bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalization of the Hausdorff Moment Problem
Authors: Borwein David , Jakimovski Amnon
Published: December 2010

Suppose throughout that {kn} is a sequence of positive integers, that

that k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying

(1)

and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.
On Some Twisted Chevalley Groups Over Laurent Polynomial Rings
Authors: Morita Jun
Published: December 2010

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.

For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relations

for all i, j,

for distinct i, j.
Valuation Rings and Rigid Elements in Fields
Authors: Ware Roger
Published: December 2010

In [20], T. A. Springer proved that if A is a complete discrete valuation ring with field of fractions F, residue class field of characteristic not 2, and uniformizing parameter π then any anisotropic quadratic form q over F has a unique decomposition as q = q1 ⊥ 〈π〉q2, where q1 and q2 represent only units of A, modulo squares in F (compare [14, Satz 12.2.2], [19, §4], [18, Theorem 8.9]). Consequently the binary quadratic form x2 + πy2 represents only elements in Ḟ2 ∪ πḞ2, where Ḟ2 denotes the set of nonzero squares in F. Szymiczek [21] has called a nonzero element a in a field F rigid if the binary quadratic form x2 + ay2 represents only elements in Ḟ2 ∪ aḞ2.
On Block-Schematic Steiner Systems: S(t, t + 1, v)
Authors: Yoshizawa Mitsuo
Published: December 2010

A Steiner system S(t, k, v) is a collection of k-subsets, called blocks, of a v-set of points with the property that any t-subset of points is contained in a unique block. We assume 1 < t < k < v. A Steiner system is called block-schematic if the blocks form an association scheme with the relations determined by size of intersection. Ito and Patton [3] proved that if S(4, 5, v) is block-schematic, then v = 11. The purpose of this paper is to extend this result, and we prove the following theorem.

THEOREM. A Steiner system S(t, t + 1, v) is block-schematic if and only if one of the following holds: (i) t = 2, (ii) t = 3, v = 8, (iii) t = 4, v = 11, (iv) t = 5, v = 12.
Distinguished Domains
Authors: Heitmann Raymond C. , McAdam Stephen
Published: December 2010

This paper introduces a class of domains which we hope to show merits some attention.

Definition. The domain R is said to be a distinguished domain if for any 0 ≠ z ∈ K, the quotient field of R, (1 : z) does not consist entirely of zero divisors modulo (1 : z–l). (Note: Here we use the fact that a zero module has no zero divisors. Thus if z–l ∈ R, so that (1 : z–l) = R, then the condition holds trivially.)

Section 1 of this paper gives numerous examples of distinguished domains, foremost among them being Krull domains and Prufer domains. In fact Prüfer domains are shown to be exactly those distinguished domains whose prime lattice forms a tree. Other distinguished domains can be constructed by the D + M construction. It is shown that distinguished domains are integrally closed but the converse fails.
Linear Isometries of Spaces of Absolutely Continuous Functions
Authors: Pathak V. D.
Published: December 2010

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequality

implies that

([7], p. 221, 223).
Some Finiteness Conditions for Orthomodular Lattices
Authors: Bruns Günter , Greechie Richard
Published: December 2010

Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:

An: L has at most n blocks,

Bn: there exists a covering of L by at most n blocks,

Cn: the set ﹛C(x)| x ∈ L﹜ has at most n elements,

Dn: out of any n + 1 elements of L at least two commute.

We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ nAn, B ⇔ ∃ nBn, C ⇔ ∃ nCn and D ⇔ ∃ nDn.
Chromatic Solutions
Authors: Tutte W. T.
Published: December 2010

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.

The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.

