26. 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 finitedimensional 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 $\check{H}^{n1}(K;G)=0$ and $\dim_G K\leq
n1$. This provides a partial answer to a question of
KallipolitiPapasoglu
whether any twodimensional homogeneous Peano continuum cannot be separated by arcs.
Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$continuum Categories:54F45, 54F15 

27. CMB 2013 (vol 57 pp. 546)
 Kalantar, Mehrdad

Compact Operators in Regular LCQ Groups
We show that a regular locally compact quantum group $\mathbb{G}$ is discrete
if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains nonzero compact operators on
$\mathcal{L}^{2}(\mathbb{G})$.
As a corollary we classify all discrete quantum groups among
regular locally compact quantum groups $\mathbb{G}$ where
$\mathcal{L}^{1}(\mathbb{G})$ has the RadonNikodym property.
Keywords:locally compact quantum groups, regularity, compact operators Category:46L89 

28. 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 nonnegative 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 

29. CMB 2012 (vol 57 pp. 326)
 Ivanov, S. V.; Mikhailov, Roman

On Zerodivisors in Group Rings of Groups with Torsion
Nontrivial pairs of zerodivisors in group rings are
introduced and discussed. A problem on the existence of nontrivial
pairs of zerodivisors in group rings of free Burnside groups of odd
exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of
zerodivisors are also found in group rings of free products of groups
with torsion.
Keywords:Burnside groups, free products of groups, group rings, zerodivisors Categories:20C07, 20E06, 20F05, , 20F50 

30. 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 nonelliptic 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 

31. 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 nonelementary 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 Category:20E45 

32. 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 

33. 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
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p1)$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 

34. CMB 2011 (vol 56 pp. 229)
 Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.

CesÃ ro Operators on the Hardy Spaces of the HalfPlane
In this article we study the CesÃ ro
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:CesÃ ro operators, Hardy spaces, semigroups, composition operators Categories:47B38, 30H10, 47D03 

35. CMB 2011 (vol 55 pp. 870)
 Wang, Hui; Deng, Shaoqiang

Left Invariant EinsteinRanders Metrics on Compact Lie Groups
In this paper we study left invariant EinsteinRanders metrics on compact Lie
groups. First, we give a method to construct left invariant nonRiemannian EinsteinRanders metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant EinsteinRanders 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:EinsteinRanders metric, compact Lie groups, geodesic, flag curvature Categories:17B20, 22E46, 53C12 

36. 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 

37. 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 coFrobenius 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
finitedimensional Hopf algebra, but also that of any
Hopf algebra with integrals (coFrobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from wellknown 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 

38. 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 

39. CMB 2011 (vol 54 pp. 283)
40. 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 

41. 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 

42. 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 

43. 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 signdeter\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 

44. CMB 2006 (vol 49 pp. 72)
 Dwilewicz, Roman J.

Additive RiemannHilbert 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 CauchyRiemann 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 

45. 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 

46. CMB 2005 (vol 48 pp. 473)
47. 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 nonnil 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 

48. 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 

49. 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 

50. 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), 440451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 
