomorphic Convexity for General Function Algebras
Authors: Rickart C. E.
Published: December 2010

In previous papers (7; 8), we have investigated certain properties of general function algebras which may be regarded as generalizations or analogues of familiar results in the theory of analytic functions of several complex variables. This investigation is continued and expanded in the present paper. The main results concern a notion of holomorphic convexity for the general situation. We also extend somewhat several of the results obtained in the earlier papers.
Orthogonal Decompositions of Multivariate Weakly Stationary Stochastic Processes
Authors: Robertson James B.
Published: December 2010

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition

1.1

and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.
On Some Non-Linear Problems
Authors: Srinivasacharyulu K.
Published: December 2010

Non-linear problems have been studied by Krasnoselski, Browder, and others; in fact Browder and independently Kirk (cf., 1; 5) have proved the following remarkable theorem: let X be a uniformly convex Banach space, U a non-expansive mapping of a bounded closed convex subset C of X into C, i.e., ||Ux — Uy|| ⩽ ||x — y|| for x, y ∊ C; then U has a fixed point in C. The aim of this paper is to give some existence theorems for non-linear functional equations in uniformly convex Banach spaces. Similar results may be found in (3 ; 6).
A Characterization of the Hyperhomology Groups of the Tensor Product
Authors: Hungerford Thomas W.
Published: December 2010

If K and L are chain complexes of abelian groups (to which we restrict ourselves throughout this paper), then denotes the graded hyperhomology group of K and L, as defined in Car tan and Eilenberg (1) by means of free double complex resolutions of K and L. Hyperhomology groups have proved convenient in proving various versions of the Künneth theorem (see, for example, (4; 1; 2)).
Estimates for General Coercive Boundary Problems on a Half-Space for a Class of Elliptic Partial Differential Operators
Authors: Greiner Peter C.
Published: December 2010

In recent years elliptic boundary value problems have been studied in great detail; see, for example, Agmon (1), Agmon, Douglis, and Nirenberg (2), Browder (4), Hormander (7), Schechter (10; 11; 12), Agranovich and Dynin (3). In all these cases the boundary problems considered were local or semilocal, i.e. the boundary operators involved are differential operators possibly having singular integral operators for coefficients (cf. (3)).
An Inclusion Theorem for Generalized Cesàro and Riesz Means
Authors: Meir A.
Published: December 2010

For a positive integer, p, a strictly increasing unbounded sequence of positive numbers {λn: n ⩾ 1} and an arbitrary sequence of complex numbers {an} let

1

2
An Axiomatic Line Geometry
Authors: Trott Stanton
Published: December 2010

In their classic treatment (5) Veblen and Young build n-dimensional projective geometry from points and lines. Naturally, each line becomes identified with the set of points with which it is incident, and many treatments build from points alone, postulating the existence of certain distinguished subsets of the set of points. From either point of view, some labour is required, even in the two-dimensional case, to establish duality; hence a considerable interest attaches to self-dual systems of axioms; cf. (2; 3).
A Decomposition of Measures
Authors: Luther Norman Y.
Published: December 2010

Let X be a set, a σ-ring of subsets of X, and let μ be a measure on . Following (1), we define μ to be semifinite if

We show (Theorem 1) that every measure can be reduced to a semifinite measure for many practical purposes. In many cases, this reduction can be made even more significantly (Theorems 2 and 3). Finally, necessary and sufficient conditions that a semifinite measure be c-finite are given as a corollary to Theorem 3.
Some Moment Problems in a Finite Interval
Authors: Leviatan D.
Published: December 2010

Let the sequence {λi} (i ≧ 0) satisfy the following conditions.

1.

2.

3.

We shall deal with the following moment problems: what are the conditions, necessary and sufficient, on a sequence {μn} (n ≧ O) in order that it should possess the representation

1.1
A Characteristic Subgroup of a p-Stable Group
Authors: Glauberman George
Published: December 2010

Let p be a prime, and let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:

(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.
The Valence of Sums and Products
Authors: Goodman A. W.
Published: December 2010

A function ƒ(z) is said to be p-valent in a region if it is regular in if the equation

1

has p distinct roots in for some particular w0, and if for each complex w0, equation (1) does not have more than p roots in . The function ƒ(z) is also said to have valence p in . In the case when p = 1, the function is said to be univalent in .
Positive Functionals and Representations of Tensor Products of Symmetric Banach Algebras
Authors: Smith Harvey A.
Published: December 2010

Except for using “algebra” rather than “ring” and “compact” rather than “bicompact”, we adopt the terminology used in (6). Every symmetric algebra, Ai, will have an identity, ei All representations will be cyclic and symmetric. Sets of functionals will carry the relative weak* topology.
Almost Convergence and Well-Distributed Sequences
Authors: Zame Alan
Published: December 2010

