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Numerical range of some doubly stochastic matricesqKristin A.Camenga a ,Patrick X.Rault b ,Daniel J.Rossi b ,Tsvetanka Sendova c ,Ilya M.Spitkovsky d ,⇑aDepartment of Mathematics and Computer Science,Houghton College,Houghton,NY 14744,USA bDepartment of Mathematics,State University of New York at Geneseo,Geneseo,NY 14454,USA cDepartment of Mathematics and Computer Science,Bennett College,Greensboro,NC 27401,USA dDepartment of Mathematics,College of William and Mary,Williamsburg,VA 23187,USAa r t i c l e i n f o Keywords:Numerical rangeDoubly stochastic matricesa b s t r a c tA classification of all possible shapes is given for numerical ranges of 4Â4doubly stochas-tic matrices.The tests determining the shape are also provided,along with illustrating examples.Ó2013Elsevier Inc.All rights reserved.1.IntroductionWith F standing for either the field R of real or C of complex numbers,we denote by F n Âm the linear space (algebra,if m ¼n )of the n Âm matrices with entries in F .We also use the standard abbreviation F n :¼F n Â1,and supply C n with the innerproduct h x ;y i ¼P n j ¼1x j y j and the respective norm k x k :¼h x ;x i1=2.The numerical range of A 2C n Ânby definition isW ðA Þ¼fh Ax ;x i :k x k ¼1g :This is a compact convex subset of C containing the spectrum r ðA Þof A ;see e.g.[1]or [2,Chapter 1]for these and other fundamental properties of W ðA Þ.We will repeatedly be using the following:(i)W ðA Þis unitarily invariant:W ðU T AU Þ¼W ðA Þfor any unitary U 2C n Ân .(ii)If A is unitarily reducible,that is,unitarily similar to the direct sum A 1ÈA 2,then W ðA Þis the convex hull ofW ðA 1Þ[W ðA 2Þ;that is W ðA Þ¼conv f W ðA 1Þ;W ðA 2Þg .From these properties it follows in particular that W ðA Þ¼conv r ðA Þwhenever A is normal.On the other hand,W ðA Þis an ellipse with the foci at the eigenvalues of A if A 2C 2Â2and is not normal (the Elliptical Range Theorem).The case n ¼3has also been treated fully,see [3]and its English translation [4]for the classification,and [5–7]for the pertinent tests.However,starting with n ¼4,the shape of W ðA Þis known only for some special classes of matrices.Here we concentrate our attention on doubly stochastic matrices,that is,matrices A 2R n Ân for which the entries are non-negative while the row and column sums are all equal to one:Xn j ¼1a ij ¼1for i ¼1;...;n ;and Xn i ¼1a ij ¼1for j ¼1;...;n :ð1:1Þ0096-3003/$-see front matter Ó2013Elsevier Inc.All rights reserved./10.1016/j.amc.2013.06.011qThe work on this paper was begun during the REUF workshop at the American Institute of Mathematics in July 2011,supported by the NSF.⇑Corresponding author.E-mail addresses:kristin.camenga@ (K.A.Camenga),rault@ (P.X.Rault),tsendova@ (T.Sendova),ilya@math. ,imspitkovsky@ (I.M.Spitkovsky).Observe that (1.1)can be interpreted as e ¼½1;...;1 T being an eigenvector of both A and A T ,corresponding to the eigenvalue one.Thus,doubly stochastic matrices are unitarily reducible to the direct sum of ð1Þwith some A 12R ðn À1ÞÂðn À1Þ:U T AU ¼ð1ÞÈA 1;ð1:2Þwhere U is an orthogonal matrix with the first column equal 1ffiffip e .Consequently,W ðA Þin this case is the convex hull of W ðA 1Þwith the point 1.Moreover,1is a corner point of W ðA Þ[8].If A (equivalently,A 1)is normal,then W ðA Þis the convex hull of r ðA Þ¼r ðA 1Þ[f 1g .Thus we are only interested in the case of non-normal A 1.This,combined with the Elliptical Range Theorem,immediately implies a complete description of numerical ranges for 3Â3doubly stochastic matrices,normal or otherwise;see [9]for details.Our aim in this paper is to realize a similar plan for n =4.According to [4],for a non-normal A 1there are a priori three possibilities:(i)W ðA 1Þis the convex hull of a point and an ellipse (with the point lying either inside or outside the ellipse),(ii)the boundary of W ðA 1Þcontains a flat portion,with the rest of it lying on a 4th degree algebraic curve,or (iii)W ðA 1Þhas an ovular shape,bounded by a 6th degree algebraic curve.Since W ðA 1Þalso is symmetric with respect to the real axis (as A 1has real entries),the ellipse in case (i)has one of its axes on R ,and the flat portion in case (ii)is vertical.In the forthcoming sections we provide criteria to distinguish between the three possibilities,show that each of them actually materializes,and describe the respective shapes of each W ðA Þ.To simplify the terminology,we label the shape of W ðA Þby the degree of a non-flat portion of its boundary.2.Numerical range of 2nd degreeIn this section we characterize the case when the non-flat portion