THE STABLE MANIFOLD THEOREM FOR SEMI-LINEAR STOCHASTIC EVOLUTION EQUATIONS AND STOCHASTIC P

t distribution 学生t分布t number t数t statistic t统计量t test t检验t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间table 表table of random numbers 随机数表table of sines 正弦表table of square roots 平方根表table of values 值表tabular 表的tabular value 表值tabulate 制表tabulation 造表tabulator 制表机tacnode 互切点tag 标签tame 驯顺嵌入tame distribution 缓增广义函数tamely imbedded 驯顺嵌入tangency 接触tangent 正切tangent bundle 切丛tangent cone 切线锥面tangent curve 正切曲线tangent function 正切tangent line 切线tangent of an angle 角的正切tangent plane 切平面tangent plane method 切面法tangent surface 切曲面tangent vector 切向量tangent vector field 切向量场tangent vector space 切向量空间tangential approximation 切线逼近tangential component 切线分量tangential curve 正切曲线tangential equation 切线方程tangential stress 切向应力tangents method 切线法tape 纸带tape inscription 纸带记录tariff 税tautology 重言taylor circle 泰勒圆taylor expansion 泰勒展开taylor formula 泰勒公式taylor series 泰勒级数technics 技术technique 技术telegraph equation 电报方程teleparallelism 绝对平行性temperature 温度tempered distribution 缓增广义函数tend 倾向tendency 瞧tension 张力tensor 张量tensor algebra 张量代数tensor analysis 张量分析tensor bundle 张量丛tensor calculus 张量演算法tensor density 张量密度tensor differential equation 张量微分方程tensor field 张量场tensor form 张量形式tensor form of the first kind 第一张量形式tensor function 张量函数tensor of torsion 挠率张量tensor product 张量乘积tensor product functor 张量乘积函子tensor representation 张量表示tensor space 张量空间tensor subspace 张量子空间tensor surface 张量曲面tensorial multiplication 张量乘法term 项term of higher degree 高次项term of higher order 高次项term of series 级数的项terminability 有限性terminable 有限的terminal decision 最后判决terminal edge 终结边terminal point 终点terminal unit 级端设备terminal vertex 悬挂点terminate 终止terminating chain 可终止的链terminating continued fraction 有尽连分数terminating decimal 有尽小数termination 终止terminology 专门名词termwise 逐项的termwise addition 逐项加法termwise differentiation 逐项微分termwise integration 逐项积分ternary 三元的ternary connective 三元联结ternary form 三元形式ternary notation 三进制记数法ternary number system 三进制数系ternary operation 三项运算ternary relation 三项关系ternary representation og numbers 三进制记数法tertiary obstruction 第三障碍tesseral harmonic 田形函数tesseral legendre function 田形函数test 检验test for additivity 加性检验test for uniform convergence 一致收敛检验test function 测试函数test of dispersion 色散检验test of goodness of fit 拟合优度检验test of hypothesis 假设检验test of independence 独立性检验test of linearity 线性检验test of normality 正规性检验test point 测试点test routine 检验程序test statistic 检验统计量tetracyclic coordinates 四圆坐标tetrad 四元组tetragon 四角形tetragonal 正方的tetrahedral 四面角tetrahedral angle 四面角tetrahedral co ordinates 四面坐标tetrahedral group 四面体群tetrahedral surface 四面曲面tetrahedroid 四面体tetrahedron 四面形tetrahedron equation 四面体方程theorem 定理theorem for damping 阻尼定理theorem of alternative 择一定理theorem of identity for power series 幂级数的一致定理theorem of implicit functions 隐函数定理theorem of mean value 平均值定理theorem of principal axes 轴定理theorem of residues 残数定理theorem of riemann roch type 黎曼洛赫型定理theorem on embedding 嵌入定理theorems for limits 极限定理theoretical curve 理论曲线theoretical model 理论模型theory of automata 自动机理论theory of cardinals 基数论theory of complex multiplication 复数乘法论theory of complexity of computations 计算的复杂性理论theory of correlation 相关论theory of differential equations 微分方程论theory of dimensions 维数论theory of elementary divisors 初等因子理论theory of elementary particles 基本粒子论theory of equations 方程论theory of errors 误差论theory of estimation 估计论theory of functions 函数论theory of games 对策论theory of hyperbolic functions 双曲函数论theory of judgment 判断论theory of numbers 数论theory of ordinals 序数论theory of perturbations 摄动理论theory of probability 概率论theory of proportions 比例论theory of relativity 相对论theory of reliability 可靠性理论theory of representations 表示论theory of sets 集论theory of sheaves 层理论theory of singularities 奇点理论theory of testing 检验论theory of time series 时间序列论theory of transversals 横断线论theory of types 类型论thermal 热的thermodynamic 热力学的thermodynamics 热力学theta function 函数theta series 级数thick 厚的thickness 厚度thin 薄的thin set 薄集third boundary condition 第三边界条件third boundary value problem 第三边界值问题third fundamental form 第三基本形式third isomorphism theorem 第三同构定理third proportional 比例第三项third root 立方根thom class 汤姆类thom complex 汤姆复形three body problem 三体问题three dimensional 三维的three dimensional space 三维空间three dimensional torus 三维环面three eighths rule 八分之三法three faced 三面的three figur 三位的three place 三位的three point problem 三点问题three series theorem 三级数定理three sheeted 三叶的three sided 三面的three sigma rule 三规则three termed 三项的three valued 三值的three valued logic 三值逻辑three valued logic calculus 三值逻辑学threshold logic 阈逻辑time interval 时程time lag 时滞time series analysis 时序分析timesharing 分时toeplitz matrix 托普利兹矩阵tolerance 容许tolerance distribution 容许分布tolerance estimation 容许估计tolerance factor 容许因子tolerance level 耐受水平tolerance limit 容许界限tolerance region 容许区域top digit 最高位数字topological 拓扑的topological abelian group 拓扑阿贝耳群topological algebra 拓扑代数topological cell 拓扑胞腔topological circle 拓扑圆topological completeness 拓扑完备性topological complex 拓扑复形topological convergence 拓扑收敛topological dimension 拓扑维topological direct sum 拓扑直和topological dynamics 拓扑动力学topological embedding 拓扑嵌入topological field 拓扑域topological group 拓扑群topological homeomorphism 拓扑同胚topological index 拓扑指数topological invariant 拓扑不变量topological limit 拓扑极限topological linear space 拓扑线性空间topological manifold 拓扑廖topological mapping 拓扑同胚topological pair 拓扑偶topological polyhedron 曲多面体topological product 拓扑积topological residue class ring 拓扑剩余类环topological ring 拓扑环topological simplex 拓扑单形topological skew field 拓扑非交换域topological space 拓扑空间topological sphere 拓扑球面topological structure 拓扑结构topological sum 拓扑和topological type 拓扑型topologically complete set 拓扑完备集topologically complete space 拓扑完备空间topologically equivalent space 拓扑等价空间topologically nilpotent element 拓扑幂零元topologically ringed space 拓扑环式空间topologically solvable group 拓扑可解群topologico differential invariant 拓扑微分不变式topologize 拓扑化topology 拓扑topology of bounded convergence 有界收敛拓扑topology of compact convergence 紧收敛拓扑topology of uniform convergence 一致收敛拓扑toroid 超环面toroidal coordinates 圆环坐标toroidal function 圆环函数torque 转矩torsion 挠率torsion coefficient 挠系数torsion form 挠率形式torsion free group 非挠群torsion group 挠群torsion module 挠模torsion of a curve 曲线的挠率torsion product 挠积torsion subgroup 挠子群torsion tensor 挠率张量torsion vector 挠向量torsionfree connection 非挠联络torsionfree module 无挠模torsionfree ring 无挠环torus 环面torus function 圆环函数torus group 环面群torusknot 环面纽结total 总和total correlation 全相关total curvature 全曲率total degree 全次数total differential 全微分total differential equation 全微分方程total error 全误差total graph 全图total image 全象total inspection 全检查total instability 全不稳定性total inverse image 全逆象total matrix algebra 全阵环total matrix ring 全阵环total order 全序total predicate 全谓词total probability 总概率total probability formula 总概率公式total regression 总回归total relation 通用关系total space 全空间total stability 全稳定性total step iteration 整步迭代法total step method 整步迭代法total stiefel whitney class 全斯蒂费尔惠特尼类total subset 全子集total sum 总和total variation 全变差totally bounded set 准紧集totally bounded space 准紧空间totally differentiable 完全可微分的totally differentiable function 完全可微函数totally disconnected 完全不连通的totally disconnected graph 完全不连通图totally disconnected groupoid 完全不连通广群totally disconnected set 完全不连通集totally disconnected space 完全不连通空间totally geodesic 全测地的totally nonnegative matrix 全非负矩阵totally ordered group 全有序群totally ordered set 线性有序集totally positive 全正的totally positive matrix 全正矩阵totally quasi ordered set 完全拟有序集totally real field 全实域totally reflexive relation 完全自反关系totally regular matrix method 完全正则矩阵法totally singular subspace 全奇异子空间totally symmetric loop 完全对称圈totally symmetric quasigroup 完全对称拟群touch 相切tournament 竞赛图trace 迹trace form 迹型trace function 迹函数trace of dyadic 并向量的迹trace of matrix 矩阵的迹trace of tensor 张量的迹tracing point 追迹点track 轨迹tractrix 曳物线trajectory 轨道transcendence 超越性transcendence basis 超越基transcendence degree 超越次数transcendency 超越性transcendental element 超越元素transcendental equation 超越方程transcendental function 超越函数transcendental integral function 超越整函数transcendental number 超越数transcendental singularity 超越奇点transcendental surface 超越曲面transfer 转移transfer function 转移函数transfinite 超限的transfinite diameter 超限直径transfinite induction 超限归纳法transfinite number 超限序数transfinite ordinal 超限序数transform 变换transformation 变换transformation equation 变换方程transformation factor 变换因子transformation formulas of the coordinates 坐标的变换公式transformation function 变换函数transformation group 变换群transformation of air mass 气团变性transformation of coordinates 坐标的变换transformation of parameter 参数变换transformation of state 状态变换transformation of the variable 变量的更换transformation rules 变换规则transformation theory 变换论transformation to principal axes 轴变换transgression 超渡transient response 瞬态响应transient stability 瞬态稳定性transient state 瞬态transient time 过渡时间transition function 转移函数transition graph 转换图transition matrix 转移矩阵transition probability 转移函数transitive closure 传递闭包transitive graph 传递图transitive group of motions 可迁运动群transitive law 可迁律transitive permutation group 可迁置换群transitive relation 传递关系transitive set 可递集transitivity 可递性transitivity laws 可迁律translatable design 可旋转试验设计translate 转移translation 平移translation curve 平移曲线translation group 平移群translation invariant 平移不变的translation invariant metric 平移不变度量translation number 殆周期translation of axes 坐标轴的平移translation operator 平移算子translation surface 平移曲面translation symmetry 平移对称translation theorem 平移定理transmission channel 传输通道transmission ratio 传输比transport problem 运输问题transportation algorithm 运输算法transportation matrix 运输矩阵transportation network 运输网络transportation problem 运输问题transpose 转置transposed inverse matrix 转置逆矩阵transposed kernel 转置核transposed map 转置映射transposed matrix 转置阵transposition 对换transversal 横截矩阵胚transversal curve 横截曲线transversal field 模截场transversal lines 截线transversality 横截性transversality condition 横截条件transverse axis 横截轴transverse surface 横截曲面trapezium 不规则四边形trapezoid 不规则四边形trapezoid formula 梯形公式trapezoid method 梯形公式traveling salesman problem 转播塞尔斯曼问题tree 树trefoil 三叶形trefoil knot 三叶形纽结trend 瞧trend line 瞧直线triad 三元组trial 试验triangle 三角形triangle axiom 三角形公理triangle condition 三角形公理triangle inequality 三角形公理triangulable 可三角剖分的triangular decomposition 三角分解triangular form 三角型triangular matrix 三角形矩阵triangular number 三角数triangular prism 三棱柱triangular pyramid 四面形triangular surface 三角曲面triangulate 分成三角形triangulation 三角剖分triaxial 三轴的triaxial ellipsoid 三维椭面trichotomy 三分法trident of newton 牛顿三叉线tridiagonal matrix 三对角线矩阵tridimensional 三维的trigammafunction 三函数trigonometric 三角的trigonometric approximation polynomial 三角近似多项式trigonometric equation 三角方程trigonometric function 三角函数trigonometric moment problem 三角矩问题trigonometric polynomial 三角多项式trigonometric series 三角级数trigonometrical interpolation 三角内插法trigonometry 三角学trihedral 三面形的trihedral angle 三面角trihedron 三面体trilateral 三边的trilinear 三线的trilinear coordinates 三线坐标trilinear form 三线性形式trinomial 三项式;三项式的trinomial equation 三项方程triplanar point 三切面重点 ?triple 三元组triple curve 三重曲线triple integral 三重积分triple point 三重点triple product 纯量三重积triple product of vectors 向量三重积triple root 三重根triple series 三重级数triple tangent 三重切线triply orthogonal system 三重正交系triply tangent 三重切线的trirectangular spherical triangle 三直角球面三角形trisecant 三度割线trisect 把...三等分trisection 三等分trisection of an angle 角的三等分trisectrix 三等分角线trivalent map 三价地图trivector 三向量trivial 平凡的trivial character 单位特贞trivial cohomology functor 平凡上同弹子trivial extension 平凡扩张trivial fibre bundle 平凡纤维丛trivial graph 平凡图trivial homogeneous ideal 平凡齐次理想trivial knot 平凡纽结trivial solution 平凡解trivial subset 平凡子集trivial topology 密着拓扑trivial valuation 平凡赋值triviality 平凡性trivialization 平凡化trochoid 摆线trochoidal 余摆线的trochoidal curve 摆线true error 真误差true formula 真公式true proposition 真命题true sign 直符号true value 真值truncated cone 截锥truncated cylinder 截柱truncated distribution 截尾分布truncated pyramid 截棱锥truncated sample 截样本truncated sequence 截序列truncation 舍位truncation error 舍位误差truncation point 舍位点truth 真值truth function 真值函项truth matrix 真值表truth set 真值集合truth symbol 真符号truth table 真值表truthvalue 真值tube 管tubular knot 管状纽结tubular neighborhood 管状邻域tubular surface 管状曲面turbulence 湍流turbulent 湍聊turing computability 图灵机可计算性turing computable 图灵机可计算的turing machine 图录机turn 转向turning point 转向点twice 再次twice differentiable function 二次可微函数twin primes 素数对twisted curve 空间曲线twisted torus 挠环面two address 二地址的two address code 二地址代码two address instruction 二地址指令two body problem 二体问题two decision problem 二判定问题two digit 二位的two dimensional 二维的two dimensional laplace transformation 二重拉普拉斯变换two dimensional normal distribution 二元正态分布two dimensional quadric 二维二次曲面two dimensional vector space 二维向量空间two fold transitive group 双重可迁群two person game 两人对策two person zero sum game 二人零和对策two phase sampling 二相抽样法two place 二位的two point distribution 二点分布two point form 两点式two sample method 二样本法two sample problem 二样本问题two sample test 双样本检验two sheet 双叶的two sided condition 双边条件two sided decomposition 双边分解two sided divisor 双边因子two sided ideal 双边理想two sided inverse 双边逆元two sided module 双边模two sided neighborhood 双侧邻域two sided surface 双侧曲面two sided test 双侧检定two stage sampling 两阶段抽样法two termed expression 二项式two valued logic 二值逻辑two valued measure 二值测度two variable matrix 双变量矩阵two way array 二向分类two way classification 二向分类twopoint boundary value problem 两点边值问题type 型type problem 类型问题typenumber 型数typical mean 典型平均。

