Canonical quantization of nonlinear many body systems
中国大陆被EI检索的期刊
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Dadi Gouzao yu Chengkuangxue1001-1552 Plasma Science & Technology (Bristol, United Kingdom)1009-0630 Diqiu Kexue1000-2383 Diqiu Wuli Xuebao0001-5733 Diqiu Xuebao1006-3021 Dixue Qianyuan1005-2321 Dizhen Dizhi0253-4967 Earthquake Engineering and Engineering Vibration1671-3664 Dianbo Kexue Xuebao1005-0388 Diangong Jishu Xuebao1000-6753 Dianji yu Kongzhi Xuebao1007-449X Dianli Xitong Zidonghua1000-1026 Dianli Zidonghua Shebei1006-6047 Dianwang Jishu1000-3673 Dianzi Keji Daxue Xuebao1001-0548 Dianzi Xuebao0372-2112 Chinese Journal of Electronics1022-4653 Dianzi yu Xinxi Xuebao1009-5896 Dongbei Daxue Xuebao ,Ziran Kexueban1005-3026 Journal of Donghua University1672-5220 Journal of Southeast University1003-7985 Dongnan Daxue Xuebao ,Ziran Kexueban1001-0505 Faguang Xuebao1000-7032 Journal of Bionic Engineering1672-6529 Communications in Nonlinear Science and Numerical Simu1007-5704 Fenzi Cuihua1001-3555 Fenmo Yejin Cailiao Kexue yu Gongcheng1673-0224 Fuhe Cailiao Xuebao1000-3851 Fuza Xiting yu Fuzaxing Kexue1672-3813 Gaodeng Xuexiao Huaxue Xuebao0251-0790 Gaodianya Jishu1003-6520 Gaofenzi Cailiao Kexue yu Gongcheng1000-7555 High Technology Letters1006-6748 Gaoxiao Huaxue Gongcheng Xuebao1003-9015Gongcheng Lixue1000-4750 Gongcheng Rewuli Xuebao0253-231X Gongneng Cailiao1001-9731Guti Huojian Jishu1006-2793Acta Mechanica Solida Sinica0894-9166 Guangdianzi Jiguang1005-0086 Optoelectronics Letters1673-1905 Guangpuxue yu Guangpu Fenxi1000-0593 Guangxue Jingmi Gongcheng1004-924X Guangxue Xuebao0253-2239Photonic Sensors1674-9251Guangzi Xuebao1004-4213 Guisuanyan Xuebao0454-5648Guofang Keji Daxue Xuebao1001-2486 International Agricultural Engineering Journal0858-2114 International Journal of Agricultural and Biological Enginee1934-6344 International Journal of Automation and Computing1476-81861006-7043Harbin Gongcheng Daxue Xuebao或Ha-erh-pin Kung Cheng Ta Hsueh Hsueh Pao Harbin Gongye Daxue Xuebao0367-6234Journal of Harbin Institute of Technology (English Edition)1005-9113Hanneng Cailiao1006-9941Hanjie Xuebao0253-360X Hangkong Dongli Xuebao1000-8055Hangkong Xuebao1000-6893Hedongli Gongcheng0258-0926Hongwai yu Haomibo Xuebao1001-9014Hongwai yu Jiguang Gongcheng1007-2276Hunan Daxue Xuebao,Ziran Kexueban1674-2974Huanan Ligong Daxue Xuebao,Ziran Kexueban1000-565X Huazhong Keji Daxue Xuebao,Ziran Kexueban1671-4512Huagong Xuebao0438-1157Journal of Environmental Sciences (Beijing, China)1001-0742Huanjing Kexue Yanjiu1001-6929Jiqiren1002-0446Jixie Gongcheng Xuebao0577-6686Jilin Daxue Xuebao,Gongxueban1671-5497Jisuanji Fuzhu Sheji yu Tuxingxue Xuebao1003-9775Jisuanji Jicheng Zhizao Xitong1006-5911Journal of Computer Science and Technology1000-90002095-2228Frontiers of Computer Science（旧名Frontiers of Computer Science in China）Jisuanji Xuebao0254-4164Jisuanji Yanjiu yu Fazhan1000-1239Jianzhu Cailiao Xuebao1007-9629Jianzhu Jiegou Xuebao1000-6869Jiaotong Yunshu Gongcheng Xuebao1671-1637Jiaotong Yunshu Xitong Gongcheng yu Xinxi1009-6744Jinshu Xuebao0412-1961Acta Metallurgica Sinica(English Letters)1006-7191 Particuology1674-2001Kongzhi Lilun yu Yingyong1000-8152Journal of Control Theory and Applications1672-6340 Kongzhi yu Juece1001-0920 International Journal of Minerals, Metallurgy and Materials1674-4799 International Journal of Mining Science and Technology2095-2686Lixue Jinzhan1000-0992Lixue Xuebao0459-1879Acta Mechanica Sinica0567-7718 Linchan Huaxue yu Gongye0253-2417 Meitan Xuebao0253-9993 Mocaxue Xuebao1004-0595 Transactions Nanjing University Aeronautics and Astronau1005-1120 Neiranji Gongcheng1000-0925 Neiranji Xuebao1000-0909 Nongye Gongcheng Xuebao1002-6819 Nongye Jixie Xuebao1000-1298 Qiche Gongcheng1000-680X Qiangjiguang yu Lizishu1001-4322 Qiaoliang Jianshe1003-4722 Tsinghua Science and Technology1007-0214 Qinghua Daxue Xuebao,Ziran Kexueban1000-0054 Ranliao Huaxue Xuebao0253-2409 Journal of Thermal Science1003-2169 Rengong Jingti Xuebao1000-985X Ruanjian Xuebao1000-9825 Shanghai Jiaotong Daxue Xuebao1006-2467 Journal of Shanghai Jiaotong University(Special Issue)1007-1172 Shengxue Xuebao0371-0025 Shiyou Diqiu Wuli Kantan1000-7210 Shiyou Kantan yu Kaifa1000-0747 Shiyou Wutan1000-1441 Shiyou Xuebao0253-2697 Shiyou Xuebao,Shiyou Jiagong1001-8719 Shiyou yu Tianranqi Dizhi0253-9985 Journal of Hydrodynamics,Series B1001-6058 Shuikexue Jinzhan1001-6791 Water Science and Engineering1674-2370Shuili Xuebao0559-9350 Sichuan Daxue Xuebao,Gongcheng Kexueban1009-3087 Taiyangneng Xuebao0254-0096 Tianjin Daxue Xuebao0493-2137 Transactions of Tianjin University1006-4982 Tianranqi Diqiu Kexue1672-1926 Tianranqi Gongye1000-09761003-9953 Journal of Natural Gas Chemistry 是旧名,新名 Journal of Energy Chemistry Tiedao Gongcheng Xuebao1006-2106 Tiedao Xuebao1001-8360 Tongxin Xuebao1000-436X Tongji Daxue Xuebao,Ziran Kexueban0253-374X Tumu Gongcheng Xuebao1000-131XTuijin Jishu1001-4055Wuji Cailiao Xuebao1000-324X Wuhan Daxue Xuebao,Xinxi Kexueban1671-8860 Journal of Wuhan University of Technology,Materials Scien1000-2413 Wuli Xuebao1000-3290 Xi'an 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Zhejiang University-SCIENCE A Applied Physics 1673-565X1869-1951 Journal of Zhejiang University-SCIENCE C Computers & Ele Zhejiang Daxue Xuebao,Gongxueban1008-973X Zhenkong Kexue yu Jishu Xuebao1672-7126 Zhendong Ceshi yu Zhenduan1004-6801 Zhendong Gongcheng Xuebao1004-4523 Zhendong yu Chongji1000-3835 Zhipu Xuebao1004-29971756-378X International Journal of Intelligent Computing and Cyberne Chinese Journal of Geochemistry1000-9426 Zhongguo Dianji Gongcheng Xuebao0258-8013 Zhongguo Gonglu Xuebao1001-7372 Zhongguo Guanxing Jishu Xuebao1005-6734 Chinese Optics Letters1671-7694 China Ocean Engineering0890-5487 China Welding1004-5341 Chinese Journal of Aeronautics1000-9361 Chinese Journal of Chemical Engineering1004-9541 Zhongguo Huanjing Kexue1000-6923Chinese Journal of Mechanical Engineering1000-9345Zhongguo Jixie Gongcheng Xuekan0257-9731Zhongguo Jiguang0258-7025Science China(Earth Sciences)1674-7313Science China(Chemistry)1674-7291Science China(Technological Sciences)1674-7321Science China(Physics,Mechanics and Astronomy）1674-7348Science China(Information Sciences）1674-733XZhongguo Kuangye Daxue Xuebao1000-1964Zhongguo Liangyou Xuebao1003-0174Zhongguo Shiyou Daxue Xuebao,Ziran Kexueban1673-5005Zhongguo Shipin Xuebao1009-7848Zhongguo Tiedao Kexue1001-4632Zhongguo Tumu Shuili Gongcheng Xuekan1015-5856Chinese Physics B1674-1056Zhongguo Yancao Xuebao 1004-5708The Journal of China University of Posts Telecommum1005-8885Zhongguo Youse Jinshu Xuebao1004-0609Transactions of Nonferrous Metals Society of China 1003-6326Zhongguo Zaochuan1000-48822095-2899Journal of Central South University（Science & Technology of Mining and Metallurgy）Zhongnan Daxue Xuebao,Ziran Kexueban1672-7207Zidonghua Xuebao0254-4156外加工京继续收录川继续收录京继续收录京继续收录京继续收录京继续收录京继续收录京继续收录京继续收录辽继续收录京继续收录辽继续收录苏继续收录京继续收录苏外加工辽新增加*粤外加工皖继续收录鄂新增加*京新增加*京新增加*京新增加*京外加工黑继续收录豫继续收录京继续收录黑继续收录苏继续收录苏继续收录京继续收录川继续收录京继续收录京继续收录京继续收录辽继续收录沪继续收录苏继续收录苏继续收录吉外加工吉外加工京新增加*甘继续收录湘继续收录京新增加*鲁新增加*吉继续收录鄂继续收录川继续收录京继续收录京继续收录京继续收录渝继续收录陕外加工鄂继续收录津外加工津继续收录京继续收录吉继续收录沪外加工川继续收录陕继续收录京继续收录湘外加工京外加工京新外加工*京继续收录黑继续收录黑继续收录黑新增加*川继续收录黑继续收录京继续收录京继续收录川继续收录沪继续收录津继续收录湘继续收录粤继续收录鄂继续收录京外加工京新增加*京继续收录辽继续收录京继续收录吉继续收录京继续收录京外加工京继续收录京继续收录京继续收录京继续收录沪继续收录京继续收录陕新增加*京继续收录辽继续收录辽外加工京外加工粤继续收录辽继续收录京继续收录苏新增加*京继续收录京外加工京新增加*苏继续收录京继续收录甘继续收录苏继续收录沪继续收录津继续收录京继续收录京继续收录京继续收录川新增加*鄂继续收录京继续收录京继续收录晋继续收录京继续收录京继续收录京继续收录沪继续收录沪继续收录京继续收录冀继续收录京新增加*苏继续收录京继续收录京新增加*鄂继续收录沪继续收录苏继续收录苏继续收录京继续收录川继续收录京继续收录津继续收录津新增加*甘新增加*川外加工辽新增加*京继续收录京继续收录京继续收录沪继续收录京继续收录鄂继续收录鄂新增加*京继续收录陕继续收录陕继续收录陕继续收录川新增加*蒙继续收录京新增加*京继续收录京继续收录陕继续收录京继续收录京继续收录京外加工京外加工京新增加*粤新增加*川继续收录晋新增加*豫继续收录鄂继续收录苏继续收录鄂继续收录京继续收录京外加工沪继续收录京继续收录京新增加*陕继续收录浙外加工浙继续收录浙继续收录京继续收录苏继续收录苏继续收录沪新增加*京外加工吉外加工贵继续收录京继续收录陕继续收录津继续收录沪外加工苏继续收录黑外加工京继续收录京外加工台继续收录沪外加工京新增加*京外加工京新外加工*京新外加工*京继续收录苏新增加*京继续收录鲁新增加*京继续收录京外加工台外加工京新增加*京继续收录京继续收录湘继续收录湘新增加*京继续收录湘继续收录湘继续收录京。
非线性动力学入门-西安交通大学教师个人主页
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另一方面梁的轴向应变的表达式也会因变形大小的不同而采用不同的表达式比如小变形时应变而当考虑大变形时可能采用的应变表达式就是进而得到的梁的振动方程将会是一个含有高度非线性项的偏微分方程组
非线性动力学入门
张新华
西安交通大学 工程力学系 2011 年 07 月
前 言
