Homotopy field theory in dimension 3 and crossed group-categories
Computer-Vision计算机视觉英文ppt
Its mainstream research is divided into three stages:
Stage 1: Research on the visual basic method ,which take the model world as the main object;
Stage 2: Research on visual model ,which is based on the computational theory;
the other is to rebuild the three dimensional object according to the two-dimensional projection images .
History of computer vision
1950s: in this period , statistical pattern recognition is most applied in computer vision , it mainly focuse on the analysis and identification of two-dimensional image,such as: optical character recognition, the surface of the workpiece, the analysis and interpretation of the aerial image.
麦克斯韦方程和规范理论的观念起源
麦克斯韦方程和规范理论的观念起源*2014－11－12收到†email：************************杨振宁1，2著汪忠1，† 译DOI：10.7693/wl20141201(1清华大学高等研究院北京100084)(2香港中文大学物理系香港沙田)早在法拉第的“电紧张态(electrotonic state)”和麦克斯韦的矢量势(vector potential) 概念中，规范自由度(gauge freedom)的存在就已经不可避免。
它如何演化成为一个支撑粒子物理标准模型的对称原理？这里有一段值得叙说的故事。
人们常说，继库仑(Charles Augustin de Cou- lomb)、高斯(Carl Friedrich Gauss)、安培(AndréMarie Ampère)、法拉第(Michael Faraday) 发现了电学和磁学的四条实验定律之后，麦克斯韦(James Clerk Maxwell)引入了位移电流，在他的麦克斯韦方程组中实现了电磁学的伟大综合。
这种说法不能说是错的，但它并没有道出微妙的几何和物理直觉之间的关联，而正是这种关联促使场论在19 世纪取代了超距作用的概念，也正是它带来了20 世纪粒子物理中非常成功的标准模型。
1 19 世纪的历史1820 年奥斯特(Hans Christian Oersted，1777—1851)发现电流能使其附近的小磁针偏转。
这一发现使整个欧洲科学界大为振奋，带来的结果之一是安培(1775—1836) 关于“超距作用(action at a* 原文已发表于Physics Today，2014 年11 月刊，第45—51 页distance)”的成功理论。
在英格兰，法拉第(1791—1867)也因为奥斯特的发现而激动不已，但他缺乏足够的数学训练，所以无法理解安培的工作。
在1822 年9 月3 日写给安培的一封信中，法拉第叹息道：“很不幸，我不具备足够的数学知识，也不具备自如地进行抽象推理的能力。
平均共形排斥子维数
Dimensions of average conformal repellerJungchao BanDepartment of mathematicsNational Hualien University of EducationHualien97003,Taiwanjcban@.twYongluo CaoDepartment of mathematicsSuzhou UniversitySuzhou215006,Jiangsu,P.R.Chinaylcao@,yongluocao@Abstract.In this paper,average conformal repeller is defined,whichis generalization of conformal ing thermodynamic formalismfor sub-additive potential defined in[5],Hausdorffdimension and boxdimension of average conformal repellers are obtained.The map f isonly needed C1,without additional condition.Key words and phrases Hausdorffdimension,Non-conformal repellers,Topological pressure.1Introduction.In the dimension theory of dynamical systems,and in particular in the study of the Hausdorffdimension of invariant sets of hyperbolic dynamics,the theory is only devel-oped to full satisfaction in the case of conformal dynamical systems(both invertible and non-invertible).Roughly speaking,these are dynamical systems for which at each point the rate of contraction and expansion are the same in every direction.Bowen02000Mathematics Subject classification:Primary37D35;Secondary37C45.[3]was thefirst to express the Hausdorffdimension of an invariant set as a solution of an equation involving topological pressure.Ruelle[13]refined Bowen’s method and obtained the following result.Assume that f is a C1+γconformal expanding map,Λis an isolated compact invariant set and f|Λis topologically mixing,then the Hausdorffdimension ofΛ,dim HΛis given by the unique solutionαof the equationP(f|Λ,−αlog D x f )=0(1.1)where P(f|Λ,·)is the topological pressure functional.The smoothness C1+γwas re-cently relaxed to C1[10].For non-conformal dynamical systems there exists only partial results.For example, the Hausdorffdimension of hyperbolic invariant sets was only computed in some special cases.Hu[12]gave an estimate of dimension of non-conformal repeller for C2map. Falconer[7,8]computed the Hausdorffdimension of a class of non-conformal repellers. Related ideas were applied by Simon and Solomyak[15]to compute the Hausdorffdimension of a class of non-conformal horseshoes in R3.For C1non-conformal repellers,in[17],the author uses singular values of the deriva-tive D x f n for all n∈Z+,to define a new equation which involves the limit of a sequence of topological pressure.Then he shows that the unique solution of the equation is an upper bounds of Hausdorffdimension of repeller.In[1],the same problem is con-sidered.The author bases on the non-additive thermodynamic formalism which was introduced in[2]and singular value of the derivative D x f n for all n∈Z+,and gives an upper bounds of box dimension of repeller under the additional assumptions for which the map is C1+γandγ-bunched.This automatically implies that for Hausdorffdimension.In[9],the author defines topological pressure of sub-additive potential un-der the condition (D x f)−1 2 D x f <1,which means that f is1−bunched.They also obtain an upper bounds of Hausdorffdimension of repeller.In[4],thefirst named author prove that the upper bounded of Hausdorffdimension for C1non-conformal repeller obtained in[1,9,17]are same and it is the unique root of Bowen equation for sub-additive topological pressure.In this paper,we introduce the notion of average conformal ing thermo-dynamic formalism for sub-additive potential defined in[5],we prove that Hausdorffdimension and box dimension of average conformal repellers is the unique root of Bowen equation for subadditive topological pressure.The map f is only needed C1,without additional condition.Meanwhile,we introduce sup-additive potential topological pres-sure and prove that for special potentials,sub-additive and sup-additive topological pressures are