Log-Sobolev Ineqaulities:Different Roles ofRic and Hess ∗Feng-Yu WangDepartment of Mathematics,University of Wales Swansea,Singleton Park,SA28PP,UKEmail:F.Y.Wang@October 24,2007AbstractLet P t be the diffusion semigroup generated by L :=∆+∇V on a completeconnected Riemannian manifold with Ric ≥−(σ2ρ2o +c )for some constants σ,c >0and ρo the Riemannian distance to a fixed point.It is shown that P t is hypercontrac-tive,or the log-Sobolev inequality holds for the associated Dirichlet form,provided −Hess V ≥δholds outside of a compact set for some constant δ>(1+√2)σ√d −1.This indicates,at least in finite dimensions,that Ric and −Hess V play quite dif-ferent roles for the log-Sobolev inequality to hold.The supercontractivity and theultracontractivity are also studied.AMS subject Classification:60J60,58G32.Keywords:Log-Sobolev inequality,Ricci curvature,Riemannian manifold,diffusion semi-group.1IntroductionLet M be a d -dimensional completed connected non-compact Riemannian manifold and V ∈C 2(M )such that Z := M e V (x )d x <∞,where d x is the volume measure on M .Let µ(d x )=Z −1e V (x )d x.Then P t ,the semigroup of the diffusion process generated by L :=∆+∇V ,is symmetric in L 2(µ).It is well-known by the Bakry-Emery criterion that (see [4])(1.1)Ric −Hess V ≥K ∗Supported in part by NNSFC(10121101),the 973-Project in China and RFDP(20040027009).for some constant K>0implies the Gross log-Sobolev inequality[15](1.2)µ(f2log f2):=Mf2log f2dµ≤Cµ(|∇f|2),µ(f2)=1,f∈C1(M)for C=2/K.This result was extended by Chen and the author[10]to the situation that Ric−Hess V is uniformly positive outside a compact set.In the case that Ric−Hess V is bounded below,sufficient concentration conditions ofµfor(1.2)to hold are presented in[20,1,21].Obviously,in a condition on Ric−Hess V the Ricci curvature and−Hess V play the same role.What can we do when Ric−Hess V is unbounded below?It seems very hard to confirm the log-Sobolev inequality with an unbounded below condition of Ric−Hess V.So,in this paper we try to clarify the roles of Ric and−Hess V in the study of the log-Sobolev inequality.Let usfirst recall the gradient estimate of P t,which is a key point in the above references to prove the log-Sobolev inequality.Let x t be the L-diffusion process starting at x,and let v∈T x M.Due to Bismut[6] and Elworthy-Li[12],under a reasonable lower bound condition of Ric−Hess V,one has∇P t f,v =E ∇f(x t),v t ,t>0,f∈C1b(M),where v t∈T xtM solves the equationD t v t:=//−1t→0dd t//t→0v t=−(Ric−Hess V)#(v t)for//t→0:T xtM→T x M the associated stochastic parallel displacement,and(Ric−Hess V)#(v t)∈T xtM with(Ric−Hess V)#(v t),X :=(Ric−Hess V)(v t,X),X∈T xtM.Thus,for the gradient of P t,which is a short distance behavior of the diffusion process,a condition on Ric−Hess V appears naturally.On the other hand,however,Ric and−Hess V play very different roles for long distance behaviors.For instance,Letρo be the Riemannian distance function to afixed point o∈M.If Ric≥−k and−Hess V≥−δfor some k≥0,δ∈R,the Laplacian comparison theorem impliesLρo≤k(d−1)cothk/(d−1)ρo+δρo.So,for largeρo the Ric lower bound leads to a bounded term while that of−Hess V provides a linear term.The same phenomena appears to the formula on distance of coupling by parallel displacement(cf.[3,(2.3),(2.4)]),which implies the above Bismut-Elworthy-Li formula by letting the initial distance tend to zero(cf.[16]).Here,k≥0is essential for our framework since the manifold has to be compact if Ric is bounded below by a positive constant.Since the log-Sobolev inequality is always available on bounded regular domains,it is more likely a long distance behavior of the diffusion process.Thus,we would conclude that Ric and −Hess V take different roles to ensure the validity of the log-Sobolev inequality.Indeed,it has been observed by the author [21]that (1.2)holds for some C >0provided Ric is bounded below and −Hess V is uniformly positive outside a compact set.So,for the validity of the log-Sobolev inequality,the positivity of −Hess V is a dominative condition,which allows the Ricci curvature has arbitrary negative constant lower bound,and hence allows Ric −Hess V to be globally negative.The first aim of this paper is to search for the weakest possibility of curvature lower bound for the log-Sobolev inequality to hold under the condition(1.3)−Hess V ≥δoutside a compact setfor some constant δ>0.This condition is reasonable as the log-Sobolev inequality implies µ(e λρ2o )<∞for some λ>0,see e.g.[2,18].According to the following Theorem 1.1and Example 1.1,we conclude that under (1.3)the optimal curvature lower bound condition for (1.2)to hold is(1.4)Ric ≥−(c +σ2ρ2o )for some constants c,σ>0.More precisely,let θ0>0be the smallest positive constant such that for any connected complete non-compact Riemannian manifold M and V ∈C 2(M ),the conditions (1.3)and (1.4)with δ>σθ0√d −1implies Z <∞and (1.2)for some C >0.Due to Theorem 1.1and Example 1.1below,we conclude thatθ0∈[1,1+√2].The exact value of θ0is however unknown.Recall that P t is called supercontractive if P t 2→4<∞for all t >0while ultracon-tractive if P t 2→∞<∞for all t >0(see [11]).In the present framework these two properties are stronger than the hypercontractivity: P t 2→4≤1for some t >0,which is equivalent to (1.2)due to Gross [15].Theorem 1.1.Assume that (1.3)and (1.4)hold for some constants c,δ,σ>0with δ>(1+√2)σ√d −1.Then (1.2)holds for some C >0.Further more,P t is supercontractive if and only if µ(exp[λρ2o ])<∞for all λ>0,while it is ultracontractive if and only if P t exp[λρ2o ] ∞<∞for all t,λ>0.Example 1.1.Let M =R 2be equipped with the rotationally symmetric metricd s 2=d r 2+ re kr 2 2d θ2under the polar coordinates (r,θ)∈[0,∞)×S 1at 0,where k >0is a constant.Then (see e.g.[14])Ric =−d 2d r2(r e kr 2)r e kr =−4k −4k 2r 2.Thus,(1.4)holds for σ=2k.Next,take V =−δ2ρ2o .By the Hessian comparison theorem and the negativity of the sectional curvature,we obtain (1.3).Since d =2ande V (x )d x =r e kr 2−δr 2/2d r d θ,one has Z <∞if and only if δ>2k =σ√d −1.Therefore,θ0≥1.Following the line of [20,21],the key point in the proof of Theorem 1.1will be a proper Harnack inequality of type(P t f (x ))α≤C α(t,x,y )P t f α(y ),t >0,x,y ∈M for any nonnegative f ∈C b (M ),where α>1is a constant and C α∈C ((0,∞),M 2)is a positive function.Such an inequality was established in [20]for Ric −Hess V bounded below and extended in [3]to a more general situation with Ric satisfying (1.4).The Harnack inequality presented in [3]contains a leading term exp[ρ(x,y )4],which is however too large to be integrability w.r.t.µ×µunder our conditions.So,to prove Theorem 1.1,we shall present a sharper Harnack inequality in Section 3by refining the coupling method introduced in [3](see Proposition 3.1below).This inequality,together with the concentration of µensured by (1.3)and (1.4),will imply the hypercontractivity of P t .To establish this new Harnack inequality,some necessary preparations are presented in Section 2.Finally,Theorem 1.1is extended in Section 4to the case that −Hess V ≥Φ◦ρo outside a compact set for an increasing positive function Φwith Φ(r )↑∞as r ↑∞.2PreparationsWe first study the concentration of µby using (1.3)and (1.4),for which we need to estimate Lρo from above according to [5]and references within.Lemma 2.1.If (1.3)and (1.4)hold then there exists a constant C 1>0such that(2.1)Lρ2o≤C 1(1+ρo )−2 δ−σ√d −1 ρ2oholds outside cut(o ),the cut-locus of o .If moreover δ>σ√d −1then Z <∞and µ(e λρ2o )<∞for all λ<12(δ−σ√d −1).Proof.By (1.4)and the Laplacian comparison theorem,∆ρo ≤ (c +σ2ρ2o )(d −1)coth(c +σ2ρ2o )/(d −1)ρo holds outside cut(o ).Thus,outside cut(o )one has∆ρ2o ≤2ρo (c +σ2ρ2o )(d −1)coth (c +σ2ρ2o )/(d −1)ρo +2≤2d +2ρo (c +σ2ρ2o )(d −1),(2.2)where the second inequality follows from the fact thatr cosh r ≤(1+r )sinh r,r ≥0.On the other hand,for x /∈cut(o )and U the unit tangent vector along the unique minimal geodesic form o to x ,by (1.3)there exists a constant c 1>0independent of x such that∇V,∇ρo (x )= ∇V,U (o )+ρo (x )0Hess V (U,U )( s )d s ≤c 1−δρo (x ).Combining this with (2.2)we prove (2.1).Finally,let δ>σ√d −1and 0<λ<12(δ−σ√d −1).By (2.1)we haveL e λρ2o ≤λe λρ2o C 1(1+ρo )−2 δ−σ√d −1 ρ2o +4λρ2o≤c 2−c 3ρ2o eλρ2o for some constants c 2,c 3>0.By [5,Proposition 3.2],this implies Z <∞andMρ2o e λρ2o d µ≤c 2c 3<∞.Lemma 2.2.Let x t be the L -diffusion process with x 0=x ∈M.If (1.3)and (1.4)hold with δ>σ√d −1,then for any δ0∈(σ√d −1,δ)there exists a constant C 2>0such thatE exp (δ0−σ√d −1)24 T 0ρo (x t )2d t ≤exp C 2T +14(δ0−σ√d −1)ρo (x )2 ,T >0,x ∈M.Proof.By Lemma 2.1,we haveLρ2o ≤C −2(δ0−σ√d −1)ρ2o outside cut(o )for some constant C >0.Then the Itˆo formula for ρo (x t )due to Kendall[17]implies that(2.3)d ρ2o (x t )≤2√2ρo (x t )d b t + C −2(δ0−σ√ρ2o (x t ) d t holds for some Brownian motion b t on R.This implies that the L-diffusion process is non-explosive so thatT n:=inf{t≥0:ρo(x t)≥n}→∞as n→∞.For anyλ>0and n≥1,it follows from(2.3)thatE exp2λδ0−σ√d−1T∧T nρ2o(x t)d t≤eλρ2o(x)+CλT E exp2√2λT∧T nρo(x t)d b t≤eλρ2o(x)+CλTE exp16λ2T∧T nρ2o(x t)d t1/2.Thus,takingλ=18(δ0−σ√d−1)we obtainE exp14δ0−σ√d−12 T∧T nρ2o(x t)d t≤exp14δ0−σ√d−1ρ2o(x)+C2Tfor some C2>0.Then the proof is completed by letting n→∞.Finally,we recall the coupling argument introduced in[3]for establishing the Harnack inequality of P t.Let T>0and x=y∈M befixed.Then the L-diffusion process starting from x can be constructed by solving the following Itˆo stochastic differential equationd I x t=√2Φt d B t+∇V(x t)d t,x0=x,where d I is the Itˆo differential on manifolds introduced in[13](see also[3]),B t is the d-dimensional Brownian motion,andΦt is the horizontal lift of x t onto the orthonormal frame bundle O(M).To construct another diffusion process y t starting from y such that x T=y T,as in[3] we add an additional drift term to the equation(as explained in[3,Section3],we may and do assume that the cut-locus of M is empty):d I y t=√2P xt,y tΦt d B t+∇V(y t)d t+ξt U(x t,y t)1{t<τ}d t,y0=y,where P xt,y t is the parallel transformation along the unique minimal geodesic from x t toy t,U(x t,y t)is the unit tangent vector of at y t,ξt≥0is a smooth function of X t to be determined,andτ:=inf {t ≥0:x t =y t }is the coupling time.Since all terms involved in the equation are regular enough,there exists a unique solution y t .Furthermore,since the additional term containing 1{t<τ}vanishes from the coupling time on,one has x t =y t for t ≥τdue to the uniqueness of solutions.Lemma 2.3.Assume that (1.3)and (1.4)hold with δ≥2σ√d −1.Then there existsa constant C 3>0independent of x,y and T such that x T =y T holds for ξt :=C 3+2σ√d −1ρo (x t )+ρ(x,y )T.Proof.According to Section 2in [3],we have(2.4)d ρ(x t ,y t )= I (x t ,y t )+ ∇V,∇ρ(·,y t ) (x t )+ ∇V,∇ρ(x t ,·) (y t )−ξt d t,t <τ,whereI Z (x t ,y t )=d −1i =1 ρ(x t ,y t )0 |∇U J i |2− R (U,J i )U,J i ( s )d s for R the Riemann curvature tensor,U the unit tangent vector of the minimal geodesic:[0,ρ(x t ,y t )]→M from x t to y t ,and {J i }d −1i =1the Jacobi fields along which,togetherwith U ,consist of an orthonormal basis of the tangent space at x t and y t and satisfyJ i (y t )=P x t ,y t J i (x t ),i =1,...,d −1.LettingK (x t ,y t )=sup {c +σ2ρ2o },we obtain from (1.4)and [22,Theorem 2.14](see also [9,8])that(2.5)I (x t ,y t )≤2 K (x t ,y t )(d −1)tanh ρ(x t ,y t )2 K (x t ,y t )/(d −1) .Moreover,(1.3)implies∇V,∇ρ(·,y t ) (x t )+ ∇V,∇ρ(x t ,·) (y t )= ρ(x t ,y t )Hess V (U,U )( s )d s ≤c 1−δρ(x t ,y t )for some constant c 1>bining this with (2.4),(2.5)and ξt =C 3+2σ√d −1ρo (x t )+ρ(x,y )T,we arrive at d ρ(x t ,y t )≤ 2 K (x t ,y t )(d −1)+c 1−δρ(x t ,y t )−C 3−2σ√d −1ρo (x t )−ρ(x,y )Td t for t <τ.Noting that K (x t ,y t )≤ c +σ2ρo (x t )+ρ(x t ,y t ) 2 1/2≤√c +σ ρo (x t )+ρ(x t ,y t ) and δ≥2σ√d −1,one has2 K (x t ,y t )(d −1)−δρ(x t ,y t )−2σ√d −1ρo (x t )≤2 c (d −1).Thus,when C 3≥c 1+2 c (d −1)we haved ρ(x t ,y t )≤−ρ(x t ,y t )T,t <τ.This implies that τ≤T and hence,x T =y T .3Harnack inequality and proof of Theorem 1.1We first prove the following Harnack inequality using results in Section 2.Proposition 3.1.Assume that (1.3)and (1.4)hold with δ>(1+√2)σ√d −1.Thenthere exist C >0and α>1such that(3.1)(P T f (y ))α≤(P T f α(x ))exp CT ρ(x,y )2+C (T +ρo (x )2) holds for all x,y ∈M,T >0and nonnegative f ∈C b (M ).Proof.According to Lemma 2.3,we takeξt =C 3+2σ√d −1ρo (x t )+ρ(x,y )Tsuch that τ≤T and x T =y T .LetR =exp 1√2 τ0 P x t ,y t Φt d B t ,ξt U (x t ,y t ) −14 τ0ξ2t d t .By the Girsanov theorem,the H¨o lder inequality,and noting that x T =y T ,we have P T f (y )=E [f (y T )R ]=E [f (x T )R ]≤(P T f α(x ))1/α(E R α/(α−1))(α−1)/α.That is,(3.2)(P T f (y ))α≤(P T f α(x ))(E R α/(α−1))α−1.Since for any exponential integrable martingale M t and any β>1one has(3.3)E e βM t −β2 M t =E e βM t −β2p 2 M t ·e β(βp −1)2 M t ≤ e βp (βp −1)2(p −1)M t (p −1)/p ,p >1,by taking β=α/(α−1)we obtain(3.4)(E R α/(α−1))α−1≤ E exp pα(pα−α+1)8(p −1)(α−1)2 T 0ξ2t d t (α−1)(p −1)/p ,p >1.Since δ>(1+√2)σ√d −1,we may take δ0∈((1+√2)σ√d −1,δ),small ε >0andlarge C 4>0,independent of T,x and y ,such thatξ2t = C 3+2σ√o (x t )+ρ(x,y )T 2≤(1−ε ) C 4+C 4ρ(x,y )2T 2+2(δ0−σ√d −1)2ρo (x t )2 holds.Moreover,since(3.5)lim p ↓1lim α↑∞pα(pα−α+1)8(p −1)(α−1)2=18,there exist p,α>1such thatpα(pα−α+1)8(p −1)(α−1)2 T 0ξ2t d t ≤C 4T +C 4ρ(x,y )2T +(δ0−σ√d −1)24 T 0ρo (x t )2d bining this with (3.4)and Lemma 2.2,we obtain (E R α/(α−1))α−1≤exp C 5T +C 5ρ(x,y )T+C 5ρo (x )2 ,T >0,x ∈M for some constant C 5>0.This completes the proof by taking large enough T >0such that C 5≤T ε.Proof of Theorem 1.1.By Proposition 3.1,let α>1and C >0such that (3.1)holds.Since δ>σ√d −1,we may take T >0such that C T ≤ε:=18δ−σ√d −1 .Then for any nonnegative f ∈C b (M )with µ(f α)=1,it follows from (3.1)that1= M P T f α(x )µ(d x )≥(P T f (y ))α M e −ερ(x,y )2−C (1+ρo (x )2)µ(d x )≥(P T f (y ))α {ρo ≤1}e −ε(1+ρo (y ))2−2C µ(d x )≥ε (P T f (y ))αexp[−2ερo (y )2],y ∈Mfor some constant ε >0.Thus,M(P T f (y ))2αµ(d y )≤1ε M e 4ερo (y )2µ(d y )<∞according to Lemma 2.1.This implies thatP T L α(µ)→L 2α(µ)<∞.Therefore,the log-Sobolev inequality (1.2)holds for some constant C >0due to the uniformly positively improving property of P t (see the proof of [21,Theorem 1.1]and[1]).Finally,due to the Harnack inequality (3.1),the assertions on supercontactivity and ultracontractivity follow from the proof of [19,Theorem 2.3]4Extension to general negative curvature lower bounds In this section we work with a stronger lower bound condition on −Hess V :(4.1)−Hess V ≥Φ◦ρo holds outside a compact subset of M,for a positive increasing function Φwith Φ(r )↑∞as r ↑∞.We aim to search for reasonable conditions on positive increasing function Ψsuch that(4.2)Ric ≥−Ψ◦ρoimplies the supercontractivity and/or ultracontractivity.Theorem 4.1.If (4.2)and (4.1)hold for some increasing positive functions Φand Ψsuch that(4.3)lim r →∞Φ(r )=lim r →∞( r 0Φ(s )d s )2Φ(r )=∞,(4.4) Ψ(r +t )(d −1)≤θ rΦ(s )d s +12 t/20Φ(s )d s +C,r,t ≥0for some constants θ∈(0,1/(1+√2))and C >0.Then P t is supercontractive.If furthermore (4.5)∞1d s √s √r0Φ(u )d u<∞,then P t is ultracontractive.More precisely,forΓ1(r ):=1√r√r 0Φ(s )d s,Γ2(r ):=∞rd s√s√sΦ(u )d u,r >0,(4.5)implies(4.6) P t 2→∞≤expc +c t1+Γ−11(c/t )+Γ−12(t/c ) <∞,t >0for some constant c >0andΓ−11(s ):=inf {t ≥0:Γ1(t )≥s },s ≥0.Proof.(a)Replacing c +ρ2o by Ψ◦ρo and noting that Hess V ≤−Φ◦ρo for large ρo ,theproof of Lemma 2.1implies (4.7)Lρ2o ≤c 1(1+ρo )−2ρoρoΦ(s )d s −Ψ◦ρo (d −1)for some constant c 1>bining this with (4.4)and noting that 1ρoρoΦ(s )d s →∞as ρo →∞,we conclude that for any λ>0L e λρ2o ≤C −2λρo √21+√2e λρ2o ρo0Φ(s )d s +4λ2ρ2oe λρ2o ≤C +C (λ)−λρo e λρ2o ρoΦ(s )d s,(4.8)where C >0is a universal constant andC (λ):=sup r>0r e λr 24λ2r −λ(1+√2)2 rΦ(s )d s=sup r 2≤Γ−114(1+√2)2λ r e λr2 4λ2r −λ(1+√2)2 r0Φ(s )d s ≤4λ2Γ−11 4(1+√2)2λ exp λΓ−11 4(1+√2)2λ≤exp 4λ+2λΓ−11 4(1+√2)2λ <∞.(4.9)Therefore,Z <∞and (4.10)µ(e λρ2o )<∞,λ>0.(b)By (4.4),(4.7)and Kendall’s Itˆo formula [17]as in the proof of Lemma 2.2,we haved ρ2o (x t )≤2√o (x t )d b t + C 1−2√2ρo (x t)(1+ε)1+√2 ρo (x t )Φ(s )d s d t for some constants ε,C 1>0,where x t and b t are in the proof of Lemma 2.2.Let(4.11)ϕ(r )=r0d s√s√sΦ(u )d u,r ≥0.We arrive atd ϕ◦ρ2o (x t )≤2√2ρo (x t )ϕ ◦ρ2o (x t )d b t +4ρ2o (x t )ϕ ◦ρ2o (x t )d t+ϕ ◦ρ2o (x t ) C 1−2√2ρo (x t)(1+ε)1+√2 ρo (x t )0Φ(s )d s d t.From (4.3)we see thatρo ϕ ◦ρ2o ϕ ◦ρ2o ρo 0Φ(s )d s≤Φ◦ρo 2( ρo 0Φ(s )d s )2which goes to zero as ρo →∞.Then there exists a constant C 2>C 1such thatd ϕ◦ρ2o (x t )≤2√2ρo (x t )Φ(s )d sd b t +C 2d t −2√21+√2 ρo (x t )0Φ(s )d s 2d t.This implies that for any λ>0E exp2√2λ1+√2 T 0 ρo (x t )Φ(s )d s 2d t ≤e C 2λT +λϕ◦ρ2o (x )E exp 2√2λ T 0ρo (x t )0Φ(s )d s d b t ≤eC 2λT +λϕ◦ρ2o (x )E exp 16λ2Tρo (x t )Φ(s )d s 2d t 1/2.Takingλ=√28(1+√2),we arrive at(4.12)E exp12(1+√2)2Tρo(x t)Φ(s)d s2d t≤e2C2T+ϕ◦ρ2o(x)√2/8(1+√2).(c)Letγ:[0,ρ(x t,y t)]→M be the minimal geodesic from x t to y t,and U its tangent unit vector.By(4.2)we have∇V,∇ρ(·,y t) (x t)+ ∇V,∇ρ(x t,·) (y t)=ρ(x t,y t)0Hess V(U s,U s)d s≤−ρ(x t,y t)/2Φ(s)d s.(4.13)To understand the last inequality,we assume,for instance,thatρo(x t)≥ρo(y t)so that by the triangle inequality,ρo(γs)≥ρo(x t)−s≥ρ(x t,y t)/2−s,s∈[0,ρ(x t,y t)/2].For the coupling constructed in Section3,one concludes from(4.13)and the proof of Lemma2.3that(4.14)dρ(x t,y t)≤2K(x t,y t)(d−1)+C3−ρ(x t,y t)/2Φ(s)d s−ξtd t,t<τholds for some constant C4>0,whereK(x t,y t):=supΨ◦ρo≤Ψ(ρo(x t)+ρ(x t,y t))and is the minimal geodesic from x t to y bining this(4.4)and(4.14),we obtaindρ(x t,y t)≤C4+2θρo(x t)Φ(s)d s−ξtd t,t<τ.So,takingξt=C4+2θρo(x t)0Φ(s)d s+ρ(x,y)T,we arrive atdρ(x t,y t)≤−ρ(x,y)Td t,t<τ.This impliesτ≤T and hence x T=y T a.s.Combining(4.4)with(3.4)and(3.5)we conclude that for the present choice ofξt there existα,p,C5>1such thatE Rα/(α−1) p/(p−1)≤E exp12(1+√2)2Tρ(x t)Φ(s)d s2d t+C5T+C5Tρ(x,y)2.Combining this with(4.12)and(3.2)we obtain(4.15)(P T f(y))α≤(P T fα(x))expCT+CTρ(x,y)2+Cϕ◦ρ2(x)holds for someα,C>1,any positive f∈C b(M)and all x,y∈M,T>0.(d)For any positive f∈C b(M)withµ(fα)=1,(4.15)implies that(P T f(y))αB(o,1)exp−CT−CTρ(x,y)2−Cϕ2(x)µ(d x)≤1.Therefore,there exists a constant C >0such that(4.16)(P T f(y))α≤expC (1+T)+CTρ(y)2,y∈M,T>0.Combining this with(4.10)we obtainP T α→pα<∞,T>0,p>1.This is equivalent to the supercontactivity by the Riesz-Thorin interpolation theorem and P t 1→1=1.Thus,thefirst assertion holds.(e)To prove(4.6),it suffices to consider t∈(0,1]since P t 2→∞is decreasing in t>0. So,below we assume that T≤1.By(4.16)and the fact that(P2T f)α≤P T(P T f)α,we have(4.17) P2T α→∞≤ P T e2C ρ2o/T ∞e C (1+T),T>0.So,by the Riesz-Thorin interpolation theorem and P t 1→1=1,for the ultracontractivity it suffices to show that(4.18) P T eλρ2o ∞<∞,λ,T>0.SinceΦis increasing,it is easy to check thatη(r):=√r√rΦ(s)d s,r≥0is convex,and so is s→sη(log sλ)forλ>0.Thus,it follows from(4.8)and the Jenseninequality thathλ,x(t):=E eλρ2o(x t)<∞,x0=x∈M,λ,t>0andd+d thλ,x(t)≤C+C(λ)−λhλ,x(t)η(λ−1log hλ,x(t)),t>0.This implies(4.18)provided(4.5)holds.This can be done by considering the following two situations.(1)Since hλ,x(t)is decreasing providedλhλ,x(t)η(λ−1log hλ,x(t))>C+C(λ),ifλhλ,x(0)η(λ−1log hλ,x(0))≤2C+2C(λ)thenhλ,x(t)≤sup{r≥1:λrη(λ−1log r)≤2C+2C(λ)}≤1λ(2C+2C(λ))+Cfor some constant C >0.(2)Ifλhλ,x(0)η(λ−1log hλ,x(0))>2C+2C(λ),then hλ,x(t)is decreasing in t up totλ:=inf{t≥0:λhλ,x(t)η(λ−1log hλ,x(t))≤2C+2C(λ)}.Indeed,d+ d t hλ,x(t)≤−λ2hλ,x(t)η(λ−1log hλ,x(t)),t≤tλ.Thus,∞hλ,x(T∧tλ)d rrη(λ−1log r)≥λ2(T∧tλ).This is equivalent toΓ2(λ−1log hλ,x(T∧tλ))≥12(T∧tλ).Hence,hλ,x(T∧tλ)≤expλΓ−1212(T∧tλ).Since it is reduced to case(1)if T>tλby regarding tλas the initial time,in conclusion we havesup x∈M hλ,x(T)≤maxexpλΓ−12T/2,C +1λ(2C+2C(λ)).Therefore,(4.6)follows from(4.17),(4.9)withλ=2C /T,and the Riesz interpolation theorem.Finally,we note that a simple example for conditions in Theorem4.1to hold isΦ(s)=sα−1,Ψ(s)=εs2αforα>1and small enoughε>0.In this case P t is ultracontractive withP t 2→∞≤exp[c(1+t−(α+1)/(α−1))],t>0for some c>0.References[1]S.Aida,Uniformly positivity improving property,Sobolev inequalities and spectralgap,J.Funct.Anal.158(1998),152–185.[2]S.Aida,T.Masuda and I.Shigekawa,Logarithmic Sobolev inequalities and exponen-tial integrability,J.Funct.Anal.126(1994),83–101.[3]M.Arnaudon,A.Thalmaier,F.-Y.Wang,Harnack inequality and heat kernel esti-mates on manifolds with curvature unbounded below,Bull.Sci.Math.130(2006),no.3,223–233.[4]D.Bakry,D.and M.Emery,Hypercontractivit´e de semi-groupes de diffusion,C.R.Acad.Sci.Paris.S´e r.I Math.299(1984),775–778.[5]V.I.Bogachev,M.R¨o ckner and Feng-Yu Wang,Elliptic equations for invariant mea-sures onfinite and infinite dimensional manifolds,J.Math.Pures Appl.80:2(2001), 177–221.[6]J.M.Bismut,Large Deviations and the Malliavin Calculus,Boston:Birkh¨a user,MA,1984.[7]J.Cheeger,M.Gromov and M.Taylor,Finite propagation speed,Laplace operator,and geometry of complete Riemannian manifolds,J.Diff.Geom.17(1982),15–53.[8]Mu-Fa Chen and F.-Y.Wang,Application of coupling method to thefirst eigenvalueon manifold,Sci.Sin.(A),1994,37(1):1–14.[9]M.Cranston,Gradient estimates on manifolds using coupling,J.Funct.Anal.99(1991),110–124.[10]M.-F.Chen and F.-Y.Wang,Estimates of logarithmic Sobolev constant:an improve-ment of Bakry-Emery criterion,J.Funct.Anal.144(1997),287–300.[11]E.B.Davies and B.Simon,Ultracontractivity and heat kernel for Schr¨o dinger oper-ators and Dirichlet Laplacians,J.Funct.Anal.59(1984),335–395. [12]K.D.Elworthy and X.-M.Li,Formulae for the derivatives of heat semigroups,J.Funct.Anal.125(1994),252–286.[13]M.Emery,Stochastic calculus in manifolds,Springer-Verlag,Berlin,1989.With anappendix by P.-A.Meyer.[14]R.E.Greene and Hong-Xi Wu,Function Theory on Manifolds Which Possess a Pole,Lecture Notes in Math.699,Springer-Verlag,1979.[15]L.Gross,Logarithmic Sobolev inequalities,Amer.J.Math.97(1976),1061–1083.[16]E.P.Hsu,Logarithmic Sobolev inequalities on path spaces over Riemannian mani-folds,Comm.Math.Phys.189(1997),9–16.[17]W.S.Kendall,The radial part of Brownian motion on a manifold:a semimartingaleproperty,Ann.Probab.15(1987),no.4,1491–1500.[18]M.Ledoux,Concentration of measure and logarithmic Sobolev inequalities,S´e minairede Probabilit´e s,XXXIII,120–216,Lecture Notes Math.1709,Springer,Berlin,1999.[19]M.R¨o ckner and F.-Y.Wang,Harnack and functional inequalities for generalizedMehler semigroups,J.Funct.Anal.203(2003),237–261.[20]F.-Y.Wang,Logarithmic Sobolev inequalities on noncompact Riemannian manifolds,Probability Theory Relat.Fields109(1997),417–424.[21]F.-Y.Wang,Logarithmic Sobolev inequalities:conditions and counterexamples,J.Operator Theory46(2001),183–197.[22]F.-Y.Wang,Functional Inequalities,Markov Properties,and Spectral Theory,SciencePress,Beijing/New York,2005.。