─1687 年，牛顿(Isaac Newton, 1643 ～ 1727)发表了《自然哲学之数学原 理》(Mathematical Principles of Natural Philosophy)，标志着经典力学(亦即牛 顿力学)的正式诞生。牛顿力学主要研究自由质点系的宏观运动规律。 ─1788 年，拉格朗日(Joseph Louis Lagrange, 1736 ～ 1813)发表了分析力 学教程(Analytical Mechanics)，标志着拉格朗日力学的诞生。Lagrange 力学属 于分析力学的主要内容之一，在位形空间中研究带有约束的质点系动力学。 ─1833 年，哈密尔顿(William Rowen Hamilton, 1805 ～ 1865)对 Lagrange 力学进行了改造，引进了相空间(2n 维空间)，对系统内在的对称性(辛对称， Symplectic)进行了刻画。狭义上的哈密尔顿力学只适用于保守系统，而广义 的哈密尔顿力学在适用于非保守系统。哈密尔顿力学也属于分析力学的主要 组成部分。在此后发展起来的量子力学中 Hamilton 力学发挥着巨大的作用。 目前在天体力学、计算 Hamilton 力学，量子力学，甚至弹性力学(即所谓的 辛弹性力学)中哈密尔顿力学依然发挥着重要作用。 ─1927 年，Birkhoff(George David Birkhoff, 1844 ～ 1944)发表了“动力系 统”(Dynamical Systems)，标志着 Birkhoff 动力学的正式问世。Birkhoff 动力 学建立了研究非完整力学的框架。 ─1892 ～ 1899， 彭加莱(Henri Poincaré, 1854 ～ 1912)发表了三卷本的“天 体力学中的新方法”(New Methods of Celestial Mechanics)，系统性地提出了 研究动力学系统的定性方法，即几何方法。经典力学的目标之一就是设法求 得系统的解析解，而 Poincaré意识到对于大多数非线性系统而言，求其解析 解是不可能的，而必须发展新的研究方法。他超越了他的时代，极富远见地 预测到了非线性系统混沌现象(系统的解对初始条件具有极端敏感依赖性)的 存在。更为重要的是，Poincaré开创了研究非线性动力系统的几何方法，当之 无愧地被誉为非线性科学之父，其影响是划时代的。 ─1892 年，李亚普诺夫(Aleksandr Mikhailovich Lyapunov, 1857 ～ 1918)在 他的博士论文“运动稳定性的一般问题”(General problem of the stability of motion )中，系统地探讨了非线性动力学系统的稳定性问题。他提出了两种研 究稳定性的方法：李亚普诺夫第一方法(间接方法)和李亚普诺夫第二方法(直 接方法)。他从代数角度出发，对动力学系统的研究开创了一个崭新的领域。 彭加莱与李亚普诺夫，前者从几何角度，后者从代数角度，开拓了非线 性科学的研究疆域和研究手段。 ─1963 年，Lorenz(Edward Norton Lorenz, 1917 ～ 2008)发表了“确定性 非周期流”(Deterministic Nonperiodic Flow)的论文，认为大气系统的性态对 初值极为敏感，从而导致准确的长期天气预报是不可能的。该文标志着人类 首次借助于计算机发现了混沌(Chaos)现象的存在。 ─1757 年，欧拉(Leonhard Euler, 1707 ～ 1783)发表了压杆稳定性的论 文，首次探讨了力学系统的分岔现象。作为分岔理论重要分支的突变理 论(Catastrophe Theory)则主要由法国数学家托姆(René Thom, 1923 ～ 2002)于 上个世纪 60 年代创立，由齐曼(Christopher Zeeman，1925 ～)在 70 年代大力 推广普及。 ─1834 年，英国的罗素(John Scott Russell, 1808 ～ 1882)骑着马在 Union 运河上散步时，发现了现在称之为孤立波(又称作孤波，Solitary wave)的 i
lindblad方程
lindblad方程Lindblad方程是量子力学中一个重要的方程，用于描述开放量子系统的演化。
它由瑞典物理学家Göran Lindblad在20世纪70年代提出，被广泛应用于量子信息、量子计算和量子测量等领域。
在量子力学中，系统的演化通常由薛定谔方程描述，但当系统与外部环境发生相互作用时，就需要考虑开放量子系统的演化。
这时，系统将不再是一个封闭的系统，而是受到外部环境的影响，可能发生退相干、能量耗散等现象。
Lindblad方程的作用就是描述这种开放量子系统的演化过程。
Lindblad方程的形式比较复杂，涉及到了密度矩阵、算符等概念，但其核心思想是描述系统的演化不再是幺正的，而是通过一系列耗散算符来描述系统与环境的相互作用。
这种非幺正演化导致了系统的密度矩阵的演化不再是线性的，而是呈现出耗散和退相干的特征。
通过Lindblad方程，我们可以更好地理解开放量子系统的演化行为，揭示系统与环境之间的相互作用对系统的影响。
在量子信息领域，Lindblad方程被广泛用于描述量子比特的退相干过程、量子纠缠的演化等现象。
通过研究Lindblad方程，我们可以设计更有效的量子纠缠保持方案，提高量子计算的性能和稳定性。
除了在量子信息领域，Lindblad方程还在量子光学、原子物理等领域有着重要的应用。
通过研究Lindblad方程，我们可以更深入地理解量子系统的演化规律，为量子技术的发展提供理论支持和指导。
总的来说，Lindblad方程作为描述开放量子系统演化的重要方程，在量子力学和量子信息领域有着广泛的应用前景。
通过深入研究Lindblad方程，我们可以揭示量子系统的奇妙行为，推动量子技术的发展，为人类带来更多的科学进步和技术创新。
Computers & Electrical Engineering样文
Adaptive control of dynamic mobile robots withnonholonomic constraintsFarzad Pourboghrat *,Mattias P.KarlssonDepartment of Electrical and Computer Engineering,Southern Illinois University,Carbondale,IL 62901-6603,USAReceived 16November 1999;accepted 22August 2000AbstractThis paper presents adaptive control rules,at the dynamics level,for the nonholonomic mobile robots with unknown dynamic parameters.Adaptive controls are derived for mobile robots,using backstepping technique,for tracking of a reference trajectory and stabilization to a fixed posture.For the tracking problem,the controller guarantees the asymptotic convergence of the tracking error to zero.For stabili-zation,the problem is converted to an equivalent tracking problem,using a time varying error feedback,before the tracking control is applied.The designed controller ensures the asymptotic zeroing of the sta-bilization error.The proposed control laws include a velocity/acceleration limiter that prevents the robot Õs wheels from slipping.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Mobile robot;Nonholonomic constraint;Dynamics level motion control;Stabilization and tracking;Adaptive control;Backstepping technique;Asymptotic stability1.IntroductionMotion control of mobile robots has found considerable attention over the past few years.Most of these reports have focused on the steering or trajectory generation problem at the ki-nematics level i.e.,considering the system velocities as control inputs and ignoring the mechanical system dynamics [1–3].Very few reports have been published on control design in the presence of uncertainties in the dynamic model [4].Some preliminary results on control of nonholonomic systems with uncertainties are given in Refs.[4–6].Two of the most important control problems concerning mobile robots are tracking of a refer-ence trajectory and stabilization to a fixed posture.The tracking problem has received solutions *Corresponding author.Tel.:+1-618-453-7026.E-mail address:pour@ (F.Pourboghrat).0045-7906/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S0045-7906(00)00053-7242 F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253including classical nonlinear control techniques[1,2,7].The basic idea is to have a reference car that generates a trajectory for the mobile robot to follow.In Refs.[1,2],nonlinear velocity control inputs were defined that made the tracking error go to zero as long as the reference car was moving.In Ref.[7],they used input–output linearization to make a mobile platform follow a desired trajec-tory.The problem of stabilization about afixed posture has been shown to be rather complicated. This is due to violating the BrockettÕs condition[8],which states that for nonholonomic systems a single equilibrium solution cannot be asymptotically stabilized using continuous static state feedback[9,10].The BrockettÕs condition essentially states that for nonholonomic systems an equilibrium solution can be asymptotically stabilized only by either a time varying,a discontin-uous,or a dynamic state feedback.In addressing the above problem,in Ref.[10]a smooth feedback control was presented for the kinematics control problem resulting in a globally marginally stable closed loop system.They also designed a smooth feedback control for a dynamical state-space model resulting in a Lagrange stable closed loop system,as defined in their paper.A two dimensional Lyapunov function was utilized in Ref.[3]to prescribe a set of desired trajectories to navigate a mobile robot to a specified configuration.The desired trajectory was then tracked using sliding mode control,resulting in discontinuous control signals.The mobile robot was shown to be exponentially stable for a class of quadratic Lyapunov functions.In Ref.[9],they formulated a reduced order state equation for a class of nonholonomic systems.Several other researchers have later used this reduced order state equation in their studies.In Ref.[4],the problem of controlling nonholonomic mechanical sys-tems with uncertainties,at the dynamics level,was ing the reduced state equation in Ref.[9],they proposed an adaptive controller for a number of important nonholonomic control problems,including stabilization of general systems to an equilibrium manifold and stabilization of differentiallyflat and Caplygin systems to an equilibrium point.In Ref.[2],they gave several examples on how the stabilization problem can be solved for a mobile robot at the kinematics level.Their solutions included time-varying control,piecewise continuous control,and time-varying piecewise continuous control.They also showed how a solution to the tracking problem could be extended to work even for the stabilization problem.Here,we present adaptive control schemes for the tracking problem and for the problem of stabilization to afixed posture when the dynamic model of the mobile robot contains unknown parameters.Our work is based on,and can be seen as an extension of,the work presented in Refs. [1,2].Using backstepping technique we derive adaptive control laws that work even when the model of the dynamical system contains uncertainties in the form of unknown constants.The assumption for the uncertainty in robotÕs parameters,particularly the mass,and hence the inertia, can be justified in real applications such as in automotive manufacturing industry and warehouses, where the robots are to move a variety of parts with different shapes and masses.In these cases,the robotÕs mass and inertia may vary up to10%or20%,justifying an adaptive control approach.2.Dynamic model of mobile robotHere,we consider a three-wheeled mobile robot moving on a horizontal plane(Fig.1).The mobile robot features two differentially driven rear wheels and a castor front wheel.The radius ofthe wheels is denoted r and the length of the rear wheel axis is 2l .Inputs to the system are two torques T 1and T 2,provided by two motors attached to the rear wheels.The dynamic model for the above wheeled-mobile robot is given by Refs.[10,11].€x ¼k m sin /þb 1u 1cos /€y ¼Àk cos /þb 1u 1sin /€/¼b 2u 28><>:ð1Þ_x sin /À_y cos /¼0ð2Þwhere b 1¼1=ðrm Þ,b 2¼l =ðrI Þ,and that m and I denote the mass and the moment of inertia of the mobile robot,respectively.Also,u 1¼T 1þT 2and u 2¼T 1ÀT 2are the control inputs,and k is the Lagrange multiplier,given by k ¼Àm _/_x cos /þ_y sin /ðÞ.Here,it is assumed that b 1and b 2are unknown constants with known signs.The assumption that the signs of b 1and b 2are known is practical since b 1and b 2represent combinations of the robot Õs mass,moment of inertia,wheel radius,and distance between the rear wheels.Eq.