bluuebunny/crossref_metadata_2025 huggingface dataset:
Some Remarks on Eisenstein Series for Metaplectic Coverings
Authors: Morris Lawrence
Published: December 2010

In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL2(A) play a crucial role.

The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L2-space of functions; this is done in Section 2 of this paper.
Diagonal Equations Over Large Finite Fields
Authors: Small Charles
Published: December 2010

We consider polynomials of the form

with non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.

We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.

It should be noted that in studying diagonal equations

we fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.
Inflectional Convex Space Curves
Authors: Bisztriczky Tibor
Published: December 2010

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.

1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.

2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.

b) Let Φ possess exactly n vertices. Then

(1)

|O(p)| ≦ nforp ∈ H(S) and

(2)

if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.
Decompositions of Submeasures
Authors: Brook Cecilia H.
Published: December 2010

In [4] we showed that one can tell whether a submeasure on a Boolean algebra has a control measure or is pathological by comparing the Fréchet-Nikodym topology it generates to the universal measure topology of Graves. We then wondered if a submeasure could be decomposed into a part with a control measure and a part which is pathological or zero. This led to the problem of finding a Lebesgue decomposition for a submeasure on an algebra of sets with respect to a Fréchet-Nikodym topology.

In [6] Drewnowski proved a Lebesgue decomposition theorem for exhaustive submeasures with respect to “additivities” and a similar theorem for exhaustive Fréchet-Nikodym topologies. He asked if an exhaustive Fréchet-Nikodym topology could be decomposed with respect to another Fréchet-Nikodym topology. In [12] Traynor showed that the answer is “yes”.
On Translation Planes which Admit Solvable Autotopism Groups Having a Large Slope Orbit
Authors: Jha Vikram
Published: December 2010

Our main object is to prove the following result.

THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.

Then the order of A is q2.

An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.
A Unified view of (Complete) Regularity and Certain Variants of (Complete) Regularity
Authors: Kohli J. K.
Published: December 2010

Regularity and complete regularity are important topological properties and several generalizations of them occur in the literature on separation axioms. The properties of certain of these variants of (complete) regularity are similar to those of (complete) regularity and their theories run, either in part or in the whole, parallel to the theory of (complete) regularity. All the more, analogies inherent in their definitions as well as the nature of results obtained in the process of their study suggest the need of formulating a coherent unified theory encompassing the theory of (complete) regularity and its generalizations. An attempt leading towards the fulfillment of this need constitutes the theme of the present paper.

Section 2 is devoted to preliminaries and basic definitions. In Section 3 we devise a framework which leads to the formulation of a unified theorv of (complete) regularity, almost (complete) regularity, ([26,], [27], [28]), (complete) s-regularity [13], (functionally) Hausdorff spaces, R1-spaces [3], and others.
Torsion Theories Induced by Tilting Modules
Authors: Assem Ibrahim
Published: December 2010

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequence

with T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories

and

of the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.
Outer Derivations and Classical-Albert-Zassenhaus lie Algebras
Authors: Winter David J.
Published: December 2010

This paper is concerned with the structure of the derivation algebra Der L of the Lie algebra L with split Cartan subalgebra H. The Fitting decomposition

of Der L with respect to ad ad H leads to a decomposition

where

This decomposition is studied in detail in Section 2, where the centralizer of ad L∞ in D0(H) is shown to be

which is Hom(L/L2, Center L) when H is Abelian. When the root-spaces La (a nonzero) are one-dimensional, this leads to the decomposition of Der L as

where T is any maximal torus of D0(H).
Basic Commutators and Minimal Massey Products
Authors: Fenn Roger , Sjerve Denis
Published: December 2010

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.

A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.
Infinitely Periodic Knots
Authors: Flapan Erica
Published: December 2010

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.
Linear Isometries of some Normed Spaces of Analytic Functions
Authors: Novinger W. P. , Oberlin D. M.
Published: December 2010

For 1 ≦ p < ∞ let Hp denote the familiar Hardy space of analytic functions on the open unit disc D and let ‖·‖ denote the Hp norm. Let Sp denote the space of analytic functions f on D such that f′ ∊ Hp. In this paper we will describe the linear isometries of Sp into itself when Sp is equipped with either of two norms. The first norm we consider is given by

(1)

and the second by

(2)

