bluuebunny/crossref_metadata_2025_split huggingface dataset:
Tensor Products of Holomorphic Discrete Series Representations
Authors: Repka Joe 
Published: December 2010

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.
A Fundamental Solution for a Nonelliptic Partial Differential Operator
Authors: Greiner Peter C. 
Published: December 2010

Let(1)and set(2)Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”(3)In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].
On the Nonstandard Duality Theory of Locally Convex Spaces
Authors: Grainger Arthur D. 
Published: December 2010

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).
Completely Regular Mappings and Homogeneous, Aposyndetic Continua
Authors: Rogers James T. 
Published: December 2010

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.
A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials
Authors: Rahman Mizan 
Published: December 2010

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)
On the Size of a Maximum Transversal in a Steiner Triple System
Authors: Brouwer A. E. 
Published: December 2010

Let (X, ) be a Steiner triple system on v = |X| points, and suppose that is a partial parallel class (transversal, clear set, set of pairwise disjoint blocks) of maximum size . We want to derive a bound on . (I conjecture that in fact r is bounded, e.g., r ≦ 4 – 4 is attained for the Fano plane, but all that has been proved so far (cf. [1], [2]) are bounds r < C.v for some C. Here we shall prove r < 5v2/3.)Define a sequence of positive real numbers by q0 = Q · r2/v, , where l is determined by ql ≧ 6, , i.e.,(The constant Q will be chosen later.) Define inductively sets Ai, Ki and collections as follows. Let
Recursive Colorings of Highly Recursive Graphs
Authors: Kierstead Henry A. 
Published: December 2010

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].
Real Flexible Division Algebras
Authors: Benkart Georgia M. , Britten Daniel J. , Osborn J. Marshall 
Published: December 2010

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.
The Associated Ultraspherical Polynomials and their q-Analogues
Authors: Bustoz Joaquin , Ismail Mourad E. H. 
Published: December 2010

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that1.1A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence1.2with1.3Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by1.4with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.
Elements of Order Coxeter Number +1 in Chevalley Groups
Authors: Chang Bomshik 
Published: December 2010

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.
Class Numbers and Biquadratic Reciprocity
Authors: Williams Kenneth S. , Currie James D. 
Published: December 2010

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,(0.1)with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by(0.2)Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by(0.3)These choices are assumed throughout.The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].
Trivial Set-Stabilizers in Finite Permutation Groups
Authors: Gluck David 
Published: December 2010

For which permutation groups does there exist a subset of the permuted set whose stabilizer in the group is trivial?The permuted set has so many subsets that one might expect that subsets with trivial stabilizer usually exist. The symmetric and alternating groups are obvious exceptions to this expectation. Another, more interesting, infinite family of exceptions are the 2-Sylow subgroups of the symmetric groups on 2n symbols, in their natural representations on 2n points.One of our main results, Corollary 1, sheds some light on this last family of groups. We show that when the permutation group has odd order, there is indeed a subset of the permuted set whose stabilizer in the group is trivial. Corollary 1 follows easily from Theorem 1, which completely classifies the primitive solvable permutation groups in which every subset of the permuted set has non-trivial stabilizer.
The Enumeration of Non-Isomorphic 2-Connected Planar Maps
Authors: Liskovets V. A. , Walsh T. R. S. 
Published: December 2010

Since Tutte initiated the systematic enumeration of planar maps [11], most of the literature on the subject has dealt with rooted maps (i.e., with maps whose automophism group has been trivialized by distinguishing a doubly-oriented edge). In particular, Tutte proved [11] that the number B′(n) of rooted planar 2-connected (i.e., non-separable) maps with n ≧ 1 edges is expressed by the formulaRecently one of the authors developed a general technique for enumerating unrooted planar maps considered up to orientationpreserving isomorphisms (see [6] and [8]). This technique, which is based on combinatorial map theory, Burnside’s lemma [3, p. 181] and the concept of a quotient map (see Section 1.4), was used to find, with little algebraic manipulation, simple counting formulae for the numbers of non-isomorphic planar maps of several types [7].
Merotopic Spaces and Extensions of Closure Spaces
Authors: Chattopadhyay K. C. , Njåstad Olav , Thron W. J. 
Published: December 2010