same.In[2,11],the authors introduce the concepts of quasi-conformal and asymptotically conformal repeller by using Markov construction and prove that its dimension is the unique root of the equation obtained by non-additive topological pressure.It is obvious that quasi-conformal and asymptotically conformal repeller are average conformal repellers,but reverse is not true.Therefore our result is a general-ization of the results in[2,11].First we recall some basic definitions and notations.Let f:X→X be a continuous map.A set E⊂X is called(n, )separated set with respect to f if x,y∈E then d n(x,y)=max0≤i≤n−1d(f i x,f i y)> .For x∈X and r>0,defineB n(x,r)={y∈X:f i y∈B(f i x,r),for all i=0,···,n−1}.Ifφis a real continuous function on X and n∈Z+,letS nφ(x)=n−1i=0φ(f i(x)).We defineP n(φ,, )=sup{x∈Eexp S nφ(x):E is a(n, )−separated subset of X}. Then the topological pressure ofφis given byP(f,φ)=lim→0lim supn→∞1nlog P n(φ, ).Next we give some properties of P(f,·):C(M,R)→R∪{∞}.Proposition 1.1.Let f:M→M be a continuous transformation of a compact metrisable space M.Ifϕ1,ϕ2∈C(X,R),then the followings are true:(1)P(f,0)=h top(f).(2)|P(f,ϕ1)−P(f,ϕ2)|≤ ϕ1−ϕ2 .(3)ϕ1≤ϕ2implies that P(f,ϕ1)≤P(f,ϕ2).Proof.See Walters book[16].Corollary1.Let f:M→M be a continuous transformation of a compact metrisable space M.Ifϕ∈C(M,R)andϕ<0then function P(α)=P(f,αϕ)is continuous and strictly decreasing inα.Proof.It easily follows from Proposition1.1.The paper is organized as follows.In Section2,we develop variational principal for sub-additive potential.In Section3,we introduce the definition of average confor-mal repeller and give related results and the main theorem.In section4,we develop sup-additive thermodynamics formalism and variational principal for sup-additive po-tential.In section5,we give the proof of main result.2A sub-additive thermodynamics formalismLet f:X→X be a continuous map.A set E⊂X is called(n, )separated set with respect to f if x,y∈E then d n(x,y)=max0≤i≤n−1d(f i x,f i y)> .A sub-additive valuation on X is a sequence of functionsφn:M→R such thatφm+n(x)≤φn(x)+φm(f n(x)),we denote it by F={φn}.In the following we will define the topological pressure of F={φn}with respect to f.We defineP n(F, )=sup{x∈Eexpφn(x):E is a(n, )−separated subset of X}.Then the topological pressure of F is given byP(f,F)=lim→0lim supn→∞1nlog P n(F, ).Let M(X)be the space of all Borel probability measures endowed with the weak* topology.Let M(X,f)denote the subspace of M(X)consisting of all f-invariant measures.Forµ∈M(X,f),let hµ(f)denote the entropy of f with respect toµ,and let F∗(µ)denote the following limitF∗(µ)=limn→∞1nφn dµ.The existence of the above limit follows from a sub-additive argument.We call F∗(µ) the Lyapunov exponent of F with respect toµsince it describes the exponentially increasing speed ofφn with respect toµ.In[5],authors proved that the following variational principalTheorem2.1.[5]Under the above general setting,we haveP(f,F)=sup{hµ(T)+F∗(µ):µ∈M(X,f)}.3Average conformal repellerLet M be a C∞Riemann manifold,dim M=m.Let U be an open subset of M and let f:U→M be a C1map.SupposeΛ⊂U is a compact invariant set,that is,fΛ=Λand there is k>1such that for all x∈Λand v∈T x M,D x fv ≥k v ,where . is the norm induced by an adapted Riemannian metric.Let M(f|Λ),E(f) denote the all f invariant measures and the all ergodic invariant measure supported onΛrespectively.By the Oseledec multiplicative ergodic theorem,for anyµ∈E(f), we can define Lyapunov exponentsλ1(µ)≤λ2(µ)≤···≤λn(µ),n=dimM.Definition3.1.An invariant repeller is called average conformal if for anyµ∈E(f),λ1(µ)=λ2(µ)=···=λn(µ)>0.It is obvious that a conformal repeller is an average conformal repeller,but reverse isn’t true.Next we will give main theorem.Theorem3.1.(Main Theorem)Let f be C1dynamical system andΛbe an average conformal repeller,then the Hausdorffdimension ofΛis zero t0of t→P(−t F),whereF={log(m(D x f n),x∈Λ,n∈N}.(3.2) where m(A)= A−1 −1The proof will be given in section5.Theorem3.2.IfΛbe an average conformal repeller,thenlim n→∞1n(log Df n(x) −log m(d f n(x)))=0uniformly onΛ.Proof.LetF n(x)=log Df n(x) −log m(d f n(x)),n∈N,x∈Λ.It is obviously that the sequence{F n(x)}is a non-negative subadditive function se-quence.That is sayF n+m(x)≤F n(x)+F m(f n(x)),x∈Λ.Suppose(3.2)is not true,then there exists 0>0,for any k∈N,there exits n k≥k and x nk∈Λsuch that1 n k F nk(x nk)≥ 0.Define measuresµnk =1n kn k−1i=0δf i(xn k).Compactness of P(f)implies there exists a subsequence ofµnkthat converges to mea-sureµ.Without loss of generality,we suppose thatµnk→µ.It is well known thatµis f-invariant.Thereforeµ∈M(f).For afixed m,we havelim k→∞M1mF m(x nk)dµnk=M1mF m(x nk)dµ.It implieslim k→∞1n kn k−1i=01mF m(f i(x nk))=M1mF m(x)dµ.For afixed m,let n k=ms+l,0≤l<m.The sub-additivity of{F n}implies that for j=0,···m−1,F nk (x nk)≤F j(x nk)+F m(f j(x nk)+···+F m(f m(s−2)f j(x nk))+F m−j+l(f m(s−1)f j(x nk))Summing j from0to m−1,we getF nk (x nk)≤1mm−1j=0s−2i=0F m(f im+j(x nk))+1mm−1j=0[F j(x nk)+F m−j+l(f(s−1)m+j(x nk))]Let C1=max i=1,···,2m−1max x∈ΛF i(x).F nk (x)≤(sm+l)−1j=01mF m(f j(x))−1msm−1j=(s−1)mF m(f j(x))+2C1≤n k−1j=01mF m(f j(x))+4C1.Hence we havelim k→∞1n kF nk(x)≤limk→∞1n kn k−1i=01mF m(f i(x))=M1mF m(x)dµ.The arbitrariness of m∈N implies thatlim k→∞1n kF nk(x)≤1mMF m(x)dµ,∀m∈N.Hencelim m→∞1mMF m(x)dµ≥ 0>0.Then ergodic