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A Short Proof of the Hajnal-Szemerédi Theorem on Equitable Coloring

A Short Proof of the Hajnal-Szemer´e di Theorem onEquitable ColoringH.A.Kierstead∗ A.V.Kostochka†September10,20061IntroductionAn equitable k-coloring of a graph G is a proper k-coloring,for which any two color classes diﬀer in size by at most one.Equitable colorings naturally arise in some scheduling,parti-tioning,and load balancing problems[1,15,16].Pemmaraju[13]and Janson and Ruci´n ski[6] used equitable colorings to derive deviation bounds for sums of dependent random variables that exhibit limited dependence.In1964Erd˝o s[3]conjectured that any graph with maxi-mum degree∆(G)≤r has an equitable(r+1)-coloring.This conjecture was proved in1970 by Hajnal and Szemer´e di[5]with a surprisingly long and complicated argument.Recently, Mydlarz and Szemer´e di[11]found a polynomial time algorithm for such coloring.In search of an easier proof,Seymour[14]strengthened Erd˝o s’conjecture by askingwhether every graph with minimum degreeδ(G)≥kk+1|G|contains the k-th power of ahamiltonian cycle.(If|G|=(r+1)(s+1)and∆(G)≤r thenδ(¯G)≥ss+1|G|;each(s+1)-interval of a s-th power of a hamiltonian cycle in¯G is an independent set in G.)The case k=1is Dirac’s Theorem and the case k=2is P´o sa’s Conjecture.Fan and Kierstead [4]proved P´o sa’s Conjecture with cycle replaced by path.Koml´o s,Sarkozy and Szemer´e di [7]proved Seymour’s conjecture for graphs with suﬃciently many(in terms of k)vertices. Neither of these partial results has a simple proof.In fact,[7]uses the Regularity Lemma, the Blow-up Lemma and the Hajnal-Szemer´e di Theorem.A diﬀerent strengthening was suggested recently by Kostochka and Yu[9,10].In the spirit of Ore’s theorem on hamiltonian cycles[12],they conjectured that every graph in which d(x)+d(y)≤2r for every edge xy has an equitable(r+1)-coloring.∗Department of Mathematics and Statistics,Arizona State University,Tempe,AZ85287,USA.E-mail address:kierstead@.Research of this author is supported in part by the NSA grant MDA904-03-1-0007†Department of Mathematics,University of Illinois,Urbana,IL,61801,USA and Institute of Mathematics, Novosibirsk,630090,Russia.E-mail address:kostochk@.Research of this author is supported in part by the NSF grant DMS-0400498.1In this paper we present a short proof of the Hajnal-Szemer´e di Theorem and present another polynomial time algorithm that constructs an equitable (r +1)-coloring of any graph G with maximum degree ∆(G )≤r .Our approach is similar to the original proof,but a discharging argument allows for a much simpler conclusion.Our techniques have paid further dividends.In another paper we will prove the above conjecture of Kostochka and Yu [9,10]in a stronger form:with 2r +1in place of 2r .They also yield partial results towards the Chen-Lih-Wu Conjecture [2]about equitable r -colorings of r -regular graphs and towards a list analogue of Hajnal-Szemer´e di Theorem (see [8]for deﬁnitions).Most of our notation is standard;possible exceptions include the following.For a vertex y and set of vertices X ,N X (y ):=N (y )∩X and d X (y )=|N X (y )|.If µis a function on edges then µ(A,B ):= xy ∈E (A,B )µ(x,y ),where E (A,B )is the set of edges linking a vertex in A to a vertex in B .For a function f :V →Z ,the restriction of f to W ⊆V is denoted by f |W .Functions are viewed formally as sets of ordered pairs.So if u /∈V then g :=f ∪{(u,γ)}is the extension of f to V ∪{u }such that g (u )=γ.2Main proofLet G be a graph with s (r +1)vertices satisfying ∆(G )≤r .A nearly equitable (r +1)-coloring of G is a proper coloring f ,whose color classes all have size s except for one small class V −=V −(f )with size s −1and one large class V +=V +(f )with size s +1.Given such a coloring f ,deﬁne the auxiliary digraph H =H (G,f )as follows.The vertices of H are the color classes of f .A directed edge V W belongs to E (H )iﬀsome vertex y ∈V has no neighbors in W .In this case we say that y is movable to W .Call W ∈V (H )accessible ,if V −is reachable from W in H .So V −is trivially accessible.Let A =A (f )denote the family of accessible classes,A := A and B :=V (G )\A .Let m :=|A|−1and q :=r −m .Then |A |=(m +1)s −1and |B |=(r −m )s +1.Each vertex y ∈B cannot be moved to A and so satisﬁesd A (y )≥m +1and d B (y )≤q −1.(1)Lemma 1If G has a nearly equitable (r +1)-coloring f ,whose large class V +is accessible,then G has an equitable (r +1)-coloring.Proof.Let P =V 1,...,V k be a path in H (G,f )from V 1:=V +to V k :=V −.This means that for each j =1,...,k −1,V j contains a vertex y j that has no neighbors in V j +1.So,if we move y j to V j +1for j =1,...,k −1,then we obtain an equitable (r +1)-coloring of G .Suppose V +⊆B .If A =V −then |E (A,B )|≤r |V −|=r (s −1)<1+rs =|B |,a contradiction to (1).Thus m +1=|A|≥2.Call a class V ∈A terminal ,if V −is reachable from every class W ∈A \{V }in the digraph H −V .Trivially,V −is non-terminal.Every non-terminal class W partitions A\{W }into two parts S W and T W =∅,where S W is the set of classes that can reach V −in H −W .Choose a non-terminal class U so that A :=T U =∅is minimal.Then every class in A is terminal and no class in A has a vertex movable to any class in (A \A )\{U }.Set t :=|A |and A := A .Thus every x ∈A satisﬁesd A (x )≥m −t.(2)2Call an edge zy with z∈W∈A and y∈B,a solo edge if N W(y)={z}.The ends of solo edges are called solo vertices and vertices linked by solo edges are called special neighbors of each other.Let S z denote the set of special neighbors of z and S y denote the set of special neighbors of y in A .Then at most r−(m+1+d B(y))color classes in A have more than one neighbor of y.Hence|S y|≥t−q+1+d B(y).(3) Lemma2If there exists W∈A such that no solo vertex in W is movable to a class in A\{W}then q+1≤t.Furthermore,every vertex y∈B is solo.Proof.Let S be the set of solo vertices in W and D:=W\S.Then every vertex in N B(S) has at least one neighbor in W and every vertex in B\N B(S)has at least two neighbors in W.It follows that|E(W,B)|≥|N B(S)|+2(|B|−|N B(S))|).Since no vertex in S is movable,every z∈S satisﬁes d B(z)≤q.By(2),every vertex x∈W satisﬁes d B(x)≤t+q. Thus,using s=|W|=|S|+|D|,qs+q|D|+2=2(qs+1)−q|S|≤|E(W,B)|≤q|S|+(t+q)|D|≤qs+t|D|It follows that q+1≤t.Moreover,by(3)every y∈B satisﬁes|S y|≥t−q+d B(y)≥1. Thus y is solo.Lemma3There exists a solo vertex z∈W∈A such that either z is movable to a class in A\{W}or z has two nonadjacent special neighbors in B.Proof.Suppose not.Then by Lemma2every vertex in B is solo.Moreover,S z is a clique for every solo vertex z∈A .Consider a weight functionµon E(A ,B)deﬁned byµ(xy):= q|S x|if xy is solo,0if xy is not solo.For z∈A we haveµ(z,B)=|S z|q|S z|=q if z is solo;otherwiseµ(z,B)=0.Thus µ(A ,B)≤q|A |=qst.On the other hand,consider y∈B.Let c y:=max{|S z|:z∈S y}, say c y=|S z|,z∈S ing that S z is a clique and(1),c y−1≤d B(y)≤q−1.So c y≤q. Together with(3)this yieldsµ(A ,y)=z∈S yq|S z|≥|S y|qc y≥(t−q+c y)qc y=(t−q)qc y+q≥t.Thusµ(A ,B)≥t|B|=t(qs+1)>qst≥µ(A ,B),a contradiction.We are now ready to prove the Hajnal-Szemer´e di Theorem.Theorem4If G is a graph satisfying∆(G)≤r then G has an equitable(r+1)-coloring.3Proof.We may assume that|G|is divisible by r+1.To see this,suppose that|G|= s(r+1)−p,where p∈[r].Let G :=G+K p.Then|G |is divisible by r+1and∆(G )≤r. Moreover,the restriction of any equitable(r+1)-coloring of G to G is an equitable(r+1)-coloring of G.Argue by induction on G .The base step G =0is trivial,so consider the induction step G ≥1.Let e=xy be an edge of G.By the induction hypothesis there exists an equitable(r+1)-coloring f0of G−e.We are done,unless some color class V contains both x and y.Since d(x)≤r,there exists another class W such that x is movable to W.Doing so yields a nearly equitable(r+1)-coloring f of G with V−(f)=V\{x}and V+(f)=W∪{x}. We now show by a secondary induction on q(f)that G has an equitable(r+1)-coloring.If V+∈A then we are done by Lemma1;in particular,the base step q=0holds. Otherwise,by Lemma3there exists a class W∈A ,a solo vertex z∈W and a vertex y1∈S z such that either z is movable to a class X∈A\{W}or z is not movable in A and there exists another vertex y2∈S z,which is not adjacent to y1.By(1)and the primary induction hypothesis,there exists an equitable q-coloring g of B−:=B\{y1}.Let A+:=A∪{y1}.Case1:z is movable to X∈A.Move z to X and y1to W\{z}to obtain a nearly equitable(m+1)-coloringϕof A+.Since W∈A (f),V+(ϕ)=X∪{z}∈A(ϕ).By Lemma 1,A+has an equitable(m+1)-coloringϕ .Thenϕ ∪g is an equitable(r+1)-coloring of G.Case2:z is not movable to any class in A.Then d A+(z)≥d A(z)+1≥m+1.Thus d B−(z)≤q−1.So we can move z to a color class Y⊆B of g to obtain a new coloring g of B∗:=B−∪{z}.Also move y1to W to obtain an(m+1)-coloringψof A∗:=V(G)\B∗. Setψ :=ψ∪g .Thenψ is a nearly equitable coloring of G with A∗⊆A(ψ ).Moreover,y2 is movable to W∗:=W∪{y1}\{z}.Thus q(ψ )<q(f)and so by the secondary induction hypothesis,G has an equitable(r+1)-coloringψ .3A polynomial algorithmOur proof clearly yields an algorithm.However it may not be immediately clear that its running time is polynomial.The problem lies in the secondary induction,where we may apply Case2O(r)times,each time calling the algorithm recursively.Lemma2is crucial here;it allows us to claim that when we are in Case2(doing lots of work)we make lots of progress.As above G is a graph satisfying∆(G)≤r and|G|=:n=:s(r+1).Let f be a nearly equitable(r+1)-coloring of G.Theorem5There exists an algorithm P that from input(G,f)constructs an equitable (r+1)-coloring of G in c(q+1)n3steps.Proof.We shall show that the construction in the proof of Theorem4can be accomplished in the stated number of steps.Argue by induction on q.The base step q=0follows immediately from Lemma1and the observation that the construction of H and the recoloringcan be carried out in14cn3steps.Now consider the induction step.In14cn3steps construct4A,A ,B,W,z,ing the induction hypothesis on the input(G[B−],f|B−),construct thecoloring g of B−in c(q(f|B−)+1)(qs)3≤cqn3steps.In14cn3steps determine whether Case1or Case2holds.If Case1holds,construct the recoloringϕ in14cn3steps.This yields an equitable(r+1)-coloring g∪ϕ in a total of34cn3+cqn3≤c(q+1)n3.If Case2holds then,by Lemma2,q+1≤t.Thus we used only18cqn3steps to constructe an additional14cn3steps to extend g toψ .Notice that W∗is non-terminal inψ .Thus we can choose A (ψ )so that A (ψ )⊆B.If Case1holds forψ then as above we canconstruct an equitable coloring in an additional14cn3+18cqn3steps.So the total numberof steps is at most c(q+1)n3.Otherwise by Lemma2q(ψ )<12q.Thus by the inductionhypothesis we canﬁnish in c qn316additional steps.Then the total number of steps is less thanc(q+1)n3.Theorem6There is an algorithm P of complexity O(n5)that constructs an equitable(r+1)-coloring of any graph G satisfying∆(G)≤r and|G|=n.Proof.As above,we may assume that n is divisible by r+1.Let V(G)={v1,...,v n}. Delete all edges from G to form G0and let f0be an equitable coloring of G0.Now,for i=1,...,n−1,do the following:(i)Add back all the edges of G incident with v i to form G i;(ii)If v i has no neighbors in its color class in f i−1,then set f i:=f i−1.(iii)Otherwise,move v i to a color class that has no neighbors of v i to form a nearly equitablecoloring fi−1of G i.Then apply P to(G i,fi−1)to get an equitable(r+1)-coloring f i of G i.Then f n−1is an equitable(r+1)-coloring of G n−1=G.Since we have only n−1stages and each stage runs in O(n4)steps,the total complexity is O(n5).Acknowledgement.We would like to thank J.Schmerl for many useful comments. References[1]J.Blazewicz,K.Ecker,E.Pesch,G.Schmidt,J.Weglarz,Scheduling computer andmanufacturing processes.2nd ed.,Berlin:Springer.485p.(2001).[2]B.-L.Chen,K.-W.Lih,and P.-L.Wu,Equitable coloring and the maximum degree,binatorics,15(1994),443–447.[3]P.Erd˝o s,Problem9,in“Theory of Graphs and Its Applications”(M.Fieldler,Ed.),159,Czech.Acad.Sci.Publ.,Prague,1964.[4]G.Fan and H.A.Kierstead,Hamiltonian square paths,bin.Theory Ser.B67(1996)167-182.[5]A.Hajnal and E.Szemer´e di,Proof of a conjecture of P.Erd˝o s,in“CombinatorialTheory and its Application”(P.Erd˝o s,A.R´e nyi,and V.T.S´o s,Eds.),pp.601-623, North-Holland,London,1970.5[6]S.Janson and A.Ruci´n ski,The infamous upper tail,Random Structures and Algorithms,20(2002),317–342.[7]J.Koml´o s,G.Sarkozy and E.Szemer´e di,Proof of the Seymour conjecture for largegraphs,Annals of Combinatorics,1(1998),43-60.[8]A.V.Kostochka,M.J.Pelsmajer,and D.B.West,A list analogue of equitable coloring,J.of Graph Theory,44(2003),166–177[9]A.V.Kostochka and G.Yu,Extremal problems on packing of graphs,Oberwolfachreports,No1(2006),55–57.[10]A.V.Kostochka and G.Yu,Ore-type graph packing problems,to appear in Combina-torics,Probability and Computing.[11]M.Mydlarz and E.Szemer´e di,Algorithmic Brooks’Theorem,manuscript.[12]O.Ore,Note on Hamilton circuits,Amer.Math.Monthly,67(1960),55.[13]S.V.Pemmaraju,Equitable colorings extend Chernoﬀ-Hoeﬀding bounds,Proceedingsof the5th International Workshop on Randomization and Approximation Techniques in Computer Science(APPROX-RANDOM2001),2001,285–296.[14]P.Seymour,Problem section,in“Combinatorics:Proceedings of the British Combina-torial Conference1973”(T.P.McDonough and V.C.Mavron,Eds.),pp.201-202, Cambridge Univ.Press,Cambridge,UK,1974.[15]B.F.Smith,P.E.Bjorstad,and W.D.Gropp,Domain decomposition.Parallel multi-level methods for elliptic partial diﬀerential equations,Cambridge:Cambridge Univer-sity Press,224p.(1996).[16]A.Tucker,Perfect graphs and an application to optimizing municipal services,SIAMReview,15(1973),585–590.6。

