Reflection Groups and Polytopes over Finite Fields, III

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高分子专业英语讲稿

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as 引出的宾语补语有如下几种表示形式： (a). 名词 We usually define energy as the ability to do work. (b). 形容词 We often regard gas as compressible. We accept the conclusion as true. (c). 介词短语 We regard the conclusion as of consequence. (具有重要意义的) of consequence: 重要, 有名 (d). 分词或分词短语（participle 分词） We treat the wire as disconnected. We consider the machine as representing the best one in our factory.
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As 引出宾语补语、主语补语 (complement 补语) 1. As 引出宾语补语 We regard the sun as the chief source of heat and light. We consider radio waves as radiant （辐射的）energy. 科技英语中，常见如下结构： “动词（或动词性词组）＋名词（或代词）＋as＋宾语补语” (a) to accept …as (b) to define… as (c) to consider…as (d) to refer to…as (e) to treat…as (f) to describe…as (h) to regard…as 把…承认为，承认…是把…解释为，对…下的定义是把…认为是把…指为（叫做）把…当作；以…来对待把…描述为把…看成
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From the viewpoint of 根据……的观点 From the point of view of From the standpoint of 根据……的观点，站在…的立场上 think of…… as……把……认为是 “主语＋被动态谓语＋as＋-ing”, “-ing”作主语补语 Water can be shown as containing admixtures (混合物). 可以证明水含有杂质 Only those substances which can be considered as being mixtures have a depressed melting point. 那些物质可以被认为是混合物，只有它们才具有较低的熔点。 * -ing 是being 时，being 常可以省略 This may be taken as (being) a result of overheating. More than one type of mistake may be thought of as (being) present.

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。

GTM研究生参考书

GTM(Graduate Texts of Mathematics)丛书目录1 Introduction to Axiomatic Set Theory, Gaisi Takeuti, W. M. Zaring2 Measure and Category, John C. Oxtoby (1997, ISBN 978-0-387-90508-2)3 Topological Vector Spaces, . Schaefer, . Wolff (1999, ISBN 978-0-387-98726-2)4 A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, ISBN 978-0-387-94823-2)5 Categories for the Working Mathematician, Saunders Mac Lane (1998, ISBN 978-0-387-98403-2)6 Projective Planes, Hughes, Piper7 A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7)8 Axiomatic Set Theory, Gaisi Takeuti, Zaring9 Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7)10 A Course in Simple-Homotopy Theory, M. M. Cohen11 Functions of One Complex Variable I, John B. Conway (1995, ISBN 978-0-387-90328-6)12 Advanced Mathematical Analysis, R. Beals (1973, ISBN 978-0-387-90065-0)13 Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, ISBN978-0-387-97845-1)14 Stable Mappings and Their Singularities, Golubitsky, Guillemin15 Lectures in Functional Analysis and Operator Theory, S. K. Berberian16 The Structure of Fields, D. Winter17 Random Processes, M. Rosenblatt18 Measure Theory, Paul R. Halmos (1974, ISBN 978-0-387-90088-9)19 A Hilbert Space Problem Book, Paul R. Halmos (1982, ISBN 978-0-387-90685-0)20 Fibre Bundles, Dale Husemoller (1994, ISBN 978-0-387-94087-8)21 Linear Algebraic Groups, James E. Humphreys (1998, ISBN 978-0-387-90108-4)22 An Algebraic Introduction to Mathematical Logic, Barnes, Mack23 Linear Algebra, Werner H. Greub (1981, ISBN 978-0-387-90110-7)24 Geometric Functional Analysis and Its Applications, Holmes25 Real and Abstract Analysis, Edwin Hewitt, Karl Stromberg (1975, ISBN 978-0-387-90138-1)26 Algebraic Theories, Manes27 General Topology, John L. Kelley (1975, ISBN 978-0-387-90125-1)28 Commutative Algebra I, Oscar Zariski, Pierre Samuel, Cohen (1975, ISBN 978-0-387-90089-6)29 Commutative Algebra II, Oscar Zariski, Pierre Samuel30 Lectures in Abstract Algebra I, Nathan Jacobson31 Lectures in Abstract Algebra II, Nathan Jacobson32 Lectures in Abstract Algebra III, Nathan Jacobson33 Differential Topology, Morris W. Hirsch34 Principles of Random Walk, Frank Spitzer35 Several Complex Variables and Banach Algebras, Herbert Alexander, John Wermer36 Linear Topological Spaces, John L. Kelley, Isaac Namioka37 Mathematical Logic, J. Donald Monk38 Several Complex Variables, Grauert, Fritzsche39 An Invitation to C * -Algebras, William Arveson40 Denumerable Markov Chains, John George Kemeny, Snell, Knapp et al.41 Modular Functions and Dirichlet Series in Number Theory, Tom M. Apostol42 Linear Representations of Finite Groups, Jean-Pierre Serre, Scott43 Rings of Continuous Functions, Gillman, Jerison44 Elementary Algebraic Geometry, K. Kendig45 Probability Theory I, M. Loève46 Probability Theory II, M. Loève47 Geometric Topology in Dimensions 2 and 3, Edwin E. Moise48 General Relativity for Mathematicians, . Sachs, H. Wu49 Linear Geometry, . Gruenberg, . Weir50 Fermat's Last Theorem, Harold M. Edwards51 A Course in Differential Geometry, Klingenberg52 Algebraic Geometry, Robin Hartshorne53 A Course in Mathematical Logic, Yu. I. Manin54 Combinatorics with Emphasis on the Theory of Graphs, Graver, Watkins55 Introduction to Operator Theory I, Brown, Pearcy56 Algebraic Topology: An Introduction, William S. Massey57 Introduction to Knot Theory, Richard H. Crowell, Ralph H. Fox (1977, ISBN 978-0-387-90272-2)58 P-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz59 Cyclotomic Fields, Serge Lang60 Mathematical Methods of Classical Mechanics, V. I. Arnold61 Elements of Homotopy Theory, George W. Whitehead62 Fundamentals of the Theory of Groups, Kargapolov, Merzljakov, Burns63 Graph Theory, Béla Bollobás64 Fourier Series I, Edwards65 Differential Analysis on Complex Manifolds, . Wells Jr.66 Introduction to Affine Group Schemes, . Waterhouse67 Local Fields, Jean-Pierre Serre, Greenberg68 Linear Operators on Hilbert Spaces, Weidmann, Szuecs69 Cyclotomic Fields II, Serge Lang70 Singular Homology Theory, William S. Massey71 Riemann Surfaces, Herschel Farkas, Irwin Kra72 Classical Topology and Combinatorial Group Theory, John Stillwell73 Algebra, Thomas W. Hungerford74 Multiplicative Number Theory, Harold Davenport, Hugh Montgomery75 Basic Theory of Algebraic Groups and Lie Algebras, G. P. Hochschild76 Algebraic Geometry, Iitaka77 Lectures on the Theory of Algebraic Numbers, Hecke, Brauer, Goldman et al.78 A Course in Universal Algebra, Burris, Sankappanavar (Online [1])79 An Introduction to Ergodic Theory, Peter Walters80 A Course in the Theory of Groups, Derek . Robinson81 Lectures on Riemann Surfaces, Forster, Gilligan82 Differential Forms in Algebraic Topology, Raoul Bott, Loring Tu83 Introduction to Cyclotomic Fields, Lawrence C. Washington84 A Classical Introduction to Modern Number Theory, Ireland, Rosen (1995, ISBN 0-387-97329-X)85 Fourier Series A Modern Introduction, R. E. Edwards86 Introduction to Coding Theory, . van Lint (3rd ed 1998, ISBN 3-540-64133-5)87 Cohomology of Groups, Kenneth S. Brown88 Associative Algebras, . Pierce89 Introduction to Algebraic and Abelian Functions, Serge Lang90 An Introduction to Convex Polytopes, Arne Brondsted91 The Geometry of Discrete Groups, Alan F. Beardon92 Sequences and Series in Banach Spaces, J. Diestel93 Modern Geometry - Methods and Applications I, Dubrovin, Fomenko, Novikov et al.94 Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner95 Probability, Shiryaev, Boas96 A Course in Functional Analysis, John B. Conway97 Introduction to Elliptic Curves and Modular Forms, Neal Koblitz98 Representations of Compact Lie Groups, Broecker, Dieck99 Finite Reflection Groups, Grove, Benson100 Harmonic Analysis on Semigroups, Berg, Christensen, Ressel101 Galois Theory, Harold M. Edwards102 Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan103 Complex Analysis, Serge Lang104 Modern Geometry - Methods and Applications II, Dubrovin, Fomenko, Novikov et al.105 SL2®, Serge Lang106 The Arithmetic of Elliptic Curves, Joseph H. Silverman107 Applications of Lie Groups to Differential Equations, Peter J. Olver108 Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range109 Univalent Functions and Teichmüller Spaces, O. Lehto110 Algebraic Number Theory, Serge Lang111 Elliptic Curves, Dale Husemöller112 Elliptic Functions, Serge Lang113 Brownian Motion and Stochastic Calculus, Ioannis Karatzas, Steven Shreve114 A Course in Number Theory and Cryptography, Neal Koblitz115 Differential Geometry, Berger, Gostiaux, Levy116 Measure and Integral, John L. Kelley, Srinivasan117 Algebraic Groups and Class Fields, Jean-Pierre Serre118 Analysis Now, Gert K. Pedersen119 An Introduction to Algebraic Topology, Joseph J. Rotman120 Weakly Differentiable Functions, William P. Ziemer121 Cyclotomic Fields I-II, Serge Lang, Karl Rubin122 Theory of Complex Functions, Remmert, Burckel123 Numbers, Lamotke, Ewing, Ebbinghaus et al.124 Modern Geometry - Methods and Applications III, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Petrovich Novikov et al. (1990, ISBN 978-0-387-97271-8)125 Complex Variables, Berenstein, Gay126 Linear Algebraic Groups, Armand Borel127 A Basic Course in Algebraic Topology, William S. Massey128 Partial Differential Equations, Jeffrey Rauch129 Representation Theory, William Fulton, Joe Harris130 Tensor Geometry, Dodson, T. Poston131 A First Course in Noncommutative Rings, . Lam132 Iteration of Rational Functions, Alan F. Beardon133 Algebraic Geometry, Joe Harris134 Coding and Information Theory, Steven Roman135 Advanced Linear Algebra, Steven Roman136 Algebra, Adkins, Weintraub137 Harmonic Function Theory, Axler, Bourdon, Ramey138 A Course in Computational Algebraic Number Theory, Henri Cohen (1996, ISBN 0-387-55640-0) 139 Topology and Geometry, Glen E. Bredon140 Optima and Equilibria, Jean-Pierre Aubin141 Gröbn er Bases, Becker, Weispfenning, Kredel142 Real and Functional Analysis, Serge Lang, (1993, ISBN 14)143 Measure Theory, . Doob144 Noncommutative Algebra, Farb, Dennis145 Homology Theory, James W. Vick146 Computability, Douglas S. Bridges147 Algebraic K-Theory and Its Applications, Jonathan Rosenberg148 An Introduction to the Theory of Groups, Joseph J. Rotman149 Foundations of Hyperbolic Manifolds, John G. Ratcliffe150 Commutative Algebra with a View Toward Algebraic Geometry, David Eisenbud 151 Advanced Topics in the Arithmetic of Elliptic Curves, Joseph H. Silverman 152 Lectures on Polytopes, Günter M. Ziegler153 Algebraic Topology, William Fulton154 An Introduction to Analysis, Brown, Pearcy155 Quantum Groups, Christian Kassel156 Classical Descriptive Set Theory, Alexander S. Kechris157 Integration and Probability, Malliavin, Airault, Kay et al.158 Field Theory, Steven Roman159 Functions of One Complex Variable II, John B. Conway160 Differential and Riemannian Manifolds, Serge Lang161 Polynomials and Polynomial Inequalities, Borwein, Erdelyi162 Groups and Representations, Alperin, Bell163 Permutation Groups, Dixon, Mortimer164 Additive Number Theory I, Melvyn B. Nathanson165 Additive Number Theory II, Melvyn B. Nathanson166 Differential Geometry, . Sharpe, Shiing-Shen Chern167 Field and Galois Theory, Patrick Morandi168 Combinatorial Convexity and Algebraic Geometry, Guenter Ewald169 Matrix Analysis, Rajendra Bhatia170 Sheaf Theory, Glen E. Bredon171 Riemannian Geometry, Peter Petersen172 Classical Topics in Complex Function Theory, Remmert, Kay173 Graph Theory, Reinhard Diestel174 Foundations of Real and Abstract Analysis, Douglas S. Bridges175 An Introduction to Knot Theory, W. B. Raymond Lickorish176 Riemannian Manifolds, John M. Lee177 Analytic Number Theory , Donald J. Newman178 Nonsmooth Analysis and Control Theory, Clarke, Ledyaev, Stern et. al179 Banach Algebra Techniques in Operator Theory, Ronald G. Douglas180 A Course on Borel Sets, . Srivastava181 Numerical Analysis, Rainer Kress182 Ordinary Differential Equations, Walter, Thompson183 An Introduction to Banach Space Theory, Robert E. Megginson184 Modern Graph Theory, Béla Bollobás185 Using Algebraic Geometry, Cox, Little, O Shea186 Fourier Analysis on Number Fields, Ramakrishnan, Valenza187 Moduli of Curves, Harris, Morrison188 Lectures on the Hyperreals, Robert Goldblatt189 Lectures on Modules and Rings, Tsit-Yuen Lam190 Problems in Algebraic Number Theory, Esmonde, Murty191 Fundamentals of Differential Geometry, Serge Lang192 Elements of Functional Analysis Hirsch, Lacombe, Levy193 Advanced Topics in Computational Number Theory, Henri Cohen (2000, ISBN 0-387-98727-4) 194 One-Parameter Semigroups for Linear Evolution Equations, Engel, Nagel195 Elementary Methods in Number Theory, Melvyn B. Nathanson196 Basic Homological Algebra, M. Scott Osborne197 The Geometry of Schemes, Eisenbud, Harris198 A Course in p-adic Analysis, Alain Robert199 Theory of Bergman Spaces, Hedenmalm, Korenblum, Zhu200 An Introduction to Riemann-Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen 201 Diophantine Geometry, Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4)202 Introduction to Topological Manifolds, John M. Lee203 The Symmetric Group, Bruce E. Sagan204 Galois Theory, Jean-Pierre Escofier205 Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-0)206 Problems in Analytic Number Theory, M. Ram Murty (2001, ISBN 978-0-387-95143-0)207 Algebraic Graph Theory, Godsil, Royle (2001, ISBN 978-0-387-95241-3)208 Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6)209 A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7)210 Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9)211 Algebra, Serge Lang212 Lectures on Discrete Geometry, Jiri Matousek213 From Holomorphic Functions to Complex Manifolds, Fritzsche, Grauert214 Partial Differential Equations, Juergen Jost215 Algebraic Functions and Projective Curves, David Goldschmidt216 Matrices, Denis Serre217 Model Theory: An Introduction, David Marker218 Introduction to Smooth Manifolds, John M. Lee (2003, ISBN 978-0-387-95448-6)219 The Arithmetic of Hyperbolic 3-Manifolds, Maclachlan, Reid220 Smooth Manifolds and Observables, Jet Nestruev221 Convex Polytopes, Branko Grünbaum (2003, ISBN 0-387-00424-6)222 Lie Groups, Lie Algebras, and Representations, Brian C. Hall223 Fourier Analysis and its Applications, Anders Vretblad224 Metric Structures in Differential Geometry, Walschap, G., (2004, ISBN 978-0-387-20430-7) 225 Lie Groups, Daniel Bump, (2004, ISBN 978-0-387-21154-1)226 Spaces of Holomorphic Functions in the Unit Ball, Zhu, K., (2005, ISBN 978-0-387-22036-9) 227 Combinatorial Commutative Algebra, Ezra Miller, Bernd Sturmfels, (2005, ISBN978-0-387-22356-8)228 A First Course in Modular Forms, Fred Diamond, J. Shurman, (2006, ISBN 978-0-387-23229-4) 229 The Geometry of Syzygies, David Eisenbud (2005, ISBN 978-0-387-22215-8)230 An Introduction to Markov Processes, Stroock, ., (2005, ISBN 978-3-540-23499-9)231 Combinatorics of Coxeter Groups, Anders Björner, Francisco Brenti, (2005, ISBN978-3-540-44238-7)232 An Introduction to Number Theory, Everest, G., Ward, T., (2005, ISBN 978-)233 Topics in Banach Space Theory, Albiac, F., Kalton, ., (2006, ISBN 978-0-387-28141-4)234 Analysis and Probability · Wavelets, Signals, Fractals, Jorgensen, (2006, ISBN978-0-387-29519-0)235 Compact Lie Groups, M. R. Sepanski, (2007, ISBN 978-0-387-30263-8)236 Bounded Analytic Functions, Garnett, J., (2007, ISBN 978-0-387-33621-3)237 An Introduction to Operators on the Hardy-Hilbert Space, Martinez-Avendano, ., Rosenthal, P., (2007, ISBN 978-0-387-35418-7)238 A Course in Enumeration, Aigner, M., (2007, ISBN 978-3-540-39032-9)239 Number Theory - Volume I: Elementary and Algebraic Methods for Diophantine Equations, Cohen, H., (2007, ISBN 978-0-387-49922-2)240 Number Theory - Volume II: Analytic and Modern Tools, Cohen, H., (2007, ISBN978-0-387-49893-5)241 The Arithmetic of Dynamical Systems, Joseph H. Silverman, (2007, ISBN 978-0-387-69903-5) 242 Abstract Algebra, Grillet, Pierre Antoine, (2007, ISBN 978-0-387-71567-4)243 Topological Methods in Group Theory, Geoghegan, Ross, (2007, ISBN 978-0-387-74611-1)244 Graph Theory, Bondy, ., Murty, (2007, ISBN 978-)245 Complex Analysis: Introduced in the Spirit of Lipman Bers,Gilman, Jane P., Kra, Irwin, Rodriguez, Rubi E. (2007, ISBN 978-0-387-74714-9)246 A Course in Commutative Banach Algebras, Kaniuth, Eberhard, (2020, ISBN 978-0-387-72475-1) 247 Braid Groups, Kassel, Christian, Turaev, Vladimir, (2008, ISBN 978-0-387-33841-5)248 Buildings Theory and Applications, Abramenko, Peter, Brown, Ken (2008, ISBN978-0-387-78834-0)249 Classical Fourier Analysis, Grafakos, Loukas, (2008, ISBN 978-0-387-09431-1)250 Modern Fourier Analysis, Grafakos, Loukas, (2008, ISBN 978-0-387-09433-5)251 The Finite Simple Groups, Wilson, Robert, Parker, Christopher W., (2009, ISBN 978-)252 Distributions and Operators, Grubb, Gerd, (2009, ISBN 978-0-387-84894-5)。

On the intrinsic limitation of the Rosenbluth method at large momentum transfer

a rXiv:h ep-ph/412216v115Dec24On the intrinsic limitation of the Rosenbluth method at large Q 2.E.Tomasi-Gustafsson DAPNIA/SPhN,CEA/Saclay,91191Gif-sur-Yvette Cedex,France ∗(Dated:February 2,2008)Abstract Correlations in the elastic electron proton scattering data show that the Rosenbluth method is not reliable for the extraction of the electric proton form factors at large momentum transfer,where the magnetic term dominates,due to the size and the ǫdependence of the radiative corrections.PACS numbers:25.30.Bf,13.40.-f,13.60.-Hb,13.88.+eThe determination of the elastic proton electromagnetic form factors(FFs)at large mo-mentum transfer is a very actual problem,due to the availability of electron beams in the GeV range,with high intensity and high polarization,large acceptance spectrometers, hadron polarized targets,hadron polarimeters.The possibility to extend the measurements of such fundamental quantities,which contain dynamical information on the nucleon struc-ture,inspired experimental programs at JLab,Frascati and at future machines,as GSI,both in the space-like and in the time-like regions.The traditional way to measure electromagnetic proton form factors consists in the de-termination of theǫdependence of the reduced elastic diﬀerential cross section,which can be written,assuming that the interaction occurs through the exchange of one-photon,as[1]:σred=ǫ(1+τ) 1+2Eα2cos2(θ/2)dσ•strong monotonical decreasing from polarization transfer measurements.R(Q2)==1−0.13(Q2[GeV2]−0.04).(2)The ratio deviates from unity,as Q2increasing,reaching a value of≃0.34at Q2≃5.5GeV2 [10].This puzzle has given rise to many speculations and diﬀerent interpretations[11,12,13], suggesting expensive experiments.In particular,it has been suggested that the2γexchange could solve this discrepancy through its interference with the the main mechanism(the1γexchange).In a previous paper[14]it has been shown that the present data do not show any evidence of the presence of the2γmechanism,in the limit of the experimental errors.The main reason is that,if one takes into account C-invariance and crossing symmetry,the2γmechanism introduces a non linear,very speciﬁcǫdependence of the reduced cross section [15,16,17],whereas the data does not show any deviation from linearity.Before analyzing the data in a diﬀerent perspective,we stress the following points:•No experimental bias has been found in both types of measurements,the experimental observables being the diﬀerential cross section on one side,and the polarization of the outgoing proton in the scattering plane(more precisely the ratio between the longitudinal and the transverse polarization),on the other side.•The discrepancy is not at the level of these observables:it has been shown that constraining the ratio R from polarization measurements and extracting G Mp from the measured cross section leads to a renormalization of2-3%with respect to the Rosenbluth data,well inside the error bars[10].•The inconsistency arises at the level of the slope of theǫdependence of the reduced cross section,which is directly related to G Ep,i.e.,the derivative of the diﬀerential cross section,with respect toǫ.The diﬀerence of such slope,derived from the two methods above,appears particularly in the last and precise data[6].One should note that the discrepancy appears in the ratio G E/G M,whereas G M,for example,decreases more than one order of magnitude from Q2=1to5GeV2.The starting point of this work is the observation of a correlation,which appears in the published FFs data extracted with the Rosenbluth method:the larger is G2E,the smallerFIG.1:Dependence of G2E/G2D versus G2M/µ2G2D:(a)for Q2≥2GeV2from Refs.[6](triangles), [7](stars)and[18](squares);(b)for Q2≤2GeV2from Refs.[19](circles),and[20](squares).G2M.This is especially visible in the most recent and precise experiments,at large Q2.The dependence of G2E/G2D versus G2M/µ2G2D is shown in Fig.1a for three recent data sets,at Q2≥2GeV2[6,7,18].In Fig.1b two data sets at low Q2(Q2≤2GeV2)are shown[19,20]. Whereas at low Q2,G2E/G2D seems constant and quite independent from G2M/µ2G2D,at large Q2an evident correlation appears.Polarization data show also a linearity of the ratio G E/G M,but with an opposite trend. In this case,the ratio is measured directly,whereas according to the Rosenbluth method, one extracts two(independent)parameters from a linearﬁt.A correlation between the two parameters could be induced by the procedure itself or could be a physical eﬀect and have a dynamical origin.In the last case,it should not depend on the experiment.In order to analyze this question in a quantitative way we have done a statistical study of the Rosenbluth data for several experiments.First of all,at large Q2,the contribution of the electric term to the cross section becomes very small,as the magnetic part is ampliﬁed by the kinematical factorτ.This is illustrated in Fig.2,where the ratio of the electric part(F E=ǫG2Ep)to the reduced cross section isshown as a function of Q2.The diﬀerent curves correspond to diﬀerent values ofǫ,assuming FFs scaling or in the hypothesis of the linear dependence(2).In the second case,one can see that,for example,forǫ=0.2,the electric contribution becomes lower than3%starting from2GeV2.This number should be compared with the uncertainty on the cross section measurement.When this contribution is larger or is of the same order,the sensitivity of the measurement to the electric term is lost and the extraction of G Ep becomes meaningless.Secondly,since theﬁrst measurements[21],the electromagnetic probes are traditionally preferred to the hadronic beams,as the electromagnetic interaction is exactly calculable in QED,and one can safely extract the information from the hadronic vertex.However,one has to introduce the radiative corrections,which become very large as the momentum transfer squared,Q2,increases.Radiative corrections wereﬁrstly calculated by Schwinger[22]and are important for the discussion of the experimental determination of the diﬀerential cross section.The measured elastic cross section,is corrected by a global factor C R,according to the prescription following[23]:(3)σred=C RσmeasredThe factor C R contains a largeǫdependence,and a smooth Q2dependence,and it is common for the electric and magnetic part.At the largest Q2considered here,this factor can reach 30-40%,getting larger when the resolution is higher.If one made a linear approximation for the uncorrected data,one might evenﬁnd a negative slope starting from Q2≥3GeV2[14].In Fig.3we show the C R dependence onǫ,for diﬀerent Q2and from diﬀerent set of data.One can see that C R increases withǫ,arising very fast asǫ→1.It may be diﬀerent in diﬀerent experiments,because its calculation requires an integration on the experimental acceptance.The Rosenbluth separation consists in a linearﬁt of the reduced cross section atﬁxed Q2, where the two parameters are G2E and G2M.Multiplying by a common factor,which depends strongly onǫ,the electric and magnetic term in(2)may induce a correlation between these two parameters.In order to test this hypothesis,we have built the error matrix for the Rosenbluthﬁts for diﬀerent sets of data available in literature.Atﬁxed Q2,the reduced cross section,normalized to G2D,has been parametrized by a linearǫdependence:σred/G2D=aǫ+b.The two parameters,a and b,have been determined for each set of data as well as their errorsσa,σb and the covariance,cov(a,b).The correlationFIG.2:Contribution of the G E dependent term to the reduced cross section(in percent)for ǫ=0.2(solid line),ǫ=0.5(dashed line),ǫ=0.8(dash-dotted line),in the hypothesis of FF scaling(thin lines)or following Eq.(2)(thick lines).coeﬃcientξ,is deﬁned asξ=cov(a,b)/σaσb and is shown in Fig.4as a function of the average of the radiative correction factor<C R>,weighted overǫ.As the radiative corrections become larger,the correlation between the two parametersFIG.3:Radiative correction factor applied to the data at Q2=3GeV2(squares)from Ref.[18], at Q2=4GeV2(triangles)and5GeV2(inverted triangles)from Ref.[5],and at Q2=0.32GeV2 from Ref.[20](circles).The lines are drawn to drive the eyes.becomes also larger,reaching values near its maximum(in absolute value).Full correlation means that the two parameters are related through a constraint,i.e.,it is possible toﬁnd a one-parameter description of the data.This does not necessarily means that the reducedcross section becomesﬂat,as a function ofǫ,but that the slope is related to a kinematical eﬀect,not to a dynamical one.The data shown here correspond to three sets of experiments,where the necessary infor-mation on the radiative corrections is available.The correlation coeﬃcient can be calculated for a larger number of data and one could plot the correlation as a function of Q2.However, diﬀerent experiments,at the same Q2,have been done at diﬀerent angles and energy,i.e., at diﬀerentǫ,and the radiative corrections which enter in the determination of a and b are diﬀerent.Such plot would be not easy to interprete.At low Q2a correlation still exists,but it is smaller.For the data from Ref.[20]the radiative corrections are of the order of15%,seldom exceed25%and correspond toǫ<0.8. This allows a more safe extraction of the FFs.Fig.4shows that,for each Q2,the extraction of FFs by a two parametersﬁt,may be biased by theǫdependence induced by the radiative corrections.Whatever the precision on the individual measurement is,the slope of the reduced cross section is not sensitive to G Ep at large Q2,which,therefore,can not be extracted from the data.The Q2dependence is therefore driven by G Mp,which follows a dipole form.For each Q2, a nonzero value of the ratio G Ep/G Mp will lead to an apparent dipole dependence of G Ep.To summarize,we reanalyzed the Rosenbluth data with particular attention to the radia-tive corrections applied to the measured cross section,and we showed,from the(published) data themselves that,at large Q2the contribution of G E to the cross section is so small that it can not be safely extracted.The method itself is biased,at large momentum transfer because the electric contribution to the measured cross section is in competition with the size of theǫdependent corrections.When plotting the reduced cross section as a function ofǫ,one,indeed,sees a nonzero slope,but it is due to theǫdependence contained in the radiative corrections,and it is no more primarily related to the inner structure of the proton.Therefore,the Rosenbluth method can not be used to extract the nucleon FFs at large momentum transfer,due to an intrinsic limitation deriving from the large size of the radia-tive corrections,compared to the electric contribution to the diﬀerential cross section,and especially to their steepǫdependence.In other words,there is a type of systematic error which becomes dominant and has never been included in the data,preventing the extrac-tion of G Ep.We conﬁrm the conclusion of a previous paper[4],whichﬁrstly suggested the polarization method for the determination of G Ep,due to the increased sensitivity of theFIG.4:Correlation coeﬃcient,ξ,as a function of the radiative correction factor<C R>,averaged overǫ,for diﬀerent sets of data:from Ref.[20](circles),from Ref.[5](triangles)and from Ref.[18](squares).cross section to the magnetic term,at large Q2:’Thus,there exist a number of polarization experiments which are more eﬀective for determining the proton charge form factor than isthe measurement of the diﬀerential cross section for unpolarized particles’.This work was inspired by stimulating discussions with M.P.Rekalo.Thanks are due to J.L.Charvet,G.I.Gakh and B.Tatischeﬀfor useful suggestions and a careful reading of the manuscript.[1]M.N.Rosenbluth,Phys.Rev.79,615(1950).[2]M.K.Jones et al.[Jeﬀerson Lab Hall A Collaboration],Phys.Rev.Lett.84,1398(2000).[3]O.Gayou et al.[Jeﬀerson Lab Hall A Collaboration],Phys.Rev.Lett.88,092301(2002).[4] A.Akhiezer and M.P.Rekalo,Dokl.Akad.Nauk USSR,180,1081(1968);Sov.J.Part.Nucl.4,277(1974).[5]L.Andivahis et al.,Phys.Rev.D50,5491(1994).[6]I.A.Qattan et al.,arXiv:nucl-ex/0410010.[7]M.E.Christy et al.[E94110Collaboration],Phys.Rev.C70,015206(2004).[8]J.Arrington,Phys.Rev.C68,034325(2003).[9]R.G.Arnold et al.,Phys.Rev.Lett.35,776(1975).[10] E.J.Brash,A.Kozlov,S.Li and G.M.Huber,Phys.Rev.C65(2002)051001.[11]P.G.Blunden,W.Melnitchouk and J.A.Tjon,Phys.Rev.Lett.91,142304(2003).[12]P.A.M.Guichon and M.Vanderhaeghen,Phys.Rev.Lett.91,142303(2003).[13]Y.C.Chen,A.Afanasev,S.J.Brodsky,C.E.Carlson and M.Vanderhaeghen,Phys.Rev.Lett.93,122301(2004).[14] E.Tomasi-Gustafsson and G.I.Gakh,arXiv:hep-ph/0412137.[15]M.P.Rekalo and E.Tomasi-Gustafsson,Eur.Phys.J.A.22,331(2004).[16]M.P.Rekalo and E.Tomasi-Gustafsson,Nucl.Phys.A740,271(2004).[17]M.P.Rekalo and E.Tomasi-Gustafsson,Nucl.Phys.A742,322(2004).[18]R.C.Walker et al.,Phys.Rev.D49,5671(1994).[19]Ch.Berger,V.Burkert,G.Knop,ngenbeck and K.Rith Phys.Lett.b1,87(1971).[20]T.Janssens,R.Hofstadter,E.B.Hughes and M.R.Yerian,Phys.Rev.142,922(1966).[21]R.Hofstadter,F Bumiller and M.Yearian,Rev.Mod.Phys.30,482(1958).[22]J.S.Schwinger,Phys.Rev.76,790(1949).[23]L.W.Mo and Y.S.Tsai,Rev.Mod.Phys.41,205(1969).。

