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376. CMB 2009 (vol 53 pp. 340)

Lusala, Tsasa; Śniatycki, Jędrzej; Watts, Jordan
Regular Points of a Subcartesian Space
We discuss properties of the regular part $S_{\operatorname{reg}}$ of a subcartesian space $S$. We show that $S_{\operatorname{reg}}$ is open and dense in $S$ and the restriction to $ S_{\operatorname{reg}}$ of the tangent bundle space of $S$ is locally trivial.

Keywords:differential structures, singular and regular points
Category:58A40

377. CMB 2009 (vol 53 pp. 118)

Lewis, Paul
The Uncomplemented Spaces $W(X,Y)$ and $K(X,Y)$
Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y.

Keywords:spaces of operators, complemented subspace, weakly compact operator, basic sequence
Categories:46B28, 46B15, 46B20

378. CMB 2009 (vol 53 pp. 295)

Guo, Boling; Huo, Zhaohui
The Global Attractor of a Damped, Forced Hirota Equation in $H^1$
The existence of the global attractor of a damped forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach.

Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness
Categories:35Q53, 35B40, 35B41, 37L30

379. CMB 2009 (vol 53 pp. 95)

Ghioca, Dragos
Towards the Full Mordell-Lang Conjecture for Drinfeld Modules
Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite.

Keywords:Drinfeld module, Mordell-Lang conjecture
Categories:11G09, 11G10

380. CMB 2009 (vol 53 pp. 263)

Feuto, Justin; Fofana, Ibrahim; Koua, Konin
Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of Hardy–Littlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.

Keywords:fractional maximal operator, fractional integral, space of homogeneous type
Categories:42B35, 42B20, 42B25

381. CMB 2009 (vol 53 pp. 58)

Dąbrowski, Andrzej; Jędrzejak, Tomasz
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves
In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height
Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15

382. CMB 2009 (vol 53 pp. 23)

Chen, Huaihui; Zhang, Minzhu
Boundedness From Below of Multiplication Operators Between $\alpha$-Bloch Spaces
In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for $f\in\mathcal B^\alpha$, we prove the following result: (i) If $0\lt \beta\lt \alpha$, or $0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below; (ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if $\liminf_{z\to\partial D}|u(z)|\gt 0$; (iii) if $1\lt \alpha\le\beta$, $M_u$ is bounded below if and only if there exist a $\delta\gt 0$ and a positive $r\lt 1$ such that for every point $z\in D$ there is a point $z'\in D$ with the property $d(z',z)\lt r$ and $(1-|z'|^2)^{\beta-\alpha}|u(z')|\ge\delta$, where $d(\cdot,\cdot)$ denotes the pseudo-distance on $D$.

Keywords:$\alpha$-Bloch function, multiplication operator
Categories:32A18, 30H05

383. CMB 2009 (vol 53 pp. 11)

Burke, Maxim R.
Approximation and Interpolation by Entire Functions of Several Variables
Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.

Keywords:entire function, complex approximation, interpolation, several complex variables
Category:32A15

384. CMB 2009 (vol 53 pp. 206)

Atçeken, Mehmet
Semi-Slant Submanifolds of an Almost Paracontact Metric Manifold
In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold.

Keywords:paracontact metric manifold, slant distribution, semi-slant submanifold, semi-slant product
Categories:53C15, 53C25, 53C40

385. CMB 2009 (vol 52 pp. 564)

Jin, Hai Lan; Doh, Jaekyung; Park, Jae Keol
Group Actions on Quasi-Baer Rings
A ring $R$ is called {\it quasi-Baer} if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer 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

386. 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

387. 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

388. CMB 2009 (vol 52 pp. 511)

Bonciocat, Anca Iuliana; Bonciocat, Nicolae Ciprian
The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value
We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

Keywords:Estimates for polynomial roots, irreducible polynomials
Categories:11C08, 11R09

389. CMB 2009 (vol 52 pp. 493)

Artebani, Michela
A One-Dimensional 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 one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Keywords:genus three curves, $K3$ surfaces
Categories:14J28, 14J50, 14J10

390. 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 \[ \prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})} =1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n \] and \[ q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})} =\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n. \] We give several more identities of this type.

Keywords:Power series expansions of certain infinite products
Categories:11E25, 11F11, 11F27, 30B10

391. 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

392. 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 side-lengths 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

393. 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 non-empty 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 Helly-type 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

394. CMB 2009 (vol 52 pp. 366)

Gévay, Gábor
A Class of Cellulated Spheres with Non-Polytopal Symmetries
We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability 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 so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.

Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope
Categories:52B11, 52B15, 52B70

395. CMB 2009 (vol 52 pp. 342)

Bezdek, K.; Kiss, Gy.
On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width
The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray 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, X-ray number, X-ray Conjecture
Categories:52A20, 52A37, 52C17, 52C35

396. 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

397. CMB 2009 (vol 52 pp. 213)

Ghenciu, Ioana; Lewis, Paul
Dunford--Pettis 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 Dunford--Pettis 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 Dunford--Pettis sets and strong Dunford--Pettis sets to extend each of the preceding theorems. The space $L_{w^*}(X^* , Y)$ of $w^*-w$ continuous operators is also studied.

Keywords:Dunford--Pettis property, Dunford--Pettis set, basic sequence, complemented spaces of operators
Categories:46B20, 46B28

398. CMB 2009 (vol 52 pp. 315)

Yi, Taishan; Zou, Xingfu
Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows
By employing the limit set dichotomy for essentially strongly order-preserving semiflows and the assumption that limit sets have infima and suprema in the state space, we prove a generic quasi-convergence principle implying the existence of an open and dense set of stable quasi-convergent points. We also apply this generic quasi-convergence 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 order-preserving semiflow, compactness, quasi-convergence
Categories:34C12, 34K25

399. 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

400. 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
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