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351. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials.

Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional
Categories:46L05, 47B99

352. CMB 2011 (vol 55 pp. 689)

Berndt, Ryan
A Pointwise Estimate for the Fourier Transform and Maxima of a Function
We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theorem
Categories:42A38, 65T99

353. CMB 2011 (vol 54 pp. 544)

Strungaru, Nicolae
Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction
In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.

Keywords:pure point diffraction, positive definite measure, Fourier transform of measures
Category:43A25

354. CMB 2011 (vol 55 pp. 176)

Spirn, Daniel; Wright, J. Douglas
Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of $t^{-5/6}$. This rate is due to competition between surface tension and gravitation at $O(1)$ wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called "slowest wave". Additionally, we combine our dispersive estimates with $L^2$ type energy bounds to prove a family of Strichartz estimates.

Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimates
Categories:76B07, 76B15, 76B45

355. CMB 2011 (vol 55 pp. 138)

Li, Benling; Shen, Zhongmin
Projectively Flat Fourth Root Finsler Metrics
In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian.

Keywords:projectively flat, Finsler metric, fourth root Finsler metric
Category:53B40

356. CMB 2011 (vol 55 pp. 73)

Dean, Andrew J.
Classification of Inductive Limits of Outer Actions of ${\mathbb R}$ on Approximate Circle Algebras
In this paper we present a classification, up to equivariant isomorphism, of $C^*$-dynamical systems $(A,{\mathbb R},\alpha )$ arising as inductive limits of directed systems $\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$ is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the $\alpha_n$s are outer actions generated by rotation of the spectrum.

Keywords:classification, $C^*$-dynamical system
Categories:46L57, 46L35

357. CMB 2011 (vol 55 pp. 164)

Pergher, Pedro L. Q.
Involutions Fixing $F^n \cup \{\text{Indecomposable}\}$
Let $M^m$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\colon M^m \to M^m$ whose fixed point set has the form $F^n \cup F^j$, where $F^n$ and $F^j$ are submanifolds with dimensions $n$ and $j$, $F^j$ is indecomposable and $ n >j$. Write $n-j=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(n-j) = 2n+p-q+1$ if $p \leq q + 1$ and $m(n-j)= 2n + 2^{p-q}$ if $p \geq q$. In this paper we show that $m \le m(n-j) + 2j+1$. Further, we show that this bound is \emph{almost} best possible, by exhibiting examples $(M^{m(n-j) +2j},T)$ where the fixed point set of $T$ has the form $F^n \cup F^j$ described above, for every $2 \le j
Keywords:involution, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic number
Category:57R85

358. CMB 2011 (vol 55 pp. 172)

Rhoades, B. E.
Hausdorff Prime Matrices
In this paper we give the form of every multiplicative Hausdorff prime matrix, thus answering a long-standing open question.

Keywords:Hausdorff prime matrices
Category:40G05

359. CMB 2011 (vol 55 pp. 81)

Divaani-Aazar, Kamran; Hajikarimi, Alireza
Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite.

Keywords:cofinite modules, generalized local cohomology modules, local cohomology modules
Categories:13D45, 13E05, 13E10

360. CMB 2011 (vol 54 pp. 726)

Ostrovskii, M. I.
Auerbach Bases and Minimal Volume Sufficient Enlargements
Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have ``exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having ``exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases.

Keywords:Banach space, Auerbach basis, sufficient enlargement
Categories:46B07, 52A21, 46B15

361. CMB 2011 (vol 54 pp. 498)

Mortad, Mohammed Hichem
On the Adjoint and the Closure of the Sum of Two Unbounded Operators
We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples "spice up" the paper.

