bluuebunny/crossref_metadata_2025_split huggingface dataset:
Quotients and Inverse Limits of Spaces of Orderings
Authors: Marshall Murray A. 
Published: December 2010

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the characterdefined byIt follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).
Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
Authors: Drake David A. 
Published: December 2010

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-netis a set ofs2µpoints together withrparallel classes of blocks. Each class consists ofsblocks of equal cardinality. Two non-parallel blocks meet in preciselyµpoints. It has been proven thatris always less than or equal to (s2µ– l) / (s– 1).
On Countability of Point-Finite Families of Sets
Authors: Junnila Heikki J. K. 
Published: December 2010

It is well known that in a separable topological space every point-finite family of open subsets is countable. In the following we are going to show that both in σ-finite measure-spaces and in topological spaces satisfying the countable chain condition, point-finite families consisting of “large” subsets are countable.Notation and terminology. Let A be a set. The family consisting of all (finite) subsets of A is denoted by . Let be a family of subsets of A. The sets and are denoted by and , respectively. We say that the family is point-finite (disjoint) if for each a ∈ A , the family has at most finitely many members (at most one member).
Not Every K1-Embedded Subspace is K0-Embedded
Authors: Mill Jan van 
Published: December 2010

All topological spaces under discussion are assumed to be Tychonoff.For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such thatThe extender κ is called a Kn-function (cf. [3]).Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.
Sets of Uniqueness for the Group of Integers of a p-Series Field
Authors: Wade William R. 
Published: December 2010

Let G denote the group of integers of a p-series field, where p is a prime ≦ 2. Thus, any element can be represented as a sequence {xi }i = 0∞ with 0 ≦ xi < p for each i ≦ 0. Moreover, the dual group {Ψm}m = 0∞ of G can be described by the following process. If m is a non-negative integer with for each k , and if then(1)where for each integer k ≧ 0 and for each x = {xi} ∈ G the functions Φk are defined by(2)In the case that p = 2, the group G is the dyadic group introduced by Fine [1] and the functions are the Walsh-Paley functions. A detailed account of these groups and basic properties can be found in [4].
A Simple C*-Algebra Generated by Two Finite-Order Unitaries
Authors: Choi Man-Duen 
Published: December 2010

We present an example which illustrates several peculiar phenomena that may occur in the theory of C*-algebras. In particular, we show that a C*-subalgebra of a nuclear (amenable) C*-algebra need not be nuclear (amenable).The central object of this paper is a pair of abstract unitary matrices,acting on a common Hilbert space. For an explicit construction, we may decompose an infinite-dimensional Hilbert space H into H = H0 ⴲ H1 , H1 = Hα ⴲ Hβ with dim H0 = dim H1 = dim Hα = dim Hβ, letting u, v Є B(H) be any two unitary operators such thatand u2 = 1, v3 = 1. Whereas many choices of u, v are possible, it might be surprising to see that C*(u, v), the C*-algebra generated by u and v, is algebraically unique; namely, if (u1,V1) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C*(u1, v1) (Theorem 2.6).
Lie Algebras with Nilpotent Centralizers
Authors: Benkart G. M. , Isaacs I. M. 
Published: December 2010

We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.
Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands
Authors: Auslander M. , Bautista R. , Platzeck M. I. , Reiten I. , Smalø S. O. 
Published: December 2010

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].
Perturbations of AF-Algebras
Authors: Phillips John , Raeburn Iain 
Published: December 2010

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.
Defining Relations in Orthogonal Groups of Characteristic Two
Authors: Güunther Georg , Nolte Wolfgang 
Published: December 2010

Introduction. It is well-known that a group is uniquely determined by a system of generators, and a set of defining relations on those generators. Clearly it is of interest to find relations that are as simple as possible. In this paper, this question is dealt with for certain orthogonal groups of characteristic 2, which are generated by involutions.Let V be a vector space over a field K of characteristic 2 (we always exclude the prime field K = G F (2)). Let Q be a quadratic form over V, and let S be the set of orthogonal transformations of (V, Q) whose path is 1-dimensional and not contained in the radical of V. Letting O* be the group generated by S, we shall show that every relation among generators in S is a consequence of relations of length 2, 3, or 4.
Union of Realcompact Spaces and Lindelöf Spaces
Authors: Kato Akio 
Published: December 2010