CCP3SURF ACESCIENCENEWSLETTERCollaborative Computational Project3on Surface ScienceNumber26-January2000ISSN1367-370XDaresbury LaboratoryContents1Editorial1 2Scientiﬁc Articles2 3SRRTNet-a new global network124High performance computing164.1Cluster-Computing Developments in the UK (16)4.2New HPC Support Mechanisms (22)5Reports on visits256Meetings,Workshops,Conferences296.1Reports on Bursaries (29)6.2Reports from Meetings (32)6.3Upcoming meetings (34)7Abstracts of forthcoming papers37 8Surface Science Related Jobs40 9Members of the working group43 Contributions to the newsletter from all CCP3members are welcome and should be sent to ccp3@eful Links:CCP3Home Page /Activity/CCP3CCP3Program Library /Activity/CCP3+896 SRRTNet /Activity/SRRTNetDLV /Activity/DLVCRYSTAL /Activity/CRYSTALCASTEP /Activity/UKCPMany useful items of software are available from the UK Distributed Computing Support web site,DISCO /Activity/DISCOEditors:Dr.Klaus Doll and Dr.Adrian Wander,Daresbury Laboratory,Daresbury, Warrington,WA44AD,UK1EditorialThe renewal of CCP3,which is due in the summer,is on all our minds,and this edition of the newsletter reminds us of our aims and achievements.Theﬂagship project supported by the post-doc is at the heart of the CCP–new programs can be developed which would not get oﬀthe ground otherwise.Over the last three years CCP3has been very lucky to have had Klaus Doll working on the development of analytic gradient methods for the CRYSTAL electronic structure package.This will lead to much more eﬃcient structure optimization,of particular beneﬁt to surface scientists where surface relaxation and reconstructions are so important.Klaus’achievements so far are described in theﬁrst article,and it is most satisfactory that tests on the CO molecule and bulk MgO have proved successful.The next step is to build in symmetry,to achieve greater computational eﬃciency,and then the new code can be released.As theﬂagship in the renewal proposal,the working group has chosen the development of methods to study the electronic structure and physical properties of large clusters.These clusters themselves possess surface-like properties, but at the same time it is proposed to study their interaction with surfaces.Such systems are a topic of active research for several of the members of the working group,both theoretical and experimental,and it is expected that their expertise will contribute greatly to the success of the project.Having Adrian Wander as a permanent member of staﬀat Daresbury supporting CCP3will lead to very welcome support for the synchrotron radiation community in the development of new surface program packages.In this newsletter he describes developments in SRRTNet,originally an American collaboration for providing sup-port for surface scientists using synchrotron radiation,which is likely to develop into an international collaboration based on the CCP model.Adrian is also in discussion with the Daresbury-based X-ray community with a view to developing new codes for the analysis of near-edge spectra in a wider range of systems than can be tackled at the moment,using the improved self-consistent electron potentials available for complex materials.This work would be based at Daresbury,and will form part of the CCP3collaboration.This issue contains short articles on our visitor programme,and by Ally Chan (Nottingham)and Yu Chen(Birmingham)who received student bursaries for participating in ECOSS-18.Please continue to apply for CCP3support!It is interesting to read in the pieces by Ally and Yu what most impressed them at ECOSS–I was struck by Ally’s comment that surface science has broadened to include nanoparticles and nanowires.Just what we thought in our choice ofﬂagship project next time round.John Inglesﬁeld12Scientiﬁc ArticlesAnalytical Hartree-Fock gradients for periodic systemsK.Doll,V.R.Saunders,and N.M.HarrisonCLRC,Daresbury Laboratory,Daresbury,Warrington,WA44AD,UKWe report on the progress of the implementation of analytic gradients in the program package CRYSTAL.The algorithm is brieﬂy summarised and tests illustrate that highly accurate analytic gradients of the Hartree-Fock energy can be obtained for molecules and periodic systems.IntroductionComputational materials science has been a fast growingﬁeld in the last years. This is mainly because methods which were developed earlier(density functional theory,molecular dynamics,Hartree-Fock and correlated quantum chemical meth-ods,Monte Carlo schemes,the GW method,etc)can now be applied to demanding realistic systems due to the increase in computational resources(faster CPUs,par-allelisation,cheaper memory and diskspace).CCP3is a collaboration in the area of surfaces and interfaces where progress de-pends on an interaction between experimental and theoretical approaches.There-fore,codes which provide a better theoretical understanding are important.One of the key issues in surface science is the determination of surface structure and adsorption energetics.From the computational point of view,a fast structural optimisation must be possible.Availability of numerical or analytical gradients facil-itatesﬁnding a minimum energy structure,and availability of analytical gradients can make optimisation algorithms more eﬃcient.As a rule of thumb,analyticalgradients are about N3times more eﬃcient than numerical gradients(with N beingthe number of variables).Also,for future developments such asﬁnding transition states,gradients are essential.Analytical gradients in quantum chemistry were pioneered by Pulay who did the ﬁrst implementation for multicentre basis sets[1].In many molecular codes based on quantum chemistry methods,analytical gradients are now implemented and gradient development has become an important task in quantum chemistry[2,3,4,5].Simi-larly,in solid-state codes such as CASTEP,WIEN,or LMTO,analytic gradients are available.Analytic Hartree-Fock gradients have already been implemented in a code for systems periodic in one dimension[6].CRYSTAL[7,8]was born in Turin and is now jointly developed in Turin and Daresbury.CRYSTAL was initially designed to deal with the exact exchange in and to solve the Hartree-Fock equations for real systems.With the modern versions of the code,density-functional2calculations or calculations using Hybrid functionals such as B3LYP with the ad-mixture of exact exchange are also possible.The target of this project,which began in October1997,was the implementation of analytical gradients in CRYSTAL and in autumn1999,theﬁrst test calculations on periodic systems were performed.In this article,we try to outline the theory and implementation of analytical gra-dients.We try to keep the mathematics at a minimum;a more formal publication is intended in the near future[9].A very comprehensive summary of the theory underpinning CRYSTAL will appear in the future[10].Total energyFirst,we want to brieﬂy summarise how the total energy is obtained.The total energy consists of•kinetic energy of the electrons•nuclear-electron attraction•electron-electron repulsion•nuclear-nuclear repulsionCRYSTAL,similar to molecular codes such as GAMESS-UK,MOLPRO(Stuttgart and Birmingham),GAUSSIAN,TURBOMOLE,etc,solves the single particle Schr¨o dinger equation and a wavefunction is calculated.The wavefunction is based on crystalline orbitalsΨi( r, k)which are linear combinations of Bloch functionsΨi( r, k)= µaµ,i( k)ψµ( r, k)(1) with the Bloch functions constructed fromψµ( r, k)= gφµ( r− Aµ− g)exp(i k g)(2) g are direct lattice vectors, Aµdenotes the coordinate of the nuclei.φµare the basis functions which are Gaussian type orbitals.For example,an s-type function centred at R a=(X a,Y a,Z a)is expressed asφ(α, r− R a,n=0,l=0,m=0)=φµ( r− R a)= Nexp(−α( r− R a)2).In molecular calculations,no mathematical problem arises from any of the interac-tions.In periodic systems,however,there are several divergent terms which have to3be dealt with:for example,in a one dimensional periodic system with lattice con-stant a and n being an index numerating the cells,the electron-electon interaction per unit cell would have contributions like:∞n=1e2na(3)This sum is divergent(similarly in two and three dimensions).Therefore,an indi-vidual treatment of this term is not possible.Instead,all the charges(nuclei and electrons)are partitioned and a scheme based on the Ewald method is used to sum the interactions[11].The Hartree-Fock equations are solved in terms of Bloch functions because the Hamiltonian becomes block-diagonal(i.e.at each k-point the equations are solved independently).The wavefunction coeﬃcients aµ,i are optimised due to this procedure and the total energy can be evaluated.For the computation of gradients,the dependence of the total energy on the nuclear coordinates must be analysed.There are three dependencies of the total energy on the nuclear coordinates:•nuclear-nuclear repulsion and nuclear-electron attraction:obviously,the coor-dinates of the nuclei enter•wavefunction coeﬃcients(or density matrix):we will obtain a diﬀerent solution with diﬀerent density matrix when moving the nuclei•basis functions:the basis functions are centred at the position of the nu-clei and therefore moving the nuclei will change integrals over the basis func-tions.These additional terms are called Pulay forces.They are missing when the Hellmann-Feynman theorem is applied and therefore Hellmann-Feynman forces often diﬀer substantially from energy derivative forces in the case of a local basis set(see[1]and references therein).Density matrix derivatives are diﬃcult to evaluate.However,for the solution of the Hartree-Fock equations,this problem can be circumvented and instead a new term is introduced,the so-called energy-weighted density matrix which is easily evaluated [12].However,this is only strictly correct for the exact Hartree-Fock solution. In practice,convergence is achieved up to a certain numerical threshold(e.g.a convergence of10−6E h of the total energy corresponding to27.2114×10−6eV).For very accurate gradient calculations,it may be necessary to make this threshold even lower.The remaining main problem is to generate all the derivatives of the integrals. In a second step,these derivatives have to be mixed with the density matrix.4Evaluation of integralsIn this section we summarise the types of integral which occur.The simplest type is the overlap integral between two basis functions at two centres:Sµν R k R l= φµ( r− R k)φν( r− R l)d3r(4) Obviously we can shift R k to the origin,and suppressing 0in the notation,we obtain:Sµν R i= φµ( r)φν( r− R i)d3r(5) with R i= R l− R k.A kinetic energy integral has the form:Tµν R i= φµ( r)(−12∆ r)φν( r− R i)d3r(6) the nuclear attraction integral has the form:Nµν R i= φµ( r)Z c| r− A c|φν( r− R i)d3r(7) and the electron-electron interaction has the form:Bµν R iτσ R j = φµ( r)φν( r− R i)φτ( r′)φσ( r′− R j)| r− r′|d3rd3r′(8)These integrals are in principle suﬃcient to deal with molecules.In the case of periodic systems,new types of integrals appear(e.g.multipolar integrals,integrals over the Ewald potential and its derivatives)[11,13,14].The fast evaluation of integrals is one of the main issues in the development of quan-tum chemistry codes.CRYSTAL uses a McMurchie-Davidson algorithm[15].Its idea is to map a product of two Gaussian type orbitals at two centres in an expan-sion of Hermite polynomials at an intermediate centre.This algorithm has proven to eﬃciently evaluate integrals,although in recent years progress in this specialised ﬁeld of quantum chemistry has been made(see for example the introduction in[16] or two recent reviews[17,18]).The expansion[15,14]looks like:5φ(α, r− A,n,l,m)φ(β, r− B,n′,l′,m′)= t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)(9)withγ=α+βand P=α A+β Bα+β.Λis a so-called Hermite Gaussian type function Λ(γ, r− P,t,u,v)= ∂∂P x t ∂∂P y u ∂∂P z v exp(−γ( r− P)2)(10)The start value E(0,0,0,0,0,0,0,0,0)=exp(−αβ( B− A)2)can be veriﬁed by inserting it in equation9.It can be derived from the Gaussian product rule[19,20]:exp(−α( r− A)2)exp(−β( r− B)2)=exp −αβα+β( B− A)2 exp −(α+β) r−α A+β Bα+β 2(11) General values E(n,l,m,n′,l′,m′,t,u,v)are obtained from recursion relations[15, 14].The E-coeﬃcients depend on the distance( B− A),but not on P or r.All the integrals can be expressed in terms of E-coeﬃcients[15,14,11,13].Evaluation of gradients of the integralsOne of the issues of the gradient project is to generalise the algorithms used to generate the energy integrals to obtain the gradients of the integrals.This madea new implementation of recursion relations necessary which are used to obtainthe coeﬃcients G in the expansion of the gradients of the integrals in Hermite polynomials.∂Φ(α, r− A,n,l,m)Φ(β, r− B,n′,l′,m′)∂A x= t,u,v G A x(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)(12)Once the coeﬃcients are known,the integration can be performed.The integrationfor the case of gradients of integrals is similar to the case of integrals for the total energy.The only diﬀerence is that,instead of the coeﬃcientsE(n,l,m,n′,l′,m′,t,u,v)which enter the energy expression,the gradient coeﬃcientsG A x(n,l,m,n′,l′,m′,t,u,v),G A y,G A z,G B x,G B y,and G B z6are used.The coeﬃcients G B x can eﬃciently be obtained together with the coeﬃ-cients G A x[21].For example,the evaluation of the overlap integral is done as follows:Sµν R i = φµ( r)φν( r− R i)d3r=t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)d3r=E(n,l,m,n′,l′,m′,0,0,0)Λ(γ, r− P,0,0,0)d3r=ME(n,l,m,n′,l′,m′,0,0,0)From theﬁrst line to the second,we have used the McMurchie-Davidson scheme, from the second to the third we exploited a property of the Hermite Gaussian type functions:all the integrals of the type Λ(γ, r− P,t,u,v)d3r with t=0or u=0or v=0vanish because of the orthogonality of these functions.The integration(fromthe third to the fourth line)is trivial.M is a normalisation constant. Calculating the gradient is easy once we know the new expansion:∂Sµν R i ∂A x =∂∂A xφµ( r)φν( r− R i)d3r=∂ t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)∂A x d3r=t,u,v G A x(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)d3r=G A x(n,l,m,n′,l′,m′,0,0,0)Λ(γ, r− P,0,0,0)d3r=MG A x(n,l,m,n′,l′,m′,0,0,0)This way,all the derivatives can be calculated!There are some integrals which involve three centres(for example nuclear attraction)where we exploit translational invariance:∂∂C x =−∂∂A x−∂∂B x(13)because the value of the integral is invariant to a simultaneous uniform translation of the three centres.Four centre integrals can be reduced to a product of two integrals over two centres which makes the calculation of gradients straightforward.As a whole,the calculation of gradients of the integrals is closely related to calcu-lating the integrals itself.This means that most of the subroutines can be used for7the gradient code.One of the main diﬀerences is that array dimensions need to be changed-dealing with gradients is similar to increasing the quantum number(a derivative of an s-function is a p-function,and so on).However,the task of adjusting the subroutines should not be underestimated.After obtaining the derivatives of the integrals,we mix them with the density ma-trix just like in the energy calculation.We have to take into account the new term which arose because we did not calculate a density matrix derivative—the energy weighted density matrix.Again,coding this additional term can be done by modi-fying existing subroutines.After this,weﬁnally obtain the forces on the individual atoms.Results from test calculationsIn this section,we summarise results from test calculations.We have considered the CO molecule which was arranged as a single molecule,as a molecule which is peri-odically reproduced with a periodicity of4˚A in one spatial direction(”polymer”), periodically reproduced with a periodicity of4˚A in two spatial directions(”slab”), and periodically reproduced with a periodicity of4˚A in three spatial directions (”solid”).Because of the large distance of4˚A,the molecules can be considered as nearly independent and the forces are quite similar.Still,the calculation of energy and gradient is completely diﬀerent and therefore this is an important test of Ewald technique and multipolar expansion.The results are given in table1.The results agree in the best case to at least6digits which is the numerical noise and in the worst case up to4digits.The diﬀerence between analytic and numerical gradi-ents in periodic systems mainly originates from an approximation made within the evaluation of the integrals[22]and from the number of k-points which aﬀects the accuracy of the energy-weighted density matrix.In table2,we display results from a MgO solid with one oxygen atom slightly distorted from the symmetrical position.Again,the forces agree well up to5digits with numerical derivatives.Future developments and ConclusionThe present version of the code is able to calculate Hartree-Fock forces for periodic systems up to a precision of4and more digits.There is no extra diskspace needed and the additional memory usage is moderate.This code will certainly be useful for structural optimisation and for future program development towards molecular dynamics or the calculation of response functions.The present version,however,is not yet ready for a release.Instead,the following steps are necessary:Firstly,the usage of symmetry must be implemented.This is of highest importance to make the code fast enough so that it can be used for practical optimisations.We expect8Table1:Force on a CO molecule with a carbon atom located at(0˚A,0˚A,0˚A)and an oxygen atom located at(0.8˚A,0.5˚A,0.4˚A).In the periodic case,the molecule is generated with a periodicity of4˚A.This means,that in one dimension,for example,there would be other molecules with a carbon atom at(n×4˚A,0˚A,0˚A)and an oxygen atom at((n×4+0.8)˚A,0.5˚A,0.4˚A),with n running overall positive and negative integers.Forces are given in E h,with E h=27.2114eVand a0=0.529177˚A.Higher ITOLs means a lower level of approximation in the evaluation of the integrals[22].ITOLs) k-points) numerical force0.3769140.37660(0.37664)0.376310.37566(0.37566) analytical force0.3769130.37663(0.37665)0.376330.37588(0.37578))on the atoms of an MgO solid.The MgO solid was chosen Table2:Forces(in E ha0to have an artiﬁcially high lattice constant of6.21˚A to make the calculation faster. Coordinates are given in fractional units,e.g.the second Mg is at0˚A,0.5×6.21˚A,0.5×6.21˚A.A normal fcc lattice would be obtained if the sixth atom(Oxygenat0.53,0,0)was at(0.5,0,0).Moving this atom from its normal position has ledto the nonvanishing forces.Mg(0.00.00.0)-0.03018-0.03019Mg(0.00.50.5)-0.00314-0.00314Mg(0.50.00.5)0.008950.00895Mg(0.50.50.0)0.008950.00895O(0.50.50.5)-0.00379-0.00379O(0.530.00.0)0.004290.00430O(0.00.50.0)0.007460.00746O(0.00.00.5)0.007460.00746that a version of the present code with symmetry will already be fast enough to compete with numerical derivatives.Further developments will be the coding of the bipolar expansion(a method to evaluate the electron-electron repulsion integrals faster),and sp-shells(s and p shells are often chosen to have the same exponentsto make the evaluation of integrals faster).Also,the newly written subroutines arenot yet optimal and they will certainly go through a technical optimisation(moreeﬃcient coding).In later stages,the code should be made applicable to metals (there is an extra term coming from the shape of the Fermi surface[23]which is notyet coded)and to magnetic systems(unrestricted Hartree-Fock gradients).Finally, pseudopotential gradients and density functional gradients should be included. References[1]P.Pulay,Mol.Phys.17,197(1969).[2]P.Pulay,Adv.Chem.Phys.69,241(1987).[3]P.Pulay,in Applications of Electronic Structure Theory,edited by H.F.Schae-fer III,153(Plenum,New York,1977).[4]H.B.Schlegel,Adv.Chem.Phys.67,249(1987).[5]T.Helgaker and P.Jørgensen,Adv.in Quantum Chem.19,183(1988)[6]H.Teramae,T.Yamabe,C.Satoko,A.Imamura,Chem.Phys.Lett.101,149(1983).[7]C.Pisani,R.Dovesi,and C.Roetti,Hartree-Fock Ab Initio Treatment of Crys-talline Systems,edited by G.Berthier et al,Lecture Notes in Chemistry Vol.48(Springer,Berlin,1988).[8]V.R.Saunders,R.Dovesi,C.Roetti,M.Caus`a,N.M.Harrison,R.Orlando,C.M.Zicovich-Wilson crystal98User’s Manual,Theoretical Chemistry Group, University of Torino(1998).[9]K.Doll,V.R.Saunders,N.M.Harrison(in preparation)[10]V.R.Saunders,N.M.Harrison,R.Dovesi,C.Roetti,Electronic StructureTheory:From Molecules to Crystals(in preparation)[11]V.R.Saunders,C.Freyria-Fava,R.Dovesi,L.Salasco,and C.Roetti,Mol.Phys.77,629(1992).[12]S.Bratoˇz,in Calcul des fonctions d’onde mol´e culaire,Colloq.Int.C.N.R.S.82,287(1958).[13]R.Dovesi,C.Pisani,C.Roetti,and V.R.Saunders,Phys.Rev.B28,5781(1983).[14]V.R.Saunders,in Methods in Computational Molecular Physics,edited by G.H.F.Diercksen and S.Wilson,1(Reidel,Dordrecht,Netherlands,1984).[15]L.E.McMurchie and E.R.Davidson,put.Phys.26,218(1978).[16]R.Lindh,Theor.Chim.Acta85,423(1993).[17]T.Helgaker and P.R.Taylor,in Modern Electronic Structure Theory.Part II,World Scientiﬁc,Singapore,725(1995)[18]P.M.W.Gill,in Advances in Quantum Chemistry,edited by P.-O.L¨o wdin,141(Academic Press,New York,1994)[19]S.F.Boys,Proc.Roy.Soc.A200,542(1950).[20]R.McWeeny,Nature166,21(1950).[21]T.Helgaker and P.R.Taylor,Theor.Chim.Acta83,177(1992).[22]The integrals Bµν R iτσ R j =Bτσ R jµν R iwhich should have the same value,are notnecessarily evaluated within the same level of approximation—this is nearly inevitable for periodic systems,as enforcing this symmetry would require a much higher computational eﬀort and much more data storage.The derivation of the equations for the analytic gradients,however,relies on these integrals be-ing equivalent.Therefore,the introduced asymmetry will lead to inaccuracies in the gradients.This can be controlled with the ITOL-parameters(tolerances as described in the CRYSTAL manual[8])which control the level of approx-imation.Higher ITOLs lead to a higher accuracy in the forces.However,the defaults appear to give forces with an accuracy up to4digits which should be good enough for most purposes.[23]M.Kertesz,Chem.Phys.Lett.106,443(1984).3SRRTNet-a new global networkFrascati’99-Birth of a NetworkScientiﬁc MeetingFrom the23rd to the25th September1999,a workshop on Theory and Computation for Synchrotron Radiation was held at the laboratory in Frascati just outside Rome, Italy.This was the third in an ongoing series of meetings on various aspects of synchrotron radiation,and follows meetings on Theory and Computation for Syn-chrotron Applications held at the Advanced Light Source in Berkeley in October 1997and Needs for a Photon Spectroscopy Theory Center held at the Argonne National Laboratory in August1998.This was an excellent meeting,featuring a variety of high quality scientiﬁc pre-sentations from both experimental and theoretical participants.Theﬁrst day was devoted to presentations concentrating on resonant x-ray processes and orbital or-dering eﬀects,particularly in V2O5.The second day then moved on to discussions of photoemission,photoelectron diﬀraction and holography,and studies of high T c superconductors.This day was concluded with an excellent conference dinner which ﬁnished rather late!Theﬁnal day then concluded with discussions of EXAFS,and x-ray spectroscopies.The overheads used in all the presentations can be viewed on line at http://wwwsis.lnf.infn.it/talkshow/srrtnet99.htmSRRTNet DiscussionsThe Friday programme also included a two hour session devoted to the idea of forming a global network concentrating on theory for synchrotron radiation re-search based research.The session began with a talk from Michel Van Hove of the Lawrence Berkeley National Laboratory who outlined the purposes and function of the proposed network.This was then followed by presentations by John Rehr of the University of Washington who highlighted moves to extend the synchrotron radia-tion research theory network(SRRTNet)in North America,by Maurizio Benfatto of the INFN Frascati,who presented the European perspective,by Kenji Makoshi of Himeji Institute of technology who discussed the Japanese eﬀorts and by Adrian Wander of the Daresbury Laboratory who presented CCP3as a possible model of how the network could be run.The concept of establishing a global network was received with enthusiasm from all present.OutcomeGiven the support of the meeting for the concept of global network of this sort,it was decided to extend SRRTNet into the global arena.The aims of the network are:•To provide a central repository for information of relevance to synchrotron radiation research•To develop theoretical methods pertaining to the experiments performed on synchrotron facilities•To provide state of the art and user friendly software for the analysis and interpretation of experiments•To provide training in the use of relevant software through workshops and site visits•To host visiting scientists•To hold periodic workshops for the dissemination of new results and method-ologiesThe directors of the network are Michel Van Hove and John Rehr.As aﬁrst step in the development of the network,Daresbury has agreed to host the web pages, and theoretical groups have been contacted and ask to provide input to this central web hub of what will grow into a globe encompassing network.If you are interested in contributing to the network and missed our e-mail announcement,the invitation letter follows;Dear Colleague,You may know of the recently established Synchrotron Radiation Research Theory Network(SRRTNet).We are contacting you to invite you,and all theorists inter-ested in this topic,to actively participate in the next phase of the network. SRRTNet aims to provide theory for experiments that use synchrotron radiation,by means of a global,web-based network linking theoretical and experimental research groups.The driving philosophy is to promote interactions between theory and exper-iment for mutual beneﬁt,by means of web-based information,workshops,exchange of theoretical methods and computer codes,as well as establishing visiting scientist programs.At the last SRRTNet workshop,conducted at Frascati near Rome in September1999, it was decided to strengthen the global character of this network by establishing a cen-tral,web-based source of information.Daresbury Laboratory is hosting this web site with Dr.Adrian Wander acting as editor.It is anticipated that all synchrotron facilities will provide direct links for their users to this web site,and consequently we expect this site to grow into an essential resource for synchrotron radiation re-searchers.An importantﬁrst function of the web site will be to provide information about theorists’research interests and links to relevant web pages.The network will be all the more valuable as this coverage becomes complete:it will thus allow theorists and experimentalists alike toﬁnd the best sources of information about the various methods for solving speciﬁc scientiﬁc problems.The purpose of this message is to ask you to provide such information and links about your group.You may visit the new web site/Activity/SRRTNetand see not only an overview of the network in general,but also the beginnings of such information about speciﬁc theoretical groups.The idea is to put a list of your research topics on the SRRTNet web site,while providing links to your own web site for more detailed and up-to-date information. If you prefer,the SRRTNet site can itself host a more complete web page covering your activities.The information we wish to present(or link to)includes as many as possible of the following items:•your topics of scientiﬁc activity related to synchrotron radiation(directly or by methodology);•your computer codes,with their capabilities and availability;•your publications,such as abstracts,papers,databases and web-presentations;•how to contact you or your group.。

