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

A Modular Sewing Kit for Entanglement Wedges

We relate the Riemann curvature of a holographic spacetime to an entanglement property of the dual CFT state: the Berry curvature of its modular Hamiltonians. The modular Berry connection encodes the relative bases of nearby CFT subregions while its bulk dual, restricted to the code subspace, relates the edge-mode frames of the corresponding entanglement wedges. At leading order in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal coordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement is a promising new tool for connecting the dynamics of entanglement and gravitation.Comment: 26 pages + Appendices, 4 figure

Killing gauge for the 0-brane on AdS2×S2AdS_2 \times S^2 coset superspace

How to gauge fix \k-symmetry for the super 0-brane action on $AdS_2 \times S^2$ in Killing gauge properly is discussed in order to find the superconformal mechanics which describes super 0-brane probes moving on $AdS_2 \times S^2$. The dependence on the coordinate frame for the proper Killing gauge is considered and the subtleties of gauge-fixing \k-symmetry in Killing gauge are analysed explicitly. It is found that the Killing gauge works indeed without the imcompatibility if the magnetic charge of the super 0-brane is nonzero.Comment: 14 pages, LaTeX2

Euclidean Path Integral, D0-Branes and Schwarzschild Black Holes in Matrix Theory

The partition function in Matrix theory is constructed by Euclidean path integral method. The D0-branes, which move around in the finite region with a typical size of Schwarzschild radius, are chosen as the background. The mass and entropy of the system obtained from the partition function contain the parameters of the background. After averaging the mass and entropy over the parameters, we find that they match the properties of 11D Schwarzschild black holes. The period \b of Euclidean time can be identified with the reciprocal of the boosted Hawking temperature. The entropy $S$ is shown to be proportional to the number $N$ of Matrix theory partons, which is a consequence of the D0-brane background.Comment: 15 pages, Late

Fine-structure diagnostics of neutral carbon toward HE 0515-4414

New high-resolution high signal-to-noise spectra of the $z=1.15$ damped Lyman $\alpha$ (DLA) system toward the quasi-stellar object HE 0515-4414 reveal absorption lines of the multiplets 2 and 3 in \ion{C}{i}. The resonance lines are seen in two components with total column densities of $\log N=13.79\pm0.01$ and $\log N=13.36\pm0.01$, respectively. The comparision of theoretical calculations of the relative fine-structure population with the ratios of the observed column densities suggests that the \ion{C}{i} absorbing medium is either very dense or exposed to very intense UV radiation. The upper limit on the local UV energy density is 100 times the galactic UV energy density, while the upper limit on the \ion{H}{i} number density is 110 cm$^{-3}$. The excitation temperatures of the ground state fine-structure levels of $T=15.7$ and $T=11.1$ K, respectively, are consistent with the temperature-redshift relation predicted by the standard Friedmann cosmology. The cosmic microwave background radiation (CMBR) is only a minor source of the observed fine-structure excitation.Comment: 5 pages, 5 figures, uses A&A macro package, gzipped tar archive, accepted by A&