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26. 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

27. 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

28. 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

29. CMB 2012 (vol 57 pp. 132)

Mubeena, T.; Sankaran, P.
Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groups

30. CMB 2012 (vol 56 pp. 630)

Sundar, S.
Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres
In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra $C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres
Categories:46L99, 20M18

31. CMB 2011 (vol 56 pp. 366)

Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term $f(t,x)$ that exhibits a $(p-1)$-superlinear growth in $x \in \mathbb{R}$ near $\pm\infty$ and near zero. A special case of the differential operator is the scalar $p$-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance
Categories:34B15, 34B18, 34C25, 58E05

32. CMB 2011 (vol 56 pp. 229)

Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.
Cesàro Operators on the Hardy Spaces of the Half-Plane
In this article we study the Cesàro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed.

Keywords:Cesàro operators, Hardy spaces, semigroups, composition operators
Categories:47B38, 30H10, 47D03

33. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature
Categories:17B20, 22E46, 53C12

34. CMB 2011 (vol 56 pp. 218)

Yang, Dilian
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations - the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups.

Keywords:functional equations, Fourier analysis, representation of compact groups
Categories:39B52, 22C05, 43A30

35. CMB 2011 (vol 55 pp. 260)

Delvaux, L.; Van Daele, A.; Wang, Shuanhong
A Note on the Antipode for Algebraic Quantum Groups
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode
Categories:16W30, 46L65

36. CMB 2011 (vol 55 pp. 882)

Xueli, Song; Jigen, Peng
Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
$L_p$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is $L_p$ stable for some $p>0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups
Categories:34D05, 47H20

37. CMB 2011 (vol 54 pp. 283)

Hillman, J. A.; Roushon, S. K.
Surgery on $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-Manifolds
We show that closed $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-manifolds are topologically rigid if $n\geq 2$, and are rigid up to $s$-cobordism, if $n=1$.

Keywords:topological rigidity, geometric structure, surgery groups
Categories:57R67, 57N16

38. CMB 2009 (vol 52 pp. 435)

Monson, B.; Schulte, Egon
Modular Reduction in Abstract Polytopes
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups.

Keywords:abstract polytopes, regular and chiral, Coxeter groups, modular reduction
Categories:51M20, 20F55

39. CMB 2007 (vol 50 pp. 588)

Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John
Cohomological Dimension and Schreier's Formula in Galois Cohomology
Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$.

Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups
Categories:12G05, 12G10

40. CMB 2006 (vol 49 pp. 371)

Floricel, Remus
Inner $E_0$-Semigroups on Infinite Factors
This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacy
Categories:46L40, 46L55

41. 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, Monodromy
Categories:57Q10, 57M27, 57M25

42. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$
In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.

Keywords:$\overline\partial$-problem, cohomology groups, line bundles
Categories:32F20, 14F05, 32C16

43. 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

44. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
Logarithms and the Topology of the Complement of a Hypersurface
This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$.

Keywords:Logarithm, homology groups and periods
Categories:32Q55, 14F45

45. CMB 2004 (vol 47 pp. 343)

Drensky, Vesselin; Hammoudi, Lakhdar
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words
Categories:16N40, 16S15, 20M05, 20M25, 68R15

46. CMB 2004 (vol 47 pp. 22)

Goto, Yasuhiro
A Note on the Height of the Formal Brauer Group of a $K3$ Surface
Using weighted Delsarte surfaces, we give examples of $K3$ surfaces in positive characteristic whose formal Brauer groups have height equal to $5$, $8$ or $9$. These are among the four values of the height left open in the article of Yui \cite{Y}.

Keywords:formal Brauer groups, $K3$ surfaces in positive, characteristic, weighted Delsarte surfaces
Categories:14L05, 14J28

47. CMB 2003 (vol 46 pp. 122)

Moon, Myoungho
On Certain Finitely Generated Subgroups of Groups Which Split
Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated.

Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental group
Categories:20E06, 20E08, 57M07

48. CMB 2002 (vol 45 pp. 231)

Hironaka, Eriko
Erratum:~~The Lehmer Polynomial and Pretzel Links
Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451.

Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
Categories:57M05, 57M25, 11R04, 11R27

49. CMB 2002 (vol 45 pp. 272)

Neusel, Mara D.
The Transfer in the Invariant Theory of Modular Permutation Representations II
In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals.

Keywords:polynomial invariants of finite groups, permutation representation, transfer

50. CMB 2001 (vol 44 pp. 440)

Hironaka, Eriko
The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
Categories:57M05, 57M25, 11R04, 11R27
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