bluuebunny/crossref_metadata_2025 huggingface dataset:
A Functional Analytic Proof of a Selection Lemma
Authors: Baggett L. W. , Ramsay Arlan
Published: December 2010

If r is a mapping of a set X onto a set Y, then a selection for r is a mapping S of Y into X for which r o s is the identity. Analogously, if F is a mapping of a set Y into the power set 2Z of a set Z, then a selection for F is a mapping f of Y into Z such that ƒ(y) is an element of F(y) for all y in Y. These two notions are formally equivalent: Given r mapping X onto Y, define F(y) = r–l(y) and Z = X. Conversely, given F mapping Y into 2Z, define X to be the subset of Y × Z consisting of the pairs (y, z) for which z belongs to F(y), and define r on X by r(y, z) = y.
Une Generalization DU Theoreme DE Ginsburg-Rose
Authors: Chaput Daniel , Sabidussi Gert
Published: December 2010

Nous présentons ici une étude relative aux applications entre monoïdes, que nous nommerons transductions, qui peuvent être réalisées par un transducteur (machine séquentielle généralisée). L'exposé s'oriente sur l'examen des conditions caractérisant de telles applications. Le point de départ est le théorème de Ginsburg-Rose [2], qui caractérise les transductions entre monoïdes libres de type fini, en termes de quatre conditions dont une, la condition de “bounded output”, implique la longueur des mots et, par conséquent, ne peut être formulée que dans un monoïde libre. A l'aide d'un langage propre aux structures algébriques en général, celle des congruences, nous étudions des conditions remplaçant ‘'bounded output”, pour généraliser le théorème connu aux monoïdes quelconques et aux monoïdes libres quelconques.
Incompressible Surfaces in the Boundary of a Handlebody–an Algorithm
Authors: Lyon Herbert C.
Published: December 2010

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.

Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.
Zero Divisors and Idempotents in Group Rings
Authors: Cliff Gerald H.
Published: December 2010

We consider the following problem: If KG is the group ring of a torsion free group over a field K,show that KG has no divisors of zero. At characteristic zero, major progress was made by Brown [2], who solved the problem for G abelian-by-finite, and then by Farkas and Snider [4], who considered Gpolycyclic-by-finite. Here we present a solution at nonzero characteristic for polycyclic-by-finite groups. We also show that if Khas characteristic p > 0 and G is polycyclic-by-finite with only p-torsion, then KG has no idempotents other than 0 or 1. Finally we show that if R is a commutative ring of nonzero characteristic without nontrivial idempotents and G is polycyclic-by-finite such that no element different from 1 in G has order invertible in R, then RG has no nontrivial idempotents. This is proved at characteristic zero in [3].
Blocking Sets and Skew Subspaces of Projective Space
Authors: Bruen Aiden A.
Published: December 2010

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.

Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.

A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].
On Homeomorphic Embeddings of Km,n in the Cube
Authors: Hartman Jehuda
Published: December 2010

Homeomorphic embeddings of Knin the m-cube were investigated in [6]. In particular, it was proved that any homeomorph of Kn+1embedded in the m-cube has at least n2edges. Furthermore, homeomorphic embeddings of Kn+1having exactly n2edges are unique up to isomorphism. In this paper a similar problem for the complete bipartite graph is considered.

We adopt the notation and terminology of [5].

All graphs considered are without loops and multiple edges.

Let x = uv be an edge of a graph G; x will be called subdivided if it is replaced by a vertex ω and by edges uω and ωv. A graph G’ is called a subdivision of G if it is obtained from G by a subdivision of an edge of G. A refinement Ĝ of G, is a graph isomorphic to a graph obtained from G by a finite sequence of subdivisions.
Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions
Authors: Dashiell F. , Hager A. , Henriksen M.
Published: December 2010

This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [6]. This condition is that every dense cozero set S in X should be C*-embedded in X (that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F-spaces of [12]).