a r X iv:mat h /2818v1[mat h.AG ]23Aug202TORELLI’S THEOREM FOR HIGH DEGREE SYMMETRIC PRODUCTS OF CUR VES NAJMUDDIN F AKHRUDDIN Abstract.We show that two smooth projective curves C 1and C 2of genus g which have isomorphic symmetric products are isomorphic unless g =2.This extends a theorem of Martens.Let k be an algebraically closed field and C 1and C 2two smooth projective curves of genus g >1over k .It is a consequence of Torelli’s theorem that if Sym g −1C 1∼=Sym g −1C 2,then C 1∼=C 2.The same holds for the d ’th symmetric products,for 1≤d <g −1as a consequence of a theorem of Martens [2].In this note we shall show that with one exception the same result continues to hold for all d ≥1,i.e.we have the following Theorem 1.Let C 1and C 2be smooth projective curves of genus g ≥2over an algebraically closed field k .If Sym d C 1∼=Sym d C 2for some d ≥1,then C 1∼=C 2unless g =d =2.It is well known that there exist non-isomorphic curves of genus 2over C with isomorphic Jacobians.Since the second symmetric power of a genus 2curve is isomorphic to the blow up of the Jacobian in a point,it follows that our result is the best possible.Proof of Theorem.Let C i ,i =1,2be two curves of genus g >1with Sym d C 1∼=Sym d C 2for some d ≥1.Since the Albanese variety of Sym d C i ,d ≥1,is iso-morphic to the Jacobian J (C i ),it follows that J (C 1)∼=J (C 2).If d ≤g −1,the theorem follows immediately from [2],since the image of Sym d C i in J (C i )(after choosing a basepoint)is W d (C i ).Note that in this case it sufices to have a birational isomorphism from Sym d C 1to Sym d C 2.Suppose g ≤d ≤2g −3.Then the Albanese map from Sym d C i to J (C i )is surjective with general fibre of dimension d −g .Interpreting the fibres as complete linear systems of degree d on C i ,it follows by Serre duality that the subvariety ofJ (C i )over which the fibres are of dimension >d −g is isomorphic to W 2g −2−d (C i ).Therefore if Sym d C 1∼=Sym d C 2,then W 2g −2−d (C 1)∼=W 2g −2−d (C 2),so Martens’theorem implies that C 1∼=C 2.Now suppose that d >2g −2and g >2.By choosing some isomorphism we identify J (C 1)and J (C 2)with a fixed abellian variety A .If φ:Sym d C 1→Sym d C 2is our given isomorphism,from the universal property of the Albanese morphism we obtain a commutative diagramSym d C 1φSym d C 2π2A12NAJMUDDIN FAKHRUDDINwhere theπi’s are the Albanese morphisms corresponding to some basepoints and f is an automorphism of A(not necessarily preserving the origin).By replacing C2 with f−1(C2)we may then assume that f is the identity.Since d>2g−2,the mapsπi,i=1,2make Sym d C i into projective bundles over A.By a theorem of Schwarzenberger[5],Sym d C i∼=P roj(E i),where E i is a vector bundle on A of rank d−g+1with c j(E i)=[W g−j(C i)],i=1,2,0≤j≤g−1,in the group of cycles on A modulo numerical equivalence.Sinceφis an isomorphism of projective bundles,it follows that there exists a line bundle L on A such that E1∼=E2⊗L.Letθi=[W g−1(C i)],so by Poincar´e’s formula[W g−j(C i)]=θj i/j!,i=1,2,i≤j≤g−1.The lemma below implies thatθg−1i =θg−12in the group of cyclesmodulo numerical equivalence on A.Sinceθg i=g!,this implies thatθ1·[C2]=g. By Matsusaka’s criterion[3],it follows that W g−1(C1)is a theta divisor for C2, which by Torelli’s theorem implies that C1∼=C2.If d=2g−2and g>2,then we can still apply the previous argument.In this case we also have that Sym d(C i)∼=P roj(E i),i=1,2but E i is a coherent sheaf which is not locally free.However on the complement of some point of A it does become locally free and the previous formula for the Chern classes remains valid.The above argument clearly does not suffice if g=2.To handle this case we shall use some properties of Picard bundles for which we refer the reader to[4]. Suppose that d>2and C i,i=1,2are two non-isomorphic curves of genus2with Sym d C1∼=Sym d ing the same argument(and notation)as the g>2case, it follows that there exist embeddings of C i,i=1,2,in A and a line bundle L on A such that E1∼=E2⊗L and L⊗d−1∼=O(C1−C2)(we identify C i,i=1,2with their images).For i≥1,let G i denote the i-th Picard sheaf associated to C2,so that P roj(G i)∼= Sym i(C2).(G i is the sheaf denoted by F2−i in[4]and G d∼=E2).There is an exact sequence([4,p.172]):0→O A→G i→G i−1→0(1)for all i>1.We will use this exact sequence and induction on i to compute the cohomology of sheaves of the form E1⊗P∼=E2⊗L⊗P,where P∈P ic0(A).Considerfirst the cohomology of G1,which is the pushforward of a line bundle of degree1on a translate of C2.Since we have assumed that C1≇C2,it followsthat C1·C2>2.Since C21=C22=2,deg(L|C2)=(C1−C2)·C2/(d−1)>0.By Riemann-Roch it follows that h j(A,G1⊗L⊗P),j=1,2is independent of P,except possibly for one P if deg(L|C2)=1,and h2(A,G1⊗L⊗P)=0since G1issupported on a curve.Now C1·C2>2also implies that c1(L)2<0.By the index theorem,it follows that h0(A,L⊗P)=h2(A,L⊗P)=0and h1(A,L⊗P)is independent of P. Therefore by tensoring the exact sequence(1)with L⊗P and considering the long exact sequence of cohomology,we obtain an exact sequence(2)0→H0(A,G i⊗L⊗P)→H0(A,G i−1⊗L⊗P)→H1(A,L⊗P)→H1(A,G i⊗L⊗P)→H1(A,G i−1⊗L⊗P)→0 and isomorphisms H2(A,G i⊗L⊗P)→H2(A,G i−1⊗L⊗P)for all i>1.By induction,it follows that H2(A,G i⊗L⊗P)=0for all i>0.Since the Euler characteristic of G i⊗L⊗P is independent of P,the above exact sequence(2)TORELLI’S THEOREM FOR HIGH DEGREE SYMMETRIC PRODUCTS OF CURVES3 along with induction shows that for all i>0and j=0,1,2,h j(G i⊗L⊗P)is independent of P,except for possibly one P.In particular,this holds for i=d hence h j(A,E1⊗P)is independent of P except again for possibly one P.We obtain a contradiction by using the computation of the cohomology of Picard sheaves in Proposition4.4of[4]:This implies that h1(A,E1⊗P)is one or zero depending on whether the point in A corresponding to P does or does not lie on a certain curve (which is a translate of C1).Lemma1.Let X be an algebraic variety of dimension g≥3and let E i,i=1,2 be vector bundles on X of rank r.Suppose c1(E i)=θi,c j(E i)=θj i/j!for i=1,2 and j=2,3(j=2if g=3),and E1∼=E2⊗L for some line bundle L on X.Then θj1=θj2for all j>1(j=2if g=3).Proof.Since E1∼=E2⊗L,c1(E1)=c1(E2)+rc1(L),hence c1(L)=(θ1−θ2)/r.For a vector bundle E of rank r and a line bundle L on any variety,we have the following formula for the Chern polynomial([1],page55):c t(E⊗L)=rk=0t k c t(L)r−k c i(E).Letting E=E1,E⊗L=E2,and expanding out the terms of degree2and3,one easily sees thatθj1=θj2for j=2and also for j=3if g>3.(Note that this only requires knowledge of c j(E i)for j=1,2,3).Since any integer n>1can be written as n=2a+3b with a,b∈N,the lemma follows.We do not know whether the theorem holds if k is not algebraically closed.Remark.S.Ramanan has suggested it may also be possible to prove the theorem when g>2by computing the Chern classes of the push forward of the tangent bundle of Sym d C to J(C).Acknowledgements.We thank A.Collino for some helpful correspondence,in par-ticular for informing us about the paper[2],and S.Ramanan for the above remark.References[1]W.Fulton,Intersection theory,Springer-Verlag,Berlin,second ed.,1998.[2]H.H.Martens,An extended Torelli theorem,Amer.J.Math.,87(1965),pp.257–261.[3]T.Matsusaka,On a characterization of a Jacobian variety,Memo.Coll.Sci.Univ.Kyoto.Ser.A.Math.,32(1959),pp.1–19.[4]S.Mukai,Duality between D(X)and D(ˆX)with its application to Picard sheaves,NagoyaMath.J.,81(1981),pp.153–175.[5]R.L.E.Schwarzenberger,Jacobians and symmetric products,Illinois J.Math.,7(1963),pp.257–268.School of Mathematics,Tata Institute of Fundamental Research,Homi Bhabha Road,Mumbai400005,IndiaE-mail address:naf@math.tifr.res.in。

冲压模具-冲压名称类英汉对照plain die简易模pierce die冲孔模forming die成型模progressive die连续模gang dies复合模shearing die剪边模riveting die铆合模pierce冲孔forming成型(抽凸,冲凸)draw hole抽孔bending折弯trim切边emboss凸点dome凸圆semi-shearing半剪stamp mark冲记号deburr or coin压毛边punch riveting冲压铆合side stretch侧冲压平reel stretch卷圆压平groove压线blanking下料stamp letter冲字(料号) shearing剪断tick-mark nearside正面压印tick-mark farside反面压印extension dwg展开图procedure dwg工程图die structure dwg模具结构图material材质material thickness料片厚度factor系数upward向上downward向下press specification冲床规格die height range适用模高die height闭模高度burr毛边gap间隙weight重量total wt.总重量punch wt.上模重量各式模具分类用语英汉对照landed plunger mold 有肩柱塞式模具burnishing die 挤光模landed positive mold 有肩全压式模具button die 镶入式圆形凹模loading shoe mold 料套式模具center-gated mold 中心浇口式模具loose detail mold 活零件模具chill mold 冷硬用铸模loose mold 活动式模具clod hobbing 冷挤压制模louvering die 百叶窗冲切模composite dies 复合模具manifold die 分歧管模具counter punch 反凸模modular mold 组合式模具double stack mold 双层模具multi-cavity mold 多模穴模具electroformed mold 电铸成形模multi-gate mold 复式浇口模具expander die 扩径模offswt bending die 双折冷弯模具extrusion die 挤出模palletizing die 叠层模family mold 反套制品模具plaster mold 石膏模blank through dies 漏件式落料模porous mold 通气性模具duplicated cavity plate 复板模positive mold 全压式模具fantail die 扇尾形模具pressure die 压紧模fishtail die 鱼尾形模具profile die 轮廓模flash mold 溢料式模具progressive die 顺序模gypsum mold 石膏铸模protable mold 手提式模具hot-runner mold 热流道模具prototype mold 雏形试验模具ingot mold 钢锭模punching die 落料模lancing die 切口模raising(embossing) 压花起伏成形re-entrant mold 倒角式模具sectional die 拼合模runless injection mold 无流道冷料模具sectional die 对合模具segment mold 组合模semi-positive mold 半全压式模具shaper 定型模套single cavity mold 单腔模具solid forging die 整体锻模split forging die 拼合锻模split mold 双并式模具sprueless mold 无注道残料模具squeezing die 挤压模stretch form die 拉伸成形模sweeping mold 平刮铸模swing die 振动模具three plates mold 三片式模具trimming die 切边模unit mold 单元式模具universal mold 通用模具unscrewing mold 退扣式模具yoke type die 轭型模模具英语解释一、入水：gate进入位：gate location水口形式：gate type大水口：edge gate细水口：pin-point gate水口大小：gate size转水口：switching runner/gate唧嘴口径：sprue diameter二、流道: runner热流道：hot runner,hot manifold热嘴冷流道: hot sprue/cold runner唧嘴直流: direct sprue gate圆形流道：round (full/half runner)流道电脑分析：mold flow analysis流道平衡:runner balance热嘴：hot sprue热流道板：hot manifold发热管：cartridge heater探针: thermocouples插头：connector plug插座：connector socket密封/封料：seal三、运水：water line喉塞：line lpug喉管：tube塑胶管：plastic tube快速接头：jiffy quick connector plug/socker 四、模具零件：mold components三板模：3-plate mold二板模：2-plate mold边钉/导边：leader pin/guide pin边司/导套：bushing/guide bushing中托司：shoulder guide bushing中托边L：guide pin顶针板：ejector retainner plate托板：support plate螺丝：screw管钉：dowel pin开模槽：ply bar scot内模管位：core/cavity inter-lock顶针：ejector pin司筒：ejector sleeve司筒针：ejector pin推板：stripper plate缩呵：movable core,return core core puller扣机（尼龙拉勾）：nylon latch lock斜顶：lifter模胚（架）：mold base上内模：cavity insert下内模：core insert行位（滑块）：slide镶件：insert压座/斜鸡：wedge耐磨板/油板：wedge wear plate压条：plate撑头: support pillar唧嘴：sprue bushing挡板：stop plate定位圈：locating ring锁扣：latch扣鸡：parting lock set推杆：push bar栓打螺丝：S.H.S.B顶板：eracuretun活动臂：lever arm分流锥：spure sperader水口司:bush垃圾钉：stop pin隔片：buffle弹弓柱：spring rod弹弓:die spring中托司：ejector guide bush中托边：ejector guide pin镶针：pin销子：dowel pin波子弹弓：ball catch喉塞: pipe plug锁模块：lock plate斜顶：angle from pin斜顶杆：angle ejector rod尼龙拉勾：parting locks活动臂：lever arm复位键、提前回杆：early return bar气阀：valves斜导边：angle pin术语：terms承压平面平衡：parting surface support balance 模排气：parting line venting回针碰料位：return pin and cavity interference模总高超出啤机规格：mold base shut hight顶针碰运水：water line interferes withejector pin料位出上/下模：part from cavith (core) side模胚原身出料位：cavity direct cut on A-plate,core direct cut on B-plate.不准用镶件：Do not use (core/cavity) insert用铍铜做镶件：use beryllium copper insert初步（正式）模图设计：preliinary (final) mold design反呵：reverse core弹弓压缩量：spring compressed length稳定性好：good stability,stable强度不够：insufficient rigidity均匀冷却：even cooling扣模：sticking热膨胀：thero expansion公差：tolorance铜公(电极）：copper electrode模具相关英语1padding block垫块stepping bar垫条upper die base上模座lower die base下模座upper supporting blank上承板upper padding plate blank上垫板spare dies模具备品spring 弹簧bolt螺栓document folder活页夹file folder资料夹to put file in order整理资料spare tools location手工备品仓first count初盘人first check初盘复棹人second count 复盘人second check复盘复核人equipment设备waste materials废料work in progress product在制品casing = containerazation装箱quantity of physical invetory second count 复盘点数量quantity of customs count会计师盘,点数量the first page第一联filed by accounting department for reference会计部存查end-user/using unit(department)使用单位summary of year-end physical inventory bills年终盘点截止单据汇总表bill name单据名称This sheet and physical inventory list will be sent to accounting department together (Those of NHK will be sent to financial department)本表请与盘点清册一起送会计部－(NHK厂区送财会部)Application status records of year-end physical inventory List and physical inventory card 年终盘点卡与清册使用－状况明细表blank and waste sheet NO.