of @W ðA Þis an elliptical arc.Theorem 1.Let A be a 4Â4doubly stochastic matrix.Then @W ðA Þconsists of elliptical arcs and line segments if and only ifl :¼tr A À1þtr A 3Àtr ðA T A 2Þtr ðA T A ÞÀtr A 2ð2:1Þis an eigenvalue of A (multiple,ifl ¼1).If,in addition,tr A À1À3l >0;ðtr A À1À3l Þ2Àtr ðA T A Þþ1þ2ðdet A Þ=l þl 2>0;ð2:2Þthen W ðA Þalso has corner points at l and 1,and thus four flat portions on the boundary.Otherwise,1is the only corner point of W ðA Þ,and @W ðA Þconsists of two flat portions and one elliptical arc.Proof.Due to (1.2),the right hand side of (2.1)coincides with tr A 1þtr A 31Àtr ðA T 1A 21Þtr ðA T 1A 1ÞÀtr A 21.From [7,Theorem 5],it then follows thatW ðA 1Þis the convex hull of an ellipse and a point if and only if l is an eigenvalue of A 1.This proves the first statement of the theorem.Further,recall from [5]that if the above condition on l is satisfied,then W ðA 1Þ¼conv f E ;l g ,where E is an ellipse withfoci at two other eigenvalues k 1;k 2of A 1and the minor axis d ¼tr A T 1A 1Àj k 1j 2Àj k 2j 2Àl 2.So,W ðA Þ¼conv f E ;l ;1g has two corner points (at 1and l )if l lies to the left of E ,and only one corner point (at 1)otherwise.Now observe that l lies to the left of E if and only if it is located in the exterior of E ,that is,ðj k 1Àl j þj k 2Àl jÞ2Àj k 1Àk 2j 2>d ;ð2:3Þand to the left of its center ðk 1þk 2Þ=2.The second condition is equivalent to3l <k 1þk 2þl ;where the right hand side is of course tr A À1.Collecting all the terms on the same side of the inequality,we arrive at the first inequality in (2.2).It remains to show the equivalence of its second inequality and (2.3).To this end,we consider separately the case of complex conjugate and real k j .Case 1.k 1¼k 2¼:a þbi .Inequality (2.3)then can be rewritten as4ða Àl Þ2þb 2Àð2b Þ2>tr ðA T 1A 1ÞÀ2ða 2þb 2ÞÀl 2:Simplifying the left side and right side individually yieldsðtr A 1À3l Þ2>tr ðA T 1A 1ÞÀ2ðdet A 1Þ=l Àl 2:Putting this in terms of A ,we arrive at the desired second inequality in (2.2).Case 2.All eigenvalues of A 1are real.Recalling that l lies to the left of the eigenvalues,inequality (2.3)becomesðk 1þk 2À2l Þ2Àðk 1Àk 2Þ2>tr ðA T 1A 1ÞÀk 21Àk 22Àl 2:K.A.Camenga et al./Applied Mathematics and Computation 221(2013)40–4741Working with the left hand side gives4l2þ4k1k2À4ðk1þk2Þl>trðA T1A1ÞÀk21Àk22Àl2:Rewriting this and collecting terms yields5l2þ2k1k2À4ðtr A1ÀlÞlþðk1þk2Þ2ÀtrðA T1A1Þ>0:After putting this in terms of A and factoring gives5l2þ2ðdet AÞ=lþðtr AÀ1ÀlÞðtr AÀ1À5lÞÀtrðA T AÞþ1>0:Noting thatðtr AÀ1À3lÞÀ2lðÞðtr AÀ1À3lÞþ2lðÞ¼ðtr AÀ1À3lÞ2À4l2completes the proof.hThe following examples demonstrate that each case considered by Theorem1actually occurs. Example1.Consider the doubly stochastic matrixA¼011510111917537019B BB@1C CC A:Using equation(2.1),we compute l¼À1=3.The characteristic polynomial of A is1=128Àð11xÞ=192Àð193x2Þ=384Àð43x3Þ=96þx4;which evaluated atÀ1=3is0.In addition,we compute the formulas in inequalities(2.2)astr AÀ1À3l¼43=96andðtr AÀ1À3lÞ2ÀtrðA T AÞþ1þ2ðdet AÞ=lþl2¼À779=9216:Since the latter is negative,Theorem1states that@WðAÞhas twoflat portions and one elliptical arc.Indeed,A reduces uni-tarily toð1ÞÈA1for some3Â3matrix A1,and in Fig.1we give WðAÞ,and the horizontal ellipse it contains,WðA1Þ. Example2.Next,we give a similar example where the elliptical arc is part of a vertical ellipse:A¼0131451213012161418143851213240124B BB@1C CC A:By coincidence,we again compute l¼À1=3,and though the characteristic polynomial is now Numerical range of A with twoflat portions and an elliptical arc on the boundary,and the ellipse it contains.The eigenvalues of A are indicated 42K.A.Camenga et al./Applied Mathematics and Computation221(2013)40–47Example3.Next,we give an example of a doubly stochastic matrix,A,where@WðAÞconsists of fourflat portions and two elliptical arcs.Let a be the second largest root of the four real roots of the polynomial À28995327þ108880894xþ143424512x2À778960896x3þ603979776x4and let b be the second smallest of the four real roots ofÀ5072þ17953xþ310274x2À1013248x3þ786432x4.WithA¼0a13Àa1011b11191Àb2Àb255Àa0À509þbþa 0B BB@1C CC A;the characteristic polynomial evaluated at l is identically0.The two formulas in inequality(2.2)come out to two positive numbers,roughly0:65and0:31respectively.Hence WðAÞis the convex hull of an ellipse and two points outside of the el-lipse.Fig.3shows WðAÞ,and that the ellipse is vertical.stly,we give an example of a doubly stochastic matrix,A,where WðAÞis the convex hull of a horizontal ellipse and two points.Let a be the second smallest root of the