In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.
Recursive Colorings of Graphs
Authors: Schmerl James H.
Published: December 2010

A graph G is an ordered pair G = (V, E) where E is a set of 2-element subsets of V. The set V is the set of vertices, and E is the set of edges. The vertices x and y are joined by an edge if {x, y} is an edge. If X is a set (of colors) and χ : V →X, then we say that χ is an X-coloring of G if whenever two vertices x and y are joined by an edge, then χ(x) ≠ x(y). A graph is X-colorable if there is an χ-coloring of it. We will identify the natural number n with the set {0, 1, …, n – 1}, and often refer to n-colorings and to graphs being n-colorable.
Akcoglu's Ergodic Theorem for Uniform Sequences
Authors: Olsen James H.
Published: December 2010

Let (X, F,) be a sigma-finite measure space. In what follows we assume p fixed, 1 < p < ∞ . Let T be a contraction of Lp(X, F, μ) (‖T‖,p ≦ 1). If ƒ ≧ 0 implies Tƒ ≧ 0 we will say that T is positive. In this paper we prove that if is a uniform sequence (see Section 2 for definition) and T is a positive contraction of Lp, then

exists and is finite almost everywhere for every ƒ ∊ Lp(X, F, μ).
Intersections Finies de Sous-Groupes Nets
Authors: Benabdallah Khalid , Robert Serge
Published: December 2010

Dans la théorie des groupes abéliens, les diverses notions de pureté de sous-groupes jouent un rôle très important. Récemment, un ouvrage entier de A. P. Mishina et L. A. Skorniakov a été consacré à ces notions et à leurs généralisations à la théorie des modules (voir [8]). Espérant faire jouer aux sous-groupes purs d'un groupe abélien un rôle analogue à celui des idéaux primaires dans la théorie des anneaux noetheriens, L. Fuchs pose le problème de caractériser les sous-groupes d'un groupe abélien qui sont des intersections de familles finies de sous-groupes purs ([4] problème 13, p. 134). Ce problème, sans l'exigence de finitude offre beaucoup moins de difficultés. Une solution en est donnée par C. Megibben dans [7] pour les familles de sous-groupes purs et par K. M. Rangaswamy dans [9] pour les familles de sous-groupes nets.
Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations
Authors: Noussair E. S. , Swanson C. A.
Published: December 2010

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.
Nonzero Symmetry Classes of Smallest Dimension
Authors: Chan G. H. , Lim M. H.
Published: December 2010

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such that

for all x1, …, xm in U.

Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizer

is a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.
Integral Means of Functions with Positive Real Part
Authors: Holland F. , Twomev J. B.
Published: December 2010

We denote by the class of functions of the form

that are regular in Δ = {z:|;z| < 1} and satisfy Re h(z) > 0 there. For 0 ≦ r < 1, we write

We note that, for , the inequality

is classical.
Hyperspaces of H-Closed Spaces
Authors: Friedler L. M. , Dickman R. F. , Krystock R. L.
Published: December 2010

A space is H(i) [R(i)] if every open [regular] filter base has a cluster point and H(ii) [R(ii)] if every open [regular] filter base with a unique cluster point converges. This terminology is due to C. T. Scarborough and A. H. Stone [11]; H(i) spaces have been studied as quasi-H-closed spaces in [10] and as generalized absolutely closed spaces in [6]. Hausdorff H(i) [H(ii)] spaces are called H-closed [minimal Hausdorff] and regular T1 R(i) [R(ii)] spaces are called R-closed [minimal regular]. For a space X, 2X is the set of all non-empty closed subsets of X with the finite topology [8]. The present study was motivated by the longstanding problem of whether or not a T3 space with every closed subset R-closed is compact, and also by the well-known result ([8] and [14]) that X is compact if and only if 2X is compact.
Continuous, Slope-Preserving Maps of Simple Closed Curves
Authors: Bisztriczky Tibor , Rival Ivan
Published: December 2010

