Analysis of a Prototypical Multiscale Method Coupling Atomistic and Continuum Mechanics： the Convex Case

In order to describe a solid which deforms smoothly in some region,but non smoothly in someother region,many multiscale methods have been recently proposed that aim at coupling an atomistic model(discrete mechanics) with a macroscopic model (continuum mechanics).We provide here a theoretical basis forsuch a coupling in a one-dimensional setting,in the case of convex energy.     

Existence and uniqueness for non-linear singular integral equations used in fluid mechanics

Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further studied for non-linear singular integral equations over a finite number of arbitrarily ordered arcs. An application to fluid mechanics theory is finally given for the determination of the form of the profiles of a turbomachine in two-dimensional flow of an incompressible fluid.     

Asymptotics of orthogonal polynomial’s entropy

This is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree. This study is motivated primarily by quantum mechanics, where the wave functions and the densities of the states of solvable quantum-mechanical systems are expressed by means of orthogonal polynomials. Moreover, the uncertainty principle, lying in the ground of quantum mechanics, is best formulated by means of position and momentum entropies. In this sense, the behavior for large values of the degree is intimately connected with the information characteristics of high energy states. But the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials. We describe some results obtained in the last 15 years, as well as sketch the ideas behind their proofs.     

From the main equation to the Klein-Gordon equation

The purpose of this paper is to bring out some connections between the work of Isaacs and physics. The first five sections are devoted to classical dynamics. The theories of Hamilton and Jacobi are revisited via the geometrical approach to optimal control and de Broglie's ideas. The other sections contain a stochastic-type generalization, leading to the Klein-Gordon equation of relativistic wave mechanics.     

Notes on ferromagnetic p‐spin and REM

In this paper we apply some of the recent mathematical techniques (mainly based on interpolation) developed in the spin glass theory to the ferromagnetic p‐spin model. We introduce two Hamiltonians and derive their thermodynamics. This is a second step toward an alternative and rigorous formulation of the statistical mechanics of simple systems on lattice. A first step has been performed in J. Stat. Phys. (2007; arXiv:0712.1344) where the techniques have been tested on the two‐body Ising model. For completeness the adaptation of the well‐known random energy model to the context of the ferromagnetism is presented. At the end a discussion on the extension of these techniques to Gaussian‐disordered p‐spin models is also briefly outlined. Copyright © 2008 John Wiley & Sons, Ltd.     

Path integral in the energy representation in quantum mechanics

We consider an alternative path integral approach to quantum mechanics. We present a resolvent of a Hamiltonian (which is the Laplace transform of the evolution operator) in a form that has the meaning of “the sum over paths” but is much better defined than the usual functional integral. We investigate this representation from different standpoints and compare such an approach to quantum mechanics with the standard approaches. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 92–111, July, 2008.     

Quantum Mechanics as a Complete Physical Theory

We show that the principles of a complete physical theory and the conclusions of the standard quantum mechanics do not irreconcilably contradict each other as is commonly believed. In the algebraic approach, we formulate axioms that allow constructing a renewed mathematical scheme of quantum mechanics. This scheme involves the standard mathematical formalism of quantum mechanics. Simultaneously, it contains a mathematical object that adequately describes a single experiment. We give an example of the application of the proposed scheme.     

Variable Programming: A Generalized Minimax Problem. Part II: Algorithms

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.This work was supported by the RGC grant CUHK 152/96H of the Hong Kong Research Grant Council.     

A Gauss-Seidel type solver for special convex programs,with application to frictional contact mechanics

An iterative method is described that solves the constrained minimization of a convex function, when the constraintsg _j(x ₁,...,x _n)0 are functions of only a few variables and can be partitioned in some way. A proof of convergence is given which is based on the fact that the function values that are generated decrease. The relation to the nonlinear equation solver TanGS is shown (Ref. 1), together with some new results for TanGS. Finally, the solver is applied to the solution of tangential traction in contact mechanics.     

