bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Characterization of Comparability Graphs and of Interval Graphs
Authors: Gilmore P. C. , Hoffman A. J. 
Published: December 2010

Let < be a non-reflexive partial ordering defined on a set P. Let G(P, <) be the undirected graph whose vertices are the elements of P, and whose edges (a, b) connect vertices for which either a < b or b < a. A graph G with vertices P for which there exists a partial ordering < such that G = G(P, <) is called a comparability graph.In §2 we state and prove a characterization of those graphs, finite or infinite, which are comparability graphs. Another proof of the same characterization has been given in (2), and a related question examined in (6). Our proof of the sufficiency of the characterization yields a very simple algorithm for directing all the edges of a comparability graph in such a way that the resulting graph partially orders its vertices.
On Matrix Commutators of Higher Order
Authors: Robinson D. W. 
Published: December 2010

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the followingProposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.
The Spin Representation of the Symmetric Group
Authors: Morris A. O. 
Published: December 2010

Let Γnbe therepresentation group or spin group(9; 4) of the symmetric groupSn. Then the irreducible representations of Γncan be allocated into two classes which we shall call (i) ordinary representations, which are the irreducible representations of the symmetric group, and (ii)spinorprojectiverepresentations.As is well known (3; 5), there is an ordinary irreducible representation [λ] corresponding to every partition (λ) = (λ1, λ2, . . . , λm) ofnwithλ1≥ λ2≥ . . . ≥ λm> 0.
Minimal Interchanges of (0, 1)-Matrices and Disjoint Circuits in a Graph
Authors: Walkup David W. 
Published: December 2010

In this paper we obtain a partial answer in graph-theoretic form to a question raised by Ryser (2, p. 68) concerning the minimal number of interchanges required to transform equivalent (0, l)-matrices into each other.For given positive integersmandnwe consider the collection ofm×n(0, 1)-matricesA= {aij}, i.e.aij= 0 or 1 for 1 ≤i≤m, 1 ≤j≤n. We say the (0, 1)-matricesA= {aij} andB= {bij} areequivalentand writeA~Bif and only if they have the same row and column sums, that is, if and only if.
A Periodic Jacobi-Perron Algorithm
Authors: Bernstein Leon 
Published: December 2010

In the first part of this paper I shall demonstrate that one irrational root of the algebraic equationcreates an algebraic number field, out of whichn— 1 irrationals can be chosen so that they yield a periodic Jacobi-Perron algorithm. The coefficients in (1) are subject to certain restrictions which will be elaborated below.
The Lototsky Transform and Bernstein Polynomials
Authors: King J. P. 
Published: December 2010

The Bernstein polynomials1associated with a function f denned on [0, 1] have been the subject of much recent research and have been generalized in several directions (1 ; 2 ; 5). The generalized Lototsky or [F, dn] matrix (3) has also been the subject of extensive research.
The Maximal Co-Rational Extension by a Module
Authors: Courter R. C. 
Published: December 2010

Modules are S-modules where S is an arbitrary ring with or without a unit element. We consider a projective module P having a submodule K such that K + Y = P implies that the submodule Y is P (P, then, is a projective cover of P/K (Definition 4 in this section)) and we define the submodule X of P byOur main result states that up to isomorphism P/X is the maximal co-rational extension over P/K (by P/K, in the more precise wording of the title).
Magic Graphs
Authors: Stewart B. M. 
Published: December 2010

In this paper we use number-theoretic properties to classify ordinary graphs that are finite and have no isolated vertices. The classification depends on whether there is an assignment of real values, usually rational integer values, to the edges of the graph, such that the set of assigned values and the set of vertex sums of these values, summed at each vertex over all the edges incident to the vertex, will be a pair of sets with prescribed properties. Then we seek corresponding graph-theoretic properties.It is possible to describe the problem in terms of a symmetric matrix having specified properties for its row sums, but in this paper we make no use of this interpretation; however, see (3).
The Isomorphism of Certain Continuous Rings
Authors: Dawkins Brian P. , Halperin Israel 
Published: December 2010