decomposition theorem[16]implies that there exists˜µ∈E(f)such thatlim m→∞1mMF m(x)d˜µ≥ 0>0.On the other hand,from Oseledec theorem and Kingman’s subadditive ergodic the-orem,we have limm→∞1mMlog Df n(x) d˜µ=λn(˜µ)and limm→∞1mMlog m(f n(x))d˜µ=λ1(˜µ).Thereforeλn(˜µ)−λ1(˜µ)≥ 0.This gives a contradiction to assumption of average conformal.4Sup-additive variational principalIn this section,wefirst give the definition of sup-additive topological principal.Then we prove the variational principal for special sup-additive potential.Let f:X→X be a continuous map.A set E⊂X is called(n, )separated set with respect to f if x,y∈E then d n(x,y)=max0≤i≤n−1d(f i x,f i y)> .A sup-additive valuation on X is a sequence of functionsϕn:M→R such thatϕm+n(x)≥ϕn(x)+ϕm(f n(x)),we denote it by F={ϕn}.In the following we will define the topological pressure of F={ϕn}with respect to f.We defineP∗n (F, )=sup{x∈Eexpϕn(x):E is a(n, )−separated subset of X}.Then the topological pressure of F is given byP∗(f,F)=lim→0lim supn→∞1nlog P n(F, ).For everyµ∈M(X,f),let F∗(µ)denote the following limitF∗(µ)=limn→∞1nϕn dµ.The existence of the above limit follows from a sup-additive argument.We call F∗(µ) the Lyapunov exponent of F with respect toµsince it describes the exponentially increasing speed ofϕn with respect toµ.Theorem4.1.Let f be C1dynamical system andΛbe an average conformal repeller, and F={ϕn(x)}={−t log Df n(x) }for t≥0be a sup-additive function sequence. Then we haveP∗(f,F)=sup{hµ(T)+F∗(µ):µ∈M(X,f)}.Proof.First we prove that for any m∈NP∗(f,F)≥P(f,ϕm m).For afixed m,let n=ms+l,0≤l<m.From the sup-additivity of{ϕn},we haveϕn(x)≥1mm−1j=0s−2i=0ϕm(f im+j(x))+1mm−1j=0[ϕj(x)+ϕm−j+l(f(s−1)m+j(x))].Let C1=min i=1,···,2m−1min x∈Xϕi(x).Then it hasϕn(x)≥(sm+l)−1j=01mϕm(f j(x))−1msm−1j=(s−1)mϕm(f j(x))+2C1≥n−1j=01mϕm(f j(x))+4C1.Hence we haveexp(ϕn(x))≥exp(n−1j=01mϕm(f j(x))+4C1).ThusP∗n (F, )=sup{x∈Eexpϕn(x):E is a(n, )−separated subset of X}≥P n(1mϕm, )×exp(4C1).It impliesP∗(f,F)≥P(f,1mϕm).From the arbitrary of m∈Z+,we haveP∗(f,F)≥P(f,1mϕm),for all m∈Z+.By the variational principal in[16],for everyµ∈M(f),we haveP∗(f,F)≥P(f,1mϕm)≥hµ(f)+M1mϕn(x)dµ,∀m∈N.Hence we have for everyµ∈M(f)P∗(f,F)≥hµ(f)+limm→∞M1mϕn(x)dµ.ThereforeP∗(f,F)≥sup{hµ(f)+limm→∞M1mϕn(x)dµ,µ∈M(f)}LetΦn(x)=−t log m(Df n(x))for t≥0.Then it is sub-additive.By the theorem in[5],we haveP(f,{Φn})=sup{hµ(f)+limm→∞M1mΦn(x)dµ,µ∈M(f)}By the definitions,−t log m(Df n(x))≥−t log Df n(x) for t≥0implies thatP∗(f,F)≤P(f,{Φn}).Theorem3.2implies that for anyµ∈M(f),it haslim m→∞M1mΦn(x)dµ=limm→∞M1mϕn(x)dµ.ThereforeP∗(f,F)=sup{hµ(f)+limm→∞M1mΦn(x)dµ,µ∈M(f)}.This completes the proof of theorem.5The proof of main theoremIn this section,we will give the proof of main theorem.First we state some known results.In[1],Barreira prove the following theorem.Theorem5.1.If f is a C1expanding map andΛis a repeller,thens1≤dim HΛ≤dim BΛ≤dim BΛ≤t1where s1and t1are the unique roots of the Bowen’s equations P(f,−t log Df(x) )=0 and P(f,−t log m(Df(x)))=0respectively.SinceΛis f-invariant,it is f n-invariant.Hence we have the following corollary.Corollary2.If f is a C1expanding map andΛis a repeller,thens n≤dim HΛ≤dim BΛ≤dim BΛ≤t nwhere s n and t n are the unique roots of the Bowen’s equations P(f n,−t log Df n(x) )= 0and P(f n,−t log m(Df n(x)))=0respectively.Next we prove that the sequences{t2k}and{s2k}are monotone.Theorem5.2.The sequence{s2k}is monotone,andlimk→∞s2k=s∗.Then we have s∗is the unique root of equation P∗(f,−t{log Df n(x) })=0.Proof.First we prove that the sequence{s2n}is monotone increasing.Letϕn=−log (Df n(x) and F={ϕn}.Then it is a sup-additive function sequence.For a fixed k∈N,P k(φ, )=sup{x∈Eexp S nφ(x):E is a(n, )−separated subset of X}.For∀ >0,by the uniformly continuity of f,there existsδ>0such that if E⊂M is an(n, )separated set of f2k+1,then E is an(2n,δ)separated set of f2k andδ→0 when →ing the subadditivity ofϕn,the Birkhoffsum S nφ2k+1ofϕ2k+1with respect to f2k+1has the following property:S nϕ2k+1(x)=ϕ2k+1(x)+ϕ2k+1(f2k+1x)+···+ϕ2k+1(f2k+1(n−1)x)≥ϕ2k(x)+ϕ2k(f2k x)+ϕ2k(f2k+1x)+ϕ2k(f2k+1f2k x)+···+ϕ2k(f2k+1(n−1)x)+ϕ2k(f2k+1(n−1)f2k x)=S2nϕ2k(x)where S2nϕ2k(x)is the Birkhoffsum ofϕ2k with respect to f2k.ThusP n(f2k+1,ϕ2k+1, )≥P2n(f2k,ϕ2k,δ).HenceP(f2k+1,ϕ2k+1)≥2P(f2k,ϕ2k).Therefore if s2k+1is the unique root of Bowen’s equation P(tϕ2k+1)=0,then we have0=P(f2k+1,s2k+1ϕ2k+1)≥2P(f2k,s2k+1ϕ2k).The monotone decreasing of the function P(f2k,tφ2k)implies that s2k≤s2k+1.The arbitrariness of k implies that the sequence{s2k}monotone decreasing.Next we prove thatP∗(f,F)≥1kP(f k,ϕk)∀k∈N.For afixed k∈N,let n=km+r,0≤r<k,and let C=min x∈M max1≤i≤kφi(x). For∀ >0,by the uniformly continuity of f,there existsδ>0such that if E⊂M is an(n, )separated set of f,then E is an(m,δ)separated set of f k andδ→0when →ing the sup-additivity ofϕn,we haveϕn(x)≥ϕk(x)+ϕk(f k(x))+···+ϕk(f(m−1)k(x))+ϕr(f mk(x)).ThusP∗n(f,F, )≥P m(f k,ϕk,δ)×e−C.HenceP∗(f,F, )≥1kP(f k,ϕk,δ).It gives thatP∗(f,F)≥1kP(f k,ϕk).ThereforeP∗(f,F)≥12kP(f2k,φ2k)∀k∈N.Let t F={tφn(x)}.Then we haveP∗(f,s2k F)≥12kP(f2k,s2kφ2k)=0∀k∈N.The monotone decreasing of P∗(f,t F)with respect to t implies that the unique root s∗of the equationP∗(f,t F)=0satisfiess∗≥s2k∀k∈N.Thuss∗≥s=limk→+∞s2k.Next we want to prove thats≥s∗.For afixed m,12mP(f2m,s2mϕ2m)=0using the variational principle,for anyµ∈M(f)⊂M(f2m),it hashµ(f)+12ms2mMϕ2m