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401. 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 non-Euclidean 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, universalityCategories:32A35, 30D50, 47B38

402. CMB 2008 (vol 51 pp. 386)

Lan, K. Q.; Yang, G. C.
 Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory The well-known Falkner--Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter involved in the equation. It is known that there exists $\lambda^{*}<0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\lambda\ge \lambda^{*}$ and has no positive solutions for $\lambda<\lambda^{*}$. The known numerical result shows $\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in [-0.4,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner--Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner--Skan equation. Keywords:Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutionsCategories:34B16, 34B18, 34B40, 76D10

403. CMB 2008 (vol 51 pp. 448)

Sasahara, Toru
 Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed. Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifoldsCategories:53C42, 53C40

404. CMB 2008 (vol 51 pp. 334)

Ascah-Coallier, I.; Gauthier, P. M.
 Value Distribution of the Riemann Zeta Function In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function. Keywords:Nevanlinna theory, deficiency, Riemann zeta functionCategory:30D35

405. CMB 2008 (vol 51 pp. 439)

Samei, Karim
 On the Maximal Spectrum of Semiprimitive Multiplication Modules An $R$-module $M$ is called a multiplication module if for each submodule $N$ of $M$, $N=IM$ for some ideal $I$ of $R$. As defined for a commutative ring $R$, an $R$-module $M$ is said to be semiprimitive if the intersection of maximal submodules of $M$ is zero. The maximal spectra of a semiprimitive multiplication module $M$ are studied. The isolated points of $\Max(M)$ are characterized algebraically. The relationships among the maximal spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown that $\Soc(M)$ is exactly the set of all elements of $M$ which belongs to every maximal submodule of $M$ except for a finite number. If $\Max(M)$ is infinite, $\Max(M)$ is a one-point compactification of a discrete space if and only if $M$ is Gelfand and for some maximal submodule $K$, $\Soc(M)$ is the intersection of all prime submodules of $M$ contained in $K$. When $M$ is a semiprimitive Gelfand module, we prove that every intersection of essential submodules of $M$ is an essential submodule if and only if $\Max(M)$ is an almost discrete space. The set of uniform submodules of $M$ and the set of minimal submodules of $M$ coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if $\Max(M)$ is the union of two disjoint open subspaces $A$ and $N$, where $A$ is almost discrete and $N$ is dense in itself. In particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ is almost discrete. Keywords:multiplication module, semiprimitive module, Gelfand module, Zariski topologCategory:13C13

406. CMB 2008 (vol 51 pp. 359)

Cho, Jong Taek; Ki, U-Hang
 Real Hypersurfaces in Complex Space Forms with Reeb Flow Symmetric Structure Jacobi Operator Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $(A)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator. Keywords:complex space form, real hypersurface, structure Jacobi operatorCategories:53B20, 53C15, 53C25

407. CMB 2008 (vol 51 pp. 378)

Izuchi, Kou Hei
 Cyclic Vectors in Some Weighted $L^p$ Spaces of Entire Functions In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in $\Lp$. Let $f\in\Lp$ and $f\poly$ be contained in the space. We show that $f$ is non-vanishing if and only if $f$ is cyclic. Keywords:weighted $L^p$ spaces of entire functions, cyclic vectorsCategories:47A16, 46J15, 46H25

408. CMB 2008 (vol 51 pp. 172)

Alkan, Emre; Zaharescu, Alexandru
 Consecutive Large Gaps in Sequences Defined by Multiplicative Constraints In this paper we obtain quantitative results on the occurrence of consecutive large gaps between $B$-free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication. Keywords:$B$-free numbers, consecutive gapsCategories:11N25, 11B05

409. CMB 2008 (vol 51 pp. 310)

Witbooi, P. J.
 Relative Homotopy in Relational Structures The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a subposet of $X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif. Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalenceCategories:55Q05, 54A05;, 18B30

410. CMB 2008 (vol 51 pp. 298)

Tocón, Maribel
 The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra. Keywords:extended affine Lie algebra, Lie torus, core, Kostrikin radicalCategories:17B05, 17B65

411. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
 The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25

412. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
 On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups An $n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field. Keywords:quantum torus, normal form, automorphisms of quantum toriCategory:16S35

413. CMB 2008 (vol 51 pp. 236)

 Konovalov, Victor N.; Kopotun, Kirill A.