1

Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.
The Analytic Character of the Birkhoff Interpolation Polynomials
Authors: Lorentz G. G.
Published: December 2010

Let E be an m × (n + 1) regular interpolation matrix with elements ei, k = (E)i, k which are zero or one, with n + 1 ones. Then for each f ∈ Cn[a, b] and each set of knots X: a ≦ x1 < … < xm ≦ b, there is a unique interpolation polynomial P(f, E, X; t) of degree ≦ n which satisfies

1

A recent paper [1] discussed the continuity of P, as a function of x1, …,xm(with coalescences allowed). We would like to study in this note the analytic character of P as a function of real or complex knots X: x1, …, xm. This is easy for the Lagrange or the Hermite interpolation. In this case P is a polynomial in x1, …, xm if f is a polynomial, and an entire function in x1, …, xm if f is entire.
On Lucas-Sets for Vector-Valued Abstract Polynomials in K-inner Product Spaces
Authors: Zaheer Neyamat
Published: December 2010

The fact that Rolle's theorem on critical points of a real differentiable function does not hold in general for analytic functions of a complex variable raises a natural question [7, p. 21] as to whether or not it can be generalized to polynomials, the simplest subclass of analytic functions. While attempting to answer this and other related problems in [4, 5, 6], Lucas proved that all the critical points of a nonconstant polynomial f lie in the convex hull of the set of zeros of f (see Theorem (6.1) in [7]). Walsh [12] has shown that Lucas’ theorem is equivalent to the following result [7, Theorem (6.2)], namely: Any convex circular region which contains all the zeros of a polynomial f also contains all the zeros of its derivative f′.
L'Enveloppe Conjuguee D'Un Demi-Groupe Inverse a Metacentre Idempotent
Authors: Pastijn Francis
Published: December 2010

Nous renvoyons le lecteur à [4] pour toutes les définitions non expliquées ci-dessous.

Un demi-groupe inverse S est un demi-groupe pour lequel pour tout élément a ∊ S il existe un seul inverse a–1, c'est-à-dire un élément a–l qui satisfait aux conditions aa–1a = a et a–1aa–1 = a–1. Une application λ : S → S, x → λx est appelée une translation à gauche si λ(xy) = (λx)y; une application ρ : S → S, x → xρ est dite translation à droite si (xy)ρ = x(yρ). Les translations à gauche (à droite) forment un demi-groupe Λ(S) [P(S)]. L'ensemble

forme un sous-demi-groupe de Λ(S) × P(S) qui est appelé l'enveloppe de translations de S. Si pour tout a ∊ S on définit λa : S → S, x → ax, et ρa : S → S, x → xa, on appelle

la partie interne de Ω(S).
Rearrangements that Preserve Rates of Divergence
Authors: Johnston Elgin H.
Published: December 2010

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.

In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.
KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank
Authors: Santharoubane L. J.
Published: December 2010

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:

1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)

2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).

The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.
A Note on Projective Capacity
Authors: Alexander H.
Published: December 2010

Introduction. In [1] we defined a capacity in Cn. Recently Molzon, Shiffman and Sibony [8] have introduced a different capacity which is useful for certain Bezout estimates. The object of this note is to apply the methods of [1] to study the capacity of [8]. We shall obtain an equivalent definition of this capacity via Tchebycheff polynomials, along the lines of [1]. Half of this equivalence was independently obtained by Sibony [9].

To establish the full equivalence of these two approaches to capacity a notion of Jensen measures in a setting more general than uniform algebras is needed. We shall consider Jensen measures for multiplicative semigroups; these are sets of functions in which only the multiplicative structure is postulated. It will also be useful to generalize the notion of polynomial hull in Cn to a hull with respect to a multiplicative semigroup of polynomials. We can then adapt the approach of [1] to these semigroups.
On Inequalities Complementary to Jensen's
Authors: Beesack Paul R.
Published: December 2010

In a paper published in 1975 [1, § 3], D. S. Mitrinovič and P. M. Vasič
used the so-called “centroid method” to obtain two new inequalities which are complementary to (the discrete version of) Jensen's inequality for convex functions. In this paper we shall present a very general version of such inequalities using the same geometric ideas used in [1] but not using the centroid method itself. At the same time we shall extend the domain of the inequalities (even in the discrete case), and clarify the value of the constant (λ) appearing in the inequality. We give applications of the theorems to some general means and also to the classical means. Our first result is given as

THEOREM 1. Let v be a nonnegative measure on a σ-algebra of subsets of a set D and let q, f be real v-measurable functions on D such that q(x) > 0, –∞ < x1 ≦ f(x) ≦ x2 < ∞ for all x ∊ D and ∫D qdv = 1.
The Hilbert Transform of Generalized Functions and Applications
Authors: Pandey J. N. , Chaudhry Muhammad Aslam
Published: December 2010

The theory of Fourier transforms of tempered distributions as developed by Laurent Schwartz [17] is quite simple and elegant and has wide variety of applications, but there does not exist a corresponding neat and simple theory for the Hilbert transform of generalized functions (distributions) having wide applications. One of the objectives of this paper is to develop such a theory for the Hilbert transform of generalized functions and indicate its applicability to a variety of problems.