Reciprocal diffusions and stochastic differential equations of second order ∗

We develop a theory of second order diffusion processes and associated stochastic differential equations of second order. We show that equations of evolution of the density, mean velocity and momentum flux are a family of first order conservation laws similar to those of continuum mechanics. We verify that the theory is satisfied for a large class of reciprocal Gaussian processes     

The effect of fractional calculus on the formation of quantum-mechanical operators

In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.     

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??This paper is based on ``Pao-Lu Hsu's lecture' (2019/3/22) at Peking University and the subsequent expansion of his reports. It begins with some recollections benefited of the author from Professor Hsu, and ends with thanking to a group of professors at Peking University for their support and help over the past decades. The middle part is the theme of the talk. It gives first an overview of personal cross research. Then, from a challenge of computing, the author reports on the study looking for a larger class of complex matrices which have real spectrum. This was done mainly in the last year. It involves the fields of computation, probability, statistical mechanics and quantum mechanics Next, the paper introduces the latest development of algorithms, which is another illustration of the intersection between probability theory and computational mathematics. As the end, it also outlines the understanding of the cross study.     

二维流体力学变分通用公式

推导得到了二维流体力学变分通用公式,该公式适用于任何二维守恒型流体力学方程,得到的泛函受约于所谓的参数约束方程(控制方程中各参数间的相互关系式)。消除参数约束,我们可以十分方便地从通用公式导得广义变分原理。几个实例证明这种方法是有效的、简单的,并具有普遍的意义。     

信息技术时代的工程力学和材料研究

论述了信息技术的重要性和发展方向，讨论了科学和工程基础研究的主要任务，建议了２１世纪力学和材料科学的研究方向     

St.Venant-Levy-Mises材料塑性力学的Lagrange形式

将Ｓｔ．Ｖｅｎａｎｔ－Ｌｅｖｙ－Ｍｉｓｅｓ材料的塑性力学问题纳入Ｌａｇｒａｎｇｅ力学的理论体系,讨论了这一种材料的塑性力学三维问题,小挠度问题以及有限变形（大挠度）问题的基本微分方程。     

Optimal capture trajectories using multiple gravity assists

Graph theoretic methods of optimal control in the presence of uncertainty are applied to a celestial mechanics problem. We find a fuel-efficient spacecraft trajectory which starts at infinity and is captured by the smaller member of a binary system, e.g., a moon of Jupiter, using multiple gravity assists.     

The simulation of offshore turbulent dispersion using seeded eddies

A way of representing turbulence in a two-dimensional situation is introduced appropriate to depth-independent offshore fluid mechanics. The turbulence is simulated by a collection of eddies, each of which has an analytically simple form but whose size, strength and position is governed by stochastically assigned variables. The problem addressed here is how contaminant is dispersed in such an eddy field. A number of experiments are performed whereby the eddies are seeded with marked particles that move with the fluid. The variance of these particles is monitored as time varies, and the results are compared with an assumed power law distribution. Although not a perfect fit, the results are in general accord with a power law with index between 1.5 and 2.5, which is in agreement with the observed power law of 2.34 due to Okubo, and a marked improvement on random walk models which give a variance directly proportional to time. Some further applications of this technique are discussed, namely the simulation of turbulent boundary layers and the simulation of the cascade of energy up turbulent length scales.     

Regularization of the backward-in-time Kuramoto-Sivashinsky equation

We are interested in solution techniques for backward-in-time evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in the recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto-Sivashinsky equation in a 1D periodic domain. Such backward problems are typical examples of ill-posed problems, where any disturbances are amplified exponentially during the backward march. Hence, regularization is required in order to solve such a problem efficiently in practice. We consider regularization approaches in which the original ill-posed problem is approximated with a less ill-posed problem obtained by adding a regularization term to the original equation. While such techniques are relatively well understood for simple linear problems, in this work we investigate them carefully in the nonlinear setting and report on some interesting universal behavior. In addition to considering regularization terms with fixed magnitudes, we also mention briefly a novel approach in which these magnitudes are adapted dynamically using simple concepts from the Control Theory.     

Quasidifferentiability in nonsmooth,nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.