401. CMB 2009 (vol 52 pp. 481)
 Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

Some Infinite Products of Ramanujan Type
In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1q^n)^5}{(1q^{5n})}
=15\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1q^{5n})^5}{(1q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 

402. CMB 2009 (vol 52 pp. 544)
 Hanafy, I. M.

Intuitionistic Fuzzy $\gamma$Continuity
This paper introduces the concepts of
fuzzy $\gamma$open sets and fuzzy $\gamma$continuity
in intuitionistic fuzzy topological spaces. After defining the fundamental
concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological
spaces, we present intuitionistic fuzzy $\gamma$open sets and
intuitionistic fuzzy $\gamma$continuity and other results related
topological concepts.
Keywords:intuitionistic fuzzy set, intuitionistic fuzzy point, intuitionistic fuzzy topological space, intuitionistic fuzzy $\gamma$open set, intuitionistic fuzzy $\gamma$\continuity, intuitionistic fuzzy $\gamma$closure ($\gamma$interior) Categories:54A40, 54A20, 54F99 

403. CMB 2009 (vol 52 pp. 535)
 Daigle, Daniel; Kaliman, Shulim

A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$
We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 

404. CMB 2009 (vol 52 pp. 511)
405. CMB 2009 (vol 52 pp. 564)
 Jin, Hai Lan; Doh, Jaekyung; Park, Jae Keol

Group Actions on QuasiBaer Rings
A ring $R$ is called {\it quasiBaer} if the right
annihilator of every right ideal of $R$ is generated by an
idempotent as a right ideal. We investigate the quasiBaer
property of skew group rings and fixed rings under a finite group
action on a semiprime ring and their applications to
$C^*$algebras.
Various examples to illustrate and
delimit our results are provided.
Keywords:(quasi) Baer ring, fixed ring, skew group ring, $C^*$algebra, local multiplier algebra Categories:16S35, 16W22, 16S90, 16W20, 16U70 

406. CMB 2009 (vol 52 pp. 493)
 Artebani, Michela

A OneDimensional Family of $K3$ Surfaces with a $\Z_4$ Action
The minimal resolution of the degree four cyclic cover of the plane
branched along a GIT stable quartic is a $K3$ surface with a non
symplectic action of $\Z_4$. In this paper
we study the geometry of the onedimensional family of $K3$ surfaces
associated to the locus of plane quartics with five nodes.
Keywords:genus three curves, $K3$ surfaces Categories:14J28, 14J50, 14J10 

407. CMB 2009 (vol 52 pp. 366)
 Gévay, Gábor

A Class of Cellulated Spheres with NonPolytopal Symmetries
We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d1}$.
We prove that these cellulations cannot be polytopal with maximal
combinatorial symmetry. Such nonrealizability phenomenon was first
described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and,
to the knowledge of the author, until now there have not been any
known examples in higher dimensions. As a starting point for the
construction, we introduce a new class of (Wythoffian) uniform
polytopes, which we call duplexes. In proving our main result,
we use some tools that we developed earlier while studying perfect
polytopes. In particular, we prove perfectness of the duplexes;
furthermore, we prove and make use of the perfectness of another
new class of polytopes which we obtain by a variant of the socalled
$E$construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope Categories:52B11, 52B15, 52B70 

408. CMB 2009 (vol 52 pp. 464)
 Stancu, Alina

Two Volume Product Inequalities and Their Applications
Let $K \subset {\mathbb{R}}^{n+1}$ be a convex body of class $C^2$
with everywhere positive Gauss curvature. We show that there exists
a positive number $\delta (K)$ such that for any $\delta \in (0,
\delta(K))$ we have $\Volu(K_{\delta})\cdot
\Volu((K_{\delta})^{\sstar}) \geq \Volu(K)\cdot \Volu(K^{\sstar}) \geq
\Volu(K^{\delta})\cdot \Volu((K^{\delta})^{\sstar})$, where $K_{\delta}$,
$K^{\delta}$ and $K^{\sstar}$ stand for the convex floating body, the
illumination body, and the polar of $K$, respectively. We derive a
few consequences of these inequalities.
Keywords:affine invariants, convex floating bodies, illumination bodies Categories:52A40, 52A38, 52A20 