Proximity spaces and contiguity spaces, and more recently nearness spaces, have been studied not just because they provide various approaches to uniform structure. Possibly of greater importance is that they can be used as a means of introducing compactifications and more general extensions of the topological spaces on which they are defined. Riesz [20] was probably the first to recognize this connection. Since then the idea was used by Freudenthal [9], Alexandroff [1], Smirnov [21], Leader [17] and Ivanov and Ivanova [13, 14, 15] among others.Recently Reed [19] using work of Bentley [2, 4] and Herrlich [11, 12] studied the 1 – 1 correspondence between the class of all cluster generated nearness spaces and the class all principal T1-extensions of a given T1-space. She succeeded in showing that the mapping induces a 1 – 1 correspondence between the contingual nearness spaces in and the compactifications in .
A Generalization of an Addition Theorem for Solvable Groups
Authors: Yuster Thomas , Peterson Bruce 
Published: December 2010

The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.
Injectivity in the Topos of Complete Heyting Algebra Valued Sets
Authors: Higgs Denis 
Published: December 2010

Let be a complete Heyting algebra (CHA). An -valued set is a pair (X, δ) where X is a set and δ is a function from X × X to such thatfor all x, y z in X. -valued sets form a category as follows: a morphism from (X, δ) to (Y, δ) is a function f from X × Y to such that(i)f(x) ∧ δ(x, x′) ≦ f(x′, y), f(x, y) ∧ δ(y, y′) ≧ f(x, y′),(ii)f(x, y) ∧ f(x, y′) ≦ δ(y, y′), and(iii)
On the Difference Property of the Class of Pointwise Discontinuous Functions and of Some Related Classes
Authors: Laczkovich M. 
Published: December 2010

Let R denote the set of real numbers. For f:R → R and h ∈ R, the difference function Δhf is defined byThe function H:R → R is called additive if it satisfies Cauchy's equationLet ℱ be a class of real valued functions defined on R. ℱ is said to have the difference property if, for every function f:R → R satisfying Δhf ∈ ℱ for every h ∈ R, there exists an additive function H such that f — H ∈ ℱIt was conjectured by P. Erdos that the class of continuous functions has the difference property. This conjecture was proved by N. G. de Bruijn in [1], where the difference property of several other classes was verified as well. (For other references, see [6].)
Modular and Admissible Semilattices
Authors: Hoo C. S. , Murty P. V. Ramana 
Published: December 2010

We correct some errors in [1] and extend some of the results there. Generally, we shall follow the terminology and notation of [1]. There is an error in the proof of Lemma 3.13 there, and consequently the subsequent results which depend on it are incorrect as stated. However, they are correct if we replace the condition “a-admissible” by “strongly a-admissible” (see [3] where this notion was introduced). We also show that the results in [1] are correct if the semilattices are assumed to be modular.We shall change the terminology in [3] slightly.Definition 1 (see [3]). Let A be a Boolean algebra and let D be a meet semilattice with 1. An admissible map f.A X D → D is called strongly admissible ifwhere a’ is the complement of a in A.
Espace des États Normaux Dun Facteur de Type IIIλ 0 < λ < 1 et D'Un Facteur de Type III0
Authors: Bion-Nadal Jocelyne 
Published: December 2010

Soit M un facteur de type IIIλ 0 ≦ λ< 1, agissant sur un espace hilbertien à base dénombrable. Le propos de ce travail est l'étude des états normaux d'un tel facteur à équivalence unitaire près. L'espace quotient n'est pas séparé; l'espace séparé associé est obtenu comme quotient de l'espace des états normaux par la relation d'équivalence R dont les classes sont les fermetures des orbites précédentes.Pour étudier cet espace, on définit un calcul fonctionnel sur M+* (espace des formes linéaires positives normales sur M) à valeurs dans M. Ce calcul fonctionnel est déterminé par la donnée d'une C* algèbre d'applications de M+ * dans M “continues” en un certain sens, notée C*(M).
Universal Pettis Integrability
Authors: Andrews Kevin T. 
Published: December 2010

Since the invention of the Pettis integral over forty years ago [11], the problem of recognizing the Pettis integrability of a function against an individual measure has been much studied [5, 6, 7, 8, 9, 20]. More recently, Riddle-Saab-Uhl [14] and Riddle-Saab [13] have considered the problem of when a function is integrable against every Radon measure on a fixed compact Hausdorff space. These papers give various sufficient conditions on a function that ensure this universal Pettis integrability. In this paper, we see how far these various conditions go toward characterizing universal Pettis integrability. We base our work on a w*-analogue of the core of a vector-valued function [8].We also give some sufficient conditions that ensure that a Banach space has the so-called universal Pettis integral property (UPIP) and consider some particular examples of spaces with this property. It is interesting that in these examples some of the special set theoretic axioms that play an important role in the study of the stronger Pettis integral property [6, 7] make an appearance.
Criteria for Biquadratic Residuacity Modulo a Prime p involving Quaternary Representations of p
Authors: Williams Kenneth S. , Friesen Christian , Howe Lawrence J. 
Published: December 2010