In problems of physics sometimes we need to find harmonic functions u(x, y) in the region y > 0 whose limit as y → 0+ does not exist in pointwise sense but does exist in the distributional sense. The theory of Hilbert transform of generalized functions that we are going to develop will provide an answer to the existence and uniqueness of this problem.
The Metric Fuglede Property and Normality
Authors: Moore R. L. , Weiss G.
Published: December 2010

In [4], H. Kamowitz considered the condition, to be satisfied by a bounded operator N on a Hilbert space , that

for all operators X on . Kamowitz discovered that such an N must be normal and its spectrum must lie on a line or a circle; that is, N must be of the form αJ + β, where α and β are complex numbers and J is either Hermitian or unitary. G. Weiss [5] showed that the Hilbert-Schmidt norm behaves differently: N need only be normal in order that

for all finite-rank operators X, and in fact this condition is equivalent to normality. Actually, the result in [5] removes the restriction that X be finite-rank, that is, if N is normal and X is any bounded operator, then
Uniform Approximation by Polynomials with Variable Exponents
Authors: Borwein Peter B.
Published: December 2010

We examine questions related to approximating functions by sums of the form

We focus on approximations to functions given by the integral transformation

where γ is a positive measure. Approximations to this class of functions (Laplace transforms in the variable — lnx) are particularly well behaved (see Theorem 1). Questions concerning existence, uniqueness and characterization of such approximations have been thoroughly examined in the equivalent setting of exponential sum approximations (see [3], [4], [6] and [9]). Less well studied is the order of convergence of the approximation. This is the problem we address. Part of the motivation for using sums of the form (1), which we shall call Gaussian sums, stems from the observation that all analytic functions with Taylor series expansion having positive coefficients are of the form (2).
On BP<1>*(K(Z, 3); Z/p)
Authors: Ucci Jack
Published: December 2010

In this paper we study the inverse limit cohomology h*(K(Z, 3)) of an Eilenberg-MacLane object K(Z, 3) for certain cohomology theories h. Our main result gives a complete description of all non-trivial differentials of the Atiyah-Hirzebruch spectral sequence (AHSS) H*(X;h*(pt)) ⇒ h*(X) for X = K(Z, 3) and h either of the complex K-theories K*( ;Z/p) and K*( ;Z(p)). This is achieved inductively using the finite symmetric product spaces SPkS3, k = pr. Identification of cycles and boundaries of each non-trivial differential leads to an explicit description of BP<1>*(K(Z, 3); Z/p) and some information about BP<1>*(K(Z, 3)).
Normal, Locally Compact, Boundedly Metacompact Spaces are Paracompact: an Application of Pixley-Roy Spaces
Authors: Daniels Peg
Published: December 2010

Let PR(X) denote the Pixley-Roy topology on the collection of all nonempty, finite subsets of a space X. For each cardinal κ, let κ* be the cardinal κ with the co-finite topology. We use PR(κ*) to obtain a partial solution in ZFC to F. Tall's question whether every normal, locally compact, metacompact space is paracompact [6]. W.S. Watson has answered this question affirmatively assuming V = L[7]. The question also has an affirmative answer if we assume either that the space is perfectly normal [1] or that it is locally connected [4].