空白与作废单号plate电镀mold成型material for engineering mold testing工程试模材料not included in physical inventory不列入盘点sample样品incoming material to be inspected进货待验description品名steel/rolled steel钢材material statistics sheet物料统计明细表meeting minutes会议记录meeting type 会别distribution department分发单位location地点chairman主席present members出席人员subject主题conclusion结论decision items决议事项responsible department负责单位pre-fixed finishing date预定完成日approved by / checked by / prepared by核准/审核/承办PCE assembly production schedule sheetPCE组装厂生产排配表model机钟work order工令revision版次remark备注production control confirmation生产确认checked by初审approved by核准department部门stock age analysis sheet库存货龄分析表on-hand inventory现有库存available material良品可使用obsolete material良品已呆滞to be inspected or reworked待验或重工total合计cause description原因说明part number/ P/N 料号type形态item/group/class类别quality品质prepared by制表notes说明year-end physical inventory difference analysis sheet年终盘点差异分析表physical inventory盘点数量physical count quantity帐面数量difference quantity差异量cause analysis原因分析raw materials原料materials物料finished product成品semi-finished product半成品packing materials包材good product/accepted goods/ accepted parts/good parts良品defective product/non-good parts不良品disposed goods处理品warehouse/hub仓库on way location在途仓oversea location海外仓spare parts physical inventory list备品盘点清单spare molds location模具备品仓skid/pallet栈板tox machine自铆机wire EDM线割EDM放电机coil stock卷料sheet stock片料tolerance工差score=groove压线cam block滑块pilot导正筒trim剪外边pierce剪内边drag form压锻差pocket for the punch head挂钩槽slug hole废料孔feature die公母模expansion dwg展开图radius半径shim(wedge)楔子torch-flame cut火焰切割set screw止付螺丝form block折刀stop pin定位销round pierce punch=die button圆冲子shape punch=die insert异形子stock locater block定位块under cut=scrap chopper清角active plate活动板模具相关英语2baffle plate挡块cover plate盖板male die公模female die母模groove punch压线冲子air-cushion eject-rod气垫顶杆spring-box eject-plate弹簧箱顶板bushing block衬套insert 入块club car高尔夫球车capability能力parameter参数factor系数phosphate皮膜化成viscosity涂料粘度alkalidipping脱脂main manifold主集流脉bezel斜视规blanking穿落模dejecting顶固模demagnetization去磁;消磁high-speed transmission高速传递heat dissipation热传rack上料degrease脱脂rinse水洗alkaline etch龄咬desmut剥黑膜D.I. rinse纯水次Chromate铬酸处理Anodize阳性处理seal封孔revision版次part number/P/N料号good products良品scraped products报放心品defective products不良品finished products成品disposed products处理品barcode条形码flow chart流程窗体assembly组装stamping冲压molding成型spare parts=buffer备品coordinate坐标dismantle the die折模auxiliary fuction辅助功能poly-line多义线heater band 加热片thermocouple热电偶sand blasting喷沙grit 砂砾derusting machine除锈机degate打浇口dryer烘干机induction感应induction light感应光response=reaction=interaction感应ram连杆edge finder巡边器concave 凹convex凸short射料不足nick缺口speck瑕疪shine亮班splay 银纹gas mark焦痕delamination起鳞cold slug冷块blush 导色gouge沟槽;凿槽satin texture段面咬花witness line证示线patent专利grit沙砾granule=peuet=grain细粒grit maker抽粒机cushion缓冲magnalium镁铝合金magnesium镁金metal plate钣金lathe车mill锉plane刨grind磨drill钻boring镗blinster气泡fillet镶;嵌边through-hole form通孔形式voller pin formality滚针形式cam driver铡楔shank摸柄crank shaft曲柄轴augular offset角度偏差velocity速度production tempo生产进度现状torque扭矩spline=the multiple keys花键quenching淬火tempering回火annealing退火carbonization碳化alloy合金tungsten high speed steel钨高速的moly high speed steel钼高速的organic solvent有机溶剂bracket小磁导liaison联络单volatile挥发性resistance电阻ion离子titrator滴定仪beacon警示灯coolant冷却液crusher破碎机模具工程类plain die简易模pierce die冲孔模forming die成型模progressive die连续模gang dies复合模shearing die剪边模riveting die铆合模pierce冲孔forming成型(抽凸,冲凸)draw hole抽孔bending折弯trim切边模具相关英语3emboss凸点dome凸圆semi-shearing半剪stamp mark冲记号deburr or coin压毛边punch riveting冲压铆合side stretch侧冲压平reel stretch卷圆压平groove压线blanking下料stamp letter冲字(料号) shearing剪断tick-mark nearside正面压印tick-mark farside反面压印冲压名称类extension dwg展开图procedure dwg工程图die structure dwg模具结构图material材质material thickness料片厚度factor系数upward向上downward向下press specification冲床规格die height range适用模高die height闭模高度burr毛边gap间隙weight重量total wt.总重量punch wt.上模重量五金零件类inner guiding post内导柱inner hexagon screw内六角螺钉dowel pin固定销coil spring弹簧lifter pin顶料销eq-height sleeves=spool等高套筒pin销lifter guide pin浮升导料销guide pin导正销wire spring圆线弹簧outer guiding post外导柱stop screw止付螺丝located pin定位销outer bush外导套模板类top plate上托板（顶板）top block上垫脚punch set上模座punch pad上垫板punch holder上夹板stripper pad脱料背板up stripper上脱料板male die公模(凸模)feature die公母模female die母模(凹模)upper plate上模板lower plate下模板die pad下垫板die holder下夹板die set下模座bottom block下垫脚bottom plate下托板(底板)stripping plate内外打(脱料板)outer stripper外脱料板inner stripper内脱料板lower stripper下脱料板零件类punch冲头insert入块(嵌入件)deburring punch压毛边冲子groove punch压线冲子stamped punch字模冲子round punch圆冲子special shape punch异形冲子bending block折刀roller滚轴baffle plate挡块located block定位块supporting block for location定位支承块air cushion plate气垫板air-cushion eject-rod气垫顶杆trimming punch切边冲子stiffening rib punch = stinger 加强筋冲子ribbon punch压筋冲子reel-stretch punch卷圆压平冲子guide plate定位板sliding block滑块模具工程常用词汇die 模具figure file, chart file图档cutting die, blanking die冲裁模progressive die, follow (-on)die连续模compound die复合模punched hole冲孔panel board镶块to cutedges=side cut=side scrap切边to bending折弯to pull, to stretch拉伸Line streching, line pulling线拉伸engraving, to engrave刻印upsiding down edges翻边to stake铆合designing, to design设计design modification设计变化die block模块folded block折弯块sliding block滑块location pin定位销lifting pin顶料销die plate, front board模板padding block垫块stepping bar垫条upper die base上模座lower die base下模座upper supporting blank上承板upper padding plate blank上垫板spare dies模具备品spring 弹簧bolt螺栓document folder文件夹file folder资料夹to put file in order整理资料spare tools location手工备品仓first count初盘人first check初盘复棹人second count 复盘人second check复盘复核人equipment设备waste materials废料work in progress product在制品casing = containerazation装箱quantity of physical invetory second count 复盘点数量quantity of customs count会计师盘,点数量the first page第一联filed by accounting department for reference会计部存查end-user/using unit(department)使用单位summary of year-end physical inventory bills年终盘点截止单据汇总表bill name单据名称This sheet and physical inventory list will be sent to accounting department together (Those of NHK will be sent to financial department)本表请与盘点清册一起送会计部－(NHK厂区送财会部) Application status records of year-end physical inventory List and physical inventory card 年终盘点卡与清册使用－状况明细表blank and waste sheet NO.空白与作废单号plate电镀mold成型material for engineering mold testing工程试模材料not included in physical inventory不列入盘点sample样品incoming material to be inspected进货待验description品名steel/rolled steel钢材material statistics sheet物料统计明细表meeting minutes会议记录meeting type 会别distribution department分发单位location地点chairman主席present members出席人员subject主题conclusion结论decision items决议事项responsible department负责单位pre-fixed finishing date预定完成日approved by / checked by / prepared by核准/审核/承办PCE assembly production schedule sheetPCE组装厂生产排配表model机锺work order工令revision版次remark备注production control confirmation生产确认checked by初审approved by核准department部门stock age analysis sheet库存货龄分析表on-hand inventory现有库存available material良品可使用obsolete material良品已呆滞to be inspected or reworked待验或重工total合计cause description原因说明part number/ P/N 料号type形态item/group/class类别quality品质prepared by制表notes说明year-end physical inventory difference analysis sheet年终盘点差异分析表physical inventory盘点数量physical count quantity帐面数量difference quantity差异量cause analysis原因分析raw materials原料materials物料finished product成品semi-finished product半成品packing materials包材good product/accepted goods/ accepted parts/good parts良品defective product/non-good parts不良品disposed goods处理品warehouse/hub仓库on way location在途仓oversea location海外仓spare parts physical inventory list备品盘点清单spare molds location模具备品仓skid/pallet栈板tox machine自铆机wire EDM线割EDM放电机coil stock卷料sheet stock片料tolerance工差score=groove压线cam block滑块pilot导正筒trim剪外边pierce剪内边drag form压锻差pocket for the punch head挂钩槽slug hole废料孔feature die公母模expansion dwg展开图radius半径shim(wedge)楔子torch-flame cut火焰切割set screw止付螺丝form block折刀stop pin定位销round pierce punch=die button圆冲子shape punch=die insert异形子stock locater block定位块under cut=scrap chopper清角active plate活动板baffle plate挡块cover plate盖板male die公模female die母模groove punch压线冲子air-cushion eject-rod气垫顶杆spring-box eject-plate弹簧箱顶板bushing block衬套insert 入块club car高尔夫球车capability能力parameter参数factor系数phosphate皮膜化成viscosity涂料粘度alkalidipping脱脂main manifold主集流脉bezel斜视规blanking穿落模dejecting顶固模demagnetization去磁;消磁high-speed transmission高速传递heat dissipation热传rack上料degrease脱脂rinse水洗alkaline etch龄咬desmut剥黑膜D.I. rinse纯水次Chromate铬酸处理Anodize阳性处理seal封孔revision版次part number/P/N料号good products良品scraped products报放心品defective products不良品finished products成品disposed products处理品barcode条码flow chart流程表单assembly组装stamping冲压molding成型spare parts=buffer备品coordinate座标dismantle the die折模auxiliary fuction辅助功能poly-line多义线heater band 加热片thermocouple热电偶sand blasting喷沙grit 砂砾derusting machine除锈机degate打浇口dryer烘干机induction感应induction light感应光response=reaction=interaction感应ram连杆edge finder巡边器concave凸convex凹short射料不足nick缺口speck瑕??shine亮班splay 银纹gas mark焦痕delamination起鳞cold slug冷块blush 导色gouge沟槽;凿槽satin texture段面咬花witness line证示线patent专利grit沙砾granule=peuet=grain细粒grit maker抽粒机cushion缓冲magnalium镁铝合金magnesium镁金metal plate钣金lathe车mill锉plane刨grind磨drill铝boring镗blinster气泡fillet镶;嵌边through-hole form通孔形式voller pin formality滚针形式cam driver铡楔shank摸柄crank shaft曲柄轴augular offset角度偏差velocity速度production tempo生产进度现状torque扭矩spline=the multiple keys花键quenching淬火tempering回火annealing退火carbonization碳化alloy合金tungsten high speed steel钨高速的moly high speed steel钼高速的organic solvent有机溶剂bracket小磁导liaison联络单volatile挥发性resistance电阻ion离子titrator滴定仪beacon警示灯coolant冷却液crusher破碎机模具工程类plain die简易模pierce die冲孔模forming die成型模progressive die连续模gang dies复合模shearing die剪边模riveting die铆合模pierce冲孔forming成型(抽凸,冲凸) draw hole抽孔bending折弯trim切边emboss凸点dome凸圆semi-shearing半剪stamp mark冲记号deburr or coin压毛边punch riveting冲压铆合side stretch侧冲压平reel stretch卷圆压平groove压线blanking下料stamp letter冲字(料号) shearing剪断tick-mark nearside正面压印tick-mark farside反面压印冲压名称类extension dwg展开图procedure dwg工程图die structure dwg模具结构图material材质material thickness料片厚度factor系数upward向上downward向下press specification冲床规格die height range适用模高die height闭模高度burr毛边gap间隙weight重量total wt.总重量punch wt.上模重量五金零件类inner guiding post内导柱inner hexagon screw内六角螺钉dowel pin固定销coil spring弹簧lifter pin顶料销eq-height sleeves=spool等高套筒pin销lifter guide pin浮升导料销guide pin导正销wire spring圆线弹簧outer guiding post外导柱stop screw止付螺丝located pin定位销outer bush外导套模板类top plate上托板（顶板）top block上垫脚punch set上模座punch pad上垫板punch holder上夹板stripper pad脱料背板up stripper上脱料板male die公模(凸模)feature die公母模female die母模(凹模)upper plate上模板lower plate下模板die pad下垫板die holder下夹板die set下模座bottom block下垫脚bottom plate下托板(底板) stripping plate内外打(脱料板) outer stripper外脱料板inner stripper内脱料板lower stripper下脱料板零件类punch冲头insert入块(嵌入件)deburring punch压毛边冲子groove punch压线冲子stamped punch字模冲子round punch圆冲子special shape punch异形冲子bending block折刀roller滚轴baffle plate挡块located block定位块supporting block for location定位支承块air cushion plate气垫板air-cushion eject-rod气垫顶杆trimming punch切边冲子stiffening rib punch = stinger 加强筋冲子ribbon punch压筋冲子reel-stretch punch卷圆压平冲子guide plate定位板sliding block滑块sliding dowel block滑块固定块active plate活动板lower sliding plate下滑块板upper holder block上压块upper mid plate上中间板spring box弹簧箱spring-box eject-rod弹簧箱顶杆spring-box eject-plate弹簧箱顶板bushing bolck衬套cover plate盖板guide pad导料块塑件&模具相关英文compre sion molding压缩成型flash mold溢流式模具plsitive mold挤压式模具split mold分割式模具cavity型控母模core模心公模taper锥拔leather cloak仿皮革shiver饰纹flow mark流痕welding mark溶合痕post screw insert螺纹套筒埋值self tapping screw自攻螺丝striper plate脱料板piston活塞cylinder汽缸套chip细碎物handle mold手持式模具常用塑料英语缩略语英文简称英文全称中文全称ABA Acrylonitrile-butadiene-acrylate 丙烯腈/丁二烯/丙烯酸酯共聚物ABS Acrylonitrile-butadiene-styrene 丙烯腈/丁二烯/苯乙烯共聚物AES Acrylonitrile-ethylene-styrene 丙烯腈/乙烯/苯乙烯共聚物AMMA Acrylonitrile/methyl Methacrylate 丙烯腈/甲基丙烯酸甲酯共聚物ARP Aromatic polyester 聚芳香酯AS Acrylonitrile-styrene resin 丙烯腈-苯乙烯树脂ASA Acrylonitrile-styrene-acrylate 丙烯腈/苯乙烯/丙烯酸酯共聚物CA Cellulose acetate 醋酸纤维塑料CAB Cellulose acetate butyrate 醋酸-丁酸纤维素塑料CAP Cellulose acetate propionate 醋酸-丙酸纤维素CE Cellulose plastics, general 通用纤维素塑料CF Cresol-formaldehyde 甲酚-甲醛树脂CMC Carboxymethyl cellulose 羧甲基纤维素CN Cellulose nitrate 硝酸纤维素CP Cellulose propionate 丙酸纤维素CPE Chlorinated polyethylene 氯化聚乙烯CPVC Chlorinated poly(vinyl chloride) 氯化聚氯乙烯CS Casein 酪蛋白CTA Cellulose triacetate 三醋酸纤维素EC Ethyl cellulose 乙烷纤维素EMA Ethylene/methacrylic acid 乙烯/甲基丙烯酸共聚物EP Epoxy, epoxide 环氧树脂EPD Ethylene-propylene-diene 乙烯-丙烯-二烯三元共聚物EPM Ethylene-propylene polymer 乙烯-丙烯共聚物EPS Expanded polystyrene 发泡聚苯乙烯ETFE Ethylene-tetrafluoroethylene 乙烯-四氟乙烯共聚物EVA Ethylene/vinyl acetate 乙烯-醋酸乙烯共聚物EVAL Ethylene-vinyl alcohol 乙烯-乙烯醇共聚物FEP Perfluoro(ethylene-propylene) 全氟（乙烯-丙烯）塑料FF Furan formaldehyde 呋喃甲醛HDPE High-density polyethylene plastics高密度聚乙烯塑料HIPS High impact polystyrene 高冲聚苯乙烯IPS Impact-resistant polystyrene 耐冲击聚苯乙烯LCP Liquid crystal polymer 液晶聚合物LDPE Low-density polyethylene plastics 低密度聚乙烯塑料LLDPE Linear low-density polyethylene 线性低密聚乙烯LMDPE Linear medium-density polyethylene 线性中密聚乙烯MBS Methacrylate-butadiene-styrene 甲基丙烯酸-丁二烯-苯乙烯共聚物MDPE Medium-density polyethylene 中密聚乙烯MF Melamine-formaldehyde resin 密胺-甲醛树脂。

高等数学英语词汇高等数学英语词汇引导语：高等数学指相对于初等数学而言，数学的对象及方法较为繁杂的'一部分。