(It is well known [1, Theorem 3.11] that f′ ∊ Hp implies continuity for f on D, the closure of D. Thus (2) actually defines a norm on Sp.) In the former case, with the norm defined by (1), we will show that an isometry of Sp induces, in a sense to be made precise in Section 2, an isometry of Hp and that Forelli's characterization [2] of the isometries of Hp can thus be used to describe the isometries of Hp.
Free Involutorial Completely Simple Semigroups
Authors: Gerhard J. A. , Petrich Mario
Published: December 2010

An involution x → x* of a semigroup S is an antiautomorphism of S of order at most 2, that is (xy)* = y*x* and x** = x for all x, y ∊ S. In such a case, S is called an involutorial semigroup if regarded as a universal algebra with the binary operation of multiplication and the unary operation *. If S is also a completely simple semigroup, regarded as an algebra with multiplication and the unary operation x → x−1 of inversion (x−1 is the inverse of x in the maximal subgroup of S containing x), then (S, −1, *), or simply S, is an involutorial completely simple semigroup. All such S form a variety determined by the identities above concerning * and

where x0 = xx−1.
On the Restricted Cesàro Summability of Multiple Orthogonal Series
Authors: Móricz Ferenc
Published: December 2010

We actually treat double orthogonal series in detail, simply for the sake of brevity in notations. Multiple orthogonal series will be shortly indicated in the concluding Section 8.

Let (X, , μ) be an arbitrary positive measure space and {ϕjk(x):i, k = 0, 1, …} an orthonormal system defined on X. We consider the double orthogonal series

(1.1)

where {ϕik:i, k = 0, 1, …} is a double sequence of real numbers (coefficients), for which

(1.2)
The largest Irreducible Character Degree of a Finite Group
Authors: Gluck David
Published: December 2010

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.

Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.
An Ergodic Theorem for Multidimensional Superadditive Processes
Authors: Çömez Doğan
Published: December 2010

The ergodic theorem for multidimensional strongly subadditive processes relative to a semigroup induced by a measure preserving point transformation on X was proved by R. T. Smythe [18]. His results have been generalized by M. A. Akçoğlu and U. Krengel [4] to the continuous parameter case. The definition of superadditivity they used is stronger than Smythe's but weaker than strong superadditivity. R. Emilion and B. Hachem [10] extended this result to strongly superadditive processes relative to a semigroup generated by a pair of commuting Markovian operators which are also L∞-contractions. The basic tool in the proof is a technique which may be referred to as “reduction of dimension“ and they used a version of it due to A. Brunei [6].

The purpose of this paper is to show that if F = {F(uv)}u>0 is a bounded strongly superadditive process with respect to a two-dimensional strongly continuous Markovian semigroup of operators on L1, then u-2F(uu) converges a.e. as u → ∞.
Extreme Positive Contractions on Finite Dimensional lp-Spaces
Authors: Grząślewicz Ryszard
Published: December 2010

In this paper we give a characterization of the extreme positive contractions on finite dimensional lp-spaces for 1 < p < ∞. This is related to the characterization of the extreme doubly stochastic operators. In Section 2 we present the basic properties of the facial structure of the set of doubly stochastic n × m matrices. In Section 3 we use these facts for description of the facial structure of the set of positive contractions on finite dimensional lp-space. Next is considered stability of the positive part of the unit ball of operators (Section 5). In Section 7 we prove that extreme positive contractions on are strongly exposed.

1. Terminology and notation. Let (X, , m) be a a-finite measure space. As usual, we denote by LP(m), 1 < p < ∞, the Banach lattice of all p-summable real-valued functions on X with standard norm and order. If X = {1, 2 , …, n) n < ∞, and m is a counting measure we write instead of LP(m). If X = [0, 1] and m is Lebesgue measure we write briefly LP.
Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups
Authors: Bami M. Lashkarizadeh
Published: December 2010

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.
The Injective Hull and the -Compactification of a Continuous Poset
Authors: Hoffmann Rudolf-E.
Published: December 2010

In [57] (2.12), D. S. Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [56], are precisely (when they are made into topological spaces via the Scott topology) the injective T0-spaces, i.e., the injective objects in the category T0 of T0-spaces and continuous maps with regard to the class of all embeddings. Moreover, the sort of morphisms between continuous lattices Scott considered are precisely the continuous maps with regard to the respective Scott topologies. These are fairly non-Hausdorff topologies. (Indeed, the Scott topology induces the partial order of the lattice L via x ≦ y if and only if x ∊ cl{j}, the “specialization order” of the topology; hence L is Hausdorff in the Scott topology if and only if L has at most one element.) In topological algebra, compact Lawson semilattices (= compact Hausdorff topological ∧-semilattices such that the ∧-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest.
On minimally Free Algebras
Authors: Bankston Paul , Schutt Richard
Published: December 2010