A space X is said to be boundedly metacompact (boundedly paracompact) provided that for each open cover of X there is a positive integer n such that has a point finite (locally finite) open refinement of order n. As the main result of this paper, we show every normal, locally compact, boundedly metacompact space is paracompact.
Two Boolean Algebras with Extreme Cellular and Compactness Properties
Authors: Bell Murray
Published: December 2010

In this paper, we construct two kinds of Boolean algebras with extreme cellular properties and nice embedding properties. The extreme cellular properties are a σ — j-linked but not σ — j + 1-linked and ccc but not σ — 2-linked. The nice embedding properties are that they are ZF-definable subalgebras of both P/F and R (see Preliminaries for notation). It is the author's opinion that R contains much of the “ZF-strength” of P/F.
Concerning the Cone = Hyperspace Property
Authors: Sherling Dorothy D.
Published: December 2010

In this paper it is shown that a sufficient condition for a continuum X to have the cone = hyperspace property is that there exists a selection for C(X)\{X} which, for some Whitney map for C(X), maps each nondegenerate Whitney level homeomorphically onto X. Also, we construct an example of a one-dimensional, nonchainable, noncircle-like continuum which has the cone = hyperspace property. The continuum is described by means of inverse limits using only one bonding map. Each factor space in the inverse limit sequence is the quotient space resulting from an upper semi-continuous decomposition of a disjoint union of simple triods. The bonding map is an adaptation of the bonding map defined by W. T. Ingram in his construction of an atriodic, tree-like continuum which is not chainable [4].

Definitions, notation, and terminology. By continuum we mean a nonempty, compact, connected metric space.
On the Resolution Diagrams of the Brieskorn Singularities (2,q,r) of Type II
Authors: Yamada Akio , Matsumoto Yukio
Published: December 2010

Let p, q, r be pairwise coprime integers with 2 ≦ p < q < r. The equation defines a complex hypersurface V(p, q, r) ⊂ C3 which has an isolated singular point at the origin. We call the singularity the Brieskorn singularity (p, q, r). An algorithm of resolving this singularity is known [1]. According to the algorithm, the resolution diagram which describes the configuration of the pre-image of the singular point in the resolved surface is a star-shaped tree Γp, q, r with three branches:

1.1

The positive integers (weights) ai, bj, ck are given as follows: Let x, y, z, b be integers satisfying

1.2
Projective Approximations
Authors: Varadarajan K.
Published: December 2010

Let R be an associative ring with 1 ≠ 0. Throughout we will be considering unitary left R-modules. Given a chain complex C over R, a free approximation of C is defined to be a free chain complex F over R together with an epimorphism τ:F → C of chain complexes with the property that H(τ):H(F) ≃ H(C). In Chapter 5, Section 2 of [3] it is proved that any chain complex C over Z has a free approximation τ:F → C. Moreover given a free approximation τ:F → C of C and any chain map f:F’ → C with F’ a free chain complex over Z, there exists a chain map φ:F’→ F with T O φ = f . Any two chain maps φ, ψ of F’ in F with T O φ = T O ψ are chain homotopic.
Endomorphism Rings and Gabriel Topologies
Authors: Khuri Soumaya Makdissi
Published: December 2010

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).
Distance-Genericity for Real Algebraic Hypersurfaces
Authors: Bruce J. W. , Gibson C. G.
Published: December 2010

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.

Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.
Supplements of Hölder's Inequality
Authors: Barnes David C.
Published: December 2010

Given vectors and (or functions f(x) and g(x)) we define the Hölder Quotient Hpq by

1

or in case of functions by

2

Here ‖·‖p and ‖·‖q are the usual Lp and Lq norms. We assume throughout that

If p and q are both greater than one then they are positive but if we allow p and q to be less than one then one of them must be positive and the other one must be negative. This may cause a problem if for example, some value ai is zero and p is negative. In this case we use the convention that and
On Vertical Order of One-Dimensional Compacta in E3
Authors: Tinsley Fred , Wright David G.
Published: December 2010

Let X be a compactum in En of dimension at most n – 2. In [9, Theorem 4.1] it was shown that there is an arbitrarily small homeomorphism h of En fixed outside any given neighborhood of X, so that h(X) has vertical order n – 1 provided n ≠ 3. If X is a 0-dimensional set or a tame 1-dimensional set in E3 then the result is still true. However, the examples of tangled continua of Bothe [2] and McMillan and Row [7] are not amenable to the techniques used in dimensions other than three. This prompted Wright [9] to make the following conjecture.
Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth
Authors: Bárány Imre , Füredi Zoltán , Pach János
Published: December 2010

A system of openly disjoint discs in the plane
is said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, then

where r(C) denotes the radius of circle C, inf and sup are taken over all C ∊
Orthogonal Polynomials with Symmetry of Order Three
Authors: Dunkl Charles F.
Published: December 2010

