In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center of the augmented barrier. This problem itself has some interes...

The critical points of the continuous series are characterized by two complex numbers l1,2(Re(l1,2) < 0), and a natural number n(n ≥ 3) which enters the string susceptibility constant through γ = −2 n−1 . The critical potentials are analytic functions with a convergence radius depending on l1 (or l2). We use the orthogonal polynomial method and solve the SchwingerDyson equations with a techniqu...