polynomialÀ301228513þ853379010xþ4421068800x2À17180393472x3þ14495514624x4and let b be the second largest root of the polynomial À10016À19031xþ1045714x2À3505152x3þ3145728x4.Just as in the previous examples,the characteristic polynomial of the matrixA¼0a13Àa1011b10951214275512Àb23Àb403512Àa0À6971536þbþaB BB@1C CC Arange A with twoflat portions and an elliptical arc on the boundary,and the contained vertical ellipse.The eigenvaluesvanishes at l .In this case the two formulas in inequality (2.2)come out to two positive numbers of about 0:77and 0:21respectively.Hence W ðA Þis the convex hull of an ellipse and two points outside of the ellipse and has four flat portions.Fig.4shows W ðA Þ,and that in this case the ellipse is horizontal.3.Numerical range of 4th degreeBy Birkhoff’s Theorem,the doubly stochastic matrices form a convex polytope in R n Ân ,its vertices being the n !permuta-tion matrices.So,it is not surprising that the set of all doubly stochastic matrices A with the fixed symmetric part H ¼ðA þA T Þ=2also forms a polytope.The latter is stated more rigorously in the following lemma.Lemma 2.Consider A 2R n Ân with given symmetric part H:A ¼H þK ;where K ¼ðA ÀA T Þ=2:ð3:1ÞThen A is doubly stochastic if and only if H is doubly stochastic,the row (and thus,column)sums of K are zero,and its upper diag-onal entries satisfyj k ij j 6h ij ;16i <j 6n :ð3:2ÞProof.Directly from (1.1)it follows that if A is doubly stochastic,then its symmetric part H is doubly stochastic as well.On the other hand,if H is doubly stochastic,then for A given by (3.1)to be doubly stochastic it is necessary and sufficient that K has zero row/column sums and h ij Æk ij P 0.The latter condition is equivalent to (3.2).hNote that antisymmetric matrices with zero row/column sums form a ðn À1Þðn À2Þ-dimensional subspace of R n Ân .So,(3.2)can be rewritten as a system of n ðn À1Þ=2linear inequalities in ðn À1Þðn À2Þ=2variables.In particular,for n ¼4we haveK ¼x 001À10000À10110À102666437775þy 010À1À10100À1011À12666437775þz 010À1À1001001À102666437775;ð3:3Þand (3.2)takes the formj x j 6h 13;j y j 6h 23;j z j 6h 24;j x þy j 6h 34;j y þz j 6h 12;j x þy þz j 6h 14:ð3:4ÞFor the following theorem,recall our notation e for the vector with all the entries equal to one.Theorem 3.Let A 2R 4Â4be a doubly stochastic matrix.Then @W ðA Þconsists of 4th degree algebraic arcs and line segments if andonly if in its representation (3.1),H ¼k I þ1Àk 4ee T Æff Tð3:5Þfor some f 2R 4orthogonal to e,and k satisfyingÀ1ð1þ4min fÆf 2i gÞ6k 61þ4min i –j fÆf i f j g ;ð3:6ÞNumerical range of A with four flat portions and elliptical arcs on the boundary,and the numerical range of the unitary reduction A 1of a horizontal ellipse and l .The eigenvalues are indicated by the points.while K is given by(3.3),satisfies(3.4),and has no eigenvectors lying in f f;e g?.The upper signs in(3.5)and(3.6)correspond to the case when@WðAÞhas threeflat portions,one of them being vertical and lying to the left of WðAÞ.Alternatively,if(3.5)and(3.6) hold with the lower signs,@WðAÞhas only twoflat portions,with1being their common endpoint.Proof.In order for@WðAÞto contain algebraic arc(s)of4th degree,the same should be true for@WðA1Þ,with A1as in(1.2). According to[5,Section3],this happens if and only if A1is unitarily irreducible and its real part H1has a multiple maximal or minimal eigenvalue k.Equivalently,k is a multiple eigenvalue of H from(3.1),which is either minimal or second in value after one.Along with the double stochasticity of H(see Lemma2),this implies(3.5)and(3.6).Conditions(3.3)and(3.4)on K follow from the description of all doubly stochastic matrices withfixed symmetric part (Lemma2again).Finally,A1is unitarily irreducible if and only if e is the only common eigenvector of H and K,that is,if K has no eigenvectors orthogonal to e and f.Suppose now that all the conditions(3.3)–(3.6)are satisfied.Then A1is unitarily irreducible and,according to[5],its numerical range has aflat portion on its boundary lying on the vertical line x¼k.Just as A reduces toð1ÞÈA1,we reduce H to ð1ÞÈH1.If k is the minimal eigenvalue of H1(that is,(3.5)and(3.6)hold with the upper signs),then this vertical portion is to the left of WðA1Þand thus of WðAÞ.Therefore,it is part of@WðAÞ,along with two otherflat portions,originating at1.On the other hand,for(3.5)and(3.6)holding with the lower signs,the vertical portion of@WðA1Þis to the right of WðA1Þand is thus absorbed by the interior of WðAÞ.hTo demonstrate that such matrices actually exist,we give examples of numerical ranges withflat portions on the bound-ary to illustrate the above theorem.Example5.Consider the following matrix.A¼20212012 03021220B BB@1C CC A:Eq.