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51. CMB 2014 (vol 57 pp. 264)

Dai, Li; Dong, Jingcheng
 On Semisimple Hopf Algebras of Dimension $pq^n$ Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $pq^n$ over an algebraically closed field of characteristic $0$. This paper proves that $H$ must possess one of the following structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$. Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld doubleCategory:16W30

52. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
 Helicoidal Minimal Surfaces in a Finsler Space of Randers Type We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids. Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers spaceCategories:53A10, 53B40

53. CMB 2014 (vol 57 pp. 231)

Bagherian, J.
 On the Multiplicities of Characters in Table Algebras In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset which is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras. Keywords:table algebra, faithful module, multiplicity of characterCategories:20C99, 16G30

54. CMB 2013 (vol 57 pp. 870)

Parlier, Hugo
 A Short Note on Short Pants It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound. Keywords:hyperbolic surfaces, geodesics, pants decompositionsCategories:30F10, 32G15, 53C22

55. CMB 2013 (vol 57 pp. 845)

Lei, Antonio
 Factorisation of Two-variable $p$-adic $L$-functions Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, he showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix. Keywords:modular forms, p-adic L-functions, supersingular primesCategories:11S40, 11S80

56. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
 Constructive Proof of Carpenter's Theorem We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem. Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theoryCategories:42C15, 47B15, 46C05

57. CMB 2013 (vol 57 pp. 585)

Lehec, Joseph
 Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the PrÃ©kopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe. Keywords:functional inequalities, Brownian motionCategories:39B62, 60J65

58. CMB 2013 (vol 57 pp. 526)

Heil, Wolfgang; Wang, Dongxu
 On $3$-manifolds with Torus or Klein Bottle Category Two A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this latter property are called $\mathcal{G}_K$-contractible. We obtain a list of the closed $3$-manifolds that can be covered by two open $\mathcal{G}_K$-contractible subsets. This is applied to obtain a list of the possible closed prime $3$-manifolds that can be covered by two open $K$-contractible subsets. Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible setsCategories:57N10, 55M30, 57M27, 57N16

59. CMB 2013 (vol 57 pp. 119)

Mildenberger, Heike; Raghavan, Dilip; Steprans, Juris
 Splitting Families and Complete Separability We answer a question from Raghavan and SteprÄns by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under $\mathfrak{s} \leq \mathfrak{a}$, partially answering a question of Shelah. Keywords:maximal almost disjoint family, cardinal invariantsCategories:03E05, 03E17, 03E65

60. CMB Online first

 Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number $3/7$, has left-orderable fundamental group if the slope $r$ satisfies $0\le r \le 4$. Keywords:left-ordering, Dehn surgeryCategories:57M25, 06F15

61. CMB Online first

 Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number $3/7$, has left-orderable fundamental group if the slope $r$ satisfies $0\le r \le 4$. Keywords:left-ordering, Dehn surgeryCategories:57M25, 06F15

62. CMB 2013 (vol 57 pp. 439)

Yang, YanHong
 The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$ We prove that for every ordinary genus-$2$ curve $X$ over a finite field $\kappa$ of characteristic $2$ with $\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist $\textrm{SL}(2,\kappa[\![s]\!])$-representations of $\pi_1(X)$ such that the image of $\pi_1(\overline{X})$ is infinite. This result produces a family of examples similar to Laszlo's counterexample to de Jong's question regarding the finiteness of the geometric monodromy of representations of the fundamental group. Keywords:vector bundle, Frobenius pullback, representation, etale fundamental groupCategories:14H60, 14D05, 14G15

63. CMB 2013 (vol 57 pp. 381)

 On Complex Explicit Formulae Connected with the MÃ¶bius Function of an Elliptic Curve We study analytic properties function $m(z, E)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $|\Im{z}|\lt 2\pi$. Keywords:L-function, MÃ¶bius function, explicit formulae, elliptic curveCategories:11M36, 11G40

64. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
 Assouad-Nagata Dimension of Wreath Products of Groups Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$. Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groupsCategories:54F45, 55M10, 54C65

65. CMB 2013 (vol 57 pp. 310)

Hakamata, Ryoto; Teragaito, Masakazu
 Left-orderable Fundamental Group and Dehn Surgery on the Knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number $3/7$, has left-orderable fundamental group if the slope $r$ satisfies $0\le r \le 4$. Keywords:left-ordering, Dehn surgeryCategories:57M25, 06F15

66. CMB 2013 (vol 57 pp. 821)

Jeong, Imsoon; Kim, Seonhui; Suh, Young Jin
 Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator In this paper we give a characterization of a real hypersurface of Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb C}^{m+2})} } }$, which means a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in ${G_2({\mathbb C}^{m+2})}$, by the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$. Keywords:real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, Reeb parallel, structure Jacobi operatorCategories:53C40, 53C15

67. CMB 2013 (vol 57 pp. 506)

Galindo, César
 On Braided and Ribbon Unitary Fusion Categories We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure. Keywords:fusion categories, braided categories, modular categoriesCategories:20F36, 16W30, 18D10

68. CMB 2013 (vol 57 pp. 401)

Perrone, Domenico
 Curvature of $K$-contact Semi-Riemannian Manifolds In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold. Keywords:contact semi-Riemannian structures, $K$-contact structures, conformally flat manifolds, Einstein Lorentzian-Sasaki manifoldsCategories:53C50, 53C25, 53B30

69. CMB 2013 (vol 57 pp. 141)

Mukwembi, Simon
 Size, Order, and Connected Domination We give a sharp upper bound on the size of a triangle-free graph of a given order and connected domination. Our bound, apart from strengthening an old classical theorem of Mantel and of TurÃ¡n , improves on a theorem of Sanchis. Further, as corollaries, we settle a long standing conjecture of Graffiti on the leaf number and local independence for triangle-free graphs and answer a question of Griggs, Kleitman and Shastri on a lower bound of the leaf number in triangle-free graphs. Keywords:size, connected domination, local independence number, leaf numberCategory:05C69

70. CMB 2013 (vol 57 pp. 449)

Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim
 ZL-amenability Constants of Finite Groups with Two Character Degrees We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009). Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groupsCategories:43A20, 20C15

71. CMB 2013 (vol 57 pp. 254)

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
 On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible. Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principleCategories:42C15, 42C40

72. CMB 2013 (vol 57 pp. 125)

Mlaiki, Nabil M.
 Camina Triples In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient. Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groupsCategory:20D15

73. CMB 2013 (vol 57 pp. 335)

Karassev, A.; Todorov, V.; Valov, V.
 Alexandroff Manifolds and Homogeneous Continua ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs. Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuumCategories:54F45, 54F15

74. CMB 2013 (vol 57 pp. 357)

Lauret, Emilio A.
 Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ are unitarily equivalent. Keywords:representation equivalent, strongly isospectrality, compact flat manifoldsCategories:58J53, 22D10

75. CMB 2013 (vol 57 pp. 364)

Li, Lei; Wang, Ya-Shu
 How Lipschitz Functions Characterize the Underlying Metric Spaces Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if $z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),$ or $z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,$ respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism. Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating mapsCategories:46E40, 54D60, 46E15
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