409. CMB 2009 (vol 52 pp. 424)
 Martini, Horst; Spirova, Margarita

Covering Discs in Minkowski Planes
We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of sidelengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 

410. CMB 2009 (vol 52 pp. 388)
 Heppes, Aladár

Transversals with Residue in Moderately Overlapping $T(k)$Families of Translates
Let $K$ denote an oval, a centrally symmetric compact convex domain
with nonempty interior. A family of translates of $K$ is said to have
property $T(k)$ if for every subset of at most $k$ translates there
exists a common line transversal intersecting all of them. The integer
$k$ is the stabbing level of the family.
Two translates $K_i = K + c_i$ and $K_j = K + c_j$ are said to be
$\sigma$disjoint if $\sigma K + c_i$ and $\sigma K + c_j$ are disjoint.
A recent Hellytype result claims that for every
$\sigma > 0 $ there exists an integer $k(\sigma)$ such that if a
family of $\sigma$disjoint unit diameter discs has property $T(k) k
\geq k(\sigma)$, then there exists a straight line meeting all members
of the family. In the first part of the paper we give the extension of
this theorem to translates of an oval $K$. The asymptotic behavior of
$k(\sigma)$ for $\sigma \rightarrow 0$ is considered as well.
Katchalski and Lewis proved the existence of a constant $r$ such that
for every pairwise disjoint family of translates of an oval $K$ with
property $T(3)$ a straight line can be found meeting all but at most
$r$ members of the family. In the second part of the paper
$\sigma$disjoint families of translates of $K$ are considered and the
relation of $\sigma$ and the residue $r$ is investigated. The
asymptotic behavior of $r(\sigma)$ for $\sigma \rightarrow 0$ is also
discussed.
Keywords:transversal, $\sigma$disjoint, $T(k)$family, Helly number, residue 

411. CMB 2009 (vol 52 pp. 342)
 Bezdek, K.; Kiss, Gy.

On the Xray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width
The Xray numbers of some classes of convex bodies are investigated.
In particular, we give a proof of the Xray Conjecture as well as
of the Illumination Conjecture for almost smooth convex bodies
of any dimension and for convex bodies of constant width of
dimensions $3$, $4$, $5$ and $6$.
Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, Xray number, Xray Conjecture Categories:52A20, 52A37, 52C17, 52C35 

412. CMB 2009 (vol 52 pp. 213)
 Ghenciu, Ioana; Lewis, Paul

DunfordPettis Properties and Spaces of Operators
J. Elton used an application of Ramsey theory to show that
if $X$ is an infinite dimensional Banach space,
then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there
is a subspace of $X$ that fails to have the DunfordPettis property.
Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$,
then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed
that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not
complemented in $L(X,Y)$. Classical results from Schauder basis theory
are used in a study of DunfordPettis sets and strong
DunfordPettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*w$ continuous operators is also studied.
Keywords:DunfordPettis property, DunfordPettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 

413. CMB 2009 (vol 52 pp. 315)
 Yi, Taishan; Zou, Xingfu

Generic QuasiConvergence for Essentially Strongly OrderPreserving Semiflows
By employing the limit set
dichotomy for essentially strongly orderpreserving semiflows and
the assumption that limit sets have infima and suprema in the
state space, we prove a generic quasiconvergence principle
implying the existence of an open and dense set of stable
quasiconvergent points. We also apply this generic quasiconvergence principle
to a model for biochemical feedback in protein
synthesis and obtain some results about the model which are of theoretical
and realistic significance.
Keywords:Essentially strongly orderpreserving semiflow, compactness, quasiconvergence Categories:34C12, 34K25 

414. CMB 2009 (vol 52 pp. 295)
 P{\l}otka, Krzysztof

On Functions Whose Graph is a Hamel Basis, II
We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, \emph{i.e.,} for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.
Keywords:Hamel basis, additive, Hamel functions Categories:26A21, 54C40, 15A03, 54C30 