For us an “algebra” is a finitary “universal algebra” in the sense of G. Birkhoff [9]. We are concerned in this paper with algebras whose endomorphisms are determined by small subsets. For example, an algebra A is rigid (in the strong sense) if the only endomorphism on A is the identity idA. In this case, the empty set determines the endomorphism set E(A). We place the property of rigidity at the bottom rung of a cardinal-indexed ladder of properties as follows. Given a cardinal number κ, an algebra A is minimally free over a set of cardinality κ (κ-free for short) if there is a subset X ⊂ A of cardinality κ such that every function f:X → A extends to a unique endomorphism ϕ ∊ E(A). (It is clear that A is rigid if and only if A is 0-free.) Members of X will be called counters; and we will be interested in how badly counters can fail to generate the algebra.
Centres of Rank-Metric Completions
Authors: Handelman David
Published: December 2010

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.

Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .

If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.
Actions of Finite Groups on Rn+k with Fixed Set Rk
Authors: Hambleton Ian , Madsen Ib
Published: December 2010

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces Rn+k which are free outside a fixed point set Rk (embedded as a vector subspace). We refer to such an action as a semi-free action on (Rn+k, Rn) and note that all our actions will be assumed orientation-preserving.

Suppose the finite group π acts semi-freely on (Rn+k, Rn), then it acts freely on (Rn+k – Rn) = Sn–l × Rk+1. Since this space is homotopy equivalent to Sn–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit space

is a finitely-dominated Poincaré complex of formal dimension n – 1 with π1W = π and as considered by Swan [41].
Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces
Authors: MacCluer Barbara D. , Shapiro Joel H.
Published: December 2010

Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator Cφ defined by Cφf = f º φ for f holomorphic on U. Our goal is to determine, in terms of geometric properties of φ, when Cφ is a compact operator on the Hardy and Bergman spaces of φ. For Bergman spaces we solve the problem completely in terms of the angular derivative of φ, and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces Hp (0 < p < ∞). We are able to use these results to provide interesting new examples and to give unified explanations of some previously discovered phenomena.
The smallest Prime Value of xn + a
Authors: McCurley Kevin S.
Published: December 2010

Let a and n be positive integers such that f(x) = xn + a is irreducible over the integers. A conjecture made by Bouniakowsky [4] in 1857 would imply that there exist infinitely many integers x such that f(x) is prime. An even stronger conjecture of Bateman and Horn [1, 2] would imply that

where π(x;f) is the number of integers m with 0 ≧ m ≧ x for which f(m) is prime, and

where w(p) is the number of solutions of the congruence

Except for the trivial case n = 1, neither of these conjectures has ever been resolved.
Sur Les Operateurs a Puissances Bornees et Le Theoreme Ergodique Ponctuel Dans Lp[0, 1], 1 < p < ∞
Authors: Assani I.
Published: December 2010

Soient Lp(X, , μ) les espaces de Banach usuels associés à un espace mesuré fini ou α-fini (X, , μ), p étant un nombre réel compris entre un et l'infini (1 < p < ∞). Notons Lp, l'espace Lp[0, 1]. Un opérateur T:Lp → Lp est dit être à puissances bornées sur Lp si

La convergence presque sure de la suite de fonctions

a été étudiée dans L2 pour T contraction [8], [1] et pour T inversible à puissances bornées dans , 1 < p ≧ 2 [9].
On C*-Diagonals
Authors: Kumjian Alexander
Published: December 2010

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.

In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.
Classification of Algebraic Surfaces with
Sectional Genus less than or Equal to Six.
II: Ruled Surfaces with dim ϕKx⊗L(x) = l
Authors: Livorni Elvira Laura
Published: December 2010

Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].

If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.