涉及非体积功的热力学基本方程

第49卷第8期2021年4月广州化工Guangzhou Chemical IndustryVol.49No.8Apr.2021涉及非体积功的热力学基本方程杨嫣，陈山川，谢娟，张改(西安工业大学材料与化工学院，陕西西安710021)摘要：热力学基本方程是物理化学课程中一个重要的知识点。

现行物理化学教材中重点阐述了仅含有体积功的热力学基本方程，而对涉及非体积功的热力学基本方程介绍很少。

对材料类专业的学生来说，学习和研究材料热力学，经常会涉及到非体积功。

因此，学习涉及非体积功的热力学基本方程非常必要。

本文重点讨论了涉及非体积功的表面系统、弹性杆、电、磁介质中的热力学基本方程的推导和应用，并总结了处理这类问题的方法。

关键词：物理化学；热力学；基本方程；非体积功中图分类号：G642文献标志码：A文章编号：1001-9677(2021)08-0139-04 Fundamental Equations in Thermodynamics Involving Non-expansion Work*YANG Yan,CHEN Shan-chuan,XIE Juan,ZHANG Gai(School of Material Science and Chemical Engineering,Xi9an Technological University,Shaanxi Xi'an710021,China) Abstract:Fundamental equation in thermodynamics is one of the key points of Physical Chemistry course・In current Physical Chemistry textbooks,the fundamental equations containing only expansionwork are mainly described, while the ones involving non-expansionwork are rarely introduced.For the students majoring in materials,non-expansionwork is often involved in the processes of studyand research.Therefore,it is necessary to study the fundamental equations in involving non-expansion work.The derivations and applications of the fundamental equations in surface systems,elastic rods,electric and magnetic media were discussed.Moreover,the way to deal with this kind of problem was summarized.Key words：Physical Chemistry;thermodynamics;the fundamental equation；non-expansion work热力学基本方程在处理平衡态热力学问题时非常有用，是物理化学课程内容的重点之一。

(完整word版)物理化学基本概念

物理化学概念及术语A B C D E F G H I J K L M N O P Q R S T U V W X Y Z概念及术语 (16)BET公式BET formula (16)DLVO理论 DLVO theory (16)HLB法hydrophile—lipophile balance method (16)pVT性质 pVT property (16)ζ电势 zeta potential (16)阿伏加德罗常数 Avogadro’number (16)阿伏加德罗定律 Avogadro law (16)阿累尼乌斯电离理论Arrhenius ionization theory (16)阿累尼乌斯方程Arrhenius equation (17)阿累尼乌斯活化能 Arrhenius activation energy (17)阿马格定律 Amagat law (17)艾林方程 Erying equation (17)爱因斯坦光化当量定律 Einstein’s law of photochemical equivalence (17)爱因斯坦－斯托克斯方程 Einstein-Stokes equation (17)安托万常数 Antoine constant (17)安托万方程 Antoine equation (17)盎萨格电导理论Onsager’s theory of conductance (17)半电池half cell (17)半衰期half time period (18)饱和液体 saturated liquids (18)饱和蒸气 saturated vapor (18)饱和吸附量 saturated extent of adsorption (18)饱和蒸气压 saturated vapor pressure (18)爆炸界限 explosion limits (18)比表面功 specific surface work (18)比表面吉布斯函数 specific surface Gibbs function (18)比浓粘度 reduced viscosity (18)标准电动势 standard electromotive force (18)标准电极电势 standard electrode potential (18)标准摩尔反应焓 standard molar reaction enthalpy (18)标准摩尔反应吉布斯函数 standard Gibbs function of molar reaction (18)标准摩尔反应熵 standard molar reaction entropy (19)标准摩尔焓函数 standard molar enthalpy function (19)标准摩尔吉布斯自由能函数 standard molar Gibbs free energy function (19)标准摩尔燃烧焓 standard molar combustion enthalpy (19)标准摩尔熵 standard molar entropy (19)标准摩尔生成焓 standard molar formation enthalpy (19)标准摩尔生成吉布斯函数 standard molar formation Gibbs function (19)标准平衡常数 standard equilibrium constant (19)标准氢电极 standard hydrogen electrode (19)标准态 standard state (19)标准熵 standard entropy (20)标准压力 standard pressure (20)标准状况 standard condition (20)表观活化能apparent activation energy (20)表观摩尔质量 apparent molecular weight (20)表面活性剂surfactants (21)表面吸附量 surface excess (21)表面张力 surface tension (21)表面质量作用定律 surface mass action law (21)波义尔定律 Boyle law (21)波义尔温度 Boyle temperature (21)波义尔点 Boyle point (21)玻尔兹曼常数 Boltzmann constant (22)玻尔兹曼分布 Boltzmann distribution (22)玻尔兹曼公式 Boltzmann formula (22)玻尔兹曼熵定理 Boltzmann entropy theorem (22)泊Poise (22)不可逆过程 irreversible process (22)不可逆过程热力学thermodynamics of irreversible processes (22)不可逆相变化 irreversible phase change (22)布朗运动 brownian movement (22)查理定律 Charle's law (22)产率 yield (23)敞开系统 open system (23)超电势 over potential (23)沉降 sedimentation (23)沉降电势 sedimentation potential (23)沉降平衡 sedimentation equilibrium (23)触变 thixotropy (23)粗分散系统 thick disperse system (23)催化剂 catalyst (23)单分子层吸附理论 mono molecule layer adsorption (23)单分子反应 unimolecular reaction (23)单链反应 straight chain reactions (24)弹式量热计 bomb calorimeter (24)道尔顿定律 Dalton law (24)道尔顿分压定律 Dalton partial pressure law (24)德拜和法尔肯哈根效应Debye and Falkenhagen effect (24)德拜立方公式 Debye cubic formula (24)德拜－休克尔极限公式 Debye-Huckel’s limiting equation (24)等焓过程 isenthalpic process (24)等焓线isenthalpic line (24)等几率定理 theorem of equal probability (24)等温等容位Helmholtz free energy (25)等温等压位Gibbs free energy (25)等温方程 equation at constant temperature (25)低共熔点 eutectic point (25)低共熔混合物 eutectic mixture (25)低会溶点 lower consolute point (25)低熔冰盐合晶 cryohydric (26)第二类永动机 perpetual machine of the second kind (26)第三定律熵 Third—Law entropy (26)第一类永动机 perpetual machine of the first kind (26)缔合化学吸附 association chemical adsorption (26)电池常数 cell constant (26)电池电动势 electromotive force of cells (26)电池反应 cell reaction (27)电导 conductance (27)电导率 conductivity (27)电动势的温度系数 temperature coefficient of electromotive force (27)电化学极化 electrochemical polarization (27)电极电势 electrode potential (27)电极反应 reactions on the electrode (27)电极种类 type of electrodes (27)电解池 electrolytic cell (28)电量计 coulometer (28)电流效率current efficiency (28)电迁移 electro migration (28)电迁移率 electromobility (28)电渗 electroosmosis (28)电渗析 electrodialysis (28)电泳 electrophoresis (28)丁达尔效应 Dyndall effect (28)定容摩尔热容 molar heat capacity under constant volume (28)定容温度计 Constant voIume thermometer (28)定压摩尔热容 molar heat capacity under constant pressure (29)定压温度计 constant pressure thermometer (29)定域子系统 localized particle system (29)动力学方程kinetic equations (29)动力学控制 kinetics control (29)独立子系统 independent particle system (29)对比摩尔体积 reduced mole volume (29)对比体积 reduced volume (29)对比温度 reduced temperature (29)对比压力 reduced pressure (29)对称数 symmetry number (29)对行反应reversible reactions (29)对应状态原理 principle of corresponding state (29)多方过程polytropic process (30)多分子层吸附理论 adsorption theory of multi—molecular layers (30)二级反应second order reaction (30)二级相变second order phase change (30)法拉第常数 faraday constant (31)法拉第定律 Faraday’s law (31)反电动势back E。

