Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

A general decomposition approach for multi-criteria decision trees

In this paper, we address the dynamic and multi-criteria decision-making problems under uncertainty, generally represented by multi-criteria decision trees. Decision-making consists of choosing, at each period, a decision that maximizes the decision-maker outcomes. These outcomes should often be measured against a set of heterogeneous and conflicting criteria. Generating the set of non-dominated solutions is a common approach considered in the literature to solve the multi-criteria decision trees, but it becomes very challenging for large problems. We propose a new approach to solve multi-criteria decision trees without generating the set of all non-dominated solutions, which should reduce the computation time and the cardinality of the solution set. In particular, the proposed approach combines the advantages of decomposition with the application of multi-criteria decision aid (MCDA) methods at each decision node. A generalization of the Bellman’s principle of decomposition to the multi-criteria context is put forward. A decomposition theorem is therefore proposed. Under the sufficient conditions stated by the theorem, the principle of decomposition will generate the set of best compromise strategies. Seven MCDA methods are then characterized (lexicographic, weighted sum, multi-attribute value theory, TOPSIS, ELECTRE III, and PROMETHEE II) against the conditions of the theorem of decomposition and against other properties (neutrality, anonymous, fidelity, dominance, independency), in order to confirm or infirm their applicability with the proposed decomposition principle. Moreover, the relationship between independency and temporal consistence is discussed as well as the effects of incomparableness, rank reversals, and use of thresholds. Two conjectures resulted from this characterization.     

Generalized CR Equation with a Singularity

This article deals with the solvability of the CR equation rw = (A₀(θ) ? O(r^α))w ? (B₀(θ) ? O(r^α)), where z = r e^iθ and where A₀, B₀ are continuous and 2π-periodic. The solutions of this equation are shown to be similar to those of the model equation rw = A₀(θ)w ? B₀(θ). The solutions of the model equation are completely characterized by using dynamical systems and Fourier techniques.     

Mach's principle and higher‐dimensional dynamics

We briefly discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP‐B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein's theory of gravitation can be locally embedded in a Ricci‐flat 5D manifold such that the 4D energy‐momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein's general relativity is presented.     

Base force element method of complementary energy principle for large rotation problems

Using the concept of the base forces, a new finite element method （base force element method, BFEM） based on the complementary energy principle is presented for accurate modeling of structures with large displacements and large rotations. First, the complementary energy of an element is described by taking the base forces as state variables, and is then separated into deformation and rotation parts for the case of large deformation. Second, the control equations of the BFEM based on the complementary energy principle are derived using the Lagrange multiplier method. Nonlinear procedure of the BFEM is then developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the BFEM.     

非线性随机动力系统的概率密度演化分析

阐述了基于概率密度演化理论进行多自由度结构非线性随机动力反应分析的基本思想.采用随机过程的正交分解或物理系统建模的思想,实现随机激励的随机函数表述.对由此获得的随机状态方程采用概率密度演化理论求解,可以获得随机动力系统反应的概率密度函数及其演化.以某剪切型框架结构的非线性随机地震反应分析为例,说明了所发展方法的可行性和有效性.     

Groupoids,Discrete Mechanics,and Discrete Variation

After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid, we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection between groupoids variation and the methods of the first and second discrete variational principles.     

The Newtonian Correspondence

It is shown that the field equations of general theory of relativity in the Einstein tensor form and the unimodular theory of gravity do not fulfill the correspondence principle commitment completely. The consistent formalisms are briefly discussed.     

Axioma for Mach's Mechanics

We present an axiomatic framework for what we call Mach's mechanics, inspired on the ideas by A. K. T. Assis and P. Graneau about relational mechanics. We show that, in contrast to a suggestion of these authors, Mach's principle does not need to be committed with any Weber's like gravitational force. Actually it is possible to settle an axiomatic framework for non-relativistic classical particle mechanics which incorporates Mach's principle and is still consistent with any force that obeys the Newtonian action-reaction principle. We make further criticisms concerning relational mechanics.