(3.1)becomes A¼HþK,where H¼ðAþA TÞ=2and K¼ðAÀA TÞ=2.H has three eigenspaces:thefirst generated by e, with eigenvalue1,the second generated by½À3;1;1;1 T with eigenvalue k2¼1,and the third a2-dimensional space gener-ated by½0;À1;0;1 T and½0;À1;1;0 T corresponding to the double eigenvalue k3¼À25¼puting HÀk3IÀ1Àk34ee T yieldsa symmetric positive semi definite matrixM¼920À320À320À320À320120120120À320120120120À320120120120B BB@1C CC Aof rank1,and the vector f from its representation M¼ff T is determined up to a sign and can be found via dividing any of theK.A.Camenga et al./Applied Mathematics and Computation221(2013)40–4745A ¼05121413111151216161411110B B B B B B @1C C C C CC A :The Hermitian part,H ,again has three eigenspaces:the first generated by e ,with eigenvalue 1,the second generated by½À3;1;1;1 T with eigenvalue k 2¼À13,and the third a 2-dimensional space generated again by ½0;À1;0;1 T and ½0;À1;1;0 Tand now corresponding to the double eigenvalue k 3¼À1¼k puting H Àk 3I À1Àk 3ee T yields a symmetric negative semi definite matrix M of rank 1.Similarly to how this was done in Example 5,M ¼Àff T ,where f ¼14ffiffi3p ½À3;1;1;1 T ,and thus H ¼k 3I þ1Àk 34ee T Àff T .Furthermore,note that inequality (3.6)is satisfied:À112 À112 1112.We conclude that when we reduce A to ð1ÞÈA 1,withA 1¼À1À1À10À1À100À1B @1C A ;@W ðA 1Þhas a vertical flat portion on the right.The convex hull of W ðA 1Þand f 1g ,which is the numerical range of A ,does not have a vertical flat portion.These numerical ranges are both illustrated in Fig.6.4.Numerical range of 6th degreeAll numerical ranges which do not fall into one of the above categories will have an ovular shaped boundary curve of de-gree 6.Example 7.In this example we will show that the following matrix does not satisfy the requirements of Theorems 1and 3.A ¼3101103103103101531015121333110B B B @1C C C A :We first compute l ¼1=55in (2.1).The characteristic polynomial of A evaluated at l is nonzero,and indeed the first of the inequalities in (2.2)does not hold.Theorem 1then implies that the numerical range of A is not the convex hull of an ellipse and two points.Next,H ¼12ðA þA T Þ¼31131171171131110B B B @1C C C A46K.A.Camenga et al./Applied Mathematics and Computation 221(2013)40–47A 1¼À111011À10À10B @1C A tells us that this ovular shape actually is W ðA 1Þ.Fig.7shows the numerical ranges of both A and A 1.References[1]K.E.Gustafson,D.K.M.Rao,Numerical Range,The Field of Values of Linear Operators and Matrices,Springer,New York,1997.[2]R.A.Horn,C.R.Johnson,Topics in Matrix Analysis,Cambridge University Press,Cambridge,1991.[3]R.Kippenhahn,Über den Wertevorrat einer matrix,Math.Nachr.6(1951)193–228.[4]R.Kippenhahn,On the numerical range of a matrix,Linear Multilinear Algebra 56(1–2)(2008)185–225(translated from the German by Paul F.Zachlin and Michiel E.Hochstenbach).[5]D.Keeler,L.Rodman,I.Spitkovsky,The numerical range of 3Â3matrices,Linear Algebra Appl.252(1997)115–139.[6]L.Rodman,I.M.Spitkovsky,3Â3matrices with a flat portion on the boundary of the numerical range,Linear Algebra Appl.397(2005)193–207.[7]P.Rault,T.Sendova,I.M.Spitkovsky,3-by-3Matrices with elliptical numerical range revisited,Electron.Linear Algebra 26(2013)158–167.[8]C.R.Johnson,An inclusion region for the field of values of a doubly stochastic matrix based on its graph,Aequationes Math.17(2–3)(1978)305–310.[9]P.Nylen,T.Y.Tam,Numerical range of a doubly stochastic matrix,Linear Algebra Appl.153(1991)161–176.。
CMmi@ Selected Technical Terms in Mechanics of Materials材料力学部分专业术语中英文对照（version 1.0, September 4, 2011）This tabulated list of selected technical terms in mechanics of materials is developed by Changwen Mi to facilitate the students in various engineering majors at the Southeast University. This file reflects part of our constant efforts in implementing bilingual teaching of a series of undergraduate and graduate mechanics courses hosted by the Department of Engineering Mechanics at the Southeast University. We had made every effort to ensure the accuracy and completeness of this file for the students’ sake. We, however, make no guarantee of the effects of using this file.Geometric properties of an area 截面几何性质centroid 形心centroidal axis 形心轴first moment of an area 静矩moment of inertia; second moment of an area 惯性矩parallel axis theorem 平行移轴定理products of inertia 惯性积polar moment of inertia 极惯性矩radius of gyration 惯性半径composite area组合截面principal centroidal axis主形心轴principal moment of inertia主惯性矩principal moments of inertia about centroidal axes 主形心惯性矩Structural