Legendrian knots and monopoles

a rX iv:mat h /41559v5[mat h.DG ]2Dec25Legendrian knots and Monopoles Tomasz Mrowka Yann Rollin Tomasz Mrowka,MIT,77Massachusetts Avenue,Cambridge MA 02139,USA Yann Rollin,Imperial College,Huxley Building,180Queen’s Gate,London SW72AZ,UK Email:mrowka@ rollin@ Abstract We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3-dimensional contact manifold (Y,ξ),under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced in [KM]is non-vanishing.The proof requires an exci-sion result for Seiberg-Witten moduli spaces;then,the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M .AMS Classiﬁcation numbers Primary:57R17;57M25;57M27;57R57Secondary:Keywords:Contact structures,Legendrian knots,Bennequin inequality,Ex-cision,Monopoles11IntroductionThis paper is a sequel to[KM]where the Seiberg-Witten invariants where gen-eralized to invariants of connected oriented smooth four-manifolds carrying a contact structure on their boundary.An oriented contact structureξ(or more generally an oriented two planeﬁeld)induces a canonical Spin c-structure sξon Y.In[KM],the Seiberg-Witten invariants were deﬁned for4-manifolds with boundary endowed with a contact structure.The domain of these in-variants is the set Spin c(M and h is an isomorphism between s|Y and sξ.The Seiberg-Witten invariant is a mapsw:Spin c(The Thurston Bennequin invariant and the rotation number of a knot in S3 generalize to invariants of the pair K andσ.The rotation number is only deﬁned up to sign until an orientation of K is chosen.Both invariants arise because a Legendrian knot has a canonical framing obtained by choosing a vectorﬁeld V transverse to the contact distribution along K.To generalize the Thurston-Bennequin invariant choose an arbitrary orientation for K give Σthe compatible orientation.Push K oﬀslightly in the V-direction,thus obtaining a disjoint oriented knot K′.Then pushΣoﬀitself to get a surface Σ′so that its boundary coincides with K′.Since theΣandΣ′are disjoint along their boundary they have a well deﬁned intersection number.The Thurston-Bennequin invariant relative toΣ,tb(K,σ)is deﬁned to be this self-intersection number.IfΣis contained in Y(so that K is null homologous in Y)thentb(K,σ)=lk(K,K′):=tb(K)Notice that tb(K)does not depend on the choice of the initial orientation for K orΣ.The generalization of the rotation number is obtained as follows.After choosing an orientation the contact distribution can be endowed with an almost complex structure Jξ,which is unique up to homotopy.Thereforeξ→Y has a structure of complex line bundle,hence it has a well deﬁned Chern class.The isomorphism h induces and isomorphism of the determinant line L s=det(W+s)of the bundle of positive spinors for the Spin c-structure s withξon the boundary If we also ﬁx an orientation for K then we get a preferred non vanishing tangent vector ﬁeld v and so by the Legendrian property a non-vanishing section of L s Then, the rotation number of K relative toΣis by deﬁnitionr(K,σ,s,h):= c1(L s,v),σwhere c1(L s,v)is the relative Chern class with respect to the trivialization of ξalong K induced by v.Notice that the rotation number depends a priori on the homology class ofΣand on the orientation of K.This deﬁnition coincides with the usual rotation number deﬁned on(R3,ξstd)as the winding number of v inξstd.1.2Main results.With these deﬁnitions and notation in place we can state our generalization of Bennequin’s Inequality.Theorem A Let(Y,ξ)be a3-dimensional closed manifold endowed with a contact structureξand letboundary Y.Suppose we have a Legendrian knot K⊂Y,andΣ⊂M,ξ(s,h)=0we haveχ(Σ)+tb(K,σ)+|r(K,σ,s,h)| 0,whereχdenotes the Euler characteristic.Notice that this result was known before in the case of compact Stein complex surfaces with pseudo-convex boundary(cf.[AM,LM]).Here are two corollaries of Theorem A.A contact manifold(Y,ξ)is called weakly symplecticallyﬁllable,if it is the boundary of a symplectic manifold(M,ξis identically zero for amanifold with an overtwisted contact boundary. Using the fact that swM,ξis not identically zero?Onthe other hand there are contact structures which are tight but not weakly sym-plecticallyﬁllable.Theﬁrst examples are due to Etnyre and Honda[EH]on certain Seifertﬁbered space and later inﬁnite families where discover by Lisca and Stipsicz(see[LS1,LS2]).4All these results rely on an excision property for Seiberg-Witten invariants. Recall that a symplectic cobordism from a contact manifolds(Y,ξ)to(Y′,ξ′)is a compact symplectic manifold(Z=−Y⊔Y′,where Y and Y′have their orientations induced by the contact structures.Y is called the concave end of the cobordism and Y′is called the convex end.In addition,it is required thatωis strictly positive onξandξ′with their induced orientations.By convention the boundary components will always be given in the order concave,convex.A symplectic cobordism,is said to be special if•the symplectic form is given in a collar neighborhood of the concave boundary by a symplectization of(Y,ξ);•the map induced by the inclusioni∗:H1(M,ξ).There is a canonical way to extend (s,h)to a Spin c structure t on M∪Z.The data of h identiﬁes the restriction s|Y with sξwhile sω|Y is identiﬁed canonically with sξ.Together these provides a gluing map and deﬁnes t.Thus we have deﬁned a canonical mapj:Spin c(M′,ξ′).The main technical result of this paper is the following.Theorem D Let Z be a special symplectic cobordism between(Y,ξ)and second contact manifold (Y′,ξ′).Let M∪Y Z onM,ξ◦j=±swM,ξ)we have:sw M,ξ(s2,h2)=swM′,ξ′◦j(s2,h2).5Remark1.2.2—The assumption(1.1)can be reformulated as follow:for any u∈Map(M.In this case the map j is not generally injective anymore.All the cobordisms of interest in this paper,will be shown to verify assump-tion1.1,that is to say1and2-handle surgeries.However,assumption(1.1) may be removed.But,the conclusion(1.2)of Theorem D has to be replaced by sw M,ξ(s,h).(1.3)This generalization is proved by reﬁning the gluing theorem E as explained in remark2.4.3.Indeed if two pairs(s1,h1)and(s2,h2)have the same image under j then we can assume upto ismorphism that s1=s2and h1=uh2 where u is an automorphism of sξwhich extends to an automorphism of sωwhich is the indentity at inﬁnity.A result of Weinstein[W]shows that,a1-handle surgery,or,a2-handle surgery along a Legendrian knot K with framing coeﬃcient−1relative to the canonical framing,on the boundary of M′given byM∪YZ,ω)is a special symplectic cobordism between(Y,ξ)and a contact boundary(Y′,ξ′)obtained by the surgery.The strategy to prove Theorem A is to capΣdoing a Weinstein surgery along K.Bennequin’s inequality is then obtained by applying the adjunction in-equality[KM2]to the resulting closed surfaceΣ′⊂M′,ξ′◦j(s,h)does not vanish.This is true un-der the assumption that swM with con-tact boundary(Y,ξ)developed by Kronheimer and theﬁrst author in[KM].61.3.1An almost K¨a hler coneWe pick a contact one-formηwith kerη=ξ.The contact form determines the Reeb vectorﬁeld R by the properties thatιR dη=0andη(R)=1.Then (0,+∞)×Y is endowed with a symplectic structure called the symplectization of(Y,η),and the symplectic form is deﬁned by the formula1ω=dη(J·,·).2Let M be the manifoldM=M.The Spin c-structure sξinduced by the contact structure on Y is canonically identiﬁed with the restriction to{T}×Y of the Spin c-structure sωinduced by the symplectic formωon C M.Therefore,an element(s,h)∈Spin c(1.3.2Monopole equationsLet A be a spin connection on W and∇A be its covariant derivative.The the Dirac operator is then deﬁned by D g A= i e i·∇A e j,where e j is an oriented orthonormal local frame on M.e i is the dual coframe and acts by Cliﬀord multiplication.It is aﬁrst order elliptic operator of order1between the space of spinorﬁelds D A:Γ(W±)→Γ(W∓).PutΨ=(1,0)∈Λ0,0⊕Λ0,2.Let B be the spin-connection in the spinor bundle W+J=Λ0,0⊕Λ0,2so that∇BΨis a section of T∗X⊗Λ0,2.The Dirac operator D can on W J is obtained by replacing∇A withˆ∇in the deﬁnition.ForΦ∈Γ(W+J),we have explicitly D canΦ=√M.The domain of the Seiberg-Witten equations on(M,g,s)is the space of con-ﬁgurationsC={(A,Φ)∈Conn(W)×Γ(W+)},where Conn(W)is the space of spin connections on W.Recall that Conn(W) is an aﬃne space modeled onΓ(iΛ1):for two spin connections A and˜A,we have˜A=A+a⊗id|W,where a is a purely imaginary1-form.We will simply write the above identity ˜A=A+a in the sequel.We introduce the curvature formF A(X,Y)=12FˆA8whereˆA is the unitary connection induced by A on the determinant line bundle1 is its usual curvature form.The Seiberg-Witten equations areL,and FˆAF+A−{Φ⊗Φ∗}0=F+B−{Ψ⊗Ψ∗}0+̟(1.5)D AΦ=0,(1.6) where̟is a self-dual purely imaginary2-form on M and{·}0represents the trace free part of an endomorphism;theﬁrst equation makes sense since purely imaginary self-dual2-forms are identiﬁed with the traceless endomorphisms of W+via Cliﬀord multiplication.The conﬁguration(B,Ψ)is clearly solution of the equations over C M.The space of solutions Z̟is acted on by a gauge group G=Map(M,S1)and the action is deﬁned on C byu·(A,Φ)=(A−u−1du,uΦ)∀u∈G.