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51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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

61. 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

62. 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

63. 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

64. 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

65. 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

66. CMB 2013 (vol 56 pp. 729)

Currey, B.; Mayeli, A.
 The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications. Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearletCategories:43A65, 42C40, 42C15

67. CMB 2013 (vol 56 pp. 449)

Akbari, S.; Chavooshi, M.; Ghanbari, M.; Zare, S.
 The $f$-Chromatic Index of a Graph Whose $f$-Core has Maximum Degree $2$ Let $G$ be a graph. The minimum number of colors needed to color the edges of $G$ is called the chromatic index of $G$ and is denoted by $\chi'(G)$. It is well-known that $\Delta(G) \leq \chi'(G) \leq \Delta(G)+1$, for any graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. Also, $G_\Delta$ is the induced subgraph on all vertices of degree $\Delta(G)$. Let $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is a coloring of the edges of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to $f$-color $G$ is called the $f$-chromatic index of $G$ and is denoted by $\chi'_{f}(G)$. It was shown that for every graph $G$, $\Delta_{f}(G)\le \chi'_{f}(G)\le \Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max_{v\in V(G)} \big\lceil \frac{d_G(v)}{f(v)}\big\rceil$. A graph $G$ is said to be $f$-Class $1$ if $\chi'_{f}(G)=\Delta_{f}(G)$, and $f$-Class $2$, otherwise. Also, $G_{\Delta_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=\Delta_{f}(G)\}$. Hilton and Zhao showed that if $G_{\Delta}$ has maximum degree two and $G$ is Class $2$, then $G$ is critical, $G_{\Delta}$ is a disjoint union of cycles and $\delta(G)=\Delta(G)-1$, where $\delta(G)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$-coloring of graphs. Also, we determine the $f$-chromatic index of a connected graph $G$ with $|G_{\Delta_f}|\le 4$. Keywords:$f$-coloring, $f$-Core, $f$-Class $1$Categories:05C15, 05C38

68. CMB Online first

Zhang, Jiao; Wang, Qing-Wen
 An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg-Zilber theorem. Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homologyCategories:19D55, 05E45

69. CMB 2013 (vol 57 pp. 210)

Zhang, Jiao; Wang, Qing-Wen
 An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg-Zilber theorem. Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homologyCategories:19D55, 05E45

70. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
 Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$. Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension functionCategory:42C40

71. CMB 2013 (vol 56 pp. 673)

Ayadi, K.; Hbaib, M.; Mahjoub, F.
 Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type $$\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}$$ where $(A, B, C)\in (\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$. In particular, we will give, under some conditions on the polynomials $A$, $B$ and $C$, well approximated elements satisfying this equation. Keywords:diophantine approximation, formal power series, continued fractionCategories:11J61, 11J70

72. CMB 2012 (vol 57 pp. 289)

Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven
 Closure of the Cone of Sums of $2d$-powers in Certain Weighted $\ell_1$-seminorm Topologies In a paper from 1976, Berg, Christensen and Ressel prove that the closure of the cone of sums of squares $\sum \mathbb{R}[\underline{X}]^2$ in the polynomial ring $\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$, the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $\ell_1$-seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem from 2001. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\ge 1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in $\mathbb{R}[\underline{X}]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$. Keywords:positive definite, moments, sums of squares, involutive semigroupsCategories:43A35, 44A60, 13J25

73. CMB 2012 (vol 56 pp. 881)

Xie, BaoHua; Wang, JieYan; Jiang, YuePing
 Free Groups Generated by Two Heisenberg Translations In this paper, we will discuss the groups generated by two Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free. Keywords:free group, Heisenberg group, complex triangle groupCategories:30F40, 22E40, 20H10

74. CMB 2012 (vol 57 pp. 326)

Ivanov, S. V.; Mikhailov, Roman
 On Zero-divisors in Group Rings of Groups with Torsion Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion. Keywords:Burnside groups, free products of groups, group rings, zero-divisorsCategories:20C07, 20E06, 20F05, , 20F50

75. CMB 2012 (vol 57 pp. 209)

Zhao, Wei
 Erratum to the Paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold" We correct two clerical errors made in the paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold". Keywords:Finsler manifold, closed geodesic, injective radiusCategories:53B40, 53C22
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