以下是店铺分享给大家的高等数学英语词汇，欢迎阅读!Aabelian group：阿贝尔群; absolute geometry：绝对几何; absolute value：绝对值; abstract algebra：抽象代数; addition：加法; algebra：代数; algebraicclosure：代数闭包; algebraic geometry：代数几何;algebraic geometry and analytic geometry：代数几何和解析几何; algebraic numbers：代数数; algorithm：算法; almost all：绝大多数; analytic function：解析函数; analytic geometry：解析几何; and：且;angle：角度; anticommutative：反交换律; antisymmetric relation：反对称关系; antisymmetry：反对称性; approximately equal：约等于; Archimedean field：阿基米德域; Archimedean group：阿基米德群; area：面积; arithmetic：算术; associative algebra：结合代数; associativity：结合律; axiom：公理; axiom of constructibility：可构造公理; axiom of empty set：空集公理;axiom of extensionality：外延公理; axiom of foundation：正则公理; axiom of pairing：对集公理; axiom of regularity：正则公理; axiom of replacement：代换公理; axiom of union：并集公理; axiom schema of separation：分离公理; axiom schema of specification：分离公理;axiomatic set theory：公理集合论; axiomatic system：公理系统;BBaire space：贝利空间; basis：基; Bézout's identity：贝祖恒等式; Bernoulli's inequality：伯努利不等式 ; Big O notation：大O符号; bilinear operator：双线性算子; binary operation：二元运算; binary predicate：二元谓词; binary relation：二元关系; Booleanalgebra：布尔代数;Boolean logic：布尔逻辑; Boolean ring：布尔环; boundary：边界; boundary point：边界点;bounded lattice：有界格;Ccalculus：微积分学; Cantor's diagonal argument：康托尔对角线方法; cardinal number：基数;cardinality：势; cardinality of the continuum：连续统的势; Cartesian coordinate system：直角坐标系; Cartesian product：笛卡尔积; category：范畴; Cauchy sequence：柯西序列; Cauchy-Schwarz inequality：柯西不等式; Ceva's Theorem：塞瓦定理; characteristic：特征;characteristic polynomial：特征多项式; circle：圆; class：类; closed：闭集; closure：封闭性或闭包; closure algebra：闭包代数; combinatorial identities：组合恒等式; commutativegroup：交换群; commutative ring：交换环; commutativity:：交换律; compact：紧致的;compact set：紧致集合; compact space：紧致空间; complement：补集或补运算; completelattice：完备格; complete metric space：完备的度量空间; complete space：完备空间; complexmanifold：复流形; complex plane：复平面; congruence：同余; congruent：全等; connectedspace：连通空间; constructible universe：可构造全集; constructions of the real numbers：实数的构造; continued fraction：连分数; continuous：连续; continuum hypothesis：连续统假设;contractible space：可缩空间; convergence space：收敛空间; cosine：余弦; countable：可数;countable set：可数集; cross product：叉积; cycle space：圈空间; cyclic group：循环群;Dde Morgan's laws：德·摩根律; Dedekind completion：戴德金完备性; Dedekind cut：戴德金分割;del：微分算子; dense：稠密; densely ordered：稠密排列; derivative：导数; determinant：行列式; diffeomorphism：可微同构; difference：差; differentiablemanifold：可微流形;differential calculus：微分学; dimension：维数; directed graph：有向图; discrete space：离散空间; discriminant：判别式; distance：距离; distributivity：分配律; dividend：被除数;dividing：除; divisibility：整除; division：除法; divisor：除数; dot product：点积;Eeigenvalue：特征值; eigenvector：特征向量; element：元素; elementary algebra：初等代数;empty function：空函数; empty set：空集; empty product：空积; equal：等于; equality：等式或等于; equation：方程; equivalence relation：等价关系; Euclidean geometry：欧几里德几何;Euclidean metric：欧几里德度量; Euclidean space：欧几里德空间; Euler's identity：欧拉恒等式;even number：偶数; event：事件; existential quantifier：存在量词; exponential function：指数函数; exponential identities：指数恒等式; expression：表达式; extended real number line：扩展的实数轴;Ffalse：假; field：域; finite：有限; finite field：有限域; finite set：有限集合; first-countablespace：第一可数空间; first order logic：一阶逻辑; foundations of mathematics：数学基础;function：函数; functional analysis：泛函分析; functional predicate：函数谓词;fundamental theorem of algebra：代数基本定理; fraction：分数;Ggauge space：规格空间; general linear group：一般线性群; geometry：几何学; gradient：梯度;graph：图; graph of a relation：关系图; graph theory：图论; greatest element：最大元;group：群; group homomorphism：群同态;HHausdorff space：豪斯多夫空间; hereditarily finite set：遗传有限集合; Heron's formula：海伦公式; Hilbert space：希尔伯特空间;Hilbert's axioms：希尔伯特公理系统; Hodge decomposition：霍奇分解; Hodge Laplacian：霍奇拉普拉斯算子; homeomorphism：同胚; horizontal：水平;hyperbolic function identities：双曲线函数恒等式; hypergeometric function identities：超几何函数恒等式; hyperreal number：超实数;Iidentical：同一的; identity：恒等式; identity element：单位元; identity matrix：单位矩阵;idempotent：幂等; if：若; if and only if：当且仅当; iff：当且仅当; imaginary number：虚数;inclusion：包含; index set：索引集合; indiscrete space：非离散空间; inequality：不等式或不等; inequality of arithmetic and geometric means：平均数不等式; infimum：下确界; infiniteseries：无穷级数; infinite：无穷大; infinitesimal：无穷小; infinity：无穷大; initial object：初始对象; inner angle：内角; inner product：内积; inner product space：内积空间; integer：整数; integer sequence：整数列; integral：积分; integral domain：整数环; interior：内部;interior algebra：内部代数; interior point：内点; intersection：交集; inverse element：逆元;invertible matrix：可逆矩阵; interval：区间; involution：回旋; irrational number：无理数;isolated point：孤点; isomorphism：同构;JJacobi identity：雅可比恒等式; join：并运算;K格式： Kuratowski closure axioms：Kuratowski 闭包公理;Lleast element：最小元; Lebesgue measure：勒贝格测度; Leibniz's law：莱布尼茨律; Liealgebra：李代数; Lie group：李群; limit：极限; limit point：极限点; line：线; line segment：线段; linear：线性; linear algebra：线性代数; linear operator：线性算子; linear space：线性空间; linear transformation：线性变换; linearity：线性性; list of inequalities：不等式列表; list oflinear algebra topics：线性代数相关条目; locally compact space：局部紧致空间; logarithmicidentities：对数恒等式; logic：逻辑学; logical positivism：逻辑实证主义; law of cosines：余弦定理; L??wenheim-Skolem theorem：L??wenheim-Skolem 定理; lower limit topology：下限拓扑;Mmagnitude：量; manifold：流形; map：映射; mathematical symbols：数学符号; mathematicalanalysis：数学分析; mathematical proof：数学证明; mathematics：数学; matrix：矩阵;matrix multiplication：矩阵乘法; meaning：语义; measure：测度; meet：交运算; member：元素; metamathematics：元数学; metric：度量; metric space：度量空间; model：模型; modeltheory：模型论; modular arithmetic：模运算; module：模; monotonic function：单调函数;multilinear algebra：多重线性代数; multiplication：乘法; multiset：多样集;Nnaive set theory：朴素集合论; natural logarithm：自然对数; natural number：自然数; naturalscience：自然科学; negative number：负数; neighbourhood：邻域; New Foundations：新基础理论; nine point circle：九点圆; non-Euclidean geometry：非欧几里德几何; nonlinearity：非线性; non-singular matrix：非奇异矩阵; nonstandard model：非标准模型; nonstandardanalysis：非标准分析; norm：范数; normed vector space：赋范向量空间; n-tuple：n 元组或多元组; nullary：空; nullary intersection：空交集; number：数; number line：数轴;Oobject：对象; octonion：八元数; one-to-one correspondence：一一对应; open：开集; openball：开球; operation：运算; operator：算子; or：或; order topology：序拓扑; ordered field：有序域;ordered pair：有序对; ordered set：偏序集; ordinal number：序数; ordinarymathematics：一般数学; origin：原点; orthogonal matrix：正交矩阵;Pp-adic number：p进数; paracompact space：仿紧致空间; parallel postulate：平行公理;parallelepiped：平行六面体; parallelogram：平行四边形; partial order：偏序关系; partition：分割; Peano arithmetic：皮亚诺公理; Pedoe's inequality：佩多不等式; perpendicular：垂直;philosopher：哲学家; philosophy：哲学; philosophy journals：哲学类杂志; plane：平面; pluralquantification：复数量化; point：点; Point-Line-Plane postulate：点线面假设; polarcoordinates：极坐标系; polynomial：多项式; polynomial sequence：多项式列; positive-definitematrix：正定矩阵; positive-semidefinite matrix：半正定矩阵; power set：幂集; predicate：谓词; predicate logic：谓词逻辑; preorder：预序关系; prime number：素数; product：积;proof：证明; proper class：纯类; proper subset：真子集; property：性质; proposition：命题; pseudovector：伪向量; Pythagorean theorem：勾股定理;QQ.E.D.：Q.E.D.; quaternion：四元数; quaternions and spatial rotation：四元数与空间旋转;question：疑问句; quotient field：商域; quotient set：商集;Rradius：半径; ratio：比; rational number：有理数; real analysis：实分析; real closed field：实闭域; real line：实数轴; real number：实数; real number line：实数线; reflexive relation：自反关系; reflexivity：自反性; reification：具体化; relation：关系; relative complement：相对补集;relatively complemented lattice：相对补格; right angle：直角; right-handed rule：右手定则;ring：环;Sscalar：标量; second-countable space：第二可数空间; self-adjoint operator：自伴随算子;sentence：判断; separable space：可分空间; sequence：数列或序列; sequence space：序列空间; series：级数; sesquilinear function：半双线性函数; set：集合; set-theoretic definitionof natural numbers：自然数的集合论定义; set theory：集合论; several complex variables：一些复变量; shape：几何形状; sign function：符号函数; singleton：单元素集合; social science：社会科学; solid geometry：立体几何; space：空间; spherical coordinates：球坐标系; squarematrix：方块矩阵; square root：平方根; strict：严格; structural recursion：结构递归;subset：子集; subsequence：子序列; subspace：子空间; subspace topology：子空间拓扑;subtraction：减法; sum：和; summation：求和; supremum：上确界; surreal number：超实数; symmetric difference：对称差; symmetric relation：对称关系; system of linearequations：线性方程组;Ttensor：张量; terminal object：终结对象; the algebra of sets：集合代数; theorem：定理; topelement：最大元; topological field：拓扑域; topological manifold：拓扑流形; topological space：拓扑空间; topology：拓扑或拓扑学; total order：全序关系; totally disconnected：完全不连贯;totally ordered set：全序集; transcendental number：超越数; transfinite recursion：超限归纳法; transitivity：传递性; transitive relation：传递关系; transpose：转置; triangleinequality：三角不等式; trigonometric identities：三角恒等式; triple product：三重积; trivialtopology：密着拓扑; true：真; truth value：真值;Uunary operation：一元运算; uncountable：不可数; uniform space：一致空间; union：并集;unique：唯一; unit interval：单位区间; unit step function：单位阶跃函数; unit vector：单位向量;universal quantification：全称量词; universal set：全集; upper bound：上界;Vvacuously true：??; Vandermonde's identity：Vandermonde 恒等式; variable：变量;vector：向量; vector calculus：向量分析; vector space：向量空间; Venn diagram：文氏图;volume：体积; von Neumann ordinal：冯·诺伊曼序数; von Neumann universe：冯·诺伊曼全集;vulgar fraction：分数;ZZermelo set theory：策梅罗集合论; Zermelo-Fraenkel set theory：策梅罗-弗兰克尔集合论; ZF settheory：ZF 系统; zero：零; zero object：零对象;下载全文。

比较定理the comparison theoremA Note on Comparison Theorems for Nonnegative MatricesIn Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.∙Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.∙Toponogov's theorem∙Myers's theorem∙Hessian comparison theorem∙Laplacian comparison theorem∙Morse–Schoenberg comparison theorem∙Berger comparison theorem, Rauch–Berger comparison theorem, M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", llinois J.Math., vol. 6 (1962) 700–712∙Berger–Kazdan comparison theorem[2]∙Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold) F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356).∙Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures (R.L.Bishop & R. Crittenden, Geometry of manifolds)∙Lichnerowicz comparison theorem∙Eigenvalue comparison theorem∙Cheng's eigenvalue comparison theoremSee also: Comparison triangleLyapunov comparison theorem qualitative analysis of large scale dynamical systemsChapter IV Systems Described by Stochastic Differential Equations【1】Lyapunov stability and practical stability of nonlinear delay stochastic systems: a unified approachTheorem 3.1【1】.Let(AI)and(A21hold.Assumethatforthe solutionprocessx(t)=w(to.#,)(t)of(2.11,EIY(t,x(t))l exists onthto-7.ThendA(t)E I Y ~ t , ~ ~ t ~ , # ~ ~ ~ t ~ ~ l ~ r ~ t ~ , # ~ ~ ~ t ~ ,trto (3.7)wheneverElV(to+s.#,(s) 1 I~,(s), SE J (3.8)Stability of linear differential equations with distributed delayDefinition 1(comparison principle) Adynamic system( A )issaidtobeaComparison Systemof adynamicsystem( B )withregardtotheproperty P(for example, stability ofits zerosolution), i fthe verification of property P for system ( A )implies the same property for system ( B ) .Remark1Note that in the linear cases such an overvaluing system constitutes a comparison system with regard to (asymptotic) stability. Indeed, if V ( x ( t ; to; 4 ) ) 5 z ( t ; to; %to),& 2 to, then any property relative to stability verified b y z(t; to; %to) is verified b y x(t; to; 4).Improved Razumikhin-Type Theorem for Input-to-State Stability of Nonlinear Time-Delay SystemsTheorem 1. Assume that there exists a continuously differentiable function V : R+ ×Rn →R+, functions α1, α2 ∈K∞,functions ρ, χ∈K, and a continuous function φ: R+ →R such that(i) α1(|x|) ≤V (t, x) ≤α2(|x|), ∀(t, x) ∈R+ × Rn;(ii) ˙V (t, ϕ(0)) ≤φ(t)V (t, ϕ(0)) for V (t, ϕ(0)) ≥max,ρ(|u|), χ−(V (t+θ, ϕ(θ)))-, ∀(t, ϕ(0)) ∈R+×Rn, and u ∈Rm;(iii) R ∞t0φ+(s)ds <∞, and there exists a positive constant ε such that, for a sufficiently large t,R tt0φ−(s)ds ≥ε(t − t0), where φ+(s) = φ(s) ∨0and φ−(s) = *−φ(s)+ ∨0; and(iv)r − Cχ−(r) ∈K∞.Then, System (1) is uniformly ISS and γ(r) = α−11(2Cρ(r)),where C = eR ∞0φ+(s)ds <∞.Small-Gain Conditions and the Comparison PrinciplePartial robust connective stability analysis for a class of interconnected power systems[1]Large-Scale Dynamic Systems: Stability and Structurecomparison theorem stochasticAsymptotic stability of stochastic impulsive system via comparison approach定积分上的比较定理性质5设f 为[a ，b]上的可积函数。