members 构件bar 杆prismatic bars 等截面直杆CMmi@ shaft 轴column 柱（只受压缩）thin-walled tubes （闭口）薄壁杆thin-walled open tubes 开口薄壁杆pressure vessel 压力容器beam 梁neutral surface 中性层neutral axis 中性轴simply supported beams 简支梁cantilever beams 悬臂梁composite beams 复合梁overhanging beams 外伸梁continuous beams 连续梁fully stressed beams; beams of constant strength 等强度梁beams of variable cross section 变截面梁wide-flange beams 工字梁web 腹板flange 翼缘fixed support; clamped support 固定端pin support 固定铰支座roller support 可动铰支座curved beams 曲梁truss 桁架frame 刚架cross-section 横截面oblique cross-sectionaxis 轴线rigid joint 刚性结点CMmi@ Loads 荷载/载荷force 力force couple 力偶moment 力矩moment of a couple 力偶矩unit load 单位力unit couple 单位力偶concentrated loads 集中力distributed loads 分布力intensity of distributed loads 分布力的集度surface force 面力body force 体积力static loads 静载dynamic loads 动载allowable loads 许用荷载reaction 反作用力internal forces 内力axial force 轴力shear force 剪力Stress, Strain and Deformation 应力、应变及变形normal stress 正应力nominal stress 名义应力true stress 真实应力average stress 平均应力maximum stress 最大应力minimum stress 最小应力allowable stress 许用应力shear stress 剪切应力pure shear 纯剪切normal strain 正应变nominal strain 名义应变true strain 真实应变shear strain 切应变deformation 变形displacement 位移deflection 挠曲Common Terms in Mechanics of Deformable Bodies 可变形体力学常用术语mechanics of materials 材料力学strength of materials 材料力学mechanics of deformable bodies 变形体力学strength 强度stiffness 刚度stability 稳定性homogeneity/homogeneous 均质/匀质的continuity/continuous 连续性/连续的isotropy/isotropic 各向同性/各向同性的infinitesimal elastic deformation 微小弹性变形elasticity 弹性elastic deformation弹性变形linearly elastic body线性弹性体mechanical properties力学性质plasticity 塑性elastoplastic materials 弹塑性材料tension 拉伸compress 压缩shearing 剪切torsion 扭转bending 弯曲buckling 失稳allowable load method 许用荷载法allowable stress 许用应力allowable stress method 许用应力法method of safety factor 安全系数法method of discount factor 折扣系数法factor of safety 安全系数stress concentration factor 应力集中因数residual stress / initial stress / prestress 残余应力初应力，预应力stress distribution 应力分布equation of equilibrium 平衡方程method of sections 截面法Other Mechanical Terms 其它力学术语dimensionless quantities 无量纲量composite material复合材料specimen 试件elastic-perfectly plastic assumption理想弹塑性假设plastic hinge塑性铰Axial Loading 轴向荷载axially loaded bars 拉压杆，轴向承载杆axial tension 轴向拉伸axial compression 轴向压缩axial forces 轴向力internal forces 内力method of section 截面法diagram of axial forces 轴力图stress tensor 应力张量longitudinal 纵向的transverse 横向的Saint-Venant’s Principle 圣维南原理stresses on oblique planes 斜截面上的应力axial deformation 轴向变形elongation 伸长量extensometer 引伸计、伸展仪、伸长计uniaxial stress 单向应力，单轴应力normal stress 正应力sign convention 符号规定transverse/lateral strain 横向应变Tension/compression rigidity 拉压刚度(EA)stress concentration factor 应力集中系数Mechanical Behavior of Materials 材料力学行为gauge length 标记长度constitutive relations 本构关系（物理方程）Hooke’s Law 胡克定律generalized Hook’s law广义胡克定律stress-strain diagram 应力应变图Hook’s law of shearing剪切胡克定律brittle 脆性brittle materials脆性材料ductile 韧性ductile materials塑性材料，韧性材料，延展性材料plastic deformation塑性变形，残余变形creep 蠕变CMmi@ relaxation 松弛proportional limit 比例极限elastic modulus; modulus of elasticity 弹性模量Young’s modulus 杨氏模量elastic limit 弹性极限yield stress 屈服应力yield strength 屈服强度offset yield stress名义屈服强度strain hardening 强化，冷作硬化ultimate strength, strength limit 强度极限ultimate stress极限应力low carbon steel 低碳钢cast iron 铸铁transversely isotropic 横向同性necking 颈缩plastic flow 塑性流动percent reduction in area 断面收缩率percent elongation 延伸率bulk modulus 大块模量，体积模量Poisson’s ratio 泊松比Shearing and bearing Stress 剪切和挤压应力Shear/shearing 剪切Shear/shearing stress 切应力bearing 挤压bearing stress挤压应力bearing surface 挤压面single shear 单剪double shear 双剪CMmi@ rod 吊杆boom 托架pin 销钉rivet 铆钉joints/connectors 连接件lap joint 搭接butt joint 对接pure shear 纯剪切theorem of conjugate shearing stress 切应力互等定理shear modulus切变模量ultimate shear stress 剪切极限应力yield shear stress 剪切屈服应力Torsion 扭转torsional moment 扭矩twisting moment 扭力矩power & torque 功率与扭矩torque diagram 扭矩图angle of twist 扭转角angle of twist per unit length 单位长度扭转角torsional rigidity 抗扭刚度，扭曲刚度section modulus in torsion 抗扭截面系数slip-lines 滑移线slip bands 滑移带，剪切带free torsion 自由扭转constrained torsion 约束扭转Bending 弯曲symmetric bending 对称弯曲symmetric longitudinal plane 纵向对称面transverse loading 横向荷载shear force 剪力shear flow 剪流shear force diagram 剪力图equation of shear forces 剪力方程bending moment 弯矩equation of bending moment 弯矩方程bending moment diagram 弯矩图pure bending纯弯曲Transverse bending 横力弯曲plane cross-section hypothesis 平面假设hypothesis of uniaxial stress 单轴应力假设neutral surface 中性层neutral axis 中性轴bending normal stress 弯曲正应力section modulus 抗弯截面系数bending shear stress弯曲切应力constant-strength beam; fully stressed beams 等强度梁deflection 挠曲，挠度angle of rotation 转角slope 