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376. CMB 2006 (vol 49 pp. 281)

Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.
 Correction to a Theorem on Total Positivity A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence. Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur functionCategories:15A48, 15A45, 15A57, 05E05

377. CMB 2006 (vol 49 pp. 270)

Occhetta, Gianluca
 A Characterization of Products of Projective Spaces We give a characterization of products of projective spaces using unsplit covering families of rational curves. Keywords:Rational curves, Fano varietiesCategories:14J40, 14J45

378. CMB 2006 (vol 49 pp. 265)

Nicholson, W. K.; Zhou, Y.
 Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring$g(x)$-clean if every element is the sum of a unit and a root of$g(x)$. If$V$is a vector space of countable dimension over a division ring$D,$they showed that$\end {}_{D}V$is$g(x)$-clean provided that$g(x)$has two roots in$C(D)$. If$g(x)=x-x^{2}$this shows that$\end {}_{D}V$is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that$\Mend {}_{R}M$is$g(x)$-clean for any semisimple module$M$over an arbitrary ring$R$provided that$g(x)\in (x-a)(x-b)C[x]$where$a,b\in C$and both$b$and$b-a$are units in$R$. Keywords:Clean rings, linear transformations, endomorphism ringsCategories:16S50, 16E50 379. CMB 2006 (vol 49 pp. 256) Neelon, Tejinder  A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for$C^{\infty }$functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a$C^{\infty }$\ function on$\mathbb{R}^{n}$that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series$F(x,y)$\ converges on a set of lines of positive capacity then$F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero. Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power seriesCategories:32A05, 26E05 380. CMB 2006 (vol 49 pp. 247) Myjak, Józef; Szarek, Tomasz; Ślȩczka, Maciej  A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces Let$X$be a Polish space. We will prove that $$\dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to } X\},$$ where$\dim_L X$and$\dim_T X$stand for the concentration dimension and the topological dimension of$X$, respectively. Keywords:Hausdorff dimension, topological dimension, LÃ©vy concentration function, concentration dimensionCategories:11K55, 28A78 381. CMB 2006 (vol 49 pp. 213) Dean, Andrew J.  On Inductive Limit Type Actions of the Euclidean Motion Group on Stable UHF Algebras An invariant is presented which classifies, up to equivariant isomorphism,$C^*$-dynamical systems arising as limits from inductive systems of elementary$C^*$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles. Keywords:classification,$C^*$-dynamical systemCategories:46L57, 46L35 382. CMB 2006 (vol 49 pp. 11) Bevelacqua, Anthony J.; Motley, Mark J.  Going-Down Results for$C_{i}$-Fields We search for theorems that, given a$C_i$-field$K$and a subfield$k$of$K$, allow us to conclude that$k$is a$C_j$-field for some$j$. We give appropriate theorems in the case$K=k(t)$and$K = k\llp t\rrp$. We then consider the more difficult case where$K/k$is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field. Keywords:$C_i$-fields, Lang's ConjectureCategories:12F, 14G 383. CMB 2006 (vol 49 pp. 55) Dubois, Jérôme  Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot. Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space,$\SU$,$\SL$, Adjoint representation, MonodromyCategories:57Q10, 57M27, 57M25 384. CMB 2006 (vol 49 pp. 3) Al-Salman, Ahmad  On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the$L^p$mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on$L^p$provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25 385. CMB 2005 (vol 48 pp. 523) Düvelmeyer, Nico  Angle Measures and Bisectors in Minkowski Planes \begin{abstract} We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve. Keywords:Radon curves, Minkowski geometry, Minkowski planes,, angular bisector, angular measure, equiframed curvesCategories:52A10, 52A21 386. CMB 2005 (vol 48 pp. 614) Tuncali, H. Murat; Valov, Vesko  On Finite-to-One Maps Let$f\colon X\to Y$be a$\sigma$-perfect$k$-dimensional surjective map of metrizable spaces such that$\dim Y\leq m$. It is shown that for every positive integer$p$with$ p\leq m+k+1$there exists a dense$G_{\delta}$-subset${\mathcal H}(k,m,p)$of$C(X,\uin^{k+p})$with the source limitation topology such that each fiber of$f\triangle g$,$g\in{\mathcal H}(k,m,p)$, contains at most$\max\{k+m-p+2,1\}$points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let$f\colon X\to Y$be a$k$-dimensional map between compact metric spaces with$\dim Y\leq m$. Then: \begin{inparaenum}[\rm(1)] \item there exists a map$h\colon X\to\uin^{m+2k}$such that$f\triangle h\colon X\to Y\times\uin^{m+2k}$is 2-to-one provided$k\geq 1$; \item there exists a map$h\colon X\to\uin^{m+k+1}$such that$f\triangle h\colon X\to Y\times\uin^{m+k+1}$is$(k+1)$-to-one. \end{inparaenum} Keywords:finite-to-one maps, dimension, set-valued mapsCategories:54F45, 55M10, 54C65 387. CMB 2005 (vol 48 pp. 580) Kot, Piotr  Exceptional Sets in Hartogs Domains Assume that$\Omega$is a Hartogs domain in$\mathbb{C}^{1+n}$, defined as$\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where$H$is an open set in$\mathbb{C}^{n}$and$\mu$is a continuous function with positive values in$H$such that$-\ln\mu$is a strongly plurisubharmonic function in$H$. Let$\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$. For a given set$E$contained in$H$of the type$G_{\delta}$we construct a holomorphic function$f\in\mathbb{O}(\Omega)$such that $E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.