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

<i>d</i>-wave superconductivity from electron-phonon interactions

I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-two-dimensional Holstein model. I use an extended Migdal-Eliashberg theory that includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb pseudopotential on the superconductivity (such as is found in contemporary superconducting materials like the cuprates), which results in the destruction of the s-wave states, while leaving the d-wave states unmodified

Fast computation of elastic and hydrodynamic potentials using approximate approximations

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order O(h2M) , M= 1 , 2 , 3 , 4 , are provided

Approximation of Uncoupled Quasi-Static Thermoelasticity Solutions Based on Gaussians

A fast approximation method to three dimensional equations in quasi-static uncoupled thermoelasticity is proposed. We approximate the density via Gaussian approximating functions introduced in the method approximate approximations. In this way the action of the integral operators on such functions is presented in a simple analytical form. If the density has separated representation, the problem is reduced to the computation of one-dimensional integrals which admit efficient cubature procedures. The comparison of the numerical and exact solution shows that these formulas are accurate and provide the predicted approximation rate 2 , 4 , 6 and 8

A high-efficiency spin-resolved phototemission spectrometer combining time-of-flight spectroscopy with exchange-scattering polarimetry

We describe a spin-resolved electron spectrometer capable of uniquely efficient and high energy resolution measurements. Spin analysis is obtained through polarimetry based on low-energy exchange scattering from a ferromagnetic thin-film target. This approach can achieve a similar analyzing power (Sherman function) as state-of-the-art Mott scattering polarimeters, but with as much as 100 times improved efficiency due to increased reflectivity. Performance is further enhanced by integrating the polarimeter into a time-of-flight (TOF) based energy analysis scheme with a precise and flexible electrostatic lens system. The parallel acquisition of a range of electron kinetic energies afforded by the TOF approach results in an order of magnitude (or more) increase in efficiency compared to hemispherical analyzers. The lens system additionally features a 90{\deg} bandpass filter, which by removing unwanted parts of the photoelectron distribution allows the TOF technique to be performed at low electron drift energy and high energy resolution within a wide range of experimental parameters. The spectrometer is ideally suited for high-resolution spin- and angle-resolved photoemission spectroscopy (spin-ARPES), and initial results are shown. The TOF approach makes the spectrometer especially ideal for time-resolved spin-ARPES experiments.Comment: 16 pages, 11 figure

Bandwidth and Electron Correlation-Tuned Superconductivity in Rb0.8_{0.8}Fe2_{2}(Se1−z_{1-z}Sz_z)2_2

We present a systematic angle-resolved photoemission spectroscopy study of the substitution-dependence of the electronic structure of Rb$_{0.8}$Fe$_{2}$(Se$_{1-z}$S$_z$)$_2$ (z = 0, 0.5, 1), where superconductivity is continuously suppressed into a metallic phase. Going from the non-superconducting Rb$_{0.8}$Fe$_{2}$(Se$_{1-z}$S$_z$)$_2$ to superconducting Rb$_{0.8}$Fe$_{2}$Se$_2$, we observe little change of the Fermi surface topology, but a reduction of the overall bandwidth by a factor of 2 as well as an increase of the orbital-dependent renormalization in the $d_{xy}$ orbital. Hence for these heavily electron-doped iron chalcogenides, we have identified electron correlation as explicitly manifested in the quasiparticle bandwidth to be the important tuning parameter for superconductivity, and that moderate correlation is essential to achieving high $T_C$

Approximation of solutions to multidimensional parabolic equations by Approximate Approximations

Abstract. We propose a fast method for high order approximations of the solution of ndimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, make our method effective also in very high dimensions. We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 up to dimension 10 7