斜率curvature 曲率radius of curvature 曲率半径deflection curve 挠曲线approximate differential equation of deflection 挠曲轴近似微分方程flexural rigidity 抗弯刚度method of successive integrations 积分法boundary condition 边界条件continuity condition 连续性条件symmetry condition 对称性条件method of superposition 叠加法linear superposition 线性叠加superposition of loads 荷载叠加superposition of rigidized structures 刚化叠加，变形叠加method of singular/discontinuity function 奇异函数法boundary values 边界值moment-area theorems 图乘法unsymmetric bending 不对称弯曲shear center弯曲中心bending strain energy 弯曲应变能Indeterminate Problems 超静定问题statically determinate problem 静定问题statically indeterminate problem 静不定问题，超静定问题degree of static indeterminacy 静不定次，超静定次数redundancy 冗余，多余redundant restraint 多余约束basic determinate system 基本静定系force method 力法equation of deformation compatibility 变形协调方程complementary equation 补充方程thermal stress 热应力coefficient of thermal expansion 线胀系数assembly stress 装配应力residual stress 残余应力thermal strain 热应变eigenstrain 特征应变CMmi@ Stress States 应力状态state of stress 应力状态damage mechanisms 破坏机制stress state of a point 一点应力状态transformation of stresses 应力变换principal stresses 主应力principal axes 主轴，主方向stress circle 应力圆Mohr’s Circle 莫尔圆state of biaxial stress 二向应力状态state of plane stress 平面应力状态state of triaxial stress 三轴（复杂）应力状态triaxial stress 三向应力experimental stress analysis 实验应力分析volumetric strain energy density 体积应变能密度distortional strain energy density 畸变能密度volumetric strain 体应变decomposition of stress tensor 应力张量分解transformation of strain 应变变换Strength Theory 强度理论strength condition 强度条件equivalent stress 相当应力maximum tensile stress theory最大拉应力理论maximum tensile strain theory最大拉应变理论maximum shear stress theory最大切应力理论maximum distortion energy theory 最大畸变能理论Mohr theory of failure莫尔强度理论measurements of strain 应变测量strain gauge 应变计strain rosette 应变花three-element rectangular rosette 三轴直角应变花three-element delta rosette 三轴等角应变花full bridge 全桥接线法half bridge 半桥接法bridge balancing 电桥平衡compensating block 补偿块Combined Loadings 组合荷载eccentric tension 偏心拉伸eccentric compression 偏心压缩core of cross-sections 截面核心Stability of Columns 压杆稳定buckling 屈曲stability condition 稳定条件Euler’s formula欧拉公式critical load 临界压力critical stress 临界应力equivalent length相当长度，有效长度coefficient of equivalent length 长度因数slenderness ratio （压杆的）柔度或长细比long columns 大柔度杆intermediate columns 中柔度杆short columns小柔度杆safety factor of stability稳定安全因数discount factor of stability 折扣安全因数Energy Methods 能量方法strain energy 应变能strain energy density 应变能密度modulus of resilience 回弹模量modulus of toughness 韧度模量principle of work and energy 功能互等定理Castigliano’s theorem 卡氏定理reciprocal theorem of displacement; Maxwell’s reciprocal theorem位移互等定理method of dummy, method of virtual forces 虚力法method of unit dummy load 单位力法Dynamic Loading 冲击荷载impact load 冲击荷载dynamic load 动荷载constant acceleration 等加速constant rotation 等角速转动horizontal impact 水平冲击vertical impact 竖直冲击statically equivalent load 静力等效荷载dynamic load factor 动荷系数Cyclic Loading and Fatigue 交变荷载及疲劳cyclic/alternate load 交变荷载cyclic stress交变应力，循环应力fatigue failure 疲劳失效stress amplitude应力幅stress scope 应力范围cycle characteristics 循环特征symmetric cycling 对称循环unsymmetric cycling 非对称循环pulse cycling 脉冲循环fatigue life疲劳寿命stress-life diagram应力-寿命曲线，S-N曲线endurance limit 疲劳极限fatigue strength 疲劳强度surface roughness 表面粗糙度surface strength 表面强度equal-amplitude fatigue 等幅疲劳fatigue strength condition 疲劳强度条件fatigue factor of safety 疲劳安全因数。
Dual problem对偶问题442,446EEconomics 经济学435,439Eigencourse 457,458Eigenvalue 特征值283,287,374,499 Eigenvalue changes 特征值变换439284,294,300 Eigenvalues of 的特征值297Eigenvalues of 的特征值362Eigenvalues of 的特征值Eigenvector basis 基底的特征向量399Eigenvectors 特征向量283,287,374Eigshow 290,368Elimination 消元法45-66,83,86,135Ellipse 椭圆290,346,366,382Energy 能量343,409Engineering 工程409,419Error 误差211,218,219,225,481,483 Error equation 误差方程477Euler angles 欧拉角474Euler’s formula 欧拉公式311,426,430,497Even 偶数113,246,258,452 Exponential 指数的314,319,327FFactorization 因式分解95,110,235,348,370,374 False proof 假证明305,338Fast Fourier Transform 快速傅立叶变换393,493,511,565Feasible set 可行集440,441FFT(see Fast见快速傅里叶变换509-514Fourier Transform)Fibonacci 菲波那契75,266,268,301,302,306,308 Finite difference 有限差分315-317,417Finite elements 有限元412,419First-order system 一阶方程组315,326Fixed-free 固定-自由410,414,417,419Force balance 平衡力412FORTRAN 16,38Forward difference 前向差分30Four Fundamental Subspaces 四个基本子空间184-199,368,424,507 Fourier series 傅立叶序列233,448,450,452Fourier Transform 傅立叶变换393,509-514Fredholm Alternative 弗雷德霍姆择一203Free 自由133,135,137,144,146,155 Full column rank 列满秩157,170,405Full row rank 行满秩159,405Function space 函数空间121,448,449 Fundamental Theorem of Linear线性代数基本定理188,198,368Algebra(see Four Fundamental