数学专业英语词汇(C)(转载)c function c类函数c manifold c廖c mapping c类映射ca set 上解析集calculability 可计算性calculable mapping 可计算映射calculable relation 可计算关系calculate 计算calculating automaton 计算自动机calculating circuit 计算电路calculating element 计算单元calculating machine 计算机calculating punch 穿孔计算机calculating register 计算寄存器calculating unit 计算装置calculation 计算calculation of areas 面积计算calculator 计算机calculus 演算calculus of approximations 近似计算calculus of classes 类演算calculus of errors 误差论calculus of finite differences 差分法calculus of probability 概率calculus of residues 残数计算calculus of variations 变分法calibration 校准canal 管道canal surface 管道曲面cancel 消去cancellation 消去cancellation law 消去律cancellation property 消去性质cancelling of significant figures 有效数字消去canonical basis 典范基canonical coordinates 标准坐标canonical correlation coefficient 典型相关系数canonical distribution 典型分布canonical ensemble 正则总体canonical equation 典型方程canonical equation of motion 标准运动方程canonical expression 典范式canonical factorization 典范因子分解canonical flabby resolution 典型松弛分解canonical form 标准型canonical function 标准函数canonical fundamental system 标准基本系统canonical homomorphism 标准同态canonical hyperbolic system 典型双曲线系canonical image 标准象canonical mapping 标准映射canonical representation 典型表示canonical sequence 标准序列canonical solution 标准解canonical system of differential equations 标准微分方程组canonical variable 典型变量canonical variational equations 标准变分方程canonical variational problem 标准变分问题cantor curve 康托尔曲线cantor discontinum 康托尔密断统cantorian set theory 经典集论cap 交cap product 卡积capacity 容量card 卡片card punch 卡片穿孔机card reader 卡片读数器cardinal 知的cardinal number 基数cardinal product 基数积cardioid 心脏线carrier 支柱carry 进位carry signal 进位信号cartan formula 嘉当公式cartan subalgebra 嘉当子代数cartan subgroup 嘉当子群cartesian coordinate system 笛卡儿坐标系cartesian coordinates 笛卡尔座标cartesian equation 笛卡儿方程cartesian folium 笛卡儿叶形线cartesian product 笛卡儿积cartesian space 笛卡儿空间cartography 制图学cascaded carry 逐位进位casimir operator 卡巫尔算子cassini oval 卡吾卵形线casting out 舍去casting out nines 舍九法catastrophe theory 突变理论categorical judgment 范畴判断categorical proposition 范畴判断categorical syllogism 直言三段论categorical theory 范畴论categoricity 范畴性category 范畴category of groups 群范畴category of modules 模的范畴category of sets 集的范畴category of topological spaces 拓扑空间的范畴catenary 悬链线catenary curve 悬链线catenoid 悬链曲面cauchy condensation test 柯微项收敛检验法cauchy condition for convergence 柯握敛条件cauchy criterion 柯握敛判别准则cauchy distribution 柯沃布cauchy filter 柯嗡子cauchy inequality 柯位等式cauchy integral 柯锡分cauchy integral formula 柯锡分公式cauchy kernel 柯嗡cauchy kovalevskaya theorem 柯慰仆吡蟹蛩箍ǘɡ眵cauchy mean value formula 广义均值定理cauchy net 柯硒cauchy principal value 柯蔚cauchy problem 柯问题cauchy process 柯锡程cauchy residue theorem 残数定理cauchy sequence 柯悟列causal relation 因果关系causality 因果律cause 原因cavity 空腔cavity coefficient 空胴系数cayley number 凯莱数cayley sextic 凯莱六次线cayley transform 凯莱变换ccr algebra ccr代数celestial body 天体celestial coordinates 天体坐标celestial mechanics 天体力学cell 胞腔cell complex 多面复形cellular approximation 胞腔逼近cellular automaton 细胞自动机cellular cohomology 胞腔上同调cellular cohomology group 胞腔上同岛cellular decomposition 胞腔剖分cellular homotopy 胞腔式同伦cellular map 胞腔映射cellular subcomplex 胞腔子复形center 中心center of a circle 圆心center of curvature 曲率中心center of expansion 展开中心center of force 力心center of gravity 重心center of gyration 旋转中心center of inversion 反演中心center of mass 质心center of pressure 压力中心center of principal curvature 助率中心center of projection 射影中心center of symmetry 对称中心centered process 中心化过程centered system of sets 中心集系centi 厘centigram 厘克centimetre 厘米central angle 圆心角central confidence interval 中心置信区间central conic 有心圆锥曲线central derivative 中心导数central difference 中心差分central difference operator 中心差分算子central divided difference 中心均差central element 中心元central extension 中心扩张central extension field 中心扩张域central limit theorem 中心极限定理central line 中线central moment 中心矩central point 中心点central processing unit 中央处理器central projection 中心射影central quadric 有心二次曲面central series 中心群列central symmetric vector field 中心对称向量场central symmetry 中心对称centralizer 中心化子centre 中心centre of a circle 圆心centre of gyration 旋转中心centre of projection 射影中心centre of similarity 相似中心centre of similitude 相似中心centrifugal force 离心力centripetal acceleration 向心加速度centroid 形心certain event 必然事件certainty 必然cesaro mean 纬洛平均cesaro method of summation 纬洛总求法chain 链chain complex 链复形chain condition 链条件chain equivalence 链等价chain equivalent 链等价的chain group 链群chain homotopic 链同伦的chain homotopy 链同伦chain index 链指数chain map 链变换chain of prime ideals 素理想链chain of syzygies 合冲链chain rule 链式法则chain transformation 链变换chainette 悬链线chamber complex 箱盒复形chance 偶然性;偶然的chance event 随机事件chance move 随机步chance quantity 随机量chance variable 机会变量change 变化change of metrics 度量的变换change of the base 基的变换change of the variable 变量的更换channel 信道channel width 信道宽度character 符号character group 特贞群character space 特贞空间characteriatic system 特寨characteristic 特征characteristic boundary value problem 特者值问题characteristic class 示性类characteristic cone 特斩characteristic conoid 特沾体characteristic curve 特怔线characteristic derivation 特阵导characteristic determinant 特招列式characteristic differential equation 特闸分方程characteristic direction 特战向characteristic equation 特战程characteristic exponent 特崭数characteristic function 特寨数characteristic functional 特蘸函characteristic group 特蘸characteristic index 特崭标characteristic initial value problem 特挣值问题characteristic linear system 特者性系统characteristic manifold 特瘴characteristic matrix 特肇阵characteristic number 特正characteristic of a logarithm 对数的首数characteristic parameter 特瘴数characteristic polynomial 特锗项式characteristic pontrjagin number 庞德里雅金特正characteristic root 特争characteristic ruled surface 特毡纹曲面characteristic series 特招characteristic set 特寨characteristic state 特宅characteristic strip 特狰characteristic subgroup 特沼群characteristic surface 特怔面characteristic value 矩阵的特盏characteristic vector 特镇量charge 电荷chart 图chebyshev function 切比雪夫函数chebyshev inequality 切比雪夫不等式chebyshev polynomial 切比雪夫多项式check 校验check digit 检验位check routine 检验程序check sum 检查和chevalley group 歇互莱群chi square distribution 分布chi squared test 检验chi squared test of goodness of fit 拟合优度检验choice function 选择函数chord 弦chord line 弦chord of contact 切弦chord of curvature 曲率弦chordal distance 弦距离christoffel symbol 克里斯托弗尔符号chromatic number 色数chromatic polynomial 色多项式cipher 数字circle 圆circle diagram 圆图circle method 圆法circle of contact 切圆circle of convergence 收敛圆circle of curvature 曲率圆circle of inversion 反演圆circle problem 圆内格点问题circuit free graph 环道自由图circuit rank 圈数circulant 循环行列式circulant matrix 轮换矩阵circular 圆的circular arc 圆弧circular cone 圆锥circular correlation 循环相关circular cylinder 圆柱circular disk 圆盘circular domain 圆形域circular frequency 角频率circular functions 圆函数circular helix 圆柱螺旋线circular measure 弧度circular motion 圆运动circular neighborhood 圆邻域circular orbit 圆轨道circular pendulum 圆摆circular permutation 循环排列circular ring 圆环circular section 圆截面circular sector 圆扇形circular segment 圆弓形circular slit domain 圆形裂纹域circular symmetry 圆对称circular transformation 圆变换circulation 循环circulation index 环粮数circulation of vector field 向量场的循环circulatory integral 围道积分circumcenter 外心circumcentre 外心circumcircle 外接圆circumcone 外切圆锥circumference 圆周circumscribe 外接circumscribed circle 外接圆circumscribed figure 外切形circumscribed polygon 外切多边形circumscribed quadrilateral 外切四边形circumscribed triangle 外切三角形circumsphere 外接球cissoid 蔓叶类曲线cissoidal curve 蔓叶类曲线cissoidal function 蔓叶类函数clairaut equation 克莱罗方程class 类class bound 组界class field 类域class field tower 类域塔class frequency 组频率class function 类函数class interval 组距class mean 组平均class number 类数class of conjugate elements 共轭元素类classical groups 典型群classical lie algebras 典型李代数classical mechanics 经典力学classical sentential calculus 经典语句演算classical set theory 经典集论classical statistical mechanics 经典统计力学classical theory of probability 经典概率论classification 分类classification statistic 分类统计classification theorem 分类定理classify 分类classifying map 分类映射classifying space 分类空间clear 擦去clifford group 克里福特群clifford number 克里福特数clockwise 顺时针的clockwise direction 顺时针方向clockwise rotation 顺时针旋转clopen set 闭开集closable linear operator 可闭线性算子closable operator 可闭算子closed ball 闭球closed circuit 闭合电路closed complex 闭复形closed convex curve 卵形线closed convex hull 闭击包closed cover 闭覆盖closed curve 闭曲线closed disk 闭圆盘closed domain 闭域closed equivalence relation 闭等价关系closed extension 闭扩张closed filter 闭滤子closed form 闭型closed formula 闭公式closed geodesic 闭测地线closed graph 闭图closed graph theorem 闭图定理closed group 闭群closed half plane 闭半平面closed half space 闭半空间closed hull 闭包closed interval 闭区间closed kernel 闭核closed linear manifold 闭线性廖closed loop system 闭圈系closed manifold 闭廖closed map 闭映射closed neighborhood 闭邻域closed number plane 闭实数平面closed path 闭路closed range theorem 闭值域定理closed region 闭域closed riemann surface 闭黎曼面closed set 闭集closed shell 闭壳层closed simplex 闭单形closed solid sphere 闭实心球closed sphere 闭球closed star 闭星形closed subgroup 闭子群closed subroutine 闭型子程序closed surface 闭曲面closed symmetric extension 闭对称扩张closed system 闭系统closed term 闭项closeness 附近closure 闭包closure operation 闭包运算closure operator 闭包算子closure property 闭包性质clothoid 回旋曲线cluster point 聚点cluster sampling 分组抽样cluster set 聚值集coadjoint functor 余伴随函子coalgebra 上代数coalition 联合coanalytic set 上解析集coarser partition 较粗划分coaxial circles 共轴圆cobase 共基cobordant manifolds 配边廖cobordism 配边cobordism class 配边类cobordism group 配边群cobordism ring 配边环coboundary 上边缘coboundary homomorphism 上边缘同态coboundary operator 上边缘算子cocategory 上范畴cochain 上链cochain complex 上链复形cochain homotopy 上链同伦cochain map 上链映射cocircuit 上环道cocommutative 上交换的cocomplete category 上完全范畴cocycle 上闭键code 代吗coded decimal notation 二进制编的十进制记数法codenumerable set 余可数集coder 编器codiagonal morphism 余对角射codifferential 上微分codimension 余维数coding 编码coding theorem 编码定理coding theory 编码理论codomain 上域coefficient 系数coefficient domain 系数域coefficient function 系数函数coefficient functional 系数泛函coefficient group 系数群coefficient of alienation 不相关系数coefficient of association 相伴系数coefficient of covariation 共变系数coefficient of cubical expansion 体积膨胀系数coefficient of determination 可决系数coefficient of diffusion 扩散系数coefficient of excess 超出系数coefficient of friction 摩擦系数coefficient of nondetermination 不可决系数coefficient of rank correlation 等级相关系数coefficient of regression 回归系数coefficient of the expansion 展开系数coefficient of thermal expansion 热膨胀系数coefficient of variation 变差系数coefficient of viscosity 粘性系数coefficient problem 系数问题coefficient ring 系数环coercive operator 强制算子cofactor 代数余子式cofiber 上纤维cofibering 上纤维化cofibration 上纤维化cofilter 余滤子cofinal set 共尾集cofinal subset 共尾子集cofinality 共尾性cofinite subset 上有限子集cofunction 余函数cogenerator 上生成元cogredient automorphism 内自同构coherence 凝聚coherence condition 凝聚条件coherent module 凝聚摸coherent ring 凝聚环coherent set 凝聚集coherent sheaf 凝聚层coherent stack 凝聚层coherent topology 凝聚拓扑coherently oriented simplex 协同定向单形cohomological dimension 上同惮数cohomological invariant 上同祷变量cohomology 上同调cohomology algebra 上同碟数cohomology class 上同掂cohomology functor 上同弹子cohomology group 上同岛cohomology group with coefficients g 有系数g的上同岛cohomology module 上同担cohomology operation 上同邓算cohomology ring 上同捣cohomology sequence 上同凋列cohomology spectral sequence 上同底序列cohomology theory 上同帝cohomotopy 上同伦cohomotopy group 上同伦群coideal 上理想coimage 余象coincidence 一致coincidence number 叠合数coincidence point 叠合点coincident 重合的coinduced topology 余导出拓扑cokernel 上核collect 收集collectionwise normal space 成集体正规空间collective 集体collinear diagram 列线图collinear points 共线点collinear vectors 共线向量collinearity 共线性collineation 直射变换collineation group 直射群collineatory transformation 直射变换collocation method 配置法collocation of boundary 边界配置collocation point 配置点colocally small category 上局部小范畴cologarithm 余对数colorable 可着色的column 列column finite matrix 列有限矩阵column matrix 列阵column rank 列秩column space 列空间column vector 列向量combination 组合combination principle 结合原理combination with repetitions 有复组合combination without repetition 无复组合combinatorial analysis 组合分析combinatorial closure 组合闭包combinatorial dimension 组合维数combinatorial geometry 组合几何学combinatorial manifold 组合廖combinatorial method 组合方法combinatorial optimization problem 组合最优化问题combinatorial path 组合道路combinatorial problem 组合最优化问题combinatorial sphere 组合球面combinatorial sum 组合和combinatorial theory of probabilities 概率组合理论combinatorial topology 组合拓朴学combinatorially equivalent complex 组合等价复形combinatories 组合分析combinatory logic 组合逻辑combinatory topology 组合拓朴学combined matrix 组合矩阵comma 逗点command 命令commensurability 可通约性commensurable 可通约的commensurable quantities 可公度量common denominator 公分母common difference 公差common divisor 公约数common factor 公因子common factor theory 公因子论common fraction 普通分数common logarithm 常用对数common measure 公测度common multiple 公倍元common perpendicular 