(1.7) The moduli space M̟(M,g,s)=Z̟/G,for suitable generic̟,is a compact smooth manifold of dimensiond= e(W+,Ψ),[M,C M] ,where e(W+,Ψ)is the relative Euler class of W+.If d=0,then the Seiberg-Witten invariant is always0;if d=0,the Seiberg-Witten invariant is the number of points of M̟(M,s)counted with signs.Following[KM]there is a trivial bundle over the conﬁguration space(the determinant line bundle of the appropriate deformation operator)which is identiﬁed with the orientation bundle of moduli space.Furthermore a trivialization of this determinant line bundle for one relative Spin c-structure determines a trivialization for all others in a canonical manner and determines a consistent orientation.Thus the set of consistent orientations is a two element set.In particular unlike the closed case the sign of the invariant cannot be pined down by a homology orientation rather the ratio of the signs of the values of the invariant for diﬀerent relative Spin c-structures is well deﬁned.It turns out that this number depends only onM,ξ); this explains the notation sw1Alternatively F A can be viewed as the curvature of the unitary connection induced by A on the virtual line bundle L1/29AcknowledgmentsThis paper obviously owes a large intellectual debt to Peter Kronheimer.The second author thanks the MIT where most of this work was done.He also thanks the Institute for Advanced Studies where he was hosted on several occa-sions.Theﬁrst author was partially supported by NSF grants DMS-0111298, DMS0206485,and FRG-0244663.The second author was partially supported by NSF grant DMS-0305130.2ExcisionThe goal of this section is to prove Theorem D.The strategy is to show,in a more general setting,that the Seiberg-Witten moduli spaces associated to M′are diﬀeomorphic for a suitable choice of metrics and perturbations. This is achieved thanks to a gluing technique in Theorem E.2.1Families of AF AK manifoldsWe set-up the analytical framework for our gluing problem.Theﬁrst step is to construct suitable families of asymptoticallyﬂat almost K¨a hler metrics.LetM. Hence we have obtained(M,g,ω,J)where g is Riemannian metric and M splits asM=(i)There is a constantκ>0,such that the injectivity radius satisﬁesκinj(x)>σ(x)for all x∈M.(ii)For each x∈M,let e x be the map e x:v→exp x(σ(x)v/κ)andγx be the metric on the unit ball in T x M deﬁned as e∗x g/σ(x)2.Then,these metrics have bounded geometry in the sense that all covariant derivatives of the curvature are bounded by some constants independent of x. (iii)For each x∈M\K,let o x similarly be the symplectic form e∗zω/σ(z)2on the unit ball.Then o x similarly approximates the translation-invariant form,along with all its derivatives.(iv)For allε>0,the function e−εσis integrable on M.(v)The symplectic formωextends as a closed form on K.Convention:When the end of M has a structure of symplectic cone as in(2.1) we chooseσ(t,y)=t on the end C M and extend it arbitrarily with the condition 0<σ(x) T onin this section,and the property(2.2)will be a consequence of the as-sumption(1.1).This property will be used for the compactness results of Section2.2.4.However this assumption is not essential as explained in remark2.2.3.•If we scale the metric byλ2,and accordingly scale T,σintoλT,λσ, then the constantκand the constants controllingγx and o x remain un-changed.In particular,we may always assume T=1in Deﬁnitions2.1.1 and2.1.2.•In Deﬁnition2.1.2,the contact formηcan be replaced by any other1-form representing the same contact structure.This is shown in the next lemma which allows us to modify the symplectic formωZ near the sharp end of the cone C Z.Lemma2.1.4Let Y be a compact3-manifold endowed with a contact struc-tureξ.Letη1andη2be two1-forms such that kerηj=ξ.Then,for every ε>0,there existα∈(0,ε)and a symplectic formωon(0,+∞)×Y such that •ω=1d(t2η2)on(ε,+∞)×Y.2Proof We writeη2=eµη1whereµis a real function on Y.We consider the exact2-form1ω=t3(2+tµf′)dt∧η∧dη.2Let c be the minimum of the functionµon Y.If c 0,we just require that f is an increasing function of t.If c<0,a suﬃcient additional condition for havingω2>0is2f′(t)<−ln(2etc],2e•f1(t)=f0(t)for t∈[ε2],12•f1(t)=1for t∈[ε4e ,ε]in such a way that we havef′ f′1on each interval where f′1is deﬁned.Therefore,the condition f′<−2MThanks to Remarks2.1.3,we may assume from now on that T=1and that the contact formηis the same in the deﬁnition of the almost K¨a hler cone C M=(T,∞)×Y and in the deﬁnition of the AFAK end2.1.2.We identify an annulus in C M⊂M with an annulus in C Z⊂Z using the dilation mapM⊃C M≃(1,+∞)×Y⊃(1,τ)×Yντ−→(1/τ,1)×Y⊂(0,1)×Y≃C Z⊂Z(t,y)−→(t/τ,y)and deﬁne the manifold Mτas the union of M∩{σM<τ}and Z∩{σZ>1/τ} and with the identify along the annuli given by the dilationντ.The operation of connected sum along Y we just deﬁned is represented in theﬁgure below.The gray regions represent the annuli,the arrows suggest that they are identiﬁed by a dilation,and the dashed regions are the parts of M and Z that are taken oﬀfor the construction of Mτ.Figure1:Construction of AFAK MτNowντ∗ωCM =τ2ωZ andσZ◦ντ=σM/τ,hence,if we scaleωZ byτ2andσZ byτ,all the structure will match on the annuli.In conclusion Mτcarries an almost K¨a hler structure(ωτ,Jτ)deﬁned outside the compact setRemark—Every compact set of K⊂M is also,by deﬁnition,a compact set of Mτprovidedτis large enough.Similarly,the structures gτ,στ,Jτ,ωτare equal on every compact set whenτis large enough.The following lemma is satisﬁed by construction.Lemma2.1.6The manifolds with almost K¨a hler structure deﬁned outside a compact set and a proper function(Mτ,gτ,Jτ,στ)satisfy Deﬁnition2.1.1uniformly,in the sense that the constantκ,ε,the bounds onγx,o x and on theintegral of e−εστcan be chosen independently ofτ.Remarks•A simple consequence of the lemma is the following:for all ε>0,there exists T k large enough such that,for everyτ,the pull-backof the almost K¨a hler structure on a unit balls in Mτ∩{στ T k},via the exponential map v→exp x v,on the tangent space T x Mτ,isε-close in.C k-norm to the euclidean structure(gτ,ωτ,Jτ)|Tx Mτ•For all the remainder of Section2,M and Mτwill denote the the man-ifolds that we just constructed at Section2.1.5together with their addi-tional structures(identiﬁcation of C M with an almost K¨a hler cone,proper functionσ,almost K¨a hler structure and Riemannian metric.2.1.7Spin c-structures on AF AK manifoldsSimilarly to the case of a manifold with a contact boundary,we deﬁne the space Spin c(X,ω)for an AFAK manifold X as the set of equivalence classes of pairs(s,h),where s is a Spin c-structure on X and h is an isomorphism, deﬁned outside a compact set K1⊂X,between sω|X\K1and s|X\K1.As we saw in Section1.3.1,there is a well deﬁned identiﬁcation of Spin c(M,ξ)→Spin c(Mτ,ωτ)when Mτis obtained by adding an AFAK end Z to the end ofand the transition map from W to W Jis given by h over the annulus{1<τστ<τ}∩Mτ⊂C M(where W Jτ=W J).At the level of equivalence classes of Spin c-structures,this procedure deﬁnes an identiﬁcation of Spin c(M,ω)with Spin c(Mτ,ωτ).We stress the fact that Wτis identiﬁed with the canonical spinor bundle W Jτfor it is,by construction,equal to it on Mτ∩{στ>1}.Moreover,the spinorbundles Wτrestricted to any compact set K⊂M are all identiﬁed providedτis large enough.In Section1.3.2we deﬁned a conﬁguration(B,Ψ)for the spinor bundle W→M.Similarly,we can deﬁne a conﬁguration(Bτ,Ψτ)for the spinor bundle Wτ→Mτ:outsideM⊂Mτ.In conclusion we have constructed a family of spinor bundles Wτ→Mτ, identiﬁed with W JoutsideτM,ξ),we consider the Spin c-structure induced on M and j(s,h)on Mτ.The Seiberg-Witten equations were intro-duced in Section1.3.2on M,which has an end modeled on an almost K¨a hler cone C M.The equations are given on Mτin the same way byF+A−{Φ⊗Φ∗}0=F+B−{Ψ⊗Ψ∗}0+̟τ(2.4)D AΦ=0,(2.5)15whereΦis a section of W+τ,A is a spin connection on Wτand̟τis a perturbation inΓ(iΛ+Mτ).As in the case of an almost K¨a hler conical end,the almost K¨a hler structure deﬁned outside2(¯∂⊕¯∂∗)and we have D gτB=D can,for B the spin connection deduced from the Levi-Civita connection on Mτand the Chern connection on Lτ≃L Jτ.Therefore(B,Ψ)solves the Seiberg-Witten equations restricted to Mτ\+1 ,(2.6)N0where N0is any number with N0 1;N0will beﬁxed later on to make the derivatives ofχτas small as required in our constructions.We deﬁne a cut-oﬀfunction on Mτby the formulaχτ(στ).By a slight abuse of notation,the latter function will be denotedχτas well.For a given perturbation of Seiberg-Witten equation̟on M,the perturbation of the equations on Mτis deﬁned by̟τ=χτ̟.