bluuebunny/crossref_metadata_2025 huggingface dataset:
Algebras of Bounded Analytic Functions containing the Disk Algebra
Authors: Izuchi Keiji , Izuchi Yuko
Published: December 2010

Let D be the open unit disk and let ∂D be its boundary. We denote by C the algebra of continuous functions on ∂d, and by L∞ the algebra of essentially bounded measurable functions with respect to the normalized Lebesgue measure m on ∂D. Let H∞ be the algebra of bounded analytic functions on D. Identifying with their boundary functions, we regard H∞ as a closed subalgebra of L∞. Let A = H∞ Pi C, which is called the disk algebra. The algebras A and H∞ have been studied extensively [5, 6, 7]. In these fifteen years, norm closed subalgebras between H∞ and L∞, called Douglas algebras, have received considerable attention in connection with Toeplitz operators [12]. A norm closed subalgebra between A and H∞ is called an analytic subalgebra. In [2], Dawson studied analytic subalgebras and he remarked that there are many different types of analytic subalgebras. One problem is to study which analytic subalgebras are backward shift invariant. Here, a subset E of H∞ is called backward shift invariant if
Projections sur des Espaces de Fonctions Holomorphes Dans des Domaines Plans
Authors: Békollé David
Published: December 2010

Soit Ω un domaine de Jordan à bord rectifiable du plan complexe C. Désignons par dλ la mesure de Lebesgue à l'intérieur de Ω, par dσ la mesure de Lebesgue sur le bord ∂Ω de Ω et par φ une représentation conforme de Ω sur le disque unité D du plan complexe.

Par définition, la classe de Bergman Ap(Ω), 0 < p ≦ +∞, est le sous-espace de Lp(dλ) formé par les fonctions holomorphes dans Ω et le projecteur de Bergman PΩ de Ω est le projecteur orthogonal de L2(dλ) sur A2(Ω); quel que soit f dans L2(dλ), on a la formule:

(1)
A Note on Orbits of Subgroups of the Permutation Groups
Authors: Leitch R. D.
Published: December 2010

In [2] we studied Milgram's Complex, C(n – 1), which was first defined in [3], in the following manner. Let Sn, the permutation group of n symbols, act on Rn in the obvious manner; put α(x) = (y), where yi = xα–1(i). Let s = (1, 2, …, n), then C(n – 1) is the convex hull of the points α(s), α ∊ Sn. Here we shall generalise this construction as follows. Let G be a subgroup of Sn, and let v ∊ Rn. Then C(G, v) is the convex hull of α(v), α ∊ G. We prove invariance over v subject to certain restrictions, give counter-examples to shew lack of invariance if we alter G, discuss how we may describe C(G, v), shew that the only “nice” case is essentially when G is Sn, and lastly give some examples.
Continuity of the Scattering Transformation for the Korteweg de Vries Equation
Authors: Gautrin Henri-François , Klemola Tapio
Published: December 2010

It is at present well known that, if q(x, t) is a solution of the Korteweg de Vries (K d V) equation

(1)

such that q(x, 0) = q0(x), where q0(x) behaves reasonably at infinity, and if

are the scattering data (see [4]) corresponding to q(x, t), then
Isotropic Immersions
Authors: Maeda Sadahiro
Published: December 2010

Recently, Ferus ([5], [6]) classified connected Riemannian manifolds with parallel second fundamental form in a real space form of constant curvature . In this paper we may restrict our attention to isotropic submanifolds with parallel second fundamental form in the Euclidean sphere Sm(k) of constant curvature k. Due to Ferus, we find that an isotropic submanifold with parallel second fundamental form in Sm is locally congruent to one of compact symmetric spaces of rank one and the immersion is locally equivalent to the second or the first standard immersion according as M is a sphere or not. In Section 2, we characterize the first standard immersion of a complex projective space into a sphere in terms of isotropic immersions.
Epimorphisms From S(X) onto S(Y)
Authors: Magill K. D. , Misra P. R. , Tewari U. B.
Published: December 2010

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for which

forms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].

The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.
Induction and Restriction of π-Special Characters
Authors: Isaacs I M.
Published: December 2010

1. Introduction. The character theory of solvable groups has undergone significant development during the last decade or so and it can now be seen to have quite a rich structure. In particular, there is an interesting interaction between characters and sets of prime numbers.