In this paper we shall prove the following two theorems (the terminology is explained in § 2 below; all rings are assumed to be associative).THEOREM 1. Suppose that is a division ring of finite order m over its centre Z and let μ(m) denote the factor sequence 1, m, m2, … , mn, … . Then the rings μ(w) and Zμ(m) are isomorphic.
Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions
Authors: Botta E. P. 
Published: December 2010

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

A formulation of a fixed point theorem, which can be applied conveniently to non-cooperative games and cooperative games, is suggested in this note.Let N1, … , Nm be m non-empty, finite disjoint sets. For k = 1, … , m we denote by Sk the simplex the coordinates of whose points are indexed by the members of Nk; thus Sk is the collection of all real functions xk defined on Nk which satisfy:1.11.2
Supermagic Complete Graphs
Authors: Stewart B. M. 
Published: December 2010

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.
Finiteness of Semigroups of Operators in Universal Algebra
Authors: Nelson Evelyn 
Published: December 2010

This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.
Expansion of Continuous Differentiable Functions in Fourier Legendre Series
Authors: Saxena R. B. 
Published: December 2010

Let1.1denote the nth partial sum of the Fourier Legendre series of a function ƒ(x). The references available to us, except (5), prove only that Sn(ƒ, x) converges uniformly to ƒ(x) in [— 1, 1] if ƒ(x) has a continuous second derivative on [—1, 1]. Very recently Suetin (5) has shown by employing a theorem of A. F. Timan (7) (which is a stronger form of Jackson's theorem) that Sn(ƒ, x) converges uniformly to ƒ(x) ƒ(x) belongs to a Lipschitz class of order greater than 1/2 in [—1, 1]. More generally he has proved the following theorem.
Finite Groups with a given Number of Conjugate Classes
Authors: Poland John 
Published: December 2010

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation*has only finitely many solutions over the positive integers.
On the Distribution of nθ Modulo 1
Authors: Graham R. L. , Van Lint J. H. 
Published: December 2010

In recent work of E. Arthurs and L. A. Shepp on a problem of H. Dym concerning the existence of an ergodic stationary stochastic process with zero entropy (cf. 1), the function dθ(n) was introduced as follows:For an irrational number θ, letbe the sequence of points {lθ}, 1 ≦ l ≦ n, (where {x} denotes x — [x], the fractional part of x) and define*dθ(n) = max(ai — ai-1), 1 ≦ in + 1.
The Bracket Function and Complementary Sets of Integers
Authors: Fraenkel Aviezri S. 
Published: December 2010

The following result is well known (as usual, [x]denotes the integral part of x):(A) Let α and β be positive irrational numbers satisfying1Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.The theorem has a converse, and the following holds:(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.
A Fixed-Point Theorem for Commuting Monotone Functions
Authors: Gray William J. 
Published: December 2010

Hamilton (1) proved that a hereditarily unicoherent, hereditarily decomposable metric continuum has the fixed-point property for homeomorphisms. In this paper we shall generalize this result by showing that if X is a hereditarily unicoherent, hereditarily decomposable Hausdorff continuum and 5 is an abelian semigroup of continuous monotone functions from X into X, then S leaves a point of X fixed.Let X be a Hausdorff continuum. X is unicoherent if, whenever X = A ∪ B, where A and B are subcontinua of X, A ∩ B is a continuum. If each subcontinuum of X is unicoherent, X is hereditarily unicoherent. X is decomposable if X is the union of two of its proper subcontinua. If each subcontinuum of X which contains more than one point is decomposable, X is hereditarily decomposable.
Stone Lattices. II. Structure Theorems
Authors: Chen C. C. , Grätzer G. 
Published: December 2010

Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice.The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple.It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).
Pascal Ovals in Projective Planes
Authors: Rigby J. F. 
Published: December 2010