代数中常用英语词汇

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

操作性DRIFT光谱表征通过非热等离子体辅助催化从空气中除去VOC中的催化剂表面上的中间物质

O R I G I N A L P A P E ROperando DRIFT Spectroscopy Characterizationof Intermediate Species on Catalysts Surface in VOCRemoval from Air by Non-thermal Plasma AssistedCatalysisAnthony Rodrigues 1•Jean-Michel Tatiboue¨t 1•Elodie Fourre ´1Received:3February 2016/Accepted:11May 2016/Published online:20May 2016ÓSpringer Science+Business Media New York 2016Abstract An innovative plasma discharge reactor was developed to ﬁt an infrared cell and to allow the in situ characterization of isopropanol (IPA)and toluene decomposition at the surface of three metal oxides (c -Al 2O 3,TiO 2and CeO 2).The impact of the plasma dis-charge on the conversion of these pollutants,with the material placed in the discharge area,was studied under real time conditions at atmospheric pressure via infrared analysis.The plasma treatment of IPA molecules led to the formation of acetone,propene,acetic acid and/or formic acid.By contrast,the toluene oxidation led to the rapid opening of the aromatic ring,followed by the total oxidation through carboxylic formation of the species arising from the toluene molecules fragmentation.Keywords In-situ characterization ÁInfrared spectroscopy ÁNon-thermal plasma ÁVOC IntroductionNon thermal atmospheric plasma (NTAP)is now recognized as an innovative technology applied to a wide range of applications (etching,deposition of thin ﬁlm,volatile organic compound oxidation,surface functionalization,biomedical …)[1].The advantageous properties of NTAP,in terms of energetic and environmental aspects,are responsible for the enhanced research in this ﬁeld.Non thermal atmospheric plasmas are produced by a variety of electrical discharges or electron beams leading to plasmas such as corona dis-charges,gliding arc discharges,dielectric barrier discharges (DBD),plasma needle,plasma jets and micro hollow cathode discharges [2].Non thermal plasmas are generated by the&Elodie Fourre ´elodie.fourre@univ-poitiers.fr1Institut de Chimie des Milieux et Mate´riaux de Poitiers (IC2MP),UMR CNRS 7285,Ecole Nationale Supe´rieure d’Inge ´nieurs de Poitiers (ENSIP),Universite ´de Poitiers,1,rue marcel Dore ´,TSA 41105,86073Poitiers Cedex 9,FrancePlasma Chem Plasma Process (2016)36:901–915DOI 10.1007/s11090-016-9718-1application of a high potential difference between two electrodes that leads to a strong electricalﬁeld,resulting in the formation of a highly reactive environment favorable to various reactions.For decades,non-thermal atmospheric plasma was used mainly for the production of ozone and water puriﬁcation[3].More recently,due to a greater concern on environmental issues,research on waste gas treatment,volatile organic compounds (VOCs)removal has been largely studied.In particular,the presence of a catalyst in the plasma discharge resulted in a synergistic effect and improved VOCs conversion[4–6]. However,if research has shown tremendous results on VOCs conversion,a few points remain unclear.It concerns essentially the interactions between the chemical species generated in the plasma discharge and a material surface.The willingness to identify active species in the plasma discharge is very challenging.It is a complexﬁeld of research,as the species concentration,nature and energetic power are dependent of the plasma reactor conﬁguration,chemical and electrical parameters.Already,research has been dedicated to in situ identiﬁcation of plasma species in the gaseous phase by spectroscopic methods[7–15].Most of these studies are dealing with the characterization of the plasma species but not with their reactivity.Rivallan et al.[11]have studied the evolution of air/CO2gas mixture under non-thermal plasma by in situ FTIR at the microsecond time-scale.They showed that the mechanism of CO2consumption is reversible and that CO2molecules are excited through a collision with excited N2molecules,where the transfer of energy occurs by a resonant effect.Other authors tried to elucidate some aspects of the ion chemistry occurring in dielectric barrier and corona discharges with benzene/air[7]or different gases such as gas mixtures including Ar,H2,N2,O2and CH4as plasma precursors[8].OH radical concentration was determined by indirect detection with salicylic acid[13]and direct measurement by laser-inducedﬂuorescence[12].Kim et al.[15]have reviewed recent progresses in understanding the interaction of NTP and catalysts.They insisted on the effects of the electrical characteristics of the catalyst on the plasma discharge gener-ation and therefore chemical reactivity.In a recent publication[14],we have shown that the decomposition of isopropanol(IPA)by non-thermal plasma on c-Al2O3,analyzed by operando Fourier Transform Infrared(FTIR)under low vacuum could proceed via oxi-dation,aldolization and fragmentation mechanisms.Such cascade reactions were only possible when c-Al2O3catalyst was placed in the discharge zone.This VOC has been studied and published in the literature with results essentially based on the analysis of the outlet gas composition.IPA conversion leads to the formation of acetone,acetic acid, formaldehyde(when the reaction is incomplete),formic acid,CO2and H2O[14–19].Al-Abduly and Christensen[20]reported a spectroscopic study of non-thermal plasma in an air-fed dielectric barrier discharge plasma jet.In situ analysis of the plasma glow and downstream gas by FTIR revealed the presence of various species such as O3,N2O5,N2O, HNO3,CO2,CO and a vibrationally excited form of CO2[i.e.CO*2(v)].Here we are dealing with a non-thermal plasma reactor set at atmospheric pressure and coupled to reﬂectance infrared spectroscopy.The test reactions studied in this work con-cerned the removal of isopropanol and toluene from c-Al2O3,TiO2and CeO2surfaces by non-thermal plasma.The impact of the plasma discharge on the conversion of these pollutants,with the material placed in the discharge area,was studied under real time conditions at atmospheric pressure via infrared analysis.Fig.1a Schematic of the IR/plasma cell.b Views of the disassembled cell(a–d)ExperimentalA non-thermal plasma at atmospheric pressure(NTPAP)reactor was developed and optimized for its combination with a DRIFT spectrometer.The spectrometer was modiﬁed in order to carry out a DBD plasma in the DRIFT cell while a direct surface analysis was carried out.The cell was made of Polytetraﬂuoroethylene(PTFE)and included a cylin-drical base(30mm diameter,29mm height)and a conical top(26mm height).The cone was equipped with two ZnSe windows,to allow the passage of the IR beam,and one glass window(Fig.1a)to view the discharge.The volumic dielectric barrier discharge plasma consisted in a tip electrode(Fig.1b,tip diameter0.5mm)held in the centre of the cone and a counter electrode(10mm910mm copper tape)placed under a dielectric(0.5mm thick in PEEK:polyether ether ketone).The sample was placed on the dielectric,under-neath the tip.The gap between the tip and dielectric surface was kept constant(1mm).The electrodes were connected,via a capacitor of known value(C m=0.56nF),to a high voltage power ampliﬁer(Trek,20/20A)coupled to a function generator(TTi TG1010A). Experiments were carried out in an airtight cell with the possibility to vary the chemical and electrical parameters.FTIR Spectrometer Frontier(PerkinElmer,UK),equipped with a Mercury Cadmium Telluride(MCT)detector,was used to record IR spectra in the 4000–900cm-1range with a resolution of8cm-1.Each spectrum is the accumulation of 50analyses allowing an acquisition time below1min.The contributions of gaseous H2O and CO2were subtracted when scanning.Before each experiment the cell was purged with dry air for10min(30mL min-1)before the initiation of the plasma discharge.IR spectra of the sample wafer were then recorded in reﬂection mode(DRIFT:Diffuse Reﬂectance Infrared Fourier Transform)as a function of the duration of DBD plasma.Scans were taken while the plasma treatment was switched on and over a period of time going from1min to 8h.All the experiments were carried out at atmospheric pressure.The DBD experiments were studied under‘‘dynamic’’conditions,aﬂow rate of 30mL min-1of gas(dry air,Air liquide)was constantly running in the cell chamber.The power injected in the reactor was determined by the analysis of the Lissajous ﬁgure(Fig.2)that reports the transferred electrical charge Q m as a function of the appliedFig.2Lissajous curve at U a=16kV and f=500Hzvoltage(U a)using the so-called‘‘Manley method’’[21].Q m was calculated from the measured voltage(U m)across the capacitor(C m)connected in series to the ground elec-trode.The energy injected,E(per cycle)was calculated from the integration of the area formed by the Lissajousﬁgure.The energy allowed the calculation of the power(in W) injected in the reactor as well as the speciﬁc energy E spe(in J L-1)following Eqs.1and2, where f is the frequency(in Hz)and D the gasﬂow rate in(L s-1):P¼EÂfð1ÞE spe¼PDð2ÞThe experiments were carried out at16kV(500Hz),where the plasma discharge was stable,homogeneous and sufﬁciently energetic to promote excited species.These param-eters correspond to a power of24mW and an energy density of48J L-1.The catalyst sample was in the form of a thin layer of powder(50–100mg)slightly pressed on the dielectric plate in order to avoid its spraying under the effect of gasﬂow rate and plasma.The three oxides were commercial oxides(Table1).The speciﬁc surface area of the samples was determined by the BET(Brunauer–Emmett–Teller)method from the nitrogen adsorption isotherms at-196°C in an automated Micromeritics Tristar3000 apparatus after drying for8h at400°C.ResultsElectrical parameters of the plasma generated in the IR cell were determined with and without the presence of a catalytic material.The inﬂuence of the material on the plasma discharge was evaluated by comparing the variation of the plasma power as a function of applied voltage(Fig.3).A slight decrease of the power is noticed when Al2O3is placed in the discharge,likely due to the slight modiﬁcation of the dielectric properties of the dielectric plate when covered by alumina.Isopropanol RemovalAll the surfaces wereﬁrst placed in contact with a gas mixture air/IPA(100ppm)at atmospheric pressure for15min.This was followed by a10minﬂow of air prior to the plasma treatment,also under airﬂow.Figure4a displays the evolution of the IR spectra of Al2O3as a function of treatment time,from1to60min.The same experiment was carried out on the two other oxide surfaces of CeO2and TiO2(Figs.5a,6a).Additionally,spectra of three likely products(acetic acid,formic acid and acetone)were recorded and are displayed in Figs.4b,5b and6b.The spectra were recorded after10min adsorption of a mixture of air/product(100ppm)at atmospheric pressure followed by a10min purge under airﬂow.The IR spectra were obtained after subtraction of the background spectrum Table1Catalystscharacteristics Oxide Supplier Surface area(m2g-1)c-Al2O3Degussa102TiO2P25Sigma Aldrich9CeO2Prolabo23of the sample under air before adsorption.For the three samples,after IPA adsorption and before plasma treatment,the predominant bands corresponding to the C–H asymmetric and symmetric stretching (m -as and m -s)were detected between 3030and 2840cm -1and between 1510and 1300cm -1for the methyl C–H asymmetric and symmetric bends (d -as and d -s).Between 1255and 1205cm -1,d –O–H from un-dissociated IPA was identiﬁed.Other components,with lower intensity,around 1150and 1075cm -1were related to the C–O stretch and C–C skeletal of isopropoxide species,formed via dissociative adsorption of the IPA molecule or/and to molecularly adsorbed IPA [16,22–25].The negative bands in the 3750–3600cm -1region,corresponding to the surface hydroxyl groups of metallic oxides show that IPA adsorption can occur either as isopropoxy group (Scheme 1a)or by coordination with surface OH groups (Scheme 1b).The remaining negative band even after oxidation of IPA by non-thermal plasma strongly suggests that the IPA is mostly adsorbed as an isopropoxy species (Scheme 1b)rather than by coordination with surface OH groups [26–28].As soon as the discharge was initiated,the bands corresponding to the methyl group stretching (2975–2840cm -1region)decreased as a function of the plasma treatment time,whatever the sample.After 60min of plasma treatment,a residual C–H contribution is still visible which results from the for-mation of decomposition compounds such as acetone,aldehydes and/or acids.A very broad band going from 3340to 2250cm -1emerged after 5min of treatment for TiO 2and CeO 2and 20min for c -Al 2O 3which corresponds to the O–H stretch of carboxylic acid.This band is prominent on c -Al 2O 3and similarly,but less evident,on TiO 2and CeO 2.In the lower wavenumber region,the analysis is more difﬁcult due to the overlapping of the bands.However,the progressive formation (from 1to 60min treatment)of a band corresponding to a C=O group (ketone,aldehyde and/or carboxylic acid)and centered at 1730,1760and 1712cm -1for c -Al 2O 3,TiO 2and CeO 2,respectively,is clearly identiﬁed.Regarding c -Al 2O 3and TiO 2,a shoulder within the ketone region band is visible at alower Fig.3Injected power in the DBD reactor as a function of the input voltage at atmospheric pressure,under a constant dry air ﬂow of 30mL min 21wavenumber (1695and 1730cm -1)and correlates closely to the C=O band of acetic and/or formic acids.CeO 2presents a much wider band,centered at 1712cm -1,encompassing the C=O contribution of the acids.Bands of methyl bending vibrations (d )evolved differently.On c -Al 2O 3surface,an increase and a broadening of the d -as C–H band (1465cm -1),a decrease/shift of d -s C–H band (IPA:1382cm -1)to 1360cm -1were observed as a function of the treatment time.The band at 1360cm -1,emerging after 5min,is related to the methyl bending vibrations of acetone [29–31].The bands at 1595and 1465cm -1could be assigned to the C=C stretching and C–H bending mode of methyl or methylene group of propene,respectively[32],according to the products formation observed by IPA oxidation at low temperature on pure c -Al 2O 3[33].Simultaneously,a small shoulder at 1225cm -1also appearedafterFig.4DRIFT Spectra on c -Al 2O 3.a In situ IR spectra of the evolution of adsorbed isopropanol as a function of plasma treatment time from bottom to top (adsorbed IPA then 1,5,10,20,30,45and 60min of plasma treatment).b Spectra of adsorbed IPA after 60min of plasma treatment and adsorbed acetone,acetic acid and formic acid (from bottom to top )20min treatment and the large band at 1730cm -1was assigned to acetone formation.The small residual contribution remaining at 1150cm -1could correspond to C–O bonds stretching and assign to formic acid formation.These observations show that isopropoxide species formed by the dissociative adsorption of isopropanol give rise to acetone and formic acid on c -Al 2O 3when treated by non-thermal plasma under air.On TiO 2,an increase and a broadening of the d -as C–H band (1470cm -1)and the total disappearance of d -s C–H band (1385cm -1),were observed as a function of the plasma treatment time.The IPA bands at lower wavenumbers:d -O–H (1253cm -1),C–O stretching and C–C skeletal elongation (1150and 1075cm -1),progressively weakened till complete disappearance as the plasma treatment time increased showing that IPA on TiO2Fig.5DRIFT Spectra on TiO 2.a In situ IR spectra of the evolution of adsorbed isopropanol as a function of plasma treatment time from bottom to top (adsorbed IPA then 1,5,10,20,30,45and 60min of plasma treatment).b Spectra of adsorbed IPA after 60min of plasma treatment and adsorbed acetone,acetic acid and formic acid (from bottom to top )Fig.6DRIFT Spectra on CeO2.a In situ IR spectra of the evolution of adsorbed isopropanol as a function of plasma treatment time from bottom to top(adsorbed IPA then1,5,10,20,30,45and60min of plasma treatment).b Spectra of adsorbed IPA after60min of plasma treatment and adsorbed acetone,acetic acidand formic acid(from bottom to top)is rapidly oxidized into acetone followed by further decomposition into CO2and gas phase water.A small acidic residue is visible from the broad band in the3000cm-1region. Additionally,at1470cm-1and in smaller extend at1590cm-1bands were identiﬁed and corresponded to the C–H bending and the COO-stretching,respectively.On CeO2,the bands corresponding to the d-as(1470cm-1)and d-s(1390cm-1) bending mode of methyl are evolving and broadening as a function of the plasma treatment time.In the same time,a large band is growing at1275cm-1which corresponds to the O–H bending vibration mode of carboxylic acids.As for the two other samples,it is difﬁcult to differentiate the different acids and their contribution due to the overlapping of the bands in this region.Cerium oxide that shows particularly broad bands makes it difﬁcult to observe the evolution of acetone formation/oxidation on this surface.Finally,an interesting feature,only visible on CeO2is the adsorption/desorption of CO2in the2300–2400cm-1 region(2365and2333cm-1),that may arise from the decomposition of IPA into CO2. However,we have to be careful with this interpretation since published work on CO2 adsorption on c-Al2O3[34]or tin oxide[35]revealed the presence of bands in the 1900–1200cm-1range relative to adsorbed CO2,which may,in this case,be hidden by other contributions at low wavenumbers.According to these observations,the steps following IPA stabilization on the surface can follow these two pathways:either its oxidation into acetone,either a C–C bond breakage of a methyl group.Oxidation reactions taken place here can either come from a radical attack, probably from OÁorÁOH,on the–OH group of IPA,either via redox reactions of the adsorbed isopropoxide species on the metal oxide surface acid sites.C–C bond breakage can be achieved byÁOH radical attack and the resulting methyl radical can react with oxygenated species to form formaldehyde,then formic acid.The other radical fragment, CH3CHÁOH,can then be oxidized in acetaldehyde,then in acetic acid.On alumina,the formation of propene from IPA dehydration can also occurs,followed by its decomposition into CO2and H2O.Toluene RemovalThe experiments regarding toluene elimination were performed in the same conditions as for IPA removal.Toluene was previously adsorbed on the catalysts(100ppm in air for 10min,30mL min-1)followed by purging with dry air before the start of the plasma treatment.The recorded spectra as a function of time are presented in Fig.7a–c.As expected,on the three oxides the main reference IR bands of adsorbed toluene are visible at around3100–3000cm-1domain corresponding to the C–H aromatic(3030and 3080cm-1)and at2930and2880cm-1corresponding to the C–H stretching of the methyl group and at1495,1460and1610cm-1,corresponding to the C–C stretching in the aromatic ring.The negative bands at1705(bending mode of hydrogen bonded surface hydroxyls)and3695cm-1(stretching mode)could be assigned to the loss of surface hydroxyl group upon toluene adsorption[36–40].On CeO2and to a lesser extent on c-Al2O3,a small contribution of in plane C–H bending at1081and1035cm-1was identiﬁed as well as weak bands at1160and1180cm-1for CeO2and c-Al2O3,respectively.Finally, a weak overtone in the2000–1800cm-1region is detected.As soon as the plasma was generated,and this independently of the catalyst sample tested,the C–H aromatic bands(3030and3080cm-1)decreased rapidly after only1min of plasma treatment and this until10min.At10min,no C–H aromatic contribution is visible,and this for the three surfaces,indicating the complete transformation of toluene.Fig.7In situ DRIFT spectra ofthe evolution of toluene on a c-Al2O3,b CeO2and c TiO2as a function of plasma treatment time from bottom to top(adsorbed toluene then1,5,10,20,30,45 and60min of plasma treatment)In the same time,a strong band in the1730–1755cm-1region evolved after5min of plasma treatment characteristic of the presence of a C=O bond.As the plasma treatment time increased,the band increased,shifted to higher wavenumber after10min of plasma to reach1775,1770and1790cm-1on c-Al2O3,CeO2and TiO2,respectively(Table2and Fig.7).The non-symmetry of the bands and apparition of a shoulder(1700–1725cm-1)at 10min for CeO2and TiO2and20min for c-Al2O3indicate the presence of a carboxylic acid in addition to aldehyde or ester[25,41].The bands at1330,1250and1290cm-1on c-Al2O3,TiO2and CeO2,respectively,can be attributed to C–O stretching.In addition,the broad shoulder of O–H stretching(2500–3500cm-1)conﬁrms the presence of carboxylic acid.Independently of the catalyst,the growing of the1790–1770cm-1band is associated with the formation of a negative signal in the OH vibration domain(3500–3700cm-1) showing that the species formed under plasma adsorb on OH surface groups.By increasing the plasma treatment duration,the main change is the shift of the C=O band for more than 10min of plasma.This shift could be assigned to the formation of bridged carbonates[42] on the catalysts surface showing the total oxidation of toluene under plasma treatment.A band at1345cm-1appeared after1min and could also correspond to primary O–H bending(d-OH)or=C–OH phenol stretching.The band increased and shifted to 1330cm-1until20min and remained constant until60min.At this wavenumber,the functional group identiﬁed is more likely C–O from carboxylic acid or=C–OH phenol stretching but also N–O symmetrical stretching from oleﬁnic compounds.However,other nitrogen related contribution were not identiﬁed.On CeO2,less intense bands appear at1414and1250–1216cm-1which could be due to the formation of carboxylic acids(O–H bending and C–O stretching,respectively)and at 1610cm-1which could be assigned to the formation of adsorbed formate species[43].The relative intensity of this band decreases from c-Al2O3,TiO2to CeO2,following the increasing oxidative property(or basicity)of these oxides.On the three surfaces,the band of C=C stretching at*1460and at*1610cm-1 broadened and increased until60min while the one at*1495cm-1decreased until complete disappearance.The increasing and broadening of the1460cm-1band could be due to various stretching behavior such as C=C of phenol or C=C from alkenes indicating a rupture of the aromatic ring.A weak shoulder in the2950–2930cm-1region,indicates the Table2Shift of the C=O band as a function of plasma treatment time and corresponding chemical groups[25,34]Plasma time(min)0151020304560Wavenumber maxima(cm-1)Al2O3–1740174017451769177517751775Corresponding carbonyl group Aldehyde/esterAld/esterAld/ester Ald/ester/acidCarb.CeO2–1736173617601783177017701770Corresponding carbonyl group Aldehyde/esterAld/esterAld/ester/acidCarb.TiO2–1750175017751782179017901790Corresponding carbonyl group –Aldehyde/esterAld/esterAld/ester/acidCarb.Ald aldehyde,carb carbonatepresence of C–H stretching from methyl group only,without aromatic contribution.The presence of bands in the2300–2400cm-1domain is associated to the presence of adsorbed CO2.Fig.8Possible mechanism pathways and intermediates of toluene oxidation and degradationAccording to the results of various authors[44–52]and ours,we can propose some probable reaction mechanisms for toluene elimination.First of all,Kohno et al.[44] showed that,from the three pathways undertaken by NTAP species for VOC elimination, the electron impact would be the favored one as it presents the lowest reaction rate constant in the order of10-6cm3s-1.The two other pathways,ion collisions and radical attack(OÁ/ OHÁ)would occur subsequently and contribute to toluene oxidation and ring opening reactions.The electronic impact would lead to the formation of benzene radical that would rapidly be oxidized in phenol.For the later,the aromatic cycle would be rapidly hydroxylated into hydroquinone,further oxidized in benzoquinone.Finally the ring would be opened by oxygen or OH radicals leading to the formation of aldehydes and carboxylic acids with different carbon chains length.Possible mechanisms are displayed in Fig.8.As long as oxidative radicals are present,in addition to electronic impact,the reaction would continue until complete fragmentation into CO2and H2O following various mechanisms that are currently not completely identiﬁed and still need further investigation.Addition-ally,methyl radical would be oxidized in formic acid followed by complete elimination in CO2/H2O.It seems that the oxidation of the methyl group on toluene,leading to ben-zaldehyde and benzoic acid,is not the favored pathway undertaken for toluene oxidation by non-thermal plasma.In fact,from our results,it seems that the band related to C=O apparition coincides with the complete disappearance of aromatic C–H bands,indicating a rapid opening of the aromatic ring prior to the oxidation of the fragments into carboxylic acid.ConclusionsThis new infrared cell allows characterizing by DRIFT the adsorbed organic species on a catalyst surface under DBD non-thermal plasma and in controlled gaseous environment at atmospheric pressure,realizing then a true operando analysis of a catalyst surface sub-mitted to the action of a non-thermal plasma.The method has proven to be so efﬁcient that it was possible to follow the evolution of secondary compounds,arising from the oxidation of isopropanol and toluene.The plasma treatment of IPA molecules adsorbed on c-Al2O3, TiO2and CeO2surfaces led to the formation of acetone,propene,acetic acid and/or formic acid.On CeO2and TiO2surface,only acids remained after60min of plasma treatment while c-Al2O3showed a small presence of acetone and propene on its surface.By contrast, the toluene oxidation by non-thermal plasma led to the rapid opening of the aromatic ring, followed by the total oxidation through carboxylic formation of the species arising from the toluene molecules fragmentation.These results should be considered as preliminary experiments probing the extended possibilities of this new IR cell to characterize in operando conditions a catalyst or any surface submitted to a non-thermal plasma and are able to improve the knowledge on plasma material interactions.Acknowledgments The authors would like to thank the French Ministry of Research for the funding of the Ph.D.Grant of A.Rodrigues.References1.Bogaerts A,Neyts E,Gijbels R,Van der Mullen J(2002)Spectrochim Acta B57:609–6582.Tendero C,Tixier C,Tristant P,Desmaison J,Leprince P(2006)Spectrochim Acta B61:2–303.Kogelschatz U,Eliasson B,Egli W(1999)Pure Appl Chem71:1819–18284.Van Durme J,Dewulf J,Leys C,Van Langenhove H(2008)Appl Catal B Environ78:324–3335.Hyun-Ha K(2004)Plasma Process Polym1:91–1106.Vandenbroucke AM,Morent R,De Geyter N,Leys C(2011)J Hazard Mater195:30–547.Tosi P,Ascenzi D,Franceschi P,Guella G(2009)Plasma Sour Sci Technol18:0340058.Tanarro I,Herrero VJ(2009)Plasma Sour Sci Technol18:0340079.Babayan SE,Ding G,Nowling GR,Yang X,Hicks RF(2002)Plasma Chem Plasma Process22:255–26910.Cruden BA,Rao MVVS,Sharma SP,Meyyappan M(2003)J Appl Phys93:5053–506211.Rivallan M,Aiello S,Thibault-Starzyk F(2010)Rev Sci 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物理光学英文总结（精选五篇）

