Two Unconditional Stable Schemes for Simulation of Heat Equation on Manifold Using DEC

To predict the heat diffusion in a given region over time, it is often necessary to find the numerical solution for heat equation. However, the computational domain of classical numerical methods are limited to flat spacetime. With the techniques of discrete differential calculus, we propose two unconditional stable numerical schemes for simulation heat equation on space manifoldand time. The analysis of their stability and error is accomplished by the use of maximum principle.     

Thermodynamics of solid surfaces

In contrast to the thermodynamics of fluid surfaces, the thermodynamics of solid surfaces was not elaborated in detail by Gibbs and other founders of surface thermodynamics. During recent decades, significant progress in this field has been achieved in both the understanding of old notions, like chemical potentials, and in formulating new areas. Applying to solid surfaces, basic relationships of classical theory of capillarity, such as the Laplace equation, the Young equation, the Gibbs adsorption equation, the Gibbs-Curie principle, the Wulff theorem and the Dupré rule, were reformulated and generalized. The thermodynamics of self-dispersion of solids and the thermodynamics of contact line phenomena were developed as well. This review provides a fresh insight into the modern state of the thermodynamics of solid surfaces. Not only a solid surface itself, both in a macroscopic body and in the system of fine particles, but also the interaction of solid surfaces with fluid phases, such as wetting phenomenon, will be analyzed. As the development of surface thermodynamics has given a powerful impetus to the creation of new experimental methods, some of these will be described as examples.     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures I：Difference Discrete Variational Principle

In this first paper of a series,we study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncommutative differential geometry.Regarding the difference as an entire geometric object,the difference discrete version of Legendre transformation can be introduced.By virtue of this variational principle,we can discretely deal with the variation problems in both the Lagrangian and Hamiltonican formalisms to get difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of the classical mechanics and classical field theory.     

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients.     

Definite paths and upper bounds on regular regions of velocity phase space

Hamilton's principle states that the path integral of the Langrangian is stationary with respect to variations of a classical path. It does not distinguish between a local minimum, a local maximum or a saddle point in path space. A simple algorithm is devised which provides strict and useful upper bounds on the region of velocity phase space occupied by paths that are either local maxima or local minima. The technique is illustrated graphically for the standard map. It is found that the bounds provide accurate numerical upper estimates for the region of velocity phase space filled by the rotational KAM surfaces at arbitrarily chosen values of the perturbation parameter.     

Champ élastique d'une dislocation rectiligne parallèle aux interfaces : une nouvelle approche

The elastic field of a crystalline defect parallel to the two free surfaces of an isotropic thin foil is studied, for plane strain, with a new approach using repeated applications of the classical solution of the Flamant's problem. The case of an edge dislocation placed at any position in the foil and with a Burgers vector parallel to the surfaces is considered in greater detail. To cite this article: R. Bonnet, C. R. Physique 4 (2003).     

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.     

An analysis of maximum clique formulations and saturated linear dynamical network

Several formulations and methods used in solving an NP-hard discrete optimization problem, maximum clique, are considered in a dynamical system perspective proposing continuous methods to the problem. A compact form for a saturated linear dynamical network, recently developed for obtaining approximations to maximum clique, is given so its relation to the classical gradient projection method of constrained optimization becomes more visible. Using this form, gradient-like dynamical systems as continuous methods for finding the maximum clique are discussed. To show the one to one correspondence between the stable equilibria of the saturated linear dynamical network and the minima of objective function related to the optimization problem, La Salle's invariance principle has been extended to the systems with a discontinuous right-hand side. In order to show the efficiency of the continuous methods simulation results are given comparing saturated the linear dynamical network, the continuous Hopfield network, the cellular neural networks and relaxation labelling networks. It is concluded that the quadratic programming formulation of the maximum clique problem provides a framework suitable to be incorporated with the continuous relaxation of binary optimization variables and hence allowing the use of gradient-like continuous systems which have been observed to be quite efficient for minimizing quadratic costs.     

Difference Discrete Variational Principles, Euler-Lagrange Cohomology and Symplectic, Multisyrnplectic Structures Ⅲ: Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and H, we have studied the difference discrete variational principle and the EulerLagrange cohomology in the framework of multi-parameter differential approach. W5 have gotten the difference discreteEulcr-Lagrangc equations and canonical ones for the difference discrete versions of classical mechanics and tield theoryas well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessaryand sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangianand Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler-Lagrangecohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonianschemes or Lagrangian ones in both the symplectic and multisymplectic algorithms arc variational integrators and theirdifference discrete symplectic structure-preserving properties can always be established not only in the solution spacebut also in the function space if and only if the related closed Euler Lagrange cohomological conditions are satisfied.     

