THE FUNCTION AND CLASSES OF ROAD MARKINGS
基于改进LMD方法的风电机组齿轮箱故障诊断研究
第42卷第3期2021年3月自动化仪表P R O C E S S A U T O M A T I O N I N S T R U M E N T A T I O NV o l.42No. 3M a r. 2021基于改进L M D方法的风电机组齿轮箱故障诊断研究李辉，邓奇(西安理工大学电气工程学院，陕西西安710048)摘要:局部均值函数的求取是局部均值分解（L M D)的关键环节。
针对局部均值函数求取存在偏差进而造成模态混叠的问题，提出 了一种基于局部积分均值的L M D风电机组齿轮箱故障诊断方法。
该方法改变了对相邻两极值点求平均值的思路，采用求取相邻两极值点的局部积分均值，再通过滑动平均法进行平滑处理，最终得到局部均值函数。
为实现风电机组齿轮箱故障诊断，首先采用改进L M D方法对信号进行降噪处理，然后采用多尺度熵提取降噪处理后信号的特征向量，最后采用极限学习机进行故障诊断。
通过仿真分析，证明了该方法能有效解决模态混叠现象，提高了 L M D的分解精度。
试验验证分析表明，该方法的故障诊断准确率为100%，通 过对比分析表明，该方法优于其他故障诊断方法，具有工程应用价值。
关键词:局部积分均值；风机齿轮箱；局部均值分解；故障诊断；极值点；多尺度熵；极限学习机；模态混叠中图分类号：T H132. 41 文献标志码：A D0I： 10. 16086/j. cnki. issn 1000-0380. 2020060025Study on Fault Diagnosis of Wind Turbine GearboxBased on Improved LMD MethodLI H u i,D E N G Qi(School of Electrical E n g i n e ering,Xi5an University of T e c h n o l o g y,X i'an 710048,China)A b s t r a c t：F i n d i n g the local m e a n function is the k e y link of the local m e a n d e c o m p o s i t i o n(L M D). A fault diagnosis for the g e a r b o x of the L M D w i n d turbine b a s e d o n the local integral m e a n is p r o p o s e d c o n s i d e r i n g its deviation a n d m o d e m ixing. T h i s m e t h o d a p p l i e d local integral m e a n v a l u e of the a d j a c e n t t w o e x t r e m e points t h e n s m o o t h i n g b y the m o v i n g a v e r a g e m e t h o d to finally obtains the local m e a n function instead of a v e r a g i n g the t w o a d j a c e n t e x t r e m e points. In or d e r to realize the fault diagnosis of w i n d turbine g e a r b o x,t h e i m p r o v e d L M D m e t h o d for d e-n o i s i n g the signal a n d multi-scale e n t r o p y for extracting the feature vector of the d e-n o i s e d signal, a n d finally the limit learning m a c h i n e is used. S i m u l a t i o n analysis p r o v e s that this m e t h o d c a n effectively solve the m o d e m i x i n g p h e n o m e n o n a n d i m p r o v e the d e c o m p o s i t i o n a c c u r a c y of L M D.T h r o u g h e x p e r i m e n t a l verification,this m e t h o d h a s a fault diagnosis a c c u r a c y rate of 100%. T h r o u g h c o m p a r a t i v e analysis,it is of e n g i n e e r i n g applicatin va l u e superior to other fault diagnosis m e t h o d s.Key w ords：L o c a l integral m e a n；W i n d turbine g e a r b o x；L o c a l m e a n d e c o m p o s i t i o n；Fault d i a g n o s i s；E x t r e m e p o i n t；Multiscale e n t r o p y；E x t r e m e learning m a c h i n e；M o d e m i x i n g〇引言近年来，由于全球环境恶化和资源短缺，使得世界 各国逐渐开始重视开发和利用可再生能源。
丘 赛 解答 几何
丘赛解答几何英文回答：Introduction.In the realm of mathematics, geometry holds a prominent position as the study of shapes, sizes, and their relationships. One of the most fundamental concepts in geometry is the quadrilateral, a polygon with four sides. Among quadrilaterals, the square is a distinctive figure characterized by its four equal sides and four right angles. In this essay, we will delve into the fascinating world of squares, exploring their properties, constructions, and applications across various disciplines.Properties of Squares.Squares possess several unique properties that distinguish them from other quadrilaterals. Firstly, as mentioned earlier, all four sides of a square are equal inlength. This property is known as equilaterality. Secondly, each of the four angles in a square measures 90 degrees, making it a rectangle. Moreover, the diagonals of a square are perpendicular bisectors of each other, dividing the square into four congruent right triangles.Construction of Squares.Constructing a square requires precision and adherence to specific steps. One common method involves using a ruler and a compass. The following steps outline the process:1. Draw a line segment of the desired length to serve as one side of the square.2. From one endpoint of the line segment, draw an arc with a radius equal to the length of the side.3. Repeat step 2 from the other endpoint of the line segment, ensuring that the arcs intersect at two points.4. Connect the intersection points to form the othertwo sides of the square.Applications of Squares.Squares find extensive applications in architecture, design, art, and even everyday life. Their inherent symmetry and aesthetics make them a popular choice for building foundations, floor plans, and decorative elements. In graphic design, squares serve as the basis for logos, icons, and user interfaces. Additionally, squares play a crucial role in grids and matrices, which are used in mathematics, computer science, and data analysis.Conclusion.Squares, with their distinct properties and versatility, hold a significant place in the realm of geometry. Their construction methods have been refined over centuries, and their applications span across multiple disciplines. From towering skyscrapers to intricate patterns in nature, squares continue to captivate and inspire mathematicians, artists, and architects alike.中文回答：概述。
无人驾驶的英语课件PPT
Other potential applications include long haul trucking, public transportation, and even self driving taxis or shared mobility services
3D Reconstruction
The creation of a 3D model of the environment from sensor data to provide more accurate representation of the scene
Path planning technology
Application scenarios for autonomous driving
Autonomous driving has the potential to revolutionize transportation, particularly in urban areas where traffic congestion and pollution are major issues
Techniques used to regulate the vehicle's velocity, acceleration, and steel angle to achieve desired performance and safety standards
Risk Assessment
The evaluation of potential hazards and their associated risks to inform decision making processes
Differential and Integral Calculus(高数)
Differential and Integral CalculusCourse Content and Basic RequirementsCourse contents: Limits and continuity; the calculus of functions of one variable; ordinary differential equation; vector algebra and analytic geometry of space; multi-function calculus; infinite series.(II) Course teaching content and Knowledge Part Arrangement1. Knowledge Unit One: The Function & The Limit (15 hours)(1) Knowledge point 1: Mapping and mapping(2) Knowledge point 2: The limit of series(3) Knowledge point 3: The limit of function(4) Knowledge point 4: Infinitesimal and Infinite(5) knowledge point 5: The limit algorithm(6) Knowledge point 6: Limitexistingcriteria Two important limits(7) Knowledge point 7: Comparison of Infinitesimals(8) knowledge point 8: Continuity and discontinuous point of the function(9) knowledge point 9: Operation propertiesof continuous function &Continuity of elementary functions(10) knowledge points 10: Property of continuous function on closed interval2. Knowledge Unit Two: Derivative and Differential (10 hours)(1) Knowledge point 1: The concept of Derivative(2) Knowledge point 2:Derivative rule of the function(3) Knowledge point 3: Derivative of higher order(4) Knowledge point 4: The derivative of the function determined byImplicit functions and Parametric equation(5) Knowledge point 5: Differential of the function3.Knowledge Unit Three: Differential Mean Value Theorem and the Applications of Derivatives (10 hours)(1) Knowledge point 1: Differential Mean Value Theorem(2) Knowledge point 2: Hospital Rule(3) Knowledge point 3: Taylor Formula(4) Knowledge point 4: Monotonicity and Concavity of the function(5) Knowledge point 5: Extremum, Maximum and Minimum of the function(6) Knowledge point 6:Delineation of function graphic(7) Knowledge point 7: Curvature4. Knowledge Unit Four: Indefinite Integral (hours 10)(1) Knowledge point 1: Concepts and Properties of the Indefinite Integral(2) knowledge point 2: Integral by Substitution(3) Knowledge point 3: Integral by Parts(4) knowledge point 4: Integration of Rational Function5. Knowledge Unit Five: Definite integral (10 hours)(1) Knowledge point 1: Concepts and Properties of the Definite Integral(2) knowledge point 2: Fundamental Theorem of Calculus(3) Knowledge point 3: Integral by Substitution and Integral by Parts of Definite Integral(4) Knowledge point 4: Improper Integral6.Knowledge Unit Six: Application of definite integral (3 hours)(1) Knowledge point 1: element method of definite integral(2) Knowledge point 2: the application of definite integral in geometry(3) Knowledge point 3: the application of definite integral in physics7. KnowledgeUnit Seven: Space analytic geometry and Vector algebra (12 hours)(1) Knowledge point 1: vector and its linear operator(2) knowledge point 2: Dot product Vector product(3) Knowledge point 3: Curved surface and its equation(4) Knowledge point 4: Space curve and its equation(5) knowledge point 5: Plane and its equation(6)Knowledge point 6: Spatial line and its equation8. Knowledge Unit Eight: multi-function differential method and its application (16 hours)(1) Knowledge point 1: Basic concepts of Multi-function(2) knowledge point 2: partial derivative(3) Knowledge point 3: total differential(4) knowledge point 4: derivative rule of composite multi-function(5) knowledge point 5: derivation formula of implicit function(6) Knowledge point 6: geometric applications of multi-function differential calculus(7) Knowledge point 7: Directional Derivatives and Gradient(8) knowledge point 8: Extremum of multi-function and its Method9. Knowledge Unit Nine: Double Integral (8 hours)(1) Knowledge point 1: concept and property of Double Integral(2) Knowledge point 2: calculation method of double integral(3) knowledge point 3: triple integral(4) Knowledge point 4: application of triple integral10.Knowledge of Unit Ten: curve integrals and surface integrals (20 hours)(1) Knowledge point 1: curve integral of arc length(2) Knowledge point 2: curve integral of coordinates(3) knowledge point 3: The green Formula and its applications(4) Knowledge point 4: surface integrals of area(5) knowledge point 5: surface integrals of coordinates(6) knowledge point 6: Gaussian formula Stokes formula11. Knowledge Unit Eleven: Infinite Series (17 hours)(1) Knowledge point 1: concept and property ofConstant series(2) knowledge point 2: inspecting convergent of constant series(3) Knowledge point 3: Power Series(4) Knowledge point 4: function expansion into power series(5) knowledge point 5: applicationsof the power series expansion function(6) Knowledge point 6: Fourier series12. Knowledge Unit Twelve: Differential Equations (13 hours)(1) Knowledge point 1: basic concepts of Differential Equations(2) Knowledge point 2: differential equation of separable variables(3) Knowledge point 3: homogeneous equation(4) Knowledge point 4: first order linear differential equations(5) Knowledge point 5: total Differential Equations(6) Knowledge point 6: reduciblehigh order differential equations(7) Knowledge point 7: High Order linear Differential Equations(8) Knowledge point 8: constant coefficient homogeneous linear differential equation(9) Knowledge point 9: constant coefficient fly homogeneous linear differential equationTotal class hours 160, include lecture class 114 hours, exercise class 30 hours, flexible 16 hoursClass Hours Associate Sheet。
Abstract
July 17, 2006
Technical Report 2Science 140 Governors Drive University of Massachusetts Amherst, Massachusetts 01003-9624
Abstract We present a novel hierarchical framework for solving Markov decision processes (MDPs) using a multiscale method called diffusion wavelets. Diffusion wavelet bases significantly differ from the Laplacian eigenfunctions studied in the companion paper (Mahadevan and Maggioni, 2006): the basis functions have compact support, and are inherently multi-scale both spectrally and spatially, and capture localized geometric features of the state space, and of functions on it, at different granularities in spacefrequency. Classes of (value) functions that can be compactly represented in diffusion wavelets include piecewise smooth functions. Diffusion wavelets also provide a novel approach to approximate powers of transition matrices. Policy evaluation is usually the expensive step in policy iteration, requiring O(|S |3 ) time to directly solve the Bellman equation (where |S | is the number of states for discrete state spaces or sample size in continuous spaces). Diffusion wavelets compactly represent powers of transition matrices, yielding a direct policy evaluation method requiring only O(|S |) complexity in many cases, which is remarkable because the Green’s function (I − γP π )−1 is usually a full matrix requiring quadratic space just to store each entry. A range of illustrative examples and experiments, from simple discrete MDPs to classic continuous benchmark tasks like inverted pendulum and mountain car, are used to evaluate the proposed framework.
