bluuebunny/crossref_metadata_2025 huggingface dataset:
Large Time Behavior for the Cubic Nonlinear Schrödinger Equation
Authors: Hayashi Nakao , Naumkin Pavel I.
Published: December 2010

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension

Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data are small and such that for some n ∈ Z, and , then the solution has an additional logarithmic timedecay in the short range region . In the far region the asymptotics have a quasilinear character.
Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions
Authors: Binding Paul , Ćurgus Branko
Published: December 2010

Form domains are characterized for regular 2n-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniformconvergence of the first n − 1 derivatives.
The Weil Character of the Unitary Group Associated to a Finite Local Ring
Authors: Gow Roderick , Szechtman Fernando
Published: December 2010

LetR/R be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote byUn(R) the unitary group of ranknassociated toR/R. The Weil representation ofUn(R) is defined and its character is explicitly computed.
Finsler Metrics with K = 0 and S = 0
Authors: Shen Zhongmin
Published: December 2010

In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.
Homotopy Decompositions Involving the Loops of Coassociative Co-H Spaces
Authors: Theriault Stephen D.
Published: December 2010

James gave an integral homotopy decomposition of ∑Ω∑X, Hilton-Milnor one for Ω(∑X ∨ ∑Y), and Cohen-Wu gave p-local decompositions of Ω∑X if X is a suspension. All are natural. Using idempotents and telescopes we show that the James andHilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative co-H spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative co-H space.
Two Algorithms for a Moving Frame Construction
Authors: Kogan Irina A.
Published: December 2010

The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.
The Maximum Number of Points on a Curve of Genus 4 over is 25
Authors: Savitt David
Published: December 2010

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.
Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators
Authors: Stessin Michael , Zhu Kehe
Published: December 2010

Every classical inner function φ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when φ(z) = z. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.
Gaussian Estimates in Lipschitz Domains
Authors: Varopoulos N. Th.
Published: December 2010

We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain.
Certain Operators with Rough Singular Kernels
Authors: Chen Jiecheng , Fan Dashan , Ying Yiming
Published: December 2010

We study the singular integral operator

defined on all test functions f, where b is a bounded function, α ≥ 0, Ω (yʻ) is an integrable function on the unit sphere Sn-1 satisfying certain cancellation conditions. We prove that, for 1 < p < ∞, TΩ,α extends to a bounded operator from the Sobolev space to the Lebesgue space Lp with Ω being a distribution in the Hardy space Hq(Sn-1) where . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TΩ,α on the Hardy spaces, as well as the boundedness for the truncated maximal operator .
Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign
Authors: Cao Xifang , Kong Qingkai , Wu Hongyou , Zettl Anton
Published: December 2010

For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.
Lie Groups of Measurable Mappings
Authors: Glöckner Helge
Published: December 2010

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space (X; Σ, μ) and (possibly infinite-dimensional) Lie group G, we construct a Lie group L∞(X; G), which is a Fréchet-Lie group if G is so. We also show that the weak direct product of an arbitrary family (Gi)i∈I of Lie groups can be made a Lie group, modelled on the locally convex direct sum .
Norms of Complex Harmonic Projection Operators
Authors: Casarino Valentina
Published: December 2010

In this paper we estimate the (Lp – L2)-norm of the complex harmonic projectors πℓ,ℓ′, 1 ≤ p ≤ 2, uniformly with respect to the indexes ℓ, ℓ′. We provide sharp estimates both for the projectors πℓ,ℓ′, when ℓ, ℓ′ belong to a proper angular sector in ℕ × ℕ, and for the projectors πℓ0 and π0ℓ. The proof is based on an extension of a complex interpolation argument by C. Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the L1 norm on the unit sphere of ℝ2n.
The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups
Authors: Katsura Takeshi
Published: December 2010

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra andK-groups of our algebras.
Euclidean Rings of Algebraic Integers
Authors: Harper Malcolm , Murty M. Ram
Published: December 2010

Let K be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.
High-Dimensional Graphical Networks of Self-Avoiding Walks
Authors: Holmes Mark , Járai Antal A. , Sakai Akira , Slade Gordon
Published: December 2010

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on ℤd in dimensions d > 4.
Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard
Authors: Kenny Robert
Published: December 2010

The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set M0 of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding ofM0 [Mor91] and estimates using convex fronts related to the derivative of the billiard ball map [Sto03] to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.
Structure of Modules Induced from Simple Modules with Minimal Annihilator
Authors: Khomenko Oleksandr , Mazorchuk Volodymyr
Published: December 2010

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.
Theta Series, Eisenstein Series and Poincaré Series over Function Fields
Authors: Pál Ambrus
Published: December 2010

We construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.
The Best Constants Associated with some Weak Maximal Inequalities in Ergodic Theory
Authors: Demeter Ciprian
Published: December 2010

We introduce a new device of measuring the degree of the failure of convergence in the ergodic theorem along subsequences of integers. Relations with other types of bad behavior in ergodic theory and applications to weighted averages are also discussed.
Infinite-Dimensional Polyhedrality
Authors: Fonf Vladimir P. , Veselý Libor
Published: December 2010

This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.

We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).
Geodesics in a Manifold with Heisenberg Group as Boundary
Authors: Ni Yilong
Published: December 2010

The Heisenberg group is considered as the boundary of a manifold. A class of hypersurfaces in this manifold can be regarded as copies of the Heisenberg group. The properties of geodesics in the interior and on the hypersurfaces are worked out in detail. These properties are strongly related to those of the Heisenberg group.
Differentiability Properties of Optimal Value Functions
Authors: Penot Jean-Paul
Published: December 2010

Differentiability properties of optimal value functions associated with perturbed optimization problems require strong assumptions. We consider such a set of assumptions which does not use compactness hypothesis but which involves a kind of coherence property. Moreover, a strict differentiability property is obtained by using techniques of Ekeland and Lebourg and a result of Preiss. Such a strengthening is required in order to obtain genericity results.
Finding and Excluding b-ary Machin-Type Individual Digit Formulae
Authors: Borwein Jonathan M. , Borwein David , Galway William F.
Published: December 2010

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, such as allowing single digits in their base b expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base b digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant κ has a Machin-type BBP formula. In particular, given b ∈ ℕ, b > 2, b not a proper power, a b-ary Machin-type BBP arctangent formula for κ is a formula of the form κ = Σmam arctan(–b–m), am ∈ ℚ, while when b = 2, we also allow terms of the form am arctan(1/(1 – 2m)). Of particular interest, we show that π has no Machin-type BBP arctangent formula when b ≠ 2. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.
Fubini's Theorem for Ultraproducts of Noncommutative Lp-Spaces
Authors: Junge Marius
Published: December 2010

Let (ℳi)i∈I, be families of von Neumann algebras and be ultrafilters in I, J, respectively. Let 1 ≤ p < ∞ and n ∈ ℕ. Let x1,… ,xn in ΠLp(ℓi ) and y1,… ,yn in be bounded families. We show the following equality

For p = 1 this Fubini type result is related to the local reflexivity of duals of C*-algebras. This fails for p = ∞.
Non-Orientable Surfaces and Dehn Surgeries
Authors: Matignon D. , Sayari N.
Published: December 2010

LetKbe a knot inS3. This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface. We look at the slopep/qof the surgery, the Euler characteristic χ() of the surface and the intersection numbersbetweenand the core of the Dehn surgery. We prove that if χ() ≥ 15 – 3q, thens= 1. Furthermore, ifs= 1 thenq≤ 4 – 3χ() orKis cabled andq≤ 8 – 5χ(). As consequence, ifKis hyperbolic and χ() = –1, thenq≤ 7.
Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form
Authors: Rouleux Michel
Published: December 2010

We prove that a Hamiltonianp∈C∞(T*Rn) is locally integrable near a non-degenerate critical point ρ0of the energy, provided that the fundamental matrix at ρ0has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in theC∞sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that whenpis holomorphic near ρ0∈T*Cn, then Repbecomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge,i.e., pmay not be integrable. These normal forms also hold in the semi-classical frame.
Cycle-Level Intersection Theory for Toric Varieties
Authors: Thomas Hugh
Published: December 2010