dµ=12m(hµ(f2m)+s2mMϕ2m dµ)≤0.Let m→∞,wehµ(f)+s limm→∞M12mϕ2m dµ≤0.Using sup-additive variational principle,we haveP∗(f,s{ϕn})≤0.Since P(f,t{ϕn})is strictly monotone decreasing with respect to t,we haves∗≤s.Lemma5.1.Ifφn(x)is a subadditive sequence,thenlim k→∞12kP(f2k,φ2k)≤limm→∞P(f,φ2m2m).Proof.For afixed k∈N.It is well known that if E⊂M is an(n, )separated set of f2k,then E is an(n2k, )separated set of f.By the definitionP(f2k,φ2k)=lim→∞lim supn→∞1nlog sup{x∈Eexp(ˆS nφ2k(x)):E is a(n, )separated set of f2k},where(ˆS nφ2k(x))=φ2k(x)+φ2k(f2k x)+···+φ2k(f(n−1)2k x).Hence for afixed m<k,let2k=2m q+r and C=max x∈M max i=1,···,2mφi(x),the subadditivity ofφn implies thatφ2k(x)≤12m2m−1j=0q−2i=0φ2m(f i2m+j(x))+12m2m−1j=0[φj(x)+φ2m−j+l(f(q−1)2m+j(x))]≤2k−1i=012mφ2m(f i(x))+4C.Thus for1≤j≤n−1,we haveφ2k(f2k j(x))≤2k−1i=012mφ2m(f i(f2k j(x))+4C.HenceˆSnφ2k(x)=φ2k(x)+φ2k(f2k x)+···+φ2k(f(n−1)2k x)≤n2k−1i=012mφ2m(f i(x))+4nC=S n2k(12mφ2m)(x)+4nC.It gives thatP n(f2k,φ2k, )≤P n2k(f,12mφ2m, )×e4nC.ThusP(f2k,φ2k)≤2k P(f,12mφ2m)+limn→∞1nlog e4nC=2k P(f,12mφ2m)+4C.Thereforelimk →∞12k P (f 2k,φ2k )≤P (f,12m φ2m )for all m ∈Z +.Hencelimk →∞12k P (f 2k,φ2k )≤lim m →∞P (f,12m φ2m ).Lemma 5.2.lim n →∞P (f,φ2k2k)≤P (f,F ).Proof.Since f :Λ→Λis expanding map,h µ(f )is an upper-semi continuous function from M (f |Λ)to R .From variational principal of topological pressure [16],we have that for every k ∈Z +there exists µ2k ∈M (f |Λ)such thatP (f |Λ,12k φ2k )=h µ2k (f )+Λ12k φ2k dµ2k .Since M (f |Λ)is compact,it implies that µ2k has a subsequence which convergence toµ∈M (f |Λ).Without loss of generality,suppose that µ2k convergence to µ.Using the subadditivity and invariant of µ2k ,then we have for every k ∈Nh µ2k (f )+Λφ2k (x )2k dµ2k ≤h µ2k (f )+ Λφ1(x )dµ2k .Furthermore for fixed s ∈N .If k >s ,from the subadditivity and invariance of µ2k ,ithash µ2k (f )+Λφ2k (x )2k dµ2k ≤h µ2k (f )+ Λφ2s (x )2s dµ2k .Since h µ(f )is a upper-semi continuous function,we havelim k →∞P (f,φ2k2k )=lim k →∞(h µ2k (f )+Λφ2k (x )2k dµ2k )≤lim k →∞(h µ2k (f )+ Λφ2s (x )2s dµ2k )≤h µ(f )+Λφ2s (x )2s dµ.Since sequence {Λφn (x )dµ}is sub-additive sequence,it haslimn →∞Λφn (x )n dµ=inf n ≥1{ Λφn (x )n dµ}.The arbitrariness of s ∈N implies thatlim k→∞P(f,φ2k2k)≤hµ(f)+lims→∞Λφ2s2s(x)dµ.Hence by variational principal of the sub-additive topological pressure in[5],we havelim k→∞P(f,φ2k2k)≤hµ(f)+lims→∞Λφ2s2s(x)dµ≤P(f,F).This completes the proof of lemma.Theorem5.3.The sequence{t n}is monotone,andlimn→∞t n=t∗where t∗is the unique root of equation P(f,−t{log m(Df n(x))})=0.Proof.First we prove that the sequence{t2n}is monotone decreasing.Letφn=−log m(Df n(x)).For afixed k∈N,P k(φ,, )=sup{x∈Eexp S nφ(x):E is a(n, )−separated subset of X}.For∀ >0,by the uniformly continuity of f,there existsδ>0such that if E⊂M is an(n, )separated set of f2k+1,then E is an(2n,δ)separated set of f2k andδ→0 when →ing the subadditivity ofφn,the Birkhoffsum S nφ2k+1ofφ2k+1with respect to f2k+1has the following property:S nφ2k+1(x)=φ2k+1(x)+φ2k+1(f2k+1x)+···+φ2k+1(f2k+1(n−1)x)≤φ2k(x)+φ2k(f2k x)+φ2k(f2k+1x)+φ2k(f2k+1f2k x)+···+φ2k(f2k+1(n−1)x)+φ2k(f2k+1(n−1)f2k x)=S2nφ2k(x)where S2nφ2k(x)is the Birkhoffsum ofφ2k with respect to f2k.ThusP n(f2k+1,φ2k+1, )≤P2n(f2k,φ2k,δ).HenceP(f2k+1,φ2k+1)≤2P(f2k,φ2k).Therefore if t2k+1is the unique root of Bowen’s equation P(tφ2k+1)=0,then we have0=P(f2k+1,t2k+1φ2k+1)≤2P(f2k,t2k+1φ2k).The monotone decreasing of the function P(f2k,tφ2k)implies that t2k≥t2k+1.The arbitrariness of k implies that the sequence{t2k}monotone decreasing.Hence limit exists and we denote the limit of this sequence by t.From the proof as above,we haveP(f2k+1,φ2k+1)2k+1≤P(f2k,φ2k)2k≤···≤P(f2,φ2)2≤P(f,φ).Next we prove thatP(f,F)≤1kP(f k,φk)∀k∈N.For afixed k∈N,let n=km+r,0≤r<k,and let C=max x∈M max1≤i≤kφi(x). For∀ >0,by the uniformly continuity of f,there existsδ>0such that if E⊂M is an(n, )separated set of f,then E is an(m,δ)separated set of f k andδ→0when →ing the subadditivity ofφn,we haveφn(x)≤φk(x)+φk(f k(x))+···+φk(f(m−1)k(x))+φr(f mk(x)).ThusP n(f,F, )≤P m(f k,φk,δ)×e C.HenceP(f,F, )≤1kP(f k,φk,δ).It gives thatP(f,F)≤1kP(f k,φk).ThereforeP(f,F)≤12kP(f2k,φ2k)∀k∈N.(5.3)Let t F={tφn(x)}.Then we haveP(f,t2k F)≤12kP(f2k,t2kφ2k)=0∀k∈N.Therefore the unique root t∗of the equationP(f,t F)=0satisfiest∗≤t2k∀k∈N.Thust∗≤t=limk→+∞t2k.Next we want to prove thatt≤t∗.From5.3and lemma5.1,5.2,we have the sequence{12k P(f2k,φ2k)}is monotone de-creasing and it converges to P(f,F).By the definition,it is easy to prove that0≤P(f2k,tφ2k)2k−P(f2k,t2kφ2k)2k≤|t−t2k|C,∀k∈N,where C=max x∈M|φ1(x)|.Let k→∞,we haveP(f,t F)=0.Hence it has,t=t∗.Theorem5.4.t∗=s∗.Proof.From theorems as above,we have functionsP(f,−t{log m(Df n(x))})andP(f,−t{log Df n(x) })coincide and both of them have unique zero points.Thereforet∗=s∗.The proof of main theorem:From Corollary2and theorems5.4as above,we havedim HΛ=dim BΛ=dim B=s∗=t∗.This completes the proof of main theorem.Acknowledgement.Author would like to thank Prof.Dejun Feng and Prof.Marcelo Viana for their discussions and suggestions.This work is partially supported by NSFC(10571130),NCET,and SRFDP of China.References[1]Barreira,L.:Dimension estimates in nonconformal hyperbolic dynamics.Non-linearity16(2003),no.5,1657–1672.[2]Barreira,L.:A non-additive thermodynamic formalism and applications to di-mension theory of hyperbolic dynamical systems Ergodic Theory Dyn.Syst.16(1996)871–927.