斯普林格数学研究生教材丛书
《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。
university and their function
•
• (P3)It enables man to construct an intellectual vision of a new world , and it preserves the zest of life by the suggestion of satisfying purposes. • People with imagination will be able to form a new outlook which is different from that of people without imagination. Imagination is capable of preserving people’s enthusiasm for life because it can show people that life has many purposes which can be pleasing.
• Alfred North Whitehead (1861– 1947) was a British mathematician, logician and philosopher best known for his work in mathematical logic and the philosophy of science. In collaboration with Bertrand Russell, he authored the landmark three-volume Principia Mathematica《数学原理》 (1910, 1912, 1913) and contributed significantly to twentieth-century logic, philosophy of science and metaphysics形而上学.
《微积分英文版》课件
Limits and continuity
Definition: A limit is the value that a function approaches as the input approaches a certain point Continuity means that the function doesn't have any breaks or jumps at any point
Course structure
03
The course is divided into several modules, each focusing on a specific topic in calculus Learners can complete the course at their own pace and in any order of the modules
Properties: One side limits, absolute continuity, uniform continuity, etc
Differentiation
Definition: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point It can be used to find the rate of change of a function
Integral definition: The integral of a function is a measure of the area under its curve It is calculated by finding the limit of the sum of areas of rectangles under the curve as the width of the rectangles approaches zero
Di
Diffusion of Dirac fermions across a topological merging transition in two dimensionsP.Adroguer,1D.Carpentier,1G.Montambaux,2and E.Orignac 11Laboratoire de Physique,´Ecole Normale Sup´e rieure de Lyon,46all´e e d’Italie,69007Lyon,France2Laboratoire de Physique des Solides,CNRS,Universit´e Paris-Sud,Universit´e Paris-Saclay,91405Orsay Cedex,France(Dated:November 3,2015)A continuous deformation of a Hamiltonian possessing at low energy two Dirac points of opposite chiralities can lead to a gap opening by merging of the two Dirac points.In two dimensions,the critical Hamiltonian possesses a semi-Dirac spectrum:linear in one direction but quadratic in the other.We study the transport properties across such a transition,from a Dirac semi-metal through a semi-Dirac phase towards a gapped ing both a Boltzmann approach and a diagrammatic Kubo approach,we describe the conductivity tensor within the diffusive regime.In particular,we show that both the anisotropy of the Fermi surface and the Dirac nature of the eigenstates combine to give rise to anisotropic transport times,manifesting themselves through an unusual matrix self-energy.I.INTRODUCTIONThe discovery of graphene has triggered a lot of work on the exotic transport properties of Dirac-like particles in solids 1.Indeed,the graphene electronic spectrum is made of two sub-bands which touch at two inequiva-lent points in reciprocal space.Near the touching points,named Dirac points,the spectrum has a linear shape and the electron dynamics is well described by a 2D Dirac equation for massless particles.Due to the structure of the honeycomb lattice,the wave functions have two com-ponents corresponding to the two inequivalent sites of the lattice,and the Hamiltonian is a 2×2matrix.To de-scribe the low energy properties,the original Hamiltonian is replaced by two copies of a 2D Dirac equationH =±c p . σ(1)where the velocity c 105m.s −1.This linearization is possible because the energy of the saddle point separating the two Dirac cones (valleys)is very large ( 3eV)com-pared to the Fermi energy and temperature scales.Other realizations of Dirac-like physics in two dimensions have been proposed in the organic conductor (BEDT-TTF)2I 3under pressure 2–5,and has been observed in artificially assembled nanostructures 6–10and ultracold atoms 11,12.Besides these two dimensional realizations,the existence and properties of semi-metallic phases in three dimen-sions have recently been studied 13,14.To go beyond and in order to account for a structure which consists in two Dirac points separated by a saddle point,one needs an appropriate low energy 2×2Hamil-tonian.Moreover,such a description is mandatory in situations where,by varying band parameters,the Dirac points can be moved in reciprocal space.Since these Dirac points are characterized by opposite topological charges,they can even annihilate each other 15–19.This merging is therefore a topological transition.It has been shown that,at the transition,the electronic dispersion is quite unusual since it is quadratic in one direction and linear in the other direction (the direction of merging).This ”semi-Dirac”20spectrum has new properties inter-mediate between a Schr¨o dinger and a Dirac spectrum.The vicinity of the topological transition can be described by the following Hamiltonian in two dimensions 18,19:H =∆+p 2x σx +c y p y σy .(2)It has been coined ”Universal Hamiltonian”since the merging scenario of two Dirac points related by time reversal symmetry is uniquely described by this Hamiltonian 18,19.The parameter ∆drives the transition (∆=0)between a semi-metallic phase (∆<0)with two Dirac points and a gapped phase (∆>0),see Figs.1,2.The evolution of several thermodynamic quantities like the specific heat and the Landau level spectrum has been studied in details 18,19.(a) (b)S M(c)x²F²F²F²F²FFigure 1.This work addresses the transport properties for anelectronic spectrum undergoing a topological merging transi-tion as depicted in this figure,and commented in more details in Fig.2.In this paper we address the evolution of the conduc-tivity tensor across the merging transition (Figs.1,2).A first objective of this work is to characterize the trans-port properties as a possible signature of the evolution ofa r X i v :1511.00036v 1 [c o n d -m a t .m e s -h a l l ] 30 O c t 20152 the underlying band structure.On a more fundamen-tal perspective,an additional interest of this problemstands from two important ingredients in the descriptionof diffusive transport.First,at low energy the electronicwave functions have a spinorial structure which leads toeffective anisotropic scattering matrix elements(similarto the case of a scalar problem with anisotropic scatter-ing due to a disorder potential withfinite range).Thisleads to a transport scattering timeτtr different from theelastic scattering timeτe,as in graphene for point-likeimpurities whereτtr=2τe.Second,the anisotropy ofthe dispersion relation leads to an additional complex-ity:the scattering times become themselves anisotropicand depend on the direction of the applied electricfield.We show that within the Green’s function formalism thisanisotropy manifests itself into a rather unusual matrixstructure of the self-energy.A comparison between aBoltzmann approach and a perturbative Green’s func-tion formalism allows for a detailed understanding of thisphysics.The outline of the paper is the following.In the nextsection,we recall the model,i.e.the Universal Hamilto-nian with coupling to impurities described by a point-likewhite noise potential.We define a directional density ofstates and derive the angular dependence of the elasticscattering time.In section III,we use the Boltzmannequation to calculate the conductivity tensor.As a re-sult of the two important ingredients mentioned above,the conductivity along a directionαis not simply propor-tional to the angular averaged squared velocity v2α(θ)because:(i)the elastic scattering time has also anangular dependence due to the angular anisotropy ofthe spectrum,so that one should consider the averagev2α(θ)τe(θ) ;(ii)since the matrix elements of the interac-tion get an angular dependence,it will lead to transporttimes different of the elastic time.These transport timesdepend on the directionαand,to obtain the conduc-tivity,we will have to consider the average v2α(θ)τtrα(θ) .These results obtained from Boltzmann equation are con-firmed by a diagrammatic calculation presented in sec-tion IV.We discuss our results in the last section.II.THE MODELA.Hamiltonian and Fermi surface parametrizationWe consider the model described by the HamiltonianH=H0+V,(3)where the disorder potential V is defined and discussedin section II C and the Hamiltonian for the pure systemis defined asH0=∆+p2x2mσx+c y p yσy.(4)In the present and the following sections(II A and II B) we start by discussing a few properties of the Hamilto-Figure2.Typical energy spectrum of the model(4)for various ∆but afixed energy >0.Dirac phase with(a)∆<− ,(S)∆=− (saddle-point),(b)− <∆<0.Critical semi-Dirac metal(M)∆=0.Gapped phase(c)∆>0.nian H0without disorder.For∆>0this Hamiltonian describes a gapped phase.When∆<0,it describes two Dirac cones with opposite chiralities,hereafter named a Dirac phase.Note that these Dirac cones are in general anisotropic with respectives velocities in the x and y di-rections c x=2|∆|/m and c y.The energy spectrum is given by2=p2x2m+∆2+(c y p y)2.(5)We will consider only the case of positive energies >0, as the situation <0can be deduced from particle-hole symmetry.Fig.2presents the different regimes discussed in this paper.Parametrization of the constant energy contours of Eq.(5)is done by taking advantage of the p x parity. For each half plane p x≶0we use the parametrization p2x2m+∆= cosθ;c y p y= sinθ;ηp=sign(p x)=±(6) whereθ∈[−θ0,θ0]is a coordinate along the constant energy contour.Its range depends on the topology of the constant energy contour,and thus on the energy , see Fig.3.Specifying the discussion to the Fermi surface associated with the Fermi energy F,we can distinguish two cases:(i)Low energy metal with two disconnected Fermi sur-faces when∆<0and F<−∆.In this caseθ0=π.This corresponds to the energy spectrum(a)of Fig.2.(ii)High energy metal with a single connected Fermi surface for F>|∆|.In this case cosθ0=∆/ F.For∆<0,θ0varies fromπfor F=−∆toπ/2for F −∆.For∆>0,θ0varies fromπ/2forF ∆to0for F→∆.This corresponds to theenergy spectra(b),M,(c)of Fig.2.The eigenstates of positive energy corresponds to wave3✓0✓0✓0=⇡(a )(b )S k xk y <0✓0✓0k xk y M(c )x y (b)and the saddle-point S defined on Fig.2.The arrow field describes the phase θparametrizing in a unambiguous way each half k x >0and k x <0of the energy contour according to Eq.(6).It also describes the relative phase between the two components of the eigenstate (7)of momentum k and energy .Right :Same quantities at the merging point ∆=0(M)and for (k x ,k y )>∆>0(c ).functions conveniently expressed with the parametriza-tion of the constant energy contourψ k( r )=1√2L1e iθke ik. r ,(7)where θ k is defined by inversion of Eq.(6),and p= k .From now on,we will set L =1.The group velocity varies along the constant energy contour according to :v x (ηp ,θ)=ηp2mcos θ√,(8a)v y (ηp ,θ)=c y sin θ,(8b)where throughout this paper we use the reduced parame-ter δ=∆/ .The evolution of the velocity along constant energy contours is shown on Fig.4.Figure 4.Velocity v ( k )along constant energy contours (here ∆<0).B.Density of StatesWe define a directional density of states along the con-stant energy contour parametrized by θfrom the equalitydp x dp y(2π )2=dk x dk y(2π)2=ρ( ,θ)d dθ,(9)withρ( ,θ)=√2m (2π )2c y 12√cos θ−δ.