Functional Integral Approach in the Theory of Color Superconductivity

In this series of lectures we present the functional integral method for studying the superconducting pairing of quarks with the formation of the diquarks as well as the quark-antiquark pairing in dense QCD. The dynamical equations for the superconducting order parameters are the nonlinear integral equations for the composite quantum fields describing the quark-quark or quark-antiquark systems. These composite fields are the bi-local fields if the pairing is generated by the gluon exchange while for the instanton induced pairing interactions they are the local ones. The expressions of the free energy densities are derived. The binding of three quarks is also discussed.Comment: 21 pages, 2 figures, Lectures at the VIth Vietnam International School in Theoretical Physics, Vung Tau, 27 December 1999 -- 08 January 200

A convergent relaxation of the Douglas-Rachford algorithm

This paper proposes an algorithm for solving structured optimization problems, which covers both the backward-backward and the Douglas-Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the algorithm is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point operators are established. When applying to nonconvex feasibility including the inconsistent case, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets which need not intersect. In this special case, we refine known linear convergence criteria for the Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems