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376. CMB 2009 (vol 52 pp. 95)

Miranian, L.
 Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel--Darboux formulas are presented for the first time. Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-DarbouxCategory:42C99

377. CMB 2009 (vol 52 pp. 66)

Dryden, Emily B.; Strohmaier, Alexander
 Huber's Theorem for Hyperbolic Orbisurfaces We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces. Keywords:Huber's theorem, length spectrum, isospectral, orbisurfacesCategories:58J53, 11F72

378. CMB 2008 (vol 51 pp. 584)

Purbhoo, Kevin; Willigenburg, Stephanie van
 On Tensor Products of Polynomial Representations We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood--Richardson coefficients that are non-zero, and a condition on Schur non-negativity. Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negativeCategories:05E05, 05E10, 20C30

379. CMB 2008 (vol 51 pp. 570)

Lutzer, D. J.; Mill, J. van; Tkachuk, V. V.
 Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$ We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces. Keywords:regular filterbase, subcompact space, function space, discrete spaceCategories:54B10, 54C05, 54D30

380. CMB 2008 (vol 51 pp. 508)

Cavicchioli, Alberto; Spaggiari, Fulvia
 A Result in Surgery Theory We study the topological $4$-dimensional surgery problem for a closed connected orientable topological $4$-manifold $X$ with vanishing second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has one end and $F(r)$ is the free group of rank $r\ge 1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups. Keywords:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly mapCategories:57N65, 57R67, 57Q10

381. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
 Universal Inner Functions on the Ball It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$, there exists an inner function $I$ such that the family of non-Euclidean translates" $(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of $H^\infty(\bn)$. Keywords:inner functions, automorphisms of the ball, universalityCategories:32A35, 30D50, 47B38