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measure

corresponds to the measure

on the triangle

(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.
On Unbounded Volterra Operators
Authors: Hughes Rhonda J.
Published: December 2010

Similarity invariants of certain families of Volterra operators acting in Lp[0, 1] were first determined by Kalisch in [4] and [5]. Subsequently, Kantorovitz made an extensive study of perturbations of the form Tα = M + αN, where M is the operation of multiplication by x, and N is the Volterra operator

both acting in Lp[0, 1], and determined that Re α is a similarity invariant for the class of Tα's (cf. [7], and earlier work cited there). This result was eventually generalized to unbounded M (cf. [6]), and one direction was proved in [2] for both M and N unbounded: if Re α = Re β, then Tα is similar to Tβ. Properties of the Riemann-Liouville semigroup of fractional integration play a key role, for in all cases the ensuing similarity turns out to be implemented by N(i(lm(α — β))), where {N(iγ)} is the boundary group of the semigroup of fractional integration.
Modularity in the Lattice of Projections of a von Neumann Algebra
Authors: Bures Donald
Published: December 2010

We say that two elements e and f of a lattice are moderately separated provided e ∧ f = 0 and both (e′, f′) and (f′, e′) are modular pairs for all e′ ≦ e and f′ ≦ f. Here (e′, f′) a modular pair means that, for all g ≧ e′,

In the lattice of projections of a factor we show that e and f, with e ∧ f = 0, are modularly separated if and only if ‖(e – k)f‖ < 1 for some finite projection k ≦ e. From there we can show that a kind of “independence property” holds for modular separation in this case: if e and f are modularly separated and if e ∨ f and g are modularly separated, then e and f ∧ g are modularly separated.
On Torsion-Free Groups of Finite Rank
Authors: Meier David , Rhemtulla Akbar
Published: December 2010

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the set

We say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.
More on Compact Hausdorff Spaces and Finitary Duality
Authors: Banaschewski B.
Published: December 2010

It is an old conjecture by P. Bankston that the category CompHaus of compact Hausdorff spaces and their continuous maps is not dually equivalent to any elementary P-class of finitary algebras (taken as a category with all homomorphisms between its members as maps), where elementary means defined by first order axioms, and a P-class is one closed under arbitrary (cartesian) products. One motivation for this conjecture is the fact that such a dual equivalence would make ultracopowers of compact Hausdorff spaces correspond to ultrapowers of finitary algebras, and one might expect this to have contradictory consequences.

As a possible step towards proving his conjecture, Bankston [2] showed that no elementary SP-class of finitary algebras can be dually equivalent to CompHaus. However, it was subsequently proved in [1] that the same holds for any SP-class of finitary algebras, using an argument independent of ultrapowers.
On basic Cycles of An, Bn, Cn and Dn
Authors: Britten D. J. , Lemire F. W.
Published: December 2010

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.

First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.
Collineations of Polar Spaces
Authors: James Donald G.
Published: December 2010

The fundamental theorem of projective geometry describes the bijective collineations between two projective spaces PV and PV′ of finite dimension (greater than one) over division rings k and k′ in terms of an isomorphism φ:k → k′ and a φ-semilinear bijective mapping between the underlying vector spaces V and V′. Tits [9, Theorem 8.611] has given an extensive generalization of this theorem to embeddable polar spaces induced by polarities coming from either (σ, )-hermitian forms or from (σ, )-quadratic forms with Witt indices at least two. In another direction, Klingenberg [7] and later André [1] and Rado [8], have generalized the fundamental theorem by considering non-injective collineations. Now the isomorphism φ must be replaced by a place φ:k → k′ ∪ ∞ and an integral structure over the valuation ring A = φminus1(k′) is induced into the projective space PV.
A Technique to Generate -Ary Free Lattices from Finitary Ones
Authors: Grätzer George , Kelly David
Published: December 2010

Letbe an infinite regular cardinal. A posetLis called an-lattice if and only if for allXLsatisfying 0 < |X| <m, ∧Xand ∨Xexist.

This paper is a part of a sequence of papers,[5], [6], [7], [8], developing the theory of-lattices. For a survey of some of these results, see [9].