415. CMB 2009 (vol 52 pp. 257)
 Ikeda, Toru

Essential Surfaces in Graph Link Exteriors
An irreducible graph manifold $M$ contains an essential torus if
it is not a special Seifert manifold.
Whether $M$ contains a closed essential surface of
negative Euler characteristic or not
depends on the difference of Seifert fibrations from the two sides
of a torus system which splits $M$ into Seifert manifolds.
However,
it is not easy to characterize geometrically the class of
irreducible graph manifolds which contain such surfaces.
This article studies this problem in the case of graph link exteriors.
Keywords:Graph link, Graph manifold, Seifert manifold, Essential surface Category:57M25 

416. CMB 2009 (vol 52 pp. 224)
 Ghiloni, Riccardo

Equations and Complexity for the DuboisEfroymson Dimension Theorem
Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.
Keywords:Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theorems Categories:14P05, 14P20 

417. CMB 2009 (vol 52 pp. 186)
 Broughan, Kevin A.

Extension of the Riemann $\xi$Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$
the symmetrized
Dedekind zeta function of the field, the inequality
$$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma
+ {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is
shown
to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of
Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$
the
inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small
and to the region $\si\ge 1+1/(\log t 5)$ for all $t\neq 0$. This
answers positively a question posed by Lagarias.
Keywords:Riemann zeta function, xi function, zeta zeros Categories:11M26, 11R42 

418. CMB 2009 (vol 52 pp. 66)
 Dryden, Emily B.; Strohmaier, Alexander

Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the
Laplace spectrum determines the length spectrum as well as the
number of singular points of a given order. The converse also holds, giving
a full generalization of Huber's theorem to the setting of
compact orientable hyperbolic orbisurfaces.
Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces Categories:58J53, 11F72 

419. CMB 2009 (vol 52 pp. 145)
 Wang, Z.; Chen, J. L.

$2$Clean Rings
A ring $R$ is said to be $n$clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$module of rank at least 2 is $2$clean and that the ring $B(R)$
of all $\omega\times \omega$ row and columnfinite matrices over
any ring $R$ is $2$clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$clean rings, $2$good rings, free modules, row and columnfinite matrix rings, group rings Categories:16D70, 16D40, 16S50 

420. CMB 2009 (vol 52 pp. 95)
 Miranian, L.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory
In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
ChristoffelDarboux formulas are presented for the first time.
Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, ChristoffelDarboux Category:42C99 

421. CMB 2009 (vol 52 pp. 18)
 Chinea, Domingo

Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds
In this paper we study holomorphic maps between almost Hermitian
manifolds. We obtain a new criterion for the harmonicity of such
holomorphic maps, and we deduce some applications to horizontally
conformal holomorphic submersions.
Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphism Categories:53C15, 58E20 

422. CMB 2008 (vol 51 pp. 508)
 Cavicchioli, Alberto; Spaggiari, Fulvia

A Result in Surgery Theory
We study the topological $4$dimensional surgery problem
for a closed connected orientable
topological $4$manifold $X$ with vanishing
second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has
one end and $F(r)$ is the free group of rank $r\ge 1$.
Our result is related to a theorem of Krushkal and Lee, and
depends on the validity of the Novikov conjecture for
such fundamental groups.
Keywords:fourmanifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map Categories:57N65, 57R67, 57Q10 

423. CMB 2008 (vol 51 pp. 481)
 Bayart, Frédéric

Universal Inner Functions on the Ball
It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the
unit ball $\bn$ of $\cn$ such that $\\phi_k(0)\$ tends to $1$,
there exists an inner function
$I$ such that the family of ``nonEuclidean translates"
$(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of
$H^\infty(\bn)$.
Keywords:inner functions, automorphisms of the ball, universality Categories:32A35, 30D50, 47B38 

424. CMB 2008 (vol 51 pp. 627)
 Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.

Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving
J.~Bourget's generalization of Bessel functions of integer order, as
well as the analogous generalizations by H.~M.~Srivastava. These series are
expressed in terms of the Riemann $\z$ function and Dirichlet
functions $\eta$, $\la$, $\b$, and can be brought into closed form in
certain cases, which means that the infinite series are represented
by finite sums.
Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Categories:33C10, 11M06, 65B10 

425. CMB 2008 (vol 51 pp. 618)
 Valmorin, V.

Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$valued $\C$linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 