物理光学英文总结（精选五篇）第一篇：物理光学英文总结1.2.3.4.5.6.7.麦克斯韦方程组（Maxwell’s equation）电感强度, electric displacement 电场强度, electric field strength 磁感强度, magnetic flux density 磁场强度，magnetic field strength 波动方程的平面简谐波解(Simple Harmonic Wave)布儒斯特定律(Brewster’s Law)Brewster’ law, in his own words, states that “when a ray of light is polarized（偏振）by reflection, the reflected（反射）ray forms a right angle with the refracted（折射）ray.On the laws which regulate the polarization of light 偏振光by reflection from transparent bodies.” 8.光波的叠加, Superposition of waves 9.驻波(Standing Wave)10.拍频（Beat frequency）11.相速度（Phase velocity）12.群速度(Group velocity)13.合成波resultant wave 14.振幅amplitude 15.干涉现象（Interference）：在两个（或多个）光波叠加的区域形成强弱稳定的光强分布的现象，称为光的干涉现象。

The term Interference refers to the phenomenon that waves, under certain conditions, intensify or weaken each other.16.相干光波（Coherent wave）相干光源，Coherent light source 17.杨氏干涉实验（Young’s interference experiment）18.干涉条纹（Interference fringes）19.Path difference（路径差）20.Phase difference（位相差）21.The order of interference（干涉级）22.The light distribution（光分布）23.A maximum amount of light(maxima)24.A minimum amount of light(minima)25.干涉条纹的可见度The visibility(contrast)of interference fringes 26.对比度（Contrast）：It can be defined as the ratio of the difference between maximum areance（面积比）Emax, and mimimumareance, Emin, to the sum of such areances:K=(Emax-Emin)/(Emax+Emin)The amount of power incident per unit area is called areance(illuminance).Visibility：K=(Imax-Imin)/(Imax+Imin)27.相干性与干涉（Coherence & interference）28.空间相干性（spatial coherence)和时间相干性（temporal coherence）29.等厚干涉（Interference of equal thickness）30.平行平板(Plane-Parallel Plates)31.等倾干涉（Interference of equal inclination）32.法布里－泊罗干涉仪（Fabry-Perot interferometer）33.分辨极限和分辨本领（Resolvance of the interferometer）34.光学系统的分辨本领（Resolving power of an optical system）35.光的衍射（Diffraction）36.衍射实验(Diffraction experiment)37.衍射现象的分类(Classification of light diffraction)（1）夫琅和费衍射(Fraunhofer diffraction)（2）菲涅耳衍射（Fresnel diffraction）38.矩孔衍射(Diffraction by a rectangular aperture)39.强度分布计算（Intensity distribution calculation）40.单缝衍射(Diffraction by a single slit)41.夫琅和费圆孔衍射(Fraunhofer diffraction by a circular aperture)42.椭圆的衍射图样(Diffraction pattern)43.光学成像系统的衍射和分辨本领Diffraction and resolving power of an optical system 44.光学系统的分辨本领（Resolving power of an optical system）45.瑞利判据（Rayleigh’s criterion）46.双缝衍射（Double-slit diffraction）47.多缝衍射（Multiple-slit diffraction）48.衍射光栅(Diffraction gratings)49.光栅方程（The grating equation）50.光栅分辨本领(Resolvance of a grating)51.光的偏振（Polarization of light）52.偏振光与自然光，Polarized light and Natural light 53.线偏振光（Linearly polarized light）54.圆偏振光（Circularly polarized light）55.椭圆偏振光（Elliptically polarized light）56.部分偏振光(Partially polarized light)57.偏振光的产生（Production of polarized light）反射和折射、二向色性、散射、双折射Polarization by reflection Polarization by transmission Polarization by dichroism Polarization by scattering Polarization by birefringence 58.马吕斯定律(Malus’ law)和消光比(Extinction ratio）59.起偏器（Polarizer）：用来产生偏振光的偏振器件。

PhysRevD.79.114009

Relativistic correction to J=c production at hadron collidersYing Fan,*Yan-Qing Ma,†and Kuang-Ta Chao‡Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University,Beijing100871,China(Received20April2009;published10June2009)Relativistic corrections to the color-singlet J=c hadroproduction at the Tevatron and LHC arecalculated up to Oðv2Þin nonrelativistic QCD(NRQCD).The short-distance coefﬁcients are obtainedby matching full QCD with NRQCD results for the subprocess gþg!J=cþg.The long-distancematrix elements are extracted from observed J=c hadronic and leptonic decay widths up to Oðv2Þ.Usingthe CTEQ6parton distribution functions,we calculate the leading-order production cross sections andrelativistic corrections for the process pþ"pðpÞ!J=cþX at the Tevatron and LHC.Weﬁnd that theenhancement of Oðv2Þrelativistic corrections to the cross sections over a wide range of large transversemomentum p t is negligible,only at a level of about1%.This tiny effect is due to the smallness of thecorrection to short-distance coefﬁcients and the suppression from long-distance matrix elements.Theseresults indicate that relativistic corrections cannot help to resolve the large discrepancy between leading-order prediction and experimental data for J=c production at the Tevatron.DOI:10.1103/PhysRevD.79.114009PACS numbers:12.38.Bx,12.39.St,13.85.Ni,14.40.GxI.INTRODUCTION Nonrelativistic QCD(NRQCD)[1]is an effectiveﬁeld theory to describe production and decay of heavy quark-onium.In this formalism,inclusive production cross sec-tions and decay widths of charmonium and bottomonium can be factored into short-distance coefﬁcients,indicating the creation or annihilation of a heavy quark pair,and long-distance matrix elements,representing the evolvement of a free quark pair into a bound state.The short-distance part can be calculated perturbatively in powers of coupling constant s,while the nonperturbative matrix elements, which are scaled as v,the typical velocity of heavy quark or antiquark in the meson rest frame,can be estimated by nonperturbative methods or models,or extracted from experimental data.One important aspect of NRQCD is the introduction of the color-octet mechanism,which allows the intermediate heavy quark pair to exist in a color-octet state at short distances and evolve into the color-singlet bound state at long distances.This mechanism has been applied success-fully to absorb the infrared divergences in P-wave[1–3] and D-wave[4,5]decay widths of heavy quarkonia.In Ref.[6],the color-octet mechanism was introduced to account for the J=c production at the Tevatron,and the theoretical prediction of production rateﬁts well with experimental data.However,the color-octet gluon frag-mentation predicts that the J=c is transversely polarized at large transverse momentum p t,which is in contradiction with the experimental data[7].(For a review of these issues,one can refer to Refs.[8–10]).Moreover,in Refs.[11,12]it was pointed out that the color-octet long-distance matrix elements of J=c production may be much smaller than previously expected,and accordingly this mayreduce the color-octet contributions to J=c production at the Tevatron.In the past a couple of years,in order to resolve the largediscrepancy between the color-singlet leading-order(LO)predictions and experimental measurements[13–15]ofJ=c production at the Tevatron,the next-to-leading-order (NLO)QCD corrections to this process have been per-formed,and a large enhancement of an order of magnitudefor the cross section at large p t is found[16,17].But thisstill cannot make up the large discrepancy between thecolor-singlet contribution and data.Similarly,the observeddouble charmonium production cross sections in eþeÀannihilation at B factories[18,19]also signiﬁcantly differfrom LO theoretical predictions[20].Much work has beendone and it seems that those discrepancies could be re-solved by including NLO QCD corrections[21–24]andrelativistic corrections[25,26].One may wonder if therelativistic correction could also play a role to some extentin resolving the long standing puzzle of J=c production at the Tevatron.In this paper we will estimate the effect of relativisticcorrections to the color-singlet J=c production based on NRQCD.The relativistic effects are characterized by therelative velocity v with which the heavy quark or antiquarkmoves in the quarkonium rest frame.According to thevelocity scaling rules of NRQCD[27],the matrix elementsof operators can be organized into a hierarchy in the smallparameter v.We calculate the short-distance part pertur-batively up to Oðv2Þ.In order to avoid model dependence in determining the long-distance matrix elements,we ex-tract the matrix elements of up to dimension-8four fermionoperators from observed decay rates of J=c[28].Weﬁnd that the relativistic effect on the color-singlet J=c produc-tion at both the Tevatron and LHC is tiny and negligible,*ying.physics.fan@†@‡ktchao@PHYSICAL REVIEW D79,114009(2009)and relativistic corrections cannot offer much help to re-solve the puzzle associated with charmonium production at the Tevatron,and other mechanisms should be investigated to clarify the problem.The rest of the paper is organized as follows.In Sec.II, the NRQCD factorization formalism and matching condi-tion of full QCD and NRQCD effectiveﬁeld theory at long distances are described brieﬂy,and then detailed calcula-tions are given,including the perturbative calculation of the short-distance coefﬁcient,the long-distance matrix elements extracted from experimental data,and the parton-level differential cross section convolution with the parton distribution functions(PDF).In Sec.III,nu-merical results of differential cross sections over transverse momentum p t at the Tevatron and LHC are given and discussions are made for the enhancement effects of rela-tivistic corrections.Finally the summary of this work is presented.II.PRODUCTION CROSS SECTION IN NRQCDFACTORIZATIONAccording to NRQCD factorization[1],the inclusive cross section for the hadroproduction of J=c can be writ-ten asd dt ðgþg!J=cþgÞ¼XnF nm d nÀ4ch0j O J=c n j0i:(1)The short-distance coefﬁcients F n describe the production of a heavy quark pair Q"Q from the gluons,which come from the initial state hadrons,and are usually expressed in kinematic invariants.m c is the mass of charm quark.Thelong-distance matrix elements h0j O J=cn j0i with mass di-mension d n describe the evolution of Q"Q into J=c.The subscript n represents the conﬁguration in which the c"c pair can be for the J=c Fock state expansion,and it isusually denoted as n¼2Sþ1L½1;8J .Here,S,L,and J standfor spin,orbital,and total angular momentum of the heavy quarkonium,respectively.Superscript1or8means the color-singlet or color-octet state.For the color-singlet3S1c"c production,there are only two matrix elements contributing up to Oðv2Þ:the leading-order term h0j O J=cð3S½1 1Þj0i and the relativistic correction term h0j P J=cð3S½1 1Þj0i.Therefore the differential cross section takes the following form:d dt ðgþg!J=cþgÞ¼Fð3S½11Þm2ch0j O J=cð3S½1 1Þj0iþGð3S½11Þm4ch0j P J=cð3S½1 1Þj0iþOðv4Þ;(2)and the explicit expressions of the matrix elements are[1]h0j O J=cð3S½1 1Þj0i¼h0j y i cða y c a cÞc y i j0i;h0j P J=cð3S½1 1Þj0i¼12y i cða y c a cÞc y iÂÀi2D$2þH:c:;(3)where c annihilates a heavy quark, creates a heavy antiquark,a y c and a c are operators creating and annihilat-ing J=c in theﬁnal state,and D$¼~DÀD.In order to determine the short-distance coefﬁcients Fð3S½1 1Þand Gð3S½1 1Þ,the matching condition of full QCD and NRQCD is needed:ddtðgþg!J=cþgÞj pert QCD¼Fð3S½11Þm2ch0j O J=cð3S½1 1Þj0iþGð3S½11Þm4ch0j P J=cð3S½1 1Þj0ij pert NRQCD:(4)The differential cross section for the production of char-monium J=c on the left-hand side of Eq.(4)can be calculated in perturbative QCD.On the right-hand side the long-distance matrix elements are extracted from ex-perimental data.Quantities on both sides of the equation are expanded at leading order of s and next-to-leading order of v2.Then the short-distance coefﬁcients Fð3S½1 1Þand Gð3S½1 1Þcan be obtained by comparing the terms with powers of v2on both sides.A.Perturbative short-distance coefﬁcientsWe now present the calculation of relativistic correction to the process gþg!J=cþg.In order to determine the Oðv2Þcontribution in Eq.(2),the differential cross section on the left-hand side of Eq.(4)or equivalently the QCD amplitude should be expanded up to Oðv2Þ.We use FeynArts[29]to generate Feynman diagrams and am-plitudes,FeynCalc[30]to handle amplitudes,and FORTRAN to evaluate the phase space integrations.A typi-cal Feynman diagram for the process is shown in Fig.1.FIG.1.Typical Feynman diagram for3S½1 1c"c hadroproduction at LO.YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D79,114009(2009)The momenta of quark and antiquark in the lab frame are [26,31,32]:12Pþq ¼L ð12P rþq r Þ;12PÀq ¼L ð12P rÀq r Þ;(5)where L is the Lorenz boost matrix from the rest frame ofthe J=c to the frame in which it moves with four momen-tum P .P r ¼ð2E q ;0Þ,E q ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃm 2c þj ~qj 2p ,and 2q r ¼2ð0;~qÞis the relative momentum between heavy quark and antiquark in the J=c rest frame.The differential cross section on the left-hand side of Eq.(4)isddtðg þg !J=c þg Þj pert QCD ¼12"X j M ðg þg !J=c þg Þj 2h 0j O J=c ð3S ½1 1Þj 0i ;(6)where h 0j O J=c ð3S ½11Þj 0i is the matrix element evaluated at tree level,and the summation/average of the color and polarization degrees of freedom for the ﬁnal/initial statehas been implied by the symbol "P.The amplitude for the color-singlet process g ðp 1Þþg ðp 2Þ!J=c ðp 3¼P Þþg ðp 4ÞisM ðg þg !J=c þg Þ¼ﬃﬃﬃﬃﬃﬃ1E q s Tr ½C ½1 Åð1ÞM am ;(7)where M am denotes the parton-level amplitude amputatedof the heavy quark spinors.The factor ﬃﬃﬃﬃ1E q q comes from the normalization of the composite state j 3S ½11i[5].Here the covariant projection operator method [33,34]is adopted.For a color-singlet state,the color projector C ½1 ¼ ijﬃﬃﬃﬃﬃcp .The covariant spin-triplet projector Åð1Þin (7)is deﬁned byÅð1Þ¼X s "s v ðs Þ"u ð"s Þ 12;s ;12;"s j 1;S z ;(8)with its explicit formÅð1Þ¼1ﬃﬃﬃ2p ðE q þm c Þ P 2Àq Àm cÂP À2E q 4E qP 2þq þm c;(9)where the superscript (1)denotes the spin-triplet state andis the polarization vector of the spin 1meson.The Lorentz-invariant Mandelstam variables are deﬁned bys ¼ðp 1þp 2Þ2¼ðp 3þp 4Þ2;t ¼ðp 1Àp 3Þ2¼ðp 2Àp 4Þ2;u ¼ðp 1Àp 4Þ2¼ðp 2Àp 3Þ2;(10)and they satisfys þt þu ¼P 2¼4E 2q ¼4ðm 2c þj ~qj 2Þ:(11)Furthermore,the covariant spinors are normalized relativ-istically as "uu¼À"vv ¼2m c .Let M be short for the amplitude M ðg þg !J=c þg Þin Eq.(7),and it can be expanded in powers of v orequivalently j ~qj .That is M ¼ M¼ M ð0Þþ1q q @2M @q @q q ¼0þO ðq 4Þ;(12)where high order terms in four momentum q have been omitted.Terms of odd powers in q vanish because the heavy quark pair is in an S-wave conﬁguration.Note that the polarization vector also depends on q ,but it only has even powers of four momentum q ,and their expressions may be found e.g.in the appendix of Ref.[35].Therefore expansion on q 2of can be carried out after amplitude squaring.The following substitute is adopted:q q¼13j ~q j 2 Àg þP P P 213j ~q j 2Á :(13)This substitute should be understood to hold in the inte-gration over relative momentum ~qand in the S-wave case.Here,j ~qj 2can be identiﬁed as [33,36]j ~qj 2¼jh 0j y ðÀi2D $Þ2c j c ð3S ½1 1Þijjh 0j y c j c ð3S 1Þij ¼h 0j P J=c ð3S ½1 1Þj 0i h 0j O J=c ð3S ½1 1Þj 0i½1þO ðv 4Þ :(14)Then the amplitude squared deﬁned in Eq.(6)up to O ðv 2Þis Xj M j 2¼Mð0ÞM Ãð0ÞXÃþ16j ~q j 2 Á @2M @q @q q ¼0M Ãð0Þþ Á @2M Ã@q @q q ¼0M ð0Þ X Ã þO ðv 4Þ:(15)The heavy quark and antiquark are taken to be on shell,which means that P Áq ¼0,and then gauge invariance is maintained.The polarization sum in Eq.(15)isXÃ ¼Àg þP PP 2:(16)RELATIVISTIC CORRECTION TO J=c PRODUCTION ...PHYSICAL REVIEW D 79,114009(2009)It is clearly seen that the polarization sum above only contains even order powers of four momentum q,therefore it will make a contribution to the relativistic correction at Oðv2Þin theﬁrst term on the right-hand side of Eq.(15)when the contraction over indices and is carried out. However,since the second term on the right-hand side of Eq.(15)already has a term proportional to q2,i.e.j~q j2,the four momentum q can be set to zero throughout the index contraction.Then we haveXj M j2¼AþB j~q j2þOðv4Þ;(17)where A and B are independent of j~q j.By comparing Eqs.(4)and(6),we obtain the short-distance coefﬁcients shown explicitly below.The leading-order one isFð3S½1 1Þm2c ¼12111c1A¼12111c1ð4 sÞ35120m c½16ðs2þtsþt2Þm4cÀ4ð2s3þ3ts2þ3t2sþ2t3Þm2cþðs2þtsþt2Þ2 =½9ðsÀ4m2cÞ2ðtÀ4m2cÞ2ðsþtÞ2 ;(18)and the relativistic correction term isGð3S½1 1Þm4c ¼116 s21641412N c13B¼116 s21641412N c13ð4 sÞ3ðÀ2560Þ½2048ð3s2þ2tsþ3t2Þm10cÀ256ð5s3À2ts2À2t2sþ5t3Þm8cÀ320ð3s4þ10ts3þ10t2s2þ10t3sþ3t4Þm6cþ16ð21s5þ63ts4þ88t2s3þ88t3s2þ63t4sþ21t5Þm4cÀ4ð7s6þ18ts5þ23t2s4þ28t3s3þ23t4s2þ18t5sþ7t6Þm2cÀstðsþtÞðs2þtsþt2Þ2 =½27m cð4m2cÀsÞ3ð4m2cÀtÞ3ðsþtÞ3 :(19)Each of the factors has its own origin:1=16 s2isproportional to the inverse square of the Møller’s invariantﬂux factor,1=64and1=4are the color average andspin average of initial two gluons,respectively,1=2N ccomes from the color-singlet long-distance matrix elementdeﬁnition in Eq.(3)with N c¼3,1=3is the spin average for total spin J¼1states,andð4 sÞ3quantiﬁes the coupling in the QCD interaction vertices.Further-more the variable u has been expressed in terms of sand t through Eq.(11).To verify our results,weﬁndthat those in Ref.[31]discussed for J=c photoproduction are consistent with ours under replacementð4 Þe2c!ð4 sÞ,and the result in Ref.[37]agrees with ours at leading order after performing the polarization summation.B.Nonperturbative long-distance matrix elements The long-distance matrix elements may be determined by potential model[25,36]or lattice calculations[38],and by phenomenological extraction from experimental data. Here weﬁrst extract the decay matrix elements from experimental data.Up to NLO QCD and v2relativistic corrections,decay widths of the color-singlet J=c to light hadrons(LH)and eþeÀcan be expressed analytically as follows[33]:À½J=c!LH ¼F LHð3S½1 1Þm2ch H j O J=cð3S½1 1Þj H iþG LHð3S½1 1Þm4ch H j P J=cð3S½1 1Þj H i;À½J=c!eþeÀ ¼F eþeÀð3S½1 1Þm2ch H j O J=cð3S½1 1Þj H iþG eþeÀð3S½1 1Þm4ch H j P J=cð3S½1 1Þj H i;(20)where the short-distance coefﬁcients are[33]F LHð3S½1 1Þ¼ðN2cÀ1ÞðN2cÀ4ÞN3cð 2À9Þ3sð2m cÞÂ1þðÀ9:46C Fþ4:13C AÀ1:161N fÞ sþ2 e2QX Nfi¼1Q2i2e1À134C Fs;G LHð3S½1 1Þ¼À5ð19 2À132Þ7293sð2m cÞ;F eþeÀð3S½1 1Þ¼2 e2Q 2e31À4C F sð2m cÞ;G eþeÀð3S½1 1Þ¼À8 e2Q 2e9:(21)YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D79,114009(2009)Then,the production matrix elements can be related to the decay matrix elements through vacuum saturation approxi-mationh 0j O J=c ð3S ½1 1Þj 0i ¼ð2J þ1Þh H j OJ=c ð3S ½11Þj H i Â½1þO ðv 4Þ ¼3h H j O J=c ð3S ½11Þj H i½1þO ðv 4Þ :(22)Combining the above equations and the experimental data in [28],i.e.,À½J=c !LH ¼81:7KeV and À½J=c !e þe À ¼5:55KeV and excluding the NLO QCD radiative corrections in (21),we get the solutions accurate at leading order in sh 0j O J=c ð3S ½1 1Þj 0i ¼0:868GeV 3;h 0j P J=c ð3S ½1 1Þj 0i ¼0:190GeV 5;(23)and the enhanced matrix elements accurate up to NLO ins can be obtained by including NLO QCD radiative corrections in (21)h 0j O J=c ð3S ½1 1Þj 0i ¼1:64GeV 3;h 0j P J=c ð3S ½11Þj 0i¼0:320GeV 5:(24)The strong coupling constant evaluated at the charm quarkmass scale is s ð2m c Þ¼0:250for m c ¼1:5GeV .The other input parameters are chosen as follows:the QCD scale parameter ÃQCD ¼392MeV ,the number of quarks with mass less than the energy scale m c is N f ¼3,color factor C F ¼4=3and C A ¼3,the electric charge of the charm quark is e Q ¼2=3,Q i are the electric charges of the light quarks and ﬁne structure constant e ¼1=137.Our numerical values for the production matrix elementsh 0j O J=c ð3S ½1 1Þj 0i and h 0j PJ=c ð3S ½11Þj 0i are accurate up to NLO in v 2with uncertainties due to experimental errors and higher order corrections.C.Cross sections for p þ pðp Þ!J=c þX and phase space integration Based on the results obtained for the subprocess g þg !J=c þg we further calculate the production cross sections and relativistic corrections in the process p þ"pðp Þ!J=c þX ,which involves hadrons as the initial states.In order to get the cross sections at the hadron level,the partonic cross section deﬁned in Eq.(6)has to be convoluted with the parton distribution function (PDF)f g=p ðx ÆÞ,where x Ædenotes the fraction of the proton or antiproton beam energy carried by the gluons.We will work in the p "p center-of-mass (CM)frame and denote the p "penergy by ﬃﬃﬃS p ,the rapidity of J=c by y C ,and that of the gluon jet by y D .The differential crosssection of p þ"pðp Þ!J=c þX can be written as [39]d 3 ðp þ"pðp Þ!J=c þX Þdp 2t dy C dy D ¼x þf g=p ðx þÞx Àf g="p ðp Þðx ÀÞd ðg þg !J=c þg Þdt;(25)wherex Æ¼m C t exp ðÆy C Þþm Dt exp ðÆy D ÞﬃﬃﬃSp ;(26)with the transverse mass m C;D t ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃm 2C;D þp 2t q ,the mesonmass m C ¼2m c ,the gluon mass m D ¼0,and the trans-verse momentum p t .The Mandelstam variables can be expressed in terms of p t ,y C ,and y Ds ¼x þx ÀS;t ¼Àp 2t Àm C t m Dt exp ðy D Ày C Þ;u ¼Àp 2t Àm C t m D t exp ðy C Ày D Þ:(27)The accessible phase space puts kinetic constraints onvariables p t ,y C ,and y D for a ﬁxed value of two colliding hadron center-of-mass energy ﬃﬃﬃSp 0 p t12ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðS;m 2C ;m 2D ÞSs ;j y C j Arcosh S þm 2C Àm 2D2ﬃﬃﬃS p m C t;Àln ﬃﬃﬃS p Àm C t exp ðÀy CÞm D t y Dln ﬃﬃﬃS p Àm C t exp ðy C Þm D t ;(28)where ðx;y;z Þ¼x 2þy 2þz 2À2ðxy þyz þzx Þ.Thedistribution over p t of the differential cross section can be obtained after phase space integration.III.NUMERICAL RESULTS AND ANALYSIS The CTEQ6PDFs [40]are used in our numerical cal-culation.We present the distribution of J=c production differential cross section d =dp t over p t at the Tevatron with ﬃﬃﬃS p ¼1:96TeV and at the LHC with ﬃﬃﬃS p ¼14TeV in Figs.2–5.The solid line represents the distribution at leading order in O ðv 2Þ,and the dotted line describes the relativistic correction at next-to-leading order in O ðv 2Þ(excluding the leading-order result).The long-distance matrix elements are accurate up to leading order in s from Eq.(23)or next-to-leading order in s from Eq.(24).The variable p t is set to be from 5GeV to 30GeV (50GeV)for the Tevatron (LHC),and the distri-butions are depicted in logarithm unit along the vertical axis.All curves decrease rather rapidly as the transverse momentum p t increases,and the leading-order d =dp t behavior is not changed by the relativistic corrections.It can be seen that the ratio of relativistic correction to leading-order term is 1%or so,and less than 2%,which is insigniﬁcant and negligible.RELATIVISTIC CORRECTION TO J=c PRODUCTION ...PHYSICAL REVIEW D 79,114009(2009)The tiny effect of relativistic corrections is partly due to the smallness of the short-distance coefﬁcient correction.In fact,the ratio of the NLO short-distance coefﬁcient to the LO one from Eqs.(18)and (19)can be expanded as a series of the small quantity m c ,as compared with ﬃﬃﬃs p ,and this series reduces to a ﬁxed small number 16if only the leading-order term is kept,i.e.,G ð3S ½11ÞF ð3S 1Þ!16;as 2mc ﬃﬃﬃs p !0;2m cﬃﬃtp !0:(29)Together with the suppression from long-distance matrix elements,the tiny effect of relativistic corrections can be accounted for.Our results for relativistic corrections in theprocess p þ"pðp Þ!J=c þX are similar to that in the J=c photoproduction process discussed in Ref.[31].These results may indicate that the nonrelativistic approxi-mation in NRQCD is good for charmonium production at high energy collisions,and relativistic corrections are not important.This is in contrast to the case of double char-monium production in e þe Àannihilation at B factories,where relativistic corrections may be signiﬁcant.IV .SUMMARYIn this paper,relativistic corrections to the color-singlet J=c hadroproduction at the Tevatron and LHC are calcu-lated up to O ðv 2Þin the framework of the NRQCD facto-rization approach.The perturbative short-distance coefﬁcients are obtained by matching the full QCDdiffer-FIG.3.The p t distribution of d ðp þ"p !J=c þX Þ=dp t(with enhanced matrix elements)at the Tevatron with ﬃﬃﬃS p ¼1:96TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.4.The p t distribution of d ðp þp !J=c þX Þ=dp t at the LHC with ﬃﬃﬃS p¼14TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.5.The p t distribution of d ðp þp !J=c þX Þ=dp t (with enhanced matrix elements)at the LHC with ﬃﬃﬃS p ¼14TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.2.The p t distribution of d ðp þ"p !J=c þX Þ=dp t at the Tevatron with ﬃﬃﬃS p ¼1:96TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D 79,114009(2009)ential cross section with the NRQCD effectiveﬁeld theorycalculation for the subprocess gþg!J=cþg.The nonperturbative long-distance matrix elements are ex-tracted from experimental data for J=c hadronic and leptonic decay widths up to Oðv2Þwith an approximate relation between the production matrix elements and decaymatrix ing the CTEQ6parton distributionfunctions,we then calculate the LO production cross sec-tions and relativistic corrections for the process pþ"pðpÞ!J=cþX at the Tevatron and LHC.Weﬁnd that the Oðv2Þrelativistic corrections to the differential cross sections over a wide range of large transverse momentum p t are tiny and negligible,only at a level of about1%.The tiny effect of relativistic corrections is due to the smallness of the short-distance coefﬁcient correction and the sup-pression from long-distance matrix elements.These 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Factoring polynomials over p-adic fields