Let G be solvable and let π be a set of primes. The “π-special” characters of G are certain irreducible complex characters (defined by D. Gajendragadkar [1]) which enjoy some remarkable properties, many of which were proved in [1]. (We shall review the definition and relevant facts in Section 3 of this paper.) Actually, we need not assume solvability: that G is π-separable is sufficient, if we are willing to use the Feit-Thompson “odd order” theorem occasionally. We shall state and prove our results under this weaker hypothesis, but we stress that anything of interest in them is already interesting in the solvable case where, of course, the “odd order” theorem is irrelevant.
The Dirichlet Problem on the Heisenberg Group III: Harmonic Measure of a certain Half-Space
Authors: Gaveau Bernard , Vauthier Jacques
Published: December 2010

0. Introduction. In this short note we give an explicit computation of the harmonic measure of a half space x > 0 in the 3-dimensional Heisenberg group in terms of a degenerate hypergeometric function. A probabilistic argument reduces the whole problem to a Hermite-type equation on a half line, that we can solve in terms of the function G(l/4, 1/2; x2).

A preliminary attempt to compute this kernel was done in [1] p. 107 and, cited by Huber [4]. Unfortunately a small mistake was made in [1] and the problem was still open until now. The first author is very grateful to Prof. Huber for having pointed out the weak argument of [1]. Since that time, other harmonic measures and even Green functions have been explicitly computed (see [2]).
A Tauberian Theorem of Exponential Type
Authors: Geluk J. L. , de Haan L. , Stadtmüller U.
Published: December 2010

1. Introduction. We will be interested in Tauberian theorems concerning the limiting behaviour of a monotone function U and its Laplace transform

A famous theorem of Karamata concerns the case in which the function U is regularly varying (i.e., U(tx)/U(t) → xα(t → ∞) for x > 0). Here we will consider functions U that grow faster, in fact our conditions will be in terms of log U rather than on U itself. So it is convenient to write the Laplace transform in terms of q = log U. For a function q:R+ → R such that exp q is locally integrable and

we define the function by the relation
On the Singularities of Plane Curves
Authors: Bisztriczky Tibor
Published: December 2010

Let Γ be a differentiable curve in a real projective plane P2 met by every line of P2 at a finite number of points. The singular points of Γ are inflections, cusps (cusps of the first kind) and beaks (cusps of the second kind). Let n1(Γ), n2(Γ) and n3(Γ) be the number of these points in Γ respectively. Then Γ is non-singular if

otherwise, Γ is singular.

We wish to determine when T is singular and then find the minimum value of n(Γ). A history and an analysis of this problem were presented in [1] and [2]. It was shown that we may assume that Γ is a curve of even order (even degree if Γ is algebraic), met by every line in P2. Then if Γ does not contain any multiple points or if Γ contains only a certain type of multiple point, Γ is singular. Presently, we complete this investigation
The Tensor Product of Operational Logics
Authors: Lock Robin H.
Published: December 2010

The concept of an operational logic has been developed by Randall and Foulis ([l]-[4], [10], [11]) as a part of a larger effort to obtain a formalism suitable for expressing, comparing, and evaluating various approaches to empirical science, statistics, and in particular, quantum mechanics. The structure of these logics is similar to that of an orthomodular partially ordered set which is often used as a model for quantum logics. However, the operational logic is a more general structure which, among other features, allows for the creation of a tensor product of logics to represent the coupling of physical systems. Randall and Foulis have shown that, given certain reasonable physical constraints, such a product is not possible within the category of orthomodular posets [12].
A Family of M*-Groups
Authors: May Coy L.
Published: December 2010

A compact bordered Klein surface of (algebraic) genus g ≦ 2 is said to have maximal symmetry [5] if its automorphism group is of order 12(g – 1), the largest possible. An M*-group acts as the automorphism group of a bordered surface with maximal symmetry. M*-groups were first studied in [6], and additional results about these groups are in [5, 7, 8].

Here we construct a new, interesting family of M*-groups. Each group G in the family is an extension of a cyclic group by the automorphism group of a torus T with holes that has maximal symmetry. Furthermore, G acts on a bordered Klein surface X that is a fully wound covering [7] of T, that is, an especially nice covering in which X has the same number of boundary components as T. The construction we use for the new family of M*-groups is a standard one that employs group automorphisms to define extensions of groups.
On the Spectra of Unbounded Subnormal Operators
Authors: McDonald G. , Sundberg C.
Published: December 2010

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.

Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),
Distance Functions and Orlicz-Sobolev Spaces
Authors: Edmunds D. E. , Edmunds R. M.
Published: December 2010

Let ∧ be a bounded, non-empty, open subset of Rn and given any x in Rn, let

let k ∊ N and suppose that p ∞ (1, ∞). It is known (c.f. e.g. [4]) that if u belongs to the Sobolev space WKp(∧) and u/dk ∊ Lp(∧), then . Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

Under appropriate smoothness conditions on the boundary of ∧, m and d are equivalent, and thus may in this case be characterized as the subspace of W1,2(∧) consisting of all functions u ∊ W1,2(∧) such that u/d ∊ L2(∧). Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that
Excising States of C*-Algebras
Authors: Akemann Charles A. , Anderson Joel , Pedersen Gert K.
Published: December 2010

A net {aα} of positive, norm one elements of a C*-algebra A excises a state f of A if

This notion has been used explicitly by the second author [4, 5, 6] for pure states, but the present paper will explore it more fully. The name is motivated by the following example. Let K be the unit disk in the complex plane, A = C(K) and f(a) = a(0). Define an(reiθ) = ϕn(r), where

Note that the sets {t ∊ K:an(t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence {an} “cuts out” the state f and, in the limit, isolates it from all other states.
Structural Properties of Elementary Operators
Authors: Apostol Constantin , Fialkow Lawrence
Published: December 2010

Let and denote complex Banach algebras and let b e a left Banach module and a right Banach -module. If

we define the bounded linear elementary operator R(A, B), acting on , by

For the case , elementary operators were introduced by Lumer and Rosenblum [19], who studied their spectral properties. In this setting many authors subsequently studied spectral, algebraic, metric, and structural properties of elementary operators, with particular attention devoted to the inner derivations δa (δa(x) = ax – xa) [25], generalized derivations τ(a, b) (τ(a, b)(x) = ax – xb) [9, 10], and elementary multiplications S(a, b) (S(a, b)(x) = axb), including left and right multiplications La and Rb [11].
Operators on L2(I) ⊕ Cr
Authors: Kemp R. R. D.
Published: December 2010

Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.

It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.
A Variation of the Koebe Mapping in a Dense Subset of S
Authors: Bshouty D. , Hengartner W.
Published: December 2010

Let H(U) be the linear space of holomorphic functions defined on the unit disk U endowed with the topology of normal (locally uniform) convergence. For a subset E ⊂ H(U) we denote by Ē the closure of E with respect to the above topology. The topological dual space of H(U) is denoted by H′(U).

Let D, 0 ∊ D, be a simply connected domain in C. The unique univalent conformal mapping ϕ from U onto D, normalized by ϕ(0) = 0 and ϕ′(0) > 0 will be called “the Riemann Mapping onto D”. Let S be the set of all normalized univalent functions
Dirac Systems with Discrete Spectra
Authors: Hinton D. B. , Shaw J. K.
Published: December 2010

In this paper we consider the one dimensional Dirac system

1.1

where αk(x) < 0, λ is a complex spectral parameter, and the remaining coefficients are suitably smooth and real valued. We regard (1.1) as regular at x = a but singular at x = b; in Section 4 we extend our result to problems having two singular endpoints.

Equation (1.1) arises from the three dimensional Dirac equation with spherically symmetric potential, following a separation of variables. For the choices p(x) = k/x, αk(x) = 1,p2(x) = (z/x) + c, p1(x) = (z/x) – c, and appropriate values of the constants, (1.1) is the radial wave equation in relativistic quantum mechanics for a particle in a field of potential V = z/x [17]. Such an equation was studied by Kalf [11] in the context of limit point-limit circle criteria, which is one of the matters we consider here.
Characters of Non-Connected, Reductive p-Abic Groups
Authors: Clozel Laurent
Published: December 2010

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.

Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.