(2)is the nonholonomic constraint,coming from the assumption that the wheels do not slip.The triplet vector function q t ðÞ¼x t ðÞ;y t ðÞ;/t ðÞ½ T denotes the trajectory (position and orientation)of the robot with respect to a fixed workspace frame.That is,at any given time,q ¼½x ;y ;/ T describes the robot Õs configuration (posture)at that time.We assume that,at any time,the robot Õs posture,q ¼½x ;y ;/ T ,as well as its derivative,_q¼½_x ;_y ;_/ T ,are available for feedback.3.Tracking problem definitionThe tracking problem consists of making the trajectory q of the mobile robot follow a reference trajectory q r .The reference trajectory q r t ðÞ¼x r t ðÞ;y r t ðÞ;/r t ðÞ½ T is generated by a reference ve-hicle/robot whose equations are_xr ¼v r cos /r _y r¼v r sin /_/r ¼x r8<:ð3ÞThe subscript ‘‘r’’stands for reference,and v r and x r are the reference translational (linear)velocity and the reference rotational (angular)velocity,respectively.We assume that v r and x r ,as well as their derivatives are available and that they all are bounded.Assumption A 1.For the tracking problem it is assumed that the reference velocities v r and x r do not both go to zero simultaneously.That is,it is assumed that at any time either lim t !1v r t ðÞ90and/or lim t !1x r t ðÞ90.The tracking problem,under the Assumption A 1,is to find a feedback control law u 1u 2 ¼u q ;_q ;q r ;v r ;x r ;_v r ;_x r ðÞsuch that lim t !1~q t ðÞ¼0,where ~q t ðÞ¼q r t ðÞÀq t ðÞis defined as the trajectory tracking error.As in Ref.[1],we define the equivalent trajectory tracking error ase ¼T ~qð4Þwhere e ¼½e 1;e 2;e 3 T ,and T ¼cos /sin /0Àsin /cos /00010@1A .Note that since T matrix is nonsingular,e is nonzero as long as ~q¼0.Assuming that the angles /r and /are given in the range ½Àp ;p ,we have the equivalent trajectory tracking error e ¼0only if q ¼q r .The purpose of the tracking controller is to force the equivalent trajectory tracking error e to 0.In the sequel we refer to e as the trajectory tracking error.Using the nonholonomic constraint (2),the derivative of the trajectory tracking error given in Eq.(4)can be written as,[1],_e1¼e 2x Àv þv r cos e 3_e 2¼Àe 1x þv r sin e 3_e3¼x r Àx 8<:ð5Þwhere v and x are the translational and rotational velocities of the mobile robot,respectively,and are expressed asv ¼_xcos /þ_y sin /x ¼_/ð6Þ4.Tracking controller designHere,the goal is to design a controller to force the tracking error e ¼½e 1;e 2;e 3 T to ing backstepping technique,since the actual control variables u 1and u 2do not appear in Eq.(5),we consider variables v and x as virtual controls.Let v d and x d denote the desired virtual controls for the mobile robot.That is,with v d and x d the trajectory tracking error e converges tozero asymptotically.Also let us define ~vand ~x as virtual control errors.Then,v and x can be written asv ¼v d þ~vx ¼x d þ~x ð7Þ244 F.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253Let us choose the virtual controls v d and x d ,asv d v r ;x r ;e 1;e 3ðÞ¼v r cos e 3þk 1v r ;x r ðÞe 1x d v r ;x r ;e 2;e 3ðÞ¼x r þk 2v r e 2þk 3v r ;x r ðÞsin e 3ð8Þwhere k 2is a positive constant and k 1ðÁÞand k 3ðÁÞare bounded continuous functions with bounded first derivatives,strictly positive on R ÂR -ð0;0Þ.Observe that our approach from here on is general for any v d and x d (with well defined first derivatives),i.e.any differentiable control law that makes the kinematics model of the mobile robot track a desired trajectory can be used instead of Eq.(8).Eq.(8)is similar to the control law proposed by Ref.[1],but with the advantage,as we are going to prove later,that it can be used to track any reference trajectory as long as As-sumption A 1holds.Now,consider the following adaptive control scheme:u 1¼^b 1ðÀc 1~v þe 1þ_v d Þu 2¼^b 2 Àc 2~x þ1k 2sin e 3þ_x d _^b 1¼Àc 1sign b 1ðÞ~v ðÀc 1~v þe 1þ_v d Þ_^b 2¼Àc 2sign b 2ðÞ~x Àc 2~x þ1k 2sin e 3þ_x d ð9Þwhere c 1,c 2,c 1,and c 2are positive constants and ^b 1is an estimate of b 1¼1=b 1and ^b 2is an estimate of b 2¼1=b 2.Result 1.If Assumption A 1holds ,then the adaptive control scheme (9)makes the origin e ¼0uniformly asymptotically stable.Proof .Consider the following Lyapunov function candidateV 1¼12e 21Àþe 22Áþ1k 21ðÀcos e 3Þð10Þwhere k 2is a positive constant.Clearly V 1is positive definite and V 1¼0only if e ¼0.Taking the time derivative of V 1,we obtain_V 1¼e 1ðÀv þv r cos e 3Þþe 2v r sin e 3þ1k 2sin e 3x r ðÀx Þð11ÞFurthermore,using Eqs.(7)and (8),we have_V 1¼Àk 1e 21Àk 3k 2sin 2e 3À~v e 1À~x 1k 2sin e 3ð12ÞIn view of Eqs.(1),(2)and (6),we find the time derivatives of ~vand ~x ,as _~v¼_v À_v d ¼€x cos /À_x sin /_/þ€y sin /þ_y cos /_/À_v d ¼b 1u 1À_v d _~x ¼_x À_x d ¼€/À_x d ¼b 2u 2À_x d ð13ÞF.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253245Consider the Lyapunov function candidateV2¼V1þ12ð~v2þ~x2Þþb1j j2c1~b21þb2j j2c2~b22ð14Þwhere~b1¼b1À^b1¼1=b1À^b1and~b2¼b2À^b2¼1=b2À^b2.Considering Eq.(9)we get:_V 2¼Àk1e21Àk3k2sin2e3Àc1~v2Àc2~x260ð15ÞSince V2is bounded from below and_V2is negative semi-definite,V2converges to afinite limit. Also,V2,as well as,e1,e2,e3,~v,~x,^b1,and^b2are all bounded.Furthermore,using Eqs.(5),(7)–(9)and(13),the second derivative of V2can be written as€V 2¼À2k1e1e2ðx rþk2v r e2þk3sin e3þ~xÞþ2k1e1ðk1e1þ~vÞÀ_k1e21þ2k3k2cos e3sin e3ðk2v r e2þk3sin e3þ~xÞÀ_k3k2sin2e3À2c1~vðb1^b1ðÀc1~vþe1þ_v dÞÀ_v dÞÀ2c2~x b2^b2Àc2~xþ1k2sin e3þ_x dÀ_x dð16Þwhich from the properties of k1,k2,and k3,the assumption that v r and x r and their derivatives are bounded,and from the above results,can be shown to be bounded,i.e.,_V2is uniformly contin-uous.Since V2ðtÞis differentiable and converges to some constant value and that€V2is bounded,by BarbalatÕs lemma,_V2tðÞ!0as t!1.This in turn implies that e1,e3,~v,and~x converge to zero [12,13].To show that e2also goes to zero,note that,using the above results,thefirst error equation can be written as_e1¼e2x rÀk1e1ð17ÞThe second derivative of e1is€e1¼_x r e2þx rðÀe1xþv r sin e3ÞÀk1e2x rðÀk1e1ÞÀ_k1e1ð18Þwhich can be shown to be bounded by once again using the properties of k1,the assumptions on v r and x r,and Eqs.(7)and(8).Since e1is differentiable and converges to zero and€e1is bounded,by BarbalatÕs lemma,_e1,and hence,e2x r tend to zero.Proceeding in the same manner,the third error equation can be written as_e3¼Àk2v r e2Àk3sin e3ð19Þand its second derivative can be shown to be bounded.Since e3is differentiable and converges to zero and€e3is bounded,again by BarbalatÕs lemma,_e3!0as t!1.Hence,k2v r e2and thus v r e2 tend to zero as t!1.Clearly,both v r e2and x r e2converge to zero.However,since v r and x r do not both tend to zero(by Assumption A1),e2must converge to zero.That is,e1,e2,e3,~v,and~x must all converge to zero.hIn Section3,we demonstrated that the system is stable if k2is a positive constant,and that k1ðÁÞand k3ðÁÞare bounded continuous functions with boundedfirst derivatives and are strictly positive on RÂR-ð0;0Þ.To get a better understanding on how the control gains affect the response of the system,we write the equations for the closed loop system when~v and~x are equal to zero as[1] 246 F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253_e¼Àk 1e 1þx r þk 2v r e 2þk 3sin e 3ðÞe 2Àx r þk 2v r e 2þk 3sin e 3ðÞe 1þv r sin e 3Àk 2v r e 2Àk 3sin e 30@1A ð20ÞBy linearizing the differential equation (20)around e ¼0,we get_e¼Ae ð21Þwhere A ¼Àk 1x r 0Àx r 0v r 0Àk 2v r Àk 30@1A ð22ÞTo simplify the analysis,we assume that v r and x r are constants.The system Õs closed loop poles are now equal to the roots of the following characteristic polynomial equation:s ðþ2nx 0Þs 2Àþ2nx 0s þx 20Áð23Þwhere n and x 0are positive real numbers.The corresponding control gains arek 1¼2nx 0k 2¼x 20Àx 2r v 2r k 3¼2nx 0ð24ÞWith a fixed pole placement strategy (n and x 0are constant),the control gain k 2increaseswithout bound when v r tends to zero.One way to avoid this is by letting the closed loop polesdepend on the values of v r and x r .As in Ref.[2],we choose x 0¼x 2r þbv 2r ÀÁð1=2Þwith b >0.The control gains then becomek 1¼2n x 2r Àþbv 2rÁ1=2k 2¼b k 3¼2n x 2r Àþbv 2r Á1=2ð25Þand the resulting control is now defined for any values of v r and x r .In the above,it is shown that the proposed algorithm works for any desired velocities,ðv d ;x d Þ.However,in practice,if the tracking errors initially are large or if the reference trajectory does not have a continuous curvature (e.g.,if the reference trajectory is a straight line connected to a circle segment),either or both of the virtual reference velocities in Eq.