(10)The density of states in then obtained by the integralρ( )=2θ0−θ0dθρ( ,θ)(11)where the extra factor 2accounts for the sign of p x .The integral givesρ( )=√2m(2π )c yI 1(δ),(12)with the functionI 1(δ)=θ0−θ0dθ√cos θ−δ,(13)where θ0=Arccos(δ)when |δ|<1and θ0=πotherwise.From Eqs.(10,12),we can rewrite ρ( ,θ)asρ( ,θ)=ρ( )2I 1(δ)√cos θ−δ.(14)4 C.Disorder Potential and Elastic Scattering TimeThe disorder part V of the Hamiltonian accounts forthe inhomogeneities in the system.This random poten-tial V( r)is assumed to describe a gaussian point-like un-correlated disorder,characterized by two cumulants=0,V( r)V( r )=γδ( r− r ).(15)where the overline denotes a statistical average over re-alizations of the random potential.The presence of thisrandom potential induces afinite lifetime for the eigen-states of momentum k of the pure model(4),called elas-tic scattering time,and obtained from the Fermi goldenrule:τe( k)=2πd2 k(2π)2δ(k− k )|A( k, k )|2,(16)where the scattering amplitude is defined byA( k, k )= ψ k|V|ψ k .(17) For uncorrelated point-like disorder,the angular depen-dence of this scattering amplitude originates from the eigenstates overlap and one has|A( k, k )|2=γ2 1+cos(θ k−θ k ) .(18) Definingτe( ,θ)=τe( k)where ,θand k are related through Eq.(6),we can express the elastic scattering time as an integralτe( ,θ)=2πγθ0−θ0dθ ρ( ,θ )[1+cos(θ−θ )].(19)Introducing the bare scattering timeτ0e( )=πγρ( ),(20)we can rewrite(19)in the formτe( ,θ)=τ0e( )1+r(δ)cosθ,(21)where the density of statesρ( )is given by(12).The denominator of this expression exactly accounts for the anisotropy of the scattering time.As a convenient parametrization of this property,we have introduced the anisotropy function r(δ)which will be used throughout this paper:r(δ)=J1(δ)/I1(δ),(22) with the function I1(δ)defined in(13)andJ1(δ)=θ0−θ0dθcosθ√cosθ−δ,(23)Figure5.Function r(δ)parametrizing the angular depen-dence of the elastic scattering timeτe plotted as a function ofδ=∆/ .It has the limits r(δ→−∞) 1/(4δ),r(−1)=−1,r(0)=2Γ(3/4)4/π2 0.456947,r(1)=1.In thisfigure,as in followingfigures,we systematically reserve the colors:blue for the Dirac phase(δ<0),black for the gapped phase(∆>0)and red for the semi-Dirac point.whereθ0=Arccos(δ)when|δ|<1andθ0=πotherwise.The function r(δ)is plotted in Fig.5.Deep in the Diracphase(∆ 0),at low energy( |∆|),one hasr(δ)→−14|δ|=−4|∆| 1,(24)so that the anisotropy can be neglected in Eq.(21)andwe recover a scattering time independent of the directionof propagation as standard for Dirac fermions.III.DIFFUSIVE REGIME FROM THEBOLTZMANN EQUATIONWe now consider the transport properties of the model(3)at afixed energy large enough so that the condi-tion kl e 1is fulfilled,l e being a typical elastic meanfree path.Therefore we will not consider the close vicin-ity of a Dirac point and the associated physics of mini-mal conductivity21,22.For a system of typical size muchlarger than this mean free path l e,this corresponds to theregime of classical diffusion.We describe this regimefirstwith a standard Boltzmann equation,before turning toa complementary but equivalent diagrammatic approachbased on Kubo formula for the conductivity.The use ofthese two approaches will reveal the physics hidden be-tween the technical specificities of the diffusive transportfor the model we consider.A.Boltzmann equationWe start from the Boltzmann equation23,24expressing the evolution of the distribution function f( k, r):d f dt +d rdt∇ r f+d kdt∇ k f=I[f],(25)where I[f]is the collision integral defined below.The po-sition r and momentum k parametrizing the distribution function f( k, r)are classical variables,whose time evolu-tions entering Eq.(25)are described by the semi-classical equations25,26d r dt = v( k)+d kdt×Fk(26)d kdt=−e E−ed rdt×B(27)with the group velocity v( k)= −1∂ ( k)/∂ k, B is a lo-cal magneticfield,and Fk =i∇k× ψ k|∇ kψ k is theBerry curvature.In the present case,we consider the re-sponse of the distribution function f( k, r)due to a uni-form weak electricfield E:we can neglect the gradient∇ r f in Eq.(25)and drop the spatial dependence of f. Due to the absence of magneticfield,we deduce fromEqs.(25,27)that a stationary out-of-equilibrium distri-bution f( k)satisfies the simpler equation−e E.∇ k f=I[f].(28)where f is now as function of k and the collision integral is expressed asI[f]=2πd2 k(2π)2δ(k− k )|A( k, k )|2f( k )−f( k).(29)By assuming the perturbation to be weak,we can expand the stationary out-of-equilibrium distribution f( k)around the equilibrium Fermi distribution f0( k)= n F(k)following the ansatz23,24f( k)=f0( k)+e∂n F∂Λ( k). E,(30)where the vector Λhas the dimension of a length,and its components correspond to transport lengths in the different spatial directions.They are related to trans-port times through the definitionΛα( k)=vα( k)τtrα( k). Eq.(30)can be rewritten as a shift of energies by the field:f( k)=n Fk+e Λ( k). E.In the case of an isotropic Fermi surface,we do not expect this shift to depend on the direction of application of thefield E: in that case a unique transport timeτtr is necessary to describe the stationary distribution24.Here,for an anisotropic Fermi surface such as(5),we generically ex-pect the response of the distribution function to depend on the direction of the electricfield E27–29.For an elec-tricfield applied in the x or y direction,this leads to the definition of different anisotropic transport timesτtr x,τtr y. From Eqs.(28,29,30),one obtainsv( k)=1∂∂ k=k=2πd2 k(2π)2δ(k− k )|A( k, k )|2Λ( k)− Λ( k ),(31)By using the parametrization(6)on the contour of constant energy ,each componentαof the velocity obeys the equation(to lighten notation,we omit the energy in the argument of the quantities in the next expressions):vα(ηp,θ)=Λα(ηp,θ)τe(θ)−πγη p=±θ0−θ0dθ ρ(θ )[1+cos(θ−θ )]Λα(η p,θ ).(32)The transport timesτtrα( ,θ)are defined asΛα( ,ηp,θ)=vα( ,ηp,θ)τtrα( ,θ).(33) We now assume the following ansatz,namely that the transport times and the elastic scattering time have the same angular dependence:τtrα( ,θ)=λα( )τe( ,θ),(34) so that the parametersλα( )are obtained from the self-consistent equation(atfixed energy )vα(ηp,θ)=λαvα(ηp,θ)−πγλαη p=±θ0−θ0dθ ρ(θ )[1+cos(θ−θ )]vα(η p,θ )τe(θ )(35)where vα(ηp,θ)is defined in Eq.(8).Then from Eq.(14)and(21),wefinally getvα(ηp,θ)=λαvα(ηp,θ)−λα1η p=±θ0−θ0dθ1+cos(θ−θ)1+r(δ)cosθvα(η p,θ )√cosθ −δ.(36)We now consider the two directionsα=x,y separately. Along the x direction,since the velocity is an odd func-tion of k x,the sum overηp in Eq.(36)vanishes:we ob-tainλx( )=1,i.e.the transport time is equal to the scattering time30τtr x( ,θ)=τe( ,θ).Along the y direction,where v y(θ)=c y sinθindepen-dent ofηp,Eq.(36)possesses a self-consistent solution, and we obtainλy(δ)=11−I2(δ)/I1(δ)(37)whereI2(δ)= θ0−θ0dθsin2θ√cosθ−δ(1+r(δ)cosθ).(38)The function I1(δ)is defined in(13).Note that the ex-pression(37)of the renormalization factor of the trans-port timeλy(δ)reflects the iterative structure of the ver-tex correction to the bare conductivity that will be ob-tained within a diagrammatic treatment in section IV B (see Eqs.66,69).The dependenceλy(δ)is plotted in Fig.6.Figure6.Dependence onδ=∆/ of the renormalization factor of the transport timeτtr y with respect to the elastic scattering time:λy(δ)=τtr y( k)/τe( k).Having obtained the transport times along the x and y directions,we now turn to the calculation of the con-ductivities.B.ConductivityWe can express the current density j occurring in re-sponse to the application of the electricfield E asj=d2 k(2π)2f( k)−n F(k)(−e v( k)).(39)By using∂n F/∂ −δ( − F)and the ansatz(30)for the distribution function f( k)we obtainj=e2ηp=±θ0−θ0dθρ( F,θ) v( F,ηp,θ)Λ(F,ηp,θ). E.(40)The symmetries of this equation imply that off-diagonal terms of the conductivity tensor vanish(σα,β=α=0) while the diagonal terms can be written asσαα=2e2θ0−θ0dθρ( F,θ)vα( F,θ)Λα( F,θ).(41)where the factor2originates from the two possible signs ofηp=±.We end up with the Einstein relationσαα=e2ρ( F)Dα,(42) with the diffusion coefficientsDα=2λα( F)θ0−θ0dθρ( F,θ)ρ( F)v2α(θ)τe(θ)(43a) =v2α( F,θ)τtrα( F,θ)θ(43b) =λα( F)v2α( F,θ)τe( F,θ)θ.(43c) where we have defined the average along the constant en-ergy contour ···θ=2θ0−θ0dθ···ρ( F,θ)/ρ( F).This corresponds to the result announced in the introduction: the diffusion coefficients Dαare obtained by an average over the Fermi surface of v2ατtrαinstead of v2ατe.With our solution of the Boltzmann equation,this difference is accounted for by a renormalization factorλα( F)of the diffusion coefficients,which does not depend on the di-rection along the Fermi surface but on the directionαof application of the electricfield.We now specify explicitly the conductivities along the two directions x and y. Along the x direction,there is no renormalization of the transport time(λx=1,τtr x=τe)and the conductivity7σxx readsσxx=2e2θ0−θ0dθρ( F,θ)v2x(θ)τe(θ)=e2πγ2mI3(∆/ )I1(∆/ ),(44)where we defineI3(δ)= θ0−θ0dθcos2θ√cosθ−δ,(45)and the function I1(δ)is given in(13).For the conductivity along the y direction,the renor-malization of the transport time is given by(37)and we obtainσyy=2e2λy(∆/ )θ0−θ0dθρ( F,θ)v2y(θ)τe(θ)=e2πγc2y I2(∆/ )I1(∆/ )−I2(∆/ ),(46)where the functions I1(δ)and I2(δ)are respectively given by Eqs.(13)and(38).Eqs.(44,46)constitute the main results of this work.We discuss them in section V.In the next section,we use a diagrammatic approach which pro-poses a complementary description of the anisotropy of transport and allows to confirm the ansatz made to solve the Boltzman equation and recover exactly the results of Eqs.(44,46).IV.DIAGRAMMATIC APPROACHAn alternative approach to describe the diffusive trans-port of electron consists in a perturbative expansion in disorder of the conductivity tensor using a diagrammatic technique31.Beyond confirming the ansatz made to solve the Boltzmann equation described above,this method al-lows for an instructive alternative treatment of the dif-ferent transport anisotropies.In the diagrammatic ap-proach,the transport coefficients of the model are ob-tained from the Kubo formula.A perturbative expansion is then used to express the transport coefficients using the average single particle Green’s function.In this for-malism,the anisotropy of scattering and transport times are cast into a unusual matrix form for the self-energy operatorΣ.Beyond the present model,such a technique allows to describe anisotropy of diffusion of Dirac fermion models due e.g.to the warping of the Fermi surface in topological insulators32or anisotropic impurity scatter-ing,the study of which goes beyond the scope of the present paper.Nevertheless our work provides a phys-ical understanding of the technicalities naturally occur-ring in these other problems.In the next subsections, wefirst discuss the self-energy and the single particle Green’s function.We then turn to the calculation of the conductivity.A.Green’s functions and self-energyThe retarded and advanced Green’s functions are de-fined by:G R/A( k, k , F)=( F∓i0)I−H0( k)δ( k− k )−V( k, k )I−1(47)In the case of the model without disorder defined by Eq.(4),the Green’s function is expressed as a2×2ma-trix:G0( k, )=I−H0( k)−1=I+2k2x2m+∆σx+c y k yσy2−2k2x2m+∆2−c2y 2k2y,(48)where I is the identity matrix.Disorder is perturbatively incorporated in the averaged Green’s¯G function through a self-energy matrixΣ( k, )such thatG R/A( k, )=( ∓i0)I−H0( k)∓i ImΣ( k, )−1.