[3]Bowen,R.:Hausdorffdimension of quasi-circles.Inst.Hautes´Etudes Sci.Publ.Math.50(1979)259–73.[4]Cao Yongluo,Dimension upper bounds estimate in non-conformal hyperbolicinvariant set.(2005)Preprint.[5]Cao Yongluo,Feng Dejun,Huang Wen:The thermodynamic formalism for sub-multiplicative potentials.(2004)Preprint.[6]Douady,A.,Oesterl,J.:Dimension de Hausdorffdes attracteurs.C.R.Acad.Sci.Paris290(1980)1135–8.[7]Falconer,K.:The Hausdorffdimension of self-affine fractals.Math.Proc.Camb.Phil.Soc.103(1988)339–50.[8]Falconer,K.:Dimensions and measures of quasi self-similar sets.Proc.Am.Math.Soc.106(1989)543–54.[9]Falconer,K.:Bounded distortion and dimension for non-conformal repellers.Math.Proc.Camb.Phil.Soc.115(1994)315–34.[10]Gatzouras,D.,Y.Peres,Y.:Invariant measures of full dimension for some ex-panding maps.Ergod.Th.&Dynam.Sys.17(1997),147–167.[11]Pesin,Y.:Dimension Theory in Dynamical Systems.Contemporary Views andApplication(Chicago,IL:University of Chicago Press),1997.[12]Hu Huyi:Dimensions of invariant sets of expanding mum.Math.Phys.176(1996),307-320.[13]Ruelle,D.:Repellers for real analytic maps.Ergodic Theory Dyn.Syst.2(1982)99–107.[14]Ruelle,D.:An inequality for the entropy of differential maps.Bol.Soc.Bras.DeMat.9(1978)83-87.[15]Simon,K.,Solomyak,B.:Hausdorffdimension for horseshoes in R3.ErgodicTheory Dyn.Syst.19(1999)1343–63.[16]Walters,P.:An introduction to ergodic theory.Berlin,Springer(1982).[17]Zhang Yingjie:Dynamical upper bounds for Hausdorffdimension of invariantsets.Ergodic Theory Dynam.Systems17(1997),no.3,739–756.。
低维拓扑简介
考研论坛»数学»低维拓扑knight51发表于2005-7-28 08:34低维拓扑<P>下面说说低维拓扑的内容：低维拓扑是微分拓扑的一部分，主要研究3，4维流形与纽结理论。
又叫几何拓扑。
主要以代数拓扑与微分拓扑为工具。
它与微分几何和动力系统关系密切。
国外搞这个方向的也几乎都搞微分几何和动力系统。
我国这个方向北大最牛，美国是伯克利和普林斯顿最牛。
比起代数几何来，它比较好入门。
初学者只需要代数拓扑，微分拓扑，黎曼几何的知识就行了。
美国这方面比较牛，几乎每个搞基础数学研究的都会低维拓扑。
</P><DIV class=postcolor>纠正一下上面的错误，美国也不是每个搞基础数的都精通低维拓扑，而是懂一些低维拓扑的知识。
如果入门后还想更加深入了解它，那还需要读一些双曲几何和拓扑动力系统的书。
</DIV><!-- THE POST --><!-- THE POST --><DIV class=postcolor>下面介绍一下这方面的牛人：Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplaybetween analysis, topology and geometry.Wall [8] goes on to describe Thurston's work in more detail:-The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).</DIV><!-- THE POST -->第2个牛人：Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaréconjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.Besides these results about closed simply connected four manifolds, Freedman also proved:(a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R.(b) There is a nonsmoothable closed four manifold.© The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:-My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.<DIV class=postcolor>介绍第3个牛人：Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case.In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:-Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold.Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-(1) Differential geometry of holomorphic vector bundles.(2) Applications of gauge theory to 4-manifold topology.and he relates his contribution to that of many others in the field.Donaldson's work in summed up by R Stern in [6]:-In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds</DIV><DIV class=postcolor>下面continue介绍第4个牛人：Robion Kirby。
单纯同伦算法的翼状伸延道路
單純同倫算法的翼狀伸延道路
王則柯
不識廬山真面目 只緣身在此山中。
2 0 0 1 2 −1
指的是它們在幾何上互不相交。 正則相處的 若干單形組成的集合 T 如果滿足下述條件, 就 叫 做 一 個單 純 複 合 形 (simplicial complex), 簡稱複形: 設 σ 是 T 的元素, 而 τ 是 σ 的面, 那麼 τ 也是 T 的元素。 複形中 維數最高的單形的維數, 就叫做複形的維數 (dimension)。 複 形是 一 個代 數 對 象, 其元 素 是 單 形。 單形作為一個幾何體是一個點集。 複形中全 體元素 (單形) 作為點集的併, 叫做該複形的 多面體 (polyhedron)。 反過來, 複形叫做它 的多面體的單純剖分。 所以, 做單純剖分, 就 是把空間 (多面體) 分割為幾何性質很好的單 形。 如果一個 p 維複形的每一個元素都是某 個 p 維元素的面, 就稱這個複形是齊次的 p 維複形。[0, 1] × Rn 的任何單純剖分, 都是一 種特別好的齊次的 n + 1 維複形。 對於齊次 複形, 我們約定單形專指它的最高維的元素, 這些單形的面和界面, 都直稱為複形的面和 界面。 這樣, [0, 1] × Rn 的任一單純剖分的 每個 ( n 維) 界面, 頂多是兩個 ( n + 1 維) 單形的面。 具體來說, [0, 1] × Rn 的單純剖 分的界面有兩種: 位於 [0, 1] × Rn 的邊界 {0, 1} × R n = {0} × R n ∪ {1} × R n 的 每個界面都只是一個單形的面, 其餘的每個 界面都恰好同時是一對單形的面 (參看 [7])。 這是 [0, 1] × Rn 的單純剖分的主要性質。 相 反, 如 果 在 一 個 p 維 複 形 中, 有 一 個 頂點 (0 維單形) 不是任何一個 p 維單形的 面, 就說這個複形是一個變維數複形。 現在我們證明, 上一節定義的映照 φ 的 零點集 φ−1 (0), 一定是一個複形的多面體:
Abel范畴上平衡对的若干注记
第38卷第4期2020年11月江苏师范大学学报(自然科学版)Journal of Jiangsu Normal University(Natural Science Edition)Vol38，No4Nov，2020文章编号：2095-4298(2020)04-0048-03Abel范畴上平衡对的若干注记何东林，李煜彥〔陇南师范高等专科学校数信学院，甘肃陇南742500)摘要：设犃是一个Abel范畴,(:r,y)是犃上的一个平衡对.利用同调代数的方法，研究平衡对(狓y)的若干性质和等价刻画，讨论与其相关的2个维数：狓分解维数(狓res.dim(U))和y余分解维数(y cores.dim(U))，其中U为犃中任意对象.证明了对于Abel范畴犃中的任意正合列(《)：0f M fN7T,如果()在函子Hom犃(狓，一)下正合且狓关于扩张封闭，那么以下说法成立：1)若M G狓,则狓res.dim(N)W狓res.dim(L)；2)若N G狓，则狓res.dim(L) W狓res.dim(M)+1;3)若L G狓且狓关于满同态的核封闭，则狓res.dim(M)=j c-res.dim(N).关键词：Abel范畴；平衡对;维数；拉回图中图分类号：O154文献标识码：A doi：103969/j issn2095-4298202004012Some notes of balanced pairs in Abel categoriesHeDonglin，LiYuyan(School of Mathematics&Information Sciences,Longnan Teachers College,Longnan742500, Gansu,China)Abstract:Let A be an Abel category,(狓，y)a balanced pair in犃.Using methods of homology algebras,some properties and equivalent characterizations of balanced pair(狓，y)are investigated in this paper,two dimensions c resolution dimension c-res.dim(犝)and y coresolution dimension y-cores.dim(犝)are discussed with U an arbitrary object of A.It is proved that for any exact sequence()：0f M f N f L f0of A,if()is exact under functors Hom A(c,―)and c is closed under extensions,then the following statements are held:1)if M G c,then c-res.dim(N)^c-res.dim(L)；2)if N G c,then c-res.dim(L)^c-res.dim(AM)+1；3)if L G c and c is closed under kernels of epimorphisms,then c-res.dim(Ad)=c-res.dim(N).Keywords：Abelcategory；balancedpair；dimension；pu l backdiagramHomotopy等价是同调代数理论研究的热点之一，许多学者先后对其进行了研究［1一5］.