our algorithm will nd approximations to factors of F(X). Elements x of K may P be written x = 1 ?m ai i , with ai 2 A. In Section 8 we discuss where this i= series can be truncated to guarantee a correct answer. Let F(X) be a monic polynomial with coe cients in OK which has no repeated factors. See Zippel 29, pp 294{295] for a simple method of removing repeated factors. Unlike Chistov's algorithm, our method does not require computing in, or even constructing, rami ed extensions of K. The algorithm is applied recursively, at each step either nding a new factor or terminating with an irreducible factor and certi cate of its irreducibility. The certi cate of irreducibility will be a generalized Eisenstein polynomial with coe cients in the maximal unrami ed (over K) sub eld of K(x), where x is a root of the irreducible factor. The p-adic Factor algorithm works by looking for a polynomial A(X) for which we can determine the factorization of R(Y ) = ResX (F(X); Y ? A(X)): (1.1) In Section 2 we show that a factor of R(Y ) lets us nd a factor of F(X), and a certi cate of irreducibility for R(Y ) also applies to F(X). Once such an A(X) is found, we apply the information to F(X) and, if necessary, recurse on remaining factors of the original polynomial. The standard \easy" method for factoring a polynomial over the p-adics, the Newton diagram method, is given in Section 3. If the Newton diagram of the polynomial is not a straight line, then Hensel's Lemma may be used to nd a factor. If the Newton diagram is a straight line with slope k=n, where n is the degree of F(X) and k is relatively prime to n, then F(X) is irreducible. Otherwise the Newton diagram method fails, and we use an extension of Hensel's Lemma given in Section 4.1. We proceed by looking at the factorization of F(X) in K. If the reduction F (X) (de ned in Section 3) has two relatively prime factors, then using Hensel's Lemma we may lift these to factors over K. If F (X) is the power of an irreducible polynomial of degree d 2, then we may factor F(X) over an unrami ed extension of degree d of K, leading to a factorization of F(X) over K. These methods form the basis of the Hensel Factor routine given in Section 4.2. The only case Hensel Factor cannot handle is when R(Y ) = an (Y r ? b s)m + terms above the Newton diagram]: (1.2)

具有反射群对称性的球面图案自动生成

具有反射群对称性的球面图案自动生成王新长;刘满凤;欧阳培昌【摘要】Equivariant mapping method is not only difficult to be implemented, but also constrained by the order of symmetry group. Drawn on the experience of the invariant theory of finite reflection group, this paper proposes an invariant mapping method to yield aesthetical spherical patterns and establishes a method to create infinite spherical patterns automatically. This method not only is easy to be implemented, but also can be extended to deal with the cases in the higher dimensional spaces.%等变映射方法在生成艺术图案中具有构造困难，受对称群阶数瓶颈限制等缺点。

借鉴有限反射群不变论的结论，提出不变映射方法生成具有正多面体反射群对称性的球面艺术图案，建立了一种可生成无穷无尽球面图案的自动化方法。

该方法不仅实施容易，且可类似地推广到高维空间中。

【期刊名称】《计算机工程与应用》【年(卷),期】2013(000)023【总页数】4页(P27-30)【关键词】有限反射群;正多面体;不变论;不变映射【作者】王新长;刘满凤;欧阳培昌【作者单位】江西财经大学信息管理学院，南昌 330013; 井冈山大学数理学院，江西吉安 343009;江西财经大学信息管理学院，南昌 330013;井冈山大学数理学院，江西吉安 343009【正文语种】中文【中图分类】TP391利用计算机技术自动生成艺术图案是一个实用的新兴课题，借助迅猛发展精工技艺（如激光喷墨、3D打印等），其研究结果可以广泛地应用到壁纸、瓷砖、包装材料、纺织等与装饰领域有关的行业，生成美观的工艺品，不仅可以满足人们对于美的追求与赏析，而且具有可观的经济价值。

哈佛大学Capasso教授小组研究用界面位相突变的超材料实现超常折射的论文

Conventional optical components rely on gradual phase shifts accumulated during light propagation to shape light beams. New degrees of freedom are attained by introducing abrupt phase changes over the scale of the wavelength. A two-dimensional array of optical resonators with spatially varying phase response and sub-wavelength separation can imprint such phase discontinuities on propagating light as it traverses the interface between two media. Anomalous reflection and refraction phenomena are observed in this regime in optically thin arrays of metallic antennas on silicon with a linear phase variation along the interface, in excellent agreement with generalized laws derived from Fermat’s principle. Phase discontinuities provide great flexibility in the design of light beams as illustrated by the generation of optical vortices using planar designer metallic interfaces. The shaping of the wavefront of light by optical components such as lenses and prisms, as well as diffractive elements like gratings and holograms, relies on gradual phase changes accumulated along the optical path. This approach is generalized in transformation optics (1, 2) which utilizesmetamaterials to bend light in unusual ways, achieving suchphenomena as negative refraction, subwavelength-focusing,and cloaking (3, 4) and even to explore unusual geometries ofspace-time in the early universe (5). A new degree of freedomof controlling wavefronts can be attained by introducingabrupt phase shifts over the scale of the wavelength along theoptical path, with the propagation of light governed byFermat’s principle. The latter states that the trajectory takenbetween two points A and B by a ray of light is that of leastoptical path, ()B A n r dr ∫r , where ()n r r is the local index of refraction, and readily gives the laws of reflection and refraction between two media. In its most general form,Fermat’s principle can be stated as the principle of stationaryphase (6–8); that is, the derivative of the phase()B A d r ϕ∫r accumulated along the actual light path will be zero with respect to infinitesimal variations of the path. We show that an abrupt phase delay ()s r Φr over the scale of the wavelength can be introduced in the optical path by suitably engineering the interface between two media; ()s r Φr depends on the coordinate s r r along the interface. Then the total phase shift ()B s A r k dr Φ+⋅∫r r r will be stationary for the actual path that light takes; k r is the wavevector of the propagating light. This provides a generalization of the laws of reflection and refraction, which is applicable to a wide range of subwavelength structured interfaces between two media throughout the optical spectrum. Generalized laws of reflection and refraction. The introduction of an abrupt phase delay, denoted as phase discontinuity, at the interface between two media allows us to revisit the laws of reflection and refraction by applying Fermat’s principle. Consider an incident plane wave at an angle θi . Assuming that the two rays are infinitesimally close to the actual light path (Fig. 1), then the phase difference between them is zero ()()()s in s in 0o i i o t t kn d x d kn d x θθ+Φ+Φ−+Φ=⎡⎤⎡⎤⎣⎦⎣⎦ (1) where θt is the angle of refraction, Φ and Φ+d Φ are, respectively, the phase discontinuities at the locations where the two paths cross the interface, dx is the distance between the crossing points, n i and n t are the refractive indices of thetwo media, and k o = 2π/λo , where λo is the vacuumwavelength. If the phase gradient along the interface isdesigned to be constant, the previous equation leads to thegeneralized Snell’s law of refraction Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and RefractionNanfang Yu ,1 Patrice Genevet ,1,2 Mikhail A. Kats ,1 Francesco Aieta ,1,3 Jean-Philippe Tetienne ,1,4 Federico Capasso ,1 Zeno Gaburro 1,51School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA. 2Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, Texas 77843, USA. 3Dipartimento di Fisica e Ingegneria dei Materiali e del Territorio, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. 4Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan and CNRS, 94235 Cachan, France. 5Dipartimento di Fisica, Università degli Studi di Trento, via Sommarive 14, 38100 Trento, Italy.o n S e p t e m b e r 1, 2011w w w .s c i e n c e m a g .o r g D o w n l o a d e d f r o m()()sin sin 2o t t i i d n n dx λθθπΦ−= (2) Equation 2 implies that the refracted ray can have an arbitrary direction, provided that a suitable constant gradient of phase discontinuity along the interface (d Φ/dx ) is introduced. Note that because of the non-zero phase gradient in this modified Snell’s law, the two angles of incidence ±θi lead to different values for the angle of refraction. As a consequence there are two possible critical angles for total internal reflection, provided that n t < n i : arcsin 2to c i i n d n n dx λθπ⎛⎞Φ=±−⎜⎟⎝⎠ (3)Similarly, for the reflected light we have ()()sin sin 2o r i i d n dx λθθπΦ−= (4) where θr is the angle of reflection. Note the nonlinear relationbetween θr and θI , which is markedly different fromconventional specular reflection. Equation 4 predicts that there is always a critical incidence angle arcsin 12o c i d n dx λθπ⎛⎞Φ′=−⎜⎟⎝⎠ (5) above which the reflected beam becomes evanescent. In the above derivation we have assumed that Φ is a continuous function of the position along the interface; thus all the incident energy is transferred into the anomalous reflection and refraction. However because experimentally we use an array of optically thin resonators with sub-wavelength separation to achieve the phase change along the interface, this discreteness implies that there are also regularly reflected and refracted beams, which follow conventional laws of reflection and refraction (i.e., d Φ/dx =0 in Eqs. 2 and 4). The separation between the resonators controls the relative amount of energy in the anomalously reflected and refracted beams. We have also assumed that the amplitudes of the scattered radiation by each resonator are identical, so that the refracted and reflected beams are plane waves. In the next section we will show by simulations, which represent numerical solutions of Maxwell’s equations, how indeed one can achieve the equal-amplitude condition and the constant phase gradient along the interface by suitable design of the resonators. Note that there is a fundamental difference between the anomalous refraction phenomena caused by phase discontinuities and those found in bulk designer metamaterials, which are caused by either negative dielectric permittivity and negative magnetic permeability or anisotropic dielectric permittivity with different signs ofpermittivity tensor components along and transverse to thesurface (3, 4).Phase response of optical antennas. The phase shift between the emitted and the incident radiation of an optical resonator changes appreciably across a resonance. By spatially tailoring the geometry of the resonators in an array and hence their frequency response, one can design the phase shift along the interface and mold the wavefront of the reflected and refracted beams in nearly arbitrary ways. The choice of the resonators is potentially wide-ranging, fromelectromagnetic cavities (9, 10), to nanoparticles clusters (11,12) and plasmonic antennas (13, 14). We concentrated on thelatter, due to their widely tailorable optical properties (15–19)and the ease of fabricating planar antennas of nanoscalethickness. The resonant nature of a rod antenna made of aperfect electric conductor is shown in Fig. 2A (20).Phase shifts covering the 0 to 2π range are needed toprovide full control of the wavefront. To achieve the requiredphase coverage while maintaining large scatteringamplitudes, we utilized the double resonance properties of V-shaped antennas, which consist of two arms of equal length h connected at one end at an angle Δ (Fig. 2B). We define twounit vectors to describe the orientation of a V-antenna: ŝalong the symmetry axis of the antenna and â perpendicular to ŝ (Fig. 2B). V-antennas support “symmetric” and “antisymmetric” modes (middle and right panels of Fig. 2B),which are excited by electric-field components along ŝ and â axes, respectively. In the symmetric mode, the current distribution in each arm approximates that of an individual straight antenna of length h (Fig. 2B middle panel), and therefore the first-order antenna resonance occurs at h ≈ λeff /2, where λeff is the effective wavelength (14). In the antisymmetric mode, the current distribution in each arm approximates that of one half of a straight antenna of length 2h (Fig. 2B right panel), and the condition for the first-order resonance of this mode is 2h ≈ λeff /2.The polarization of the scattered radiation is the same as that of the incident light when the latter is polarized along ŝ or â. For an arbitrary incident polarization, both antenna modes are excited but with substantially different amplitude and phase due to their distinctive resonance conditions. As a result, the scattered light can have a polarization different from that of the incident light. These modal properties of the V-antennas allow one to design the amplitude, phase, and polarization state of the scattered light. We chose the incident polarization to be at 45 degrees with respect to ŝ and â, so that both the symmetric and antisymmetric modes can be excited and the scattered light has a significant component polarized orthogonal to that of the incident light. Experimentally this allows us to use a polarizer to decouple the scattered light from the excitation.o n S e p t e m b e r 1, 2011w w w .s c i e n c e m a g .o r g Do w n l o a d e d f r o mAs a result of the modal properties of the V-antennas and the degrees of freedom in choosing antenna geometry (h and Δ), the cross-polarized scattered light can have a large range of phases and amplitudes for a given wavelength λo; see Figs. 2D and E for analytical calculations of the amplitude and phase response of V-antennas assumed to be made of gold rods. In Fig. 2D the blue and red dashed curves correspond to the resonance peaks of the symmetric and antisymmetric mode, respectively. We chose four antennas detuned from the resonance peaks as indicated by circles in Figs. 2D and E, which provide an incremental phase of π/4 from left to right for the cross-polarized scattered light. By simply taking the mirror structure (Fig. 2C) of an existing V-antenna (Fig. 2B), one creates a new antenna whose cross-polarized emission has an additional π phase shift. This is evident by observing that the currents leading to cross-polarized radiation are π out of phase in Figs. 2B and C. A set of eight antennas were thus created from the initial four antennas as shown in Fig. 2F. Full-wave simulations confirm that the amplitudes of the cross-polarized radiation scattered by the eight antennas are nearly equal with phases in π/4 increments (Fig. 2G).Note that a large phase coverage (~300 degrees) can also be achieved using arrays of straight antennas (fig. S3). However, to obtain the same range of phase shift their scattering amplitudes will be significantly smaller than those of V-antennas (fig. S3). As a consequence of its double resonances, the V-antenna instead allows one to design an array with phase coverage of 2π and equal, yet high, scattering amplitudes for all of the array elements, leading to anomalously refracted and reflected beams of substantially higher intensities.Experiments on anomalous reflection and refraction. We demonstrated experimentally the generalized laws of reflection and refraction using plasmonic interfaces constructed by periodically arranging the eight constituent antennas as explained in the caption of Fig. 2F. The spacing between the antennas should be sub-wavelength to provide efficient scattering and to prevent the occurrence of grating diffraction. However it should not be too small; otherwise the strong near-field coupling between neighboring antennas would perturb the designed scattering amplitudes and phases.A representative sample with the densest packing of antennas, Γ= 11 µm, is shown in Fig. 3A, where Γ is the lateral period of the antenna array. In the schematic of the experimental setup (Fig. 3B), we assume that the cross-polarized scattered light from the antennas on the left-hand side is phase delayed compared to the ones on the right. By substituting into Eq. 2 -2π/Γ for dΦ/dx and the refractive indices of silicon and air (n Si and 1) for n i and n t, we obtain the angle of refraction for the cross-polarized lightθt,٣= arcsin[n Si sin(θi) – λo/Γ] (6) Figure 3C summarizes the experimental results of theordinary and the anomalous refraction for six samples with different Γ at normal incidence. The incident polarization isalong the y-axis in Fig. 3A. The sample with the smallest Γcorresponds to the largest phase gradient and the mostefficient light scattering into the cross polarized beams. We observed that the angles of anomalous refraction agree wellwith theoretical predictions of Eq. 6 (Fig. 3C). The same peak positions were observed for normal incidence withpolarization along the x-axis in Fig. 3A (Fig. 3D). To a good approximation, we expect that the V-antennas were operating independently at the packing density used in experiments (20). The purpose of using a large antenna array (~230 µm ×230 µm) is solely to accommodate the size of the plane-wave-like excitation (beam radius ~100 µm). The periodic antenna arrangement is used here for convenience, but is notnecessary to satisfy the generalized laws of reflection and refraction. It is only necessary that the phase gradient isconstant along the plasmonic interface and that the scattering amplitudes of the antennas are all equal. The phaseincrements between nearest neighbors do not need to be constant, if one relaxes the unnecessary constraint of equal spacing between nearest antennas.Figures 4A and B show the angles of refraction and reflection, respectively, as a function of θi for both thesilicon-air interface (black curves and symbols) and the plasmonic interface (red curves and symbols) (20). In therange of θi = 0-9 degrees, the plasmonic interface exhibits “negative” refraction and reflection for the cross-polarized scattered light (schematics are shown in the lower right insetsof Figs. 4A and B). Note that the critical angle for totalinternal reflection is modified to about -8 and +27 degrees(blue arrows in Fig. 4A) for the plasmonic interface in accordance with Eq. 3 compared to ±17 degrees for thesilicon-air interface; the anomalous reflection does not exist beyond θi = -57 degrees (blue arrow in Fig. 4B).At normal incidence, the ratio of intensity R between the anomalously and ordinarily refracted beams is ~ 0.32 for the sample with Γ = 15 µm (Fig. 3C). R rises for increasingantenna packing densities (Figs. 3C and D) and increasingangles of incidence (up to R≈ 0.97 at θi = 14 degrees (fig.S1B)). Because of the experimental configuration, we are notable to determine the ratio of intensity between the reflected beams (20), but we expect comparable values.Vortex beams created by plasmonic interfaces. To demonstrate the versatility of the concept of interfacial phase discontinuities, we fabricated a plasmonic interface that isable to create a vortex beam (21, 22) upon illumination by normally incident linearly polarized light. A vortex beam hasa helicoidal (or “corkscrew-shaped”) equal-phase wavefront. Specifically, the beam has an azimuthal phase dependenceexp(i lφ) with respect to the beam axis and carries an orbitalonSeptember1,211www.sciencemag.orgDownloadedfromangular momentum of L l=h per photon (23), where the topological charge l is an integer, indicating the number of twists of the wavefront within one wavelength; h is the reduced Planck constant. These peculiar states of light are commonly generated using a spiral phase plate (24) or a computer generated hologram (25) and can be used to rotate particles (26) or to encode information in optical communication systems (27).The plasmonic interface was created by arranging the eight constituent antennas as shown in Figs. 5A and B. The interface introduces a spiral-like phase delay with respect to the planar wavefront of the incident light, thereby creating a vortex beam with l = 1. The vortex beam has an annular intensity distribution in the cross-section, as viewed in a mid-infrared camera (Fig. 5C); the dark region at the center corresponds to a phase singularity (22). The spiral wavefront of the vortex beam can be revealed by interfering the beam with a co-propagating Gaussian beam (25), producing a spiral interference pattern (Fig. 5E). The latter rotates when the path length of the Gaussian beam was changed continuously relative to that of the vortex beam (movie S1). Alternatively, the topological charge l = 1 can be identified by a dislocated interference fringe when the vortex and Gaussian beams interfere with a small angle (25) (Fig. 5G). The annular intensity distribution and the interference patterns were well reproduced in simulations (Figs. D, F, and H) by using the calculated amplitude and phase responses of the V-antennas (Figs. 2D and E).Concluding remarks. Our plasmonic interfaces, consisting of an array of V-antennas, impart abrupt phase shifts in the optical path, thus providing great flexibility in molding of the optical wavefront. This breaks the constraint of standard optical components, which rely on gradual phase accumulation along the optical path to change the wavefront of propagating light. We have derived and experimentally confirmed generalized reflection and refraction laws and studied a series of intriguing anomalous reflection and refraction phenomena that descend from the latter: arbitrary reflection and refraction angles that depend on the phase gradient along the interface, two different critical angles for total internal reflection that depend on the relative direction of the incident light with respect to the phase gradient, critical angle for the reflected light to be evanescent. We have also utilized a plasmonic interface to generate optical vortices that have a helicoidal wavefront and carry orbital angular momentum, thus demonstrating the power of phase discontinuities as a design tool of complex beams. The design strategies presented in this article allow one to tailor in an almost arbitrary way the phase and amplitude of an optical wavefront, which should have major implications for transformation optics and integrated optics. We expect that a variety of novel planar optical components such as phased antenna arrays in the optical domain, planar lenses,polarization converters, perfect absorbers, and spatial phase modulators will emerge from this approach.Antenna arrays in the microwave and millimeter-waveregion have been widely used for the shaping of reflected and transmitted beams in the so-called “reflectarrays” and “transmitarrays” (28–31). There is a connection between thatbody of work and our results in that both use abrupt phase changes associated with antenna resonances. However the generalization of the laws of reflection and refraction wepresent is made possible by the deep-subwavelengththickness of our optical antennas and their subwavelength spacing. It is this metasurface nature of the plasmonicinterface that distinguishes it from reflectarrays and transmitarrays. The last two cannot be treated as an interfacein the effective medium approximation for which one canwrite down the generalized laws, because they typicallyconsist of a double layer structure comprising a planar arrayof antennas, with lateral separation larger than the free-space wavelength, and a ground plane (in the case of reflectarrays)or another array (in the case of transmitarrays), separated by distances ranging from a fraction of to approximately one wavelength. In this case the phase along the plane of the array cannot be treated as a continuous variable. This makes it impossible to derive for example the generalized Snell’s lawin terms of a phase gradient along the interface. This generalized law along with its counterpart for reflectionapplies to the whole optical spectrum for suitable designer interfaces and it can be a guide for the design of new photonic devices.References and Notes1. J. B. Pendry, D. Schurig, D. R. Smith, “Controllingelectromagnetic fields,” Science 312, 1780 (2006).2. U. Leonhardt, “Optical conformal mapping,” Science 312,1777 (2006).3. W. Cai, V. Shalaev, Optical Metamaterials: Fundamentalsand Applications (Springer, 2009)4. N. Engheta, R. W. Ziolkowski, Metamaterials: Physics andEngineering Explorations (Wiley-IEEE Press, 2006).5. I. I Smolyaninov, E. E. Narimanov, Metric signaturetransitions in optical metamaterials. Phys. Rev. Lett.105,067402 (2010).6. S. D. Brorson, H. A. Haus, “Diffraction gratings andgeometrical optics,” J. Opt. Soc. Am. B 5, 247 (1988).7. R. P. Feynman, A. R. Hibbs, Quantum Mechanics andPath Integrals (McGraw-Hill, New York, 1965).8. E. Hecht, Optics (3rd ed.) (Addison Wesley PublishingCompany, 1997).9. H. T. Miyazaki, Y. Kurokawa, “Controlled plasmonnresonance in closed metal/insulator/metal nanocavities,”Appl. Phys. Lett. 89, 211126 (2006).onSeptember1,211www.sciencemag.orgDownloadedfrom10. D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil,“Flat dielectric grating reflectors with focusing abilities,”Nature Photon. 4, 466 (2010).11. J. A. Fan et al., “Self-assembled plasmonic nanoparticleclusters,” Science 328, 1135 (2010).12. B. Luk’yanchuk et al., “The Fano resonance in plasmonicnanostructures and metamaterials,” Nature Mater. 9, 707 (2010).13. R. D. Grober, R. J. Schoelkopf, D. E. Prober, “Opticalantenna: Towards a unity efficiency near-field opticalprobe,” Appl. Phys. Lett. 70, 1354 (1997).14. L. Novotny, N. van Hulst, “Antennas for light,” NaturePhoton. 5, 83 (2011).15. Q. Xu et al., “Fabrication of large-area patternednanostructures for optical applications by nanoskiving,”Nano Lett. 7, 2800 (2007).16. M. Sukharev, J. Sung, K. G. Spears, T. Seideman,“Optical properties of metal nanoparticles with no center of inversion symmetry: Observation of volume plasmons,”Phys. Rev. B 76, 184302 (2007).17. P. Biagioni, J. S. Huang, L. Duò, M. Finazzi, B. Hecht,“Cross resonant optical antenna,” Phys. Rev. Lett. 102,256801 (2009).18. S. Liu et al., “Double-grating-structured light microscopyusing plasmonic nanoparticle arrays,” Opt. Lett. 34, 1255 (2009).19. J. Ginn, D. Shelton, P. Krenz, B. Lail, G. Boreman,“Polarized infrared emission using frequency selectivesurfaces,” Opt. Express 18, 4557 (2010).20. Materials and methods are available as supportingmaterial on Science Online.21. J. F. Nye, M. V. Berry, “Dislocations in wave trains,”Proc. R. Soc. Lond. A. 336, 165 (1974).22. M. Padgett, J. Courtial, L. Allen, “Ligh’'s orbital angularmomentum,” Phys. Today 57, 35 (2004).23. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P.Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys.Rev. A, 45, 8185 (1992).24. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen,J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321 (1994).25. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White,“Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221 (1992).26. H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angularmomentum to absorptive particles from a laser beam witha phase singularity,” Phys. Rev. Lett. 75, 826 (1995).27. G. Gibson et al, “Free-space information transfer usinglight beams carrying orbital angular momentum,” Opt.Express 12, 5448 (2004). 28. D. M. Pozar, S. D. Targonski, H. D. Syrigos, “Design ofmillimeter wave microstrip reflectarrays,” IEEE Trans.Antennas Propag. 45, 287 (1997).29. J. A. Encinar, “Design of two-layer printed reflectarraysusing patches of variable size,” IEEE Trans. AntennasPropag. 49, 1403 (2001).30. C. G. M. Ryan et al., “A wideband transmitarray usingdual-resonant double square rings,” IEEE Trans. AntennasPropag. 58, 1486 (2010).31. P. Padilla, A. Muñoz-Acevedo, M. Sierra-Castañer, M.Sierra-Pérez, “Electronically reconfigurable transmitarrayat Ku band for microwave applications,” IEEE Trans.Antennas Propag. 58, 2571 (2010).32. H. R. Philipp, “The infrared optical properties of SiO2 andSiO2 layers on silicon,” J. Appl. Phys. 50, 1053 (1979).33. R. W. P. King, The Theory of Linear Antennas (HarvardUniversity Press, 1956).34. J. D. Jackson, Classical Electrodynamics (3rd edition)(John Wiley & Sons, Inc. 1999) pp. 665.35. E. D. Palik, Handbook of Optical Constants of Solids(Academic Press, 1998).36. I. Puscasu, D. Spencer, G. D. Boreman, “Refractive-indexand element-spacing effects on the spectral behavior ofinfrared frequency-selective surfaces,” Appl. Opt. 39,1570 (2000).37. G. W. Hanson, “On the applicability of the surfaceimpedance integral equation for optical and near infraredcopper dipole antennas,” IEEE Trans. Antennas Propag.54, 3677 (2006).38. C. R. Brewitt-Taylor, D. J. Gunton, H. D. Rees, “Planarantennas on a dielectric surface,” Electron. Lett. 17, 729(1981).39. D. B. Rutledge, M. S. Muha, “Imaging antenna arrays,”IEEE Trans. Antennas Propag. 30, 535 (1982). Acknowledgements: The authors acknowledge helpful discussion with J. Lin, R. Blanchard, and A. Belyanin. Theauthors acknowledge support from the National ScienceFoundation, Harvard Nanoscale Science and EngineeringCenter (NSEC) under contract NSF/PHY 06-46094, andthe Center for Nanoscale Systems (CNS) at HarvardUniversity. Z. G. acknowledges funding from theEuropean Communities Seventh Framework Programme(FP7/2007-2013) under grant agreement PIOF-GA-2009-235860. M.A.K. is supported by the National ScienceFoundation through a Graduate Research Fellowship.Harvard CNS is a member of the NationalNanotechnology Infrastructure Network (NNIN). TheLumerical FDTD simulations in this work were run on theOdyssey cluster supported by the Harvard Faculty of Artsand Sciences (FAS) Sciences Division ResearchComputing Group.onSeptember1,211www.sciencemag.orgDownloadedfrom。