(49) The real part of the self-energy has been neglected.The elastic scattering rates will be defined below from the imaginary part of the self-energy.To lowest order in the disorder strengthγ,this self-energy,solution of a Dyson equation,readsΣ( k, )=d k(2π)2V( k )V(− k )G0( k− k , ).(50) Its imaginary part is then obtained as−ImΣ( )=πγθ0−θ0dθρ( ,θ)[I−cosθσx](51)=2τ0e( )[I+r(δ)σx].(52)The densities of statesρ( ,θ)andρ( ),the bare scat-tering timeτ0e( )and the anisotropy factor r(δ)have been defined in Eqs.(10,12,20,22).It is worth noting that this self-energy acquires an unusual matrix struc-ture in pseudo-spin space:this manifests within the dia-grammatic approach the anisotropy of the scattering time τe( ,θ),which was described in Eq.(21)previously.In-deed,in the Green function formalism,the direction of propagation of eigenstates of the Hamiltonian(4)is en-coded into their spinor structure(the relative phase be-tween their components,see Eq.(7)).Hence the scatter-ing time in the corresponding direction will be obtained as the matrix element of the above self-energy in the as-sociated spinor eigenstate.8GRab( k )abGAcd( k )dcV2( k )=γFigure 7.Conventions for the diagrammatic representation of perturbation theory of transport.G R)Figure 8.Diagrammatic representation of the classical con-ductivity with the conventions of Fig.7.The renormalized current operator is defined in Fig.9.B.Conductivity 1.Kubo formulaThe longitudinal conductivity can be deduced from the Kubo formula (α=x,y ):σαα=2πL Tr j α( k )G R ( k, k , F )j α( k )G A ( k , k, F ) ,(53)where Tr corresponds to a trace over the pseudo-spin and momentum quantum numbers :Tr =tr k L 2tr d k/(2π)2and tr is a trace over the pseudo-spin indices only.For clarity,throughout this section on trans-port coefficients,we will omit the dependence on the Fermi energy F of various quantities.The current den-sity operators are also operators acting on both spin and momentum spaces.They are deduced from the Hamilto-nian (4)as:j x ( k )=−ek x σx ;j y (k )=−ec y σy .(54)Note that j x is linear in momentum while j y depends only on spin quantum numbers.Perturbation in the dis-order amplitude of the conductivity (53)is obtained by expanding the Green’s function in the disorder potential V before averaging over the gaussian distribution.In the classical diffusive limit,the dominant terms which deter-mine the averaged classical conductivity are represented diagrammatically on Fig.8and lead toσαα= 2πL 2Tr J αG R j αG A (55)where J αis the renormalized current density operator.The discrepancy between J αand the bare current opera-tor j αaccounts for the appearance of transport time τtrαin the Boltzmann approach 33due to the anisotropy ofscattering.This renormalized current operator is easier to define diagrammatically,as shown on Fig.9.2.Conductivity along xIn this direction,the current operator is linear in k x ,while the averaged Green’s functions G R ( k ),G A( k )are even functions of k x .Hence all the terms in the expres-sion of the renormalized current J x with at least a Green’s function vanish by k x →−k x symmetry,andJ x ( k )=j x ( k )=−emk x σx .(56)There is no renormalization of the current operator,inagreement with the result τtrx =τe from the Boltzmann equation approach.In the x direction,the expression (55)reduces toσxx = e m22πL 2Tr k 2x σx G R ( k )σx G A ( k ) .(57)Using L −2kρ( ,θ)d dθand the parametrization defined in Eq.(6)of the contours of constant energy we perform the integration over energy to obtainσxx =e 2τ0e F m+θ0−θ0dθρ( ,θ)(cos θ−δ)1+r (δ)cos θ×tr σx [I +cos θσx +sin θσy ]σx [I +cos θσx +sin θσy ] .Performing the spin trace first,we obtain σxx=4e 2τ0e F m+θ0−θ0dθρ( ,θ)(cos θ−δ)1+r (δ)cos θcos 2θ.(58)By using eq.(14)for the directional density of states we recover exactly the integral expression for the result (44)of Boltzmann approach:σxx=e 2 πγ2 m I 3(∆/ )I 1(∆/ ).(59)3.Renormalized current operator along yIn the y direction,the current operator is renormal-ized :the bare current operator j y is independent of the momentum k and the symmetry argument used for the x direction does not hold anymore.This renormalized current operator satisfies a Bethe-Salpeter equation rep-resented in Fig.9:J y =j y +J y Πγ(60)9=++++···ab=jα( k)a+G R( k )G A( k )γFigure9.Schematic representation of renormalized current operator[Jα]ab( k)as the infinite sum of vertex corrections to the bare current operator(top),and corresponding recursive equation satisfied by Jα(bottom).where tensor product in spin space are assumed andΠ( ,∆)=d k(2π)2G R( k, )⊗G A( k, )T.(61)Due to the spinorial structure of the wave functions,this propagator is here an operator acting as the ten-sor product of two spin12spaces.The notation···Tcorresponds to a transposition of spin ingthe parametrization defined in Eq.(6)of the contours ofconstant energy we perform the integration over energyto obtain forΠ( ,∆)≡Π(δ=∆/ ):Π(δ)=πτ0e+θ0−θ0dθρ( ,θ)×[I+cosθσx+sinθσy]⊗[I+cosθσx−sinθσy].(62)The expression(14)for the directional density of statesallows to rewrite it asΠ(δ)=12γI1(δ)I1(δ)I⊗I+(I1(δ)−I2(δ))σx⊗σx−I2(δ)σy⊗σy+J1(δ)(I⊗σx+σx⊗I),(63)where we introduced the functions:I1(δ)=θ0−θ0dθ1√cosθ−δ(1+r(δ)cosθ),(64)J1(δ)=θ0−θ0dθcosθ√cosθ−δ(1+r(δ)cosθ),(65)whereas I1and I2are defined in Eqs.(13,38).The inversion of the Bethe-Salpeter equation(60)isdone in the appendix A and wefindJ y=j y(I⊗I−γΠ(δ))−1=1−I2(δ)I1(δ)−1j y.(66)4.Conductivity along yFollowing the formula(55),the average conductivityalong y is expressed asσyy=2πtr[J y.Π(δ).j y].(67)From the eq.(60),we express J y.Π(δ)=γ−1(J y−j y)toobtain from(67):σyy=2πγtr[(J y−j y)j y].(68)The expression for the renormalized current operator(66)leads to thefinal resultσyy=e2πγc2y I2(δ)I1(δ)−I2(δ),(69)which is precisely the result(46)obtained within theBoltzmann equation approach.This concludes the derivation of the conductivity ten-sor within the diagrammatic approach.In doing so,wehave identified the encoding of the anisotropic scatteringrates through the matrix self-energy(52),while the cor-responding transport times are hidden into the renormal-ization of vertex operators(56,66).Comparison with theBoltzmann approach allows to unveil the physical mean-ing of these technical structures,which we believe to beapplicable to other situations of anisotropic transport ofDirac-like states.V.RESULTS AND DISCUSSIONWe now turn to a discussion of our results for varioussituations corresponding to energy spectra represented inFig.2.。
2017 ap calculusab 微积分
2017 ap calculusab 微积分英文版2017 AP Calculus AB: A Journey Through MicrocalculusAs the sun rose over the horizon, students across the globe began their journey into the world of mathematics with the 2017 AP Calculus AB exam. This exam, known for its depth and breadth, tests the student's understanding of the fundamental concepts of calculus.The exam began with a gentle reminder of the basic derivative rules, followed by questions that required students to apply these rules to real-world scenarios. One such question dealt with the optimization of a profit function, testing the student's ability to identify the maximum or minimum value of a function. This question highlighted the practical applications of calculus in real-life situations.As the exam progressed, the questions became more complex, delving into the realm of integration. Students werechallenged to evaluate integrals using various techniques, such as substitution and integration by parts. One notable question dealt with the concept of areas between curves, requiring students to apply their knowledge of integration to find the enclosed area.The exam also included questions on sequences and series, testing the student's understanding of convergence and divergence. Questions on infinite series were particularly challenging, as they required students to analyze the behavior of the series as it approached infinity.Towards the end of the exam, students were presented with a challenging question on differential equations. This question tested their ability to understand and manipulate differential equations, ultimately finding a solution that satisfied the given conditions.The 2017 AP Calculus AB exam was not just a test of mathematical knowledge; it was a testament to the students' dedication, perseverance, and understanding of the beauty andpower of calculus. As the sun set, students closed their books, satisfied that they had done their best, and hoped that their efforts would be rewarded with a smile on the faces of their teachers and parents.中文版2017 AP微积分AB：微积分之旅随着太阳从地平线上升起，全球的学生们开始了他们的数学之旅，参加了2017年的AP微积分AB考试。
城市轨道交通规划选线重点思路与方法彭伟
城市轨道交通规划选线重点思路与方法彭伟发布时间：2021-08-24T09:13:50.510Z 来源：《防护工程》2021年15期作者：彭伟[导读] 根据轨道交通规划选线的复杂性，在充分理解规划选线目的和意义的前提下，从规划、建设管理、选线人员素质方面的问题及分析条件下，提出了以线路功能为指引，充分认识规划选线基础资料的重要性，并处理好规划选线过程中与行车组织、车站、区间、信号与轨道等专业重点接口，才能做好以规划为引领的选线设计，提出了前期工作对后续工程建设的重要性。
彭伟昆明轨道交通集团有限公司云南昆明 650000摘要：根据轨道交通规划选线的复杂性，在充分理解规划选线目的和意义的前提下，从规划、建设管理、选线人员素质方面的问题及分析条件下，提出了以线路功能为指引，充分认识规划选线基础资料的重要性，并处理好规划选线过程中与行车组织、车站、区间、信号与轨道等专业重点接口，才能做好以规划为引领的选线设计，提出了前期工作对后续工程建设的重要性。
关键词：规划选线；基础资料；功能定位；接口处理【Abstract】According to the complexity of rail transit planning and route selection，on the premise of fully understanding the purpose and significance of planning and route selection，and from the aspects of planning，construction management，quality of route selection personnel and analysis conditions，this paper puts forward that taking the line function as the guide，fully understanding the importance of basic data of planning and route selection，and dealing with the problems in the process of planning and route selection，such as train operation organization，station，interval，traffic control，etc In order to do a good job of route selection design guided by planning，the importance of preliminary work for subsequent engineering construction is proposed.[Key words]Planning and route selection，Basic data，Function orientation，Interface processing序言城市轨道交通规划选线是一项“牵扯面广、考虑因素多、劳动强度大、责任重大、总体性强”的系统工程，应遵循“符合规划、兼顾工程、节能环保、有利运营”的原则。
GammaFunction：伽玛函数
