Mei’s symmetry theorem for time scales nonshifted mechanical systems

We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities.Firstly,the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle.Secondly,the definitions of Mei symmetry on time scales are given and its criterions are deduced.Finally,Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems,time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved,and new conserved quantities of above systems are obtained.Results are illustrated with two examples.

Birkhoff’s Theorem and Lie Symmetry Analysis

Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.

Framing Noether’s Theorem

We discuss Noether’s theorem from a new perspective and show that the spatial continuous symmetries of a system are on one hand symmetries of the space and on the other hand are dictated by the system’s potential energy. The Noether’s charges arising from an infinitesimal motion, or a Killing vector field, of the space, are conserved if the Lie derivative of the potential energy by this vector field vanishes. The possible spatial symmetries of a mechanical system are listed according to the potential energy of the external forces.

Global Static Solutions of the Spherically Symmetric Vlasov-Einstein-Maxwell (VEM) System for Low Charge

We consider the VEM system in the context of spherical symmetry and we try to establish a global static solutions with isotropic pressure that approaches Minkowski spacetime at infinity and have a regular center. To be in accordance with numerical investigation we take here low charge particles.

Hubble Scale Dark Energy Meets Nano Scale Casimir Energy and the Rational of Their T-Duality and Mirror Symmetry Equivalence

We establish that ordinary energy, Casimir energy and dark energy are not only interlinked but are basically the same thing separated merely by scale and topology. Casimir energy is essentially a nano scale spacetime phenomenon produced by the boundary condition of the two Casimir plates constituting the Casimir experimental set up for measuring the Casimir force. By contrast dark energy is the result of the cosmic boundary condition, i.e. the boundary of the universe. This one sided M?bius-like boundary located at vast cosmic distance and was comparable only to the Hubble radius scales of the universe. All the Casimir energy spreads out until the majority of it reaches the vicinity of the edge of the cosmos. According to a famous theorem due to the Ukrainian-Israeli scientist I. Dvoretzky, almost 96% of the total energy will be concentrated at the boundary of the universe, too far away to be measured directly. The rest of the accumulated Casimir energy density is consequently the nearly 4% to 4.5%, the existence of which is confirmed by various sophisticated cosmic measurements and observations. When all is said and done, the work is essentially yet another confirmation of Witten’s T-duality and mirror symmetry bringing nano scale and Hubble scale together in an unexpected magical yet mathematically rigorous way.

时间尺度上非迁移Birkhoff系统的Mei对称性定理

研究并证明时间尺度上非迁移Birkhoff系统的Mei对称性定理.首先,建立任意时间尺度上Pfaff-Birkhoff原理和广义Pfaff-Birkhoff原理,由此导出时间尺度上非迁移Birkhoff系统(包括自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统)的动力学方程.其次,基于非迁移Birkhoff方程中的动力学函数经历变换后仍满足原方程的不变性,给出了时间尺度上Mei对称性的定义,导出了相应的判据方程.再次,建立并证明了时间尺度上非迁移Birkhoff系统的Mei对称性定理,得到了时间尺度上Birkhoff系统的Mei守恒量.并通过3个算例说明了结果的应用.

离散完整力学系统的Mei对称性共形不变性

基于离散完整系统的差分Euler-Lagrange方程,研究离散完整力学系统的Mei对称性共形不变性和守恒量.提出了该系统Mei对称性共形不变性的定义和确定方程.结合规范函数和共形因子,得到在无限小单参数点变换群作用下系统的共形不变性导致的守恒量形式.举例说明结果的应用.

Extension of Noether's theorem in PT-symmetry systems and its experimental demonstration in an optical setup

Noether’s theorem is one of the fundamental laws in physics,relating the symmetry of a physical system to its constant of motion and conservation law.On the other hand,there exist a variety of non-Hermitian parity-time(PT)-symmetric systems,which exhibit novel quantum properties and have attracted increasing interest.In this work,we extend Noether’s theorem to a class of significant PT-symmetry systems for which the eigenvalues of the PT-symmetry Hamiltonian HPTchange from purely real numbers to purely imaginary numbers,and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics.We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the PT-symmetry unbroken regime,or a chiral symmetry in the PT-symmetry broken regime.In addition,we experimentally investigate the extended Noether’s theorem in PT-symmetry single-qubit and two-qubit systems using an optical setup.Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the PT-symmetry of a system is broken.Furthermore,a novel phenomenon of masking quantum information is first observed in a PT-symmetry two-qubit system.This study not only contributes to full understanding of the relation between symmetry and conservation law in PT-symmetry physics,but also has potential applications in quantum information theory and quantum communication protocols.