This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan inN, the choice consists of giving an inner product or a complete flag forMQ= ℚ ⊗ Hom(N, ℤ), or more generally giving for each cone σ in the fan a linear subspace ofMQcomplementary to σ⊥, satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. IfXis simplicial (for instance, ifXis non-singular), we obtain a commutative ring structure to the invariant cycles ofXwith rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.
The Fourier Algebra for Locally Compact Groupoids
Authors: Paterson Alan L. T.
Published: December 2010

We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
Equivariant Formality for Actions of Torus Groups
Authors: Scull Laura
Published: December 2010

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use S1-equivariant minimal models to give a number of examples of S1-spaces illustrating the properties of the various definitions.
Second Order Operators on a Compact Lie Group
Authors: Fegan H. D. , Steer B.
Published: December 2010

We describe the structure of the space of second order elliptic differential operators on a homogenous bundle over a compact Lie group. Subject to a technical condition, these operators are homotopic to the Laplacian. The technical condition is further investigated, with examples givenwhere it holds and others where it does not. Since many spectral invariants are also homotopy invariants, these results provide information about the invariants of these operators.
On a Conjecture of Birch and Swinnerton-Dyer
Authors: Kuo Wentang , Murty M. Ram
Published: December 2010

Let E/Q be an elliptic curve defined by the equation For a prime p, define

As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c,

Let αp and βp be the eigenvalues of the Frobenius at p. Define
Reducibility of Generalized Principal Series
Authors: Muić Goran
Published: December 2010

In this paper we describe reducibility of non-unitary generalized principal series for classical p-adic groups in terms of the classification of discrete series due to Moeglin and Tadić.
Hermitian Yang–Mills–Higgs Metrics on Complete Kähler Manifolds
Authors: Zhang Xi
Published: December 2010

In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact Kähler manifolds.
Deformations ofG2and Spin(7) Structures
Authors: Karigiannis Spiro
Published: December 2010

We consider some deformations ofG2-structures on 7-manifolds. We discover a canonical way to deform aG2-structure by a vector field in which the associated metric gets “twisted” in some way by the vector cross product. We present a system of partial differential equations for an unknown vector fieldwwhose solution would yield a manifold with holonomyG2. Similarly we consider analogous constructions for Spin(7)-structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the Spin(7) case.
Power Residues of Fourier Coefficients of Modular Forms
Authors: Weston Tom
Published: December 2010

Let ρ: GQ → GLn(Qℓ) be a motivic ℓ-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under ρ is an m-th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of ρ is open. We further conjecture that for such ρ the set of these primes p is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem(which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.
Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups
Authors: Dungey Nick
Published: December 2010

LetKbe a function on a unimodular locally compact groupG, and denote byKn=K*K* · · · *Kthen-th convolution power ofK. Assuming thatKsatisfies certain operator estimates inL2(G), we give estimates of the normsandfor largen. In contrast to previous methods for estimating, we do not need to assume that the functionKis a probability density or nonnegative. Our results also adapt for continuous time semigroups onG. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
Authors: Zhitomirskii M.
Published: December 2010

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset N of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to N of the contact structure. This is a generalization of the Darboux-Givental’ theoremfor smooth submanifolds of a contactmanifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold N in terms of the algebraic restriction of the contact structure to N. In particular, the algebraic restriction is zero if and only if N is contained in a smooth Legendrian submanifold of M.
C*-Algebras of Irreversible Dynamical Systems
Authors: Exel R. , Vershik A.
Published: December 2010

We show that certain C*-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.
Infinite Dimensional Representations of Canonical Algebras
Authors: Reiten Idun , Ringel Claus Michael
Published: December 2010

The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to themore general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.
Stark's Conjecture and New Stickelberger Phenomena
Authors: Snaith Victor P.
Published: December 2010