Since X is ruled, h2,0(X) = 0 and

see [4] and [12, p. 390].
The Ideal Generation Conjecture for 28 Points in P3
Authors: Roberts Leslie G.
Published: December 2010

The ideal generation conjecture has recently been proved for general points in , (k a field) [1], [6]. The proof in [1] is by induction. One of the starting points for the induction (called A(5) in [1]) is proved in [6]. The theoretical proof of A(5) in [6] seems to be very difficult, apparently even more difficult than the induction. Because of this, and also because [6] is not publically available, I feel it is worth knowing that A(5) can be proved numerically with modest readily available computing facilities. In this note I discuss the computation involved, and give a few explicit examples. In the course of working out these examples I found 26 points in , that satisfy the ideal generation conjecture, but which cannot be extended to 27 or 28 points satisfying the ideal generation conjecture. This phenomenon can be interpreted combinatorially, leading to an infinite number of similar examples.
Four Integers whose Twelve Quotients Sum to Zero
Authors: Leech John
Published: December 2010

This paper is devoted to the study of sets of four unequal integers ni such that the twelve quotients ni/nj(i ≠ j) of pairs of distinct members sum to zero. (Without the restriction i ≠ j the condition would be equivalent to (Σ ni)(Σ l/ni) = 0, of no great interest.) Constructions for these sets are given, and relations between them are studied. It is found that each set belongs to an orbit of six related sets, and that each such orbit is related to four neighbours, each of which is another orbit of six sets. A study is made of the graph formed by assigning a node to each orbit of six solutions and joining it to the nodes assigned to its four neighbours. This appears to comprise one component containing an infinity of cycles together with an infinite forest of infinite trees.
A Classification of Reflexive Graphs: The use of “Holes”
Authors: Jawhari El Moustafa , Pouzet Maurice , Rival Ivan
Published: December 2010

The purpose of this article is to develop aspects of a classification theory for reflexive graphs. A first important step was already taken in [2]; throughout we follow, at least the spirit, of the classification theory for ordered sets initiated in [1].

For a graph G let V(G) denote its vertex set and E(G) ⊆ V(G) × V(G) its edge set. A graph K is a subgraph of G if V(K) ⊆ V(G) and for a, b ∊ V(K), (a, b) ∊ E(K) just if (a, b) ∊ E(G). The subgraph K of G is a retract of G, and we write K ◅ G, if there is an edge-preserving map g of V(G) to V(K) satisfying g(v) = v for each v ∊ V(K); g is called a retraction. A reflexive graph is an undirected graph with a loop at every vertex. The reason for a loop at a vertex is that an edge-preserving map can send the two vertices of an adjacent pair to it. The concept is illustrated in Figure 1. From here on, though, we shall for convenience suppress the illustration of the loops in the figures of reflexive graphs.
Behavior of Coefficients of Inverses of α-Spiral Functions
Authors: Libera Richard J. , Złotkiewicz Eligiusz J.
Published: December 2010

If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form

(1.1)

then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.

(1.1)

serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .
Semisimple Algebras of Infinite Valued Logic and Bold Fuzzy Set Theory
Authors: Belluce L. P.
Published: December 2010

In classical two-valued logic there is a three way relationship among formal systems, Boolean algebras and set theory. In the case of infinite-valued logic we have a similar relationship among formal systems, MV-algebras and what is called Bold fuzzy set theory. The relationship, in the latter case, between formal systems and MV-algebras has been known for many years while the relationship between MV-algebras and fuzzy set theory has hardly been studied. This is not surprising. MV-algebras were invented by C. C. Chang [1] in order to provide an algebraic proof of the completeness theorem of the infinitevalued logic of Lukasiewicz and Tarski. Having served this purpose (see [2]), the study of these algebras has been minimal, see for example [6], [7]. Fuzzy set theory was also being born around the same time and only in recent years has its connection with infinite-valued logic been made, see e.g. [3], [4], [5]. It seems appropriate then, to take a further look at the structure of MV-algebras and their relation to fuzzy set theory.
Hopf Algebras and Projective Representations of G ≀ Sn AND G ≀ An
Authors: Hoffman Peter N. , Humphreys John F.
Published: December 2010

In 1911, Schur published a rather formidable paper [9] in which he determined all the complex projective characters for the symmetric group (denoted Σn here, despite the title), and for the alternating group An (A pronounced “alpha”). As far as we know, the construction of the modules involved is still an unsolved problem. The results of Schur can be expressed in terms of certain induced representations whose characters form a basis for the group of virtual characters, plus formulae expressing the irreducible characters in terms of these induced characters. Here we give a new formulation of the above induced characters in the spirit of the well known “induction algebra” approach to the linear representations of Σn. We use some Hopf algebra techniques inspired by [5] to give new proofs of Schur's results, and to determine the extra structure which we define.
The Hausdorff Dimension Distribution of Finite Measures in Euclidean Space
Authors: Cutler Colleen D.
Published: December 2010

Let E be a Borel set of RN. The α-outer Hausdorff measure of E has been defined to be

where

and each Bi is a closed ball. d(Bi) denotes the diameter of Bi.

