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376. CMB 2006 (vol 49 pp. 55)

Dubois, Jérôme
 Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot. Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space, $\SU$, $\SL$, Adjoint representation, MonodromyCategories:57Q10, 57M27, 57M25

377. CMB 2006 (vol 49 pp. 3)

 On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the $L^p$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on $L^p$ provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25

378. CMB 2005 (vol 48 pp. 523)

Düvelmeyer, Nico
 Angle Measures and Bisectors in Minkowski Planes \begin{abstract} We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve. Keywords:Radon curves, Minkowski geometry, Minkowski planes,, angular bisector, angular measure, equiframed curvesCategories:52A10, 52A21

379. CMB 2005 (vol 48 pp. 614)

Tuncali, H. Murat; Valov, Vesko
 On Finite-to-One Maps Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that for every positive integer $p$ with $p\leq m+k+1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that each fiber of $f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most $\max\{k+m-p+2,1\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f\colon X\to Y$ be a $k$-dimensional map between compact metric spaces with $\dim Y\leq m$. Then: \begin{inparaenum}[\rm(1)] \item there exists a map $h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$; \item there exists a map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to Y\times\uin^{m+k+1}$ is $(k+1)$-to-one. \end{inparaenum} Keywords:finite-to-one maps, dimension, set-valued mapsCategories:54F45, 55M10, 54C65

380. CMB 2005 (vol 48 pp. 580)

Kot, Piotr
 Exceptional Sets in Hartogs Domains Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$, defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where $H$ is an open set in $\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $-\ln\mu$ is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$. For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function $f\in\mathbb{O}(\Omega)$ such that $E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.$ Keywords:boundary behaviour of holomorphic functions,, exceptional setsCategory:30B30

381. CMB 2005 (vol 48 pp. 561)

Foth, Philip
 A Note on Lagrangian Loci of Quotients We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group. Keywords:Quotients, involutions, real forms, Lagrangian lociCategory:53D20

382. CMB 2005 (vol 48 pp. 547)

Fehér, L. M.; Némethi, A.; Rimányi, R.
 Degeneracy of 2-Forms and 3-Forms We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation $\Lambda^k\C^n$. Keywords:Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theoryCategories:14N10, 57R45

383. CMB 2005 (vol 48 pp. 505)

Bouikhalene, Belaid
 On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$. Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.Categories:39B32, 39B42, 22D10, 22D12, 22D15

384. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
 The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by non-Euclidean translates" of $I$. Keywords:universal inner functionsCategories:32A35, 30D50, 47B38

385. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
 Short Geodesics of Unitaries in the $L^2$ Metric Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$ Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.Categories:46L51, 58B10, 58B25

386. CMB 2005 (vol 48 pp. 161)

Betancor, Jorge J.
 Hankel Convolution Operators on Spaces of Entire Functions of Finite Order In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals. Keywords:Hankel transform, convolution, entire functions, finite orderCategory:46F12

387. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
 Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20

388. CMB 2005 (vol 48 pp. 267)

Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.
 Continuous Adjacency Preserving Maps on Real Matrices It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective. Keywords:adjacency of matrices, continuous preservers, affine transformationsCategories:15A03, 15A04.

389. CMB 2005 (vol 48 pp. 260)

Oberlin, Daniel M.
 A Restriction Theorem for a \\$k$-Surface in $\mathbb R ^n$ We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in $\mathbb R ^n$, where $n=k(k+3)/2$. Keywords:Fourier restrictionCategory:42B10

390. CMB 2005 (vol 48 pp. 195)

Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
 On Suslinian Continua A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set. Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finiteCategories:54F15, 54D15, 54F50

391. CMB 2005 (vol 48 pp. 180)

Cynk, Sławomir; Meyer, Christian
 Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds We study Calabi--Yau manifolds constructed as double coverings of $\mathbb{P}^3$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi--Yau manifolds and 14 families with $h^{1,2}=1$. The modularity conjecture is verified for all the rigid examples. Keywords:Calabi--Yau, double coverings, modular formsCategories:14G10, 14J32

392. CMB 2005 (vol 48 pp. 147)

 Baker-Type Estimates for Linear Forms in the Values of $q$-Series We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field~$\II$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincar\'e-type equations and the connection between the solutions of these functional equations and the generalized Heine series. Keywords:measure of linear independence, $q$-seriesCategories:11J82, 33D15

393. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
 Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc$ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$. Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fieldsCategories:11A55, 11D09, 11R11

394. CMB 2004 (vol 47 pp. 624)

Zhang, Xi
 A Compactness Theorem for Yang-Mills Connections In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of curvature is compact in $C^{\infty}$ topology. Keywords:Yang-Mills connection, vector bundle, gauge transformationCategories:58E20, 53C21

395. CMB 2004 (vol 47 pp. 530)

Iranmanesh, A.; Khosravi, B.
 A Characterization of $PSU_{11}(q)$ Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PSU_{11}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PSU_{11}(q)$ are obtained. Keywords:Prime graph, order component, finite group,simple groupCategories:20D08, 20D05, 20D60

396. CMB 2004 (vol 47 pp. 481)

Bekjan, Turdebek N.
 A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space. Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic functionCategories:46B20, 52A07, 60G44

397. CMB 2004 (vol 47 pp. 321)

Bullejos, M.; Cegarra, A. M.
 Classifying Spaces for Monoidal Categories Through Geometric Nerves The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories. Keywords:monoidal category, pseudo-simplicial category,, simplicial set, classifying space, homotopy typeCategories:18D10, 18G30, 55P15, 55P35, 55U40

398. CMB 2004 (vol 47 pp. 332)

Charette, Virginie; Goldman, William M.; Jones, Catherine A.
 Recurrent Geodesics in Flat Lorentz $3$-Manifolds Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely classified when $\Gamma$ is cyclic. This implies that for any pair of periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$. Keywords:geometric structures on low-dimensional manifolds, notions of recurrenceCategories:57M50, 37B20

399. CMB 2004 (vol 47 pp. 417)

Niu, Pengcheng; Han, Yanwu; Han, Junqiang
 A Hopf Type Lemma and a CR Type Inversion for the Generalized Greiner Operator In this paper we establish a Hopf type lemma and a CR type inversion for the generalized Greiner operator. Some nonlinear Keywords:Hopf type lemma, CR inversion, Liouville type, theorem generalized Greiner operator

400. CMB 2004 (vol 47 pp. 398)

McKinnon, David
 A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating rational curves such that the density of rational points on $U-Z_U$ is strictly less than the density of rational points on $Z_U$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets $Z_U$ for successively smaller sets $U$. In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, heightCategories:11G35, 14G05
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