Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d (≥ 1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in 2 dimensional case.

A 1-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in ℝN, is formulated by the use of subdifferentials of a singular energy—the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result, a local-in-time solution of 1-harmonic map flow equation is constructed as a limit of the solutions of p-harmonic (p>1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition

We consider the numerical analysis of evolution variational inequalities which are derived from Maxwell's equations coupled with a nonlinear constitutive relation between the electric field and the current density and governing the magnetic field around a type-II bulk superconductor located in three dimensional space. The nonlinear Ohm's law is formulated using the sub-differential of a convex energy so the theory is applied to the Bean critical state model, a power law model and an extended Bean critical state model. The magnetic field in the nonconductive region is expressed as a gradient of a magnetic scalar potential in order to handle the curl free constraint. The variational inequalities are discretized in time implicitly and in space by Nédélec's curl conforming finite element of lowest order. The non-smooth energies are smoothed with a regularization parameter so that the fully discrete problem is a system of nonlinear algebraic equations at each time step. We prove various convergence results. Some numerical simulations under a uniform external magnetic field are presented.Keywords: macroscopic models for superconductivity, variational inequality, Maxwell's equations, edge finite element, convergence, computational electromagnetism.

For a class of tight-binding many-electron models on hyper-cubic lattices the equal-time correlation functions at non-zero temperature are proved to decay exponentially in the distance between the center of positions of the electrons and the center of positions of the holes. The decay bounds hold in any space dimension in the thermodynamic limit if the interaction is sufficiently small depending on temperature. The proof is based on the U (1)-invariance property and volume-independent perturbative bounds of the finite dimensional Grassmann integrals formulating the correlation functions.