It is easily seen that the same value Hα(E) is obtained if we consider coverings of E by open balls or by balls which may be either open or closed.

By dim(E) we will mean the usual Hausdorff-Besicovitch dimension of E, where
On Basis-Conjugating Automorphisms of Free Groups
Authors: McCool J.
Published: December 2010

Let X = {x1, … xn} be a free generating set of the free group Fn and let H be the subgroup of Aut Fn consisting of those automorphisms α such that α(xi) is conjugate to xi for each i = 1, 2 , …, n. We call H the Z-conjugating subgroup of Aut Fn. In [1] Humphries found a generating set for the isomorphic copy H1 of H consisting of Nielsen transformations

where each is conjugate to ui (see remark 1 below). The purpose of this paper is to find a presentation of H (and hence of H1).

Let i ≠ j be elements of {1, 2, …, n}. We denote by (xi; xj) the automorphism of Fn which sends xi to and fixes xk if k ≠ i. Let S be the set of all such automorphisms.
Colour Switching and Homeomorphism of Manifolds
Authors: Ferri Massimo
Published: December 2010

Throughout this paper, we work in the PL and pseudosimplicial categories, for which we refer to [17] and [10] respectively. For the graph theory involved see [9].

An h-coloured graph (Γ, γ) is a multigraph Γ = (V(Γ), E(Γ)) regular of degree h, endowed with an edge-coloration γ by h colours. If is the colour set, for each we set

For each set . For n ∊ Z, n ≧ 1, set

Δn will be mostly used to denote the colour set for an (n + 1)-coloured graph.
Analytic Subalgebras of Von Neumann Algebras
Authors: Muhly Paul S. , Saito Kichi-Suke
Published: December 2010

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to be

where H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such that

is σ-weakly dense in M, and such that
Idempotents in Complex Banach Algebras
Authors: Hile G. N. , Pfaffenberger W. E.
Published: December 2010

The concept of the spectrum of A relative to Q, where A and Q commute and are elements in a complex Banach algebra with identity I, was developed in [1]. A complex number z is in the Q-resolvent set of A if and only if is invertible in otherwise, z is in the Q-spectrum of A, or spectrum of A relative to Q. One result from [1] was the following.

THEOREM. Suppose no points in the ordinary spectrum of Q have unit magnitude. Let C be a simple closed rectifiable curve which lies in the Q-resolvent of A, and let

*

where P is defined asxs

•
On The Factorization of Partial Differential Equations
Authors: Brownawell W. Dale
Published: December 2010

In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.

In this note we generalize Steinmetz’ work to show the following:

a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, and

b) the function g(z) can itself be allowed to be a function of several variables.
Uniqueness of Free Actions on S3 Respecting a Knot
Authors: Boileau Michel , Flapan Erica
Published: December 2010

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.

THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.
Uniformly Closed Algebras Generated by Boolean Algebras of Projections in Locally Convex Spaces
Authors: Ricker Werner
Published: December 2010

The theory of operator algebras in Banach spaces generated by Boolean algebras of projections is by now well known. It is systematically exposed in the penetrating studies of W. Bade, [1], [2] and [6, Chapter XVII]. Many of these results, a priori independent on normability of the underlying space, have recently been extended to the setting of locally convex spaces; see [3], [4], [5], [11] and [15], for example.

However, one of Bade's fundamental results, stating that the closed algebra generated by a complete Boolean algebra in the uniform operator topology is the same as the closed algebra that it generates in the weak operator topology, has remained remarkably resistant in attempts to extend it to locally convex spaces. Recently however, a class of Boolean algebras in non-normable spaces, called boundedly σ-complete Boolean algebras, was exhibited in which the analogue of Bade's result is valid, [14; Theorem 5.3].
Commutator Theory on Hilbert Space
Authors: Robinson Derek W.
Published: December 2010

Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.
Harmonic Mappings onto Convex Domains
Authors: Abu-Muhanna Yusuf , Schober Glenn
Published: December 2010

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.

