351. CMB 2009 (vol 53 pp. 206)
 Atçeken, Mehmet

SemiSlant Submanifolds of an Almost Paracontact Metric Manifold
In this paper, we define and study the geometry of semislant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semislant submanifold to be semislant product and obtain integrability conditions for the distributions involved in the definition of a semislant submanifold.
Keywords:paracontact metric manifold, slant distribution, semislant submanifold, semislant product Categories:53C15, 53C25, 53C40 

352. CMB 2009 (vol 53 pp. 218)
 Biswas, Indranil

Restriction of the Tangent Bundle of $G/P$ to a Hypersurface
Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 

353. CMB 2009 (vol 53 pp. 378)
 Zhou, Sizhong

A New Sufficient Condition for a Graph To Be $(g,f,n)$Critical
Let $G$ be a graph of order $p$, let $a$,
$b$, and $n$ be nonnegative integers with $1\leq a\lt b$, and let $g$
and $f$ be two integervalued functions defined on $V(G)$ such
that $a\leq g(x)\lt f(x)\leq b$ for all $x\in V(G)$. A $(g,f)$factor
of graph $G$ is a spanning subgraph $F$ of $G$ such
that $g(x)\leq d_F(x)\leq f(x)$ for each $x\in V(F)$. Then a graph
$G$ is called $(g,f,n)$critical if after deleting any $n$
vertices of $G$ the remaining graph of $G$ has a $(g,f)$factor.
The binding number $\operatorname{bind}(G)$ of $G$ is the minimum value of
${N_G(X)}/{X}$ taken over all nonempty subsets $X$ of
$V(G)$ such that $N_G(X)\neq V(G)$. In this paper, it is proved
that $G$ is a $(g,f,n)$critical graph if
\[
\operatorname{bind}(G)\gt \frac{(a+b1)(p1)}{(a+1)p(a+b)bn+2}
\quad\text{and}\quad p\geq
\frac{(a+b1)(a+b2)}{a+1}+\frac{bn}{a}.
\]
Furthermore, it is
shown that this
result is best possible in some sense.
Keywords:graph, $(g,f)$factor, $(g,f,n)$critical graph, binding number Category:05C70 

354. CMB 2009 (vol 53 pp. 133)
355. CMB 2009 (vol 53 pp. 122)
 Mo, Xiaohuan; Zhou, Linfeng

A Class of Finsler Metrics with Bounded Cartan Torsion
In this paper, we find a class of $(\alpha,\beta)$ metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.
Keywords:Finsler manifold, $(\alpha,\beta)$ metric, Cartan torsion, Rquadratic, flag curvature Category:58E20 

356. CMB 2009 (vol 53 pp. 340)
357. 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 

358. 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 

359. 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 

360. 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 

361. CMB 2009 (vol 52 pp. 511)
362. 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 

363. 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 

364. 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 

365. 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 

366. 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 

367. 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 

368. 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 

369. 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 

370. 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 

371. 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 

372. 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 

373. 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 

374. 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 

375. 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 