Adsorption of polydisperse polymer chains

a r X i v :c o n d -m a t /9901014v 1 [c o n d -m a t .s o f t ] 4 J a n 1999Adsorption of polydisperse polymer chainsRichard P.SearDepartment of Physics,University of Surrey Guildford,Surrey GU25XH,United Kingdomemail:r.sear@February 1,2008AbstractThe adsorption of polydisperse ideal polymer chains is shown to be sensitive to the large N tail of the distribution of chains.If and only if the number of chains decays more slowly than exponentially then there is an adsorption transition like that of monodisperse inﬁnite chains.If the number decays exponentially the adsorption density diverges continuously at a temperature which is a function of the mean chain length.At low coverages,chains with repulsive monomer–monomer interactions show the same qualitative behaviour.PACS:36.20.-r,67.70.+n,36.20.CwSynthetic polymers are almost inevitably polydisperse.The polymer chains are not all of the same length,and so a polymer is not a single component but a mixture of chains of diﬀerent lengths.However,almost all of the large number of theoretical studies of polymer adsorption have considered monodisperse polymer chains,where the chains are all of the same length.This is done to simplify matters and is a reasonable approximation when,as is often the case,the width of the distribution of lengths of polymer chain is small.Here we study polymers in which there are polymer chains with a very wide range of lengths,paying particular attention to the longest polymers in the distri-bution.Our motivation is not just the experimental fact that synthetic polymers are polydisperse yet are almost invariably treated within theory as being monodisperse.Polydisperse polymers may,and we show that they do,ex-hibit behaviour which is qualitatively diﬀerent from that of monodisperse polymers.We study a very highly idealised model of polymer adsorption:ideal chains adsorbing onto a wall due to a short-ranged attraction between the wall and the monomers.This problem has been extensively studied for monodisperse chains [2–6],but not as far as the author is aware for polydisperse chains.The adsorption of an ideal chain is a textbook prob-lem.It is highly idealised:almost invariably the densityof polymer adsorbed onto a surface is too high for interac-tions between the polymer segments to be neglected.How-ever,it has served well as a simple ﬁrst model of adsorption of monodisperse polymer chains [2]and we shall use it as such for polydisperse chains.There has been previous work within the Scheutjens–Fleer theory [7],on the adsorption of polydisperse polymer chains but not,as far as the author is aware,on ideal chains.Roefs et al.[7]did not consider the large N tail of the distribution and so did not ﬁnd the behaviour we will describe below.Note that in ref.[7]the ratio between the volume of the solution and the surface area of the wall was only of the order of the radius of gyra-tion of the polymers.By contrast here we study a wall in contact with a bulk polymer solution so the ratio between the volume and surface area is inﬁnity.We start by brieﬂy reviewing the behaviour of monodis-perse chains,with particular emphasis on the dependence of the adsorption on the length of the chain.The chains are ideal and each consists of a linear chain of N monomers of length a .The chains are at a non-zero number density ρin the bulk.The bulk polymer solution is in contact with a wall.This wall attracts monomers via a short-ranged attraction;the range is taken to be a for simplicity.The strength of attraction is ǫ.Having discussed monodisperse chains we then generalise the theory to describe an arbi-trary polydisperse mixture of chains of diﬀering lengths.We show that not all polydisperse mixtures behave in the same way:there are three qualitatively diﬀerent behaviours possible.Which behaviour a mixture exhibits depends on the large N tail of the distribution.This is also true of the cloud-point curve of polydisperse polymers,as was shownby ˇSolc [8].Recent work by the author on the bulk phasebehaviour of polydisperse hard spheres [9]has found a sim-ilar sensitivity to the tail of the distribution.Ideal chains do not interact with each other and so ﬁnite ideal chains are independent systems with a ﬁnite number of degrees of freedom.They therefore cannot exhibit a phase transition.For ideal chains there is only an adsorp-tion transition in the limit that the number of monomers N1tends to inﬁnity.We denote the transition temperature of inﬁnite chains by T a.Forﬁnite chains there is only a steep increase in the adsorbed density near T a.Most distribu-tion functions for polydisperse mixtures tend to zero only at inﬁnity(e.g.,the Schulz and distribution[1,10]),which is to say that they contain an inﬁnitesimal density of in-ﬁnitely large chains.The question then arises:as there are inﬁnitely long chains present is there a phase transition?Below the adsorption temperature T a large chains,N→∞are adsorbed onto the wall.The chains form a layer of height D which can be easily estimated using a scaling ar-gument due to de Gennes[2].We restrict ourself to the weak-adsorption regime[2]where D≫a.The free en-ergy of adsorption of a single ideal chain has two competing parts.Theﬁrst is the entropy change when a polymer chain is conﬁned to a layer of height D,this scales as−Na2/D2. We take Boltzmann’s constant to be equal to unity.The second is the energy change due to adsorption,this is−Nǫtimes the fraction of monomers within the range of wall’s attraction a/D.So,the free energy of adsorption∆F isgiven by∆F∼T Na2D.(1)The width D of the layer at equilibrium is found by min-imising∆FD∼a(T/ǫ),(2) which gives an adsorption free energy of∆F∼−Nǫ2/T.(3) An exact treatment of adsorbed ideal chains yields[3–6]∆F(N)But the most interesting case is that of an exponential decay.There the behaviour is not like that of any monodis-perse polymer.Consider a simple exponentially decreasing distribution functionx(N)=N)(10)whereN d N(11)∼ ρ/1+N−1/2.The divergence is also present for a distribution func-tion x(N)which is an exponential times a power law,as the Schulz distribution function is[1,10].Indeed the behaviour is qualitatively unchanged if the exponential is multiplied by any function of N which varies more slowly than expo-nentially,allowing the exponential dependence to dominate at large N.See ref.[1]for diﬀerent distribution functions found for linear polymers.All the above is for ideal polymers.What about poly-mers with monomers which repel each other,the standard model of a polymer in a good solvent?The free energy of adsorption of isolated polymer chains with monomers which repel each other again has a dominant linear term in N al-though the dependence on T is diﬀerent[4,6].An exact ﬁeld-theory treatment yields for long chains with repulsive monomer–monomer interactions[4,6]∆F(N)N,the normalisation of the distribution is almost unaﬀected by the cutoﬀand we can estimateρa by simply replacing∞by N c as the upper limit of integration in eq.(11)ρa∼ ρ/N−1 N c T<T a,(15)which can also be written asρa1−N(ǫ/T−ǫ/T a)2−1 N c N T<T a.(16)The adsorption only depends on two parameters,N.There is no transition but for not too smallN.In fact polydisperse chains have a much higher adsorbed density than monodisperse chains with the same average lengthN(solid curve).Physically,what is happening is that most of the ad-sorbed monomers are from the largest chains of the distri-bution.That most of the adsorbed monomers are from the longest chains can be seen if we recall that the contribution to the adsorbed density of chainsρa from chains of length N is given by the integrand of eq.(11).For a cutoﬀdistri-bution below the T e of a non-cutoﬀdistribution with the same(A)x(N)decays more slowly than exponentially in the N→∞limit,then the mixture behaves like inﬁnite monodisperse chains;(B)x(N)decays exponentially in which case the adsorption density diverges continuously, and(C)either x(N)decays more rapidly than exponen-tially or is cutoﬀat some maximum chain length,in which case the mixture behaves likeﬁnite monodisperse chains.To conclude,we have been able to classify polydisperse polymer solutions into three classes,depending on their adsorption behaviour.The case of an exponentially decay-ing distribution of chain lengths is especially interesting as it behaves in a way which is qualitatively diﬀerent from that of monodisperse chains.Sharp distinctions between monodisperse and the diﬀerent polydisperse distributions can only be drawn when there are inﬁnite chains present. Only then is a phase transition possible whose presence or absence can be used to assign unambiguously a distribution to one of the classes.However,at least when the polydis-perse distribution function x(N)decays exponentially,even when the distribution is truncated beyond someﬁnite value N c there is a remnant of the behaviour of the untruncated distribution.Near the temperature T e where the adsorbed density of the untruncated distribution would diverge there is changeover in behaviour:above T e the shortest chains contribute most to the adsorbed densityρa,below it the longest chains contribute most.References[1]FLORY P.J.,Principles of Polymer Chemistry(Cor-nell University Press,Ithaca)1953.[2]DE GENNES P.-G.,Scaling Concepts in PolymerPhysics(Cornell University Press,Ithaca)1979.[3]RUBIN R.,J.Chem.Phys.,43(1965)2392.[4]EISENRIEGLER E.,KREMER K.and BINDER K.,J.Chem.Phys.,77(1982)6296.[5]EISENRIEGLER E.,J.Chem.Phys.,79(1983)1052.[6]EISENRIEGLER E.,Polymers near Surfaces(WorldScientiﬁc Press,Singapore)1993.[7]ROEFS S.P.F.M.,SCHEUTJENS J.M.H.M.andLEERMAKERS F.A.M.,Macromolecules,27(1994) 4810.[8]ˇSOLC K.,Macromolecules,3(1970)665.[9]SEAR R.P.,cond-mat/9811150().[10]SALACUSE J.J.and STELL G.,J.Chem.Phys.77(1982)3714.Figure CaptionFig. 1.The adsorbed densityρa at the wall is plotted as a function of the attraction energy over the thermal en-ergy minus its value at the adsorption transition for in-ﬁnitely long chains.The solid and dot-dashed curves are for monodisperse chains of lengths N=50and150,re-spectively.The dashed curve is for polydisperse chains dis-tributed according to an exponential distribution of sizes with an average chain length。

斯普林格数学研究生教材丛书

高分子的良溶剂和不良溶剂的两种定义辩析

作者简介:胡文兵(1966-),男,浙江人,复旦大学高分子科学系讲师,博士,1998年赴德国符赖堡大学物理系从事博士后研究,现主要从事高分子物理的教学和科研工作,研究方向集中于高分子凝聚态物理的基本问题。

高分子的良溶剂和不良溶剂的两种定义辩析胡文兵(复旦大学高分子科学系,上海200433) 摘要:介绍了目前国际学术界存在的对高分子溶液体系的溶剂优劣性的两种判断标准,即相互作用参数ς=1 2和ς=0,前者基于混合自由能,后者基于混合热。

作者阐明了其倾向于采用后者的观点,而后者在目前国内所有高分子物理学教科书中均未被介绍。

关键词:高分子溶液;良溶剂;不良溶剂判断溶剂对某种高分子溶质的作用是研究高分子溶液的重要内容之一,这种作用主要表现在两个概念层面上,一个是可溶性问题(so lub ility ),这主要从高分子及溶剂的溶解度参数的相近程度上来判断[1],也即所谓的‘相似相容’原则;另一个是亲和性问题(affin ity ),对此普遍采用良溶剂(good so lven t )和不良溶剂(poo r so lven t )的说法来区分溶剂的优劣性,但是目前国际学术界对良溶剂和不良溶剂的划分标准却存在两种定义,有必要在有关高分子溶液的教学和科研中加以澄清。

第一种定义是目前国内所有高分子物理学教科书中所介绍的,它考察溶剂分子与整个高分子之间混合时相互作用的变化大小,也就是从热力学混合自由能的角度,来判断溶剂的优劣。

例如,F lo ry 2K rigboum 稀溶液理论[2],假定单链线团内链单元沿径向满足高斯分布,以F lo ry 2H uggin s 溶液平均场理论为基础,将其外推到稀溶液渗透压展开式中,算出使第二维利系数A 2=0的结果,即相互作用参数ς=1 2,这正好是F lo ry 2H uggin s 理论预测当链长无穷大时相分离的临界点,此时的溶剂称为theta 溶剂,作为判断溶剂优劣性的分界点,当ς<1 2时,称良溶剂,当ς>1 2时,则称不良溶剂。