The-latticeis described in [6]; γ denotes the zero and γ′ the unit of. In particular, formulas for-joins and meets are given. (We repeat the essentials of this description in Section 4.)

In [6] we proved the theorem stated below. Our proof was based on characterization of(the free-lattice onP) due to [1]; as a result, our proof was very computational.
The Method of Upper and Lower Solutions for Some Nonlinear Boundary Value Problems in Unbounded Domains
Authors: Các Nguyên Phuong
Published: December 2010

Let D be a bounded domain in the Euclidean space RN(N ≦ 2) and let where is the closure of D. We assume that the boundary ∂G of G is smooth. Consider the boundary value problem (abbreviated to BVP in the sequel).

(1)

(1)

where A is a nonlinear elliptic differential operator in divergence form of Leray-Lions type, ∇u = grad u, f is a distribution on G and p(x, t, η) is a function defined on G × R × RN. Among other hypotheses we shall, roughly speaking, assume that p has completely unrelated growth rates in the first and the third variables. In this paper we prove the solvability of the BVP (1), (2) under the assumption that it has both an upper solution ψ and a lower solution φ with φ ≧ ψ.
Beyond the Enveloping Algebra of sl3
Authors: Flath Daniel E. , Biedenharn L. C.
Published: December 2010

The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl3-modules V and W. For sl2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl3.

For each dominant integral weight α let Vα be an irreducible representation of sl3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ in Hom(Vα, Vα+μ) is bounded by the multiplicity of μ in Vλ, with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ) which is valid for all a.
Application of a Selection Theorem to Hyperspace Contractibility
Authors: Curtis D. W.
Published: December 2010

For X a metric continuum, 2X denotes the hyper space of all nonempty subcompacta, with the topology induced by the Hausdorff metric H, and C(X) ⊂ 2X the hyperspace of subcontinua. These hyperspaces are continua, in fact are arcwise-connected, since there exist order arcs between each hyperspace element and the element X. They also have trivial shape, i.e., maps of the hyperspaces into ANRs are homotopic to constant maps. For a detailed discussion of these and other general hyperspace properties, we refer the reader to Nadler's monograph [4].

The question of hyperspace contractibility was first considered by Wojdyslawski [8], who showed that 2X and C(X) are contractible if X is locally connected. Kelley [2] gave a more general condition (now called property K) which is sufficient, but not necessary, for hyperspace contractibility. The continuum X has property K if for every there exists δ > 0 such that, for every pair of points x, y with d(x, y) < δ and every subcontinuum M containing x, there exists a subcontinuum N containing y with .
Approximately Periodic Functionals on C*-Algebras and von Neumann Algebras
Authors: Quigg John C.
Published: December 2010

In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.
Almost all Convex Polyhedra are Asymmetric
Authors: Bender Edward A. , Wormald Nicholas C.
Published: December 2010

Although many types of rooted planar maps have been enumerated (see [8] for example), not much has been done on enumeration of entirely unrooted planar maps. Yet in virtually all cases of interest, it has appeared that comparatively very few of the maps are symmetric (have non-trivial automorphisms). This suggests that an asymptotic formula for the numbers of unrooted maps of a particular type on n edges can be obtained by dividing the numbers of rooted maps of that type on n edges by 4n, where 4n is the number of potentially distinct rootings of an asymmetric n-edged map. The assertion that almost all maps of a given type are asymmetric has previously been proved in only two non-trivial cases: for 3-connected planar triangulations by Tutte [9] and for all n-edged 3-connected planar maps in [5]. We prove here that it is also true for 3-connected planar maps with a given number of vertices and faces, uniformly as either parameter approaches infinity.
Projections on Tree-Like banach Spaces
Authors: Andrew A. D.
Published: December 2010

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.

In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.
Characterizations for Prime Semilattices
Authors: Shum K. P. , Chan M. W. , Lai C. K. , So K. Y.
Published: December 2010

Throughout this paper when we refer to a semilattice S we shall mean that S is a meet semilattice. We shall denote the infimum of two elements a, b of S by a ∧ b, and the supremum, if it exists, by a ∨ b. A prime semilattice is a meet semilattice such that the infimum distributes over all existing finite suprema, in the sense that if x1 ∨ x2 … ∨ xn exists then (x ∧ x1) ∨ (x ∧ x2) … ∨ (x ∧ xn) exists for any x and equals x ∧ (x1 ∨ x2 … ∨ xn). Such semilattices were first studied by Balbes [1]
and we use his terminology.