Subspaces)GGaussian elimination 高斯消元法45,49,135Gaussian probability distribution 高斯概率分布455Gauss-Jordan 高斯-约当83,84,91,469Gauss-Seidel 高斯-塞德尔481,484,485,489Gene expression data 基因表达数据457Geometric series 等比数列436Gershgorin circles 格尔什戈林圆491Gibbs phenomenon 吉布斯效应451Givens rotation 吉文斯旋转471Google 谷歌368,369,434Gram-Schmidt 格拉姆-施密特223,234,236,241,370,469 Graph 图表74,143,307,311,420,422,423 Group 群119,354HHalf-plane 半平面7Heat equation 热方程式322,323Heisenberg 海森堡305,310Hilbert space 希尔伯特空间447,449Hook e’s Law 虎克定律410,412Householder reflections 镜像变换237,469,472Hyperplane 超平面30,42IIll-conditioned matrix 病态矩阵371,473,474Imaginary 虚数289Independent 独立的26,27,134,168,200,300 Initial value 初值313Inner product 内积11,56,108,448,502,506 Input and output basis 基底输入输出399Integral 积分24,385,386Interior point method 内点法445Intersection of spaces 交空间129,183Inverse matrix 逆矩阵24,81,27082Inverse of的逆Invertible 可逆的86,173,200,248Iteration 迭代481,482,484,489,492JJacobi 雅可比481,483,485,489Jordan form 约当型356,357,358,361,482 JPEG 364,373KKalman filter 卡尔曼滤波器93,214Kernel 核377,380Kirchhoff’s Laws 基尔霍夫定律143,189,420,424-427 Krylov 克雷洛夫491,492L225,480norm 和范数Lagrange multiplier 拉格朗日乘子445Lanczos method 兰索斯方法490,492LAPACK 线性代数软件包98,237,486Leapfrog method 跳步法317,329Least squares 最小平方218,219,236,405,408,453184,186,192,425Left nullspace左零空间Left-inverse 左逆的81,86,154,405Length 长度12,232,447,448,501Line 线34,40,221,474Line of springs 线弹簧411Linear combination 线性组合1,3Linear equation 线性方程23Linear programming 线性规划440Linear transformation 线性变换44,375-398Linearity 线性关系44,245,246Linearly independent 线性独立26,134,168,169,200 LINPACK 线性系统软件包465Loop 环路307,425,426Lower triangular 下三角9598,100,474Lucas numbers 卢卡斯数306MMaple 38,100Mathematica 38,100MATLAB 17,37,237,243,290,337,513 Matrix(see full page 570) 矩阵22,384,387Matrix exponential 矩阵指数314,319,327Matrix multiplication 矩阵乘法58,59,67,389----。
［推荐］［名词委审定］汉英力学名词（1993）［翻译与翻译辅助工具］［回复］［引用回复］［表格型］［跟帖］［转发到Blog］［关闭］［浏览930次］用户名：westbankBZ反应||Belousov-Zhabotinski reaction, BZ reactionFPU问题||Fermi-Pasta-Ulam problem, FPU problemKBM 方法||KBM method, Krylov-Bogoliubov-Mitropolskii methodKS动态］熵||Kolmogorov-Sinai entropy, KS entropyKdV 方程||KdV equationU 形管||U-tubeWKB 方法||WKB method, Wentzel-Kramers-Brillouin method［彻］体力||body force［单］元||eleme nt［第二类］拉格朗日方程||Lagra nge equati on ［of the seco nd kin d］［叠栅］云纹||moir e fringe;物理学称"叠栅条纹”。
［叠栅］云纹法||moir e method［抗］剪切角||a ngle of shear resista nee［可］变形体||deformable body［钱］币状裂纹||penny-shape crack［映］象||image［圆］筒||cyli nder［圆］柱壳||cylindrical shell［转］轴||shaft［转动］瞬心||instantaneous center ［of rotation］［转动］瞬车由||instantaneous axis ［of rotation］［状］态变量||state variable［状］态空间||state space［自］适应网格||［self-］adaptive meshC 0 连续问题||C0-continuous problemC 1 连续问题||C1-continuous problemCFL条件||Courant-Friedrichs-Lewy condition, CFL conditionHRR场||Hutchinson-Rice-Rosengren fieldJ 积分||J-integralJ 阻力曲线||J-resistance curveKAM定理||Kolgomorov-Arnol'd-Moser theorem, KAM theoremKAM环面||KAM torush收敛||h-c on verge ncep收敛||p-c on verge ncen 定理||Buckingham theorem, pi theorem阿尔曼西应变||Almansis strain阿尔文波||Alfven wave阿基米德原理||Archimedes principle阿诺德舌［头川Arnol'd tongue阿佩尔方程||Appel equation阿特伍德机||Atwood machine埃克曼边界层||Ekman boundary layer埃克曼流||Ekman flow 埃克曼数||Ekman number 埃克特数||Eckert number 埃农吸引子||Henon attractor 艾里应力函数||Airy stress function 鞍点"saddle [point] 鞍结分岔||saddle-node bifurcation 安定[性]理论||shake-down theory 安全寿命||safe life 安全系数||safety factor 安全裕度||safety margin 暗条纹||dark fringe 奥尔-索末菲方程||Orr-Sommerfeld equation 奥辛流||Oseen flow 奥伊洛特模型||Oldroyd model 八面体剪应变||octohedral shear strain 八面体剪应力||octohedral shear stress 八面体剪应力理论||octohedral shear stress theory 巴塞特力||Basset force 白光散斑法||white-light speckle method 摆||pe ndulum 摆振||shimmy 板||plate 板块法||panel method 板元||plate element 半导体应变计||semic on ductor stra in gage 半峰宽度||half-peak width 半解析法||semi-analytical method 半逆解法||semi-inverse method 半频进动||half frequency precession 半向同性张量||hemitropic tensor 半隐格式||semi-implicit scheme 薄壁杆||thin-walled bar 薄壁梁||thin-walled beam 薄壁筒||thin-walled cylinder 薄膜比拟||membrane analogy 薄翼理论||thin-airfoil theory保单调差分格式||monotonicity preserving differenee scheme 保守力||conservative force 保守系||conservative system 爆发||blow up 爆高||height of burst 爆轰||detonation; 又称"爆震”。