As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.
The L-Theory of Twisted Quadratic Extensions
Authors: Ranicki Andrew
Published: December 2010

For surgery on codimension 1 submanifolds with non-trivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN∗(π′ → π), with π a group and π′ a subgroup of index 2, such that there is defined an exact sequence involving the ordinary L-groups of rings with involution

with the superscript w signifying a different involution on Z[π]. Geometry was used in [13] to identify

with (α, u) an antistructure on Z[π′] in the sense of Wall [14]. The main result of this paper is a purely algebraic version of this identification, for any twisted quadratic extension of a ring with antistructure.
End Behaviour and Ergodicity for Homeomorphisms of Manifolds with Finitely Many Ends
Authors: Alpern S. , Prasad V.
Published: December 2010

The recent paper of Berlanga and Epstein [5] demonstrated the significant role played by the “ends” of a noncompact manifold M in answering questions relating homeomorphisms of M to measures on M. In this paper we show that an analysis of the end behaviour of measure preserving homeomorphisms of a manifold also leads to an understanding of some of their ergodic properties, and allows results previously obtained for compact manifolds to be extended (with qualifications) to the noncompact case. We will show that ergodicity is typical (dense Gδ) with respect to various compact-open topology closed subsets of the space consisting of all homeomorphisms of a manifold M which preserve a measure μ. It may be interesting for topologists to note that we prove when M is a σ-compact connected n-manifold, n≧ 2, then M is the countable union of an increasing family of compact connected manifolds. If M is a PL or smooth manifold, this is well known and easy. If M is just, however, a topological n-manifold then we apply the recent results [9] and [12] to prove the result. The Borel measure μ, is taken to be nonatomic, locally finite, positive on open sets, and zero for the manifold boundary of M.
A New Result on Comma-Free Codes of Even Word-Length
Authors: Tang Betty , Golomb Solomon W. , Graham Ronald L.
Published: December 2010

Comma-free codes were first introduced in [1] in 1957 as a possible genetic coding scheme for protein synthesis. The general mathematical setting of such codes was presented in [3], and the biochemical and mathematical aspects of the problem were later summarized and extended in [4].

Using the notation of [3], a set D of k-tuples or k-letter words, (a1a2 … ak), where

for fixed positive integers k and n, is said to be a comma-free dictionary if and only if, whenever (a1a2 … ak) and (b1b2 … bk) are in D, the “overlaps”

are not in D. This precludes codewords having a subperiod less than k; and two codewords which are cyclic permutations of one another cannot both be in D.
MOD-C Postnikov Approximation of a 1-Connected Space
Authors: Behera A. , Nanda S.
Published: December 2010

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context [4]. They have also shown that if the set of morphisms is saturated then the Adams completion of an object is characterized by a certain couniversai property. We want to prove a stronger version of this result by dropping the saturation assumption on the set of morphisms; we also prove that the canonical map from an object to its Adams completion is an element of the set of morphisms under very moderate assumptions. These two results are fairly general in nature and are applicable to most cases of interest. Further using these two results and introducing “modulo a Serre class C of abelian groups” [9] we have obtained the mod-C Postnikov approximation of a 1-connected based CW-complex, with the help of a suitable set of morphisms.
Absolute Retracts and Varieties of Reflexive Graphs
Authors: Hell Pavol , Rival Ivan
Published: December 2010

For a graph G, let V(G) denote its vertex set and E(G) its edge set. Here we shall only consider reflexive graphs, that is graphs in which every vertex is adjacent to itself. These adjacencies, i.e., the loops, will not be depicted in the figures, although we always assume them present. For graphs G and H, an edge-preserving map (or homomorphism) of G to H is a mapping of V(G) to V(H) such that f(g) is adjacent to f(g′) in H whenever g is adjacent to g′ in G. Because our graphs are reflexive, an edge-preserving map can identify adjacent vertices, i.e., possibly f(g) = f(g′) for some g adjacent to g′, cf. Figure 1(a).
Matrix Summability of Geometrically Dominated Series
Authors: Fricke G. H. , Fridy J. A.
Published: December 2010

One of the common uses of summability theory is found in its applications to power series. A partial listing such as [l]-[5], [13], and [15]-[17] might serve to remind us of the many instances of summability theory applied to power series. In some studies, the summability transformation is applied to the sequence of partial sums of the power series, while in others it is applied to the general term akzk as a series-to-series transformation. In [2] and [5] the transformation is applied to the coefficient sequence of the Taylor series. In the present study we investigate matrix transformations that are applied to the sequence {akzk), but we are not concerned with the usual preservation of convergence or sums. At a point within its disc of convergence, a power series exhibits more than ordinary convergence; it converges very rapidly, being dominated by a convergent geometric series.
Existence, Unicite et Multiplicite de Solutions Periodiques D'Equations Differentielles de Duffing Non-Lineaires Avec Dissipation
Authors: Nkashama M. N.
Published: December 2010

Nous nous intéressons à l'existence, l'unicité et la multiplicité de solutions de l'équation différentielle forcée:

1.1

vérifiant les conditions périodiques

1.2

où e ∊ L1(0, 2π), c ∊ R, c arbitraire, g:[0, 2π] × R → R satisfait les conditions de Carathéodory i.e.,g(·, x) est measurable pour tout x ∊ R, g(t, ·) est continu pour presque tout t ∊ [0, 2π].