How many of the continuous maps of a simple closed curve to itself are slope-preserving? For the unit circle S1 with centre (0, 0), a continuous map σ of S1 to S1 is slope-preserving if and only if σ is the identity map [σ(x, y) = (x, y)] or σ is the antipodal map [σ(x, y) = (–x, –y)]. Besides the identity map, more general simple closed curves can also possess an “antipodal” map (cf. Figure 1).
How to Construct Almost Free Groups
Authors: Mekler Alan H.
Published: December 2010

Almost free groups were introduced in [9] as groups all of whose “small” subgroups are free. Here “small” means generated by fewer elements than the cardinality of the group. This concept is a generalization of locally free. Suppose κ is a cardinal > ω. A group is κ-free if every subgroup generated by fewer than κ elements is free. A group of cardinality κ which is κ-free is almost free. There are two related concepts which are closer approximations to freeness.
Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree
Authors: Bondy J. A. , Entringer R. C.
Published: December 2010

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].

Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,
Invariance of Torsion and the Borsuk Conjecture
Authors: Chapman T. A.
Published: December 2010

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.

THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.

THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.

If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.
Local-Global Criteria for Outer Product Rings
Authors: Estes Dennis R. , Matijevic Jacob R.
Published: December 2010

Let R be a commutative ring with multiplicative identity. We say that R is an outer product ring (OP-ring) if each vector v in the exterior power ⋀nRn+1 is decomposable; i.e. v = v1 ⋀ v2 ⋀ … ⋀ vn with vi ∈ Rn+1 (alternatively, each n + 1 tuple of elements in R is the tuple of n × n minors of some n × (n + 1) matrix with entries in R).

Lissner initiated the study of which commutative rings are OP-rings, showing that any Dedekind domain is an OP-ring [13]. Towber classified the local Noetherian OP-rings as those with maximal ideal generated by two elements, and Hinohara's reformulation of Towber's result extended the context to semi-local rings [18, 11]. Assuming the stronger condition that all Pliicker vectors are decomposable and designating such rings as Towber rings, Lissner and Geramita gave necessary conditions for a Noetherian ring to be a Towber ring [14].
Arithmetical Semigroup Rings
Authors: Hardy Bonnie R. , Shores Thomas S.
Published: December 2010

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].
The Homotopy of Simplicial Algebras
Authors: Lakser Harry
Published: December 2010

In [6] Walter Taylor investigated the relationship between the algebraic structure of a topological algebra A and the group structure of its fundamental group π1(A) and of the higher homotopy groups πn(A),n > 1. The main result is that a variety satisfies a group law λ in homotopy (that is, π1) if and only if every group in the idempotent reduct of obeys λ. (The relevant definitions are in [6] and also § 2 of this paper.) A similar result is stated for the higher homotopy groups. As Taylor points out in the introduction, the hard part of the theorem is constructing a topological algebra in whose fundamental group may fail to obey λ; indeed, in [6] this is only done in detail for the commutative law, and the proof is rather computational.
On 3-Connected Matroids
Authors: Oxley James G.
Published: December 2010

This paper extends several graph-theoretic results to matroids. The main result of Tutte's paper [10] which introduced the theory of n-connection for matroids was a generalization of an earlier result of his [9] for 3-connected graphs. The latter has since been strengthened by Halin [3] and in Section 3 of this paper we prove a matroid analogue of Halin's result. Tutte used his result for 3-connected graphs to deduce a recursive construction of all simple 3-connected graphs having at least four vertices. In Section 4 we generalize this by giving a recursive construction of all 3-connected matroids of rank at least three. Section 2 contains a generalization to minimally n-connected matroids of a result of Dirac [2] for minimally 2-connected graphs.
A Note on Young's Raising Operator
Authors: Thomas Glânffrwd P.
Published: December 2010

Consider the following formula due to Young [7] for the calculation of the homogeneous product sum, hλ, in terms of Schur functions;

where the operation Srs is defined as follows:

Y1: Srs, where r < s, “represents the operation of moving one letter from the s-th row up to the r-th row; and the resulting term is regarded as zero, when any row becomes less than a row below it, or when letters from the same row overlap; as, for instance, happens when λ1 = λ2 in the case of S13S23.“

The following example of the above is given by Robinson [4].