In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equation

where a is analytic and |a| < 1, such that f(U) ⊂ D and
On Fixed Points and Multiparameter Ergodic Theorems in Banach Lattices
Authors: Millet Annie , Sucheston Louis
Published: December 2010

We present here multiparameter results about positive operators acting on a weakly sequentially complete Banach lattice. Sections 1, 2 and 3 generalize results obtained by M. A. Akcoglu and the second author in the case of a contraction. Even in that case, the classical L1 theory extends to Banach lattices only under an additional monotonicity assumption (C), introduced in [3], without which the TL (or stochastic) ergodic theorem fails. The example proving this in [4] also shows that, without (C), the decomposition of the space into the “positive” part P, the largest support of a T-invariant element, and the “null” part N on which the TL limit is zero (see, e.g., [22], p. 141), also fails. If T is not a contraction but only mean-bounded, then the space decomposes into the “remaining” part Y, the largest support of a T*-invariant element, and the “disappearing part“ Z (see, e.g., [22], p. 172). Here we obtain, for Banach lattices and in the multiparameter case, a unified proof of both decompositions, and of the TL ergodic theorem.
The Heat Equation for the -Neumann Problem, II
Authors: Beals Richard , Stanton Nancy K.
Published: December 2010

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.

THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,

(0.1)

(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)

Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.
Regularite De Fonctions Holomorphes Sur Des Wedges
Authors: Coupet Bernard
Published: December 2010

Nous traitons de la régularité d'une fonction holomorphe sur un ouvert de Cn à partir de celle de la fonction ou de la partie réelle connue sur une sous-variété totalement réelle maximale (de dimension n) incluse dans le bord de l'ouvert de définition.

Un théorème classique de Privalov assure qu'une fonction continue sur un disque fermé de C, holomorphe sur le disque ouvert, de partie réelle lipschitzienne d'ordre α sur le bord du disque vérifie une condition de même ordre α sur tout le disque. De façon analogue, le module de continuité d'une fonction holomorphe sur un ouvert borné de Cn, continue sur l'adhérence est contrôlé par celui de la fonction restreinte au bord [2] et [5], Le bord de l'ouvert peut être remplacé par la frontière de Shilov dans le cas d'un polyèdre analytique [4].

E. L. Stout [9] étudie la régularité sur une sous-variété totalement réelle maximale M de classe Cp, d'une fonction holomorphe f définie sur un ouvert Ω de classe Cq dont le bord contient M et telle que M soit l'ensemble sur lequel |f| est maximum; il démontre que f|M est Cp−0 lorsque q > p et Cp−1 lorsque q = p pour p supérieur ou égal à 3.
Pointwise Convergence of Alternating Sequences
Authors: Akcoglu M. A. , Sucheston L.
Published: December 2010

Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).

Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.

(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2)

exists a.e. for all.
Tensor Products of Fundamental Representations
Authors: Kempf George , Ness Linda
Published: December 2010

Let G be a reductive group over a field of characteristic zero. Fix a Borel subgroup B of G which contains a maximal torus T. For each dominant weight X we have an irreducible representation V(X) of G with highest weight X. For two dominant representation X1 and X2 we have a decomposition

This decomposition is determined by the element

of the group ring of the group of characters of T.

The objective of this paper is to compute r(X1, X2) for all pairs X1 and X2 of fundamental weights. This will be used to compute the equations for cones over homogeneous spaces. This problem immediately reduces to the case when G has simple type; An, Bn, Cn, Dn, E6, E7, E8, F4 and G2. We will give complete details for the classical types. For the case An we will work with GLn.
On the Selmer Group of Twists of Elliptic Curves with Q-Rational Torsion Points
Authors: Frey G.
Published: December 2010

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.

Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one has

with k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.
The Quantificational Tangent Cones
Authors: Ward Doug
Published: December 2010

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.

Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.

In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)
Some Differential Equations Related to Iteration Theory
Authors: Aczél J´nos , Gronau Detlef
Published: December 2010

In connection with the translation equation

(T)

three differential equations arise together with a differential initial condition. They are satisfied by the differentiable solutions of (T) and of the initial condition

(I)

These equations are attributed in [9] to E. Jabotinsky who seems to have been the first who treated these equations in connection with the theory of analytic iteration (see [6], cf. [7, 8], but see also [1, 2, 3]).