$ Keywords:boundary behaviour of holomorphic functions,, exceptional setsCategory:30B30 388. CMB 2005 (vol 48 pp. 561) Foth, Philip  A Note on Lagrangian Loci of Quotients We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group. Keywords:Quotients, involutions, real forms, Lagrangian lociCategory:53D20 389. CMB 2005 (vol 48 pp. 547) Fehér, L. M.; Némethi, A.; Rimányi, R.  Degeneracy of 2-Forms and 3-Forms We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation$\Lambda^k\C^n$. Keywords:Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theoryCategories:14N10, 57R45 390. CMB 2005 (vol 48 pp. 505) Bouikhalene, Belaid  On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group Let$G$be a compact group. Let$\sigma$be a continuous involution of$G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where$f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on$G$such that$f$is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group$G$. Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.Categories:39B32, 39B42, 22D10, 22D12, 22D15 391. CMB 2005 (vol 48 pp. 409) Gauthier, P. M.; Xiao, J.  The Existence of Universal Inner Functions on the Unit Ball of$\mathbb{C}^n$It is shown that there exists an inner function$I$defined on the unit ball${\bf B}^n$of${\mathbb C}^n$such that each function holomorphic on${\bf B}^n$and bounded by$1$can be approximated by non-Euclidean translates" of$I$. Keywords:universal inner functionsCategories:32A35, 30D50, 47B38 392. CMB 2005 (vol 48 pp. 340) Andruchow, Esteban  Short Geodesics of Unitaries in the$L^2$Metric Let$\M$be a type II$_1$von Neumann algebra,$\tau$a trace in$\M$, and$\l2$the GNS Hilbert space of$\tau$. We regard the unitary group$U_\M$as a subset of$\l2$and characterize the shortest smooth curves joining two fixed unitaries in the$L^2$metric. As a consequence of this we obtain that$U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of$\l2$Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.Categories:46L51, 58B10, 58B25 393. CMB 2005 (vol 48 pp. 260) Oberlin, Daniel M.  A Restriction Theorem for a \\$k$-Surface in$\mathbb R ^n$We establish a sharp Fourier restriction estimate for a measure on a$k$-surface in$\mathbb R ^n$, where$n=k(k+3)/2$. Keywords:Fourier restrictionCategory:42B10 394. CMB 2005 (vol 48 pp. 283) Thibault, Lionel; Zagrodny, Dariusz  Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions$f$and$g$have the subdifferential of$f$included in the$\gamma$-enlargement of the subdifferential of$g$, then the difference of those functions is$ \gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20 395. CMB 2005 (vol 48 pp. 267) Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.  Continuous Adjacency Preserving Maps on Real Matrices It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form$ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix$R$. The result extends previously known theorems where the map was assumed to be also injective. Keywords:adjacency of matrices, continuous preservers, affine transformationsCategories:15A03, 15A04. 396. CMB 2005 (vol 48 pp. 195) Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.  On Suslinian Continua A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most$\omega_1$and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set. Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finiteCategories:54F15, 54D15, 54F50 397. CMB 2005 (vol 48 pp. 180) Cynk, Sławomir; Meyer, Christian  Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds We study Calabi--Yau manifolds constructed as double coverings of$\mathbb{P}^3$branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over$\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi--Yau manifolds and 14 families with$h^{1,2}=1$. The modularity conjecture is verified for all the rigid examples. Keywords:Calabi--Yau, double coverings, modular formsCategories:14G10, 14J32 398. CMB 2005 (vol 48 pp. 161) Betancor, Jorge J.  Hankel Convolution Operators on Spaces of Entire Functions of Finite Order In this paper we study Hankel transforms and Hankel convolution operators on spaces of entire functions of finite order and their duals. Keywords:Hankel transform, convolution, entire functions, finite orderCategory:46F12 399. CMB 2005 (vol 48 pp. 147) Väänänen, Keijo; Zudilin, Wadim  Baker-Type Estimates for Linear Forms in the Values of$q$-Series We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field~$\II$, in particular of the values of$q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincar\'e-type equations and the connection between the solutions of these functional equations and the generalized Heine series. Keywords:measure of linear independence,$q$-seriesCategories:11J82, 33D15 400. CMB 2005 (vol 48 pp. 121) Mollin, R. A.  Necessary and Sufficient Conditions for the Central Norm to Equal$2^h$in the Simple Continued Fraction Expansion of$\sqrt{2^hc}$for Any Odd$c>1$We look at the simple continued fraction expansion of$\sqrt{D}$for any$D=2^hc $where$c>1$is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be$2^h$. At the end of the paper, we also address the case where$D=c$is odd and the central norm of$\sqrt{D}$is equal to$2\$. Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fieldsCategories:11A55, 11D09, 11R11
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