A projective plane consists of a set of points and a set of lines, together with a relation of incidence between points and lines, such that(i) any two distinct points P, Q are incident with a unique line PQ,(ii) any two distinct lines p, q are incident with a unique point p ∩ q,(iii) there exist four points, no three of which are incident with the same line.We shall use the usual geometrical terminology.An oval in a projective plane π is a set of points of π such that:(i) no three points of are collinear; this means that a line of π is either a secant of , containing two points of , or a tangent of , containing just one point of , or a non-secant of , containing no point of ;
Coefficients of Functions with Bounded Boundary Rotation
Authors: Robertson M. S. 
Published: December 2010

For fixed k ≧ 2, let Vk denote the class of normalized analytic functionssuch that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and1Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjectureholds for functions convex in one direction.
Cell Complexes, Valuations, and the Euler Relation
Authors: Perles M. A. , Sallee G. T. 
Published: December 2010

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.
Products of Reflections in an Affine Moufang Plane
Authors: Götzky K. Martin 
Published: December 2010

Let be a Moufang plane. By specializing one line ω, the line at infinity, weobtain an affine Moufang plane . The group generated by the shears of is called the equiaffine group. Veblen [9, § 52] asked whether every equiaffinity is a product of two affine reflections. He gave a proof which will work in an affine Pappian plane, using the following two properties.Property 1. If an equiaffinity fixes two distinct proper points of , it fixes every point collinear with them.Property 2. Let e be an equiaffinity and P a point such that ppe2pe3 is a triangle. Then the lines PePe2 and PPe3 are parallel.Without using these properties, it will be proved that the answer to Veblen's question is “yes“ if and only if the Moufang plane is Pappian.
Systems of equations and generalized characters in groups
Authors: Isaacs I. M. 
Published: December 2010

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.
The Structure of the Algebra of Hankel Transforms and the Algebra of Hankel-Stieltjes Transforms
Authors: Schwartz Alan 
Published: December 2010

Let M be the space of all bounded regular complex-valued Borel measures defined on I = [0, ∞). M is a Banach space with ‖μ‖ = ∫d|μ|(x) (μ ∈ M). (Integrals in this paper extend over all of I unless otherwise specified.) Let v be a fixed real number no smaller than and let if z ≠ 0 and , where Jv, is the Bessel function of the first kind of order v and cv =[2vΓ(v + 1)]–1; is an entire function, as can be seen from the power series definition ofThe Hankel-Stieltjes transform of order v is given by .
The Maximal p-Extension of a Local Field
Authors: Marshall Murray A. 
Published: December 2010

1. Let k denote a local field, that is, a complete discrete-valued field with perfect residue class field . Let G denote the Galois group of the maximal separable algebraic extension M of k, and let g denote the corresponding object over . For a given prime integer p, let G(p) denote the Galois group of the maximal p-extension of k. The dimensions of the cohomology groupsconsidered as vector spaces over the prime field Z/pZ, are equal, respectively, to the rank and the relation rank of the pro-p-group G(p); see [4; 9]. These dimensions are well known in many cases, especially when k is finite [6; 3; (Hoechsmann) 2, pp. 297-304], but also when k has characteristic p, or when k contains a primitive pth root of unity [4, p. 205].
On Class Sums inp-Adic Group Rings
Authors: Sehgal Sudarshan K. 
Published: December 2010

In this note we prove that an isomorphism ofp-adic group rings of finitep-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of thep-adic group ring of a finitep-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of thep-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.Gdenotes a finitep-group.Zdenotes the ring of (rational) integers.ZPdenotes the ring ofp-adic integers.Qpdenotes thep-adic number field.
Going Down in Polynomial Rings
Authors: McAdam Stephen 
Published: December 2010

In this paper, R ⊂ T will be commutative domains having a common identity.Definition. Suppose that R is a subdomain of T.(i) If P is a prime ideal of R and Q is a prime ideal of T, we say that Q lies over P if Q ∩ R = P.(ii) If every prime of R has a prime of T lying over it, we say that R ⊂ T has lying over.(iii) If there is a unique prime of T lying over P in R, we say that P is unibranched in T.(iv) If every prime of R is unibranched in T we say that R ⊂ T is unibranched.
Counting Coloured Graphs. III
Authors: Wright E. M. 
Published: December 2010

