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76. 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, shearlet
Categories:43A65, 42C40, 42C15

77. 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

78. 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 homology
Categories:19D55, 05E45

79. 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 homology
Categories:19D55, 05E45

80. 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 function

81. 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 \begin{equation} \alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}} \end{equation} 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 fraction
Categories:11J61, 11J70

82. 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 radius
Categories:53B40, 53C22

83. 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 semigroups
Categories:43A35, 44A60, 13J25

84. 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 group
Categories:30F40, 22E40, 20H10

85. 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-divisors
Categories:20C07, 20E06, 20F05, , 20F50

86. CMB 2012 (vol 57 pp. 105)

Luca, Florian; Shparlinski, Igor E.
On the Counting Function of Elliptic Carmichael Numbers
We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss several possible ways for further improvements.

Keywords:elliptic Carmichael numbers, applications of sieve methods
Categories:11Y11, 11N36

87. CMB 2012 (vol 57 pp. 240)

Bernardes, Nilson C.
Addendum to ``Limit Sets of Typical Homeomorphisms''
Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f : X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$ (respectively to the alpha limit set $\alpha(f,x)$) is topologically conjugate to the universal odometer.

Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets
Categories:37B20, 54H20, 28C15, 54C35, 54E52

88. CMB 2012 (vol 56 pp. 570)

Hoang, Giabao; Ressler, Wendell
Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups
In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in any Hecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.

Keywords:Hecke groups, conjugacy class, quadratic forms
Categories:11F06, 11E16, 11A55

89. CMB 2012 (vol 57 pp. 194)

Zhao, Wei
A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold
In this paper, we obtain a lower bound for the length of closed geodesics on an arbitrary closed Finsler manifold.

Keywords:Finsler manifold, closed geodesic, injective radius
Categories:53B40, 53C22

90. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
2-dimensional Convexity Numbers and $P_4$-free Graphs
For $S\subseteq\mathbb R^n$ a set $C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\subseteq\mathbb R^2$ without an uncountable $3$-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in $\mathbb R^2$ without uncountable 3-cliques in terms of clopen, $P_4$-free graphs on Polish spaces.

Keywords:convex cover, convexity number, continuous coloring, perfect graph, cograph
Categories:52A10, 03E17, 03E75

91. CMB 2012 (vol 57 pp. 178)

Rabier, Patrick J.
Quasiconvexity and Density Topology
We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then $\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while $\inf_{U}f=\operatorname{ess\,inf}_{U}f$ if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second) property is typical of lsc (usc) functions and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions. This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on ``large'' subsets, which may be of relevance in a variety of issues. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

Keywords:density topology, quasiconvex function, approximate continuity, point of continuity
Categories:52A41, 26B05

92. CMB 2012 (vol 56 pp. 683)

Nikseresht, A.; Azizi, A.
Envelope Dimension of Modules and the Simplified Radical Formula
We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.

Keywords:envelope dimension, simplified radical formula, prime submodule
Categories:13A99, 13C99, 13C13, 13E05

93. CMB 2012 (vol 57 pp. 42)

Fonf, Vladimir P.; Zanco, Clemente
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls
e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.

Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces
Categories:46B20, 46C05, 52C17

94. CMB 2012 (vol 57 pp. 12)

Aribi, Amine; Dragomir, Sorin; El Soufi, Ahmad
On the Continuity of the Eigenvalues of a Sublaplacian
We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology.

Keywords:CR manifold, contact form, sublaplacian, Fefferman metric
Categories:32V20, 53C56

95. CMB 2012 (vol 56 pp. 477)

Ayadi, Adlene
Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$
We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.

Keywords:affine, hypercyclic, dense, orbit, affine group, abelian
Categories:37C85, 47A16

96. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraïssé theory and Ramsey theory. This work generalizes some of the known results about countable structures.

Keywords:universal minimal flows, ultrafilter flows, Ramsey theory
Categories:37B05, 03E02, 05D10, 22F50, 54H20

97. CMB 2012 (vol 57 pp. 145)

Mustafayev, H. S.
The Essential Spectrum of the Essentially Isometric Operator
Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right) $ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $ C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma $ is the unit circle. Some related problems are also discussed.

Keywords:Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculus
Categories:47A10, 47A53, 47A60, 47B07

98. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
On the Average Number of Square-Free Values of Polynomials
We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture.

Keywords:polynomials, square-free numbers

99. CMB 2012 (vol 57 pp. 113)

Madras, Neal
A Lower Bound for the End-to-End Distance of Self-Avoiding Walk
For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists.

Keywords:self-avoiding walk, critical exponent
Categories:82B41, 60D05, 60K35

100. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
Subadditivity Inequalities for Compact Operators
Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities
Categories:47A63, 15A45
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