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraicK–groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related tol–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension whenr= –2,–4,–6, … the Coates–Sinnott conjecturemerely predicts that zero annihilatesK–2rof the ring ofS–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.
On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself
Authors: Dijkstra Jan J. , Mill Jan van
Published: December 2010

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line ℝ, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers ℚ onto itself is homeomorphic to the infinite power of ℚ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of ℚ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ℝn for n ≥ 2, by linking the groups in question with Erdős space.
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II
Authors: Greither Cornelius , Kučera Radan
Published: December 2010

We prove, for a field K which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of K by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin–Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part. The present work builds on and strengthens previous work of the authors; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.
Centralizers and Twisted Centralizers: Application to Intertwining Operators
Authors: Yu Xiaoxiang
Published: December 2010

The equality of the centralizer and twisted centralizer is proved based on a case-by-case analysis when the unipotent radical of a maximal parabolic subgroup is abelian. Then this result is used to determine the poles of intertwining operators.
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
Authors: Chiang Yik-Man , Ismail Mourad E. H.
Published: December 2010

We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.
Distributions invariantes sur les groupes réductifs quasi-déployés
Authors: Courtès François
Published: December 2010

Soit F un corps local non archimédien, et G le groupe des F-points d’un groupe réductif connexe quasi-déployé défini sur F. Dans cet article, on s’intéresse aux distributions sur G invariantes par conjugaison, et à l’espace de leurs restrictions à l’algèbre de Hecke ℋ des fonctions sur G à support compact biinvariantes par un sous-groupe d’Iwahori I donné. On montre tout d’abord que les valeurs d’une telle distribution sur ℋ sont entièrement déterminées par sa restriction au sous-espace de dimension finie des éléments de ℋ à support dans la réunion des sous-groupes parahoriques de G contenant I. On utilise ensuite cette propriété pour montrer, moyennant certaines conditions sur G, que cet espace est engendré d’une part par certaines intégrales orbitales semi-simples, d’autre part par les intégrales orbitales unipotentes, en montrant tout d’abord des résultats analogues sur les groupes finis.
The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates
Authors: Bownik Marcin , Speegle Darrin
Published: December 2010

The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.
Intégrales orbitales tordues sur GL(n, F) et corps locaux proches : applications
Authors: Henniart Guy , Lemaire Bertrand
Published: December 2010

Soient F un corps commutatif localement compact non archimédien, G = GL(n, F) pour un entier n ≥ 2, et κ un caractère de F× trivial sur (F×)n. On prouve une formule pour les κ-intégrales orbitales régulières sur G permettant, si F est de caractéristique > 0, de les relever à la caractéristique nulle. On en déduit deux résultats nouveaux en caractéristique à 0 : le “lemme fondamental” pour l’induction automorphe, et une version simple de la formule des traces tordue locale d’Arthur reliant κ-intégrales orbitales elliptiques et caractères κ-tordus. Cette formule donne en particulier, pour une série κ-discrète de G, les κ-intégrales orbitales elliptiques d’un pseudo-coefficient comme valeurs du caractère κ-tordu.
The General Structure of G-Graded Contractions of Lie Algebras I. The Classification
Authors: Weimar-Woods Evelyn
Published: December 2010

We give the general structure of complex (resp., real) G-graded contractions of Lie algebras where G is an arbitrary finite Abelian group. For this purpose, we introduce a number of concepts, such as pseudobasis, higher-order identities, and sign invariants. We characterize the equivalence classes of G-graded contractions by showing that our set of invariants (support, higher-order identities, and sign invariants) is complete, which yields a classification.
Some Results on the Schroeder–Bernstein Property for Separable Banach Spaces
Authors: Ferenczi Valentin , Galego Elόi Medina
Published: December 2010

We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is 2ℵ0. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.
On the Convergence of a Class of Nearly Alternating Series
Authors: Foster J. H. , Serbinowska Monika
Published: December 2010