In an earlier paper [4], we found an asymptotic expansion for Mn = Mn(k), the number of coloured graphs on n labelled nodes, when n is large. Such a graph is a set of n distinguishable objects called nodes, and a set of “edges”, that is, undirected pairs of nodes. The nodes are mapped onto k colours. Every pair of nodes of different colours may or may not be joined by an edge, but no edge can join a pair of nodes of the same colour.We write mn for the number of these graphs which are connected, Fn for the number which use all k colours (i.e., at least one node in each graph is mapped onto each of the k colours), and fn for the number of connected graphs which use all k colours.
The Cohomological Dimension of a Directed Set
Authors: Mitchell Barry 
Published: December 2010

Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functoris left exact, and its kth right derived functor is denoted by colimk. The R-cohomological dimension of C is defined byIf there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.
A Characterization of Some Geometries of Chains
Authors: Chen Yi 
Published: December 2010

The geometries considered here are the Möbius plane
M() (W. Benz [1]),
the Laguerre plane L() (W. Benz and H.
Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G.
Kaerlein [18]) over a field . All of them are
geometries of an algebra with identity over a field.
The characterization of the projective plane over a field by the proposition
of Pappus first gave a close relation between algebraic and geometric structures.
B. L. v. d. Waedern and L. J. Smid [28] presented a further example by
characterizing the Möbius and Laguerre plane with incidence axioms and the
"complete" proposition of Miquel.
Symmetrized Kronecker Products of Group Representations
Authors: Butler P. H. , King R. C. 
Published: December 2010

Certain phases are associated with the Kronecker squares and cubes of
representations of the finite and of the compact semi-simple groups. These phases
are important in giving the symmetry properties of the 1 —
jm and 3 — jm symbols of
the groups [4; 9]. It is our primary purpose to evaluate these phases.
The Frobenius-Schur invariant [12, p. 142] for an irreducible representation
of group
G

(1.1)
On Odd Functions of Bounded Boundary Rotation
Authors: Leach Ronald J. 
Published: December 2010

Let VK denote the class of functionsthat are analytic in the unit disc U, satisfy f′(z) ≠ 0 in U, and map U onto a domain with boundary rotation at most Kπ (for a definition of this concept, see [9]). V. Paatero [9] showed that f(z) ∊ VK if and only if1.1
Sur La Convergence Presque Partout des Suites de Fonctions Mesurables
Authors: Bucchioni D. , Goldman A. 
Published: December 2010

L'objet de cet article est de donner quelques résultats concernant la
structure des suites de fonctions mesurables sur un espace mesuré abstrait
(X, ∑, μ), le théorème principal étant le
suivant:
Theoreme (A). Soit (ƒn) une suite de fonctions
mesurables sur un espace mesuré (X, ∑, μ) dont aucune
sous-suite ne converge presque partout. Il existe alors un élément Y ∊ ∑, μ
(Y) &gt; 0, deux nombres r ∊ R, δ &gt; 0 et
une partie infinie M de N tels
que, pour tout A ∊ ∑, ⊂ C
F, μ(A) &gt; 0 et pour toute
partie infinie L CP.S. M (c'est-à-dire L\M est fini), on puisse trouver
x et y ∊
A vérifiant ƒn(x) &gt; r +
δ et ƒn(y) &lt; r pour une
infinité d'indices n ∊ L.
Suites D'Interpolation Pour les Classes de Bergman de la Boule et du Polydisque de Cn
Authors: Amar Eric 
Published: December 2010

Soit Dn = ﹛z = (z1, . . . , zn) ∈ Cn, |zi| <> 1﹜ le polydisque de Cn et ƛn la mesure de Lebesgue de Cn normalisée sur Dn. Pour b > 0, .on définit les espaces de Bergman Ap(ƛn) de la manière suivante:Ap(ƛn) est l'espace des fonctions analytiques dans Dn telles que:
Analytic Sets, Baire Sets and the Standard Part Map
Authors: Henson C. Ward 
Published: December 2010

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.
Stable Conjugacy: Definitions and Lemmas
Authors: Langlands R. P. 
Published: December 2010

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.