原子核物理专业英语

原子核物理专业词汇中英文对照表absorption cross-section吸收截面activity radioactivity放射性活度activity活度adiabatic approximation浸渐近似allowed transition容许跃迁angular correlation角关联angular distribution角分布angular-momentum conservation角动量守恒anisotropy各项异性度annihilation radiation湮没辐射anomalous magnetic moment反常极矩anti neutrino反中微子antiparticle反粒子artificial radioactivity人工放射性atomic mass unit原子质量单位atomic mass原子质量atomic nucleus原子核Auger electron俄歇电子backbending回弯bag model口袋模型baryon number重子数baryon重子binary fission二分裂变binging energy结合能black hole黑洞bombarding particle轰击粒子bottom quark底夸克branching ration 分支比bremsstrahlung轫致辐射cascade radiation级联辐射cascade transition级联跃迁centrifugal barrier离心势垒chain reaction链式反应characteristic X-ray特征X射线Cherenkov counter切连科夫计数器coincidence measurement符合剂量collective model集体模型collective rotation 集体转动collective vibration集体震动color charge色荷complete fusion reaction全熔合反应complex potential复势compound-nucleus decay复合核衰变compound-nucleus model复合核模型compound nucleus复合核Compton effect康普顿效应Compton electron康普顿电子Compton scattering康普顿散射cone effect圆锥效应conservation law守恒定律controlled thermonuclear fusion受控热核聚变cosmic ray宇宙射线Coulomb barrier库仑势垒Coulomb energy库伦能Coulomb excitation库仑激发CPT theorem CPT定理critical angular momentum临界角动量critical distance临界距离critical mass临界质量critical volume临界体积daily fuel consumption 燃料日消耗量dalitz pair 达立兹对damage criteria 危害判断准则damage 损伤damped oscillations 阻尼震荡damped vibration 阻尼震荡damped wave 阻尼波damper 减震器damping factor 衰减系数damping 衰减的damp proof 防潮的damp 湿气danger coefficient 危险系数danger dose 危险剂量danger range 危险距离danger signal 危险信号dark current pulse 暗电瘤冲dark current 暗电流data acquisition and processing system 数据获得和处理系统data base 数据库data communication 数据通信data processing 数据处理data reduction equipment 数据简化设备data 数据dating 测定年代daughter atom 子体原子daughter element 子体元素daughter nuclear子核daughter nucleus 子体核daughter nuclide 子体核素daughter 蜕变产物dd reaction dd反应dd reactor dd反应器deactivation 去活化dead ash 死灰尘dead band 不灵敏区dead space 死区dead time correction 死时间校正dead time 失灵时间deaerate 除气deaeration 除气deaerator 除气器空气分离器deaquation 脱水debris activity 碎片放射性debris 碎片de broglie equation 德布罗意方程de broglie frequency 德布罗意频率de broglie relation 德布罗意方程de broglie wavelength 德布罗意波长de broglie wave 德布罗意波debuncher 散束器debye radius 德拜半径debye scherrer method 德拜谢乐法debye temperature 德拜温度decade counter tube 十进计数管decade counting circuit 十进制计数电路decade counting tube 十进管decade scaler 十进位定标器decagram 十克decalescence 相变吸热decalescent point 金属突然吸热温度decanning plant 去包壳装置decanning 去包壳decantation 倾析decanter 倾析器decanting vessel 倾析器decan 去掉外壳decarburization 脱碳decascaler 十进制定标器decatron 十进计数管decay chain衰变链decay coefficient 衰变常数decay constant 衰变常数decay constant衰变常量decay energy衰变能decay factor 衰变常数decay fraction衰变分支比decay heat removal system 衰变热去除系统decay heat 衰变热decay kinematics 衰变运动学decay out 完全衰变decay period 冷却周期decay power 衰减功率decay rate 衰变速度decay scheme衰变纲图decay series 放射系decay storage 衰变贮存decay table 衰变表decay time 衰变时间decay 衰减decelerate 减速deceleration 减速decigram 分克decimeter wave 分米波decladding plant 去包壳装置decladding 去包壳decommissioning 退役decompose 分解decomposition temperature 分解温度decomposition 化学分解decontaminability 可去污性decontamination area 去污区decontamination factor 去污因子decontamination index 去污指数decontamination plant 去污装置decontamination reagent 去污试剂decontamination room 去污室decontamination 净化decoupled band 分离带decoupling 去耦解开decrease 衰减decrement 减少率deep dose equivalent index 深部剂量当量指标deep inelastic reaction深度非弹性反应deep irradiation 深部辐照deep therapy 深部疗deep underwater nuclear counter 深水放射性计数器deep water isotopic current analyzer 深海水连位素分析器de excitation 去激发de exemption 去免除defecation 澄清defective fuel canning 破损燃料封装defective fuel element 破损元件defect level 缺陷程度defectoscope 探伤仪defect 缺陷defence 防护deficiency 不足define 定义definite 确定的definition 分辨deflagration 爆燃deflecting coil 偏转线圈deflecting electrode 偏转电极deflecting field 偏转场deflecting plate 偏转板deflecting system 偏转系统deflecting voltage 偏转电压deflection angle 偏转角deflection plate 偏转板deflection system 偏转系统deflection 负载弯曲deflector coil 偏转线圈deflector field 致偏场deflector plate 偏转板deflector 偏转装置deflocculation 解凝defoamer 去沫剂defoaming agent 去沫剂defocusing 散焦deformation bands 变形带deformation energy 变形能deformation of irradiated graphite 辐照过石墨变形deformation parameter形变参量deformation 变形deformed nucleus 变形核deformed region 变形区域deform 变形degassing 脱气degas 除气degeneracy 简并degenerate configuration 退化位形degenerate gas 简并气体degenerate level 简并能级degenerate state 简并态degeneration 简并degradation of energy 能量散逸degradation 软化degraded spectrum 软化谱degree of acidity 酸度degree of anisotropic reflectance 蛤异性反射率degree of burn up 燃耗度degree of cross linking 交联度degree of crystallinity 结晶度degree of degeneration 退化度degree of dispersion 分散度degree of dissociation 离解度degree of enrichment 浓缩度degree of freedom 自由度degree of hardness 硬度degree of ionization 电离度degree of moderation 慢化度degree of polymerization 聚合度degree of purity 纯度dehumidify 减湿dehydrating agent 脱水剂dehydration 脱水deionization rate 消电离率deionization time 消电离时间deionization 消电离dejacketing 去包壳delay circuit 延迟电路delayed alpha particles 缓发粒子delayed automatic gain control 延迟自动增益控制delayed coincidence circuit 延迟符合电路delayed coincidence counting 延迟符合计数delayed coincidence method 延迟符合法delayed coincidence unit 延迟符合单元delayed coincidence 延迟符合delayed criticality 缓发临界delayed critical 缓发临界的delayed fallout 延迟沉降物delayed fission neutron 缓发中子delayed gamma 延迟性射线delayed neutron detector 缓发中子探测器delayed neutron emitter 缓发中子发射体delayed neutron failed element monitor 缓发中子破损燃料元件监测器delayed neutron fraction 缓发中子份额delayed neutron method 缓发中子法delayed neutron monitor 缓发中子监测器delayed neutron precursor 缓发中子发射体delayed neutron 缓发中子delayed proton缓发质子delayed reactivity 缓发反应性delay line storage 延迟线存储器delay line 延迟线delay system 延迟系统delay tank 滞留槽delay time 延迟时间delay unit 延迟单元delay 延迟delineation of fall out contours 放射性沉降物轮廓图deliquescence 潮解deliquescent 潮解的delivery dosedose 引出端delta electron 电子delta metal 合金delta plutonium 钚delta ray 电子demagnetization 去磁demagnetize 去磁dematerialization 湮没demineralization of water 水软化demineralization 脱盐demonstration reactor 示范反应堆demonstration 示范dempster mass spectrograph 登普斯特质谱仪denaturalization 变性denaturant 变性剂denaturation of nuclear fuel 核燃料变性denaturation 变性denature 变性denaturize 变性denitration 脱硝dense plasma focus 稠密等离子体聚焦dense 稠密的densimeter 光密度计densimetry 密度测定densitometer 光密度计densitometry 密度计量学density analog method 密度模拟法density bottle 密度瓶density effect 密度效应density gradient instability 密度梯度不稳定性density of electrons 电子密度deoxidation 脱氧deoxidization 脱氧departure from nucleate boiling ratio 偏离泡核沸腾比departure from nucleate boiling 偏离泡核沸腾dependability 可靠性dependence 相依dependency 相依dephlegmation 分凝酌dephlegmator 分馏塔depilation dose 脱毛剂量depilation 脱毛depleted fraction 贫化馏分depleted fuel 贫化燃料depleted material 贫化材料depleted uranium shielding 贫铀屏蔽depleted uranium 贫化铀depleted water 贫化水depleted zone 贫化区域deplete uranium tail storage 贫化铀尾料储存depletion layer 耗尽层depletion 贫化;消耗depolarization 去极化depolymerization 解聚合deposit dose 地面沉降物剂量deposited activity 沉积的放射性deposition 沉积deposit 沉淀depression 减压depressurization accident 失压事故depressurizing system 降压系统depth dose 深部剂量depth gauge 测深计depth of focus 焦点深度depthometer 测深计derby 粗锭derivant 衍生物derivate 衍生物derivative 衍生物derived estimate 导出估价值derived unit 导出单位derived working limit 导出工撰限desalinization 脱盐desalting 脱盐descendant 后代desensitization 脱敏desensitizer 脱敏剂desiccation 干燥desiccator 干燥器防潮器design basis accident 设计依据事故design basis depressurization accident 设计依据卸压事故design basis earthquake 设计依据地震design dose rate 设计剂量率design of the safeguards approach 保障监督方法设计design power 设计功率design pressure 设计压力design safety limit 设计安全限design temperature rise 设计温度上升design transition temperature 设计转变温度design 设计desmotropism 稳变异构desmotropy 稳变异构desorption 解吸desquamation 脱皮destruction test 破坏性试验destructive distillation 干馏detailed balance principle细致平衡原理detailed decontamination 细部去污detectable activity 可探测的放射性detectable 可检测的detection efficiency 探测效率detection efficiency探测效率detection limit 探测限detection of neutrons from spontaneous fission 自发裂变中子探测detection of radiation 辐射线的探测detection probability 探测概率detection time 探测时间detection 探测detector 1/v 1/v探测器detector efficiency 探测僻率detector foil 探测骗detector noise 探测齐声detector shield 探测屏蔽detector tube 检波管detector with internal gas source 内气源探测器detector 探测器敏感元件detect 探测;检波detergent 洗涤剂determination 确定deterrence of diversion 转用制止detonating gas 爆鸣气detonation altitude 爆炸高度detonation point 爆炸点detonation yield 核爆炸威力detonation 爆炸detoxifying 净化detriment 损害detted line 点线deuteride 氘化物deuterium alpha reaction 氘反应deuterium critical assembly 重水临界装置deuterium leak detector 重水检漏器deuterium moderated pile low energy 低功率重水慢化反应堆deuterium oxide moderated reactor 重水慢化反应堆deuterium oxide 重水deuterium pile 重水反应堆deuterium sodium reactor 重水钠反应堆deuterium target 氘靶deuterium tritium fuel 氘氚燃料deuterium tritium reaction 氘氚反应deuterium 重氢deuteron alpha reaction 氘核反应deuteron binding energy 氘核结合能deuteron induced fission 氘核诱发裂变deuteron neutron reaction 氘核中子反应deuteron proton reaction 氘核质子反应deuteron stripping 氘核涎deuterum moderated pile 重水反应堆deuton 氘核development of uranium mine 铀矿开发development 发展deviation from the desired value 期望值偏差deviation from the index value 给定值偏差deviation 偏差dewatering 脱水dewindtite 水磷铅铀矿dew point 露点dextro rotatory 右旋的diagnostic radiology 诊断放射学diagnostics 诊断diagram 线图dialkyl phosphoric acid process 磷酸二烷基酯萃取法dialysis 渗析dial 度盘diamagnetic effect 抗磁效应diamagnetic loop 抗磁圈diamagnetic substance 抗磁体diamagnetic susceptibility 抗磁化率diamagnetism of the plasma particles 等离子体粒子反磁性diamagnetism 反磁性diamagnet 抗磁体diameter 直径diamond 稳定区;金刚石diaphragm gauge 膜式压力计diaphragm type pressure gauge 膜式压力计diaphragm 薄膜diapositive 透谬片diascope 投影放影器投影仪diathermance 透热性diathermancy 透热性diatomic gas 双原子气体diatomic molecule 二原子分子dibaryon 双重子diderichite 水菱铀矿dido type heavy water research reactor 迪多型重水研究用反应堆dido 重水慢化反应堆dielectric after effect 电介质后效dielectric constant 介电常数dielectric hysteresis 电介质滞后dielectric polarization 电介质极化dielectric strain 电介质变形dielectric strength 绝缘强度dielectric 电介质diesel engine 柴油机diesel oil 柴油difference ionization chamber 差分电离室difference linear ratemeter 差分线性计数率计difference number 中子过剩difference of potential 电压difference scaler 差分定标器differential absorption coefficient 微分吸收系数differential absorption ratio 微分吸收系数differential albedo 微分反照率differential control rod worth 控制棒微分价值differential cross section 微分截面differential cross-section微分截面differential discriminator 单道脉冲幅度分析器differential dose albedo 微分剂量反照率differential energy flux density 微分能通量密度differential particle flux density 粒子微分通量密度differential pressure 压差differential range spectrum 射程微分谱differential reactivity 微分反应性differential recovery rate 微分恢复率differential scattering cross section 微分散射截面differentiator 微分器diffraction absorption 衍射吸收diffraction analysis 衍射分析diffraction angle 衍射角diffraction grating 衍射光栅diffraction instrument 衍射仪diffraction pattern 衍射图diffraction peak 衍射峰值diffraction scattering 衍射散射diffraction spectrometer 衍射谱仪diffraction spectrum 衍射光谱diffraction 衍射diffractometer 衍射仪diffusate 扩散物diffuse band 扩散带diffused junction semiconductor detector 扩散结半导体探测器diffused 散射的diffuseness parameter 扩散性参数diffuse reflection 漫反射diffuser 扩散器diffuse scattering 漫散射diffuse 扩散diffusion approximation 扩散近似diffusion area 扩散面积diffusion barrier 扩散膜diffusion cascade 扩散级联diffusion chamber 扩散云室diffusion coefficient for neutron flux density 中子通量密度扩散系数diffusion coefficient for neutron number density 中子数密度扩散系数diffusion coefficient 扩散系数diffusion column 扩散塔diffusion constant 扩散常数diffusion cooling effect 扩散冷却效应diffusion cooling 扩散冷却diffusion cross section 扩散截面diffusion current density 扩散淋度diffusion current 扩散电流diffusion energy 扩散能diffusion equation 扩散方程diffusion factory 扩散工厂diffusion kernel 扩散核diffusion layer 扩散层diffusion length 扩散长度diffusion length扩散长度diffusion mean free path 扩散平均自由程diffusion plant 扩散工厂diffusion pump 扩散泵diffusion rate 扩散速率diffusion stack 务马堆diffusion theory 扩散理论diffusion time 扩散时间diffusion 扩散diffusivity 扩散系数digital analog converter 数模转换器digital computer 数字计算机digital data acquisition and processing system 数字数据获取与处理系统digital data handling and display system 数字数据处理和显示系统digital recorder 数字记录器digital time converter 数字时间变换器dilation 扩胀dilatometer 膨胀计diluent 稀释剂dilute solution 稀溶液dilute 冲淡dilution analysis 稀释分析dilution effect 稀释效应dilution method 稀释法dilution ratio 稀释比dilution 稀释dimensional change 尺寸变化dimension 尺寸diminishing 衰减dimorphism 双晶现象di neutron 双中子dineutron 双中子dingot 直接铸锭dip counter tube 浸入式计数管dipelt 双重线dipole dipole interaction 偶极子与偶极子相互酌dipole layer 偶极子层dipole momentum 偶极矩dipole moment 偶极矩dipole radiation 偶极辐射dipole transition 偶极跃迁dipole 偶极子di proton 双质子dirac electron 狄拉克电子dirac equation 狄拉克方程dirac quantization 狄拉克量子化dirac theory of electron 狄拉克电子论direct and indirect energy conversion 直接和间接能量转换direct contact heat exchanger 直接接触式换热器direct conversion reactor study 直接转换反应堆研究direct conversion reactor 直接转换反应堆direct current 直流direct cycle integral boiling reactor 直接循环一体化沸水堆direct cycle reactor 直接循环反应堆direct cycle 直接循环direct digital control 直接数字控制direct energy conversion 能量直接转换direct exchange interaction 直接交换相互酌direct exposure 直接辐照direct fission yield 原始裂变产额direct interaction 直接相互酌directional correlation of successive gamma rays 连续射线方向相关directional counter 定向计数器directional distribution 方向分布directional focusing 方向聚焦directional 定向的direction 方向direct isotopic dilution analysis 直接同位素稀释分析directly ionizing particles 直接电离粒子directly ionizing radiation 直接电离辐射direct measurement 直接测量direct radiant energy 直接辐射能direct radiation proximity indicator 直接辐射接近指示器direct radiation 直接辐射direct reaction 直接反应direct reaction直接反应direct use material 直接利用物质direct voltage 直羚压direct x ray analysis 直接x射线分析dirft tube 飞行管道dirt column 尘土柱dirty bomb 脏炸弹disadvantage factor 不利因子disagreement 不一致disappearence 消失discharge chamber 放电室discharge current 放电电流discharge in vacuo 真空放电discharge potential 放电电压discharge tube 放电管discharge voltage 放电电压discharge 放电discomposition 原子位移discontinuity 非连续性discontinuous 不连续的disc operating system 磁盘操椎统discrepancy 差异discrete energy level 不连续能级discrete spectrum 不连续光谱discrete state 不连续态discrete 离散的discrimination coefficient 甄别系数discriminator 鉴别器disinfectant 杀菌剂disintegrate 蜕衰disintegration chain 放射系disintegration constant 衰变常数disintegration curve 衰变曲线disintegration energy 衰变能disintegration heat 衰变热disintegration of elementary particles 基本粒子衰变disintegration particle 衰变粒子disintegration probability 衰变概率disintegration product 蜕变产物disintegration rate 衰变速度disintegration scheme 蜕变图disintegration series 蜕变系disintegrations per minute 衰变/分disintegrations per second 衰变/秒disintegration 蜕变disk source 圆盘放射源dislocation edge 位错边缘dislocation line 位错线dislocation 位错dismantling 解体disorder scattering 无序散射disorder 无序dispersal effect 分散效应dispersal 分散disperser 分散剂dispersing agent 分散剂dispersion fuel element 弥散体燃料元件dispersion fuel 弥散体燃料dispersion 分散dispersive medium 色散媒质displacement current 位移电流displacement kernel 位移核displacement law of radionuclide 放射性核素位移定律displacement law 位移定律displacement spike 离位峰displacement 替换displace 位移;代替disposal of radioactive effluents 放射性瘤液处置disposition 配置disproportionation 不均disruption 破坏disruptive instability 破裂不稳定性disruptive voltage 哗电压dissipation of energy 能消散dissipation 耗散dissociation constant 离解常数dissociation energy 离解能dissociation pressure 离解压dissociation 离解dissociative ionization 离解电离dissolution 溶解dissolver gas 溶解气体dissolver heel 溶解泣滓dissolver 溶解器distance control 遥控distant collision 远距离碰撞distillate 蒸馏液distillation column 蒸馏塔distillation method 蒸馏法distillation tower 蒸馏塔distillation 蒸馏distilled water 蒸馏水distiller 蒸馏器distilling apparatus 蒸馏器distilling flask 蒸馏瓶distorted wave Born approximation,DWBA扭曲波波恩近似distorted wave impulse approximation 畸变波冲动近似distorted wave theory 畸变波理论distorted wave 畸变波distortionless 不失真的distortion 畸变distributed ion pump 分布式离子泵distributed processing 分布式处理distributed source 分布源distribution coefficient 分配系数distribution factor 分布因子distribution function 分布函数distribution law 分配定律distribution of dose 剂量分布distribution of radionuclides 放射性核素分布distribution of residence time 停留时间分布distribution ratio 分配系数distribution 分布distrubited constant 分布常数disturbance 扰动disturbation 扰动diuranium pentoxide 五氧化二铀divergence of ion beam 离子束发散divergence problem 发散问题divergence 发散divergent lens 发射透镜divergent reaction 发散反应diversing lens 发射透镜diversion assumption 转用假定diversion box 转换箱diversion hypothesis 转用假设diversion path 转用路径diversion strategy 转用战略diversion 转向divertor 收集器divider 分配器division of operating reactors 反应堆运行部division 刻度djalmaite 钽钛铀矿document information system 文献情报体系doerner hoskins distribution law 德尔纳霍斯金斯分配定律dollar 元domain 磁畴dome 圆顶水柱dominant mutation 显性突变donut 环形室doping control of semiconductors 半导体掺杂物第Dopper effect多普勒效应doppler averaged cross section 多普勒平均截面doppler broadening 多普勒展宽doppler coefficient 多普勒系数doppler effect 多普勒效应doppler free laser spectroscopy 无多普勒激光光谱学doppler shift method 多普勒频移法doppler width 多普勒宽度dosage measurement 剂量测定dosage meter 剂量计dosage 剂量dose albedo 剂量反照率dose build up factor 剂量积累因子dose commitment 剂量负担dose effect curve 剂量效应曲线dose effect relationship 剂量效应关系dose equivalent commitment 剂量当量负担dose equivalent index 剂量当量指标dose equivalent limit 剂量当量极限dose equivalent rate 剂量当量率dose equivalent 剂量当量dose equivalent剂量当量dose fractionation 剂量分割dose limit 剂量极限dose measurement 剂量测量dose meter 剂量计dose modifying factor 剂量改变系数dose of an isotope 同位素用量dose prediction technique 剂量预报技术dose protraction 剂量迁延dose rate meter 剂量率测量计dose ratemeter 剂量率表dose rate 剂量率dose reduction factor 剂量减低系数dose response correlation 剂量响应相关dose unit 剂量单位dose 剂量dosifilm 胶片剂量计dosimeter charger 剂量计充电器dosimeter 剂量计dosimetry applications research facility 剂量测定法应用研究设施dosimetry 剂量测定法dotted line 点线double beam 双射束double beta decay 双衰变double bond 双键double charged 双电荷的double clad vessel 双层覆盖容器double compton scattering 双康普顿散射double container 双层容器double contingency principle 双偶然性原理double decomposition 复分解double differential cross section 二重微分截面double focusing mass spectrometer 双聚焦质谱仪double focusing 双聚焦double-humped barrier双峰势垒double ionization chamber 双电离室double precision 双倍精度double probe 双探针double pulse 双脉冲double resonance spectroscopy 双共振光谱学double resonance 双共振double scattering method 双散射法doublet splitting 双重线分裂doublet 电子对double walled heat exchanger 双层壁换热器doubling dose 加倍剂量doubling time meter 倍增时间测量计doubling time 燃料倍增时间doubly charged 双电荷的doubly closed shell nuclei 双闭合壳层核doughnut 环形室downcomer 下降管down quark下夸克down time 停机时间downwards coolant flow 下行冷却剂流downwind fall out 下风放射性沉降物draft 通风drain tank 排水槽draught 通风drell ratio 多列尔比dressing of uranium ore 铀矿石选矿dressing 选矿drier 干燥器drift instability 漂移不稳定性drift mobility 漂移率drift speed 漂移速度drift transistor 漂移晶体管drift velocity 漂移速度driven magnetic fusion reactor 从动磁核聚变反应堆driver fuel 驱动燃料drive voltage 控制电压drop reaction 点滴反应drop 点滴dry active waste 干放射性废物dry analysis 干法分析dry box 干箱dry criticality 干临界dry distillation 干馏dryer 干燥器dry friction 干摩擦dry ice 干冰drying oil 干性油drying oven 烘干炉drying 干燥dry out 烧干dry reprocessing 干法再处理dry way process 干法过程dry well 干井dt fuel cycle dt燃料循环dt reactor dt反应堆dual cycle boiling water reactor system 双循环沸水反应堆系统dual cycle reactor 双循环反应堆dual decay 双重放射性衰变dual energy use system 能量双重利用系统duality 二重性dual purpose nuclear power station 两用核电站dual purpose reactor 两用反应堆dual temperature exchange separation process 双温度交换分离法dual temperature exchange 双温度交换duant d形盒ductile brittle transition temperature 延性脆性转变温度ductility 延伸性duct 管dummy load 仿真负载dumontite 水磷铀铅矿dump condenser 事故凝汽器dump tank 接受槽dump valve 事故排放阀dump 烧毁元件存放处dunkometer 燃料元件包壳破损探测器duplet 电子对duration of a scintillation 闪烁持续时间duration 持续时间dust chamber 集尘室dust cloud 尘埃云dust collector 集尘器dust cooled reactor 粉尘冷却反应堆dust monitor 灰尘监测器dust sampler 灰尘取样器dust trap 集尘器dye laser 染料激光器dynamical friction 动摩擦dynamic behaviour 动态dynamic characteristic 动特性dynamic equilibrium ratio 动态平衡比dynamic equilibrium 动态平衡dynamic pressure 动压dynamic process inventory determination 动态过程投料量测定dynamic stabilization 动力稳定dynamic viscosity 动力粘滞系数dynamitron 地那米加速器并激式高频高压加速器dynamometer 测力计dynamo 发电机dyne 达因dynode 倍增电极dysprosium 镝dystectic mixture 高熔点混合物elastic scattering cross-section弹性散射截面elastic scattering弹性散射electronic stopping电子阻止elementary particle基本粒子EMC effect EMC效应endothermic reaction吸能反应energy conservation能量守恒energy loss能量损失energy resolution能量分辨率evaporation model蒸发模型even-even nucleus偶偶核exchange force交换力excitation curve激发曲线excitation function 激发函数excited state激发态exothermic reaction放能反应experimental Q-wave实验Q值exposure照射量fabrication 制造facility attachment 设施附属文件facility practice 设施实行facility safeguards approach 设施的保障监督方法facility 设施factor of porosity 孔隙率factor of stress concentration 应力集中因数factor 系数fading 阻尼failed can detection 破损燃料探测failed element indicator 破损元件指示器failed element monitor 破损元件监测器failed element 破损元件failed fuel detection and location 破损燃料探测和定位failed fuel detection 破损燃料探测failed fuel detector 破损燃料探测器fail safe instrument 故障时安全运行的仪器fail safe operation 安全运行failsafe 故障自动保险的failure checking 故障检查failure free operation 无故障运行failure mode 故障种类failure of parity conservation 宇称守恒的破坏failure prediction 故障预测fall back 回落falling stream method 降哩fallout density 放射性沉降物密度fallout monitoring 沉降物监测fallout particle 沉降粒子fallout pattern 沉降物分布型式fallout radioactive material 放射性沉降物fallout sampling network 沉降物取样网fallout shelter 沉降物掩蔽所fall out 放射性沉降fall time 下降时间false alarm probability 假报警几率false curvature 假曲率false scram 错误信号紧急停堆family 系fano's theorem 法诺定理faraday cage 法拉第笼faraday constant 法拉第常数faraday cup 法拉第笼farad 法拉far field 远场far infra red radiation 远红外辐射far ultraviolet radiation 远紫外辐射farvitron 线振质谱仪fast acting control rod 快动棕制棒fast advantage factor 快中子有利因子fast amplifier 宽频带放大器fast and thermal reactor burnup computer code 快和热反应堆燃耗计算机代码fast breeder reactor 快中子增殖反应堆fast breeder 快中子增殖反应堆fast burst reactor facility 快中子脉冲反应堆装置fast burst reactor 快中子脉冲反应堆fast ceramic reactor 陶瓷燃料快堆fast chamber 快速电离室fast chopper 快中子选择器fast coincidence unit 快符合单元fast coincidence 快符合fast compression cloud chamber 快压缩云室fast conversion 快中子转换fast cosmic ray neutron 宇宙射线的快中子fast critical assembly 快中子临界装置fast cross section 快中子截面fast detector 快速探测器fast effect 快中子倍增效应fast electron 快电子fast exponential experiment 快中子指数实验装置fast fissionability 快中子致裂变性fast fission effect factor 快中子裂变效应系数fast fission region 快中子裂变区fast fission 快中子裂变fast flux test facility 快中子通量试验装置fast flux 快中子通量fast fragment 快碎片fast killing dose 快速杀伤剂量fast leakage factor 快中子泄漏因子fast mean free path 快中子平均自由程fast medium 快中子介质fast multiplication effect 快中子倍增效应fast multiplication factor 快中子倍增因子fast neutron activation method 快中子活化法fast neutron breeder reactor 快中子增殖反应堆fast neutron breeding 快中子增殖fast neutron calibration 快中子刻度fast neutron collimator 快中子准直器fast neutron counter tube 快中子计数管fast neutron cycle 快中子增殖循环fast neutron detector 快中子探测器fast neutron diffusion length 快中子扩散长度fast neutron dose equivalent 快中子剂量当量fast neutron dosimeter 快中子剂量计fast neutron fission cross section 快中子裂变截面fast neutron fission increase rate 快中子裂变增加率fast neutron fluence 快中子积分通量fast neutron generator 快中子发生器fast neutron non leakage probability 快中子不泄漏几率fast neutron range 快中子区fast neutron reaction 快中子反应fast neutron reactor 快中子裂变反应堆fast neutron selector 快中子选择器fast neutron spectrometer 快中子谱仪fast neutron 快中子fast plutonium reactor 快中子钚反应堆fast radiochemistry 快速放射化学fast reaction 快速核反应fast reactor core test facility 快堆堆芯试验装置fast reactor physics 快速反应堆物理学fast reactor test assembly 快堆试验装置fast reactor thermal engineering facility 快堆热工程研究设施fast reactor 快中子裂变反应堆fast region 快中子区fast setback 迅速下降fast slow coincidence circuit 快慢符合电路fast sub critical assembly 快中子次临界装置fast test reactor 快中子试验反应堆fast thermal coupled reactor 快热耦合反应堆fast zero power reactor 快中子零功率反应堆fatal dose 致命剂量fatalities 死亡事故fatigue fracture 疲劳断裂fatigue limit 疲劳极限fatigue test 疲劳试验fatigue 疲劳faulted condition 损伤状态faulty fuel assembly 破损燃料组件fault 故障favorable geometry 有利几何条件fb 快中子增殖反应堆fcc 核燃料循环成本fcf 核燃料循环设施feather analysis 费塞分析feather's empirical formula 费瑟经验公式feather's rule 费瑟规则feed adjustment tank 进料蝶槽feedback circuit 反馈回路feedback control 反馈控制feedback loop 反馈回路feedback ratio 反馈比feedback signal 反馈信号feedback 反馈feed end 加料端feed material 给料物质feed plant 核燃料生产工厂feed pump 给水泵feed stage 给料段feed water control system 给水控制系统feedwater equipment 给水设备feedwater flow control 给水量控制feed water 给水feed 供给ferganite 水钒铀矿fermat's principle 费马原理fermi acceleration 费米加速fermi age equation 费米年龄方程fermi age theory 费米年龄理论fermi age 费米年龄fermi beta decay theory 费米衰变理论fermi characteristic energy level 费米能级fermi constant 费米常数fermi dirac gas 费米狄拉克气体fermi dirac statistics 费米狄拉克统计学fermi distribution function 费米狄拉克分布函数fermi distribution 费米分布fermi energy 费米能级fermi function 费米函数Fermi function费米函数fermi gas model 费米气体模型fermi gas 费米气体Fermi interaction F相互作用fermi interaction 费米相互酌fermi intercept 散射长度fermi level 费米能级fermi limit 费米能级fermion 费米子fermi particle 费米子fermi perturbation 费米微扰fermi plot 费米线图fermi potential 费米势fermi reactor 费米中子反应堆fermi resonance 费米共振fermi selection rules 费米选择定则fermi's golden rule 费米黄金法则fermi spectrum 费米谱fermi statistics 费米统计fermi surface 费米面fermi temperature 费米温度fermi theory of cosmic ray acceleration 费米宇宙射线加速理论fermi transition 费米跃迁fermium 镄fermi 费米。