Let C be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If α is of the first type and (ck) ∈ C, then ∑(—1)⎿ck⏌ converges if and only if ck log k → 0. If α is of the second type and (ck) ∈ C, then ∑(—1)⎿ck⏌ converges if and only if ∑ ck/k converges. An example of a quadratic irrational of the first type is and an example of the second type is . The analysis of this problem relies heavily on the representation of α as a simple continued fraction and on properties of the sequences of partial sums and double partial sums .
On Two Exponents of Approximation Related to a Real Number and Its Square
Authors: Roy Damien
Published: December 2010

For each real number ξ, let denote the supremum of all real numbers λ such that, for each sufficiently large X, the inequalities |x0| ≤ X, |x0ξ – x1| ≤ X–λ and |x0ξ2 – x2| ≤ X–λ admit a solution in integers x0, x1 and x2 not all zero, and let denote the supremum of all real numbers ω such that, for each sufficiently large X, the dual inequalities |x0 + x1ξ + x2ξ2| ≤ X–ω, |x1| ≤ X and |x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponent where ξ ranges through all real numbers with [ℚ(ξ):ℚ] > 2 form a dense subset of the interval while, for the same values of ξ, the dual exponents form a dense subset of . Part of the proof rests on a result of V. Jarník showing that for any real number ξ with [ℚ(ξ):ℚ] > 2.
Harmonic Analysis on Metrized Graphs
Authors: Baker Matt , Rumely Robert
Published: December 2010

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.
Weighted Inequalities for Hardy–Steklov Operators
Authors: Bernardis A. L. , Martín-Reyes F. J. , Ortega Salvador P.
Published: December 2010

We characterize the pairs of weights (v, w) for which the operator with s and h increasing and continuous functions is of strong type (p, q) or weak type (p, q) with respect to the pair (v, w) in the case 0 < q < p and 1 < p < ∞. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on s and h and we obtain that the strong type inequality (p, q), q < p, is characterized by the fact that the function

belongs to Lr(gqw), where 1/r = 1/q – 1/p and the supremum is taken over all c and d such that c ≤ x ≤ d and s(d) ≤ h(c).
Computations of Elliptic Units for Real Quadratic Fields
Authors: Dasgupta Samit
Published: December 2010

Let K be a real quadratic field, and p a rational prime which is inert in K. Let α be amodular unit on Γ0(N). In an earlier joint article with Henri Darmon, we presented the definition of an element u(α, τ) ∊ Kxp attached to α and each τ ∈ K. We conjectured that the p-adic number u(α, τ) lies in a specific ring class extension of K depending on τ , and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of u(α, τ). This article provides computational evidence for these conjectures. We present an efficient algorithm for computing u(α, τ), and implement this algorithm with the modular unit α(z) = Δ(z)2Δ(4z)/Δ(2z)3. Using p = 3, 5, 7, and 11, and all real quadratic fields K with discriminant D < 500 such that 2 splits in K and K contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define u(α, τ) is shown to be Z-valued rather than only Zp ∩ Q-valued; this is an improvement over our previous result and allows for a precise definition of u(α, τ), instead of only up to a root of unity.
Cardinal Invariants of AnalyticP-Ideals
Authors: Hernández-Hernández Fernando , Hrušák Michael
Published: December 2010

We study the cardinal invariants of analyticP-ideals, concentrating on the idealof asymptotic density zero. Among other results we prove min {b, cov ()} ≤ cov*() ≤ max{b, non()}.
Jet Modules
Authors: Billig Yuly
Published: December 2010

In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of suchmodules is given by spaces of jets of tensor fields.
A Weighted L2-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds
Authors: Finster Felix , Kraus Margarita
Published: December 2010

We derive a weighted L2-estimate of theWitten spinor in a complete Riemannian spin manifold (Mn, g) of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of M enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of M.
On the Restriction to 𝒟* × 𝒟* of Representations of p-adic GL2(𝒟)
Authors: Raghuram A.
Published: December 2010

Let be a division algebra over a nonarchimedean local field. Given an irreducible representation π of GL2(), we describe its restriction to the diagonal subgroup × . The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that π is × -distinguished if and only if π admits a Shalika model. We further prove that if is a quaternion division algebra then the twisted Jacquetmodule is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to × in the quaternionic case.
Some Remarks on a Paper of McCarthy
Authors: Erdös P.
Published: December 2010

As usual we denote the number of integers not exceeding n and relatively prime to n by Euler's ϕ function ϕ(n). Lehmer calls the ϕ(n) integers

the totatives of n.