(8)might become too large for a real robot to attain in practice.Hence,the translational/rotational acceleration might become too large causing the robot to slip [1].In order to prevent the mobile robot from slipping,in a real application,a simple velocity/acceleration limiter may be implemented [1],as shown in Fig.2.This limits the virtual reference velocities ðv d ;x d Þby constants ðv max ;x max Þand the virtual referenceaccelerations ða ;a Þby constants ða max ;a max Þ,where a ¼_vd and a ¼_x d are the virtual reference accelerations.In practice,these parameters must be determined experimentally as the largest values with which the mobile robot never slips.F.Pourboghrat,M.P.Karlsson /Computers and Electrical Engineering 28(2002)241–253247An important advantage of adding the limiter is that it lowers the control gains indirectly only when the tracking errors are large,i.e.,when too high a gain could cause the robot to slip,while for small tracking errors it does not affect the performance at all.Thus,by using the limiter one can have higher control gains for small tracking errors to allow for better tracking,while letting the limiter to‘‘scale down’’the gains,indirectly,for large tracking errors,to prevent the robot from slipping.5.Simulation results for tracking control problemHere,the results of computer simulation,using MATLAB/SIMULINK,are presented for a mobile robot with the proposed tracking control and with the velocity/acceleration limiter.The computer simulations for the above controller without the limiter,although not shown here,produce similar results,but with somewhat different transient characteristics.All simulations have the common parameters of c1¼c2¼100,c1¼c2¼10and b¼250.Also selected are,the damping factor n¼1,v max¼1:5m/s,x max¼3rad/s,a max¼5m/s2and a max¼25rad/s2.Moreover,the robotÕs dy-namic parameters are chosen as b1¼b2¼0:5,which are assumed to be unknown to the con-troller,but with known signs.Simulation results for the case where the reference trajectory is a straight line are shown in Figs. 3and4for t2½0;10 .The reference trajectory is given by x rðtÞ¼0:5t,y rðtÞ¼0:5t and/rðtÞ¼p=4, defining a straight line,starting from q rð0Þ¼½x rð0Þ;y rð0Þ;/rð0Þ T¼½0;0;p=4 T.The mobile robot, however,is initially at qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½1;0;0 T,where/¼0indicates that the robot is heading toward positive direction of x.As it can be seen from thesefigures,first the robot backs up and then heads toward the virtual reference robot moving on the straight line.Figs.5and6show the simulation results for tracking a circular trajectory.The reference trajectory is a point moving counter clockwise on a circle of radius1,starting at q rð0Þ¼½x rð0Þ;y rð0Þ;/rð0Þ T¼½1;0;p=2 T.The reference velocity is kept constant at v rðtÞ¼0:5m/s.The initial conditions for the mobile robot,however,is taken as qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½0;0;0 T.Again,as it is seen from thesefigures,the robot immediately heads toward the reference robot,which is moving on the circle.It then reaches it quickly and continues to track it.6.Stabilization problem definitionThe stabilization problem,given an arbitrary desired posture q d ,is to find a feedback control law,u 1u 2¼u q Àq d ;_q ;t ðÞ,such that lim t !1q t ðÞÀq d ðÞ¼0,for any arbitrary initial robot posture q ð0Þ.Without loss of generality,we may take q d ¼½0;0;0 T.6.1.Stabilization controller designRecall that there is no continuous static state feedback that can asymptotically stabilize a nonholonomic system about a fixed posture [8–10].The approach to the problem taken here is the dynamic extension of that in Ref.[2]where a kinematics model of the mobile robot is used.In-stead of designing a new controller for the stabilization problem the same controller as for the tracking problem is used.The idea is to let the reference vehicle move along a path that passes through the point ðx d ;y d Þwith heading angle /d .The stabilization to a fixed posture problem isnow equivalent to,and can be treated as,a tracking problem(convergence of the tracking errors to zero)with the additional requirement that the reference vehicle should itself be asymptotically stabilized about the desired posture.As in Ref.[2],we let the reference vehicle move along the x-axis,i.e.y rðtÞ¼0and/rðtÞ¼0,for all values on t.The design method is the same as derived for the tracking case.However,in this casev r¼_x r¼Àk4x rþgðe;tÞ;ð26Þwithgðe;tÞ¼k e k sin tð27Þwhere k4>0.Different time-varying functions gðe;tÞhave also been suggested in the literature,see Refs.[2,11]and the references therein.Since,from the Section5,the tracking errors e1,e2,and e3are bounded,the time-varying function gðe;tÞis bounded.Therefore v r and the state x r also remain bounded.By taking the time derivative of Eq.(26),it can be shown in the same way that_v r is bounded.Since v r and_v r are bounded,the assumptions made in Section3concerning the reference velocity are fulfilled.If v r is not equal to zero,then e must converge to zero.When e tends to zero,gðe;tÞalso tends to zero. Therefore,the robotÕs position x must track x r,which converges to zero and hence lead the mobile robot to the desired posture.6.2.Simulation results for stabilization control problemHere,the simulation results for the stabilization problem are shown in Figs.7and8.The control parameters and system parameters are the same as for the simulations shown for the tracking problem and k4¼1.The mobile robot is initially at qð0Þ¼½xð0Þ;yð0Þ;/ð0Þ T¼½0;1;0 T.As it is seen from thefigures,the stabilization about thefinal posture at the origin is achieved quite satisfactorily.Note,in this case,that the robot actually turns around and backs up into the final posture.7.ConclusionsTwo important control problems concerning mobile robots with unknown dynamic parameters have been considered,namely,tracking of a reference trajectory and stabilization to afixed posture.An adaptive control law has been proposed for the tracking problem and has been ex-tended for the stabilization problem.A simple velocity/acceleration limiter was added to the controller,for practical applications,to avoid any slippage of the robotÕs wheels,and to improve the tracking performance.Several simulation results have been included to demonstrate the performance of the proposed adaptive control law.References[1]Kanayama Y,Kimura Y,Miyazaki F,Noguchi T.A stable tracking control method for an autonomous mobilerobot.vol.1.Proceedings of IEEE International Conference on Robotics and Automation,Cincinnati,Ohio,1990, p.384–9.[2]Canudas de Wit C,Khennouf H,Samson C,Sordalen OJ.Nonlinear control design for mobile robots.In:ZhengYF,editor.Recent trends in Mobile robots,World Scientific,1993.p.121–56.[3]Guldner J,Utkin VI.Stabilization of nonholonomic mobile robot using Lyapunov functions for navigation andsliding mode control.Control-Theory Adv Technol1994;10(4):635–47.[4]Colbaugh R,Barany E,Glass K.Adaptive Control of Nonholonomic Mechanical Systems.Proceedings of35thConference on Decision and Control,Kobe,Japan,1996.p.1428–34.[5]Fierro R,Lewis FL.Control of nonholonomic mobile robot:backstepping kinematics into dynamics.J Robot Sys1997;14(3)149-163.[6]Jiang ZP,Pomet bining backstepping and time-varying techniques for a new set of adaptive controllers.Proceedings of33rd IEEE Conf on Decision and Control,Lake Buena Vista,FL,1994.p.2207–12.[7]Sarkar N,Yun X,Kumar V.Control of mechanical systems with rolling constraints:application to dynamiccontrol of mobile robots.Int J Robot Res1994;13(1):55–69.[8]Brockett RW.Asymptotic stability and feedback stabilization.In:Brockett RW,Millman RS,Sussmann HJ,editors.Differential Geometric Control Theory,Boston,MA:Birkhauser;1983.p.181–91.[9]Bloch AM,Reyhanoglu MR,McClamroch NH.Control and stabilization of nonholonomic dynamic systems.IEEE Trans Automat Contr1992;37(11):1746–56.[10]Campion G,d’Andrea-Novel B,Bastin G.Controllability and state feedback stabilization of nonholonomicmechanical systems.Canudas de Wit C,editor.Advanced Robot Control,Berlin:Springer;1991.p.106–24. [11]Kolmanovsky I,McClamroch NH.Developments in nonholonomic control problems.IEEE Contr Sys Magaz1995;15(6):20–36.[12]Krstic M,Kanellakopoulos I,Kokotovic P.Nonlinear and Adaptive Control Design,New York:Wiley;1995.[13]Karlsson MP.Control of nonholonomic systems with applications to mobile robots.Master Thesis,SouthernIllinois University,Carbondale,IL62901,USA,1997.Farzad Pourboghrat received his Ph.D.degree in Electrical Engineering from the University of Iowa in1984. He is now with the Department of Electrical and Computer Engineering at Southern Illinois University at Carbondale(SIU-C)where he is an Associate Professor.His research interests are in adaptive and slidingcontrol with applications to DSP embedded systems,mechatronics,flexible structures andMEMS.Mattias Karlsson received the B.S.E.E.degree from the University of Bor a s,Sweden and the M.S.E.E.degree from Southern Illinois University,Carbondale,IL,in1995and1997,respectively.He has been employed at Orian Technology since1997.He is currently an on-site consultant at Caterpillar Inc.Õs Technical Center, Mossville,IL.His current interests include control algorithm development for mechanical and electrical systems and software development for embedded systems.F.Pourboghrat,M.P.Karlsson/Computers and Electrical Engineering28(2002)241–253253。
