Defect free global minima in Thomson's problem of charges on a sphere

Given $N$ unit points charges on the surface of a unit conducting sphere, what configuration of charges minimizes the Coulombic energy $\sum_{i>j=1}^N 1/r_{ij}$? Due to an exponential rise in good local minima, finding global minima for this problem, or even approaches to do so has proven extremely difficult. For \hbox{$N = 10(h^2+hk+k^2)+ 2$} recent theoretical work based on elasticity theory, and subsequent numerical work has shown, that for $N \sim >500$--1000 adding dislocation defects to a symmetric icosadeltahedral lattice lowers the energy. Here we show that in fact this approach holds for all $N$, and we give a complete or near complete catalogue of defect free global minima.Comment: Revisions in Tables and Reference

Cosmological singularity theorems for f(R)f(R) gravity theories

In the present work some generalizations of the Hawking singularity theorems in the context of $f(R)$ theories are presented. The assumptions are of these generalized theorems is that the matter fields satisfy the conditions $\bigg(T_{ij}-\frac{g_{ij}}{2} T\bigg)k^i k^j\geq 0$ for any generic unit time like field, that the scalaron takes bounded positive values during its evolution, and that the resulting space time is globally hyperbolic. Then, if there exist a Cauchy hyper surface $\Sigma$ for which the expansion parameter $\theta$ of the geodesic congruence emanating orthogonally from $\Sigma$ satisfies some specific conditions, it may be shown that the resulting space time is geodesically incomplete. Some mathematical results of reference \cite{fewster} are very important for proving this. The generalized theorems presented here apply directly some specific models such as the Hu-Sawicki or Starobinsky ones \cite{especif3}, \cite{capoziello4}. However, for other scenarios, some extra assumptions should be implemented for the geodesic incompleteness to take place. However, the negation of the hypothesis of these results does not necessarily imply that a singularity is absent, but that other mathematical results should be considered to prove that.Comment: An improved version is published in JCAP 05 (2016) 02

Geometric Mean Neutrino Mass Relation

Present experimental data from neutrino oscillations have provided much information about the neutrino mixing angles. Since neutrino oscillations only determine the mass squared differences $\Delta m^2_{ij} = m^2_i - m^2_j$, the absolute values for neutrino masses $m_i$ can not be determined using data just from oscillations. In this work we study implications on neutrino masses from a geometric mean mass relation $m_2=\sqrt{m_1 m_3}$ which enables one to determined the absolute masses of the neutrinos. We find that the central values of the three neutrino masses and their $2\sigma$ errors to be $m_1 = (1.58\pm 0.18){meV}$, $m_2 = (9.04\pm 0.42){meV}$, and $m_3 = (51.8\pm 3.5){meV}$. Implications for cosmological observation, beta decay and neutrinoless double beta decays are discussed.Comment: 7 pages. Talk given at COSPA06. A reference adde

Image quality assessment based on harmonics gain/loss information

We present an objective reduced-reference image quality assessment method based on harmonic gain/loss information through a discriminative analysis of local harmonic strength (LHS). The LHS is computed from the gradient of images, and its value represents a relative degree of the appearance of blockiness on images when it is related to energy gain within an image. Furthermore, comparison between local harmonic strength values from an original, distortion-free image and a degraded, processed, or compressed version of the image shows that the LHS can also be used to indicate other types of degradations, such as blurriness that corresponds with energy loss. Our simulations show that we can develop a single metric based on this gain/loss information and use it to rate the quality of images encoded by various encoders such as DCT-based JPEG, wavelet-based JPEG 2000, or various processed images. We show that our method can overcome some limitations of the traditional PSNR

A cosmologically motivated reference formulation of numerical relativity

The application of numerical relativity to cosmological spacetimes is providing new insights into the behavior of Einstein's equations, beyond common approximations. In order for simulations to be performed as accurately and efficiently as possible, we investigate a novel formulation of Einstein's equations. This formulation evolves differences from a "reference" solution describing the dominant behavior of the spacetime, which mitigates error due to both truncation and approximate finite difference calculations. We find that the error in solutions obtained using the reference formulation can be smaller by an order of magnitude or more, with the level of improvement depending on how well the reference solution approximates the exact solution.Comment: 15 pages, 5 figures, planning to submit to CQ

Renormalized Second-order Perturbation Theory for The Electron Correlation Energy: Concept, Implementation, and Benchmarks

We present a renormalized second-order perturbation theory (rPT2), based on a Kohn-Sham (KS) reference state, for the electron correlation energy that includes the random-phase approximation (RPA), second-order screened exchange (SOSEX), and renormalized single excitations (rSE). These three terms all involve a summation of certain types of diagrams to infinite order, and can be viewed as "renormalization" of the 2nd-order direct, exchange, and single excitation (SE) terms of Rayleigh-Schr\"odinger perturbation theory based on an KS reference. In this work we establish the concept of rPT2 and present the numerical details of our SOSEX and rSE implementations. A preliminary version of rPT2, in which the renormalized SE (rSE) contribution was treated approximately, has already been benchmarked for molecular atomization energies and chemical reaction barrier heights and shows a well balanced performance [Paier et al, New J. Phys. 14, 043002 (2012)]. In this work, we present a refined version of rPT2, in which we evaluate the rSE series of diagrams rigorously. We then extend the benchmark studies to non-covalent interactions, including the rare-gas dimers, and the S22 and S66 test sets. Despite some remaining shortcomings, we conclude that rPT2 gives an overall satisfactory performance across different chemical environments, and is a promising step towards a generally applicable electronic structure approach.Comment: 16 pages, 11 figure