特别地, Chen［］引入了平衡对的概念，并研究了基于平衡对的Homotopy等价，作为应用，证明了在左Goren-stein环上，Gorenstein投射模的Homotopy范畴与Gorenstein内射模的Homotopy范畴之间存在一个三角等价.Li等［7］引入并讨论了由平衡对(c,y)导出的余挠理论，并证明当y的c分解维数有限时,y 的有界Homotopy范畴包含在c中.基于以上研究背景，本文将讨论平衡对(c,y)的若干性质和等价刻画，并进一步研究与其相关的c分解维数和y余分解维数，以及短正合列中各项的c分解维数与y 余分解维数之间的关系1基本知识和定义文中的A均指Abel范畴，子范畴均指A的关于同构和直和因子封闭的加法全子范畴.P(A)和1(A)分别表示A的所有投射对象和内射对象组成的子范畴.设c是A的一个子范畴，且XGc,M是A中任意对象,称同态aXfM是对象M的右c逼近旧，如果对任意同态p：X'fM(X'Gc),都存在同态7：XfX',使得Y=a.对偶地，可定义M的左c 逼近.如果A中每个对象都存在右c逼近，那么称子范畴c是反变有限的;如果A中每个对象都存在左c逼近,那么称子范畴c是共变有限的.设c是A的一个反变有限子范畴,y是A的一个共变有限子范畴，如果存在复形X2f X j f X l M f0(其中X i G c),且该复形在函子Hom A(c,—)下正合，则称该复形为对象M的一个c分解［7］，记作X°f M.如果存在复形0f M f犢0f 犢1f犢2f…(其中犢G y),且该复形在函子Hom A(—,y)下正合，则称该复形为对象M的一个y余分解［7］，记作M f Y°.记c-res dim(M)=收稿日期：2019-11-10基金项目：甘肃省高等学校创新基金项目(2020A-277),甘肃省高等学校创新能力提升项目(2019B-224) 作者简介：何东林，女，讲师，硕士，主要从事同调代数方面的研究.第4期何东林，等：Abel范畴上平衡对的若干注记49inf{n|存在Hom A(c,—)下正合的复形0f X“f…f X i f X0f M f0(其中&G c)},y-cores.dim(M) =inf{m|存在Hom A(—,y)下正合的复形0—M f 犢0f Y1f…f Y犿f0(其中0G y)},分别称为对象M的狓分解维数和y余分解维数.定义1[]设cy是Abel范畴A的子范畴，称(c,y)是一个平衡对，如果以下条件成立：1)狓在A中是反变有限的狔在A中是共变有限的；2) 对任意MGA,都存在Hom A(—,y)下正合的狓分解---X2f X1f X0f M f0，其中X2G c；3)对任意MG A,都存在Hom A(c,—)下正合的y余分解0fMfY0f Y1f Y2f…，其中0G y.例11)设Abel范畴A具有足够的投射对象和内射对象，则(犘(A),1(A))是一个平衡对.2)设R是一个环犘犘(犚)和P i(犚)分别表示所有纯投射模(即关于纯正合列投射的模)和纯内射模(即关于纯正合列内射的模)组成的左R模范畴的子范畴，则(P p(R)P i(R))是一个平衡对.3)设R是一个n-Gorenstein环，即R是双边Noether环并且双边自内射维数不超过某个非负整数”G p(R)和G i(R)分别表示所有Gorenstein投射模和Gorenstein内射模组成的左R模范畴的子范畴，则(G p(R)G i(R))是一个平衡对.引理1[7]设(c,y)是A上的一个平衡对M 和N是A中任意2个对象，且X°f M和N f Y°分别为M的c分解和N的y余分解，则Ext(M,N)=Ext y(M,N),其中Ext(M,N)=H,(Hom A(X°,N)),Ex t；(M,N)= H,(Hom A(M9Y°)).为了方便，不妨将Abel群Ext(M,N)和Ex t；(M,N)均记为Ex t；(M,N).对任意对象丁GA,如果Ex佇】(犜，犜)=0,则称犜是关自正交的.2主要结果定理1设(c,y)是A上的一个平衡对，则对任意K G c Pl y,都有Exe1(K,K)=0.证由引理1知，对任意对象A G A,X G c和Y G y,有Ex t；(X,A)=0=Ex i(A,Y).从而对任意K G c Q y和任意i>1,都有Ex i(K,K)=0,即ExL(K,K)=0.定理2设(c,y)是A上的一个平衡对，且丁是兴自正交的，则丁的任意直和因子也是兴自正交的.证设犜是犜的任意直和因子，且犜=犜。
gtm158 Field Theory - Steven Roman
158
Editorial Board S. Axler K.Ats in Mathematics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTI]ZARING. Introduction to Axiomatic Set Theory. 2nd ed. OXTOBY.Measure and Category. 2nd ed. SCHAEFER. Topological Vector Spaces. 2nd ed. HILTON/STAMMBACH. A Course in Homological Algebra. 2nd ed. MAC LANE. Categories for the Working Mathematician. 2nd ed. HUGHES/PIPER.Projective Planes. J.-P. SERRE. A Course in Arithmetic. TAKEUTI/ZARING. Axiomatic Set Theory. HUMPHREYS.Introduction to Lie Algebras and Representation Theory. COHEN.A Course in Simple Homotopy Theory. CONWAY.Functions of One Complex Variable I. 2nd ed. BEALS.Advanced Mathematical Analysis. ANDERSON/FULLER.Rings and Categories of Modules. 2nd ed. GOLUBITSKY/GUILLEMIN. Stable Mappings and Their Singularities. BERBERIAN.Lectures in Functional Analysis and Operator Theory. WINTER.The Structure of Fields. ROSENBLATr.Random Processes. 2nd ed. HALMOS.Measure Theory. HALMOS.A Hilbert Space Problem Book. 2nd ed. HUSEMOLLER.Fibre Bundles. 3rd ed. HUMPHREYS.Linear Algebraic Groups. BARNES/MACK.An Algebraic Introduction to Mathematical Logic. GREUB. Linear Algebra. 4th ed. HOLMES.Geometric Functional Analysis and Its Applications. HEWITT/STROMBERG. Real and Abstract Analysis. MANES.Algebraic Theories. KELLEY.General Topology. ZARISKI]SAMUEL. Commutative Algebra. Vol. I. ZARISKI/SAMUEL. Commutative Algebra. Vol. II. JACOBSON.Lectures in Abstract Algebra I. Basic Concepts. JACOBSON.Lectures in Abstract Algebra II. Linear Algebra. JACOBSON.Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. HIRSCH. Differential Topology. 34 35 36 37 38 39 40 41 SPITZER.Principles of Random Walk. 2nd ed. ALEXANDER/WERMER.Several Complex Variables and Banach Algebras. 3rd ed. KELLEY/NAMIOKA et al. Linear Topological Spaces. MONK. Mathematical Logic. GRAUERT/FRITZSCHE.Several Complex Variables. ARVESON.An Invitation to C*-Algebras. KEMENY/SNELL/KNAPP.Denumerable Markov Chains. 2nd ed. APOSTOL.Modular Functions and Dirichlet Series in Number Theory. 2nd ed. J.-P. SERRE. Linear Representations of Finite Groups. GILLMAN/JERISON. Rings of Continuous Functions. KENDIG.Elementary Algebraic Geometry. LOEVE.Probability Theory I. 4th ed. LOEVE.Probability Theory II. 4th ed. MOISE.Geometric Topology in Dimensions 2 and 3. SACHS/WU.General Relativity for Mathematicians. GRUENBERG/WEIR.Linear Geometry. 2nd ed. EDWARDS.Fermat's Last Theorem. KLINGENBERG. A Course in Differential Geometry. HARTSHORNE.Algebraic Geometry. MANIN.A Course in Mathematical Logic. GRAVER/WATKINS. Combinatorics with Emphasis on the Theory of Graphs. BROWN/PEARCY.Introduction to Operator Theory I: Elements of Functional Analysis. MASSEY.Algebraic Topology: An Introduction. CROWELL/FOX.Introduction to Knot Theory. KOBLITZ. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed. LANG.Cyclotomic Fields. ARNOLD.Mathematical Methods in Classical Mechanics. 2nd ed. WHITEHEAD.Elements of Homotopy Theory. KARGAPOLOV/MERIZJAKOV. Fundamentals of the Theory of Groups. BOLLOBAS.Graph Theory.