3 章 流体输运方程 - 中南大学能源与工程学院
大 系统的特点是： （1）系统将随系统内质点一起运动，系统内的质点始终包含在系统内，系统边界的形
南 状和所围空间的大小，则可随运动而变化； （2）系统与外界无质量的交换，但可以有力的相互作用以及能量（热和功）交换。
中如果研究对象是系统，由于力学中的一些基本定律是建立在质点、质点系上的，因此，
流体力学这些力学定律可直接用原始数学形式表达出来。
与 场该项为零）；另一项 u ⋅ ∇φ 称为位变导数项（Carrier derivative），它表示在非均匀的φ 场
中（有梯度 ∇φ ），由空间位置变化（由 u ）引起的物理量的变化率。
学 为了说明这一点，我们取时间间隔（ t ，t + Δt ），位于（ x, y, z ）处的流体质点，将沿
u( x, y, z, t) 的方向移动距离近似为 | u | Δt ，由于在 u( x, y, z, t) 方向单位长度上的φ 的改变
2
自同一流体质点，而非取自同一空间点（ x, y, z ））。由于该流体质点是运动的，即 x, y, z 是
随时间变化的。若以 a, b, c 表示该点的拉格朗日坐标，则 x, y, z 将依式（3-2）变化。从而
φ = φ (x, y, z, t) 的变化应按复合函数求导法则处理。因此，物理量φ = φ (x, y, z, t) 的随体导
科 量 为 (u / | u |) ⋅ ∇φ ， 于 是 在 此 方 向 上 ， 在 距 离 | u | Δt 上 的 φ 的 改 变 量 为
| u | Δt((u / | u |) ⋅ ∇φ) = u ⋅ ∇φΔt ，因此在单位时间内φ 的变化率为 u ⋅ ∇φ 。
源 显然，若（1） u = 0 ，即流体静止；（2）φ 是均匀场，这时 ∇φ = 0 （但φ 可随时间变
JCR分区(2011年版)
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ANAL1040-7294应用数学Journal of Dynamics and Differential EquJ DYN DIFFER EQU1435-9855应用数学J EUR MATH SOCJOURNAL OF THE EUROPEAN MATHEMATICAL SOCJ FOURIER ANAL A1069-5869应用数学JOURNAL OF FOURIER ANALYSIS AND APPLICATJ GLOBAL OPTIMJOURNAL OF GLOBAL OPTIMIZATION0925-5001应用数学JOURNAL OF MATHEMATICAL ANALYSIS AND APP0022-247X应用数学J MATH ANAL APPLJ MOD DYNAM Journal of Modern Dynamics1930-5311应用数学Journal of Noncommutative Geometry1661-6952应用数学J NONCOMMUT GEOMJ NONLINEAR SCI J OURNAL OF NONLINEAR SCIENCE0938-8974应用数学J SCI COMPUT JOURNAL OF SCIENTIFIC COMPUTING0885-7474应用数学KINET RELAT MOD K inetic and Related Models1937-5093应用数学。
Squeezed light
Squeezed lightA. I. LvovskyInstitute for Quantum Information Science, University of Calgary, Calgary, Canada, T2N 1N4 and Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow region, 143025, Russia∗ (Dated: January 17, 2014) The squeezed state of the electromagnetic field can be generated in many nonlinear optical processes and finds a wide range of applications in quantum information processing and quantum metrology. This article reviews the basic properties of single-and dual-mode squeezed light states, methods of their preparation and detection, as well as their quantum technology applications.I.WHAT IS SQUEEZED LIGHT? A. Single-mode squeezed lightIn squeezed states of light, the noise of the electric field at certain phases falls below that of the vacuum state. This means that, when we turn on the squeezed light, we see less noise than no light at all. This apparently paradoxical feature is a direct consequence of quantum nature of light and cannot be explained within the classical framework. The basic idea of squeezing can be understood by considering the quantum harmonic oscillator, familiar from undergraduate quantum mechanics. Its vacuum state wavefunction in the dimensionless position basis is given by1 1 −X 2 /2 e , π 1/4 which in the momentum basis corresponds to ψ0 (X ) = ˜0 (P ) = √1 ψ 2π+∞(1)e−iP X ψ0 (X )dX =−∞1 π 1 /4e −P2/2(2) (so the vacuum state wavefunction is the same in the position and momentum bases). The variance of the position and momentum observables in the vacuum state equals 0| ∆X 2 |0 = 0| ∆P 2 |0 = 1/2. The wavefunction of the squeezed-vacuum state |sqR with the squeezing parameter R > 0 is obtained from that of the vacuum state by means of scaling transformation: √ 2 R ψR (X ) = 1/4 e−(RX ) /2 , (3) π and 2 1 ˜R (P ) = √ e−(P/R) /2 ψ (4) 1 / 4 π R in the position and momentum bases, respectively. In this state, the variances of the two canonical observables are ∆X 2 = 1/(2R2 ) and ∆P 2 = R2 /2. (5)∗ 1lvov@ucalgary.ca ˆ P ˆ ] = i for the quadrature observables. We use convention [X,If R > 1, the position variance is below that of the vacuum state, so |sqR is position-squeezed ; for R < 1 the state is momentum-squeezed. In other words, if we prepare multiple copies of |sqR , and perform a measurement of the squeezed observable on each copy, our measurement results will exhibit less variance than if we performed the same set of measurements on multiple copies of the vacuum state. More generally, we say that a state of a single harmonic oscillator exhibits (quadrature) squeezing if the variance of the position, momentum, or any other quadraˆθ = X ˆ cos θ + P ˆ sin θ (where θ is a real number ture X known as quadrature angle ) in that state exhibits variance below 1/2. In accordance with the uncertainty principle, both position and momentum observables, or any two quadratures associated with orthogonal angles, cannot be squeezed at the same time. For example, in state (1) the product ∆X 2 ∆P 2 = 1/4 is the same as that for the vacuum state. Squeezing is best visualized by means of the Wigner function — the quantum analogue of the phase-space probability density. Figure 1(c,d) display the Wigner functions of the position- and momentum-squeezed vacuum states, respectively. The squeezing feature becomes apparent when these Wigner functions are compared with that of the vacuum state [Fig. 1(a)]. Figure 1(e,f) shows squeezed coherent states, which are analogous to the squeezed vacuum except that their Wigner function is displaced from the phase space origin akin to the coherent state [Fig. 1(b)]. The state shown in Fig. 1(f) is particularly interesting because it exhibits, as a consequence of momentum squeezing, phase squeezing — reduction of the uncertainty in the phase with respect to a coherent state of the same amplitude. Because the Schr¨ odinger evolution under the standard harmonic oscillator Hamiltonian corresponds to clockwise rotation of the phase space around the origin point, the phase squeezing property is preserved under this evolution. In the same context, the state in Fig. 1(e) is sometimes called amplitude squeezed. According to the quantum theory of light, the Hilbert space associated with a mode of the electromagnetic field is isomorphic to that of the mechanical harmonic oscillator. The role of the position and momentum observables in this context is played by the electric field magnitudes measured at specific phases. For example, the fieldarXiv:1401.4118v1 [quant-ph] 15 Jan 20142 at phase zero (with respect to a certain reference) corresponds to the position observable, that at phase π/2 to the momentum observable, and so on. Accordingly, phase-sensitive measurements of the field in an electromagnetic wave are affected by quantum uncertainties. For the coherent and vacuum states, this uncertainty is ω/2ε0 V (the standard phase-independent and equals quantum limit, or SQL), where ω is the optical frequency and V is the quantization volume [1]. But squeezed optical states exhibit uncertainties below SQL at certain phases. Dependent on whether the mean coherent amplitude of the state is zero, squeezed optical states are classified into squeezed vacuum and (bright) squeezed light. Squeezed coherent states form a subset of bright squeezed light states. zero while its variance equals ∆X 2 = ψ | (ˆ a+a ˆ † )2 1 |ψ = − s, 2 2 (7)so for state |ψ is position squeezed for positive s.a)pump crystalb)pump photon pair crystal photon pairFIG. 