382. CMB 2008 (vol 51 pp. 487)

Betancor, Jorge J.; Mart\'{\i}nez, Teresa; Rodr\'{\i}guez-Mesa, Lourdes
 Laplace Transform Type Multipliers for Hankel Transforms In this paper we establish that Hankel multipliers of Laplace transform type are bounded from $L^p(w)$ into itself when $1 Keywords:Hankel transform, Laplace transform, multiplier, CalderÃ³n--ZygmundCategory:42 383. CMB 2008 (vol 51 pp. 627) Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.  Summation of Series over Bourget Functions In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann$\z$function and Dirichlet functions$\eta$,$\la$,$\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums. Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functionsCategories:33C10, 11M06, 65B10 384. CMB 2008 (vol 51 pp. 618) Valmorin, V.  Vanishing Theorems in Colombeau Algebras of Generalized Functions Using a canonical linear embedding of the algebra${\mathcal G}^{\infty}(\Omega)$of Colombeau generalized functions in the space of$\overline{\C}$-valued$\C$-linear maps on the space${\mathcal D}(\Omega)$of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one. Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functionsCategories:32A60, 45P05, 46F30 385. CMB 2008 (vol 51 pp. 593) Ros{\l}anowski, Andrzej; Stepr\={a}ns, Juris  Chasing Silver We show that limits of CS iterations of the$n$-Silver forcing notion have the$n$-localization property. Keywords:$n$-localization property, the Silver forcing, CS iterationsCategories:03E40, 03E35 386. CMB 2008 (vol 51 pp. 359) Cho, Jong Taek; Ki, U-Hang  Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type$(A)$in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator. Keywords:complex space form, real hypersurface, structure Jacobi operatorCategories:53B20, 53C15, 53C25 387. CMB 2008 (vol 51 pp. 448) Sasahara, Toru  Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed. Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifoldsCategories:53C42, 53C40 388. CMB 2008 (vol 51 pp. 439) Samei, Karim  On the Maximal Spectrum of Semiprimitive Multiplication Modules An$R$-module$M$is called a multiplication module if for each submodule$N$of$M$,$N=IM$for some ideal$I$of$R$. As defined for a commutative ring$R$, an$R$-module$M$is said to be semiprimitive if the intersection of maximal submodules of$M$is zero. The maximal spectra of a semiprimitive multiplication module$M$are studied. The isolated points of$\Max(M)$are characterized algebraically. The relationships among the maximal spectra of$M$,$\Soc(M)$and$\Ass(M)$are studied. It is shown that$\Soc(M)$is exactly the set of all elements of$M$which belongs to every maximal submodule of$M$except for a finite number. If$\Max(M)$is infinite,$\Max(M)$is a one-point compactification of a discrete space if and only if$M$is Gelfand and for some maximal submodule$K$,$\Soc(M)$is the intersection of all prime submodules of$M$contained in$K$. When$M$is a semiprimitive Gelfand module, we prove that every intersection of essential submodules of$M$is an essential submodule if and only if$\Max(M)$is an almost discrete space. The set of uniform submodules of$M$and the set of minimal submodules of$M$coincide.$\Ann(\Soc(M))M$is a summand submodule of$M$if and only if$\Max(M)$is the union of two disjoint open subspaces$A$and$N$, where$A$is almost discrete and$N$is dense in itself. In particular,$\Ann(\Soc(M))=\Ann(M)$if and only if$\Max(M)$is almost discrete. Keywords:multiplication module, semiprimitive module, Gelfand module, Zariski topologCategory:13C13 389. CMB 2008 (vol 51 pp. 386) Lan, K. Q.; Yang, G. C.  Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory The well-known Falkner--Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to$\lambda\pi/2$, where$\lambda\in \mathbb R$is a parameter involved in the equation. It is known that there exists$\lambda^{*}<0$such that the equation with suitable boundary conditions has at least one positive solution for each$\lambda\ge \lambda^{*}$and has no positive solutions for$\lambda<\lambda^{*}$. The known numerical result shows$\lambda^{*}=-0.1988$. In this paper,$\lambda^{*}\in [-0.4,-0.12]$is proved analytically by establishing a singular integral equation which is equivalent to the Falkner--Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner--Skan equation. Keywords:Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutionsCategories:34B16, 34B18, 34B40, 76D10 390. CMB 2008 (vol 51 pp. 378) Izuchi, Kou Hei  Cyclic Vectors in Some Weighted$L^p$Spaces of Entire Functions In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in$\Lp$. Let$f\in\Lp$and$f\poly$be contained in the space. We show that$f$is non-vanishing if and only if$f$is cyclic. Keywords:weighted$L^p$spaces of entire functions, cyclic vectorsCategories:47A16, 46J15, 46H25 391. CMB 2008 (vol 51 pp. 334) Ascah-Coallier, I.; Gauthier, P. M.  Value Distribution of the Riemann Zeta Function In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function. Keywords:Nevanlinna theory, deficiency, Riemann zeta functionCategory:30D35 392. CMB 2008 (vol 51 pp. 217) Filippakis, Michael E.; Papageorgiou, Nikolaos S.  A Multivalued Nonlinear System with the Vector$p$-Laplacian on the Semi-Infinity Interval We study a second order nonlinear system driven by the vector$p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis$\mathbb{R}_+.$Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval$[0,n]$and then via a diagonalization method we produce a solution for the original infinite time-horizon system. Keywords:semi-infinity interval, vector$p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous mapCategory:34A60 393. CMB 2008 (vol 51 pp. 310) Witbooi, P. J.  Relative Homotopy in Relational Structures The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs$(X,A)$where$X$is a poset and$A$is a subposet of$X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif. Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalenceCategories:55Q05, 54A05;, 18B30 394. CMB 2008 (vol 51 pp. 298) Tocón, Maribel  The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra. Keywords:extended affine Lie algebra, Lie torus, core, Kostrikin radicalCategories:17B05, 17B65 395. CMB 2008 (vol 51 pp. 283) Ravindra, G. V.  The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in$\mathbb P^{2n+1}$of degree at least$2+2/n$, the$n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25 396. CMB 2008 (vol 51 pp. 261) Neeb, Karl-Hermann  On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups An$n$-dimensional quantum torus is a twisted group algebra of the group$\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational$n$-dimensional quantum tori over any field. Moreover, we show that for$n = 2$the natural exact sequence describing the automorphism group of the quantum torus splits over any field. Keywords:quantum torus, normal form, automorphisms of quantum toriCategory:16S35 397. CMB 2008 (vol 51 pp. 195) Chen, Huaihui; Gauthier, Paul  Boundedness from Below of Composition Operators on$\alpha$-Bloch Spaces We give a necessary and sufficient condition for a composition operator on an$\alpha$-Bloch space with$\alpha\ge 1$to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen. Keywords:Bloch functions, composition operatorsCategories:32A18, 30H05 398. CMB 2008 (vol 51 pp. 236)  Konovalov, Victor N.; Kopotun, Kirill A. 399. CMB 2008 (vol 51 pp. 172) Alkan, Emre; Zaharescu, Alexandru  Consecutive Large Gaps in Sequences Defined by Multiplicative Constraints In this paper we obtain quantitative results on the occurrence of consecutive large gaps between$B$-free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication. Keywords:$B$-free numbers, consecutive gapsCategories:11N25, 11B05 400. CMB 2008 (vol 51 pp. 205) Duda, Jakub  On GÃ¢teaux Differentiability of Pointwise Lipschitz Mappings We prove that for every function$f\from X\to Y$, where$X$is a separable Banach space and$Y$is a Banach space with RNP, there exists a set$A\in\tilde\mcA$such that$f$is G\^ateaux differentiable at all$x\in S(f)\setminus A$, where$S(f)$is the set of points where$f$is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every$K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to$\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if$X$is Asplund,$f\from X\to\R$cone monotone,$g\from X\to\R$continuous convex, then there exists a point in$X$, where$f$is Hadamard differentiable and$g\$ is Fr\'echet differentiable. Keywords:GÃ¢teaux differentiable function, Radon-NikodÃ½m property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functionsCategories:46G05, 46T20
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