结合快速鲁棒性特征改进ORB的特征点匹配算法

针对定向二进制简单描述符(ORB)算法不具备尺度不变性的问题,提出一种结合快速鲁棒性特征(SURF)算法和ORB的改进算法。首先,利用Hessian矩阵检测特征点的方法,使得提取出的特征点具有尺度不变性;然后,用ORB生成特征描述子;接着采用K-近邻算法进行粗匹配;最后,通过比率测试、对称测试、最小平方中值(LMed S)定理进行提纯。尺度变化时,该算法比ORB的匹配精度提高了74.3个百分点,比SURF的匹配精度提高了4.8个百分点;旋转变化时,该算法比ORB的匹配精度提高了6.6个百分点;匹配时间高于SURF低于ORB。实验结果表明,改进算法不仅保持了ORB的旋转不变性,而且具备了尺度不变性,在不失速度的前提下,匹配精度得到较大提高。

无耗网络幺正性条件的等价表述

本文将无耗网络的广义模对称定理加以推广,将其遗漏的相位信息补充完整,并证明了推广形式的定理与S矩阵的幺正性条件完全等价。文中还给出了应用这一定理分析无耗网络特性的实例。

薛定谔方程的U(1)对称性与连续方程

通过量子力学的场论表述指出:薛定谔方程U(1)对称性的守恒流方程是连续方程,守恒荷是全空间几率,量子力学中的几率守恒是薛定谔方程的结果。

Bloch定理与散射平面波的干涉

从波动学的角度分析空晶格模型中的自由电子在周期性晶体势微扰作用下转变为晶体中Bloch电子的过程,揭示出了这一过程的物理实质.结果表明,在晶体势微扰作用下,代表空晶格模型中自由电子定态波函数的行进平面波在晶体内各点处产生了各种许可波矢的散射平面波,Bloch定理是晶体微观结构的平移对称性(即周期性)使得这些散射平面波产生干涉的结果.

NOETHER’S THEOREM AND ITS INVERSE OF NONHOLONOMIC NONCONSERVATIVE DYNAMICAL SYSTEMS

Noether’s theorem and its inverse of nonconservative systems subject to first-ordernonlinear nonholonomic constraints are obtained by introducing the generalized quasi-symmetry of the infinitesimal transformation of the group of transformations G_t for thegiven dynamical systems.

泊松括号与Noether定理

从泊松括号出发 ,利用分析力学知识导出Noether定理 ,进而讨论了能量。

基于广义分数阶算子Birkhoff系统Noether定理

本文就广义分数阶导数算子,提出两个新的“变换公式”,将其应用于分数阶Birkhoff系统,并导出分数阶Birkhoff系统的Noether定理.这个定理提供了一个计算分数阶Birkhoff系统运动常数的方法,克服了先前文献研究分数阶动力学系统守恒量的一些缺陷,并在文末给出两个推论.

一类带有对数非线性项的拟线性椭圆方程解的存在性及多重性

讨论一类带有对数非线性项的拟线性椭圆方程解的存在性和多重性问题.对主项系数A(x,t)提出合适的条件,利用山路引理证明该问题存在山路解,进一步利用对称山路定理证明该问题存在无穷多个非平凡解.

Noether＇s Theorem of Nonholonomic Systems in Optimal Control

In this paper, we extend Noether＇s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control problems with nonholonomic constraints, which are invariant under the parameter groups of infinitesimal transformations that change all （time, state, control） variables. Meanwhile, the Noether equalities corresponding to the conservation laws are given. Then, we obtain a new version of Noether＇s theorem to optimal control systems. An example is given to illustrate the application of these results.

一类二维p-Laplace算子方程解的对称性

为研究二维带p-Laplace算子的拟线性方程解的对称性,利用Liouville型定理,在u∈L~∞(R^2)且_1u> 0的假设下,证明u是一维对称解.该结果推广了一类二维拟线性方程解的对称性质.

The Noether's Theory of Birkhoffian Systems~*

By using the generalized quasi-symmetry of the infinitesimal transformation of the transformation group G_r, the Noether’s theory for Birkhoffian systems (including Noether’s theorem and its inverse) has been established. An example is given.

我国大宗商品出口需求价格弹性与最优出口关税——以稀土为例

出口税作为最常用的一种出口限制措施近来在各国贸易政策中屡见不鲜。现有文献对依据"勒纳对称定理"得到的最优关税结论存有异议。文章首先研究了大国作为唯一出口国情况下最优出口关税的决定,认为已有文献关于最优出口税结果不同的根源在于假设差异所致,然后,分析了非唯一出口国情况下大国最优出口关税的决定,最优出口关税和出口商品的需求价格弹性、本国出口所占世界市场份额和其他出口国的供给弹性有关,最后,以稀土为例,采用面板数据模型测算了我国稀土出口的需求价格弹性和需求收入弹性,并得到了我国稀土出口的最优关税。