The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.Comment: Proceedings of the Workshop "Numerical Analysis of Multiscale Computations" at Banff International Research Station, December 200

A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime

Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

Differentiable production and condition indices of premigrant eels (Anguilla anguilla) in two Atlantic coastal catchments of France

This paper assesses potential production of premigrant European eels Anguilla anguilla based on analysis of sedentary eel populations in two small river systems in western France that are in close proximity. Abundance and biological characteristics were evaluated from electrofishing surveys conducted in three years in September and October, before the catadromous migration of silver eels. Mean density and biomass density of the eel population differed greatly between the systems (39 ± 6 ind.100 m ± 2 and 1352 ± 171 g.100 m ± 2 in the Frémur River and 3 ± 0.32 ind.100 m ± 2 and 385 ± 42 g.100 m ± 2 in the Oir River). Premigrants were dominated by males in the Frémur (85.8%) and by females in the Oir (79.0%). Estimated premigrant biomass density was 4.5-fold higher in the Frémur (254.5 g.100 m ± 2.year ± 1) than in the Oir (56.0 g.100 m ± 2.year ± 1). Mean Fulton’s K condition factor was significantly higher for both sexes in the Oir (0.20 ± 0.004 and 0.20 ± 0.003 for males and females, respectively) than in the Frémur (0.17 ± 0.002 and 0.17 ± 0.004, respectively). The large differences in densities and biological characteristics of eels from neighboring catchments suggest that huge variability of both quantity and quality of silver eel production can be expected at the scale of the European stock

Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.Comment: 23 pages, 27 figure