Finite XXZ chain with double boundaries is considered at critical regime −1 < ∆ < 1. We construct the eigenvectors of finite Hamiltonian by means of the vertex opertors and the quasi-boundary states. Using the free field realizations of the vertex operators and the quasi-boundary states, integral representations for the correlation functions are derived.

We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains (L<16) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the mode...

A model of Landau-Ginsburg-Wilson type is used to describe the dependence of the critical amplitude (associated with coexistence curve, or correlation length, or Susceptibility) on molar mass for chain-molecule solutions of both small molecules and polymers. Experimental methods for measurements of the critical volume fraction, coexistence curve, and turbidity are discussed. Some experimental r...

This paper provides an analysis of Supply Chain Network (SCN) design problems under uncertainty, and it presents a critical review the optimization models proposed in the literature to help design robust value-creating SCN. It points out some drawbacks and missing links in the literature, and provides motivations for the development of a comprehensive SCN design methodology. The paper reviews k...

closed-loop supply chain network design is a critical issue due to its impact on both economic and environmental performances of the supply chain. in this paper, we address the problem of designing a multi-echelon, multi-product and capacitated closed-loop supply chain network. first, a mixed-integer linear programming formulation is developed to maximize the total profit. the main contribution...