Search Site12,720 entries Calculus and Analysis > Special Functions > Gamma Functions Calculus and Analysis > Special Functions > Named Integrals Calculus and Analysis > Special Functions > Product Functions Gamma FunctionThe (complete) gamma function is defined to be an extension of the factorial to arguments. It is related to the factorial bya slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8). everywhere except at, -1, -2, ..., and the residue atisThere are no points at which.There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write . for(Euler's integral form)The complete gamma function can be generalized to the upper and lower incomplete gamma function .Min MaxRe Register for Unlimited Interactive Examples >>Im ReplotPlots of the real and imaginary parts of in the complex plane are illustrated above.Wolfram Research Demonstrations Site Integrator Tones Functions Site Wolfram Science more…Download Complete Documentation >>Find 171 formulas about the Gamma Function at -55-55(8)(9)If is an integer, 2, 3, ..., then(10)(11)so the gamma function reduces to the factorial for a positive integer argument.A beautiful relationship betweenand the Riemann zeta functionis given by(12)for(Havil 2003, p. 60).The gamma function can also be defined by an infinite product form (Weierstrass form )(13)where is the Euler-Mascheroni constant (Krantz 1999, p. 157; Havil 2003, p. 57). This can be written(14)where(15)(16)for, where is the Riemann zeta function (Finch 2003). Taking the logarithm of both sidesof (◇),(17)Differentiating,(18)(19)(20)(21)(22)(23)(24)(25)where is the digamma function and is the polygamma function . th derivatives are given in terms of the polygamma functions , , ..., .The minimum valueoffor real positiveis achieved when(26)(27)This can be solved numerically to give (Sloane's A030169; Wrench 1968), which has continued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (Sloane's A030170). At ,achieves the value 0.8856031944... (Sloane's A030171), which has continued fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (Sloane's A030172).The Euler limit form is(28)so(29)(30)(31)(32)(Krantz 1999, p. 156).One over the gamma functionis an entire function and can be expressed as(33)where is the Euler-Mascheroni constant and is the Riemann zeta function (Wrench 1968). An asymptotic series for is given by(34) Writing(35) the satisfy(36) (Bourguet 1883, Davis 1933, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically computed the coefficients for the series expansion about 0 of(37) The Lanczos approximation gives a series expansion for for in terms of an arbitrary constant such that .The gamma function satisfies the functional equations(38)(39) Additional identities are(40)(41)(42)(43)Using (40), the gamma function of a rational number can be reduced to a constant times or . For example,(44)(45)(46)(47) For ,(48) Gamma functions of argument can be expressed using the Legendre duplication formula(49) Gamma functions of argument can be expressed using a triplication formula(50) The general result is the Gauss multiplication formula(51) The gamma function is also related to the Riemann zeta function by(52) For integer , 2, ..., the first few values of are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (Sloane's A000142). For half-integer arguments, has the special form(53)where is a double factorial. The first few values for , 3, 5, ... are therefore(54)(55)(56), , ... (Sloane's A001147 and A000079; Wells 1986, p. 40). In general, for a positive integer, 2, ...(57)(58)(59)(60) Simple closed-form expressions of this type do not appear to exist for for a positive integer . However, Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots and elliptic integral singular values, i.e., elliptic moduli such that(61)where is a complete elliptic integral of the first kind and is the complementary integral. M. Trott (pers. comm.) has developed an algorithm for automatically generating hundreds of such identities.(62)(63)(64)(65)(66)(67)(68)(69)(70)(71)(72)(73)(74)(75)(76)(77)(78)(79)(80)(81)(82)Several of these are also given in Campbell (1966, p. 31).A few curious identities include(83)(84)(85)(86)(87)(88)(89) of which Magnus and Oberhettinger 1949, p. 1 give only the last case,(90) and(91)(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:(92)(93) where(94)(95)(Berndt 1994).Ramanujan gave the infinite sums(96)(97) and(98)(99)(Hardy 1923; Hardy 1924; Whipple 1926; Watson 1931; Bailey 1935; Hardy 1999, p. 7).The following asymptotic series is occasionally useful in probability theory (e.g., theone-dimensional random walk):(100) (Graham et al. 1994). This series also gives a nice asymptotic generalization of Stirling numbers of the first kind to fractional values.It has long been known that is transcendental (Davis 1959), as is (Le Lionnais 1983; Borwein and Bailey 2003, p. 138), and Chudnovsky has apparently recently proved thatis itself transcendental (Borwein and Bailey 2003, p. 138).There exist efficient iterative algorithms for for all integers (Borwein and Bailey 2003, p. 137). For example, a quadratically converging iteration for (Sloane'sA068466) is given by defining(101)(102) setting and , and then(103)(Borwein and Bailey 2003, pp. 137-138).No such iteration is known for (Borwein and Borwein 1987; Borwein and Zucker 1992; Borwein and Bailey 2003, p. 138).SEE ALSO:Bailey's Theorem, Barnes G-Function, Binet's Fibonacci Number Formula, Bohr-Mollerup Theorem, Digamma Function, Double Gamma Function, Fransén-Robinson Constant Gauss Multiplication Formula, Incomplete Gamma Function, Knar's Formula, Lambda Function, LanczosApproximation, Legendre Duplication Formula, Log Gamma Function, Mellin's Formula, Mu Function, Nu Function, Pearson's Function, Polygamma Function, Regularized Gamma Function, Stirling's Series, Superfactorial. [Pages Linking Here]RELATED WOLFRAM SITES:/GammaBetaErf/Gamma/,/GammaBetaErf/LogGamma/REFERENCES:Abramowitz, M. and Stegun, I. A. (Eds.). "Gamma (Factorial) Function" and "Incomplete Gamma Function." §6.1 and 6.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 255-258 and 260-263, 1972.Arfken, G. "The Gamma Function (Factorial Function)." Ch. 10 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 339-341 and 539-572, 1985.Artin, E. The Gamma Function. New York: Holt, Rinehart, and Winston, 1964.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 334-342, 1994.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 6, 1987.Borwein, J. M. and Zucker, I. J. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind." IMA J. Numerical Analysis12, 519-526, 1992.Bourguet, L. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes." Acta Math.2, 261-295, 1883. Campbell, R. Les intégrales eulériennes et leurs applications. Paris: Dunod, 1966.Davis, H. T. Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933.Davis, P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly66, 849-869, 1959.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Gamma Function." Ch. 1 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 1-55, 1981.Finch, S. R. "Euler-Mascheroni Constant." §1.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 28-40, 2003.Gauss, C. F. "Disquisitiones Generales Circa Seriem Infinitametc. Pars Prior." Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1812. Reprinted in Gesammelte Werke, Bd. 3, pp. 123-163 and 207-229, 1866.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer to Problem 9.60 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc.21, 492-503, 1923.Hardy, G. H. "Some Formulae of Ramanujan." Proc. London Math. Soc. (Records of Proceedings at Meetings) 22,xii-xiii, 1924.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. "The Gamma Function." Ch. 6 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 53-60, 2003.Isaacson, E. and Salzer, H. E. "Mathematical Tables--Errata: 19. J. P. L. Bourget, 'Sur les intégrales Eulériennes et quelques autres fonctions uniformes,' Acta Mathematica, v. 2, 1883, pp. 261-295.' " Math. Tab. Aids Comput.1, 124, 1943.Koepf, W. "The Gamma Function." Ch. 1 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 4-10, 1998.Krantz, S. G. "The Gamma and Beta Functions." §13.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 155-158, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983.Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949.Nielsen, N. "Handbuch der Theorie der Gammafunktion." Part I in Die Gammafunktion. New York: Chelsea, 1965. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients" and "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.1 and 6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209 and 209-214, 1992.Sloane, N. J. A. Sequences A000079/M1129, A000142/M1675, A001147/M3002, A030169, A030170, A030171,A030172, and A068466 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Gamma Function " and "The Incomplete Gamma and Related Functions." Chs. 43 and 45 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 411-421 and 435-443, 1987. Watson, G. N. "Theorems Stated by Ramanujan (XI)." J. London Math. Soc.6, 59-65, 1931.Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 210, 266-276, 1939.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986. Whipple, F. J. W. "A Fundamental Relation between Generalised Hypergeometric Series." J. London Math. Soc.1,138-145, 1926.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput.22, 617-626, 1968.LAST MODIFIED:December 26, 2005CITE THIS AS:Weisstein, Eric W. "Gamma Function." From MathWorld--A Wolfram Web Resource./GammaFunction.html© 1999 CRC Press LLC, © 1999-2007 Wolfram Research, Inc. | Terms of Use。
MTI-开题答辩PPT
Content
➢ Project Introduction ➢ Topic Basis ➢ Research Significance
01 translation project About 13,000 words Academic research No Chinese version
Totally, the translation project is aimed at bringing the concept of ACEA to Chinese readers, especially the stakeholders of road transport. Above all, since the practitioner in the automotive industry are eager to learn more about the regulations of passenger car emission in Europe, the report can provide some useful information for Chinese readers who want to learn the experience of CO2 reduction and how it paved the way for Chinese counterparts to step up their progress to achieve carbon peak and realize green transportation. Moreover, since CO2 challenge is intensifying day by day especially in transport field in China, the translation of Joining Forces to Tackle the Road Transport CO2 Challenge-A multi-stakeholder initiative is also aimed to popularize some general knowledge of this field, and the proposal of the ACEA. At last, it is also helpful in conducting further research on how to reduce CO2 emission. This translation may also help to solve some practical issues in the English sci-tech text translation. This translation project also aims to summarize the experience of translation, and propose some feasible methods of translating English report, especially how to cope with English long and complicated sentences.
Balanced vehicular traffic at a bottleneck
2.1 Background
Piccoli and Garavello [2] appear to have been the first to propose an intersection modeling by using the Aw-Rascle “second order” model of traffic flow [1]. In their approach, only the mass flux is conserved but not the pseudomomentum. In [8], Herty and Rascle proposed another approach in which mass flux and pseudo-momentum are both conserved. But they maximized the mass fluxes at the intersection with some arbitrary given homogenization coefficients. In [7], the latter approach has been generalized by maximizing the total mass flux at the junction without fixing any condition. In fact, the homogenization coefficients are not arbitrary but obtained directly from the mass flux maximization. Another approach also based on the mass flux maximization is given in [6]. Our approach in the current paper is similar to the latter one, in particular we maximize the total flux at the junction, at the same time conserving the pseudo-momentum of the original “Aw-Rascle” system. Here, we are particularly concerned with the BVT model [10]. We note that our treatment of the homogenization problem which naturally arises in a merge junction is different from the one in [8],[7], see Remark 1.
高数双语课件section9.2
Example
Show that
f
(
x,
y
)
x
2
xy
y2
,
x2 y2 0
0,
x2 y2 0
is not continuous at the origin.
Proof If we take y kx, then
lim
x0
xy x2 y2
lim
x0
x2
kx 2 k2x2
y0
y kx
is a product of
u x2 y2
1 and v x2 y2 .
Since u x2 y2
1
is a continuous function,
v x2 y2
is a continuous function on the plane R2 expect at the point (0, 0).
in the domain of f,
| f ( x, y) A | , holds for all 0 ( x x0)2 ( y y0)2 .
We write
lim f ( x, y) A,
( x, y)( x0 , y0 )
and assume ( x0, y0 ) is an accumulation point of the domain of f.
lim f ( x, y) A,
( x, y)( x0 , y0 )
and we say, “The limit of f as ( x, y) approaches ( x0 , y0 ) equals A.”
This is like the limit of a function of one variable, except that two independent variables are involved instead of one, complicating the issue of “closeness.”