All spaces in this paper are completely regular Hausdorff and all maps are continuous onto, unless otherwise stated. The purpose of this paper is to investigate the realcompactness of a space X which contains a Lindelöf space L such that every zero-set Z (in X) disjoint from L is realcompact. We show in § 2 that such a space X is very close to being realcompact (Theorems I, II and III). But in general such a space fails to be realcompact. Indeed, in §§ 3 and 4 the following questions of Mrówka [18, 19] are answered, both in the negative:(Q. 1) If X = L ∪ G where L is Lindelöf closed and G is E-compact, then is X E-compact?(Q. 2) Suppose f:X → Y is a perfect map such that the set M(f) = {y ∊ Y| |f−l(y)| > 1} of multiple points of f is Lindelöf (especially, countable) closed. If X is E-compact, is Y also E-compact?
An Inequality Characterizes the Trace
Authors: Gardner L. Terrell 
Published: December 2010

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |ϕ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional ϕ on the C*-algebra satisfies(†)then ϕ satisfies also the equivalent conditions (i) ϕ(xy) = ϕ(yx) for all x, y in ;(ii) ϕ(x*x) = ϕ(xx*) for all x in ;(iii) ϕ(x) = ϕ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element.
Some Structural Properties of the Set of Remote Points of a Metric Space
Authors: Gates Catherine L. 
Published: December 2010

A remote point of a metric space X is a point in βX |Xnot in the βX-closure of any discrete subset of X. Remote points have been studied by Fine and Gillman [2], Plank [4], Robinson [6], Woods [9, 10], Van Douwen [7] and others. The main results concerning the existence of remote points are listed in Section 1. In this paper we determine some structural properties of the set of remote points of a metric space which has no isolated points. The notation of Gillman and Jerison [3] and Walker [8] will be used. All spaces are assumed to be Tychonoff.Let X be a space. The general remote points of X, denoted TX, are those points in βX |X which are not in the βX-closure of any nowhere dense sets of X.
On Asymptotic Centers and Fixed Points of Nonexpansive Mappings
Authors: Lim Teck-Cheong 
Published: December 2010

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define
Groupings of Metabelian Groups and Extension Categories
Authors: Roggenkamp K. W. 
Published: December 2010

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall proveTHEOREM. If RG ≌ RH as R-algebras, then G ≌ HThe question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules
A Generalization of Degree Two Simple Finite-Dimensional Noncommutative Jordan Algebras
Authors: Conlon Mary Ellen 
Published: December 2010

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose1Then is noncommutative Jordan.The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.
The Largest Class of Hereditary Systems Defining a C0 Semigroup on the Product Space
Authors: Delfour M. C. 
Published: December 2010

The object of this paper is to characterize the largest class of autonomous linear hereditary differential systems which generates a strongly continuous semigroup of class C0 on the product space Mp = Rn × Lp(-h, 0), 1 ≦ p < ∞, 0 < h ≦ + ∞ (R is the field of real numbers and Lp(– h, 0) is the space of equivalence classes of Lebesgue measurable maps x:[ – h, 0] ⌒ R → Rn which are p-integrable in [ –h, 0] ⌒R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing “finite memory” (h < + ∞ ) and those of [14], [5] and [6] in the “infinite memory case (h = + ∞ )”.Consider the autonomous linear hereditary differential equation(1.1)where x(t) ∊ Rn, x:[–h, 0] ⌒ R → Rn is defined as xt(θ) = x(t + θ), C(–h, 0) is the space of bounded continuous functions [–h, 0] ⌒ R → Rn and L:C(–h, 0) → Rn is a continuous linear map.
Capitulation in Class Field Extensions of Type (p, p)
Authors: Chang S. M. , Foote R. 
Published: December 2010

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).
Uniform Approximation on the Graph of a Smooth Map in Cn
Authors: Weinstock Barnet M. 
Published: December 2010

Let X be a compact set in Cn, and let ƒ1, …, ƒm, m ≧ n, be continuous, complex-valued functions on X which have C1 extensions to some neighborhood of X. We wish to describe the algebra A of continuous complex-valued functions on X which can be approximated uniformly by polynomials in the functions z1 …, zn, ƒ1 …, ƒm. For this purpose we introduce the setsand
Congruence Subgroups of the Picard Group
Authors: Fine Benjamin 
Published: December 2010