Calculation by other means shows that the above analysis of h(3,2,1) is correct; however, it will be noticed that the operator S123S23 does not appear in the above yet it is not specifically excluded by the rule Y1.
Generators of Orthogonal Groups over Valuation Rings
Authors: Ishibashi Hiroyuki
Published: December 2010

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.

Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .
Markov's and Bernstein's Inequalities on Disjoint Intervals
Authors: Borwein Peter B.
Published: December 2010

In 1889, A. A. Markov proved the following inequality:

INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n then

where ‖ ‖A denotes the supremum norm on A.

In 1912, S. N. Bernstein established

INEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n then

for x ∈ (a, b).

In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.

THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Then

for x ∈ (a, b).
Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators
Authors: Ahlbrandt Calvin D. , Hinton Don B. , Lewis Roger T.
Published: December 2010

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l by

where y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfying

with inner product

where . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.
Essentially Convexoid Operators
Authors: Furuta Takayuki
Published: December 2010

Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.
Absolute Approximate Retracts and AR-Spaces
Authors: Martin John R.
Published: December 2010

A subset A of a topological space X is an approximateretract of X if for every neighborhood U of A in X there is a retract R of X such that A ⊂ R ⊂ U. A compactum X is an absolute approximate retract (AAR-space) if whenever X is embedded as a subset of a compactum Z, then X is an approximate retract of Z. These concepts were first defined in [2] where it is shown that every AAR-space is a contractible Peano continuum. In [3] an example is given to show that there exists a contractible LC∞ compactum which is not an AAR-space.
Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings
Authors: Heinicke A. G.
Published: December 2010

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.

In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.
Signatures of Covering Links
Authors: Gordon C. Mca. , Litherland R. A. , Murasugi K.
Published: December 2010

A (tame) knotkninS3is said to haveperiod nif there exists a homeomorphismϕ:S3→S3, necessarily orientationpreserving, such that

(i) the fixed point set ofϕis a circle disjoint fromkn

(ii)ϕ(kn) =kn;

(iii)ϕhas ordern.

Several necessary conditions for a knot to have periodnhave already been established in the literature; see[3] [11] [14] [18]. Here we establish the following further condition, involving the signatureσ(kn) of (kn).
Lying-Over Pairs of Commutative Rings
Authors: Dobbs David E.
Published: December 2010

(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.
Embeddings of L-Groups
Authors: Shelstad D.
Published: December 2010

To a real reductive group G there is attached a family of (real) groups, each of lower dimension but sharing Cartan subgroups with G (cf. [8]). The purpose of these groups is to provide “building blocks” (in a specific sense (cf. [11])) for analysis on G. Their définition is via an L-group construction; the connected component of the identity, LH0, in the L-group of such a group H is naturally a subgroup of LG0 but the requirement that H “share” Cartan subgroups with G precludes defining LH the full L-group of H, as a subgroup of LG. Nevertheless, the principle of functoriality in the L-group suggests that the embeddings of LH in LG will play a role in analysis. In this paper, we study the embeddings of LH in LG in order to resolve a problem about the normalization of orbital integrals.
Spaces of Maps into Eilenberg-Maclane Spaces
Authors: Hansen Vagn Lundsgaard
Published: December 2010

In this note we provide alternative and unified proofs for two theorems on the homotopy groups of spaces of (continuous) maps into Eilenberg-MacLane spaces. The first theorem is due to Thorn, and independently Federer, and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, n) for n ≧ 1 with π abelian. The second theorem is due to Gottlieb and deals with spaces of maps into Eilenberg- MacLane spaces of type (π, 1) with π nonabelian. As a main tool we shall use the homotopy sequences for certain fibrations of spaces of maps.

