On the limit configuration of four species strongly competing systems

We analysed some qualitative properties of the limit configuration of the solutions of a reaction-diffusion system of four competing species as the competition rate tends to infinity. Large interaction induces the spatial segregation of the species and only two limit configurations are possible: either there is a point where four species concur, a 4-point, or there are two points where only three species concur. We characterized, for a given datum, the possible 4-point configuration by means of the solution of a Dirichlet problem for the Laplace equation

Through a Lattice Darkly -- Shedding Light on Electron-Phonon Coupling in the High Tc_c Cuprates

With its central role in conventional BCS superconductivity, electron-phonon coupling has appeared to play a more subtle role in the phase diagram of the high temperature superconducting cuprates. The added complexity of the cuprates with potentially numerous competing phases including charge, spin, orbital, and lattice ordering, makes teasing out any unique phenomena challenging. In this review, we present our work using angle resolved photoemission spectroscopy (ARPES) to explore the role of the lattice and its effect on the valence band electronic structure in the cuprates. We provide an introduction to the ARPES technique and its unique ability to the probe the effect of bosonic renormalization (or "kink") on the near-E$_F$ band structure. Our survey begins with the establishment of the ubiquitous nodal cuprate kink leading to the way isotope substitution has shed a critical new perspective on the role and strength of electron-phonon coupling. We continue with recently published work on the connection between the phonon dispersion as seen with inelastic x-ray scattering (IXS) and the location of the kink as observed by ARPES near the nodal point. Finally, we present very recent and ongoing ARPES work examining how induced strain through chemical pressure provides a potentially promising avenue for understanding the broader role of the lattice to the superconducting phase and larger cuprate phase diagram.Comment: 17 pages, 20 figures, Review Articl

Fast cubature of high dimensional biharmonic potential based on Approximate Approximations

We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast and effective also in very high dimensions. Numerical tests show that the formulas are accurate and provide the predicted approximation rate (O(h^8)) up to the dimension 10^7

Tensor product approximations of high dimensional potentials

The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations.Comment: 20 page

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

We propose a fast method for high order approximation of potentials of the Helmholtz type operator Delta+kappa^2 over hyper-rectangles in R^n. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and kappa^2=100

Approximate Approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.Comment: 29 pages, 17 figure

Computation of volume potentials over bounded domains via approximate approximations

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.Comment: 18 page

Fast cubature of volume potentials over rectangular domains

In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $O(h^6)$ up to dimension $10^8$.Comment: 17 page

Educational Achievement of Second Generation Immigrants: An International Comparison

This paper investigates the educational achievements of second generation immigrants in several OECD countries in a comparative perspective. We first show that the educational achievement (measured as test scores in PISA achievement tests) of children of immigrants is quite heterogeneous across countries, and strongly related to achievements of the parent generation. The disadvantage considerably reduces, and even disappears for some countries, once we condition on parental background characteristics. Second, we provide novel analysis of cross-country comparisons of test scores of children from the same country of origin, and compare (conditional) achievement scores in home and host countries. The focus is on Turkish immigrants, whom we observe in several destination countries. We investigate both mathematics and reading test scores, and show that the results vary according to the type of skills tested. For mathematics, in most countries and even if the test scores achievement of the children of Turkish immigrants is lower than that of their native peers, it is still higher than that of children of their cohort in the home country - conditional and unconditional on parental background characteristics. The analysis suggests that higher school quality relative to that in the home country is important to explain immigrant children’s educational advantageEducation, Second-Generation Immigrants