A non-empty subset F of S is a filter provided that x ∧ y ∊ F if and only if x ∊ F and y ∊ F.
Directed Graphs and the Jacobi-Trudi Identity
Authors: Goulden I. P.
Published: December 2010

Let |aij|n×n denote the n × n determinant with (i, j)-entry aij, and hk = hk(x1, …, xn) denote the kth-homogeneous symmetric function of x1, …, xn defined by

where the summation is over all m1, …, mn ≧ 0 such that m1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α1 ≧ α2 … ≧ αn ≧ 0, the Jacobi-Trudi identity (see [6], [7]) states that

In this paper we give a combinatorial proof of an equivalent identity, Theorem 1.1, obtained by moving the denominator on the RHS to the numerator on the LHS.
Graded Complexes Over Power Series Rings
Authors: Roberts Paul
Published: December 2010

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.
Extending Algebras to Model Congruence Schemes
Authors: Berman J. , Grätzer G. , Platt C. R.
Published: December 2010

This paper is concerned with the description of principal congruence relations. Given elementsaandbof a universal algebra, let θ(a, b) denote the smallest congruence relation oncontaining the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says thatc≡d(θ(a, b))if and only if there exists a sequencez0,z1, …,znof elements ofand a sequencef1,f2, …,fnof unary algebraic functions such thatc=z0,d=zn, and for eachi= 1, …,n,

Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice ofa, b, candd. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.
Solution of a Problem of L. Fuchs Concerning Finite Intersections of Pure Subgroups
Authors: Göbel R. , Vergohsen R.
Published: December 2010

L. Fuchs states in his book “Infinite Abelian Groups” [6, Vol. I, p. 134] the following

Problem 13. Find conditions on a subgroup of A to be the intersection of a finite number of pure (p-pure) subgroups of A.

The answer to this problem will be given as a special case of our theorem below. In order to find a better setting of this problem recall that a subgroup S ⊆ E is p-pure if pnE ∩ S = pnS for all natural numbers. Then S is pure in E if S is p-pure for all primes p. This generalizes to pσ-isotype, a definition due to L. J. Kulikov, cf. [6, Vol. II, p. 75] and [11, pp. 61, 62]. If α is an ordinal, then S is pσ-isotype if
Generators of Chevalley Groups over Z
Authors: Chang Bomshik
Published: December 2010

Let be the universal Chevalley group ([1], p. 197) of type over a field K and where Ф is the set of roots of . Let n = {α1, α2, …, an} be a fundamental system of roots of and put

Then we know from [2] (p. 950) that

where h is the Coxeter number of We call an element of conjugate to W a Coxeter element and an element conjugate to M a Kac element.
Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
Authors: Borwein J. M. , Strojwas H. M.
Published: December 2010

As various types of tangent cones, generalized derivatives and subgradients prove to be a useful tool in nonsmooth optimization and nonsmooth analysis, we witness a considerable interest in analysis of their properties, relations and applications.

Recently, Treiman [18] proved that the Clarke tangent cone at a point to a closed subset of a Banach space contains the limit inferior of the contingent cones to the set at neighbouring points. We provide a considerable strengthening of this result for reflexive spaces. Exploring the analogous inclusion in which the contingent cones are replaced by pseudocontingent cones we have observed that it does not hold any longer in a general Banach space, however it does in reflexive spaces. Among the several basic relations we have discovered is the following one: the Clarke tangent cone at a point to a closed subset of a reflexive Banach space is equal to the limit inferior of the weak (pseudo) contingent cones to the set at neighbouring points.
The Geometry of GF(q3)
Authors: Sherk F. A.
Published: December 2010

1. Introduction. Inversive geometry involves as basic entities points and circles [2, p. 83; 4, p. 252]. The best known examples of inversive planes (the Miquelian planes) are constructed from a field K which is a quadratic extension of some other field F. Thus the complex numbers yield the Real Inversive Plane, while the Galois field GF(q2)(q = pe, p prime) yields the Miquelian inversive plane M(q) [2, chapter 9; 4, p. 257]. The purpose of this paper is to describe an analogous geometry of M(q) which derives from GF(q3), the cubic extension of GF(q).