科学文献

Decomposition of polytopes and polynomials⋆Shuhong Gao1and Alan Lauder21Department of Mathematical Sciences,Clemson University,Clemson,SC 29634-0975,USA.E-mail:sgao@.2Mathematical Institute,Oxford University,Oxford OX13LB,U.K.E-mail: lauder@.Abstract.Motivated by a connection with the factorization of multivariable poly-nomials,we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum.Weﬁrst show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial time algorithm for decomposing polygons.For higher dimensional polytopes,we give a heuristic algorithm which is based upon projections and uses randomization.Applications of our algorithms include absolute irreducibility testing and factorization of polyno-mials via their Newton polytopes.1IntroductionIt is well-known that the theory of convex polytopes has many applications across mathematics and computer science[2,9,12,28].One such application is to polynomial factorization,and motivated by this connection we discuss de-composition algorithms for polytopes.Given a multivariable polynomial one may associate with it,in a way we shall fully explain in Section2,an integral polytope called its Newton polytope.It was observed by Ostrowski in1921 that if the polynomial factors then its Newton polytope decomposes,in the sense of the Minkowski sum,into the Newton polytopes of the factors.The ramiﬁcations of this simple observation are two-fold.Firstly,criteria which ensure polytope indecomposability can be used to construct families of ir-reducible,indeed absolutely irreducible,polynomials.Secondly,algorithms which test whether a polytope is decomposable and construct decomposi-tions may be useful in factoring polynomials.Of course,such criteria and algorithms are also of independent interest and may have other applications. Indecomposability conditions were explored by theﬁrst author in[4]and will be discussed further in Section3.Our main focus will be,however,on the second application,that is on algorithms for decomposing polytopes.Weﬁrst show that the problem of testing whether a polytope is indecom-posable is NP-complete even in dimension two,so there does not exist,unless NP=P,a genuinely eﬃcient algorithm for decomposing polytopes.However,⋆Theﬁrst author was supported in part by NSF under Grant#DMS9970637and NSA under Grant#MDA904-00-1-0048.The second author gratefully acknowl-edges the support of the Marr Educational Trust and Wolfson College,Oxford, and thanks Dominic Welsh for his help and encouragement.we present a“pseudo-polynomial”time algorithm(see[6])for testing inde-composability in dimension two and a modiﬁed version which also allows oneto count the number of decompositions andﬁnd summands.We also discuss aheuristic algorithm which uses randomization for testing higher dimensionalpolytopes for indecomposability.In Section5,we describe applications of ouralgorithms to polynomials with respect to their irreducibility and factoriza-tion.In particular,we touch upon an open problem in polynomial factoriza-tion which we now describe.In his survey paper on polynomial factorization[16],Kaltofen concludes with several open problems one of which,due to B.Sturmfels,is stated as follows:“From the support vectors(e j1,...,e jn)of a sparse polynomial t j=1a e j1,...,e jn X e j11···X e jn n,compute by geometric con-siderations the support vectors of all possible factorizations”.This problemcan be attacked by our polytope method,although it must be noted that weare unable to give a complete solution.The basic idea runs as follows:Givena bivariable polynomial,we can compute its Newton polytope and thenﬁndall the integral summands of this polytope.The summands correspond to theNewton polytopes of all the possible factors of the polynomial.The integralpoints in a summand give the support vectors of the factor corresponding tothe summand.The remainder of the paper is organized in the following way.Section2contains the necessary background material on the theory of convex polytopesand in Section3we discuss some preliminary results on polytope indecom-posability which shall be useful to us but are also of independent interest.Section4is devoted to algorithms and is further divided into two parts:InSection4.1we present algorithms for both testing polygons for decompos-ability and counting and constructing decompositions of polygons.Section4.2contains a heuristic randomized algorithm for higher dimensional poly-topes based upon projections down to dimension two.Finally,in Section5we discuss applications of these algorithms to absolute irreducibility testingand polynomial factorization.2Polynomials and Newton polytopes2.1Background geometry and algebraBefore describing the connection between polynomials and polytopes,we re-call some terminology and results from the theory of convex polytopes([13]).Let I R denote theﬁeld of real numbers and I R n the Euclidean n-space.Aconvex set in I R n is a set such that the points on the line segment joiningany two points of the set lie in the set;the convex hull of a set of points isthe smallest convex set which contains them;and the convex hull of aﬁniteset of points is called a convex polytope.A point of a polytope is called avertex(or extreme point)if it does not belong to the interior of any line seg-ment contained in the polytope.A polytope is always the convex hull of its2vertices.A hyperplane cuts a polytope if both of the open half spaces deter-mined by it contain points of the polytope.A hyperplane which does not cut a polytope,but has a non-empty intersection with it is called a supporting hyperplane .The intersection of a supporting hyperplane and a polytope is a (proper)face ,and the union of all (proper)faces is the boundary .One may equivalently deﬁne a vertex to be a 0-dimensional face,and 1-dimensional faces are known as edges .For two subsets A and B in I R n ,deﬁne their Minkowski sum to be A +B ={a +b |a ∈A,b ∈B }.We call A and B the summands of A +B .It is easy to show that the Minkowski sum of two convex polytopes is a convex polytope.Let f ∈K [X 1,...,X n ]be a nonconstant polynomial where K is an ar-bitrary ﬁeld.We call f absolutely irreducible over K if it has no non-trivial factors over the algebraic closure of K .Supposef = a i 1...i n X i 11···X i n n .For each term with a i 1...i n =0,the corresponding exponent vector (i 1,...,i n ),viewed in I R n ,is called a support vector of f .Deﬁne Supp (f )to be the set of all support vectors of f ,i.e.,Supp (f )={(i 1,...,i n )|a i 1...i n =0}.Note that Supp (f )is empty if f =0.The total degree of f ,where f =0,is the maximum value of 1≤j ≤n i j over all (i 1,...,i n )∈Supp (f ).The convex hull of the set Supp (f ),denoted P f ,is known as the Newton polytope of f .The following lemma was observed by Ostrowski [21]in 1921(see also [22,Theorem VI,p.226]).Lemma 1.Let f,g,h ∈K [X 1,...,X n ]with f =gh .Then P f =P g +P h .An integral polytope is a polytope whose vertices have integer coordi-nates,and we say that an integral polytope is integrally decomposable ,or simply decomposable ,if it can be written as a Minkowski sum of two integral polytopes,each of which has more than one point.A summand in an integral decomposition is called an integral summand .We say an integral polytope is integrally indecomposable ,or simply indecomposable ,if it is not decompos-able.The Newton polytope of a polynomial is certainly integral and if the polynomial factors into two polynomials each of which has at least two terms,then by Lemma 1its Newton polytope must be decomposable.Thus we have the following simple irreducibility criterion from [4].Corollary 2(Irreducibility Criterion).Let f ∈K [X 1,...,X n ]with f not divisible by any X i for 1≤i ≤n .If the Newton polytope of f is integrally indecomposable,then f is absolutely irreducible.In Section 3,we shall discuss in more detail constructions of indecom-posable polytopes and show how to get indecomposable polytopes of high3dimension from those of lower dimensions.From these indecomposable poly-topes one can easily give explicitly many inﬁnite families of polynomials which are absolutely irreducible when considered over anyﬁeld.2.2Relevant computational problemsFrom a computational point of view,the following problem is of interest. Problem3.Given an integral polytope,say as its list of vertices,decide whether it is integrally indecomposable.This problem is not only pertinent to the study of polynomial factoriza-tion,but is a natural problem to consider and as such may be useful in other applications.Here the input size is the length of the binary representation of the coordinates of the vertices.Note that in our applications the polytope will be presented as the convex hull of a set of integral points.There is a large literature on computing the convex hull of anyﬁnite set of points in I R n;see[9,pages361–375].In particular,the convex hull of t points in a plane can be computed in time O(t log t)[10].Any of these algorithms can be used to compute the vertices of the Newton polytope of a given polynomial and we shall ignore this computational problem in the presentation of our algorithms.As mentioned before,the above problem is NP-complete,thus we shall be contented with algorithms that are“eﬃcient”in terms of some more generous measure,say the volume of polytopes.In Section4we give such an algorithm for polytopes in I R2and we also present a heuristic algorithm for higher dimensional polytopes which uses randomization.It is an open problem to develop an“eﬃcient”deterministic or even randomized algorithm for testing general integral polytopes for indecomposability.For a decomposable integral polytope,it is desirable toﬁnd all of its integral summands.Here we should identify polytopes that are translations of each other.Problem4.Given an integral polytope,say as its list of vertices,ﬁnd all of its integral summands.Again,this problem seems hard,but we shall give in Section4an algo-rithm for polytopes of dimension two which is“best possible”in the sense that the running time is linearly related to the number of decompositions.2.3Some preliminary resultsWe shall need more properties of the Minkowski sum.The next result from [4]describes how the faces decompose in a Minkowski sum of polytopes;for its proof,see Ewald[2,Theorem1.5],Gr¨u nbaum[13,Theorem1,p.317],or Schneider[24,Theorem1.7.5].4Lemma5.Let P=Q+R where Q and R are polytopes in I R n.Then(a)Each face of P is a Minkowski sum of unique faces of Q and R.(b)Let P1be any face of P and c1,...,c k all of its vertices.Suppose thatc i=a i+b i where a i∈Q and b i∈R for1≤i≤k.LetQ1=conv(a1,...,a k),R1=conv(b1,...,b k).Then Q1and R1are faces of Q and R,respectively,and P1=Q1+R1.A polytope of dimension two is called a polygon.(We refrain from using the term Newton polygon for a2-dimensional Newton polytope as in num-ber theory this term is used to refer to the lower boundary of the“Newton polyhedron”of certain power series.)The only proper faces of a polygon are its vertices and edges.For polygons,the above lemma can be rephrased as follows.Corollary6.Let P,Q and R be convex polygons(in I R n)with P=Q+R. Then every edge of P decomposes uniquely as the sum of an edge of Q and an edge of R,possibly one of them being a point.Conversely,any edge of Q or R is a summand of exactly one edge of P.3Indecomposable polytopesFirst of all,we mention the following two constructions of indecomposable polytopes from[4].Theorem7.Let Q be any integral polytope in I R n contained in a hyperplane H and v∈I R n an integral point lying outside of H.Suppose that v1,...,v k are all the vertices of Q.Then the polytope conv(v,Q)is integrally indecom-posable iﬀgcd(v−v1,...,v−v k)=1.Here and hereafter the gcd of a collection of integral vectors is deﬁned to be the gcd of all their coordinates together.Theorem8.Let Q be an indecomposable integral polytope in I R n that is contained in a hyperplane H and has at least two points,and let v∈I R n be a point(not necessarily integral)lying outside of H.Let S be any set of integral points in the polytope conv(v,Q).Then the polytope conv(S,Q)is integrally indecomposable.Theﬁrst construction shows that an integral line segment conv(v0,v1)is indecomposable iﬀgcd(v0−v1)=1,and an integral triangle conv(v0,v1,v2) is integrally indecomposable iﬀgcd(v0−v1,v0−v2)=1.The second construc-tion gives many indecomposable polygons with more than three edges.These5two constructions can be used iteratively to get indecomposable polytopes of any higher dimension.In the following,we give a new construction based on a projection.Intu-itively,one hopes that if a projection of a polytope is indecomposable then the polytope is indecomposable itself.Unfortunately,this is not true in gen-eral;consider for example a square and project it along one of its edges.The following lemma,however,gives a suﬃcient condition.We say that a linear mapπ:I R n−→I R m is integral if it maps integral points in I R n to integral points in I R m.It is straightforward to see that the image of any integral polytope under an integral linear map is still an integral polytope.Lemma9.Let P be any integral polytope in I R n andπ:I R n−→I R m any integral linear map.Ifπ(P)is integrally indecomposable and each vertex of π(P)has only one preimage in P then P must be integrally indecomposable.Proof.It suﬃces to show thatπ(P)is decomposable if P is decomposable. Suppose that P=A+B for some integral polytopes A and B in I R n each with at least two points.Thenπ(P)=π(A)+π(B).We need to show that bothπ(A)andπ(B)have at least two points.Suppose otherwise,sayπ(A) has only one point.Let w0be any vertex of P such thatπ(w0)is a vertex of π(P).Since P=A+B,there are unique vertices u0∈A and v0∈B such that w0=u0+v0.As A has at least two points,it has another vertex u1 such that u0u1is one of its edges.Then,by Lemma5,P has an edge w0w1 that starts at w0and is parallel to u0u1where w1is a vertex of P diﬀerent from w0.The latter property implies that w1−w0=t(u1−u0)for some real number t.Henceπ(w1)−π(w0)=π(w1−w0)=π(t(u1−u0))=t(π(u1)−π(u0))=0,asπ(A)has only one point and u1,u0∈A.This means thatπmaps two vertices of P to one vertex ofπ(P),contradicting our assumption.Corollary10.Let P be any integral polytope in I R n andπ:I R n−→I R m any integral linear map that is injective on the vertices of P.Ifπ(P)is integrally indecomposable then so must be P.Theorem11.Let Q be any integrally indecomposable polytope in I R m and π:I R n−→I R m any integral linear map.Let S be any set of integral points inπ−1(Q)having exactly one point inπ−1(v)for each vertex v of Q.Then the polytope conv(S)in I R n is integrally indecomposable.Proof.It follows directly from Lemma9.Remark.Theorem8can be viewed as a special case of Theorem11in the case that Q has suﬃciently many integral points besides its vertices,since it seems likely that there is an integral linear map that projects integral points6in the cone conv(v,Q)to integral points in its base Q.Such a projection is impossible if Q has no integral points other than its vertices.In concluding this section,we would like to discuss the relationship of integral decomposibility with a diﬀerent concept of decomposibility of poly-topes deﬁned in Gr¨u nbaum[13,Chapter15].Let P,Q be polytopes in I R n (not necessarily integral).We say that Q is homothetic to P if there is a real number t≥0and a vector a∈I R n such thatQ=tP+a={tb+a:b∈P}.A polytope P is called homothetically indecomposable if it is the case that whenever P=P1+P2for any polytopes P1and P2,then P1or P2is homoth-etic to P.Otherwise,P is called homothetically decomposable.Indecompos-able polytopes in this sense have been extensively studied in the literature [3,14,19,20,25–27].Homothetic decomposability is not directly comparable with integral de-composability.On the one hand,the only homothetically indecomposable polytopes in the plane are line segments and triangles so any polygon with more than3edges is homothetically decomposable[13,24].On the other hand, we saw above that some triangles can be integrally decomposable and many polygons with more than3edges are integrally indecomposable!The next result,however,shows that homothetic indecomposability implies integral indecomposability under a simple condition.Proposition12.Let Q be an integral polytope in I R n with vertices v i,where 0≤i≤k.If Q is homothetically indecomposable andgcd(v0−v1,···,v0−v k)=1,then Q is integrally indecomposable.Proof.Suppose that Q=T+S for some integral polytopes T and S.Then T or S is homothetic to Q,say T.This means that there is a real number r≥0and a∈I R n such that T=rQ+a.Hence the vertices of T areu i:=rv i+a,i=0,1,...,k.Since T is integral,all the vertices u0,u1,...,u k are integral and in particularu0−u i=r(v0−v i),i=1,...,kare integral.So r must be a rational number and the denominator of r divides gcd(v0−v1,···,v0−v k)=1;hence r is an integer.As0≤r≤1,we have r=0or1.In either case,T is a trivial summand of Q.Therefore Q is integrally indecomposable.By the above theorem,the homothetically indecomposable polytopes con-structed in[3,14,19,20,26,27]give many integrally indecomposable polytopes.74Decomposing polytopesIn this section we present our algorithms for both testing polytopes for in-decomposability and constructing summands of polytopes.We restrict our attention to polygons in Section4.1before considering the more general case in Section4.2.4.1PolygonsGiven a convex polygon in the Euclidean plane,one may form aﬁnite se-quence of vectors associated with it as follows.Let v0,v1,...,v m−1be the vertices of the polygon ordered cyclically in a clockwise direction.The edges of P are represented by the vectors E i=v i−v i−1=(a i,b i)for1≤i≤m, where a i,b i∈Z Z and the indices are taken modulo m.We call each E i an edge vector.A vector v=(a,b)∈Z Z2is called a primitive vector if gcd(a,b)=1. Let n i=gcd(a i,b i)and deﬁne e i=(a i/n i,b i/n i).Then E i=n i e i where e i is a primitive vector,1≤i≤m.Each edge E i contains precisely n i+1integral points including its end points.The sequence of vectors{n i e i}1≤i≤m,which we call the edge sequence or a polygonal sequence,uniquely identiﬁes the poly-gon up to translation determined by v0,and will be the input to our polygon decomposition algorithm.It will be convenient to identify sequences with those obtained by extending the sequence by inserting an arbitrary number of zero vectors.We may thus assume that the edge sequence of a summand of a polygon P has the same length as that of P.As the boundary of the polygon is a closed path,we have that 1≤i≤m n i e i=(0,0).Lemma13.Let P be a polygon with edge sequence{n i e i}1≤i≤m where e i∈Z Z2are primitive vectors.Then an integral polygon is a summand of P iﬀits edge sequence is of the form{k i e i}1≤i≤m,0≤k i≤n i,with 1≤i≤m k i e i= (0,0).Proof.Let{e′i}1≤i≤m be the edge sequence of an integral summand Q of P. By theﬁnal statement in Corollary6,each edge of Q occurs as the summand of some edge ne of P where e is a primitive vector,and it is easily seen that its corresponding edge vector must be of the form ke with0≤k≤n.The sum is zero simply because the boundary of Q is a closed path.Conversely, any sequence of this form will determine a closed path.Since{n i e i}1≤i≤m is a polygonal sequence,{k i e i}1≤i≤m must deﬁne the boundary of a convex polygon.It will be a summand of P,with the other summand having edge sequence{(n i−k i)e i}1≤i≤m.Given as input a sequence of edge vectors{n i e i}1≤i≤m of a polygon P,our polygon decomposition algorithm will check for the existence of a sequence of integers k i with0≤k i≤n i,1≤i≤m,such that 1≤i≤m k i e i=(0,0), k m=n m,and not all k i=0.(If P is decomposable then at least one of8its summands has k m=n m.)Thus the decision problem underlying ouralgorithm isPolygon Decomposability(PolyDecomp)Input:The egde sequence{n i e i}1≤i≤m of an integral convex polygon P.Question:Does P have a proper integral decomposition?The input size of an instance of this problem is O(m(log N+log E))where N=max{n1,...,n m}and E the maximum of absolute values of thecoordinates of e i,1≤i≤m.The next result puts the diﬃculty of thisproblem in context.Proposition14.PolyDecomp is NP-complete.Proof.Certainly the language associated with PolyDecomp lies in NP as wemay use a proper decomposition of P to verify membership of the language.We give a polynomial reduction of Partition to PolyDecomp which proves,since Partition is NP-complete[6],that PolyDecomp is NP-complete.Recall that the input to Partition is a sequence{s i}1≤i≤m of positiveintegers which we may take to be non-decreasing.Thus s1≤s2≤...≤s m.Let t= 1≤i≤m s i.The question in Partition is whether there is a subsequence of{s i}with sum t/2.Observe that we may assume that t iseven,for otherwise the question is easily answered.Consider now the followinginstance of PolyDecomp:the edge sequence(s1,1),(s2,1),...(s m,1),m(0,−1),(−t/2,−1),(−t/2,1)where all n i=1.Firstly,it is easy to check that this is indeed a polygonalsequence.Secondly,any polygon associated with the polygonal sequence has aproper decomposition if and only if the sequence{s i}1≤i≤m has a subsequencewith sum t/2.Thus we have a polynomial reduction,which completes theproof.Since it is widely believed that NP=P,it seems unreasonable to attempttoﬁnd a genuinely eﬃcient algorithm for solving PolyDecomp;however,we shall present an algorithm below whose running time is polynomial in thelength of the sides of the polygon rather than the logarithm of the lengths.In the parlance of[6],this is an example of a“pseudopolynomial-time”algo-rithm.In Section5we shall indicate how this algorithm may be used to testbivariable polynomials for absolute irreducibility;the algorithm thus obtainedis eﬃcient in terms of the total degree of the polynomial,rather than the num-ber of non-zero terms.Thus the distinction between genuinely eﬃcient algo-rithms for deciding polytope decomposability and“pseudopolynomial-time”algorithms is mirrored to a certain extent in that between eﬃcient algorithmsfor polynomials in terms of their sparse and dense representations.9Algorithm15(PolyDecomp)Input:The edge sequence{n i e i}1≤i≤m of an integral convex polygon P start-ing at a vertex v0where e i∈Z Z2are primitive vectors.Output:Whether P is decomposable.Step1:Compute the set IP of all the integral points in P,and set A0=∅. Step2:For i from1up to m−1,compute the set A i of points in IP that are reachable via the vectors e1,...,e i:2.1For each0<k≤n i,if v0+ke i∈IP then add it to A i;2.2For each u∈A i−1and0≤k≤n i,if u+ke i∈IP then add it to A i.Step3:Compute the last set A m:For each u∈A m−1and0≤k<n m,ifu+ke m∈IP then add it to A m.Step4:Return“Indecomposable”if v0∈A m and“Decomposable”otherwise. Theorem16.The above algorithm decides decomposability correctly in O(tmN) vector operations where t is the number of integral points in P,m the numberof edges and N the maximum number of integral points on an edge.Proof.The running time is easy to see as each set A i has size at most t. (Note that by a vector operation we mean addding two vectors,multiplyinga vector by a scalar,or adjoining a point to a set.)To prove the correctness, observe that all the points in A m are of the form v0+ m i=1k i e i,0≤k i≤n i. Step2.1ensures that k i=0for some i<m and Step3insists that k m<n m (note that v0+ke m∈IP for all k>0).If one of the points in A m is equal to v0then m i=1k i e i=(0,0),and so the sequence{k i e i}forms the edge sequence of a proper integral summand of P.On the other hand,for any proper integral summand Q of P,Q can be“slid”into P at v0,that is,Q canbe translated so that v0is a vertex of Q and Q lies inside P.Hence all the vertices of Q must lie in P and thus in IP.Consequently its edge sequence will be detected by our algorithm.We next give a simple generalisation of the above algorithm which not only outputs the number of proper decompositions of the polygon,but also outputs an array.The array may then be used to recover all decompositions,a single“recovery”requiring linear time.Thus the total time taken to recoverall decompositions is essentially linearly related to the number of decompo-sitions.This is the best that one can expect;however,it does not yield a “pseudopolynomial-time”algorithm as the number of decompositions maybe exponential in the area of the polygon.For example,consider the polygon with edge sequence(1,1),(2,1),...,(m,1),m(0,−1),t(−1,0)10where t=(m+1)m/2.The polygon has area less than12+22+···+m2= O(m3)while the number of integral summands is exactly2m.Algorithm17(PolyDecompNum)Input:The edge sequence{n i e i}1≤i≤m of an integral convex polygon P start-ing at a vertex v0where e i∈Z Z2are primitive vectors.Output:The number of integral summands of P including the trivial ones, and an array A.Each cell in A contains a pair(u,S)where u is a non-negative integer and S is a subset of{(k,i):1≤k≤n i,1≤i≤m}.Step1:Compute the set IP of all the integral points in P(so v0∈IP); say IP has t points.Initialize a t-array A0indexed by the points in IP.Set A0[v]:=(0,∅)for all v∈IP except the cell A0[v0]which is set to(1,∅). Step2:For i from1up to m,compute the t-array A i from A i−1:2.1First copy the contents of all the cells of A i−1into A i(this step is fork=0).2.2For each v∈IP with theﬁrst number of the cell A i−1[v]nonzero,andfor each0<k≤n i,if v′=v+ke i∈IP then update the cell A i[v′]as follows:if(u1,S1)is the value of A i−1[v]and(u2,S2)the current value of A i[v′]then the new value of A i[v′]is(u1+u2,S2∪{(k,i)}).Step3:Return the number u and the array A=A m,where(u,S)is the content of cell A m[v0].Theorem18.The integer output by Algorithm17is the total number of integral summands of the polygon P.Proof.Supposing v=v0+k1e1+···+k i e i,we may view the vector sum as a path from v0to v,so the number of such paths is equal to the sum of the numbers of paths from v0to v−ke i for0≤k≤n i,using e1,...,e i−1. Hence the numbers of paths can be computed iteratively as described in the algorithm:the number u in A i[v]records the number of paths from v0to v using e1,...,e i and the set S records all the pairs(k,j),j≤i,for which a path reaches v with its last edge being ke j with k>0.Thus the integer in cell A m[v0]is the total number of closed paths 1≤i≤m k i e i starting at v0. By Lemma13this is the number of integral summands of P.The signiﬁcance of the array A output by the algorithm is that it may be used to recover all decompositions of the polygon P.We show how a single decomposition can be recovered:Suppose the cell A[v0]contains the pair (u,S).Choose any(k,i)∈S.The line segment ke i will be the“ﬁnal edge”(counting clockwise)in our summand of P.Let(u′,S′)be the contents of cell B[v0−ke i].Pick any(k′,i′)∈S′with i′<i.The line segment k′e i′will be the“penultimate edge”in our summand of P.We continue in this way,11and as our sequence of i’s is decreasing we shall eventually return to the cell A[v0].At that point we will have recovered one summand in a decomposition of P.With regard to the running time,each cell in the array can be updated at most mN times,thus the running time is O(tmN)“cell updates”.The data in each cell is a pair(u,S)where S is a set of size at most mN and u an integer less than N m(an upper bound on the number of summands). Updating the integer u involves integer addition and this has a bit complexity of O(log N m)=O(m log N).Updating the set S simply involves unioning it with an element(k,i).Ignoring logarithmic factors,we can consider this a sin-gle bit operation.Thus the running time of PolyDecompNum is O(tm2N) bit operations,ignoring logarithmic factors.4.2Higher dimensional polytopesThe problem of testing higher dimensional polytopes for decomposability appears to be signiﬁcantly more diﬃcult.Certainly it is NP-complete as it includes that of polygons as a special case.It would be interesting to inves-tigate whether this problem was“strongly NP-complete”in the sense of[6]; this essentially means that the problem remains“NP-complete”when one bounds running time by the lengths,instead of logarithm of the lengths,of the edge vectors.If this more general problem is“strongly NP-complete”then it is unlikely there is an algorithm for determining whether a convex polytope of arbitrary dimension is indecomposable whose running time is polynomial in terms of the volume of the polytope.In this section,we present a heuristic“randomized algorithm”based on the projections considered in Lemma9.The algorithm has running time poly-nomial in the lengths of the edges of the polytope,thus is“eﬃcient”in the sense which we have been considering.The idea is to choose a random inte-gral linear map that projects a polytope into a polygon in a plane and then test the decomposability of the polygon.If the polygon is indecomposable and the condition of Lemma9is satisﬁed then the original polytope is inde-composable.We will show that the condition of Lemma9is always satisﬁed with high probability,but we do not know how to prove a good bound on the probability that the projected polygon be indecomposable when the original polytope is indecomposable.We now describe the details of our algorithm.Let S⊂I R n be anyﬁnite set of integral points,which will be the input to our algorithm,and P=conv(S). We want to decide whether P is integrally indecomposable.Note that P can be computed from S by any of the algorithms in[9,10];however,our algorithm does not require that the vertices,which are all in S,of P be known in advance but detects them automatically.This is because the points of S that are mapped to vertices of a polygon will be vertices of P,provided each vertex of the polygon has only one preimage in S.12。

Monte Carlo smoothing for non-linear time series

t t
p (x1:t , y1:t ) = f (x1 )
i=2
f (xi |xi−1 )
i=1
g (yi|xi )
where f (x1 ) is the distribution of the initial state. Here x1:t = (x1 , ..., xt ) and y1:t = (y1 , ..., yt ) denote collections of observations and states from time 1 through t. In proving the validity of our proposed smoothing algorithm a more formal deﬁnition of the state space model will be required. This is presented in Appendix A. A primary concern in many state-space inference problems is sequential estimation of the ﬁltering distribution p(xt |y1:t ). Updating of the ﬁltering density can be achieved in principle using the standard ﬁltering recursions p(xt+1 |y1:t) = p(xt+1 |y1:t+1 ) = p(xt |y1:t)f (xt+1 |xt )dxt g (yt+1 |xt+1 )p(xt+1 |y1:t ) . p(yt+1 |y1:t )

营销团队管理攻略

Langmuir 2008, 24, 10851-10857
Highly transparent superhydrophobic coatings
(PDDA-silicate/PAA)*12 + (SiO2/PAH)*4+ POTS
100
(3)
(2)
95
(1)
Transmittance (%)
90
LbL assembled free-standing films
Free-standing films refer to films exist without solid substrates. ➢ (i) To broaden further the application of LbL assembled multilayer films. ➢ (ii) To investigate directly the elastomeric properties of the LbL assembled films.
400
500
600
700
800
Wavelength /nm
(1) Bare quartz (2) Transparent superhydrophobic coating (3) Highly transparent superhydrophobic coating
CA：157°，SA: 1°
Chem. Commun. 2009, 2730.
Highly transparent superhydrophobic coatings
Outdoor photovoltaic and displaying devices
Transparent superhydrophobic coatings

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∗ †
Supported by NSERC of Canada Grant # 4818 Supported by NSA-grants H98230-05-1-0027 and H98230-07-1-0005
Байду номын сангаас
1
to complete a description of the locally toroidal polytopes provided by our construction (see Section 4). To that end, in Section 2 we establish some new structural theorems concerning the ‘polytopality’ of orthogonal groups. As a test case, we also apply our methods in Section 3 to an interesting family of polytopes of general rank n. Let us begin with a review of the basic set up and key results from parts I and II ([14] and [15], respectively). In [14], we ﬁrst surveyed some of the essential properties of an abstract regular polytope P , referring to [12] for details. Crucially, for each such P the automorphism group Γ(P ) is equipped with a natural list of involutory generators and is further a very special quotient of a certain Coxeter group G. (We say that Γ(P ) is a string C-group.) Since P can be uniquely reconstructed from Γ(P ), we may therefore shift our focus. Throughout, then, G = r0 , . . . , rn−1 will be a possibly inﬁnite, crystallographic Coxeter group [p1 , p2 , . . . , pn−1 ] with a string Coxeter diagram ∆c (G) (with branches labeled p1 , p2 , . . . , pn−1 , respectively), obtained from the corresponding abstract Coxeter group Γ = ρ0 , . . . , ρn−1 via the standard representation on real n-space V . (Very often G will be inﬁnite.) For any odd prime p, we may reduce G modulo p to obtain a subgroup Gp of GLn (Zp ) generated by the modular images of the ri ’s. We shall abuse notation by referring to the modular images of objects by the same name (such as ri , bi , B = [bi · bj ], V , etc.). In p particular, {bi } will denote the standard basis for V = Zn p . In any event, G is a subgroup of the orthogonal group O (Zn p ) of isometries for the (possibly singular) symmetric bilinear form x · y , the latter being deﬁned on Zn p by means of the Gram matrix B . Likewise, each ri remains a reﬂection, although we may write ri (x) = x − 2 x · bi bi bi · bi
Reﬂection Groups and Polytopes over Finite Fields, III
B. Monson∗ University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 and Egon Schulte† Northeastern University Boston, Massachussetts, USA, 02115 February 1, 2008
arXiv:0707.4007v1 [math.CO] 26 Jul 2007
With best wishes for our friend and colleague, J¨ org Wills
Abstract When the standard representation of a crystallographic Coxeter group Γ is reduced modulo an odd prime p, one obtains a ﬁnite group Gp acting on some orthogonal space over Zp . If Γ has a string diagram, then Gp will often be the automorphism group of a ﬁnite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of Gp . Building on this we investigate the class of ‘3–inﬁnity’ groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction. Key Words: reﬂection groups, abstract regular polytopes AMS Subject Classiﬁcation (2000): Primary: 51M20. Secondary: 20F55.
only if b2 i := bi ·bi ≡ 0 mod p. Concerning this situation, we now make a convenient deﬁnition: if p ≥ 5, or p = 3 but no branch of ∆c (G) is marked 6, then we say that p is generic for G. Indeed, in such cases, no node label b2 i of the diagram ∆(G) (for a basic system) is zero mod p, and the corresponding root bi is anisotropic. Also, a change in the underlying basic system for G has the eﬀect of merely conjugating Gp in GLn (Zp ). On the other hand, in the non-generic case, in which p = 3 and ∆c (G) has some branch marked 6, the group Gp may depend essentially on the actual diagram ∆(G) taken for the reduction mod p. (Note that p generic does not necessarily mean that p ∤ |G|, or that certain subspaces of V are non-singular, etc.) Now we confront two questions: what exactly is the ﬁnite reﬂection group Gp and when is it a string C -group (i.e. the automorphism group of a ﬁnite, abstract regular n-polytope P = P (Gp ))? To help answer the ﬁrst question, we recall from [14, Thm. 3.1] that an irreducible group Gp of the above sort, generated by n ≥ 3 reﬂections, must necessarily be one of the following: • an orthogonal group O (n, p, ǫ) = O (V ) or Oj (n, p, ǫ) = Oj (V ), excluding the cases O1 (3, 3, 0), O2 (3, 5, 0), O2 (5, 3, 0) (supposing for these three that disc(V ) ∼ 1), and also excluding the case Oj (4, 3, −1); or • the reduction mod p of one of the ﬁnite linear Coxeter groups of type An (p ∤ n + 1), Bn , Dn , E6 (p = 3), E7 , E8 , F4 , H3 or H4 . 2

Reflection Groups and Polytopes over Finite Fields, III

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