414. CMB 2008 (vol 51 pp. 217)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
 A Multivalued Nonlinear System with the Vector $p$-Laplacian on the Semi-Infinity Interval We study a second order nonlinear system driven by the vector $p$-Laplacian, with a multivalued nonlinearity and defined on the positive time semi-axis $\mathbb{R}_+.$ Using degree theoretic techniques we solve an auxiliary mixed boundary value problem defined on the finite interval $[0,n]$ and then via a diagonalization method we produce a solution for the original infinite time-horizon system. Keywords:semi-infinity interval, vector $p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous mapCategory:34A60

415. CMB 2008 (vol 51 pp. 205)

Duda, Jakub
 On GÃ¢teaux Differentiability of Pointwise Lipschitz Mappings We prove that for every function $f\from X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is G\^ateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\from X\to\R$ cone monotone, $g\from X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fr\'echet differentiable. Keywords:GÃ¢teaux differentiable function, Radon-NikodÃ½m property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functionsCategories:46G05, 46T20

416. CMB 2008 (vol 51 pp. 195)

Chen, Huaihui; Gauthier, Paul
 Boundedness from Below of Composition Operators on $\alpha$-Bloch Spaces We give a necessary and sufficient condition for a composition operator on an $\alpha$-Bloch space with $\alpha\ge 1$ to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen. Keywords:Bloch functions, composition operatorsCategories:32A18, 30H05

417. CMB 2008 (vol 51 pp. 146)

Zhou, Xiaowen
 Stepping-Stone Model with Circular Brownian Migration In this paper we consider the stepping-stone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow on the circle and the marginal distribution of this model. We then give a new representation for the stepping-stone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computations. In particular, we find the distribution for the first time when there is only one type left across the circle. Keywords:stepping-stone model, circular coalescing Brownian motion, Arratia flow, duality, entrance lawCategories:60G57, 60J65

418. CMB 2008 (vol 51 pp. 100)

Petkov, Vesselin
 Dynamical Zeta Function for Several Strictly Convex Obstacles The behavior of the dynamical zeta function $Z_D(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function $\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re s = s_2$, where $s_2$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$. Keywords:dynamical zeta function, periodic raysCategories:11M36, 58J50

419. CMB 2008 (vol 51 pp. 86)

Nakazato, Hiroshi; Bebiano, Natália; Providência, Jo\ ao da
 The Numerical Range of 2-Dimensional Krein Space Operators The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained. Keywords:numerical range, generalized numerical range, indefinite inner product spaceCategories:15A60, 15A63, 15A45

420. CMB 2008 (vol 51 pp. 3)

 Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

421. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
 Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications. Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsionCategories:14F30, 14J30

422. CMB 2007 (vol 50 pp. 579)

Kot, Piotr
 $p$-Radial Exceptional Sets and Conformal Mappings For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary of the unit disc $\partial\mathbb D$ we construct a holomorphic function $f\in\mathbb O(\mathbb D)$ such that $\int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty$ and$E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .$ In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$. Keywords:boundary behaviour of holomorphic functions, exceptional setsCategories:30B30, 30E25

423. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
 Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$. Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10

424. CMB 2007 (vol 50 pp. 598)

Lorestani, Keivan Borna; Sahandi, Parviz; Yassemi, Siamak
 Artinian Local Cohomology Modules Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i Keywords:local cohomology module, Artinian module, reflexive moduleCategories:13D45, 13E10, 13C05 425. CMB 2007 (vol 50 pp. 588) Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John  Cohomological Dimension and Schreier's Formula in Galois Cohomology Let$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\$-groupsCategories:12G05, 12G10
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