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351. CMB 2011 (vol 54 pp. 680)

Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés
 $2$-Local Isometries on Spaces of Lipschitz Functions Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $\| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\| f\| _\infty +L(f),$ where $L(f)$ is the Lipschitz constant of $f.$ It is said that the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical if every surjective linear isometry of $\mathop{\textrm{Lip}}(X)$ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry of $\mathop{\textrm{Lip}}(X)$ is a surjective linear isometry. Furthermore, we give a complete description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is bounded. Keywords:isometry, local isometry, Lipschitz functionCategories:46B04, 46J10, 46E15

352. CMB 2011 (vol 54 pp. 472)

Iacono, Donatella
 A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps We study infinitesimal deformations of holomorphic maps of compact, complex, KÃ¤hler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain. Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebrasCategories:13D10, 14D15, 14B10

353. CMB 2011 (vol 54 pp. 338)

Nakazi, Takahiko
 SzegÃ¶'s Theorem and Uniform Algebras We study SzegÃ¶'s theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra. Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman spaceCategories:32A35, 46J15, 60G25

354. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
 On an Identity due to Bump and Diaconis, and Tracy and Widom A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity. Keywords:Toeplitz matrices, Jacobi-Trudi identity, SzegÅ limit theorem, Heine identity, Wiener-Hopf factorizationCategories:47B35, 05E05, 20G05

355. CMB 2011 (vol 54 pp. 249)

Dattori da Silva, Paulo L.
 A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic. Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)Categories:35A01, 58Jxx

356. CMB 2011 (vol 54 pp. 311)

Marzougui, Habib
 Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows We give some extension to theorems of JimÃ©nez LÃ³pez and Soler LÃ³pez concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces. Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder Categories:37B20, 37E35

357. CMB 2011 (vol 54 pp. 330)

Mouhib, A.
 Sur la borne infÃ©rieure du rang du 2-groupe de classes de certains corps multiquadratiques Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que $p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q} (\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le $2$-groupe de classes de $k$. A. FrÃ¶hlich a dÃ©montrÃ© que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article, nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le rang de $\operatorname{Cl}_2(k)$ est toujours supÃ©rieur ou Ã©gal Ã  $2$. Nous dÃ©montrons aussi, que la valeur $2$ est optimale pour une famille infinie de corps $k$. Keywords:class group, units, multiquadratic number fieldsCategories:11R29, 11R11

358. CMB 2011 (vol 54 pp. 302)

Kurka, Ondřej
 Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$. Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class Categories:46B20, 54H05, 46B10

359. CMB 2010 (vol 54 pp. 193)

Bennett, Harold; Lutzer, David
 Measurements and $G_\delta$-Subsets of Domains In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D.~K. Burke to show that there is a Scott domain $P$ for which $\max(P)$ is a $G_\delta$-subset of $P$ and yet no measurement $\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also correct a mistake in the literature asserting that $[0, \omega_1)$ is a space of this type. We show that if $P$ is a Scott domain and $X \subseteq \max(P)$ is a $G_\delta$-subset of $P$, then $X$ has a $G_\delta$-diagonal and is weakly developable. We show that if $X \subseteq \max(P)$ is a $G_\delta$-subset of $P$, where $P$ is a domain but perhaps not a Scott domain, then $X$ is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain $P$ such that $\max(P)$ is the usual space of countable ordinals and is a $G_\delta$-subset of $P$ in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space. Keywords:domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$-diagonal, Äech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-completeCategories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99

360. CMB 2010 (vol 54 pp. 538)

Srinivasan, Gopala Krishna; Zvengrowski, P.
 On the Horizontal Monotonicity of $|\Gamma(s)|$ Writing $s = \sigma + it$ for a complex variable, it is proved that the modulus of the gamma function, $|\Gamma(s)|$, is strictly monotone increasing with respect to $\sigma$ whenever $|t| > 5/4$. It is also shown that this result is false for $|t| \leq 1$. Keywords:Gamma function, modulus, monotonicityCategory:33B15

361. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
 On the Dichotomy of the Evolution Families: A Discrete-Argument Approach We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family ${\cal U}=\{U(t,s)\}_{t \geq s\geq 0}$ acting on a Banach space $X$ is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$ admits a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of proof effectively eliminates the continuity hypothesis on the evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$ or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively $[0,t]$). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended. Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibilityCategories:34D05, 47D06, 93D20