(2.7) 2.2.2Linear theoryThe Study of Seiberg-Witten equations requires introducing suitable Sobolev spaces rather than using naive smooth objects deﬁned in Section1.3.2.We recall very quickly the results of[KM]in this section.The conﬁguration(B,Ψ)is a solution of Seiberg-Witten equations on the al-most K¨a hler end of Mτ;hence we study solutions(A,Φ)with the same asymp-totic behavior.We introduce the conﬁguration spaceC l(Mτ)={(A,Φ)∈Conn(Wτ)×Γ(W+τ)/A−B∈L2l(gτ),andΦ−Ψ∈L2l(gτ,B)},16the gauge groupG l(Mτ)={u:Mτ→C/|u|=1,and1−u∈L2l+1(gτ)},acting on C l by u·(A,Φ)=(A−u−1du,uΦ),and,for someﬁxedε0>0,the perturbation spaceN(Mτ)=e−ε0στC r(i su(W+τ)),equipped with the norm̟ Nτ= eε0στ̟ C r(gτ).The L2k(gτ)-norm is the usual L2norm with k derivatives on Mτdeﬁned using the metric gτ.In order to deﬁne a similar norm on the spinorﬁelds,a unitary connection A is needed.We putφ 2L2l(gτ,A)= Mτ |φ|2+|∇Aφ|2+···+|∇l Aφ|2 vol gτ,and deﬁne L2l(gτ,A)as the completion of the space of smooth sections for this norm.For two diﬀerent connections,A and A′with A−A′∈L2l(gτ), Sobolev multiplication theorems show that the norms L2l(A,gτ)and L2l(A′,gτ) are commensurate.Remark—For any pairτ,τ′,it is easy to construct a diﬀeomorphism f: Mτ→M′τcovered by an isomorphism F between Wτand Wτ′which are a dilations near inﬁnity.Therefore F∗C l(Mτ′)=C l(Mτ)and f∗G l(Mτ′)= G l(Mτ).In this sense,the spaces C l and G l are in fact independent ofτ. However,the fact that the norms depend onτwill become crucial for analyzing the compactness properties of the family of moduli spaces on Mτ.Of course the choice of l 2is actually perfectly arbitrary thanks to elliptic regularity.However it will be chosen with l 4so that we have the inclusion L2l⊂C1.The Sobolev multiplication theorem shows that G l is a Hilbert Lie group acting smoothly on the Hilbert aﬃne space C l.Furthermore action of G l is free:if we have u·(A,Φ)=(A,Φ),then du=0hence u must be constant. Now u−1∈L2l+1therefore u=1.Let Zτl be the space of conﬁgurations(A,Φ)∈C l which verify the Seiberg-Witten equations(2.4)on Mτwith perturbation̟τ.Then Zτl is invariant under the gauge group action and we deﬁneM l(Mτ)=Zτl/G l.17We drop the reference to the indexτat the moment,for simplicity of notations. All of what we say in the rest of Section2.2.2holds for M and Mτor indeed any AFAK manifold.The linearized action of the gauge group at an arbitrary conﬁguration(A,Φ)∈C l is given by a diﬀerential operatorδ1,(A,Φ):L2l+1(i R)−→L2l(iΛ1)⊕L2l,B(W+)v−→(−dv,vΦ)and its formal adjoint is given byδ∗1,(A,Φ)(a,φ)=−d∗a+i Im Φ,φ ;notice that with our convention,the Hermitian product ·,· is anti-complex linear in theﬁrst variable.A tangent vector(a,φ)is L2-orthogonal to the orbit through(A,Φ)if and only ifδ∗1(a,φ)=0;the orbit space C l/G l is a smooth Hilbert manifold,and its tangent space at(A,Φ)is identiﬁed with kerδ∗1,(A,Φ).The linearized Seiberg-Witten equations at(A,Φ)are given as well by a diﬀer-ential operatorδ2,(A,Φ):L2l+1(iΛ1)⊕L2l+1,B(W+)→L2l(i su W+)⊕L2l,B(W−)(a,φ)→(d+a−{Φ⊗φ∗+φ⊗Φ∗}0,D Aφ+a·Φ).Remark—There is a slight inconsistency in the conventions of[KM].The linear theory studied there is exactly the one presented in the current paper. However,this is not the one of the equations written in[KM]where F A isreplaced by the curvature FˆA of the unitary connectionˆA induced by A onthe determinant line bundle L.Since F A=2FˆA ,the corresponding linearizedequations should beδ2,(A,Φ)(a,φ)=(2d+a−{Φ⊗φ∗+φ⊗Φ∗}0,D Aφ+a·Φ). At at solution of Seiberg-Witten equations(A,Φ),the operators verifyδ2,(A,Φ)◦δ1,(A,Φ)=0,so we have an elliptic complex0→L2l+2(i R)δ1,(A,Φ)−→L2l+1(iΛ1)⊕L2l+1,B(W+)δ2,(A,Φ)−→L2l(i su W+)⊕L2l,B(W−)−→0; H0=0since the action of the gauge group is free,H1=kerδ2/Imδ1is the virtual tangent space to the Seiberg-Witten moduli space at(A,Φ),and H2= cokerδ2is the obstruction space.Equivalently H1can be viewed as the kernel of the elliptic operator18D(A,Φ)=δ∗1,(A,Φ)⊕δ2,(A,Φ).(2.8) Facts:•The moduli space M l is compact.•By elliptic regularity,M l=M l+1=M l+2=···=M r+1;this is precisely why the choice of l does not matter.So the moduli space is simply referred to by M.•By Sard-Smale theory,we may always assume that H2=0after choosinga suitable generic perturbation̟τ.Then M is unobstructed;it is asmooth manifold of dimension equal to the virtual dimensiond= e(W+,Ψ),[M be a manifold with a contact boundary(Y,ξ)and an element(s,h)∈Spin c(M with Z or Z′along Y,together with the Riemannian metrics gτand g′τand the particular Seiberg-Witten equations with perturbation̟τconstructed in Section2.2.1.Then,forτlarge enough,the moduli spaces M(Mτ)and M(M′τ)are generic, and there is a diﬀeomorphismG:M̟τ(Mτ)→M̟τ(M′τ).Furthermore there is a canonical identiﬁcation of the set of consistent orienta-tions for Mτand M′τ.Using this canonical identiﬁcation the above diﬀeomor-phism becomes orientation preserving.Remark 2.2.3—If we remove the assumption(2.2)for the cobordisms Z and Z′,then the extension maps j:Spin c(M,ξ)→Spin c(M′τ,ω′τ)are not injective in general.Then we may still prove a generalization of Theorem E:assume that Z is just the symplectization of(Y,ξ)and that Z′is an AFAK end as before,but without19assuming the property(2.2).To discuss the generalization we need to make the notation more precise.We denote by M̟τ(Mτ,j(s,h))the moduli space for some choice of(s,h)∈Spin c(M,ξ) j′(s,h)=(s′,h′)M̟τ(Mτ,j(s,h)).(2.9)Then the conclusion of Theorem E is the same if we replace M̟τ(M′τ)byM̟τ(M′τ,(s′,h′))and M̟τ(Mτ)by M̟τ(Mτ,(s′,h′)).In particular,there isan orientation preserving diﬀeomorphismG: M̟τ(Mτ,(s′,h′))→M̟τ(M′τ,(s′,h′)).The rest of Section2is devoted to construct the map G by a gluing technique and to show that it is a diﬀeomorphism.2.2.4CompactnessIn this section,we reﬁne the result of compactness for oneﬁxed moduli space M(Mτ),by showing that a sequence of solutions(Aτ,Φτ)of Seiberg-Witten equations on Mτconverge in some sense asτ→∞,up to extraction of a sub-sequence,and modulo gauge transformations,to a solution of Seiberg-Witten equations on M.We review the arguments proving the compactness of one particular moduli space in[KM]and explain how to apply them to the family Mτ.Lemma2.2.5There exist constantsκ1,κ2such that for everyτand every solution of Seiberg-Witten equations(A,Φ)on Mτwe have the estimateΦ C0 κ1+κ2 ̟τ C0.Proof By construction of the moduli space,Φ−Ψ∈L23;now the Sobolev inclusion C0⊂L23together with Lemma2.1.6tells us that the pointwise norm |Φ−Ψ|C0→0near inﬁnity on Mτ.Hence,either|Φ| 1,and we are done,either this is not true,and|Φ|must have a local maximum at a point x∈Mτ.We apply the maximum principle20at x:14|Φ2|where the last identity follows from the Lichnerowicz ing Seiberg-Witten equations,we have0 −14|Φ|2.The lemma follows from the fact that the pointwise norm of s,F B,Ψis bounded independently ofτby Lemma2.1.6.As we saw,the AFAK structure induces a Chern connectionˆ∇and and a spin connection B on the bundle Wτrestricted to Mτ\Mbyˆ∇AΦ:=ˆ∇Φ+a⊗Φ;in particularˆ∇B:=ˆ∇with these notations.Notice that∇B=ˆ∇B unless the almost complex structure is integrable.Proposition2.2.6There exists a compact K⊂M large enough andδ>0 such that for every integer k there is a constant c k>0so that,for everyτlarge enough and every solution of Seiberg-Witten equations(A,Φ)on Mτ,we have the pointwise estimate on Mτ\K|1−|β|2−|γ|2|,|γ|,|ˆ∇AΦ|,|ˆ∇2AΦ|,······,|ˆ∇k AΦ| c k e−δστ,(2.10) whereΦ=(β,γ)∈Λ0,0⊕Λ0,2.Remark—The quantities(2.10)controlled by the lemma are gauge invari-ant.Proof The lemma was proved in[KM,Proposition3.15]forτﬁxed.It is readily checked that it extends as stated for the family Mτ.We recall what the ingredients of the proof are.A conﬁguration(A,Φ)on Mτhas an energy which is a gauge invariant quantity deﬁned byEτ(A,Φ)= Mτ\K0 (|β|2+|γ|2−1)2+|γ|2+|ˆ∇AΦ|2 vol gτ, where all the norms,connections are taken with respect to the structures deﬁned on Mτand K0is a compact in M containingLemma2.2.7There exist a compact K0⊂M large enough,and some con-stantsκ3andκ4,such that for everyτlarge enough and every solution of Seiberg-Witten equations(A,Φ)on Mτ,we haveEτ(A,Φ) κ3+κ4 ̟τ 2Nτ.Proof The proof of this lemma is the same proof than for[KM,Lemma3.17]. The fact thatκ3andκ4do not depend onτis insured by Lemma2.1.6.If we look carefully at the proof,using the notations of[KM,page232],we read the claim that K3da∧ωcan be controlled,for K3a compact domain large enough,ωa closed form extending the symplectic form on the whole manifold as in Deﬁnition2.1.1,and a=A−B decaying exponentially fast.No explanation of this is given and we provide one now.Pick an arbitrarily smallε>0.We haveK3da∧ω ε2ε K3|ω|2 ε2ε K3|ω|2.Using the exponential decay of a,we have as in the case of a compact manifoldM|da|2=2 M|d+a|2.ThereforeK3da∧ω ε M\K3|d+a|2+ε K3|d+a|2+1。