The resulting space, is three-dimensional, involving a class of surfaces which include planes, some quadric surfaces, and some cubic surfaces. We explore these surfaces, giving particular attention to the number of points they contain, and their intersections with lines and planes of the space .
Equations de Cauchy-Riemann Sur SU(2) et Leurs Enveloppes D'Holomorphie
Authors: Debiard Amédée , Gaveau Bernard
Published: December 2010

Le problème de la construction d'enveloppes d'holomorphie globales ou d'enveloppes polynomialement convexes n'a été que très peu abordé et peu d'exemples explicites sont connus. Il semble donc qu'il soit intéressant de se limiter d'abord au calcul d'enveloppes d'ensembles homogènes, par l'action d'un groupe agissant par transformations holomorphes. C'est le cas des sphères unité de Cn, des polytores par exemple. En s'inspirant des actions de groupes discutés par Ehrenpreis [2] ou par Hua [3], nous considérons ici l'action des groupes SU(2) et U(2) sur C3 identifié à l'ensemble des matrices symétriques complexes 2 × 2. L'action de SU(2) sur C3 est en fait l'action canonique de SU(2) sur une famille de quadriques dégénérant vers la quadrique singulière lorsque A tends vers 0. L'étude des orbites est équivalente à l'étude des équations de Cauchy Riemann sur SU(2), invariantes à gauche. Le spectre de l'algèbre des fonctions annulant une telle équation sur SU(2), qui sont invariantes par l'application antipodale est alors un domaine d'holomorphie de l'une de ces quadriques qui est d'ailleurs topologiquement non trivial.
A Voronovskaya Theorem for Variation-Diminishing Spline Approximation
Authors: Marsden M. J.
Published: December 2010

In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.

Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by

(1.1)

where

and the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.
Spaces on which every Continuous Map into a given Space is Constant
Authors: Iliadis S. , Tzannes V.
Published: December 2010

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.

Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].
Theoremes de Bornage Pour L'Operateur de Nemyckii Dans Les Espaces Ideaux
Authors: Appell Jürgen , de Pascale Espedito
Published: December 2010

Soit Ω un domaine borné de RN, et soit f:Ω × R → R une fonction satisfaisant à la condition de Carathéodory (i.e., f(s, ·) est continue pour presque tout s ∊ Ω, et f (·, u) est mesurable pour tout u ∊ R). Considérons l'opérateur de la superposition

(1.1)

(encore appelé opérateur de Nemyckii), engendré par la fonction f. Cet opérateur joue un grand rôle dans la théorie des équations intégrales, différentielles (ordinaires et aux dérivées partielles), et fonctionnelles-différentielles, où il est important de connaître les propriétés analytiques et topologiques de F dans certains espaces de fonctions mesurables, intégrables, continues, différentiables, analytiques etc., les propriétés les plus importantes étant : théorèmes de transfert, de continuité, de bornage, et de compacité. Par exemple, on connaît de nombreux résultats sur l'opérateur (1) dans les espaces de Lebesgue L (voir [10] pour une présentation assez complète); en effet, si l'opérateur (1) envoie une partie de L , d'intérieur non vide, dans L, alors, il est automatiquement continu et borné sur chaque boule.
Frobenius Induction for Higher Whitehead Groups
Authors: Nicas Andrew J.
Published: December 2010

The theory of induced representations has served as a powerful tool in the computations of algebraicK-theory andL-theory ([2], [7], [4, 5], [9], [10, 11, 12, 13], [14], [17], [18]). In this paper we show how to apply this theory to obtain induction theorems for the higher Whitehead groups of Waldhausen. The same technique applies to the analogs of Whitehead groups in unitaryK-theory and inL-theory.

For any ringAwith unit, letK(A) be the spectrum of the algebraicK-theory ofA([8, p. 343]). Given a discrete group Γ and a subringRof the rational numbers, Loday defines a map of spectra:

*

where (BΓ) is the classifying space of Γ union with a disjoint base point andRΓ is the group-ring of Γ overR.