On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.     

Goldfish Geodesics and Hamiltonian Reduction of Matrix Dynamics

We describe the Hamiltonian reduction of a time-dependent real-symmetric N×N matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars–Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.      

The Relativistic Geometry and Dynamics of Electrons

Atiyah and Sutcliffe (Proc R Soc Lond Ser A 458:1089–1115, 2002) made a number of conjectures about configurations of N distinct points in hyperbolic 3-space, arising from ideas of Berry and Robbins (Proc R Soc Lond Ser A 453:1771–1790, 1997). In this paper we prove all these conjectures, purely geometrically, but we also provide a physical interpretation in terms of Electrons.     

Dynamical Trajectories of Simple Mechanical Systems as Geodesics in Space with an Extra Dimension

We show the advantages of representing thedynamics of simple mechanical systems described by anatural Lagrangian, in terms of geodesics of aRiemannian (or pseudo-Riemannian) space with anadditional dimension. We demonstrate how generaltrajectories of simple mechanical systems can be putinto one-toone correspondence with the geodesics of asuitable manifold. Two different ways in which thegeometry of the configuration space can be obtained froma higher dimensional model are presented and compared:(1) by a straightforward projection, and (2) as a spacegeometry of a quotient space obtained by the action of the timelike Killing vectorgenerating a stationary symmetry of a background spacegeometry with an additional dimension. The second modelis more informative and coincides with the so-called optical model of the line-of-sight geometry. Onthe base of this model we study the behavior of nearbygeodesics to detect their sensitive dependence oninitial conditions — the key ingredient ofdeterministic chaos. The advantage of such a formulation isits invariant character.     

An Alternative Approach to the Classical Dynamics of an Extended Charged Particle

In this paper the analysis of the classical dynamics of a charged particle is carried out without considering that the electromagnetic field necessarily goes to zero at infinity. A quite general non-linear equation of motion is obtained for an extended charged particle valid for any distribution of charge in the particle and for an electromagnetic field satisfying any boundary conditions. Some common approximations are analyzed with detail to determine how the usual difficulties arise.     

The Equations of Grand Unified Field Theory in Terms of the Maurer-Cartan Structure Relations of Differential Geometry

The first and second Maurer-Cartan structure relations are combined with the Evans field equation [1] for differential forms to build a grand unified field theory based on differential geometry. The tetrad or vielbein plays a central role in this theory, and all four fields currently thought to exist in nature can be described by the same equations, the tangent space index of the tetrad in general relativity being identified with the tetrad's internal (gauge group) index guage theory.     

The Dynamics of the 3D Radial NLS with the Combined Terms

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms $$iu_{t} + \Delta{u} = -|u|^{4}u + |u|^{2}u \qquad \qquad \qquad \qquad {\rm (CNLS)}$$ in the energy space ${{H^{1}(\mathbb{R}^{3})}}$ . The threshold is given by the ground state W for the energy-critical NLS: iu _t +  Δu =  ?|u|⁴ u. This problem was proposed by Tao, Visan and Zhang in (Comm PDEs 32:1281–1343, 2007). The main difficulty is lack of the scaling invariance. Illuminated by (Ibrahim et al., in Analysis & PDE 4(3):405–460, 2011), we need to give the new radial profile decomposition with the scaling parameter, then apply it to the scattering theory. Our result shows that the defocusing, ${{\dot{H}^{1}}}$ -subcritical perturbation |u|² u does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.     

Optimal Dynamics of a Spherical Squirmer in Eulerian Description

JETP Letters - The problem of optimization of a cycle of tangential deformations of the surface of a spherical object (micro-squirmer) self-propelling in a viscous fluid at low Reynolds numbers is...     

Path Integrals and Alternative Effective Dynamics in Loop Quantum Cosmology

The alternative dynamics of loop quantum cosmology is examined by the path integral formulation.We consider the spatially flat FRW models with a massless scalar field,where the alternative quantizations inherit more features from full loop quantum gravity.The path integrals can be formulated in both timeless and deparameterized frameworks.It turns out that the effective Hamiltonians derived from the two different viewpoints are equivalent to each other.Moreover,the first-order modified Friedmann equations are derived and predict quantum bounces for contracting universe,which coincide with those obtained in canonical theory.     

Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Abstract

We consider the general properties of the replicator dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The generalization of replicator dynamics to the case of multi-agent systems is introduced. We propose a new mathematical model to describe the multi-agent interaction in complex system.     

锂电池负极材料CuSn的电子和几何结构

使用基于混合基表示的第一原理赝势法 ,研究了CuSn化合物的电子与几何结构性质 .得出CuSn二元化合物在NaCl结构、CsCl结构、闪锌矿结构、WC结构、NiAs结构和四角结构 (在CsCl结构计算的基础上 ,再沿C轴畸变 )下的体系“能量 体积”的关系 ,即能量与结构相图 ;还给出了最稳定相的能带结构、电子态密度以及电荷密度分布等性质 ,也讨论了CuSn在最稳定的NiAs结构下电子键合性质的特点 .计算得到的CuSn能量最低结构为NiAs结构 ,与实验结果一致     

Geometry of an Accelerated Rotating Disk

We analyze the geometry of a rotating disk with a tangential acceleration in the framework of the theory of Special Relativity, using the kinematic linear differential system that verifies the relative position vector of time-like curves in a Fermi reference. A numerical integration of these equations for a generic initial value problem is made up and the results are compared with those obtained in other works.     

Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.     

Electromagnetic Modes in Schwarzschild Geometry in an Annular Region

The fundamental electromagnetic modes in a spherical annular cavity with perfectly conducting walls at r = a and r = b are calculated, in the presence of a weak Schwarzschild gravitational field. Explicit expressions are given for the case of a thin shell, (b - a)/a 1. The modified angular frequency for mode ,m can be written as = = ₀ \{ 1 - (GM/a)[2 - ( + 1)/(₀ a)²] where ₀ is the nongravitational frequency. This formula (being independent of m) holds for the magnetic as well as for the electric modes.     

电子在α-磁铁中的运动描述

 由于α-磁铁具有能量选择和束团脉宽压缩的功能，它是RF-gun注入器中一个非常重要的组成部分。描述了α-磁铁的结构特点，讨论了电子在α-磁铁中运动的归一化方程。采用数值计算的方法给出了与电子运动能量无关的理想轨道运动方程，并导出了α-磁铁的主要性质。     

电子在α-磁铁中的运动描述

由于α-磁铁具有能量选择和束团脉宽压缩的功能,它是RF-gun注入器中一个非常重要的组成部分。描述了α-磁铁的结构特点,讨论了电子在α-磁铁中运动的归一化方程。采用数值计算的方法给出了与电子运动能量无关的理想轨道运动方程,并导出了α-磁铁的主要性质。