2. Basic notation and some preliminary remarks. For any pair of connected CW-complexes X and Y with base points, we denote by M(X, Y), respectively F(X, Y), the space of free maps, respectively based maps, of X into Y.
Open and Proper Maps Characterized by Continuous Setvalued Maps
Authors: Colebunders Eva Lowen-
Published: December 2010

In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .
The Linearization of the Product of Continuous q-Jacobi Polynomials
Authors: Rahman Mizan
Published: December 2010

The problem of linearizing the product of two Jacobi polynomials, Pm(α, β)(x)Pn(α, β)(x), and to establish the conditions for the non-negativity of the coefficients has been of considerable interest for many years. Explicit non-negative representations were sought and found by many authors [7, 8, 13, 14], but only in the special case α = β, although Hylleraas [14] succeeded in finding a formula in another case α = β + 1. Gasper [9, 10] found the necessary and sufficient conditions for the non-negativity of the linearization coefficients by exploiting a recurrence relation obtained by Hylleraas for the above-mentioned product. Koornwinder [16] approached the same problem from a different point of view and managed to find a non-negative integral expression to these coefficients when . However, an exact formula in a hypergeometric series form for general α, β has been very elusive so far, in spite of the fact that all computation of special cases seemed to indicate that such a formula should exist.
The Translation Planes of Dempwolff
Authors: Johnson N. L.
Published: December 2010

In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.

Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.

Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.
Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces
Authors: Elliott Joanne
Published: December 2010

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.

A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if

(0.1)

for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and

(0.2)

it is conservative if

(0.3)
Isometries of Bergman Spaces over Bounded Runge Domains
Authors: Kolaski Clinton J.
Published: December 2010

1.1. The isometries of the Hardy spaces Hp(0 < p < ∞, p ≠ 2) of the unit disc were determined by Forelli in [2]. Generalizations to several variables: For the polydisc the isometries of Hp onto itself were characterized by Schneider [9]. For the unit ball the case p > 2 was then done by Forelli [3]; Rudin [8] removed the restriction p > 2 by proving a theorem on equimeasurability. Finally, Koranyi and Vagi [6] noted that the methods developed by Forelli, Rudin and Schneider applied to bounded symmetric domains.

In this note it will be shown that their methods also apply to the Bergman spaces over bounded Runge domains. The isometries which are onto are completely characterized; the special cases of the ball and polydisc are particularly nice and are given separately.
Weak Solutions for Semi-Martingales
Authors: Pellaumail J.
Published: December 2010

The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.

For such an equation, the existence of a “weak solution” is stated; actually, the notion of weak solution here considered is more precise than this one introduced by Strook and Varadhan (cf. [30], [31], [23]).

Namely, this weak solution is a probability, so-called “rule,” defined on (DH × Ω), DH being the classical Skorohod space of all the cadlag sample paths and Ω is the initial space which Z is defined on: the marginal distribution of R on Ω is the given probability P on Ω. This concept of rule is defined in Section 3.
Which Ordered Sets have a Complete Linear Extension?
Authors: Pouzet Maurice , Rival Ivan
Published: December 2010

It is a well known and useful fact [4] that every (partially) ordered set P has a linear extension L (that is, a totally ordered set (chain) on the same underlying set as P and satisfying a ≦ b in L whenever a ≦ b in P). It is just as well known that an ordered set P can be embedded in an ordered set P′ which, in turn, has a complete linear extension L′ (that is, a linear extension in which every subset has both a supremum and an infimum); just take L′ to be the “completion by cuts” of L. However, an arbitrary ordered set P need not, itself, have a complete linear extension (for example, if P is the chain of integers or, for that matter, if P is any noncomplete chain). It is natural to ask which ordered sets have a complete linear extension?
Induced Quaternion Algebras in the Schur Group
Authors: Mollin Richard A.
Published: December 2010