Le problème (1.1)-(1.2) a été étudié par plusieurs auteurs. Nous mentionerons les travaux [22], [18], [21], [20], [19], [7], [29], [25] et la bibliographie contenue dans ces articles.
On the Intersection of a Class of Maximal Subgroups of a Finite Group
Authors: Mukherjee N. P. , Bhattacharya Prabir
Published: December 2010

Of late there has been considerable interest in the study of analogs of the Frattini subgroup of a finite group and the investigation of their properties, particularly their influence on the structure of the group, see [2-11], [14-16] and [18]. Gaschütz [11] and more recently Bechtell [2] and Rose [18] have considered extensively the intersection of the family of all non-normal, maximal subgroups of a finite group. Deskins [8] has discussed the intersection of the family of all maximal subgroups of a finite group whose indices are not divisible by a given prime. Bhatia [7] considered the intersection of the class of all maximal subgroups of a given group whose indices are composites. In this paper we investigate the intersection of another class of maximal subgroups and its relationship with the structure of the group. The subgroup we consider here contains the Frattini subgroup and also the two subgroups introduced in [8] and [7].
On Major and Minor Branches of Rooted Trees
Authors: Meir A. , Moon J. W.
Published: December 2010

Let denote a rooted tree with n nodes. (For definitions not given here, see, e.g. [4]). For any node v of , let B(v) denote the subtree of determined by v and all nodes u such that v is between u and the root of ; node v serves as the root of B(v). The branches of are the subtrees B(v) such that node v is joined to the root of . A branch B with i nodes is a primary branch of if n/2 ≦ i ≦ n – 1; if has a primary branch B with i nodes, then a branch C with j nodes is a secondary branch if (n – i)/2 ≦ j ≦ n – 1 ≦ i; if has a primary branch B with i nodes and a secondary branch C with j nodes, then a branch D with h nodes is a tertiary branch if
Relations Between Mahler's Measure and Values of L-Series
Authors: Ray Gary Alan
Published: December 2010

Mahler's measure is a natural generalization of Jensen's formula to polynomials in several variables. Its definition is as follows:

The importance of Mahler's measure for polynomials in several variables lies in its connection to Lehmer's classical question which can be phrased in terms of Mahler's measure for polynomials in one variable:

Given , are there any polynomials p with integer coefficients in one variable for which ?

Surprisingly, Boyd [1] has shown that to answer this question, it is necessary to investigate the larger question involving polynomials in several variables.
Explicit Formulas for the Coefficients of α-Convex Functions, α ≧ 0
Authors: Todorov Pavel G.
Published: December 2010

Let the function

1

be analytic in the unit disk Δ = {z│ │z│ ≤ 1), with

there, and let α be a real number. Then f(z) is said to be α-convex in Δ if and only if the inequality

holds in Δ. The class of α-convex functions was introduced in [8] and was studied in detail in the series [5]–[10], where in particular it is shown that α-convex functions are univalent and starlike for all α (−∞≦ α ≦ + ∞), that is, the inequality

holds in Δ.
Simple Type III Self-Injective Rings and Rings of Column-Finite Matrices
Authors: O'Meara K. C.
Published: December 2010

Relatively little is known about simple, Type III, right self-injective rings Q. This is despite their common occurrence, for example as Qmax(R) for any prime, nonsingular, countable-dimensional algebra R without uniform right ideals. (In particular Q can be constructed with a given field as its centre.) As with their directly finite, SP(1), right self-injective counterparts, division rings, there are few obvious invariants apart from the centre.

One reason perhaps why little interest has been shown in their structure is that the usual construction of such Q, namely as a suitable Qmax(R), is not concrete enough; in general R sits far too loosely inside Q and not enough information transfers to Q from R. Thus, for example, taking R to be a non-right-Ore domain and Q = Qmax(R) tells us little about Q (although it has been conjectured that all Q arise this way).
Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV
Authors: Salzmann Helmut
Published: December 2010

Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.