2. Spontaneous parametric down-conversion. a) Degenerate configuration, leading to single-mode squeezed vacuum. b) Non-degenerate configuration, leading to two-mode squeezed vacuum.a)2 -2 -2b)P-2 0 2 4 6X-2 0 2P2 -2 -2P-2 0 2 4 6X-2 0 2P246XDj 246Xc)2 -2 -2d)P-2 0 2 4 6X6-2 0 2P-22P-2 0 2 4 6X6-2 0 2P24X2 -24Xe)2 -2 -2f)P-2 0 2 4 6X6-2 0 2P-22P24X-2 0 2 4 6 -2 0 2 Dj X 2 4 6XP-2FIG. 1. Wigner functions of certain single-oscillator states. a) Vacuum state. b) coherent state. c,d) Position- and momentum-squeezed vacuum states. e,f) Position- and momentum-squeezed coherent states with real amplitudes. Panels (b) and (f) show the phase uncertainties of the respective states to emphasize the phase squeezing of state (f). Insets show wavefunctions in the position and momentum bases.This result illustrates one of the primary methods of producing squeezing. Spontaneous parametric downconversion (SPDC) is a nonlinear optical process in which a photon of a powerful laser field propagating through a second-order nonlinear optical medium may split into two photons of lower energy. The frequencies, wavevectors and polarizations of the generated photons are governed by phase-matching conditions. Single-mode squeezing, such as that in the above example, is obtained when SPDC is degenerate : the two generated photons are indistinguishable in all their parameters: frequency, direction, and polarization. The quantum state of the optical mode into which the photon pairs are emitted exhibits squeezing [Fig. 2(a)]. Aside from being an interesting physical entity by itself, squeezed light has a variety of applications. One of the primary applications of single-mode squeezed light is in precision measurements of distances. Such measurements are typically done by means of interferometry. Quantum phase noise poses an ultimate limit to interferometry, and the application of squeezing (in particular, the phase squeezed state discussed above) permits expanding this limit beyond a fundamental boundary. For example, squeezing is employed in the new generation of gravitational wave detectors — GEO 600 in Europe and LIGO in the United States.B. Two-mode squeezed lightHow can one generate optical squeezed states in experiment? Consider the state s |ψ = |0 − √ |2 , 2 (6)where |0 and |2 are photon number (Fock) states and s is a real positive number. We assume s to be small, so the norm of state (6) is close to one. √ The mean value of ˆ = (ˆ the position operator X a+a ˆ† )/ 2 in this state isA state that is closely related to the single-oscillator squeezed vacuum in its theoretical description and experimental procedures, but quite different in properties is the two-mode squeezed vacuum (TMSV), also known as the twin-beam state. As the name suggests, this is a state of not one, but two mechanical or electromagnetic oscillators. We introduce this state by first analyzing the tensor product |0 ⊗ |0 of vacuum states of the two oscillators. In the position basis, its wavefunction [Fig. 3(a)],2 2 1 Ψ00 (Xa , Xb ) = √ e−Xa /2 e−Xb /2 π(8)3 can be rewritten as2 2 1 Ψ00 (Xa , Xb ) = √ e−(Xa −Xb ) /4 e−(Xa +Xb ) /4 . πboth Alice’s and Bob’s observables: (9) 1 −(Pa −Pb )2 /(4R2 ) −R2 (Pa +Pb )2 /4 ˜ R (Pa , Pb ) = √ Ψ e e . (11) π We see that for R > 1 Alice’s and Bob’s momenta are √ anticorrelated, i.e. the variance of the sum (Pa + Pb )/ 2 is below the level expected from two vacuum states [Fig. 3(d)]. The two-mode squeezed vacuum does not imply squeezing in each individual mode. On the contrary, Alice’s and Bob’s position and momentum observables in TMSV obey a Gaussian probability distribution with variance2 2 2 2 ∆Xa = ∆Xb = ∆ Pa = ∆ Pb =Here, Xa and Xb are the position observables of the two oscillators which are traditionally associated with fictional experimentalists Alice and Bob. The meaning √ of Eq. (9) √ is that the observables (Xa − Xb )/ 2 and (Xa + Xb )/ 2 have a Gaussian distribution with variance 1/2. This is not surprising because in the double-vacuum state Alice’s and Bob’s position observables are uncorrelated and both of them have variance 1/2. The behavior of the momentum quadratures in this state is analogous to that of the position.a)4 2 -4 -2 -2 -4XB4 2 2 4PB1 + R4 . 4R2(12)XA-4-2 -2 -424PAthat exceeds that of the vacuum state for any R = 1. In other words, each mode of a TMSV considered individually is in the thermal state. With increasing R > 1, the uncertainty of individual quadratures increases while that of the difference of Alice’s and Bob’s position observables as well as the sum of their momentum observables decreases. In the extreme case of R → ∞, the wavefunctions of the two-modes squeezed state take the form ΨR (Xa , Xb ) ∝ δ (Xa − Xb ) ˜ R (Pa , Pb ) ∝ δ (Pa + Pb ) Ψ (13) (14)b)4 2 -4 -2 -2 -4XB4 2 2 4PBXA-4-2 -2 -424PAFIG. 3. Wavefunctions (not Wigner functions!) of two-mode states in the position (left) and momentum (right) bases. a) Double-vacuum state is uncorrelated in both bases. b) The two-mode squeezed state with position observables correlated, and momentum observables anticorrelated beyond the standard quantum limit.The wavefunction of the two-mode squeezed vacuum state |TMSVR is given by2 2 2 2 1 ΨR (Xa , Xb ) = √ e−(Xa +Xb ) /(4R ) e−R (Xa −Xb ) /4 , π (10) where R, as previously, is the squeezing parameter [Fig. 3(c)]. In contrast to the double-vacuum, TMSV is an entangled state, and Alice’s and Bob’s position observables are nonclassically correlated thanks to that √ entanglement. For R > 1, the variance of (Xa − Xb )/ 2 is less than 1/2, i.e. below the value for the double vacuum state. The wavefunction of TMSV in the momentum basis is obtained from Eq. (10) by means of Fourier transform byBoth Alice’s and Bob’s positions are completely uncertain, but at the same time precisely equal, whereas the momenta are precisely opposite. This state is the basis of the famous quantum nonlocality paradox in its original formulation of Einstein, Podolsky and Rosen (EPR) [2]. EPR argued that by choosing to perform either a position or momentum measurement on her portion of the TMSV, Alice remotely prepares either a state with a certain position or one with a certain momentum at Bob’s location. But according to the uncertainty principle, certainty of position implies complete uncertainty of momentum, and vice versa. In other words, by choosing the setting of her measurement apparatus, Alice can instantly and remotely, without any interaction, prepare at Bob’s station one of two mutually incompatible physical realities. This apparent contradiction to basic principles of causality has lead EPR to challenge quantum mechanics as complete description of physical reality and triggered a debate that continues to this day. Experimental realization of TMSV is largely similar to that of single-mode squeezing. SPDC is the primary method; however, in contrast to the single-mode case, it is implemented in the non-degenerate configuration. The photons is each generated pair are emitted into two distinguishable modes that become carriers of the TMSV state [Fig. 2(b)]. In order to understand how non-degenerate SPDC leads to squeezing, consider the two-mode state |Ψ = |0 ⊗ |0 + s |1 ⊗ |1 , (15)4 i.e. a pair of photons has been emitted into Alice’s and Bob’s modes with amplitude s. Now √ if we evaluate the variance of the observable (Xa − Xb )/ 2, we find 1 1 1 ∆(Xa − Xb )2 = Ψ| (ˆ a+a ˆ† − ˆ b−ˆ b† )2 |Ψ = − s, 2 4 2 (16) i.e. Alice’s and Bob’s position observables are correlated akin to TMSV. A similar calculation shows anticorrelation of Alice’s and Bob’s momentum observables. Both the single-mode and two-mode squeezed vacuum states are valuable resources in quantum optical information technology. TMSV, in particular, is useful for generating heralded single photons and unconditional quantum teleportation.II. SALIENT FEATURES OF SQUEEZED STATES A. The squeezing operatorIf this evolution continues for time t, we will have ˆ (t) = S ˆ † (r )X ˆ (0)S ˆ (r ) = X ˆ (0)e−r ; X ˆ (t) = S ˆ † (r )P ˆ (0)S ˆ(r) = P ˆ (0)er , P (24a) (24b)which corresponds to position squeezing by factor R = er and corresponding momentum antisqueezing (Fig. 4). If the initial state is vacuum, the evolution will result in a squeezed vacuum state; coherent states will yield squeezed light [3]. As a self-check, we find the factor of quadrature squeezing in state (18), in analogy to Eq. (7): R= 0|∆X 2 |0 = ˆ† (r)∆X 2 S ˆ(r)|0 0|S 1/2 ≈1+r 1/2 − rwhich is in agreement with R = er for small r. The corresponding transformation of the creation and annihilation operators is given by a ˆ(t) = a ˆ(0) cosh r − a ˆ† (0) sinh r; a ˆ† (t) = a ˆ† (0) cosh r − a ˆ(0) sinh r, known as Bogoliubov transformation. (25a) (25b)We now proceed to a more rigorous mathematical description of squeezing. Single-mode squeezing occurs under the action of operator ˆ(ζ ) = exp[(ζ a S ˆ2 − ζ ∗ a ˆ†2 )/2], (17)Pwhere ζ = reiφ is the squeezing parameter, with r and φ being real numbers, upon the vacuum state. Phase φ determines the angle of the quadrature that is being squeezed. In the following, we assume this phase to be zero so ζ = r. Note that, for a small r, the squeezing operator (17) acting on the vacuum state, generates state √ ˆ(r) |0 ≈ [1+(ra S ˆ2 −r a ˆ†2 )/2] |0 = |0 −(r/ 2) |2 , (18) which is consistent with Eq. (6) for s = r. The action of the squeezing operator can be analyzed as fictitious evolution under Hamiltonian ˆ = i α[ˆ H a2 − (ˆ a† )2 ]/2 (19)Xˆ )t ˆ(r) = e−i(H/ for time t = r/α (so that S ). Analyzing this evolution in the Heisenberg picture, we use [ˆ a, a ˆ† ] = 1 to find that˙ = i [H, ˆ a a ˆ ˆ] = −αa ˆ† and ˙ † = −αa a ˆ ˆ.