Denote by ϕ(k,l,n) the number of a's satisfying

If nl ≡ 0 (mod k) or n(l + 1) ≡ 0 (mod k) then, since n > k, (n - l/k, n) > 1 and (n(l + 1)/k, n)>1 respectively. Thus ϕ(k, l, n) is the number of totatives of n satisfying
On a Normal Form of the Orthogonal Transformation II
Authors: Zassenhaus Hans
Published: December 2010

In this section we continue to study the indecomposable matrix pairs adopting the same notation as in part I of this paper.

If the matrix A is regular and if it is symmetric or anti-symmetric such that

3.1

and if the matrix pair (X, A) is indecomposable then the corresponding representation space either is indecomposable or it is the direct sum of two indecomposable invariant subspaces. These are operator isomorphic if and only if the minimal polynomial mX of X is equal to (x-δ)μ where

3.2

at any rate there is even a decomposition of the representation space into the direct sum of two isotropic indecomposable invariant subspaces provided the characteristic of F is not 2.
Outline of an Introduction to Mathematical Logic II
Authors: Robinson Abraham
Published: December 2010

It is the purpose of the calculus described in the preceding section (i) to yield as theorems all tautologies which can be formulated in terms of the given variables, and (ii) to yield no other formulae as theorems. We shall establish presently that these two conditions are indeed satisfied. However, there are questions of a more general character which can be solved both with regard to the present calculus and with regard to many other calculi, including some with a more comprehensive vocabulary.
On the Distribution of Primitive Lattice Points in the Plane
Authors: Chalk J.H.H. , Erdos P.
Published: December 2010

Let 1,θ1, θ2, …,θn be real numbers linearly independent over the rational field and let α1, α2,…, αn be arbitrary real numbers. Then, to each N > 0 and ε > 0, there correspond integers

which satisfy the set of inequalities

A
Discontinuous Functions with the Darboux Property
Authors: Halperin Israel
Published: December 2010

If f(x) is real-valued and continuous, it has the property that it takes on all intermediate values when it passes from one value to another. This means that whenever f(x1) and f(x2) are different and u is any number between them, then f(x) = u for at least one x between x1 and x2. We shall call this the Darboux property.
Solutions of the Functional Equation (f(x))2 - f(x2) = h(x)
Authors: Baker I. N.
Published: December 2010

In a recent paper [2] Lambek and Moser have introduced the functional equations

1

and

2

in connection with some problems of number theory, in particular in dealing with the sums by pairs of sets of integers. The second may be put into the same form as (1) by the substitutions x = In z, f(ln z) = F(z), h(ln z) = H(z).
Reflexive Homomorphic Relations
Authors: Findlay G. D.
Published: December 2010

It is well known that a symmetric and transitive relation on a set is reflexive wherever it is defined. In this note we show that a converse is true for homomorphic relations in certain classes of algebras.

Consider a class of similar algebras which contains the sub-algebras and quotient algebras of each of its members. Assume also that the direct product A x B of each pair A, B in is also an algebra belonging to . The algebras of , being similar, have the same set of operations. We observe that other operations, called compound operations, may be obtained by composition from the assigned operations.
On a Problem of Erdös and Szekeres
Authors: Atkinson F. V.
Published: December 2010

Write

1

2

where the maximum is over all real θ, and the lower bound is over all sets of positive integers a1 ≤ a2 ≤ … ≤ an. The problem of the order of magnitude of f(n) was posed by Erdös and Szekeres [1], side by side with a number of other interesting questions. Writing g(n) = log f(n), it is obvious that g(n) is sub-additive, in the sense that g(m+n) ≤ g(m) + g(n), and also that g(1) = log 2, so that g(n) ≤ n log 2.
Some Extensions of the Hausdorff-Young and Paley Theorems
Authors: Bullen P. S.
Published: December 2010