三体中的名词术语英文刘宇昆

三体中的名词术语英文刘宇昆The Three-Body Problem by Liu Cixin has captivated readers around the world with its intricate plot and rich scientific concepts. One of the unique aspects of this acclaimed science fiction novel is the author's introduction of various specialized terms and jargon related to the fictional "Three-Body" universe. These terms not only add depth and authenticity to the narrative but also challenge readers to delve deeper into the complex ideas presented. In this essay we will explore some of the key English terminology used in The Three-Body Problem and examine their significance within the context of the story.One of the most prominent terms in the novel is "Trisolaris" which refers to the fictional three-star planetary system that serves as the primary setting for much of the narrative. The word "Trisolaris" is a combination of the prefix "tri-" meaning three and the word "solaris" meaning related to the sun. This term encapsulates the core premise of the story where a planet orbits not one but three suns, leading to extreme climatic instability and unpredictability. The chaotic nature of the Trisolaris system is a key driver of the plot as it sets the stagefor the extraterrestrial civilization's attempts to flee their doomed homeworld and seek refuge on Earth.Another important term is "sophon" which describes the mysterious and highly advanced microscopic probes sent by the Trisolarians to infiltrate and monitor human civilization. The word "sophon" is derived from the Greek word "sophos" meaning wise or intelligent. This term effectively conveys the incredible technological capabilities of the Trisolarian civilization, as these sophons are able to manipulate the fundamental particles of the universe to gather information and even influence human affairs. The sophons serve as a constant threat and source of tension throughout the narrative, highlighting the asymmetry of power between the Trisolarians and humanity.A crucial concept introduced in the novel is that of "the Sphere" which represents the boundary of the Trisolaris system's influence. This term refers to an imaginary three-dimensional sphere within which the Trisolarian civilization can effectively exert its control and disrupt the normal laws of physics. The Sphere is a crucial plot device as it defines the limits of the Trisolarians' technological superiority and the extent to which they can intervene in human affairs. The constant struggle to expand or maintain the Sphere is a central theme in the novel, as both the Trisolarians and humanity attempt to gain the upper hand in this cosmic chess match.Another notable term is "dark forest" which describes the author's vision of a hostile and unforgiving universe where civilizations must remain hidden and silent to avoid annihilation. This metaphor is used to convey the idea that in a universe teeming with intelligent life, the safest strategy for survival is to avoid drawing attention to oneself, much like a hunter hiding in a dark forest. The "dark forest" concept is a key driver of the Trisolarians' decision to seek out and destroy humanity, as they believe that by remaining hidden, they can avoid the fate that has befallen countless other civilizations.The term "cosmic sociology" is also introduced in the novel to describe the study of the relationships and interactions between different civilizations on a cosmic scale. This field of study is central to the Trisolarians' understanding of the universe and their attempts to navigate the treacherous landscape of interstellar politics. The term "cosmic sociology" highlights the novel's exploration of the broader implications of humanity's first contact with an alien civilization, and the potential challenges and consequences that such an encounter may bring.Another intriguing term is "supra-quantum computer" which refers to the Trisolarians' most advanced computing technology. This term suggests a level of computational power that transcends the limitations of traditional quantum computers, allowing theTrisolarians to perform feats of calculation and simulation that are beyond the reach of human technology. The supra-quantum computer is a crucial plot device, as it enables the Trisolarians to model and manipulate the behavior of human civilization with uncanny precision.Finally, the term "Wallfacer" is used to describe a select group of individuals tasked with devising strategies to defend Earth against the Trisolarian invasion. The word "Wallfacer" evokes the image of a defensive wall or barrier, suggesting the critical role these individuals play in protecting humanity from the extraterrestrial threat. The Wallfacers are imbued with immense power and resources, but their true plans and motivations remain shrouded in mystery, adding to the suspense and intrigue of the narrative.These are just a few of the many specialized terms and concepts that are integral to the world-building and storytelling in The Three-Body Problem. Each of these terms serves to deepen the reader's understanding of the complex scientific and philosophical ideas that underpin the novel's narrative. By exploring these terms and their significance, we gain a deeper appreciation for the richness and depth of Liu Cixin's imaginative vision, and the ways in which he has seamlessly blended scientific speculation with captivating storytelling.。

Arrhenius equation

Arrhenius equationFrom Wikipedia, the free encyclopediaThe Arrhenius equation is a simple, but remarkably accurate, formula for the temperaturedependence of the reaction rate constant, and therefore, rate of a chemical reaction.[1] Theequation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation for it.Nowadays it is best seen as an empirical relationship.[2] It can be used to model thetemperature-variance of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions.A historically useful generalization supported by the Arrhenius equation is that, for many commonchemical reactions at room temperature, the reaction rate doubles for every 10 degree Celsius increase in temperature.∙∙∙∙∙[edit]OverviewIn short, the Arrhenius equation gives "the dependence of the rate constant k of chemicalreactions on the temperature T (in absolute temperature, such as kelvins or degrees Rankine) and activation energy[3]E a", as shown below:[1]where A is the pre-exponential factor or simply the prefactor and R is the gas constant. Theunits of the pre-exponential factor are identical to those of the rate constant and will varydepending on the order of the reaction. If the reaction is first order it has the units s−1, and for that reason it is often called the frequency factor or attempt frequency of the reaction. Mostsimply, k is the number of collisions that result in a reaction per second, A is the total numberof collisions (leading to a reaction or not) per second and is the probability that any given collision will result in a reaction. When the activation energy is given in molecular units instead of molar units, e.g., joules per molecule instead of joules per mole, the Boltzmann constant is used instead of the gas constant. It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in rate of reaction.Given the small temperature range kinetic studies occur in, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor isnegligible compared to the temperature dependence of the factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.Some authors define a modified Arrhenius equation,[4] that makes explicit the temperature dependence of the pre-exponential factor. If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem (i.e., determining the prefactor and activation energy from experimental data) becomes singular. The modified equation is usually of the formwhere T0 is a reference temperature and allows n to be a unitless power. Clearly the original Arrhenius expression above corresponds to n = 0. Fitted rate constants typically lie in the range -1<n<1. Theoretical analyses yield various predictions for n. It has been pointed out that "it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T½ dependence of the pre-exponential factor isobserved experimentally."[2] However, if additional evidence is available, from theory and/or from experiment (such as density dependence), there is no obstacle to incisive tests of the Arrhenius law.Another common modification is the stretched exponential formwhere β is a unitless number of order 1. This is typically regarded as a fudgefactor to make the model fit the data, but can have theoretical meaning, forexample showing the presence of a range of activation energies or in specialcases like the Mott variable range hopping.Taking the natural logarithm of the Arrhenius equation yields:So, when a reaction has a rate constant that obeys the Arrhenius equation, a plot of ln(k) versus T−1 gives a straight line, whose slope and intercept can be used to determine E a and A. This procedure has become so common in experimental chemical kinetics that practitioners have taken to using itto define the activation energy for a reaction. That is the activation energy is defined to be (-R) times the slope of a plot of ln(k) vs. (1/T ):[edit]Kinetic theory's interpretation of Arrhenius equationArrhenius argued that for reactants to transform into products, theymust first acquire a minimum amount of energy, called the activationenergy E a. At an absolute temperature T, the fraction of molecules that have a kinetic energy greater than E a can be calculated fromthe Maxwell-Boltzmann distribution of statistical mechanics, and turns out to be proportional to . The concept of activationenergy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories.[edit]Collision theoryMain article: Collision theoryOne example comes from the "collision theory" of chemical reactions, developed by Max Trautz and William Lewis in the years 1916-18. In this theory, molecules are supposed to react if they collide with arelative kinetic energy along their lines-of-center that exceeds E a. This leads to an expression very similar to the Arrhenius equation.[edit]Transition state theoryAnother Arrhenius-like expression appears in the "transition statetheory" of chemical reactions, formulatedby Wigner, Eyring, Polanyi and Evans in the 1930s. This takes various forms, but one of the most common iswhere ΔG‡ is the Gibbs free energy ofactivation, k B is Boltzmann's constant, and h is Planck's constant.At first sight this looks like an exponential multiplied by a factor that is linear in temperature. However, one must remember that free energy is itself a temperature dependent quantity. The free energy of activation is the difference of an enthalpy term and an entropy term multiplied by the absolute temperature. When all of the details are worked out one ends up with an expression that again takes the form of an Arrhenius exponential multiplied by a slowly varying function of T. The precise form of the temperature dependence depends upon the reaction, and can be calculated using formulas from statistical mechanics involving the partition functions of the reactants and of the activated complex.[edit]Limitations of the idea of Arrhenius Activation EnergyBoth the Arrhenius activation energy and the rate constant k are experimentally determined, and represent macroscopic reaction-specific parameters that are not simply related to threshold energies and the success of individual collisions at the molecular level. Consider a particular collision (an elementary reaction) between molecules A and B. The collision angle, the relative translational energy, the internal (particularly vibrational) energy will all determine the chance that the collision will produce a product molecule AB. Macroscopic measurements of E and k are the result of many individual collisions with differing collision parameters. To probe reaction rates at molecular level, experiments have to be conducted under near-collisional conditions and this subject is often called molecular reaction dynamics (see Levine).'[edit]See also▪Accelerated aging▪Arrhenius plot▪Eyring equation▪Q10 (temperature coefficient)[edit]Notes and references1. ^ a b Arrhenius equation - IUPAC Goldbook definition2. ^ a b Kenneth Connors, Chemical Kinetics, 1990, VCHPublishers3. ^Arrhenius activation energy - IUPAC Goldbook definition4. ^IUPAC Goldbook definition of modified Arrheniusequation▪Laidler, K. J. (1997) Chemical Kinetics,Third Edition, Benjamin-Cummings▪Laidler, K. J. (1993) The World of Physical Chemistry, Oxford University Press▪Levine, R.D. (2005) Molecular Reaction Dynamics, Cambridge University Press[edit]External links▪Carbon Dioxide solubility in Polyethylene - Using Arrhenius equation for calculating species solubility in polymersArrhenius plotFrom Wikipedia, the free encyclopediaAn Arrhenius plot displays the logarithm of kinetic constants (ln(k), ordinate axis) plotted against inverse temperature (1 / T, abscissa). Arrhenius plots are often used to analyze the effect of temperature on the rates of chemical reactions. For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.Conventional plot:k against TArrhenius plot:ln(k) against 1/TThe Arrhenius equation given in the form: can be written equivalently as:Where:k = Rate constantA = Pre-exponential factorE a = Activation energyR = Gas constantT = Absolute temperature, KWhen plotted in the manner described above, the value of the "y-intercept" will correspond to ln(A), and the gradient of the line will be equal to −E a / R.The pre-exponential factor, A, is a constant of proportionality that takes into account a number of factors such as the frequency of collision between and the orientation of the reacting particles.The expression −E a / RT represents the fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature.[edit]See also▪Arrhenius equation。