362. CMB 2010 (vol 54 pp. 207)

Chen, Jiecheng; Fan, Dashan
 A Bilinear Fractional Integral on Compact Lie Groups As an analog of a well-known theorem on the bilinear fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein, we establish the similar boundedness property for a bilinear fractional integral on a compact Lie group. Our result is also a generalization of our recent theorem about the bilinear fractional integral on torus. Keywords:bilinear fractional integral, $L^p$ spaces, Heat kernelCategories:43A22, 43A32, 43B25

363. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
 Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation. Keywords:evolution families, exponential instability, Lyapunov equationCategories:34D05, 47D06

364. CMB 2010 (vol 54 pp. 270)

Dow, Alan
 Sequential Order Under PFA It is shown that it follows from PFA that there is no compact scattered space of height greater than $\omega$ in which the sequential order and the scattering heights coincide. Keywords:sequential order, scattered spaces, PFACategories:54D55, 03E05, 03E35, 54A20

365. CMB 2010 (vol 54 pp. 381)

Velušček, Dejan
 A Short Note on the Higher Level Version of the Krull--Baer Theorem Klep and Velu\v{s}\v{c}ek generalized the Krull--Baer theorem for higher level preorderings to the non-commutative setting. A $n$-real valuation $v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section of $\overline{v}$ is a crucial ingredient of the construction of a complete preordering on the base field $D$ such that its projection on the residue skew field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give a proof of the existence of the section of $\overline{v}$, which was left as an open problem by Klep and Velu\v{s}\v{c}ek, and thus complete the generalization of the Krull--Baer theorem for preorderings. Keywords:orderings of higher level, division rings, valuationsCategories:14P99, 06Fxx

366. CMB 2010 (vol 54 pp. 12)

Bingham, N. H.; Ostaszewski, A. J.
 Homotopy and the Kestelman-Borwein-Ditor Theorem The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by translation in (measure/category) large'' sets has two generalizations. Miller replaces the translated sequence by a sequence homotopic to the identity''. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows from the functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version. Keywords:measure, category, measure-category duality, differentiable homotopyCategory:26A03

367. CMB 2010 (vol 54 pp. 159)

 Hardy Inequalities on the Real Line We prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line. Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spacesCategories:42A05, 42A99

368. CMB 2010 (vol 54 pp. 147)

Nelson, Sam
 Generalized Quandle Polynomials We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants that further generalize the quandle counting invariant. Keywords:finite quandles, finite biquandles, link invariantsCategories:57M27, 76D99

369. CMB 2010 (vol 54 pp. 180)

Spurný, J.; Zelený, M.
 Additive Families of Low Borel Classes and Borel Measurable Selectors An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a $\sigma$-discrete refinement. We confirm the conjecture for point-countable $\mathbf\Pi_3^0$-additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol. Keywords:$\sigma$-discrete refinement, Borel-additive family, measurable selectionCategories:54H05, 54E35

370. CMB 2010 (vol 53 pp. 587)

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq
 Hulls of Ring Extensions We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls $\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of $B$ are strongly Morita equivalent as $C^*$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided. Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebraCategories:16N60, 16D90, 16S99, 16S50, 46L05

371. CMB 2010 (vol 54 pp. 39)

Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.
 Elements in a Numerical Semigroup with Factorizations of the Same Length Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions. Keywords:numerical monoid, numerical semigroup, non-unique factorizationCategories:20M14, 20D60, 11B75

372. CMB 2010 (vol 53 pp. 690)

Puerta, M. E.; Loaiza, G.
 On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of $\ell_p$ spaces. In a previous paper, an interpolation space, defined via the real method and using $\ell_p$ spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm. Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spacesCategories:46M05, 46M35, 46A32

373. CMB 2010 (vol 53 pp. 684)

Proctor, Emily; Stanhope, Elizabeth
 An Isospectral Deformation on an Infranil-Orbifold We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon. Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifoldCategories:58J53, 53C20

374. CMB 2010 (vol 53 pp. 674)

Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba
 Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments. Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutionsCategories:58J05, 35P30

375. CMB 2010 (vol 53 pp. 667)

Khashyarmanesh, Kazem
 On the Endomorphism Rings of Local Cohomology Modules Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also prove that, for a nonnegative integer $n$ such that $H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for all $i >0$ and $z \in \mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic image of $R$, where $R_z$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{z^j \mid j \geqslant 0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z \in \mathfrak{a}$, then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$ is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$. Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequenceCategories:13D45, 13D07, 13D25
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