the road to learn react pdf 2023 中文
the road to learn react pdf 2023 中文1. 引言1.1 概述本文旨在介绍《The Road to Learn React PDF 2023》一书，并对React技术进行概述。
React作为一种流行的JavaScript库，被广泛应用于前端开发中，拥有很多优势和应用场景。
通过本文的阅读，读者将了解这本书的内容概览以及学习React的关键要点。
1.2 文章结构文章共分为五个部分：引言、React概述、《The Road to Learn React PDF 2023》内容概览、学习React的关键要点以及结论与展望。
每个部分都包含具体的子节标题，方便读者查找所需信息。
1.3 目的本文主要目的是向读者介绍React技术并推荐《The Road to Learn React PDF 2023》这本书。
通过阅读该书，读者可以系统地学习和掌握React开发所需的知识和技巧。
同时，在文章中还会提供一些React的相关资源和未来发展方向的展望，帮助读者更好地理解这门技术并在实践中取得成果。
以上是“1. 引言”部分内容。
2. React概述2.1 React简介React是一个用于构建用户界面的JavaScript库。
它由Facebook开发并于2013年首次发布，现已成为使用最广泛的前端开发技术之一。
与传统的基于模板的开发方式不同，React采用了组件化思想，通过将UI拆分为独立可复用的组件来简化开发流程。
2.2 React的优势React具有以下几个主要优势：- 高效：React通过使用虚拟DOM（Virtual DOM）实现了高性能的渲染机制。
它只更新需要改变的部分，而不是整个页面，从而提升了应用程序的性能。
- 组件化：React支持将UI划分为多个独立、可复用的组件。
这种组件化思想使得代码更加模块化和易于维护。
- 单向数据流：在React中，数据是单向流动的。
父组件可以传递数据给子组件，但子组件无法直接修改父组件中的数据。
外文文献中国作者格式
外文文献中国作者格式一、参考文献的.类型参考文献(即引文出处)的类型以单字母方式标识,具体如下:M——专著C——论文集N——报纸文章J——期刊文章 D——学位论文R ——报告对于不属于上述的文献类型,采用字母“Z”标识。
对于英文参考文献,还应注意以下两点:①作者姓名采用“姓在前名在后”原则,具体格式是: 姓,名字的首字母. 如: Malcolm Richard Cowley 应为:Cowley, M.R.,如果有两位作者,第一位作者方式不变,&之后第二位作者名字的首字母放在前面,姓放在后面,如:Frank Norris 与Irving Gordon应为:Norris, F. & I.Gordon.;②书名、报刊名使用斜体字,如:Mastering English Literature,English Weekly。
二、参考文献的格式及举例1.期刊类【格式】[序号]作者.篇名[J].刊名,出版年份,卷号(期号):起止页码.【举例】[1] 王海粟.浅议会计信息披露模式[J].财政研究,2004,21(1):56-58.[2] 夏鲁惠.高等学校毕业论文教学情况调研报告[J].高等理科教育,2004(1):46-52.[3] Heider, E.R.&D.C.Oliver. The structure of color space in naming and memory of two languages [J]. Foreign Language Teaching and Research, 1999, (3): 62 – 67.2.2.专著类【格式】[序号]作者.书名[M].出版地:出版社,出版年份:起止页码.【举例】[4]葛家澍,林志军.现代西方财务会计理论[M].厦门:厦门大学出版社,2001:42.[5] Gill, R. Mastering English Literature [M]. London: Macmillan, 1985: 42-45.3.报纸类【格式】[序号]作者.篇名[N].报纸名,出版日期(版次).【举例】[6]李大伦.经济全球化的重要性[N].光明日报,1998-12-27(3).[7] French, W. Between Silences: A Voice from China[N]. Atlantic Weekly, 1987-8-15(33).4.论文集【格式】[序号]作者.篇名[C].出版地:出版者,出版年份:起始页码.【举例】[8] 伍蠡甫.西方文论选[C]. 上海:上海译文出版社,1979:12-17.[9] Spivak,G. “Can the Subaltern Speak?”[A]. In C.Nelson & L. Grossberg(eds.). Victory in Limbo: Imigism [C]. Urbana: University of Illinois Press, 1988, pp.271-313.[10] Almarza, G.G. Student foreign language teacher’s knowledge growth [A]. In D.Freeman and J.C.Richards (eds.). Teacher Learning in Language Teaching [C]. New York: Cambridge University Press. 1996. pp.50-78.5.学位论文【格式】[序号]作者.篇名[D].出版地:保存者,出版年份:起始页码.【举例】[11]张筑生.微分半动力系统的不变集[D].北京:北京大学数学系数学研究所, 1983:1-7.6.研究报告【格式】[序号]作者.篇名[R].出版地:出版者,出版年份:起始页码.【举例】[12] 冯西桥.核反应堆压力管道与压力容器的LBB分析[R].北京:清华大学核能技术设计研究院, 1997:9-10.7.条例【格式】[序号]颁布单位.条例名称.发布日期【举例】[15] 中华人民共和国科学技术委员会.科学技术期刊管理办法[Z].1991—06—058.译著【格式】[序号]原著作者. 书名[M].译者,译.出版地:出版社,出版年份:起止页码.三、注释注释是对论文正文中某一特定内容的进一步解释或补充说明。
timescales
The u-invariant of the function fields of p-adic curves
a rXiv:078.3128v1[mat h.N T]23Aug27The u -invariant of the function fields of p -adic curves R.Parimala and V.Suresh Abstract The u -invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field.It is an open question whether the u -invariant of function fields of p -aidc curves is 8.In this paper,we answer this question in the affirmative for function fields of non-dyadic p -adic curves.Introduction It is an open question ([L],Q.6.7,Chap XIII)whether every quadratic form in at least nine variables over the function fields of p -adic curves has a non-trivial zero.Equivalently,one may ask whether the u -invariant of such a field is 8.The u -invariant of a field F is defined as the maximal dimension of anisotropic quadratic forms over F .In this paper we answer this question in the affirmative if the p -adic field is non-dyadic.In ([PS],4.5),we showed that every quadratic form in eleven variables over the function field of a p -adic curve,p =2,has a nontrivial zero.Themain ingredients in the proof were the following:Let K be the function field of a p -adic curve X and p =2.1.(Saltman ([S1],3.4))Every element in the Galois cohomology group H 2(K,Z /2Z )is a sum of at most two symbols.2.(Kato [K],5.2)The unramified cohomology group H 3nr(K/X ,Z /2Z (2))is zero for a regular projective model X of K .1If K is as above,we proved([PS],3.9)that every element in H3(K,Z/2Z)is a symbol of the form(f)·(g)·(h)for some f,g,h∈K∗and f may be chosen to be a value of a given binary form<a,b>over K.If further givenζ=(f)·(g)·(h)∈H3(K,Z/2Z)and a ternary form<c,d,e>,one canchoose g′,h′∈K∗such thatζ=(f)·(g′)·(h′)with g′a value of<c,d,e>, then,one is led to the conclusion that u(K)=8(cf4.3).We in fact provethat such a choice of g′,h′∈K∗is possible by proving the following local global principle:Theorem.Let K=k(X)be the functionfield of a curve X over a p-adic field k.Let l be a prime not equal to p.Assume that k contains a primitive l th root of unity.Givenζ∈H3(K,µ⊗2l)andα∈H2(K,µl)corresponding to a degree l central division algebra over K,satisfyingζ=α∪(h v)for some(h v)∈H1(K v,µl),for all discrete valuations of K,there exists(h)∈H1(K,µl)such thatζ=α∪(h).In fact one can restrict the hypothesis to discrete valuations of K centered on codimension one points of a regular model X,projective over the ring of integers O k of k.A key ingredient towards the proof of the theorem is a recent result of Saltman([S3])where the ramification pattern of prime degree central simple algebras over functionfields of p-adic curves is completely described.We thank J.-L.Colliot-Th´e l`e ne for helpful discussions during the prepa-ration of this paper and for his critical comments on the text.1.Some PreliminariesIn this section we recall a few basic facts from the algebraic theory of quadratic forms and Galois cohomology.We refer the reader to([C])and([Sc]).Let F be afield and l a prime not equal to the characteristic of F.Let µl be the group of l th roots of unity.For i≥1,letµ⊗i l be the Galois module given by the tensor product of i copies ofµl.For n≥0,let H n(F,µ⊗i l)be the n th Galois cohomology group with coefficients inµ⊗i l.We have the Kummer isomorphism F∗/F∗l≃H1(F,µl).For a∈F∗, its class in H1(F,µl)is denoted by(a).If a1,···,a n∈F∗,the cup prod-uct(a1)···(a n)∈H n(F,µ⊗n l)is called a symbol.We have an isomorphism H2(F,µl)with the l-torsion subgroup l Br(F)of the Brauer group of F.We2define the index of an elementα∈H2(F,µl)to be the index of the corre-sponding central simple algebra in l Br(F).Suppose F contains all the l th roots of unity.Wefix a generatorρfor the cyclic groupµl and identify the Galois modulesµ⊗i l with Z/l Z.Thisleads to an identification of H n(F,µ⊗ml )with H n(F,Z/l Z).The element inH n(F,Z/l Z)corresponding to the symbol(a1)···(a n)∈H n(F,µ⊗n l)through this identification is again denoted by(a1)···(a n).In particular for a,b∈F∗, (a)·(b)∈H2(K,Z/l Z)represents the cyclic algebra(a,b)defined by the relations x l=a,y l=b and xy=ρyx.Let v be a discrete valuation of F.The residuefield of v is denoted byκ(v).Suppose char(κ(v))=l.Then there is a residue homomorphism∂v:H n(F,µ⊗ml )→H n−1(κ(v),µ⊗(m−1)l).Letα∈H n(F,µ⊗ml).We saythatαis unramified at v if∂v(α)=0;otherwise it is said to be ramified at v.If F is complete with respect to v,we denote the kernel of∂v by H n nr(F,µ⊗ml).Supposeαis unramified at v.Letπ∈K∗be a parameter at vandζ=α∪(π)∈H n+1(F,µ⊗(m+1)l ).Letαis independent of the choice of the parameterπand is called the specialization ofαat v.We say thatαspecializes tou)·(we have ∂(ζ)=(v )for some units u,v ∈R .Since ∂is an isomorphism,we have ζ=(u )·(v )·(π).Thus every element in H 3(K,µ⊗3l )is a symbol.2Corollary 1.2Let k be a p -adic field and K the function field of an integral curve over k .Let l be a prime not equal to p .Let K v be the completion of K at a discrete valuation of K .Then H 3nr (K v ,µ⊗3l )=0.Suppose further that K contains a primitive l th root of unity.Then every element in H 3(K v ,µ⊗3l )is a symbol.Proof.Let v be a discrete valuation of K and K v the completion of K at v .The residue field κ(v )at v is either a p -adic field or a function field of a curve over a finite field of characteristic p .In either case,the cohomological dimension of κ(v )is 2and hence H n (κ(v ),µ⊗3l )=0for n ≥3.By (1.1),H 3nr (K v ,µ⊗3l )=0.If κ(v )is a local field,by local class field theory,every finite dimension central division algebra over κ(v )is split by an unramified (cyclic)exten-sion.If κ(v )is a function field of a curve over finite field,then by a classical theorem of Hasse-Brauer-Neother-Albert,every finite dimensional central di-vision algebra over κ(v )is split by a cyclic extension.Since κ(v )contains a primitive l th of unity,every element in H 2(κ(v ),Z /l Z )is a symbol.By (1.1),every element in H 3(K v ,Z /l Z )is a symbol.2Let X be a regular integral scheme of dimension d ,with field of fractions F .Let X 1be the set of points of X of codimension 1.A point x ∈X 1gives rise to a discrete valuation v x on F .The residue field of this discrete valuation ring is denoted by κ(x )or κ(v x ).The corresponding residue homomorphism is denoted by ∂x .We say that an element ζ∈H n (F,µ⊗m l )is unramified at x if ∂x (ζ)=0;otherwise it is said to be ramified at x .We define the ramification divisor ram X (ζ)= x as x runs over X 1where ζis ramified.The unramified cohomology on X ,denoted by H n nr (F/X ,µ⊗m l ),is defined as the intersection of kernels of the residue homomorphisms ∂x :H n (F,µ⊗m l )→H n −1(κ(x ),µ⊗(m −1)l ),x running over X 1.We say that ζ∈H n (F,µ⊗m l )is unramified on X if ζ∈H n nr (F/X ,µ⊗m l ).If X =Spec(R ),then we also say that ζis unramified on R if it is unramified on X .Suppose C is an irreducible subscheme of X of codimension 1.Then the generic point x of C belongs to X 1and we set ∂x =∂C .If α∈H n (F,µ⊗m l )is unramified at x ,then we say4thatαis unramified at C.Let k be a p-adicfield and K the functionfield of a smooth,projective, geometrically integral curve X over k.By the resolution of singularities for surfaces(cf.