The Picard group Γ = PSL2 (Z [i]) is the group of linear fractional transformationswith ad – bc = ± 1 and a, b, c, d Gaussian integers.Γ is of interest as an abstract group and in automorphic function theory. In an earlier paper [1], a decomposition of Γ as a free product with amalgamated subgroup was given and this was utilized to investigate Fuchsian subgroups. Karrass and Solitar used a similar decomposition to characterize abelian and nilpotent subgroups. Maskit [6], Mennicke [7] and Fine [2], used Γ to generate faithful representations of Fundamental Groups of Riemann Surfaces while more recently Wielenberg [10] represented certain knot and link groups as subgroups of Γ. In this paper, we will examine the structure of the congruence subgroups of Γ. Our technique will be to use the decomposition cited above [1], together with the Karrass-Solitar subgroup structure theory for free products with amalgamations [3]. Finally, we give a conjecture and some results concerning Fuchsian subgroups which are contained in congruence subgroups.
Injective Sheaves of Abelian Groups: A Counterexample
Authors: Banaschewski B. 
Published: December 2010

It has been claimed that a sheaf of abelian groups on a Hausdorff space in which the compact open sets form a basis is injective in the category of all such sheaves whenever its group of global elements is divisible (Dobbs [1]). The purpose of this note is to present an optimal counterexample to this by showing, more generally, that on any nondiscrete T0-space there exists a sheaf of the type in question which is not injective.Recall that a sheaf A of abelian groups on a space X assigns to each open set U in X an abelian group AU and to each pair U, V of open sets in X such that V ⊆ U a group homomorphism, denoted s ⟿ s|V, satisfying the familiar sheaf conditions ([3, p. 246]) which make A a special type of contravariant functor from the category given by the inclusion relation between the open sets of X into the category Ab of abelian groups, and that a map between sheaves A and B of abelian groups is a natural transformation h:A → B, with component homomorphisms hu:AU → BU. In the following, AbShX will be the category with these A as objects and these h:A → B as maps (= morphisms).
The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods
Authors: Jackson D. M. , Goulden I. P. 
Published: December 2010

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.
Isometries of sp(α)
Authors: Fleming R. J. , Jamison J. E. 
Published: December 2010

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which(1.1)In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.
σ4-Actions On Homotopy Spheres
Authors: Liang Chao-Chu 
Published: December 2010

Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.
Diamond Principles, Ideals and the Normal Moore Space Problem
Authors: Taylor Alan D. 
Published: December 2010

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such thatThe sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.
Expansions of Arbitrary Analytic Functions in Series of Exponentials
Authors: Dickson D. G. 
Published: December 2010

Let ϕ ≠ 0 be an entire function of one complex variable and of exponential type. Let B denote the set of all monomial exponentials of the form zneζ where ζ is a zero of ϕ of order greater than h. If R is a simply connected plane region and H(R) denotes the space of functions analytic in R with the topology of uniform convergence on compacta, then ϕ can be considered as an element of the topological dual H′(R) if the Borel transform of ϕ is analytic on , the complement of R. The duality is given bywhere C is a simple closed curve in the common region of analyticity of ƒ and , and C winds once around the complement of a set in which is analytic.
Wallman Compactification and Representation
Authors: Hegde Shankar 
Published: December 2010

Let X be any set and A be a uniformly closed algebra of bounded real valued functions on X which contains the constants and separates the points. For a lattice ℒ of subsets of X (we assume throughout that ∅ and X belong to ℒ), let MR(ℒ) denote the space of all finite, finitely additive,ℒ-regular measures defined on the field of sets generated by ℒ . Generalizing the notion of an integral representation, in [5] Kirk and Crenshaw define a standard representation of A*, the Banach dual of A, in MR(ℒ) to be a linear map I of A* into MR(ℒ) with the property that if 0 ≦ ϕ ∈ A*, thenfor every W in ℒ. The space MR(ℒ) is said to represent A* if there exists a (unique) standard representation I of A* onto MR(ℒ) which is a Banach lattice isomorphism.
Counting Coloured Graphs of High Connectivity
Authors: Bollobás Béla 
Published: December 2010

Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.