Projective geometry and special relativity

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane – or rest space – with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre‐metric theory is discussed in the context of complex projective geometry, and ultimately, it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.     

A Continuous Transition Between Quantum and Classical Mechanics. II

Examples are worked out using a new equation proposed in the previous paper to show that it has new physical predictions for mesoscopic systems.     

A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

     

Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties

This is the first part of a series devoted to the study of thermodynamic behavior of large dynamical systems with the use of a family of fully-discrete and conservative models named elementary reversible cellular automata (ERCAs). In this paper, basic properties such as conservation laws and phase space structure are investigated in preparation for the later studies. ERCAs are a family of one-dimensional reversible cellular automata having two Boolean variables on each site. Reflection and Boolean conjugation symmetries divide them into 88 equivalence classes. For each rule, additive conserved quantities written in a certain form are regarded as a kind of energy, if they exist. By the aid of the discreteness of the variables, every ERCA satisfies the Liouville theorem or the preservation of phase space volume. Thus, if an energy exists in the above sense, statistical mechanics of the model can formally be constructed. If a locally defined quantity is conserved, however, it prevents the realization of statistical mechanics. The existence of such a quantity is examined for each class and a number of rules which have at least one energy but no local conservation laws are selected as hopeful candidates for the realization of thermodynamic behavior. In addition, the phase space structure of ERCAs is analyzed by enumerating cycles exactly in the phase space for systems of comparatively small sizes. As a result, it is revealed that a finite ERCA is not ergodic, that is, a large number of orbits coexist on an energy surface. It is argued that this fact does not necessarily mean the failure of thermodynamic behavior on the basis of an analogy with the ergodic nature of infinite systems.     

On Brennan's Conjecture for a Special Class of Functions

In this paper, we prove Brennan's conjecture for conformal mappings f of the disk {z : | z| < 1} assuming that the Taylor coefficients of the function log(zf′(z)/f(z)) at zero are nonnegative. We also obtain inequalities for the integral means over the circle |z| = r of the squared modulus of the function zf′(z)/f(z).     

非完整约束奇异广义力学系统的Poincaré-Cartan积分

对受高阶微商非完整约束并用奇异Lagrange量描述的广义力学系 统，基于广义Apell-Четаев约束条件，并考虑到系统的内在约束，导出了该非完整 约 束奇异广义力学系统的广义Poincaré-Cartan积分不变量. 并证明了该不变量与非完整约束 奇异广义力学系统的广义正则方程等价.     

In-Situ Intrinsic Interface Strength Measurement at Elevated Temperatures and its Relationship to Interfacial Structure

A previously developed laser spallation technique has been modified to measure the tensile strength of thin film interfaces in-situ at temperatures up to 1100°C. Tensile strengths of Nb/A-plane sapphire, FeCrAl/A-plane sapphire and FeCrAlY/A-plane sapphire were measured up to 950°C. The measured strengths at high temperatures were substantially lower compared with their corresponding strengths at ambient temperature. For example, at 850°C, the interface tensile strength for the Nb/sapphire (151 ± 17 MPa), FeCrAl/sapphire (62 ± 8 MPa) and FeCrAlY/sapphire (82 ± 11 MPa) interface systems were lower by factors of approximately, 3, 5, and 8, respectively, over their corresponding ambient values. These results underscore the importance of using such in-situ measured values under operating conditions as the failure criterion in any life prediction or reliability models of such coated systems where local interface temperature excursions are expected. The results on alloy film interfaces also demonstrate that the presence of Y increases the strength of FeCrAl/Al₂O₃ interfaces.     

广义C-R组的性质及两个边值问题

本文考虑广义C-R组^[1](H)的解f=u iv jw∈C^2的一些性质，提出与之相关的两个边值问题，用积分方程方法和调和函数性质证明了边值问题解的存在唯一性，并写出解的积分表达式．     

A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-steppingmethods to preserve the statistical properties of systems inmolecular dynamics. We consider a family of deterministic systemsconsisting of a finite number of particles interacting on acompact interval. The particles are given random initial conditionsand interact through instantaneous energy- and momentum-conservingcollisions. As the number of particles, the particle density,and the mean particle speed go to infinity, the trajectory ofa tracer particle is shown to converge to a stationary Gaussianstochastic process. We approximate this system by one describedby a system of ordinary differential equations and provide numericalevidence that it converges to the same stochastic process. Wesimulate the latter system with a variety of numerical integrators,including the symplectic Euler method, a fourth-order Runge-Kuttamethod, and an energyconserving step-and-project method. Weassess the methods' ability to recapture the system's limitingstatistics and observe that symplectic Euler performs significantlybetter than the others for comparable computational expense.     

非Chetaev型变质量非完整系统的Lie对称性与Noether对称性

研究非Chetaev型变质量非完整系统的Lie对称性与Noether对称性以及其间的 关系,给出Lie对称性导致Noether对称性以及Noether对称性导致Lie对称性的条件.