科技英语翻译1

驱动这些机器的动力装置是一台50马力的感应电动机。
► 2）通顺易懂 ► 译文的语言符合译语语法结构及表达习惯，容易为读者所理解和接受。
► A. When a person sees, smells, hears or touches something, then he is perceiving.
2. Cramped(狭窄的) conditions means that passengers’ legs cannot move around freely.
空间狭窄，旅客的两腿就不能自由活动。
3. All bodies are known to possess weight and occupy space.
忠实、通顺（普遍观点）
► 科技英语文章特点：(well-knit structure;tight logic;various styles)结构严谨，逻辑严密，文体多样
1. 科技翻译的标准：准确规范，通顺易懂，简洁明晰 1）准确规范
所谓准确，就是忠实地，不折不扣地传达原文的全部信息内容。所谓规范，就是译文要符合所涉及的科学技或某个专业领域的专业语言表
实验结果等，而不是介绍这是这些结果，理论或现象是由谁发明或发现的。
► In this section, a process description and a simplified process flowsheet are given for each DR process to illustrate the types of equipment used and to describe the flow of materials through the plant. The discussion does not mention all the variations of the flowsheet which may exist or the current status of particular plants. In the majority of the DR processes described in this section, natural gas is reformed in a catalyst bed with steam or gaseous reduction products from the reduction reactor. Partial oxidation processes which gasify liquid hydrocarbons, heavy residuals and coal are also discussed. The reformer and partial oxidation gasifier are interchangeable for several of the DR processes.

Ornstein–Uhlenbeck process - Wikipedia, the f

Ornstein–Uhlenbeck process - Wikipedia,the f...Ornstein–Uhlenbeck process undefinedundefinedFrom Wikipedia, the free encyclopediaJump to: navigation, searchNot to be confused with Ornstein–Uhlenbeck operator.In mathematics, the Ornstein–Uhlenbeck process (named after LeonardOrnstein and George Eugene Uhlenbeck), is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The process is stationary, Gaussian, and Markov, and is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.[1] Over time, the process tends to drift towards its long-term mean: such a process is called mean-reverting.The process x t satisfies the following stochastic differential equation:where θ> 0, μ and σ> 0 are parameters and W t denotes the Wiener process. Contents[hide]1 Application in physical sciences2 Application in financialmathematics3 Mathematical properties4 Solution5 Alternative representation6 Scaling limit interpretation7 Fokker–Planck equationrepresentation8 Generalizations9 See also10 References11 External links[edit] Application in physical sciencesThe Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. Consider for example a Hookean spring with spring constant k whose dynamics is highly overdamped with friction coefficient γ. In the presence of thermal fluctuations with temperature T, the length x(t) of the spring will fluctuate stochastically around the spring rest length x0; its stochastic dynamic is described by an Ornstein–Uhlenbeck process with:where σ is derived from the Stokes-Einstein equation D = σ2 / 2 = k B T / γ for theeffective diffusion constant.In physical sciences, the stochastic differential equation of an Ornstein–Uhlenbeck process is rewritten as a Langevin equationwhere ξ(t) is white Gaussian noise with .At equilibrium, the spring stores an averageenergy in accordance with the equipartition theorem.[edit] Application in financial mathematicsThe Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter μ represents the equilibrium or mean value supported by fundamentals; σ the degree of volatility around it caused by shocks, and θ the rate by which these shocks dissipate and the variable reverts towards the mean. One application of the process is a trading strategy pairs trade.[2][3][edit] Mathematical propertiesThe Ornstein–Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast tothe Wiener process; the difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas for the Ornstein–Uhlenbeck process it is dependent on the current value of the process: if the current value of the process is less than the (long-term) mean, the drift will be positive; if the current valueof the process is greater than the (long-term) mean, the drift will be negative. In other words, the mean acts as an equilibrium level for the process. This gives the process its informative name, "mean-reverting." The stationary (long-term) variance is given byThe Ornstein–Uhlenbeck process is the continuous-time analogue ofthe discrete-time AR(1) process.three sample paths of different OU-processes with θ = 1, μ = 1.2, σ = 0.3:blue: initial value a = 0 (a.s.)green: initial value a = 2 (a.s.)red: initial value normally distributed so that the process has invariant measure [edit] SolutionThis equation is solved by variation of parameters. Apply Itō–Doeblin's formula to thefunctionto getIntegrating from 0 to t we getwhereupon we seeThus, the first moment is given by (assuming that x0 is a constant)We can use the Itōisometry to calculate the covariance function byThus if s < t (so that min(s, t) = s), then we have[edit] Alternative representationIt is also possible (and often convenient) to represent x t (unconditionally, i.e.as ) as a scaled time-transformed Wiener process:or conditionally (given x0) asThe time integral of this process can be used to generate noise with a 1/ƒpower spectrum.[edit] Scaling limit interpretationThe Ornstein–Uhlenbeck process can be interpreted as a scaling limit of a discrete process, in the same way that Brownian motion is a scaling limit of random walks. Consider an urn containing n blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let X n be the number of blueballs in the urn after n steps. Then converges to a Ornstein–Uhlenbeck process as n tends to infinity.[edit] Fokker–Planck equation representationThe probability density function ƒ(x, t) of the Ornstein–Uhlenbeck process satisfies the Fokker–Planck equationThe stationary solution of this equation is a Gaussian distribution with mean μ and variance σ2 / (2θ)[edit ] GeneralizationsIt is possible to extend the OU processes to processes where the background driving process is a L évy process . These processes are widely studied by OleBarndorff-Nielsen and Neil Shephard and others.In addition, processes are used in finance where the volatility increases for larger values of X . In particular, the CKLS (Chan-Karolyi-Longstaff-Sanders) process [4] with the volatility term replaced by can be solved in closed form for γ = 1 / 2 or 1, as well as for γ = 0, which corresponds to the conventional OU process.[edit ] See alsoThe Vasicek model of interest rates is an example of an Ornstein –Uhlenbeck process.Short rate model – contains more examples.This article includes a list of references , but its sources remain unclear because it has insufficient inline citations .Please help to improve this article by introducing more precise citations where appropriate . (January 2011)[edit ] References^ Doob 1942^ Advantages of Pair Trading: Market Neutrality^ An Ornstein-Uhlenbeck Framework for Pairs Trading ^ Chan et al. (1992)G.E.Uhlenbeck and L.S.Ornstein: "On the theory of Brownian Motion", Phys.Rev.36:823–41, 1930. doi:10.1103/PhysRev.36.823D.T.Gillespie: "Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral", Phys.Rev.E 54:2084–91, 1996. PMID9965289doi:10.1103/PhysRevE.54.2084H. Risken: "The Fokker–Planck Equation: Method of Solution and Applications", Springer-Verlag, New York, 1989E. Bibbona, G. Panfilo and P. Tavella: "The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise", Metrologia 45:S117-S126,2008 doi:10.1088/0026-1394/45/6/S17Chan. K. C., Karolyi, G. A., Longstaff, F. A. & Sanders, A. B.: "An empirical comparison of alternative models of the short-term interest rate", Journal of Finance 52:1209–27, 1992.Doob, J.L. (1942), "The Brownian movement and stochastic equations", Ann. of Math.43: 351–369.[edit] External linksA Stochastic Processes Toolkit for Risk Management, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares TrikiSimulating and Calibrating the Ornstein–Uhlenbeck process, M.A. van den Berg Calibrating the Ornstein-Uhlenbeck model, M.A. van den BergMaximum likelihood estimation of mean reverting processes, Jose Carlos Garcia FrancoRetrieved from ""。