A sequence (xn) of real numbers is said to be well-distributed modulo 1 (abbreviated w.d.) if for each subinterval I = [a, b] of [0, 1] we have that

where χI is the characteristic function of I modulo 1. A sequence (rn) of positive numbers is lacunary if
On Isomorphisms of Meromorphic Function Fields
Authors: Kelleher James J.
Published: December 2010

In this work we deal with algebraic properties of some fields of functions meromorphic in the complex plane with a view to determining the possible isomorphisms between two such fields. Interest in problems of this type began with a paper by Bers (2), in which it was shown that the algebraic structure of the ring of functions analytic on a plane region determines the conformai structure of the region to within conformai or anti-conformal equivalence, and this result was later extended to arbitrary non-compact Riemann surfaces by Nakai (7).
Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism
Authors: Gross Fletcher
Published: December 2010

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.
The Measure Algebra as an Operator Algebra
Authors: Ramirez Donald E.
Published: December 2010

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).

Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.
A Class of Singular Functions
Authors: Shapiro Harold S.
Published: December 2010

Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).
A Presentation of the Groups PSL(2, p)
Authors: Behr H. , Mennicke J.
Published: December 2010

In the present paper, we shall prove the following result.

Theorem A. The groups PSL(2, p) can be presented by the following system of generators and relations:

1.1

This theorem considerably improves earlier results of Bussey, Frasch, and Todd (cf. 2, pp. 93-96). The presentation of Frasch reads as follows:

1.2
On the Extensions of Lie Algebras
Authors: Block Richard E.
Published: December 2010

In this paper we give some results on the extensions of Lie algebras, with emphasis on the case of prime characteristic, although part of the paper is also of interest at characteristic 0. An extension of a Lie algebra L is a pair (E, π), where £ is a Lie algebra and π is a homomorphism of E onto L. The kernel K of the extension is ker π.
Reidemeister Projective Planes
Authors: Kallaher Michael J.
Published: December 2010

By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).
A Note on Involutions with a Finite Number of Fixed Points
Authors: Miller John D.
Published: December 2010

LetMbe a smooth, closed, simply connected manifold of dimension greater than 5. LetTbe an involution onMwith a positive, finite number of fixed points. Our aim in this paper is to prove the following theorem (which is somewhat like that of Wasserman (7)).
On the Homotopy-Commutativity of Loop-Spaces and Suspensions
Authors: Hoo C. S.
Published: December 2010

Let X be a space. We are interested in the homotopy-commutativity of the loop-space ΩX and the suspension ΣX, that is, in the question whether or not nil X ≦ 1, conil X ≦ 1, respectively. Let c: ΩX× ΩX ⟶ ΩX, c': ΣX ⟶ ΣX V ΣX be the commutator and co-commutator maps, respectively.
Projective Modules Over Central Separable Algebras
Authors: DeMeyer F. R.
Published: December 2010

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.
Two-Dimensional Linear Groups Over Local Rings
Authors: Lacroix Norbert H. J.
Published: December 2010

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.

Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .
Normal Completions of Small Categories
Authors: Kennison J. F.
Published: December 2010

In (3), Isbell proposed a stronger definition for the term “complete category” and obtained many nice theorems for the resulting notion of a completion. In particular, he showed (3, Theorem 3.20) that completions of small categories satisfy a strong normality condition.

In this paper we shall always use the term “complete” in the weaker sense of Freyd (1). (In (3), Isbell used the term “small-complete” for this weaker notion.) We shall prove that the completions, in the sense of Freyd, of small categories also enjoy the same normality condition, provided they admit at least one bicategory structure. (The complete categories in the sense of Isbell always admit bicategory structures; see the remark following Proposition 2.4.)

In what follows, we let mean that is a full subcategory of . Moreover, if , then means that each object of is equivalent to an object in .
Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements
Authors: Lamoreaux Jack W.
Published: December 2010

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.

In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.

In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.
Sur les Ensembles de Sommets Indépendants Dans les Graphes Chromatiques Minimaux
Authors: Brown W. G. , Moon J. W.
Published: December 2010

f(n, k) sera l'entier maximum t tel qu'il existe un graphe G ayant les propriétés suivantes:

(a) G possède n sommets;

(b) le nombre chromatique de G est égal à k;

(c) Gest minimal par rapport à la propriété (b); c'est-à-dire, la suppression ede n'importe quelle arête rend G(k – 1)-colorable;

(d) il existe t sommets indépendants de G, c'est-à-dire dont nulle paire ne se joigne par une arête.

Un graphe sera k-minimal s'il possède les propriétés (b) et (c). Puisque les graphes 3-minimaux sont tous des polygones impairs, il s'ensuit que f(n, 3) = [n/2] (n = 3, 5, 7, …), Il y a quelque temps T. Gallai a posé la conjecture:

1.1

M. Simonovits a réfuté l'inégalité stricte dans (1.1) en prouvant que

1.2