(20)FIG. 4. Transformation of quadratures under the action of the squeezing Hamiltonian (19) with α > 0. Grey areas show examples of Wigner function transformations with r = αt = ln 2.(21)Now using the expressions for quadrature observables √ √ ˆ = (ˆ ˆ = (ˆ X a+a ˆ† )/ 2 and P a−a ˆ† )/ 2i, (22) we rewrite Eqs. (20) and (21) as ˙ ˆ X = −αX ; ˙ ˆ P = αP. (23a) (23b)Two-mode squeezing is treated similarly. The twomode squeezing operator is ˆ2 (ζ ) = exp[(−ζ a S ˆˆ b + ζ ∗a ˆ†ˆ b† )]. (26)Assuming, again, a real ζ = r, introducing the fictitious Hamiltonian and recalling that the creation and annihilation operators associated with different modes commute,5 we find a ˆ(t) = a ˆ(0) cosh r + ˆ b(0)† sinh r; ˆ b(t) = ˆ b(0) cosh r + a ˆ(0)† sinh r; and hence ˆ a (t) ± X ˆ b (t) = [X ˆ a (0) ± X ˆ b (0)]e±r ; X ˆa (t) ± P ˆb (t) = [P ˆa (0) ± P ˆb (0)]e∓r . P (28a) (28b) (27a) (27b) Decomposing the exponent in right-hand side of the above equation into the Taylor series with respect to α, we obtain α 2m . m! n=0 m=0 (33) Because this equality must hold for any real α, each term of the sum in the left-hand side must equal its counterpart in the right-hand side that contains the same power of α. Hence n = 2m and 2R 1 + R2 2m |sqR = 1 − R2 2R 1 + R2 2(1 + R2 )m ∞αn n |sqR √ = n!∞1 − R2 2(1 + R2 )mInitially, Alice’s and Bob’s modes are in vacuum states, and the quadrature observables in these modes are uncorrelated. But as the time progresses, Alice’s and Bob’s position observables become correlated while the momentum observables become anticorrelated.(2m)! . m!(34)Since R = er , we haveB. Photon number statistics1 2R = 1 + R2 cosh rand1 − R2 = − tanh r, 1 + R2(35)An important component in the theoretical description of squeezed light is its decomposition in the photon number basis, i.e. calculating the quantities n |sqR for the single-mode squeezed state and mn |TMSVR for the two-mode state. Due to non-commutativity of the photon creation and annihilation operators, this calculation turns out surprisingly difficult even for basic squeezed vacuum states, let alone squeezed coherent states and the states that have been affected by losses. Possible approaches to this calculation include the disentangling theorem for SU(1,1) Lie algebra [4], direct calculation of the wavefunction overlap in the position space [5] or transformation of the squeezing operator [6]. Here we derive the photon number statistics of single- and twomode squeezed vacuum states by calculating their inner product with coherent states. The wavefunction of a coherent state with real amplitude α is ψα (X ) = 1 π 1/4 e−(X −α√ 2)2 /2so Eq. (34) can be rewritten as |sqR = √ 1 cosh r∞(− tanh r)mm=0(2m)! |2m . 2m m!(36)We stop here for a brief discussion. First, we note that that for r 1, Eq. (36) becomes √ |sqR = |0 − (r/ 2) |2 + O(r2 ), (37),(29)so its inner product with the position squeezed state (3) equalsR2 2R − 1+ α2 R2 . e 2 1+R −∞ (30) Now we recall that the coherent state is decomposed into the Fock basis according to+∞α |sqR =ψα (X )ψR (X )dX =∞|α =n=0e −α2/2αn √ |n , n!(31)consistently with Eq. (18). Second, note that the squeezed vacuum state (36) contains only terms with even photon numbers. This is a fundamental feature of this state; in fact, one of the earlier names for squeezed states has been “two-photon coherent states” [7]. This feature follows from the nature of the squeezing operator (17): in its decomposition into the Taylor series with respect to r, creation and annihilation operators occur only in pairs. Pairwise emission of photons is also a part of the physical nature of SPDC: due to energy conservation a pump photon can only split into two photons of half its energy. We now turn to finding the photon number decomposition of the two-mode squeezed state. We first notice, by looking at Eq. (26), that |RAB must only contain terms with equal photon numbers in Alice’s and Bob’s modes. This circumstance allows us to significantly simplify the algebra. We proceed along the same route as outlined above, calculating the overlap of |RAB with the tensor product |αα of identical coherent states |α in Alice’s and Bob’s channels using Eqs. (10) and (29): αα|TMSVR+∞so we have∞= α n |sqR √ = n!nψα (Xa )ψα (Xb )ΨR (Xa , Xb )dXa dXb−∞n=02R e 1 + R21−R2 α2 2(1+R2 )(32)=2R − 1+2R2 α2 e . 1 + R2(38)6 Decomposing the coherent states in the left-hand side into the Fock basis according to Eq. (31) and keeping only the terms with equal photon numbers, we have∞−R2 2 2R − 1 α2n α e 1+R2 nn| TMSVR √ = 2 1+R n!(39)n=0Now writing the Taylor series for the right-rand side and using Eq. (35), we obtain |TMSVR = 1 tanhn r |nn . cosh r n=0∞(40)FIG. 5. Experimentally reconstructed photon number statistics of the squeezed vacuum state. For low photon numbers, the even terms are greater than the odd terms due to pairwise production of photons, albeit the odd term contribution is nonzero due to loss. Reproduced from Ref. [10].position-squeezed vacuum ˆ¢(t ) bˆ¢(t ) a momentum-squeezed vacuumSimilarly to the single-mode squeezing, it is easy to verify that result is consistent with state (15) for small r. On the other hand, in contrast to the single-mode case, the energy spectrum of TMSV follows Boltzmann distribution with mean photon number in each mode n = sinh2 r. This is in agreement with our earlier observation that Alice’s and Bob’s portions of TMSV considered independently of their counterpart are in the thermal state, i.e. the state whose photon number distribution obeys Boltzmann statistics with the temperature given by e− ω/kT = tanh r. While the present analysis is limited to pure squeezed vacuum states, photon number decompositions of squeezed coherent states and squeezed states that have undergone losses can be found in the literature [8, 9]. In contrast to pure squeezed vacuum states, these decompositions have nonzero terms associated to non-paired photons. The origin of these terms is easily understood. If a one- or two-mode squeezed vacuum state experiences a loss, it may happen that one of the photons in a pair is lost while the other one remains. If the squeezing operator acts on a coherent state, the odd photon number terms will appear in the resulting state because they are present initially. Photon statistics of both classes of squeezed states have been tested experimentally, as discussed in Section III below. An example is shown in Fig. 5.ˆ0 a fictitious input vacuum ˆ0 bˆ(0) b input vacuum2-mode squeezerˆ(0) aˆ(t ) a two-mode squeezed vacuum ˆ(t ) bFIG. 6. Interconversion of the two-mode squeezed vacuum and two single-mode squeezed vacuum states. Dashed lines show a fictitious beam splitter transformation of a pair of vacuum states such that the modes a ˆ (t), ˆ b (t) are explicitly single-mode squeezed with respect to modes a ˆ 0, ˆ b 0.In accordance with the definition (22) of quadrature observables, Eqs. (41) apply in the same way to the position and momentum of the input and output modes. Applying this to Eqs. (28), we find √ ˆ a,b = [X ˆ a (t) ∓ X ˆ b (t)]/ 2 X √ ˆ a (0) ∓ X ˆ b (0)]/ 2 = e ∓r [ X (42) for the output positions and √ ˆa,b = [P ˆa (t) ∓ P ˆb (t)]/ 2 P √ ˆa (0) ∓ P ˆb (0)]/ 2 = e ±r [ PC.Interconversion between single- and two-mode squeezing(43)If the modes of the TMSV are overlapped on a symmetric beam splitter, two unentangled single-mode vacuum states will emerge in the output (Fig. 6). To see this, we recall the beam splitter transformation a ˆ = τa ˆ − ρˆ b; ˆ ˆ b = τ b + ρa ˆ, (41a) (41b)for the momenta. In order to understand what state this corresponds to, let us assume, for the sake of the argument, that vacuum modes a ˆ and ˆ b at the SPDC input have been obtained from another pair of modes by means of another symmetric beam splitter: √ a ˆ0 = [ˆ a(0) − ˆ b(0)]/ 2 (44) √ 0 ˆ ˆ b = [ˆ a(0) + b(0)]/ 2. (45) Of course, since modes a ˆ(0) and ˆ b(0) are in the vacuum 0 0 ˆ state, so are a ˆ and b . We then have:0 ˆ a,b = e∓r X ˆ a,b X ; ±r ˆ 0 ˆ Pa,b = e Pa,b ,where τ and ρ are the beam splitter amplitude transmissivity and reflectivity, respectively. For a symmetric √ beam splitter, τ = ρ = 1/ 2. In writing Eqs. (41), we neglected possible phase shifts that may be applied to individual input and output modes [5].