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351. 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 $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. 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

352. 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

353. 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

354. 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

355. 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

356. 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

357. 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

358. 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

359. 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

360. 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

361. 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

362. 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

363. 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

364. 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

365. 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

366. 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

367. CMB 2009 (vol 52 pp. 224)

Ghiloni, Riccardo
Equations and Complexity for the Dubois--Efroymson 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

368. 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

369. 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

370. 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 column-finite 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 column-finite matrix rings, group rings
Categories:16D70, 16D40, 16S50

371. 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 Christoffel--Darboux formulas are presented for the first time.

Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-Darboux
Category:42C99

372. 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

373. CMB 2008 (vol 51 pp. 584)

Purbhoo, Kevin; Willigenburg, Stephanie van
On Tensor Products of Polynomial Representations
We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood--Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative
Categories:05E05, 05E10, 20C30

374. CMB 2008 (vol 51 pp. 570)

Lutzer, D. J.; Mill, J. van; Tkachuk, V. V.
Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$
We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

Keywords:regular filterbase, subcompact space, function space, discrete space
Categories:54B10, 54C05, 54D30

375. 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:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map
Categories:57N65, 57R67, 57Q10
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