In 1958, Hasse [10, p. 236], in connection with his work on the 2n-th power character of 2 in the cyclotomic field Q(exp(2πi/2n)), proved that for every prime p ≡ 1 (mod 16) the pair of equationsis always solvable in integers x, u, v, w. Later in 1972 Giudici, Muskat, and Robinson [7, p. 388] showed in their work on Brewer's character sums that Hasse's system is also solvable for primes p ≡ 7 (mod 16). Moreover they also showed [7, p. 345] that for primes p ≡ 1 (mod 5) the pair of equationsis solvable in integers x, u, v, w. In this paper we consider a pair of diophantine equations (involving a prime p and an integer m) which includes, the above two systems as the special cases when m = 2 and m = 5. The system is then used to give criteria for m to be a biquadratic residue modulo p.
Opérations En K-Théorie Algébrique
Authors: Soulé Christophe 
Published: December 2010

C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K0(X)), construit à l'aide des -modules localement libres sur un schéma X[14]. La somme directe et le produit tensoriel de modules font de K0(X) un anneau, et les opérations de puissances extérieures lui fournissent une structure supplémentaire, que Grothendieck appelle λ-anneau. Un λ-anneau est muni d'une filtration décroissante, la γ-filtration, et un des principaux résultats de Grothendieck est que, si X est lisse sur un corps, le groupeest isomorphe, à la torsion près, au groupe de Chow CH1(X) des cycles de codimension i sur X, modulo l'équivalence linéaire.
A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds
Authors: Frenzen C. L. , Wong R. 
Published: December 2010

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:(1.1)where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).
Banach Envelopes of Non-Locally Convex Spaces
Authors: Kalton N. J. 
Published: December 2010

Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .
Antichains and Finite Sets that Meet all Maximal Chains
Authors: Ginsburg J. , Rival I. , Sands B. 
Published: December 2010

This paper is inspired by two apparently different ideas. Let P be an ordered set and let M(P) stand for the set of all of its maximal chains. The collection of all sets of the formandwhere x ∊ P, is a subbase for the open sets of a topology on M(P). (Actually, it is easy to check that the B(x) sets themselves form a subbase.) In other words, as M(P) is a subset of the power set 2|p| of P, we can regard M(P) as a subspace of 2|p| with the usual product topology. M. Bell and J. Ginsburg [1] have shown that the topological space M(P) is compact if and only if, for each x ∊ P, there is a finite subset C(x) of P all of whose elements are noncomparable to x and such that {x} ∪ C(x) meets each maximal chain.
Paracompactness in Locally Lindelöf Spaces
Authors: Balogh Zoltán 
Published: December 2010

This paper contains a set of results concerning paracompactness of locally nice spaces which can be proved by (variations on) the technique of “stationary sets and chaining” combined with other techniques available at the present stage of knowledge in the field. The material covered by the paper is arranged in three sections, each containing, in essence, one main result.The main result of Section 1 says that a locally Lindelöf, submeta-Lindelöf ( = δθ-refinable) space is paracompact if and only if it is strongly collectionwise Hausdorff. Two consequences of this theorem, respectively, answer a question raised by Tall [7], and strengthen a result of Watson [9]. In the last two sections, connected spaces are dealt with. The main result of the second section can be best understood from one of its consequences which says that under 2ωl > 2ω, connected, locally Lindelöf, normal Moore spaces are metrizable.
Estimating the Size of Context-Free Tiling Languages
Authors: Holladay Kenneth 
Published: December 2010

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.
Higher Derivations and Distinguished Subfields
Authors: Deveney James K. , Heerema Nickolas 
Published: December 2010

Let L be a finitely generated extension of a field K having characteristic p ≠ 0. A rank t higher derivation on L over K is a sequenceof additive maps of K into K such thatd0 is the identity map and dt(x) = 0, i > 0, x ∊ K. [6] contains the relevant background material on higher derivations. By Zorn's Lemma, there are maximal separable extensions of K in L. A maximal separable extension D of K in L is called distinguished ifDieudonné [4] established that any finitely generated extension always has distinguished subfields. L has the same dimension over any distinguished subfield [5], and this dimension is called the order of inseparability of L/K. The least n such that K(LP)n is separable over K is called the inseparable exponent of L/K, inex(L/K).