Let K be a finite, imaginary and abelian extension of the rational number field Q, and let M be the maximal real subfield of K. It is well known that each element of order 2 in S(K), the Schur group of K, is induced from an element of order 2 in B(M), the Brauer group of M; i.e., if D is a quaternion division algebra central over K such that its class [D] in B(K) is in fact in S(K) then [D] = [B ⊗MK] where B is a quaternion division algebra with [B] ∈ B(M). A natural question to ask is: “When is every element of S(K) of order 2 induced from S(M)?” The main result of this paper is to provide necessary and sufficient conditions for this to occur when G(L/K), the Galois group of L over K, is cyclic where L is the smallest root of unity field containing K.
Separating Closed Sets by Continuous Mappings into Developable Spaces
Authors: Brandenburg Harald
Published: December 2010

A topological space X is called developable if it has a development, i.e., a sequence of open covers of X such that for each x ∈ X the collection is a neighbourhood base of x, where

This class of spaces has turned out to be one of the most natural and useful generalizations of metrizable spaces [23]. In [4] it was shown that some well known results in metrization theory have counterparts in the theory of developable spaces (i.e., Urysohn's metrization theorem, the Nagata-Smirnov theorem, and Nagata's “double sequence theorem”). Moreover, in [3] it was pointed out that subspaces of products of developable spaces (i.e., D-completely regular spaces) can be characterized in much the same way as subspaces of products of metrizable spaces (i.e., completely regular T1-spaces).
Some Obstacles to Duality in Topological Algebra
Authors: Bankston Paul
Published: December 2010

0. Introduction. Functors form an equivalence of categories (see [8,]) if Γ(Φ(A)) ≅ A and Φ (Γ(B)) ≅ B naturally for all objects A from and B from . Letting denote the opposite of we say that and are dual if there is an equivalence between and .

Let τ be a similarity type of finitary operation symbols. We let Lτ denote the first order language (with equality) using nonlogical symbols from τ, and consider the class of all algebras of type τ as a category by declaring the morphisms to be all homomorphisms in the usual sense (i.e., those functions preserving the atomic sentences of Lτ). If is a class in (i.e., and is closed under isomorphism), we view as a full subcategory of , and we define the order of to be the number of symbols occurring in τ.
Eisenstein Series for Reductive Groups over Global Function Fields I.
Authors: Morris L. E.
Published: December 2010

Let G be the Lie group SL(2, R) and Γ a discrete subgroup of arithmetic type. The homogeneous space Γ\G can be equipped with an invariant measure so that there is a Hilbert space of square integrable functions, denoted L2(Γ\G), on which G acts by right translations. If Γ\G is compact then this Hilbert space breaks up into a countable direct sum of irreducible representations of G, each occurring with finite multiplicity. Quite often however Γ\G is not compact, but of finite volume; in this case L2(Γ\G) splits into a discrete spectrum Ld2 which behaves as if Γ\G were compact, and a continuous spectrum Lc2 which is described by the so called theory of Eisenstein series. These are generalized eigenfunctions of the Casimir operator of G, which are parametrized by a right half plane in C, and as such are analytic functions on this half-plane; in the course of describing the continuous spectrum Lc2 however, one analytically continues them to meromorphic functions over all of C, and shows them to satisfy functional equations.
Integral Group Rings of Some p-Groups
Authors: Ritter Jürgen , Sehgal Sudarshan
Published: December 2010