Keywords:unbounded operators, sum and products of operators, Hilbert and Banach adjoints, self-adjoint operators, closed operators, closure of operators
Category:47A05

362. CMB 2011 (vol 55 pp. 60)

Coons, Michael
Extension of Some Theorems of W. Schwarz
In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C} [\mkern-3mu[ z]\mkern-3mu] $ satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$ with $A(z)\neq 0$ and $\deg A(z),\deg B(z)
Keywords:functional equations, transcendence, power series
Categories:11B37, 11J81

363. CMB 2011 (vol 54 pp. 396)

Cho, Jong Taek; Inoguchi, Jun-ichi; Lee, Ji-Eun
Parabolic Geodesics in Sasakian $3$-Manifolds
We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.

Keywords:parabolic geodesics, pseudo-Hermitian geometry, Sasakian manifolds
Category:58E20

364. CMB 2011 (vol 54 pp. 411)

Davidson, Kenneth R.; Wright, Alex
Operator Algebras with Unique Preduals
We show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-$*$ closed unital operator algebra containing a weak-$*$ dense subalgebra of compact operators has a unique Banach space predual.

Keywords:unique predual, free semigroup algebra, CSL algebra
Categories:47L50, 46B04, 47L35

365. CMB 2011 (vol 55 pp. 114)

Kon, S. H.; Loo, Tee-How
On Characterizations of Real Hypersurfaces in a Complex Space Form with $\eta$-Parallel Shape Operator
In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

Keywords:complex space form, Hopf hypersurfaces, ruled real hypersurfaces, $\eta$-parallel shape operator
Categories:53C40, 53C15

366. CMB 2011 (vol 54 pp. 645)

Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
An Extension of Craig's Family of Lattices
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields
Categories:11H31, 11H55, 11H50, 11R18, 11R04

367. CMB 2011 (vol 54 pp. 757)

Sun, Qingfeng
Cancellation of Cusp Forms Coefficients over Beatty Sequences on $\textrm{GL}(m)$
Let $A(n_1,n_2,\dots,n_{m-1})$ be the normalized Fourier coefficients of a Maass cusp form on $\textrm{GL}(m)$. In this paper, we study the cancellation of $A (n_1,n_2,\dots,n_{m-1})$ over Beatty sequences.

Keywords:Fourier coefficients, Maass cusp form on $\textrm{GL}(m)$, Beatty sequence
Categories:11F30, 11M41, 11B83

368. CMB 2011 (vol 54 pp. 580)

Baoguo, Jia; Erbe, Lynn; Peterson, Allan
Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales
Consider the second order superlinear dynamic equation \begin{equation*} (*)\qquad x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$} \end{equation*} where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for $x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly known as the Emden--Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

Keywords:Oscillation, Emden-Fowler equation, superlinear
Categories:34K11, 39A10, 39A99

369. CMB 2011 (vol 54 pp. 716)

Okassa, Eugène
Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds
Categories:13N05, 53D05, 53D10

370. CMB 2011 (vol 54 pp. 422)

Pérez, Juan de Dios; Suh, Young Jin
Two Conditions on the Structure Jacobi Operator for Real Hypersurfaces in Complex Projective Space
We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Keywords:complex projective space, real hypersurface, structure Jacobi operator, two conditions
Categories:53C15, 53B25

371. CMB 2011 (vol 54 pp. 566)

Zhou, Xiang-Jun; Shi, Lei; Zhou, Ding-Xuan
Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

Keywords:multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence rates
Categories:68T05, 62J02

372. CMB 2011 (vol 54 pp. 693)

Lusala, Tsasa; Śniatycki, Jędrzej
Stratified Subcartesian Spaces
We show that if the family $\mathcal{O}$ of orbits of all vector fields on a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$ is locally closed, then $\mathcal{O}$ defines a smooth Whitney A stratification of $P$. We also show that the stratification by orbit type of the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth manifold $M$ is given by orbits of the family of all vector fields on $M/G$.

Keywords:Subcartesian spaces, orbits of vector fields, stratifications, Whitney Conditions
Categories:58A40, 57N80

373. 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 function
Categories:46B04, 46J10, 46E15

374. 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, Kähler 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 algebras
Categories:13D10, 14D15, 14B10

375. 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, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
Categories:47B35, 05E05, 20G05
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