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a r X i v :c o n d -m a t /0009113v 1 [c o n d -m a t .s t a t -m e c h ] 7 S e p 2000Is there Ornstein-Zernike equation in the canonical ensemble?J.A.White and S.VelascoDepartamento de F´ısica Aplicada,Facultad de Ciencias Universidad de Salamanca,37008Salamanca,Spain(February 1,2008)A general density-functional formalism using an extended variable-space is presented for classical ﬂuids in the canonical ensemble (CE).An exact equation is derived that plays the role of the Ornstein-Zernike (OZ)equation in the grand canonical ensemble (GCE).When applied to the ideal gas we obtain the exact result for the total correlation function h N .For a homogeneous ﬂuid with N particles the new equation only diﬀers from OZ by 1/N and it allows to obtain an approximate expression for h N in terms of its GCE counterpart that agrees with the expansion of h N in powers of 1/N .PACS number(s)61.20.Gy,05.20.GgThe Ornstein-Zernike (OZ)equation is an essential ingredient of any theory of the equilibrium microscopic structure of classical ﬂuids since it provides an exact relation between correlation functions at the two-particle level.In the world of the integral equation theory [1,2],most integral equations are based on this relation closed with an additional approximate relation between the correlation functions.Closures like the Percus-Yevick approximation,the hypernet-ted chain approximation,or the mean spherical approximation,together with the OZ relation have led to well-known theories of classical ﬂuids.In the world of density functional theory (DFT)[3–5],one makes explicit approximations for the free-energy functional and,after using a variational principle,one obtains the direct correlation function via functional diﬀerentiation.Inserting the direct correlation function into the OZ equation allows to obtain the pair correlation function.Furthermore,as we shall see below,the OZ equation is an identity that arises naturally from the DFT formalism [3–5].When using the OZ equation it is important to notice that it has been formulated for the grand canonical ensemble (GCE)and thus care must be taken when considering it in other statistical mechanics ensembles.Of particular interest is the canonical ensemble (CE),not only from a fundamental point of view but also because of its relevance in the analysis of systems with a ﬁxed,ﬁnite number of particles N .The aim of this Letter is to study the possibility of establishing an OZ relation in the CE [6].We shall see how the ﬁxed-N constraint precludes the direct application of the GCE-OZ relation,this problem will be solved in a DFT context by using an extended variable-space in which,together with the familiar inhomogeneous density,a Lagrange multiplier is considered as a variable.The OZ equation for a nonuniform system with ﬁxed chemical potential µand temperature T ,i.e.,in the GCE,in the presence of an external potential V ext (r ),is usually expressed as the following relation between the total correlation function h and the direct correlation function c (2)h (r 1,r 2)=c (2)(r 1,r 2)+ρ(r 3)c (2)(r 1,r 3)h (r 3,r 2)d r 3.(1)where ρ(r )is the inhomogeneous density,and,in terms of functional derivatives,h and c (2)are deﬁned (ﬁxed T ,µ)byβ−1δρ(r 1)δρ(r 2)=c (2)(r 1,r 2)−δ(r 1−r 2)δV ext (r 3)δV ext (r 3)grand canonical ensemble(GCE)where the one-to-one relation betweenρ(r)and V ext(r)is a key theorem(see,e.g., [3])and the OZ equation follows as a corollary[4].We now focus our attention on the canonical ensemble(CE)where the number of particles N isﬁxed.This means that the CE inhomogeneous densityρN is normalized to N,i.e.,ρN(r)d r=N.(5) Furthermore,from the usual deﬁnition of the two particle densityρ(2)N(see, e.g.,[1])it is direct to obtain ρ(2)N(r1,r2)d r2=(N−1)ρN(r1),and thus,taking into account thatρ(2)N(r1,r2)=ρN(r1)ρN(r2) h N(r1,r2)+1 , the following constraint for h N arises in the CE:ρN(r1)h N(r1,r2)d r1=−1,(6)which,as pointed out by Ashcroft[5],in the context of electronic systems leads to the notion of exchange-correlation hole.We remark that Eq.(6)is a result of considering a system in the CE where the number of particles cannot ﬂuctuate.An important consequence of this equation is that the OZ equation(1)is no longer valid in the CE,since it is inconsistent with(6).Furthermore,since the OZ equation is associated to the one-to-one relation betweenρ(r) and V ext(r)it would seem that there is not a CE counterpart of this one-to-one relation,which would be of serious concern in DFT of the CE.The goal of this Letter is to clarify this point and to introduce a CE-OZ equation similar to(1).It is clear that,ﬁxed N and T,the CE densityρN is determined by the external potential V ext,but this potential is not(strictly)unique:any potential that diﬀers from V ext by an additive constant leads to the sameρN and thus the inhomogeneous density is not uniquely determined by the external potential.This does not contradict theﬁrst Hohenberg-Kohn theorem for DFT[7]because it allows for this additive constant.However,this(strict) nonuniqueness means that the(Legendre)transform from the functional variable V ext to the functional variableρN is not invertible and thusδρN(r1)/δV ext(r2)is a singular matrix,which in turn implies that Eq.(4)cannot hold forthe pair V ext,ρN.We want to stress that this is due to theﬁxed-N constraint which implies that0=δN/δV ext(r2)= δρN(r1)/δV ext(r2)d r1which shows that the matrix is singular[8].Our proposal for solving this diﬃculty is to‘bypass’the constraint by means of an extended variable space where instead of V ext alone we consider the set{V ext,N}and, instead ofρN,the set{ρN,λN}whereλN will be deﬁned later and identiﬁed with the Lagrange multiplier related to theﬁxed-N constraint.The role played by the diﬀerent variables is clariﬁed by rebuilding the main DFT results in this extended space.Our derivation is based on Legendre transform approach of Ref.[9].The CE Helmholtz free energy F is a functional of the external potential V ext and a function of the number ofparticles N and the temperature T,i.e.,F=F(T,N,[V ext]).The densityρN and the‘chemical potential’λN(diﬀerent fromµ!)are functions of T and N and functionals of V ext,and are obtained from the following derivatives:ρN(r,[V ext])= δF∂N V ext(8) where the explicit dependence on T has been been omitted.We next consider the Legendre transform of F to the new variablesρN andλN,ˆF(λN,[ρN])=F(λN,[ρN])− ρN(r)V ext(r,λN,[ρN])d r−λN ρN(r)d r,(9)where we have used N= ρN(r)d r[Eq.(5)]and we deﬁne F(λN,[ρN])≡F( ρN(r)d r,[V ext(λN,[ρN])]).In order to introduce a variational principle we‘revert’this transform by obtaining a functional that depends on both the new variables{ρN,λN}and the old ones{V ext,N}:F Vext ,N(λN,[ρN])=ˆF(λN,[ρN])+ ρN(r)V ext(r)d r+λN N.(10)The minimization of this functional w.r.t.{ρN ,λN }leads toδˆF(λN ,[ρN ])∂λNρN+N =0.(12)Substituting (9)into (12)and takingintoaccountEq.(5)onehas thatthefunctionalF ≡ˆF +λN ρN (r )d r =F (λN ,[ρN ])−ρN (r )V ext (r ,λN ,[ρN ])d r (13)does not depend on λN ,that is F =F [ρN ]and thus,from (11)we obtainδF [ρN ]ρN (r 2)∂ρN (r 2)∂N=ρN (r 2)∂Nc (2)N (r 2,r 3)−c (2)N (r 1,r 3)d r 1d r 3,(17)where in analogy with the GCE we have introduced the CE direct correlation function c (2)N asc (2)N (r 1,r 2)=−βδ2(F [ρN ]−F id [ρN ])N(19)where |r i −r j |≡r ij and we have taken into account that the uniformﬂuid is both translationally and rotationally invariant.This equation should be comparedwiththe GCE-OZ equation (1),that,in the uniform limit,ρ(r )→ρ=ρN ,reduces toh (r 12)=c (2)(r 12)+ρc (2)(r 13)h (r 32)d r 3.(20)Then,for uniform ﬂuids,both equations only diﬀer by the 1/N term which vanishes in the thermodynamic limit.It is well-known [11,12]that the GCE radial distribution function g (r )=h (r )+1can be related to g N (r )=h N (r )+1by means of a series expansion in powers of 1/N .Using this expansion one has for the total correlation function (with ρN =ρ)h N (r )≈h (r )−S (0)2N ∂2N ],(21)where S (k )≡1+ρ˜h(k )is the structure factor,being ˜h (k )=h (r )exp(i k ·r )d r the Fourier transform of h (r ).In what follows we shall indicate how one can rederive this result by means of the OZ equations (19)and (20)and anapproximate expression for c (2)N in terms of c (2)(r )(conversely,Eq.(21)and Eqs.(19)-(20)can be used to obtain anapproximate expression for c (2)N ).Since the OZ equations (19)and (20)involve convolutions it is convenient to work in Fourier space.Subtracting (20)from (19),after some algebra,we obtain the exact relation∆˜h(k )=−8π3N −1S (k )δ(k )+∆˜c (k )S 2(k )N+S (k )NρS (k )∆˜c (k ).(23)We note that,to zeroth order,this equation reduces to ∆˜h(k )≈−8π3S (k )δ(k )/N and thus ∆h (r )≈−S (0)/N ,i.e.,the ﬁrst contribution to Eq.(21).In order to evaluate (23)we need an expression (valid to ﬁrst order)for ∆c (2),this expression can be obtained from the above-mentioned saddle point approximation for the CE free-energy functional [10]which leads to∆c(2)(r 12)≈−1δρ(r 1)δρ(r 2)log∆2(N )[1]J.P.Hansen and I.R.McDonald,Theory of Simple Liquids (Academic Press,London,1986).[2]J.R.Henderson,in Fundamentals of Inhomogeneous Fluids edited by D.Henderson (Dekker,New York,1992),p.23.[3]R.Evans,Adv.in Phys.28,143(1979).[4]R.Evans,in Fundamentals of Inhomogeneous Fluids (Ref.[2]),p.85.[5]N.W.Ashcroft,in Density Functional Theory edited by E.K.U.Gross and R.M.Dreizler (Plenum Press,New York,1995),p.581.[6]After completing this work we were aware of an eprint by Hernando and Blum;arXiv:cond-mat/0008291;in which the authors address similar issues,with a diﬀerent approach.[7]W.Kohn,in Density Functional Theory (Ref.[5]),p.3.[8]An equivalent demonstration follows from noting that (2)is also valid in the canonical ensemble,which together with Eq.(6)leads to −β−1 δρN (r 1)/δV ext (r 2)d r 1=0.[9]N.Argaman and G.Makov,Am.J.Phys.68,69(2000).[10]J.A.White,A.Gonz´a lez,F.L.Rom´a n,and S.Velasco,Phys.Rev.Lett.84,1220(2000).[11]J.L.Lebowitz and J.K.Percus,Phys.Rev.122,1675(1961).[12]J.J.Salacuse,A.R.Denton and P.A.Egelstaﬀ,Phys.Rev E 53,2382(1996).[13]J.A.White and S.Velasco,Phys.Rev.E 62,4427(2000);.[14]A.Gonz´a lez,J.A.White,F.L.Rom´a n,S.Velasco,and R.Evans,Phys.Rev.Lett.79,2466(1997);A.Gonz´a lez,J.A.White,F.L.Rom´a n,and R.Evans,J.Chem.Phys.109,3637(1998);R.Evans in New Approaches to Problems in Liquid State Theory edited by C.Caccamo et al (Kluwer,Amsterdam,1999),p.153.。