公有垂线common point 公共点common ratio 公比common tangent of two circles 二圆公切线communality 公因子方差communication channel 通讯通道commutant 换位commutation law 交换律commutation relation 交换关系commutative 可换的commutative diagram 交换图表commutative group 交换群commutative groupoid 阿贝耳广群commutative law 交换律commutative lie ring 交换李环commutative ordinal numbers 交换序数commutative ring 交换环commutativity 交换性commutator 换位子commutator group 换位子群commute 交换compact 紧的compact convergence 紧收敛compact group 紧群compact open topology 紧收敛拓扑compact operator 紧算子compact set 紧集compact space 紧空间compact subgroup 紧子群compact support 紧支柱compactification 紧化compactification theorem 紧化定理compactness 紧性compactness theorem 紧性定理compactum 紧统comparability of cardinals 基数的可比较性comparable curve 可比曲线comparable function 可比的函数comparable topology 可比拓扑comparable uniformity 可比一致性comparison function 比较函数comparison method 比较法comparison series 比较用级数comparison test 比较检验comparison theorem 比较定理compass 两脚规compatibile condition 相容性条件compatibility 一致性compatibility condition 相容性条件compatible system of algebraic equations 相容代数方程组compatible topology 相容拓扑学compensate 补偿compensating method 补偿法compensation 补偿compensation of error 误差的补偿compiler 编译程序compiling routine 编译程序complanar line 共面线complele induction 数学归纳法complement 补集complement of an angle 余角complementary 补的complementary angle 余角complementary degree 余次数complementary divisor 余因子complementary event 余事件complementary function 余函数complementary graph 余图complementary ideal 余理想complementary laws 补余律complementary module 补模complementary modulus 补模数complementary set 补集complementary space 补空间complementary submodules 补子模complementary subset 余子集complementary subspace 补子空间complemented lattice 有补格complete abelian variety 完备阿贝耳簇complete accumulation point 完全聚点complete axiom system 完备公理系统complete category 完全范畴complete class 完备类complete continuity 完全连续性complete disjunction 完全析取complete elliptic integral 完全椭圆积分complete field 完全域complete field of sets 集的完全域complete graph 完全图complete group 完全群complete group variety 完备群簇complete homomorphism 完全同态complete induction 数学归纳法complete integral 完全积分complete intersection 完全交叉complete lattice 完全格complete linear system 完备线性系统complete local ring 完全局部环complete measure 完全测度complete measure space 完备测度空间complete metric space 完备度量空间complete normality axiom 完全正规性公理complete ordered field 全序域complete orthogonal sequence 完全正交序列complete orthogonal set 完全正交系complete orthogonal system 完全正交系complete orthonormal sequence 完备标准正交序列complete orthonormal system 完备标准正交系complete probability space 完全概率空间complete quadrangle 完全四点形complete quadrilateral 完全四边形complete reducibility theorem 完全可约性定理complete regularity separation axiom 完全正则性分离公理complete reinhardt domain 完全赖因哈耳特域complete set 完全集complete solution 完全积分complete space 完备空间complete subcategory 完全子范畴complete system 完备系complete system of functions 函数完备系complete system of fundamental sequences 完全基本序列系complete system of invariants 完全的不变量系complete tensor product 完全张量积completed shell 闭壳层completely additive 完全加性的completely additive family of sets 完全加性集族completely additive measure 完全加性测度completely compact set 完全紧集completely continuous function 完全连续函数completely continuous linear operator 完全连续线性算子completely continuous mapping 全连续映射completely continuous operator 全连续映射completely distributive lattice 完全分配格completely homologous maps 完全同党射completely independent system of axioms 完全独立公理系统completely integrable 完全可积的completely integrable system 完全可积组completely integrally closed 完全整闭的completely mixed game 完全混合对策completely monotone 完全单的completely monotonic function 完全单弹数completely monotonic sequence 完全单凋列completely multiplicative 完全积性的completely multiplicative function 完全积性函数completely primary ring 完全准素环completely reducible 完全可约的completely reducible group 完全可约群completely regular filter 完全正则滤子completely regular space 完全正则空间completely regular topology 完全正则拓扑completely separated sets 完全可离集completely specified automaton 完全自动机completely splitted prime ideal 完全分裂素理想completely transitive group 全可迁群completeness 完全性completeness theorem 完全性定理completion 完备化complex 复形complex analytic fiber bundle 复解析纤维丛complex analytic manifold 复解析廖complex analytic structure 复解析结构complex cone 线丛的锥面complex conjugate 复共轭的complex conjugate matrix 复共轭阵complex curve 复曲线complex curvelinear integral 复曲线积分complex domain 复域complex experiment 析因实验complex field 复数域complex flnction 复值函数complex fraction 繁分数complex group 辛群complex line 复线complex line bundle 复线丛complex manifold 复廖complex multiplication 复数乘法complex number 复数complex number plane 复数平面complex plane with cut 有割的复平面complex quantity 复量complex root 复根complex series 复级数complex sphere 复球面complex surface 线丛的曲面complex unit 单位复数complex valued function 复值函数complex variable 复变量complex vector bundle 复向量丛complex velocity potential 复速度位势complexity 复杂性complication 复杂化component 分量component of variance 方差的分量componentwise convergence 分量方式收敛composable 组成的compose 组成composite 合成composite divisor 合成除数composite function 合成函数composite functor 合成函子composite group 合成群composite hypothesis 复合假设composite number 合成数composite probability 复合概率composition 合成composition algebra 合成代数composition factor 合成因子composition homomorphism 合成同态composition of vector subspaces 向量子空间的合成composition operator 合成算子composition series 合成列compound determinant 复合行列式compound event 复合事件compound function 合成函数compound number 合成数compound probability 合成概率compound proportion 复比例compound rule 复合规则computable function 可计算函数computation 计算computational error 计算误差computational formula 计算公式computational mistake 计算误差compute 计算computer 计算机computing center 计算中心computing element 计算单元computing machine 计算机computing time 计算时间comultiplication 上乘法concave 凹的concave angle 凹角concave convex game 凹击对策concave curve 凹曲线concave function 凹函数concave polygon 凹多边形concavity 凹性concavo convex 凹击的concentration 集中;浓度concentration ellipse 同心椭圆concentric circles 同心圆concept 概念conchoid 蚌线conchoidal 蚌线的conclusion 结论concomitant variable 相伴变量concrete number 名数concurrent form 共点形式concurrent planes 共点面concyclic points 共圆点condensation of singularities 奇点的凝聚condensation point 凝聚点condensation principle 凝聚原理condition equation 条件方程condition for continuity 连续性条件condition number 条件数condition of connectedness 连通性条件condition of positivity 正值性条件conditional convergence 条件收敛conditional definition 条件定义conditional density 条件性密度conditional distribution 条件分布conditional entropy 条件熵conditional equation 条件方程conditional event 条件性事件conditional gradient method 条件梯度法conditional inequality 条件不等式conditional instability 条件不稳定conditional instruction 条件指令conditional jump 条件转移conditional mathematical expectation 条件数学期望conditional probability 条件概率conditional probability measure 条件概率测度conditional proposition 条件命题conditional sentence 条件命题conditional stability 条件稳定性conditional transfer of control 条件转移conditionally compact set 条件紧集conditionally complete 条件完备的conditionally convergent 条件收敛的conditionally convergent series 条件收敛级数conditionally well posed problems 条件适定的问题conditioned observation 条件观测conditioning number 条件数conditions of similarity 相似条件conduction 传导conductivity 传导率conductor 导体;前导子conductor ramification theorem 前导子分歧定理cone 锥cone of a complex 复形锥面cone of a simplex 单形锥面confidence belt 置信带confidence coefficient 置信系数confidence ellipse 置信椭圆confidence ellipsoid 置信椭面confidence interval 置信区间confidence level 置信水平confidence limit 置信界限confidence region 置信区域configuration 布局configuration space 构形空间confinal 共尾的confinality 共尾性confirmation 证实confluent divided difference 合六差confluent hypergeometric equation 合镣超几何微分方程confluent hypergeometric function 合连几何函数confluent hypergeometric series 合连几何级数confluent interpolation polynomial 汇合内插多项式confocal conic sections 共焦二次曲线confocal conics 共焦二次曲线confocal quadrics 共焦二次曲面conformable matrices 可相乘阵conformal 保角的conformal curvature tensor 保形曲率张量conformal differential geometry 保形微分几何学conformal geometry 保形几何conformal mapping 保角素示conformal projection 保形射影conformal representation 保角素示conformal transformation 保角映射conformally connected manifold 保形连通廖conformally geodesic lines 保形测地线confounding 混杂confrontation 比较confusion 混乱congruence 同余式congruence group 同余群congruence method 同余法congruence of lines 线汇congruence relation 同余关系congruence subgroup 同余子群congruence zeta function 同余函数congruent 同余的congruent mapping 合同映射congruent number 同余数congruent transformation 合同映射conic 圆锥曲线conic function 圆锥函数conic section 圆锥曲线conical helix 圆锥螺旋线conical surface 锥面conics 圆锥曲线论conjugate 共轭的conjugate axis 共轭轴conjugate class 共轭类conjugate complex 共轭复形conjugate complex number 共轭复数conjugate convex function 共轭击函数conjugate curve 共轭曲线conjugate curve of the second order 共轭二次曲线conjugate diameter 共轭直径conjugate direction 共轭方向conjugate dyad 共轭并向量conjugate element 共轭元素conjugate exponent 共轭指数conjugate field 共轭域conjugate foci 共轭焦点conjugate function 共轭函数conjugate gradient method 共轭梯度法conjugate hyperbola 共轭双曲线conjugate latin square 共轭拉丁平conjugate line 共轭直线conjugate number 共轭数conjugate operator 共轭算子conjugate points 共轭点conjugate quaternion 共轭四元数conjugate root 共轭根conjugate ruled surface 共轭直纹曲面conjugate series 共轭级数conjugate space 共轭空间conjugate transformation 共轭变换conjugate vector 共轭向量conjugation map 共轭映射conjugation operator 共轭算子conjunction 合取conjunctive normal form 合取范式connected 连通的connected asymptotic paths 连通渐近路线connected automaton 连通自动机connected category 连通范畴connected chain 连通链connected complex 连通复形connected component 连通分支connected curve 连通曲线connected domain 连通域connected graph 连通图connected group 连通群connected sequence of functors 函子的连通序列connected set 连通集connected space 连通空间connected sum 连通和connectedness 连通性connecting homomorphism 连通同态connecting morphism 连通同态connecting path 连接道路connection 联络connection component 连通分量connectivity 连通性connex 连通conoid 劈锥曲面conormal 余法线conormal image 余法线象conrol chart technique 控制图法consequence 后承consequent 后项conservation law 守恒律conservation of angular momentum 角动量守恒conservation of energy 能量守恒conservation of mass 质量守恒conservation of momentum 动量守恒conservative extension 守恒扩张conservative field of force 保守力场conservative force 保守力conservative measurable transformation 守恒可测变换conservative vector field 守恒向量场consistency 相容性consistency conditions 相容条件consistency of equations 方程组的相容性consistency problem 相容性问题consistencyproof 相容性的证明consistent axiom system 相容性公理系consistent equations 相容方程组consistent estimator 相容估计consistent system of equations 相容方程组consistent test 相容检验constancy of sign 符号恒性constant 常数constant coefficient 常系数constant field 常数域constant function 常值函数constant mapping 常值映射constant of integration 积分常数constant of proportionality 比例系数constant of structure 构造常数constant pressure chart 等压面图constant pressure surface 等压面constant sheaf 常数层constant sum game 常和对策constant term 常数项constant value 定值constituent 组分constitutional diagram 组分图constrained game 约束对策constrained maximization 约束最大化constrained minimization 约束最小化constrained optimization 约束最优化constraint 约束construct 准constructibility 可构成性constructible 可构成的constructible map 可构成映射constructible set 可构成集construction 构成construction problem 准题constructive dilemma 构造二难推论constructive existence proof 可构造存在证明constructive mathematics 可构造数学constructive ordinal number 可构造序数consumer's risk 用户风险contact 接触contact angle 接触角contact point 接触点contact surface 接触面contact transformation 切变换content 含量context sensitive grammar 上下文有关文法contiguity 接触contiguous confluent hypergeometric function 连接合连几何函数contiguous hypergeometric function 连接超几何函数contiguous map 连接映射contingency 随机性contingency table 列contingent 偶然事故continuability 可延拓性continuation method 连续法continued equality 连等式continued fraction 连分数continued fraction expansion 连分式展开式continued proportion 连比例continuity 连续性continuity axiom 连续性公理continuity condition 连续性条件continuity equation 连续方程continuity in the mean 均方连续性continuity interval 连续区间continuity method 连续法continuity of function 函数的连续性continuity on both sides 双边连续性continuity on the left 左连续性continuity on the right 右连续性continuity principle 连续性原理continuity theorem 连续性定理continuous 连续的continuous analyzer 连续分析器continuous approximation 连续近似continuous curve 连续曲线continuous differentiability 连续可微性continuous distribution 连续分布continuous distribution function 连续分布函数continuous dynamical system 连续动力系统continuous function 连续函数continuous function in the mean 均方连续函数continuous game 连续对策continuous geometry 连续几何continuous group 拓扑群continuous homology 连续同调continuous homology group 连续同岛continuous image 连续象continuous in x 依x连续的continuous limit 连续极限continuous map 连续映射continuous on the left 左方连续的。