[Li1]and[Li2]),there exists a regular,projective model X of X over the ring of integers O k of k.We call such an X a regular projective model of K.Since the genericfibre X of X is geometrically integral,it follows that the specialfibrethe following graph:The set of vertices is the set of irreducible curves in ram X(α)and there is an edge between two vertices if there is a chilli point in the intersection of the two irreducible curves corresponding to the vertices.A loop in this graph is called a chilli loop.Proposition1.4([S3],2.6,2.9)There exists a projective model X of K such that there are no chilli loops and no cool points on X forα.Let F be afield of characteristic not equal to2.The u-invariant of F, denoted by u(F),is defined as follows:u(F)=sup{rk(q)|q an anisotropic quadratic form over F}.For a1,···,a n∈F∗,we denote the diagonal quadratic form a1X21+···+a n X2n by<a1,···,a n>.Let W(F)be the Witt ring of quadratic forms over F and I(F)be the ideal of W(F)consisting of even dimension forms.Let I n(F) be the n th power of the ideal I(F).For a1,···,a n∈F∗,let<<a1,···,a n>> denote the n-fold Pfister form<1,a1>⊗···⊗<1,a n>.The abelian group I n(F)is generated by n-fold Pfister forms.The dimension modulo2gives an isomorphism e0:W(F)/I(F)→H0(F,Z/2Z).The discriminant gives an isomorphism e1:I(F)/I2(F)→H1(F,Z/2Z).The classical result of Merkurjev([M]),asserts that the Clifford invariant gives an isomorphism e2:I2(F)/I3(F)→H2(F,Z/2Z).Let P n(F)be the set of isometry classes of n-fold Pfister forms over F. There is a well-defined map([A])e n:P n(F)→H n(F,Z/2Z)given by e n(<1,a1>⊗···<1,a n>)=(−a1)···(−a n)∈H n(F,Z/2Z).A quadratic form version of the Milnor conjecture asserts that e n induces a surjective homomorphism I n(F)→H n(F,Z/2Z)with kernel I n+1(F).This conjecture was proved by Voevodsky,Orlov and Vishik.In this paper we are interested infields of2-cohomological dimension at most3.For suchfields, the above Milnor’s conjecture is already proved by Arason,Elman and Jacob ([AEJ],Corollary4and Theorem2),using the theorem of Markurjev([M]).Let q1and q2be two quadratic forms over F.We write q1=q2if they represent the same class in the Witt group W(F).We write q1≃q2,if q1 and q2are isometric quadratic forms.We note that if the dimensions of q1 and q2are equal and q1=q2,then q1≃q2.62.Divisors on Arithmetic SurfacesIn this section we recall a few results from a paper of Saltman([S3])on divisors on arithmetic surfaces.Let Z be a connected,reduced scheme offinite type over a Noetherian ring.Let O∗Z be the sheaf of units in the structure sheaf O Z.Let P be afinite set of closed points of Z.For each P∈P,letκ(P)be the residue field at P andιP:Spec(κ(P))→Z be the natural morphism.Considerthe sheaf P∗=⊕P∈P ι∗Pκ(P)∗,whereκ(P)∗denotes the group of units inκ(P).Then there is a surjective morphism of sheaves O∗Z→P∗given bythe evaluation at each P∈P.Let O∗(1)Z,P be its kernel.When there is noambiguity we denote O∗(1)Z,P by O∗(1)P.Let K be the sheaf of total quotient rings on Z and K∗be the sheaf of groups given by units in K.Every element γ∈H0(Z,K∗/O∗)can be represented by a family{U i,f i},where U i are open sets in Z,f i∈K∗(U i)and f i f−1j∈O∗(U i∩U j).We say that an elementγ={U i,f i}of H0(Z,K∗/O∗)avoids P if each f i is a unit at P for all P∈U i∩P.Let H0P(Z,K∗/O∗)be the subgroup of H0(Z,K∗/O∗) consisting of thoseγwhich avoid P.Let K∗=H0(Z,K∗)and K∗P be the subgroup of K∗consisting of those functions which are units at all P∈P. We have a natural inclusion K∗P→H0P(Z,K∗/O∗)⊕(⊕P∈Pκ(P)∗).We have the followingProposition2.1([S3],1.6)Let Z be a connected,reduced scheme offinite type over a Noetherian ring.ThenH1(Z,O∗(1)P)≃H0P(Z,K∗/O∗)⊕(⊕P∈Pκ(P)∗)X be the reduced specialfibre ofη.Assume thatX is connected if the genericfibre is geometrically integral.Let P be a finite set of closed points in X.Since every closed point of X is inX.Let m be an integer coprime with p.7Proposition2.2([S3],1.7)The canonical map H1(X,O∗(1)X,P)→H1(X,P) induces an isomorphismH1(X,O∗(1)X,P)X,O∗(1).mH1(X,P)Let X be as above.Suppose thatX.Such a curve exists by([S3],1.1).We note that any closed point on X is a point of codimension2and there is a unique closed point on any geometric curve on X(cf.§1).The following is extracted from([S3],§5).Proposition2.3Let X,P,E,Q i,D i be as above.For each closed pointQ i,let m i be the intersection multiplicity of the support of E and the specialfibreγ∈P ic(X.In particular,we havethe m i’s are intersection multiplicities of E andX,K∗/O∗)is m i Q i,the image X,O∗P)is zero. By(2.2),we haveγ′∈lH1(X,O∗P).Using(2.1),there exists(E′,(λP))∈H0P(Z,K∗/O∗)⊕(⊕P∈Pκ(P)∗)such that(−E+ m i D i,1)=l(E′,(λP))= (lE′,(λl P))modulo K∗P.Thus there existsν∈K∗P⊂K∗such that(ν)= (−E+ m i D i,1)−(lE′,(λl P)).i.e.(ν)=−E+ m i D i−lE′andν(P)=λl P for each P∈P.23.A local-global principleLet k be a p-adicfield,O k be its ring of integers and K the functionfield of a smooth,projective,geometrically integral curve over k.Let l be a prime not equal to p.Throughout this section,except in3.6,we assume that k contains a primitive l th root of unity.Wefix a generatorρforµl and identify µl with Z/l Z.Lemma3.1Letα∈H2(K,µl).Let X be a regular projective model of K.Assume that the ramification locus ram X(α)is a union of regular curves {C1,···,C r}with only normal crossings.Let T be afinite set of closed points of X including the points of C i∩C j,for all i=j.Let D be an irreducible curve on X which is not in the ramification locus ofαand does not pass through any point in T.Then D intersects C i at points P where∂Ci(α)is unramified.Suppose further that at such points P,∂Ci (α)specializes to0in H1(κ(P),Z/l Z).Thenαis unramified at D and specializes to0in H2(κ(D),µl).Proof.Since k contains a primitive l th of unity,wefix a generatorρforµl and identify the Galois modulesµ⊗j l with Z/l Z.Let P be a point in the intersection of D and the support of ram X(α).Since D does not pass through the points of T and T contains all the points of intersection of distinct C j,the point P belongs to a unique curve C i in the support of ram X(α).We have([S1],1.2)α=α′+(u,π),whereα′is unramified on O X,P,u∈O X,P is a unit andπ∈O X,P is a prime defining the curve C i at P.Therefore ∂Ci(α)=(not in the ramification locus ofα,αis unramified at D.Letαis zero,by classfield theory it is enough to show thatαonαis unramified at R.Suppose P is on the ramification locus ofα.As before,we haveα=α′+(u,π),whereα′is unramified on O X,P,u∈O X,P is a unit andπ∈O X,P(α)=is a prime defining the curve C i at P.Therefore∂Ciα=u,αat R is(u(P))ν(αis unramified at R.2Proposition3.2Let K and l be as above.Letα∈H2(K,µl)with index l.Let X be a regular projective model of K such that the ramification locus ram X(α)and the specialfibre of X are a union of regular curves with only normal crossings andαhas no cool points and no chilli loops on X(cf.1.4). Let s i be the corresponding coefficients(cf.§1).Let F1,···,F r be irreducible regular curves on X which are not in ram X(α)={C1,···,C n}and such that{F1,···,F r}∪ram X(α)have only normal crossings.Let m1,···,m r be integers.Then there exists f∈K∗such thatdiv X(f)= s i C i+ m s F s+ n j D j+lE′,where D1,···,D t are irreducible curves which are not equal to C i and F s for all i and s andαspecializes to zero at D j for all j and(n j,l)=1.Proof.Let T be afinite set of closed points of X containing all the points of intersection of distinct C i and F s and at least one point from each C i and F s.By a semilocal argument,we choose g∈K∗such that div X(g)=10s i C i+ m s F s+G where G is a divisor on X whose support does not contain any of C i or F s and does not intersect T.Sinceαhas no cool points and no chilli loops on X,by([S3],Prop.4.5), there exists u∈K∗such that div X(ug)= s i C i+ m s F s+E,where E is a divisor on X whose support does not contain any C i or F s,does not pass through the points in T and either E intersect C i at a point P where(α)is0or the intersection multiplicity(E·C i)P is a the specialization of∂Cimultiple of l.Suppose C i for some i is a geometric curve on X.Since every closed point of X is on the specialfibreX. Let Q1,···,Q t be the points of intersection of the support of the divisor E and the specialfibre with intersection multiplicity n j at Q j coprime with l, for1≤j≤r.In particular we have∂C(α)=0for1≤j≤r.For eachjQ j,let D j be a regular geometric curve on X such that Q j is the multiplicity one intersection of D j andu∈κ(v)∗\κ(v)∗l.Suppose further that if αis ramified at v,∂v(α)=[L]∈H1(κ(v),Z/l Z),where L=K(u1u∈κ(v)∗l,there is a unique discrete valuation˜v of L extending the valuation v of K,which is unramified with residual degree l.In particular v(N L/K(g))is a multiple of l.Thus ifα′∈H2(K v,Z/l Z)is unramified at v, then(N L/K(g))·α′∈H3(K v,Z/l Z)is unramified.Since H3nr(K v,Z/l Z)=011(cf. 1.2),we have(N L/K(g))·α′=0for anyα′∈H2(K v,Z/l Z)which is unramified at v.In particular,ifαis unramified at v,thenα·(N L/K(g))=0.Suppose thatαis ramified at v.Then by the choice of u,we haveα=α′+(u)·(πv),whereπv is a parameter at v andα′∈H2(K v,Z/l Z)is unramified at v.Thus we have(N L/K(g))·α=(N L/K(g))·α′+(N L/K(g))·(u)·(πv)=(N L/K(g))·(u)·(πv)∈H3(K v,Z/l Z).Since L v=K v(u1u=∂C(α)∈H1(κ(C i),Z/l Z)for alliC i∈ram X(α),νF i(u)=m i,whereνF i is the discrete valuation at F i andu∈κ(C j)∗l for any C j∈ram X(ζ)\ram X(α).Let L=K(u1is a union of regular curves with only normal crossings and there are no cool points and no chilli loops forαL on˜Y(cf1.4).We denote the strict transforms of˜F i by˜F i again.By(3.2),there exists g∈L∗such thatdiv˜Y(g)=C+ −m i˜F i+ n j D j+lD,where the support of C is contained in ram˜Y (αL)and D j’s are irreduciblecurves which are not in ram˜Y (αL)andαL specializes to zero at all D j’s.We now claim thatζ=α∪(fN L/K(g)).Since the group H3nr(K/X,Z/l Z)=0([K],5.2),it is enough to show thatζ−α∪(fN L/K(g))is unramified onX.Let S be an irreducible curve on X.Since the residue map∂S factors through the completion K S,it suffices to show thatζ−α∪(fN L/K(g))=0over K S.Suppose S is not in ram X(α)∪ram X(ζ)∪Supp(fN L/K(g)).Then eachofζandα·(fN L/K(g))is unramified at S.Suppose that S is in ram X(α)∪ram X(ζ).Then by the choice of f we have(f)=(f v)∈H1(K v,Z/l Z)where v is the discrete valuation associatedto S.Henceζ=α∪(f)over the completion K S of K at the discretevaluation given by S.It follows from(3.3)that(N L/K(g))∪α=0over K S andζ=α∪(fN L/K(g))over K S.Suppose that S is in the support of div X(fN L/K(g))and not in ram X(α)∪ram X(ζ).Thenαis unramified at S.We show that in this caseα∪(fN L/K(g))=0over K S.We havediv X(fN L/K(g))=div X(f)+div X(N L/K(g))=C′+ m i F i+lE+η∗π∗(C+ −m i˜F i+ n j D j+lD)=C′+η∗π∗(C)+ n jη∗π∗(D j)+lη∗π∗(E).We note that if D j maps to a point,thenη∗π∗(D j)=0.Since the support ofC is contained in ram˜Y (αL),the support ofη∗π∗(C)is contained in ram X(α).Thus S is in the support ofη∗π∗(D j)for some j or S is in the support of lη∗π∗(E).In the later case,clearlyα·(fN L/K(g))is unramified at S and henceα·(fN L/K(g))=0over K S.Suppose S is in the support ofη∗π∗(D