THEOREM. If dim Σ > 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).

By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].
On the Sieve of Eratosthenes
Authors: Murty M. Ram , Saradha S.
Published: December 2010

Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the number of natural numbers not exceeding x which fail to satisfy the inequality

1

is o(x) as x → ∞. A very simple proof of this was subsequently given by Turán. He showed that

2
Analysis on Root Systems
Authors: Debiard Amédée , Gaveau Bernard
Published: December 2010

A great part of mathematical analysis relies directly on the methods of separation of variables and on the successive reduction of several variables problems to one-dimensional equations and to the theory of classical special functions; for example, the theory of elliptic or parabolic equations with regular coefficients (even with non constant coefficients) can be done because we know explicitly the fundamental solutions of the Laplace operator or of the heat equation; these fundamental solutions are functions of one variable; pseudodifferential or parametrices methods are thus basically small perturbations of an explicitly known problem in one variable.

On the other hand, there are many problems which are not of this type: they are related to the questions of operators with singular coefficients and to the global behaviour of the solutions; in that case, the local model cannot be reduced to a one variable problem but is fundamentally a several variables problem which cannot be treated in a detailed way by one variable methods or perturbation analysis of a one variable problem.
Congruences and Norms of Hermitian Matrices
Authors: Pierce Stephen , Rodman Leiba
Published: December 2010

Two complex hermitian n × n matrices A and B are congruent if S*AS = B for some invertible n × n matrix S (with complex entries). The matrix S is called a congruence matrix. Given congruent hermitian matrices A and B, a congruence matrix is, of course, not unique. For instance, if A = B then one can take S = αI with |α| = 1, as well as any other matrix satisfying S*AS = A. However, here the choice S = I seems naturally to be best possible in the sense that when applied to an n-dimensional column vector it produces no distortion or movement of the vector at all. We shall measure the distortion (or movement) of the vector x ∊ Cn under an n × n invertible matrix A in terms of ║x − Ax║, where the norm is euclidean. Then the distortion produced by A is ║I − A║, with the induced operator norm.
Hodge Theory on Compact Two-Dimensional Spacetimes and the Uniqueness ofgijwith a SpecifiedRij
Authors: Ihrig Edwin
Published: December 2010

The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see[4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determinesg. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general thatgmust inherit the symmetries of Ric. It is not even clear that there is a Ric such that everygwith this Ricci tensor is known.
Tubes, Cohomology with Growth Conditions and an Application to the Theta Correspondence
Authors: Kudla Stephen S. , Millson John J.
Published: December 2010

In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point of this paper is to prove an integral formula, Theorem 2.1, which will allow us to generalize the results obtained in the above papers to the finite volume case (the previous papers treated only the compact case). We then apply our integral formula to certain finite volume quotients of symmetric spaces of orthogonal groups. The main result obtained is Theorem 4.2 which is described below. We let (,) denote the bilinear form associated to a quadratic form with integer coefficients of signature (p, q). We assume that the fundamental group Γ ⊂ SO(p, q) of our locally symmetric space is the subgroup of the integral isometries of (,) congruent to the identity matrix modulo some integer N. We assume that N is chosen large enough so that Γ is neat (the multiplicative subgroup of C* generated by the eigenvalues of the elements of Γ has no torsion), Borel [2], 17.1 and that every element in Γ has spinor norm 1, Millson-Raghunathan [15], Proposition 4.1. These conditions are needed to ensure that our cycles Cx (see below) are orientable. The methods we will use apply also to unitary and quaternion unitary locally symmetric spaces, see [13].
Generalized Variation and Functions of Slow Growth
Authors: Berman Robert D.
Published: December 2010

Many of the basic results of HP theory on the disk Δ = {|z| < 1} are proved using the Cauchy-Stieltjes representation