Gronau (see [9]) asked whether, conversely, it is true that all solutions of each of these “Jabotinsky differential equations”, possibly with some further initial conditions added, are also solutions of the translation equation. In this paper we give counter examples but also partial positive answers to these questions. First we show how one obtains the Jabotinsky equations from (T) and (I).
Strong Morita Equivalence for Heisenberg C*-Algebras and the Positive Cones of
Their K 0-Groups
Authors: Packer Judith A.
Published: December 2010

In [14] we began a study of
C*-algebras corresponding to projective
representations of the discrete Heisenberg group, and classified these
C*-algebras up to *-isomorphism. In this sequel to
[14] we continue the study of these so-called Heisenberg
C*-algebras, first concentrating our study on the
strong Morita equivalence classes of these C*-algebras.
We recall from [14] that a Heisenberg
C*-algebra is said to be of class
i, i ∊ {1, 2, 3}, if the
range of any normalized trace on its K
0 group has rank i as a subgroup
of R; results of Curto, Muhly, and Williams
[7] on strong Morita equivalence for crossed products
along with the methods of [21] and
[14] enable us to construct certain strong Morita
equivalence bimodules for Heisenberg
C*-algebras.
On the Hankel and Some Related Transformations
Authors: Heywood P. , Rooney P. G.
Published: December 2010

The transformations we will discuss in this paper are the Hankel
transformation Hυ
defined for f ∊ C
0, the collection of continuous functions compactly supported
in (0, ∞), by

(1.1)

and the and
transformations
defined for such f by

(1.2)

and

(1.3)

where Jv
>and Yv
are the Bessel functions of the first and second kinds
respectively, and H

v
is the Struve function; for the theory of these functions see
[1, Chapter VII].

These transformations were studied extensively by one of us in
[5] and [6] on the spaces
defined in
[7; Sections 1 & 5]. In those papers the
boundedness of the three transformations was fully given on the spaces
for 1 <
p < ∞, but not for
p = 1. Also inversion formulae were given for the
transformations only for portions of their respective ranges of
boundedness.
On the Askey-Wilson and Rogers Polynomials
Authors: Ismail Mourad E. H. , Stanton Dennis
Published: December 2010

The q-shifted factorial (a)n or (a; q)n is

and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials

1.1

where

1.2

and

1.3

We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].
On the Existence of Positive Decaying Entire Solutions for a Class of Sublinear Elliptic Equations
Authors: Furusho Yasuhiro , Kusano Takaŝi
Published: December 2010

In recent years there has been a growing interest in the existence and asymptotic behavior of entire solutions for second order nonlinear elliptic equations. By an entire solution we mean a solution of the elliptic equation under consideration which is guaranteed to exist in the whole Euclidean N-space RN, N ≧ 2. For standard results on the subject the reader is referred to the papers [2-7, 9-21].

The study of entire solutions, which at an early stage was restricted to simple equations of the form Δu + f(x, u) = 0, x ∊ RN, Δ being the N-dimensional Laplacian, has now been extended and generalized to elliptic equations of the type

A

where
On VMOA for Riemann Surfaces
Authors: Aulaskari Rauno
Published: December 2010

Let Δ = {z│ │z│ < 1} be the unit disk and f an analytic function in Δ. The Dirichlet integral DΔ(f) of f on Δ is defined by

and we denote by AD(Δ) the space of all functions f analytic on Δ for which DΔ(f) < ∞. We denote by BMOA(Δ) the space of analytic functions f in Δ for which

and by VMOA(Δ) the space of those analytic functions f in BMOA(Δ) satisfying the condition
Existence of Positive Global Solutions of Mixed Sublinear-Superlinear Problems
Authors: Allegretto W. , Huang Y. X.
Published: December 2010

Consider the elliptic quasilinear problem:

1

in Rn, n ≧ 3, where

We are interested in establishing sufficient conditions on f for the existence of positive solutions u(x) with specified behaviour at ∞. Of special interest to us are criteria which guarantee that u(x) decays at least as fast as |x|−α for some α ≧ 0, given below, in the case f(x, u, ∇u) contains terms of type

That is: f is of mixed sublinear-super linear type. Our main result is Theorem 3 below which explicitly states sufficient conditions for the existence of such solutions.