j) for some j.In this case,if D j lies over an inert curve,thenη∗π∗(D j)is a multiple of l and we are done.Suppose that D j lies over a split curve.Since αL specializes to zero at D j,it follows thatαspecializes to zero atη∗π∗(D j) and we are done.213Theorem3.5.Let k be a p-adicfield and K a functionfield of a curve over k.Let l be a prime not equal to p.Suppose that all the l th roots of unity are in K.Then every element in H3(K,µ⊗3l)is a symbol.Proof.We again identify the Galois modulesµ⊗jl with Z/l Z.Let v be a discrete valuation of K and K v the completion of K at v.By (1.2),every element in H3(K v,Z/l Z)is a symbol.Letζ∈H3(K,Z/l Z)and X be a regular projective model of K.Let v be a discrete valuation of K corresponding to an irreducible curve in ram X(ζ). Then we haveζ=(f v)·(g v)·(h v)for some f v,g v,h v∈K∗v.By weak approximation,we canfind f,g∈K∗such that(f)=(f v)and(g)=(g v)in H1(K v,Z/l Z)for all discrete valuations v corresponding to the irreducible curves in ram X(ζ).Let v be a discrete valuation of K corresponding to an irreducible curve C in X.If C is in ram X(ζ),then by the choice of f and g we haveζ=(f)·(g)·(h v)∈H3(K v,Z/l Z).If C is not in the ram X(ζ),thenζ∈H3nr(K v,Z/l Z)≃H3(κ(v),Z/l Z)=0.In particular we haveζ=(f)·(g)·(1)∈H3(K v,Z/l Z).Letα=(f)·(g)∈H2(K,Z/l Z). Then we haveζ=α·(h′v)∈H3(K v,Z/l Z)for some h′v∈K∗v for each discrete valuation v of K associated to any point of X1.By(3.4),there exists h∈K∗such thatζ=α·(h)=(f)·(g)·(h)∈H3(K,Z/l Z).2Remark3.6.We remark that all the results of this section can be extended to the situation where k does not necessarily contain a primitive l th root of unity.This can be achieved by going to the extension k′of k obtained by adjoining a primitive l th of unity to k and noting that the extension k′/k is unramified of degree l−1.We do not use this remark in the sequel.4.The u-invariantIn(4.1)and(4.2)below,we give some necessary conditions for afield k to have the u-invariant less than or equal to8.If K is the functionfield of a curve over a p-adicfield and K v is the completion of K at a discrete valuation v of K,then the residuefieldκ(v)of K v,which is either a globalfield of positive characteristic or a p-adicfield,has u-invariant4.By a theorem of Springer, u(K v)=8and we use(4.1)and(4.2)for K v.14Proposition4.1Let K be afield of characteristic not equal to2.Suppose that u(K)≤8.Then I4(K)=0and every element in I3(K)is a3-foldPfister form.Further ifφis a3-fold Pfister form and q2a rank2quadratic form over K,then there exists f,g,h∈K∗such that f is a value of q2and φ=<1,f><1,g><1,h>.Proof.Suppose that u(K)=8.Then every4-fold Pfister form is isotropic and hence hyperbolic;in particular,I4(K)=0.Letφ1=<1,f1><1,g1>< 1,h1>andφ2=<1,f2><1,g2><1,h2>be two anisotropic3-fold Pfister forms.Since u(K)≤8,the Witt index ofφ1−φ2is at least4.In particular φ1−φ2is isotropic,i.e.φ1andφ2represent a common value a∈K.Sinceφ1 andφ2are anisotropic,a=0.We haveφ1=<a>⊥φ′1andφ2=<a>⊥φ′2for some quadratic formφ1andφ2over K([Sc],p.7,Lemma3.4).Since the Witt index ofφ1−φ2is at least4,the Witt index ofφ′1−φ′2is at least3.Repeating this process,we see that there exists a quadratic form <a,b,c,d>over K which is a subform of bothφ1andφ2.Let C be a conic given by the quadratic form abc<a,b,c>=<bc,ac,ab>.Since abc<a,b,c>is isotropic over the functionfield K(C)of C(cf.[Sc],p.154, Remark5.2(iv))and abc<a,b,c>is a subform of abcφ1and abcφ2,φ1and φ2are isotropic over K(C).Sinceφ1andφ2are Pfister forms,they are hyperbolic over k(C)([Sc],p.144,Cor.1.5).Letψ=<1,bc><1,ac>and k(ψ)be the functionfield of the quadratic formψ.Then k(C)and k(ψ) are birational(cf.[Sc],p.154)and henceφ1andφ2are hyperbolic over k(ψ). Thereforeφ1≃<1,bc><1,ac><a1,b1>andφ2≃<1,bc><1,ac><a2,b2> for some a1,a2,b1,b2∈K∗(cf.[Sc],p.155,Th.5.4).Since I4(K)=0,we haveφ1≃<1,bc><1,ac><1,a1b1>andφ2≃<1,bc><1,ac><1,a2b2>. We haveφ1+φ2=φ1−φ2=<1,bc><1,ac><1,a1b1,−1,−a2b2>=<1,bc><1,ac><a1b1,−a2b2>=a1b1<1,bc><1,ac><1,−a1a2b1b2>=<1,bc><1,ac><1,−a1a2b1b2>.Thus the sum of any two3-fold Pfister forms is a Pfister form in I3(K)and every element in I3(K)is the class of a3-fold Pfister form.Letφ=<1,a><1,b><1,c>be a3-fold Pfister form andφ′be its puresubform.Let q2be a quadratic form over K of dimension2.Since dim(φ′)=715and u(K)≤8,the quadratic formφ′−q2is isotropic.Therefore there exists f∈K∗which is a value of q2andφ′≃<f>+φ′′for some quadratic form φ′′over K.Hence by([Sc],p.143),φ=<1,f><1,b′><1,c′>for some b′,c′∈K∗.2Proposition4.2.Let K be afield of characteristic not equal to2.Suppose that u(K)≤8.Letφ=<1,f><1,a><1,b>be a3-fold Pfister form over K and q3a quadratic form over K of dimension3.Then there exist g,h∈K∗such that g is a value of q3andφ=<1,f><1,g><1,h>.Proof.Letψ=<1,f><a,b,ab>.Since u(K)≤8,the quadratic form ψ−q3is isotropic.Hence there exists g∈K∗which is a common value ofq3andψ.Thus,ψ≃<g>+ψ1for some quadratic formψ1over K.Since√ψis hyperbolic over K(Proof.Let q be a quadratic form over K of dimension9.Since every element in H2(K,µ2)is a sum of at most2symbols,as in([PS],proof of4.5),wefind a quadratic form q5=λ<1,a1,a2,a3,a4>over K such that φ=q+q5∈I3(K).By the assumptions2,3and4,there exist f,g,h∈K∗such thatφ=<1,f><1,g><1,h>and f is a value of<a1,a2>andg is a value of<fa1a2,a3,a4>.We have<a1,a2>≃<f,fa1a2>and <fa1a2,a2,a3>≃<g,g1,g2>for some g1,g2∈K∗.Since I4(K)=0,we haveλφ=φandλq=λq+λq5−λq5=λφ−λq5=φ−λq5=<1,f><1,g><1,h>−<1,a1,a2,a3,a4>=<1,f><1,g><1,h>−<1,f,g,g1,g2>=<gf>+<1,f><h,gh>−<g1,g2>.Since the dimension ofλq is9and the dimension of<gf>⊥<1,f><h,gh>−<g1,g2>is7,it follows thatλq and hence q is isotropic over K.2Proposition4.4.Let k be a p-adicfield,p=2and K a functionfield of a curve over k.Letφbe a3-fold Pfister form over K and q2a quadratic form over K of dimension2.Then there exist f,a,b∈K∗such that f is a value of q2andφ=<1,f><1,a><1,b>.Proof.Letζ=e3(φ)∈H3(K,µ2).Let X be a projective regular model of K.Let C be an irreducible curve on X and v the discrete valuation given by C.Let K v be the completion of K at v.Since the residuefieldκ(v)=κ(C)is either a p-adicfield or a functionfield of a curve over afinitefield,u(κ(v))=4 and u(K v)=8([Sc],p.209).By(4.1),there exist f v,a v,b v∈K∗v such that f v is a value of q2over K v andφ=<1,f v><1,a v><1,b v>over K v. By weak approximation,we canfind f,a∈K∗such that f is a value of q2over K and f=f v,a=a v modulo K∗2v for all discrete valuations v corresponding to the irreducible curves C in the support of ram X(ζ).Let C be any irreducible curve on X and v the discrete valuation of K given by C.If C is in the support of ram X(ζ),then by the choice of f and a,we haveζ=e3(φ)=(−f)·(−a)·(−b v)over K v.If C is not in the support of ram X(ζ),thenζ∈H3nr(K v,µ2)≃H3(κ(v),µ2)=0.In particular we17haveζ=(−f)·(−a)·(1)over K v.Letα=(−f)·(−a)∈H2(K,µ2).By (3.4),there exists b∈K∗such thatζ=α·(−b)∈H3(K,µ2).Since e3: I3(K)→H3(K,µ2)is an isomorphism,we haveφ=<1,f><1,a><1,b> as required.2.There is a different proof of the Proposition4.4in([PS],4.4)!Proposition4.5Let k be a p-adicfield,p=2and K a functionfield of a curve over k.Letφ=<1,f><1,a><1,b>be a3-fold Pfister form over K and q3a quadratic form over K of dimension3.Then there exist g,h∈K∗such that g is a value of q3andφ=<1,f><1,g><1,h>.Proof.Letζ=e3(φ)=(−f)·(−a)·(−b)∈H3(K,µ2).Let X be a projective regular model of K.Let C be an irreducible curve on X and v the discrete valuation of K given by C.Let K v be the completion of K at v.Then as in the proof of(4.4),we have u(K v)=8.Thus by(4.2), there exist g v,h v∈K∗v such that g v is a value of the quadratic form q3and φ=<1,f><1,g v><1,h v>over K v.By weak approximation,we canfindg∈K∗such that g is a value of q3over K and g=g v modulo K∗2v for all discrete valuations v of K given by the irreducible curves C in ram X(ζ).Let C be an irreducible curve on X and v the discrete valuation of K given by C. By the choice of g it is clear thatζ=e3(φ)=(−f)·(−g)·(−h v)for all the discrete valuations v of K given by the irreducible curves C in the support of ram X(ζ).If C is not in the support of ram X(ζ),then as is the proof of(4.4), we haveζ=(−f)·(−g)·(1)over K v.Letα=(−f)·(−g)∈H2(K,µ2).By (3.4),there exists h∈K∗such thatζ=α∪(−h)=(−f)·(−g)·(−h).Since e3:I3(K)→H3(K,µ2)is an isomorphism,φ=<1,f><1,g><1,h>.Theorem4.6Let K be a functionfield of a curve over a p-adicfield k.If p=2,then u(K)=8.Proof.Let K be a functionfield of a curve over a p-adicfield k.Assume that p=2.By a theorem of Saltman([S1],3.4,cf.[S2]),every element in H2(K,µ2)is a sum of at most2symbols.Since the cohomological dimension of K is3,we also have I4(K)≃H4(K,µ2)=0([AEJ]).Now the theorem follows from(4.4),(4.5)and(4.3).2.18。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
THE FUNCTION AND CLASSES OF ROAD MARKINGS
Road markings may be defined as markings on the surface of the road for the control, warning, guidance or information of road users. They may be used to supplement upright signs, or they may be used alone.
Road markings have the limitation that they may be obscured by snow, leaves or debris on the carriageway. Their conspicuity is impaired when wet or dirty and their durability depends largely on their exposure to traffic wear. Nevertheless, they serve a very important function in conveying to drivers information and requirements which might not otherwise be possible by the use of upright signs. They have the advantage that they can often be seen when an upright sign is obscured and, unlike such signs, they can provide a continuing message to the moving driver.
The continued increase in the volume of traffic using the roads makes the extensive use of road markings essential to ensure that full advantage is taken of the available road space. In particular, widespread use of lane markings is desirable. Enhancing lane discipline adds to the safety of traffic, besides improving traffic flows.
Road markings should be considered in detail at design stage in respect of new or improved roads and junctions. The markings for existing roads are best considered on plan before the work is undertaken.
Road markings may be classified as follows:
a. Transverse markings, which are at right-angles (or thereabouts) to the centre line of the carriageway;
b. Longitudinal markings (including double line systems);
c. Hatched markings;
d. Worded and diagrammatic markings;
e. Markings indicating parking and loading requirements;
f. Bus and tram markings;
g. Cycle track markings;
h. Yellow box markings; and
i. Road studs.。