1.1

and the Poisson-Stieltjes representation

1.2

Here, μ:R → C is a complex-valued function of a real variable that is of bounded variation on [0, 2π] such that μ(t + 2π) = μ(t) + μ(2t) — μ(0), t ∊ R,

is the Cauchy kernel, and

is the Poisson kernel. It is therefore natural to generalize these representations in such a way that some of the basic properties and results carry over. Such a generalization occurs when the assumption that μ is of bounded variation on [0, 2μ] is replaced by the requirement that it is measurable and bounded on [0, 2μ] (cf. [9]). The integrals in (1.1) and (1.2) are then defined by a formal integration by parts. After some preliminaries in Section 2, we catalogue a variety of results which remain valid in Section 3.
An Index Theory for Semigroups of *-Endomorphisms of and Type II1 Factors.
Authors: Powers Robert T.
Published: December 2010

In this paper we study unit preserving *-endomorphisms of and type II1 factors. A *-endomorphism α which has the property that the intersection of the ranges of αn for n = 1 , 2 , … consists solely of multiples of the unit are called shifts. In Section 2 it is shown that shifts of can be characterized up to outer conjugacy by an index n = ∞ 1, 2 , …. In Section 3 shifts of R the hyperfinite II1 factor are studied. An outer conjugacy invariant of a shift of R is the Jones index [R: α(R)]. In Section 3 a class of shifts of index 2 are studied. These are called binary shifts. It is shown that there are uncountably many binary shifts which are pairwise non conjugate and among the binary shifts there are at least a countable infinity of shifts which are pairwise not outer conjugate.
Shape Equivalences of Whitney Continua of Curves
Authors: Kato Hisao
Published: December 2010

By a compactum, we mean a compact metric space. A continuum is a connected compactum. A curve is a 1-dimensional continuum. Let X be a continuum and let C(X) be the hyperspace of (nonempty) subcontinua of X, C(X) is metrized with the Hausdorff metric (e.g., see [12] or [18]). One of the most convenient tools in order to study the structure of C(X) is a monotone map ω:C(X) → [0, ω(X)] defined by H. Whitney [25]. A map ω:C(X) → [0, ω(X)] is said to be a Whitney map for C(X) provided that

The continua {ω−1} (0 < t < ω(X)) are called the Whitney continua of X. We may think of the map ω as measuring the size of a continuum. Note that ω−1(0) is homeomorphic to X and ω−1(ω(X)) = {X}. Naturally, we are interested in the structures of ω−1(t)(0 < t < ω(X)). In [14], J. Krasinkiewicz proved that if X is a circle-like continuum and ω is any Whitney map for C(X), then for any 0 < t < ω(X)ω−1(t) is shape equivalent to X, i.e., Sh ω−1(t) = Sh X (e.g., see [1] or [17]). In [8], we proved the following: If one of the conditions (i) and (ii) is satisfied, then the shape morphism

which is defined in [7] and [8], is a shape equivalence.
On the Analytic Continuation of c-Functions
Authors: Ringseth Paul F.
Published: December 2010

Let G be a reductive Lie group; Γ a nonuniform lattice in G. Then the theory of Eisenstein series plays a major role in the spectral decomposition of L2(G/Γ) (cf. [5]). One of the most difficult aspects of the subject is the analytic continuation of the Eisenstein series along with its associated c-function. This was originally done by Langlands using some very difficult analysis (cf. [5]). Later Harish-Chandra was able to simplify somewhat the most difficult part of the continuation, the continuation to zero, by the introduction of the Maas-Selberg relation. The purpose of this note is to give a simplified account of this particular part of the theory.

Our chief tool will be the truncation operator of Arthur (cf. [1] and [8]), the systematic utilization of which has the effect of streamlining the earlier accounts, especially in so far as continuation to zero is concerned, which is reduced to an elementary manipulation.
Conjugates of Infinite Measure Preserving Transformations
Authors: Alpern S. , Choksi J. R. , Prasad V. S.
Published: December 2010

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we prove

THEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such that

Then there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.

The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).

Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.
Maximal Homotopy Lie Subgroups of Maximal Rank
Authors: Frohliger John A.
Published: December 2010

Let G be a compact connected Lie group with
H a connected subgroup of maximal rank. Suppose
there exists a compact connected Lie subgroup K with
H ⊂ K ⊂
G. Then there exists a smooth fiber bundle
G/H → G/K with
K/H as the fiber. (See for example
[13].) This can be incorporated into a diagram involving
the classifying spaces as follows:

(1)

Here π, ϕ, ϕ
1 and ϕ
2 denote fibrations. We also know that the homogeneous spaces
and the Lie groups, which are homotopy equivalent to the loop spaces of their
respective classifying spaces, are homotopy equivalent to connected finite
complexes.

Now suppose H is a maximal subgroup. Can there
still exist spaces, which we will call BK, K/H, and
G/K, and fibrations so that diagram (1) is still
valid? This paper will show that in many cases either
G/K or K/H will be homo
topically trivial.
Notions De Base Pour L'Arthmetique De Fq((1/t))
Authors: Hellegouarch Yves
Published: December 2010

Le but de ce travail est de définir un certain nombre d'objets de
F

q
((1/t) et de fonctions

qui sont analogues à des objets classiques de R(ou
C) et à des fonctions classiques
R → R (ou
C → C) : par exemple
ζ(l), ζ(2), etc.,
fractions continues, domaine fondamental du “demi-plan” de Poincaré, séries
d'Eisenstein, fonctions multiplement périodiques.

Un tel travail avait été entrepris autrefois par E. Artin
[2], mais dans un autre esprit, et la fonction zéta que
j'ai définie dans [7] n'est pas celle d'Artin, mais celle
de Carlitz [3].

L'intersection de la thèse d'Artin et de ce travail n'est donc pas vide, mais
elle n'est pas non plus très étendue bien que je me sois limité aux définitions et
aux propriétés les plus élémentaires. En revanche, par son esprit, ce travail se
rapproche davantage du point de vue de Carlitz.
Linear Algebra of Curvature Tensors and Their Covariant Derivatives
Authors: Strichartz Robert S.
Published: December 2010

Fix a point on a Riemannian manifold, and consider the tangent space V at the point equipped with inner product g. The Riemann curvature tensor R and its first covariant derivative ∇R at the point are tensors in and . If we take all the known symmetries of these tensors we can define subspaces and such that R ∊ Curv and ∇R ∊ ∇Curv. Also, the orthogonal group O(g) acts naturally on all these spaces. The two fundamental problems of the linear algebra of the spaces Curv and ∇Curv are: (1) find the decomposition into irreducible representations of O(g), with corresponding projection operators, (2) give a description of the structure of the O(g) orbits, by means of orbit invariant functions and a canonical form for elements of each orbit.
Certain Invariant Subspaces of H2 and L2 on a Bidisc
Authors: Nakazi Takahiko
Published: December 2010

We let T2 be the torus that is the cartesian product of 2 unit circles in C. The usual Lebesgue spaces, with respect to the Haar measure m of T2, are denoted by Lp = Lp(T2), and Hp = Hp(T2) is the space of all f in LP whose Fourier coefficients

are 0 as soon as at least one component of (j, ℓ) is negative.

A closed subspace M of L2 is said to be invariant if

Whenever this is the case, it follows that fM ⊂ M for every f in H∞. One can ask for a classification or an explicit description (in some sense) of all invariant subspaces of L2, but this seems out of reach.
Elementary Abelian Cartesian Groups Cartesian Groups
Authors: Hayden John L.
Published: December 2010

Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if

(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.

By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.
Products of Normal Operators
Authors: Wu Pei Yuan
Published: December 2010

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.

If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).
The Modular Group Algebras of P-Groups of Maximal Class
Authors: Bagiński C. , Caranti A.
Published: December 2010

The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.

The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than pp+1, having an abelian maximal subgroup, for odd primes p.

What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p4. In Passman's paper the case of groups of maximal class is actually the most difficult one.
Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials
Authors: Máté Attila , Nevai Paul
Published: December 2010

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that
Canonical Extensions of Harish-Chandra Modules to Representations of G
Authors: Casselman W.
Published: December 2010

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to [38].
Starlike Univalent Functions Bounded on the Real Axis
Authors: Fournier Richard
Published: December 2010

We denote by E the open unit disc in C and by H(E) the class of all analytic functions f on E with f(0) = 0. Let (see [3] for more complete definitions)

S = {ƒ ∈ H(E)|ƒ is univalent on E}

S0 = {ƒ ∈ H(E)|ƒ is starlike univalent on E}

TR = {ƒ ∈ H(E)|ƒ is typically real on E}.

The uniform norm on (— 1, 1) of a function ƒ ∈ H(E) is defined by
Rotundity In Köthe Spaces of Vector-Valued Functions
Authors: Kamińska A. , Turett B.
Published: December 2010

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].
Finite Spaces of Signatures
Authors: Powers Victoria
Published: December 2010

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.