Minimum Entropy Production of Neutrino Radiation in the Steady State

A thermodynamic minimum principle valid for photon radiation is shown to hold for arbitrary geometries. It is successfully extended to neutrinos, in the zero mass and chemical potential case, following a parallel development of photon and neutrino statistics. This minimum principle stems more from that of Planck than that of classical Onsager–Prigogine irreversible thermodynamics. Its extension from bosons to fermions suggests that it may have a still wider validity.     

Complexity for Extended Dynamical Systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.     

On a more precise statement of Hamilton's principle

It has been recognized in the literature of the calculus of variations that the classical statement of the principle of least action (Hamilton's principle for conservative systems) is not strictly correct. Recently, mathematical proofs have been offered for what is claimed to be a more precise statement of Hamilton's principle for conservative systems. According to a widely publicized version of this more precise statement, the action integral for conservative systems is a minimum for discrete systems for small time intervals only and is never minimum for continuous systems. In this paper, two contradictions to this more precise statement are demonstrated, one for a discrete system and one for a continuous system.     

有限空间中经典场的正则量子化

从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.     

Position versus momentum

Weaknesses of the classical uncertainty principle inequality $\text{ΔqΔp ?}\text{1}\text{2}$ are described. Two methods of overcoming them are presented, one via local uncertainty principles and the other via the dual notions of mean width and mean constancy. Several new inequalities are obtained.     

Fundamental notions of classical and quantum statistics: A root approach

A new method of solving classical and quantum problems of statistical data analysis based on the symbiosis of notions of quantum theory and mathematical statistics is considered. Particular attention is given to the specificity of quantum problems, determined by mutually complementary measurements (according to the Bohr complementarity principle), when, for example, a spatial-temporal picture is complemented by a momentum-energy one. The possibility of construction of multiparametric statistical models admitting a stable reconstruction of the parameters from observations (the inverse statistical problem) is studied. In this case, the only universal model of such a kind is the root model, based on the representation of the probability density as the square of the modulus of some function (called the psi function by analogy with quantum mechanics). The psi function is represented as an expansion in terms of an orthonormal basis, with the expansion coefficients being estimated by the maximum likelihood technique. The root approach makes it possible to represent the Fisher information matrix, covariance matrix, and statistical properties of the estimates of the reconstructed states in the simplest and a universal form. Being asymptotically efficient, the method allows one to reconstruct the states with an accuracy close to the theoretically attainable accuracy. It is shown that the requirement for the expansion to be of a root kind can be considered as a quantization condition, which makes it possible to choose, from among all the statistical models, which, on the average, are consistent with the laws of classical mechanics, those systems that are described by quantum mechanics.     

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality. Received: 23 May 2000/ Accepted: 21 November 2000     

Quantum theory as an indication of a new order in physics. Part A. The development of new orders as shown through the history of physics

In this paper, we discuss the general significance of order in physics, as a first step toward the development of new notions of order. We begin with a brief historical discussion of the notions of order underlying ancient Greek views, and then go on to show how these changed in key ways with the rise of classical physics. This leads to a broader view of the significance of order, which helps to indicate what is to be meant by a change of our general notions of order in physics. We then go into relativity and quantum theory, showing how these developments actually did bring in further new notions of order, which are however inconsistent and otherwise inadequate in certain ways. Finally, using these inconsistencies and inadequacies as clues or indications for yet a further new concept of order, we make some proposals for novel directions of inquiry (to be discussed in some detail in later papers) which could lead to theories as different from relativity and quantum theory as these are from classical physics.     

Inhomogeneous plane wave and the most energetic complex ray

This paper presents a study on the wave surfaces of anisotropic solids. In addition to the classical and real rays, which are defined by the normal to the slowness surfaces, it is obtained complex rays, which are associated to specific inhomogeneous plane waves. Referring to the complex Christoffel's equation and to the Fermat's principle, an intrinsic equation can be associated to these complex rays. Limiting the study to principal planes and plotting the associated complex wave surfaces, it can be shown that four energetic rays always exist in any directions for both quasi-isotropic and anisotropic media (even beyond the cusp). Consequently, it is always possible to define four closed wave surfaces (real or not).