Solution space structure of random constraint satisfaction problems with growing domains

In this paper we study the solution space structure of model RB, a standard prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using rigorous the first and the second moment method, we show that in the solvable phase close to the satisfiability transition, solutions are clustered into exponential number of well-separated clusters, with each cluster contains sub-exponential number of solutions. As a consequence, the system has a clustering (dynamical) transition but no condensation transition. This picture of phase diagram is different from other classic random CSPs with fixed domain size, such as random K-Satisfiability (K-SAT) and graph coloring problems, where condensation transition exists and is distinct from satisfiability transition. Our result verifies the non-rigorous results obtained using cavity method from spin glass theory, and sheds light on the structures of solution spaces of problems with a large number of states.Comment: 8 pages, 1 figure

From Petrov-Einstein-Dilaton-Axion to Navier-Stokes equation in anisotropic model

In this paper we generalize the previous works to the case that the near-horizon dynamics of the Einstein-Dilaton-Axion theory can be governed by the incompressible Navier-Stokes equation via imposing the Petrov-like boundary condition on hypersurfaces in the non-relativistic and near-horizon limit. The dynamical shear viscosity $\eta$ of such dual horizon fluid in our scenario, which isotropically saturates the Kovtun-Son-Starinet (KSS) bound, is independent of both the dilaton field and axion field in that limit.Comment: 13 pages,no figures; v2: 15 page, Equation.(33), some discussions and references added, minor corrections , Version accepted for publication in Physics Letters

A High-Precision Single Shooting Method for Solving Hypersensitive Optimal Control Problems

Solving hypersensitive optimal control problems is a long-standing challenge for decades in optimization engineering, mainly due to the possible nonexistence of the optimal solution to meet the required error tolerance under double-precision arithmetic and the hypersensitivity of the optimal solution with respect to the initial conditions. In this paper, a new high-precision single shooting method is presented to address the above two difficulties. Multiple-precision arithmetic and Taylor series method are introduced to provide the accurate optimal solution with arbitrary higher significant digits and arbitrary higher integral accuracy, respectively. Besides, a new modified bidirectional single shooting method is developed, which fully utilizes the three-segment structure of the hypersensitive optimal control problems and provides appropriate initial guess that is close to the optimal solutions. Numerical demonstrations in a typical hypersensitive optimal control problem are presented to illustrate the effectiveness of this new method, which indicates that the accurate optimal solution of this challenging problem can be easily solved by this simple single shooting method within several iterations