Orthonormal sequences, o.n. s., {ϕn} defined on [0,1] and satisfying

1

have been studied in [3] and [1]. One of the objects of this paper is to indicate that the methods used to study such o. n. s. can be used for a much wider class, and that, although there seems to be no super theorem to cover all cases, a knowledge of the results and methods of proof in some fairly broad special cases enables one to state and prove theorems for other classes of o. n. s.
L'Esperance Mathematique du Nombre de Nombres Premiers Aleatoires Inferieurs ou Egaux A x
Authors: Lachapelle Benoît
Published: December 2010

D. Hawkins [1] propose un modèle stochastique simple de la distribution des nombres premiers. II construit un crible aléatoire analogue à celui d' Eratosthène. La règle pour obtenir une suite de nombres premiers aléatoires est la suivante: 2 est un nombre premier aléatoire, que nous désignerons par P1; chaque entier plus grand que P1 est ensuite criblé avec probabilité 1/P1 d' être éliminé par le crible; si P2 est le premier entier plus grand que P1 qui n' a pas été éliminé par le crible, on crible chaque nombre plus grand que p2, qui n' a pas été éliminé par le crible à la première opération, avec probabilité 1/P2 d'être élimine; à la niéme étape, si Pn est le premier nombre plus grand que Pn, qui n' a pas été éliminé par les n - i premières opérations du crible, on crible chaque nombre plus grand que Pn, qui n' a pas été éliminé par les n - 1 premières opérations du crible, avec probabilité l / Pn d' être éliminé. On obtient ainsi une suite { Pn} n = 1, 2, 3, … de nombres premiers aléatoires.
Those Stirling Numbers Again
Authors: Mendelsohn N. S.
Published: December 2010

In his book [1] Combinatorial Analysis, J. Riordan (p. 32) refers to the continual rediscovery of the Stirling numbers. The author of this note has been surprised on many occasions by the number of different environments in which these numbers make a natural appearance and, in fact, this article is concerned with just such an occurrence. The connection is made in a study of the exponential generating function of nr.
Right and Left Orthogonality
Authors: Wild Jonathan
Published: December 2010

Let V be a vector space over an arbitrary field F. In V a bilinear form

is given. If f is symmetric [(x, y) ≡ (y, x)] or skew-symmetric [(x, y) + (y, x) ≡ 0], then

1

Thus right and left orthogonality coincide. It is well known that (1) implies conversely that f is either symmetric or skew-symmetric in V. We wish to give a simple proof of this result.
On a Problem in Geometrical Probability
Authors: Melzak Z.A.
Published: December 2010

We consider the following problem. Let A = (aij) be a symmetric n x n matrix of non-negative numbers with aii = 0 for all i, and let n points x1, x2, …, xn be chosen at random from the interval [0, L]. What is the probability P = P(n, A, L) that for all i and j, |xi - xj| ≥ aij?
On the Representation of Boolean Algebras
Authors: Bruns Günter
Published: December 2010

Let B be a Boolean algebra and let ℳ and n be two systems
of subsets of B, both containing all finite subsets of B. Let us assume
further that the join ∨M of every set M∊ℳ and the meet ∧N of every set
N∊n exist. Several authors have treated the question
under which conditions there exists an isomorphism φ between B and a field δ
of sets, satisfying the conditions:
An Analogue of the Multinomial Theorem
Authors: Narayana T. V.
Published: December 2010

Let xi, yi (i=1,2,…, t) and n be non-negative
integers. A function (n; x1, …, xt) may be defined
recursively as follows: let(0;0,…, 0)=1 and

1
On the Score Sequence of an N-Partite Tournament
Authors: Moon J.W.
Published: December 2010

Let there be given n(≥l) distinct sets of points
Pi=(Pi1,…, Pini), with ni≥1, for i=1,…, n. If joining each pair of
points not in the same set is a line oriented towards one, and only one,
point of the pair the resulting configuration will be called an n-partite
tournament. If the line joining Pij and Pkl is
oriented towards the latter point we shall indicate this by Pij
→Pkl, and similarly if the orientation is in the opposite
sense.