(46)7 where superscript 0 associates the quadrature with modes a ˆ0 and ˆ b0 . We see that modes a ˆ and ˆ b are re0 0 ˆ lated to vacuum modes a ˆ and b by means of position and momentum squeezing transformations, respectively. Because the beam-splitter transformation is reversible, it can also be used to obtain a TMSV from two singlemode squeezed vacuum states with squeezing in orthogonal quadratures. This technique has been used, for example, in the experiment on continuous-variable quantum teleportation [11].E. Effect of lossesD.Squeezed vacuum and squeezed lightSqueezed vacuum and bright squeezed light are readily converted between each other by means of the phasespace displacement operator [5], whose action in the Heisenberg picture can be written as ˆ † (α)ˆ ˆ (α) = a D a† D ˆ + α. (47)Squeezed states that occur in practical experiments necessarily suffer from losses present in sources, transmission channels and detectors. In order to understand the effect of propagation losses on a single-mode squeezed vacuum state, we can use the model in which a lossy optical element with transmission T is replaced by a beam splitter (Fig. 8). At the other input port of the beam splitter there is a vacuum state. The interference of the signal mode a ˆ with the vacuum mode v ˆ will produce a mode with operator a ˆ = τa ˆ − ρv ˆ (with τ 2 = T and ρ2 = 1 − T being the beam splitter transmissivity and reflectivity) in the beam splitter output. Accordingly, we have ˆ θ,out = τ X ˆ a,θ − ρX ˆ v,θ . X (52)This means, in particular, that the position and momentum transform according to √ ˆ →X ˆ + Re α 2; (48) X √ ˆ ˆ P → P + Im α 2, (49) ˆ (α), the entire phase space disso, under the action of D places itself, thereby changing the coherent amplitude of the squeezed state without changing the degree of squeezing.Because the quadrature observable of the signal and vacuum states are uncorrelated, and since ∆(Xθ )2 = 1/2, it follows that2 2 ∆Xθ, ∆(Xa,θ )2 + ρ2 ∆(Xv,θ )2 out = τ= T ∆(Xa,θ )2 + (1 − T )/2.(53)Analyzing Eqs. (41) we see that the optical loss alone, no matter how significant, cannot eliminate the property of squeezing completely.ˆ alow-reflectivity beam splitterˆ - rb aˆ aˆ b b1signalˆout aoutputFIG. 7. Implementation of phase-space displacement. ρ is the beam splitter’s amplitude reflectivity.ˆ vacuum vFIG. 8. The beam splitter model of loss.Phase-space displacement can be implemented experimentally by overlapping the signal state with a strong coherent state |β on a low-reflectivity beam splitter (Fig. 7). Applying the beam splitter transformation (41), we find for the signal mode a ˆ = τa ˆ − ρˆ b (50)Given that mode ˆ b is in a coherent state (i.e. an eignestate of ˆ b) and that ρ 1 (i.e. τ ∼ 1), we have a ˆ =a ˆ − ρβ (51)in analogy to Eq. (47). The displacement operation has been used to change the amplitude of squeezed light in many experiments, for example, in Ref. [12].Ideal squeezed-vacuum and coherent states have the minimum-uncertainty property: the product of uncer2 2 tainties ∆Xout ∆Pout reaches the theoretical minimum of 1/4. But this is no longer the case in the presence of losses. The deviation of the uncertainty from the minimum can be used to estimate the preparation quality of a squeezed state. Suppose a measurement of a squeezed state yielded the minimum and maximum quadrature un2 2 and ∆Xmax , respectively. certainty values of ∆Xmin One can assume that the state has been obtained from an ideal (minimum-uncertainty) squeezed state with squeezing R by means of loss channel with transmissivity T . Using Eq. (5) and solving Eqs. (53), one finds T [13], which can then be compared with the values expected from the setup at hand.。
1汉英力学名词(1993)
BZ反应||Belousov-Zhabotinski reaction, BZ reactionFPU问题||Fermi-Pasta-Ulam problem, FPU problemKBM方法||KBM method, Krylov-Bogoliubov-Mitropolskii method KS[动态]熵||Kolmogorov-Sinai entropy, KS entropyKdV 方程||KdV equationU形管||U-tubeWKB方法||WKB method, Wentzel-Kramers-Brillouin method[彻]体力||body force[单]元||element[第二类]拉格朗日方程||Lagrange equation [of the second kind] [叠栅]云纹||moiré fringe; 物理学称“叠栅条纹”。
[叠栅]云纹法||moiré method[抗]剪切角||angle of shear resistance[可]变形体||deformable body[钱]币状裂纹||penny-shape crack[映]象||image[圆]筒||cylinder[圆]柱壳||cylindrical shell[转]轴||shaft[转动]瞬心||instantaneous center [of rotation][转动]瞬轴||instantaneous axis [of rotation][状]态变量||state variable[状]态空间||state space[自]适应网格||[self-]adaptive meshＣ0连续问题||C0-continuous problemＣ1连续问题||C1-continuous problemＣＦＬ条件||Courant-Friedrichs-Lewy condition, CFL condition ＨＲＲ场||Hutchinson-Rice-Rosengren fieldＪ积分||J-integralＪ阻力曲线||J-resistance curveＫＡＭ定理||Kolgomorov-Arnol'd-Moser theorem, KAM theoremＫＡＭ环面||KAM torusｈ收敛||h-convergenceｐ收敛||p-convergenceπ定理||Buckingham theorem, pi theorem阿尔曼西应变||Almansis strain阿尔文波||Alfven wave阿基米德原理||Archimedes principle阿诺德舌[头]||Arnol'd tongue阿佩尔方程||Appel equation阿特伍德机||Atwood machine埃克曼边界层||Ekman boundary layer埃克曼流||Ekman flow埃克曼数||Ekman number埃克特数||Eckert number埃农吸引子||Henon attractor艾里应力函数||Airy stress function鞍点||saddle [point]鞍结分岔||saddle-node bifurcation安定[性]理论||shake-down theory安全寿命||safe life安全系数||safety factor安全裕度||safety margin暗条纹||dark fringe奥尔-索末菲方程||Orr-Sommerfeld equation奥辛流||Oseen flow奥伊洛特模型||Oldroyd model八面体剪应变||octohedral shear strain八面体剪应力||octohedral shear stress八面体剪应力理论||octohedral shear stress theory巴塞特力||Basset force白光散斑法||white-light speckle method摆||pendulum摆振||shimmy板||plate板块法||panel method板元||plate element半导体应变计||semiconductor strain gage半峰宽度||half-peak width半解析法||semi-analytical method半逆解法||semi-inverse method半频进动||half frequency precession半向同性张量||hemitropic tensor半隐格式||semi-implicit scheme薄壁杆||thin-walled bar薄壁梁||thin-walled beam薄壁筒||thin-walled cylinder薄膜比拟||membrane analogy薄翼理论||thin-airfoil theory保单调差分格式||monotonicity preserving difference scheme 保守力||conservative force保守系||conservative system爆发||blow up爆高||height of burst爆轰||detonation; 又称“爆震”。
Graph Regularized Nonnegative Matrix
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1 INTRODUCTION
HE
techniques for matrix factorization have become popular in recent years for data representation. In many problems in information retrieval, computer vision, and pattern recognition, the input data matrix is of very high dimension. This makes learning from example infeasible [15]. One then hopes to find two or more lower dimensional matrices whose product provides a good approximation to the original one. The canonical matrix factorization techniques include LU decomposition, QR decomposition, vector quantization, and Singular Value Decomposition (SVD). SVD is one of the most frequently used matrix factorization techniques. A singular value decomposition of an M Â N matrix X has the following form: X ¼ UÆVT ; where U is an M Â M orthogonal matrix, V is an N Â N orthogonal matrix, and Æ is an M Â N diagonal matrix with Æij ¼ 0 if i 6¼ j and Æii ! 0. The quantities Æii are called the singular values of X, and the columns of U and V are called
【5A文】关于序列二次规划(SQP)算法求解非线性规划问题研究
关于序列二次规划（SQP）算法求解非线性规划问题研究兰州大学硕士学位论文关于序列二次规划（SQP）算法求解非线性规划问题的研究姓名：石国春申请学位级别：硕士专业：数学、运筹学与控制论指导教师：王海明20090602兰州大学2009届硕士学位论文摘要非线性约束优化问题是最一般形式的非线性规划NLP问题，近年来，人们通过对它的研究，提出了解决此类问题的许多方法，如罚函数法，可行方向法，Quadratic及序列二次规划SequentialProgramming简写为SOP方法。
本文主要研究用序列二次规划SOP算法求解不等式约束的非线性规划问题。
SOP算法求解非线性约束优化问题主要通过求解一系列二次规划子问题来实现。
本文基于对大规模约束优化问题的讨论，研究了积极约束集上的SOP 算法。
我们在约束优化问题的s一积极约束集上构造一个二次规划子问题，通过对该二次规划子问题求解，获得一个搜索方向。
利用一般的价值罚函数进行线搜索，得到改进的迭代点。
本文证明了这个算法在一定的条件下是全局收敛的。
关键字：非线性规划，序列二次规划，积极约束集Hl兰州人学2009届硕二t学位论文AbstractNonlinearconstrainedarethemostinoptimizationproblemsgenericsubjectsmathematicalnewmethodsareachievedtosolveprogramming．Recently,Manyasdirectionit，suchfunction，feasiblemethod，sequentialquadraticpenaltyprogramming??forconstrainedInthisthemethodspaper,westudysolvinginequalityabyprogrammingalgorithm．optimizationproblemssequentialquadraticmethodaofSQPgeneratesquadraticprogrammingQPsequencemotivationforthisworkisfromtheofsubproblems．OuroriginatedapplicationsinanactivesetSQPandSQPsolvinglarge-scaleproblems．wepresentstudyforconstrainedestablishontheQPalgorithminequalityoptimization．wesubproblemsactivesetofthesearchdirectionisachievedQPoriginalproblem．AbysolvingandExactfunctionsaslinesearchfunctionsubproblems．wepresentgeneralpenaltyunderobtainabetteriterate．theofourisestablishedglobalconvergencealgorithmsuitableconditions．Keywords：nonlinearprogramming，sequentialquadraticprogrammingalgorithm，activesetlv兰州大学2009届硕士学位论文原创性声明本人郑重声明：本人所呈交的学位论文，是在导师的指导下独立进行研究所取得的成果。