1. Introduction. The group of units, , of the integral group ring of a finite non-abelian group G is difficult to determine. For the symmetric group of order 6 and the dihedral group of order 8 this was done by Hughes-Pearson [3] and Polcino Milies [5] respectively. Allen and Hobby [1] have computed , where A4 is the alternating group on 4 letters. Recently, Passman-Smith [6] gave a nice characterization of where D2p is the dihedral group of order 2p and p is an odd prime. In an earlier paper [2] Galovich-Reiner-Ullom computed when G is a metacyclic group of order pq with p a prime and q a divisor of (p – 1). In this note, using the fibre product decomposition as in [2], we give a description of the units of the integral group rings of the two noncommutative groups of order p3, p an odd prime. In fact, for these groups we describe the components of ZG in the Wedderburn decomposition of QG.
A Construction for Vertex-Transitive Graphs
Authors: Alspach Brian , Parsons T. D.
Published: December 2010

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.

In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.
A Theorem on Derivations of Prime Rings with Involution
Authors: Herstein I. N.
Published: December 2010

In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.)

In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.
Traces On Jordan Algebras
Authors: Pedersen Gert Kjærgård , Størmer Erling
Published: December 2010

In the theory of Jordan algebras one encounters several definitions of the trace, and it is sometimes unclear whether the different notions are equivalent or not. If we restrict attention to the so–called JB–algebras studied in [2] and their weakly closed analogues JBW–algebras [8], we shall in the present note show that the different concepts are all equivalent for JBW–algebras, and that the conditions not involving projections are equivalent for JB–algebras. Among the seven equivalent conditions we shall consider, the second (ii) was used by Alfsen and Shultz [1] to show that if the JBW–algebra, is the self–adjoint part of a von Neumann algebra, then the condition characterizes traces on the enveloping von Neumann algebra. Condition (iii) appears in Robertson's paper [7] together with the implication (ii) ⇒ (iii). The inequality (iv) is a Jordan analogue of Gardner's inequality | ϕ(x) ≦ ϕ(|x|)|, [3], characterizing traces on Cast;–algebras.
Properties of a Class of (0,1)-Matrices Covering a given Matrix
Authors: Anstee R. P.
Published: December 2010

We wish to consider the class of (0,1)-matrices with prescribed row and column sums. LetR= (r1,r2, …,rm) andS= (s1,s2, …,sn) be vectors with nonnegative integral entries andr1+r2+ … +rm=s1+s2+ … +sn. We define the classto be the set ofm×n(0, 1)-matrices withithrow sumriandjthcolumn sumsjfor 1 ≦i≦mand 1 ≦j≦n.

Gale and Ryser independently found simple necessary and sufficient conditions forto be nonempty [9,14]. FromR, we form anm×n(0, 1)-matrixĀas follows. Theith row sum ofĀisriand the 1‘s are as far to the left as possible. Letbe thejthcolumn sum ofA. We define the sequenceto be theconjugateof the sequence (ri).
A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem
Authors: Poljak Svatopluk , Turzík Daniel
Published: December 2010

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,

for a connected graph G of n vertices and m edges, where

and ﹛x﹜ denotes the smallest integer ≧ x.

We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.

Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.
The Generalized Orthocompletion and Strongly Projectable Hull of a Lattice Ordered Group
Authors: Ball Richard N.
Published: December 2010

The central result is the existence and uniqueness for an arbitrary l-group G of two hulls, Ḡ and Ḡω, which in the representable case coincide with the orthocompletion and strongly protectable hull of G. This is done by introducing two notions of extension, written ≼ and ≼ω, and proving that each G has a maximal ≼ extension Ḡ and a maximal ≼ω extension Ḡω. Two constructions of Ḡ and Ḡω are-given: an l-permutation construction leads to descriptive structural information, and a construction by “consistent maps” leads to a strong universal mapping property.