物质和辐射的Cardassian模型
物质和辐射的Cardassian模型孙昌波【摘要】研究了Cardassian宇宙的动力学.在物质优势时期,给出了几个典型模型的精确解.当同时考虑物质和辐射时,存在3个临界点,宇宙从辐射优势时期演化到Cardassian项优势时期.宇宙的命运将对应于3种吸引子:quintessence吸引子,big rip吸引子或者de Sitter吸引子.%In this paper, we investigate the cosmological dynamics of the Cardassian universe. We find exact solutions of some typical models for the matter dominated epoch. When both matter and radiation are included, there are three critical points and the universe evolves from the radiation dominated epoch to the Cardassian term dominated epoch. The fate of the universe corresponds to quintessence attractor, big rip attractor or de Sitter attractor.【期刊名称】《上海师范大学学报（自然科学版）》【年(卷),期】2008(037)001【总页数】6页(P40-45)【关键词】相空间方法;Cardassian模型;异宿轨线;临界点;吸引子【作者】孙昌波【作者单位】上海师范大学,天体物理中心,上海,200234【正文语种】中文【中图分类】P1421 IntroductionAstronomical observations from SNeIa, CMB anisotropy and galaxy power spectra depicted that our universe is spatially flat and currently undergoing accelerated cosmic expansion. Lots of explanations including a variety of dark energy models have been explored: a cosmological constant, Quintessence[1], Phantom[2], Chaplygin gas[3], and so on. An alternative explanation is a modification to the Friedmann eqution called Cardassian[4～8] in which the universe is flat, matter dominated at the current epoch, and accelerating. The advantage of this model is that it doesn′t have to introduce a new dark energy component. The Friedmann equation is modified to become(1)where ρ contains only matter and radiation, H is the Hubble param eter. The function g(ρ) returns to ρ at early epochs and takes a different form that drives an accelerated expansion at z～ O(1).The dynamical system analysis has been used for exploring the power law Cardassian model filled with only baryonic matter and the model with a scalar field component[8]. In this paper, we study the dynamics of three kinds of typical models (Power Law model, Modified Polytropic model and Exponential model) filled with both matter and radiation.2 Basic equations of Cardassian ModeUsing the fluid description of the Cardassian universe[4], the first law ofthermodynamics Td(sV) = d(ρV) + pdV, adiabatic expansion d(sV) = 0, mass conservation d(ρm V) = 0, and radiation conservation d(ργ V4 / 3) = 0, we get the total pressure in the universe(2)The Friedmann equation and the evolution equation of Hubble parameter H are(3)(4)The energy conservation equation is(5)We can show that Eq.(5) is equivalent to particle number conservation equation and radiation conservation equation. Eqs.(3)～(5) are the basic equations of the Cardassian model and the following analyses are based on them.3 The formalism in general caseThe total energy density g(ρ) includes two parts ρ′ = ρ+ f(ρ) where ρ is the normal energy density and f(ρ) is the Carda ssian term. Eqs.(2)～(4) can be written as(6)(7)(8)With defining(9)and the dimensionless variables x = y = κ2 = 8π G) and N = ln a, the basic equations can be reduced to a system of equations(10)In the following, we roughly give the analysis process of the Power Law Model(PL) and Modified Polytropic Model(MP) and only the results of Exponential model(Exp).4 Exact solutions and fits of current dataPower Law (PL) model is the simplest version of Cardassian[4]. At present time, its Friedmann equation and the evolution equation of Hubble parameter H are(11)(12)where b and n are constants, n < 2 / 3. Introducing the dimensionless variables x = and N = ln a, the basic equations can be reduced to anequation(13)This differential equation is integrable. From x = , we can get the exact solution(14)where Ωm,0 and a0 are present cosmic parameter for matter and present scale factor of the universe. The equation of state in PL is w≈n - 1. Current observation shows - 1.14 < w < - 0.93, so the range of n is- 0.14 < n < 0.07.(15)In a flat universe today, the relation between and n is(16)where is the redshift at the matter-Cardassian term equality epoch(Ωm = Ωcar ), Ωm,0 and Ωcar,0 are the cosmic parameters for matter and Cardassian t erm today. From current observation Ωm,0 = 0.27, Ωcar,0 = 0.73 and (15), we can get the range of(17)The MP model was suggested[4](18)where ρcar,eq is the energy density of Cardassian term at which Ωm =Ωcar . q and ν are constants, q > 0, ν > Sim ilarly, we get an exact solution(19)The equation of state of MP model is(20)which means ν ～1. The relation of , q and ν is(21)Table 1 lists some typical values of q, ρcar,eq and for a given ν = 1.2. Table 1 For a given ν = 1.2 in MP, the typical values of q,ρcar,eq and ρcrit (g · cm - 3)q z'eq3.18× 10 - 300.60.475 4.72× 10 - 300.80.3745.69× 10 - 301.00.3196.04× 10 - 301.10.2995 PL and MP filled with both matter and radiationFor PL, if the radiation can not be ignored, the autonomous system is(22)Table 2 The critical points and their properties of system (22) Critical pointsEigenvaluesStabilityEpochFate of the universe(0,1)12 , 4 - 4n unstableradiation dominated(1,0) - 12 , 3 - 3n unstablematter dominated(0,0)3n2 - 2 , 4n - 32 stableCardassian dominated n > 0, quintessence attractor n = 0, de Sitter attractor n < 0, big rip attractor In TABLE 2, we list the critical points and their properties of system (22). The evolution of cosmic parameters for matter, radiation and Cardassianterm is shown in Fig.1, and the phase graph is shown in Fig.2. We can see that the phase graph consists of a set of heteroclinic orbits which connect (0,1) with (0,0).For MP model, the autonomous system is(23)The critical points and their properties are contained in Table III. We plot the dynamical evolution of cosmic parameters for this model in Fig.3. Table 3 The critical points and their properties of system (23) Critical pointsEigenvaluesStabilityEpochFate of the universe(0,1)12 , 4qν unstable radiation dominated(1,0) - 12 , 3qν unstable matter dominated(0,0) - 12 - 3ν 2 , - 3ν 2 stableCardassian dominated ν > 1, quintessence attractor ν≈ 1,de Sitter attractor ν < 1, big rip attractor6 The dynamics of ExpThe Exp model is a new kind of Cardassian model[7]. Here we consider two typical models: Exp model I(24)and Exp model II(25)where ρcar,eq is a characteristic constant energy density and q and n are two dimensionless positive constants. Consider the case these models filled with only matter, we give the exact solution of Exp model I(26)The plot begins with Ωm,i = Ωγ ,i = 0.49995 and Ω car = 0.0001. We choose ν = 1.2and q = 0.6,0.7,0.8,0.9,1.0 respectively.Figure 