von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

Fluid‐structure coupling of linear elastic model with compressible flow models

Cavitation erosion is caused in solids exposed to strong pressure waves developing in an adjacent fluid field. The knowledge of the transient distribution of stresses in the solid is important to understand the cause of damaging by comparisons with breaking points of the material. The modeling of this problem requires the coupling of the models for the fluid and the solid. For this purpose, we use a strategy based on the solution of coupled Riemann problems that has been originally developed for the coupling of 2 fluids. This concept is exemplified for the coupling of a linear elastic structure with an ideal gas. The coupling procedure relies on the solution of a nonlinear equation. Existence and uniqueness of the solution is proven. The coupling conditions are validated by means of quasi‐1D problems for which an explicit solution can be determined. For a more realistic scenario, a 2D application is considered where in a compressible single fluid, a hot gas bubble at low pressure collapses in a cold gas at high pressure near an adjacent structure.     

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.