The notion of a strongly projectable hull has a lengthy history. The concept of an orthocompletion, together with the first proof of its existence and uniqueness, is due to Bernau [4]. Conrad summarized and extended all these results in an important paper [10].
The Measure Spectrum of a Uniform Algebra and Subharmonicity
Authors: Kumagai Donna
Published: December 2010

Let A be a uniform algebra on a compact Hausdorff
space X. The spectrum, or the maximal ideal space, MA, of A is given by

We define the measure spectrum, SA, of A by

SA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfying

The concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.
Homology Operations Revisited
Authors: Fiedorowicz Z. , May J. P.
Published: December 2010

There are two principal kinds of input data for infinite loop space theory, namelyE∞spaces à la Boardman-Vogt [3] and May [7] and Γ-spaces à la Segal [14]. May and Thomason [13] introduced a common generalization and used it to prove the equivalence of the output obtained from these two kinds of input.

This suggests that any invariants of one kind of input should have analogs for the other. Homology operations are among the most basic invariants ofE∞spaces, and we here establish the analogous invariants for Γ-spaces. The definition is transparently obvious from the point of view of the common generalization but is at first sight rather surprising and unnatural from the point of view of Γ-spaces alone. Probably for this reason, there is no hint of the possibility of a direct definition of homology operations for Γ-spaces in the literature.
Inequalities for Polynomials with a Prescribed Zero
Authors: Aziz Abdul
Published: December 2010

If P(z) is a polynomial of degree n, then the inequality

1

is trivial. It was asked by Callahan [1], what improvement results from supposing that P(z) has a zero on |z| = 1 and he answered the question by showing that if P( l ) = 0, then

2

Donaldson and Rahman [3] have shown that if P(z) is a polynomial of degree n such that P(β) = 0 where β is an arbitrary non-negative number, then

3

whereas if the polynomial P(z) is such that P(l) = 0, then [4]

4
Rings whose Indecomposable Injective Modules are Uniserial
Authors: Hill David A.
Published: December 2010

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].

This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.
Ford and Dirichlet Regions for Fuchsian Groups
Authors: Beardon A. F. , Nicholls P. J.
Published: December 2010

There has recently been some interest in a class of limit points for Fuchsian groups now known as Garnett points [5], [8]. In this paper we show that such points are intimately connected with the structure of Dirichlet regions and the same ideas serve to show that the Ford and Dirichlet regions are merely examples of one single construction which also yields fundamental regions based at limit points (and which properly lies in the subject of inversive geometry). We examine in the general case how the region varies continuously with the construction. Finally, we consider the linear measure of the set of Garnett points.

2. Hyperbolic space.Let Δ be any open disc (or half-plane) in the extended complex planeC∞: usually Δ will be the unit disc or the upper half-plane. We may regard Δ as the hyperbolic plane in the usual way and the conformai isometries of Δ are simply the Moebius transformations of Δ onto itself.
Intersection Patterns of Families of Convex Sets
Authors: Katchalski M. , Liu A.
Published: December 2010

In this paper, we study the intersection pattern of families of convex sets. Since we only consider finite families, we may assume that the sets are also compact.

As an example, we consider families of 5 convex sets in R2 such that every two intersect and no three intersect. One such family that comes immediately to mind is that of 5 lines in general position. However, this is not the only family which exhibits this intersection pattern. Fig. 1 shows a family of 3 lines (sides of the large triangle) and 2 triangles (inscribed in the large triangle) that has this property.
Chromatic Solutions, II
Authors: Tutte W. T.
Published: December 2010

This paper is a continuation of the Waterloo Research Report CORR 81-12, (see [1]) referred to in what follows as I. That Report is entitled “Chromatic Solutions”. It is largely concerned with a power series h in a variable z2, in which the coefficients are polynomials in a “colour number” λ. By definition the coefficient of z2r, where r > 0, is the sum of the chromatic polynomials of the rooted planar triangulations of 2r faces. (Multiple joins are allowed in these triangulations.) Thus for a positive integral λ the coefficient is the number of λ-coloured rooted planar triangulations of 2r faces. The use of the symbol z2 instead of a simple letter t is for the sake of continuity with earlier papers.

In I we consider the case

(1)

where n is an integer exceeding 4.