3 Evolution of cosmic parameters for matter, radiation and Cardassian term of MP model. The heteroclinic orbits connect (0,1)with (0,0). We choose n = 0.6, Ω m,0 = 0.3.Figure 4 Phase graph of Exp model I for different initial x，y and z( = 1 - x2 - y2).Using the dynamical analysis method, we get the quite similar results to PL and MP models: there are also three critical points(0,1), (1,0)and (0,0). In the phase graph, a set of heteroclinic orbits connect (0,1) with (0,0), that means the universe evolves from the radiation dominated epoch to the Cardassian term dominated epoch. The fate of the universe corresponds to quintessence attractor, big rip attractor or de Sitter attractor. In Fig.4, we give the phase diagram of Exp I.7 ConclusionUsing the phase space method, one can get the main properties and the evolution track of a given cosmological model. In above sections, we give the dynamics of the Cardassian universe. In the case the models are filled with both matter and radiation, all of them have three critical points and the phase graph consists of a set of heteroclinic orbits. The universeevolves from the radiation dominated epoch to the Cardassian term dominated epoch. The fate of the universe corresponds to quintessence attractor, big rip attractor or de Sitter attractor. We also find that some of the models have exact solutions at the matter dominated epoch. References:[1] LI X Z, HAO J G, LIU D J. Quintessence with O(N) symmetry[J]. Class Quant Grav, 2002, 19: 6049-6058.[2] CALDWELL R R. A Phantom menace[J]. Phys Lett B, 2002, 545: 23-29.[3] LIU D J, LI X Z. CMBR constraint on a modified Chaplygin gas model[J]. Chin Phys Lett, 2005, 22: 1600-1603.[4] FREESE K, LEWIS M. Cardassian expansion: A Model in which the universe is flat, matter dominated, and accelerating[J]. Phys Lett B, 2002, 540: 1-8.[5] FREESE K. Cardassian Expansion: Dark Energy Density from Modified Friedmann Equations[J]. New Astron Rev, 2005, 49: 103-109.[6] WANG Y, FREESE K, GONDOLO P, et al. Future Type IA supernova data as tests of dark energy from modified Friedmann equations[J]. Astrophys J, 2003, 594: 25-32.[7] LIU D J, SUN C B, LI X Z. Exponential cardassian universe[J]. Phys Lett B, 2006, 634: 442-447.[8] SZYDLOWSKI M, CZAJA W. Modified Friedmann cosmologies-Theory and observations[J]. Annals Phys, 2005, 320: 261-281.。
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Homotopy field theory in dimension 3 and crossed group-categories Vladimir Turaev
Abstract
arXiv:math/0005291v1 [math.GT] 31 May 2000
A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group π , we introduce a notion of a modular crossed π -category and show that such a category gives rise to a 3-dimensional HQFT with target space K (π, 1). This includes numerical invariants of 3-dimensional π -manifolds and a 2-dimensional homotopy modular functor. We also introduce and discuss a parallel notion of a quasitriangular crossed Hopf π -coalgebra.
characterization of 2-dimensional HQFT’s whose target space is the EilenbergMacLane space K (π, 1) determined by a group π . In this paper we focus on 3-dimensional HQFT’s with target space K (π, 1). A manifold M endowed with a homotopy class of maps M → K (π, 1) is called a π -manifold. The homotopy classes of maps M → K (π, 1) classify principal π -bundles over M and (for connected M ) bijectively correspond to the homomorphisms π1 (M ) → π . A 3-dimensional HQFT with target space K (π, 1) comprises two ingredients: a homotopy modular functor assigning K -modules to π -surfaces and an invariant of 3-dimensional π -manifolds taking value in the module associated with the boundary. In particular, the HQFT provides numerical invariants of closed 3-dimensional π -manifolds. Our main aim is to introduce an algebraic technique allowing to construct 3-dimensional HQFT’s. Our approach is based on a deep connection between the theory of braided categories and invariants of knots, links and 3-manifolds. This connection has been essential in the construction of “quantum” invariants of knots and 3-manifolds from quantum groups, see [RT], [Tu2], [KRT]. Here we extend these ideas to links ℓ ⊂ S 3 endowed with homomorphisms π1 (S 3 \ℓ) → π and to 3-dimensional π manifolds. To this end we introduce a notion of a crossed π -category and study braidings and twists in such categories. This leads us to the notion of a modular crossed π -category. We show that each modular crossed π -category gives rise to a 3-dimensional HQFT with target K (π, 1). In the case π = 1 we recover the usual construction of 3-dimensional TQFT’s from modular categories, see [Tu2]. The crossed π -categories are quite delicate algebraic objects. We discuss a few general methods producing such categories. In particular we introduce quasitriangular Hopf π -coalgebras and show that they give rise to crossed π -categories. Other methods are based on a study of self-equivalences of a braided category, a study of quasi-abelian cohomology of π , and a transfer-type construction. This gives several examples of modular crossed π -categories. However, the problem of systematic finding of modular crossed π -categories is largely open. It would be most interesting to extend the quantum groups associated with semisimple finite dimensional complex Lie algebras to quasitriangular Hopf π -coalgebras. The content of the paper is as follows. In Sections 1 and 2 we introduce crossed π -categories and various additional structures on them (braiding, twist etc.). In Sections 3 and 4 we introduce π -links, π -tangles and π -graphs in R3 colored over a ribbon crossed π -category C . We also define their canonical functorial invariant taking values in C . In Section 5 we study traces of morphisms in C . In Section 6 we introduce modular crossed π -categories. They are used in Section 7 to define invariants of 3-dimensional π -manifolds. In Sections 8-10 we introduce the 2-dimensional and 3-dimensional HQFT’s derived from a modular crossed π -category. In Sections 11-13 we discuss algebraic constructions of crossed π -categories. The paper is ended with three appendices. In Appendix 1 we briefly discuss quasi-abelian 3-cohomology of groups and their relations to crossed categories. In Appendix 2 we outline a state sum approach to invariants of 3-dimensional π -manifolds. In Appendix 3 we discuss a few open problems. Throughout the paper, the symbol K denotes a commutative ring with unit. 2