Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

In this paper we prove the following geometric inequality in the hyperbolic space \H^n ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{array}{rcl} \ds \int_\Sigma \s_4 d \mu\ge \ds\vs C_{n-1}^4\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 12 + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array} provided that $\Sigma$ is a horospherical convex hypersurface. Equality holds if and only if $\Sigma$ is a geodesic sphere in \H^n.Comment: 18page

A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $\tau$ satisfies $\tau > \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.Comment: 32 pages. arXiv:1211.7305 was integrated into this new version as an applicatio

The Gauss-Bonnet-Chern mass of conformally flat manifolds

In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.Comment: 17 pages, references added, the statement of Prop. 4.6 correcte

Thioglycolic acid on the gold (111) surface and Raman vibrational spectra

The interaction of thioglycolic acid with the Au(111) surface is investigaged, and it is found that at the low coverage the molecule lies down on the substrate. If the mercaptan-hydrogen atom is eliminated, the resulting SCH_2COOH molecule is randomly oriented on the surface. If the carboxylic acid group in the HSCH_2COOH molecule is deprotonated instead, the HSCH_2COO^ molecule lies down on the surface. However, when the mercaptan-hydrogen atom in the HSCH_2COO^- molecule is removed, the resulting SCH_2COO^- molecule rises up to a certain level on the substrate. The calculated Raman vibrational spectra decipher which compounds and atomic displacements contribute to the corresponding frequencies. We thus propose a consistent mechanism for the deposition of thioglycolic acid on the Au(111) surface.Comment: 18 pages, 5 figures, submitted to J. Chem. Phy

[Colored solutions of Yang-Baxter equation from representations of U_{q}gl(2)]

We study the Hopf algebra structure and the highest weight representation of a multiparameter version of $U_{q}gl(2)$. The commutation relations as well as other Hopf algebra maps are explicitly given. We show that the multiparameter universal ${\cal R}$ matrix can be constructed directly as a quantum double intertwiner, without using Reshetikhin's transformation. An interesting feature automatically appears in the representation theory: it can be divided into two types, one for generic $q$, the other for $q$ being a root of unity. When applying the representation theory to the multiparameter universal ${\cal R}$ matrix, the so called standard and nonstandard colored solutions $R(\mu,\nu; {\mu}', {\nu}')$ of the Yang-Baxter equation is obtained.Comment: [14]pages, latex, no figure

A rescaled method for RBF approximation

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties

A rescaled method for RBF approximation

A new method to compute stable kernel-based interpolants has been presented by the second and third authors. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties. First, we prove that the method is an instance of the Shepard\u2019s method, when certain weight functions are used. In particular, the method can reproduce constant functions. Second, it is possible to define a modified set of cardinal functions strictly related to the ones of the not-rescaled kernel. Through these functions, we define a Lebesgue function for the rescaled interpolation process, and study its maximum - the Lebesgue constant - in different settings. Also, a preliminary theoretical result on the estimation of the interpolation error is presented. As an application, we couple our method with a partition of unity algorithm. This setting seems to be the most promising, and we illustrate its behavior with some experiments