材料专业英语常见词汇The saying "the more diligent, the more luckier you are" really should be my charm in2006.材料专业英语常见词汇一Structure 组织Ceramic 陶瓷Ductility 塑性Stiffness 刚度Grain 晶粒Phase 相Unit cell 单胞Bravais lattice 布拉菲点阵Stack 堆垛Crystal 晶体Metallic crystal structure 金属性晶体点阵 Non-directional 无方向性Face-centered cubic 面心立方Body-centered cubic体心立方 Hexagonal close-packed 密排六方 Copper 铜Aluminum 铝Chromium 铬 Tungsten 钨Crystallographic Plane晶面 Crystallographic direction 晶向 Property性质 Miller indices米勒指数 Lattice parameters 点阵参数Tetragonal 四方的Hexagonal 六方的Orthorhombic 正交的Rhombohedra 菱方的Monoclinic 单斜的Prism 棱镜 Cadmium 镉 Coordinate system 坐 Point defec点缺陷Lattice 点阵 Vacancy 空位Solidification 结晶Interstitial 间隙Substitution 置换Solid solution strengthening 固溶强化Diffusion 扩散Homogeneous 均匀的Diffusion Mechanisms 扩散机制Lattice distortion 点阵畸变Self-diffusion 自扩散Fick’s First Law 菲克第一定律 Unit time 单位时间Coefficient 系数Concentration gradient 浓度梯度Dislocations 位错Linear defect 线缺陷Screw dislocation 螺型位错Edge dislocation 刃型位错Vector 矢量Loop 环路Burgers’vector 柏氏矢量Perpendicular 垂直于Surface defect 面缺陷Grain boundary 晶界Twin boundary 晶界 Shear force 剪应力Deformation 变形Small or low angel grain boundary 小角度晶界Tilt boundary 倾斜晶界Supercooled 过冷的Solidification 凝固Ordering process 有序化过程Crystallinity 结晶度Microstructure 纤维组织Term 术语Phase Diagram 相图Equilibrium 平衡Melt 熔化Cast 浇注Crystallization 结晶Binary Isomorphous Systems 二元匀晶相图Soluble 溶解Phase Present 存在相Locate 确定Tie line 连接线Isotherm 等温线Concentration 浓度Intersection 交点The Lever Law 杠杆定律Binary Eutectic System 二元共晶相图Solvus Line 溶解线Invariant 恒定Isotherm 恒温线Cast Iron 铸铁Ferrite 珠光体Polymorphic transformation 多晶体转变Austenite 奥氏体Revert 回复Intermediate compound 中间化合物Cementite 渗碳体Vertical 垂线Nonmagnetic 无磁性的Solubility 溶解度Brittle 易脆的Eutectic 共晶Eutectoid invariant point 共析点Phase transformation 相变Allotropic 同素异形体Recrystallization 再结晶Metastable 亚稳的Martensitic transformation 马氏体转变Lamellae 薄片Simultaneously 同时存在Pearlite 珠光体Ductile 可塑的Mechanically 机械性能Hypo eutectoid 过共析的Particle 颗粒Matrix基体Proeutectoid 先共析Hypereutectoid 亚共析的Bainite 贝氏体Martensite 马氏体Linearity 线性的Stress-strain curve 应力-应变曲线Proportional limit 比例极限Tensile strength 抗拉强度Ductility 延展性Percent reduction in area 断面收缩率Hardness 硬度Modulus of Elasticity 弹性模量Tolerance 公差Rub 摩擦Wear 磨损Corrosion resistance 抗腐蚀性Aluminum 铝Zinc 锌Iron ore 铁矿Blast furnace 高炉Coke 焦炭Limestone 石灰石Slag 熔渣Pig iron 生铁Ladle 钢水包Silicon 硅Sulphur 硫Wrought 可锻的Graphite 石墨Flaky 片状Low-carbon steels 低碳钢Case hardening 表面硬化Medium-carbon steels 中碳钢Electrode 电极As a rule 通常Preheating 预热Quench 淬火Body-centered lattice 体心晶格Carbide 碳化物Hypereutectoid过共晶Chromium 铬Manganese 锰Molybdenum 钼Titanium 钛Cobalt 钴Tungsten 钨Vanadium 钒Pearlitic microstructure 珠光体组织Martensitic microstructure 马氏体组织Viscosity 粘性Wrought 锻造的Magnesium 镁Flake 片状Malleable 可锻的Nodular 球状Spheroidal 球状Superior property 优越性Galvanization 镀锌Versatile 通用的Battery grid 电极板Calcium 钙Tin 锡Toxicity 毒性Refractory 耐火的Platinum铂Polymer 聚合物Composite 混合物Erosive 腐蚀性Inert 惰性Thermo chemically 热化学Generator 发电机Flaw 缺陷Variability 易变的Annealing 退火Tempering回火Texture 织构Kinetic 动力学Peculiarity 特性Critical point 临界点Dispersity 弥散程度Spontaneous 自发的Inherent grain 本质晶粒Toughness 韧性Rupture 断裂Kinetic curve of transformation 转变动力学曲线Incubation period 孕育期Sorbite 索氏体Troostite 屈氏体Disperse 弥散的Granular 颗粒状Metallurgical 冶金学的Precipitation 析出Depletion 减少Quasi-eutectoid 伪共析Superposition 重叠Supersede 代替Dilatometric 膨胀Unstable 不稳定Supersaturate 使过饱和Tetragonality 正方度Shear 切变Displacement 位移Irreversible 不可逆的金属材料工程专业英语acid-base equilibrium酸碱平衡 acid-base indicator酸碱指示剂 acid bath酸槽 acidBessemerconverter 酸性转炉 acid brick酸性耐火砖 acid brittleness酸洗脆性、氢脆性 acid burden酸性炉料acid clay酸性粘土 acid cleaning同pickling酸洗 acid concentration酸浓度 acid converter酸性转炉 acid converter steel酸性转炉钢 acid content酸含量 acid corrosion酸腐蚀 acid deficient弱酸的、酸不足的 acid dip酸浸acid dip pickler沉浸式酸洗装置 aciddiptank酸液浸洗槽acid drain tank排酸槽acidless descaling无酸除鳞acid medium酸性介质acid mist酸雾acid-proof paint耐酸涂料漆acid-proof steel耐酸钢acid-resistant耐酸钢acid-resisting vessel耐酸槽acid strength酸浓度acid supply pump供酸泵acid wash酸洗acid value酸值acid wash solution酸洗液acieration渗碳、增碳Acm point Acm转变点渗碳体析出温度acorn nut螺母、螺帽acoustic absorption coefficient声吸收系数acoustic susceptance声纳actifier再生器action line作用线action spot作用点activated atom激活原子activated bath活化槽activated carbon活性碳activating treatment活化处理active corrosion活性腐蚀、强烈腐蚀active area有效面积active power有功功率、有效功率active product放射性产物active resistance有效电阻、纯电阻active roll gap轧辊的有效或工作开口度active state活性状态active surface有效表面activity coefficient激活系数、活度系数actual diameter钢丝绳实际直径actual efficiency实际效率actual error实际误差actual time实时actual working stress实际加工应力actuating device调节装置、传动装置、起动装置actuating lever驱动杆、起动杆actuating mechanism 动作机构、执行机构actuating motor驱动电动机、伺服电动机actuating pressure作用压力actuation shaft起动轴actuator调节器、传动装置、执行机构acute angle锐角adaptive feed back control自适应反馈控制adaptive optimization自适应最优化adaptor接头、接合器、连结装置、转接器、附件材料科学基础专业词汇:第一章晶体结构原子质量单位 Atomic mass unit amu 原子数 Atomic number 原子量 Atomic weight波尔原子模型 Bohr atomic model 键能 Bonding energy 库仑力 Coulombic force共价键 Covalent bond 分子的构型 molecular configuration电子构型electronic configuration 负电的 Electronegative 正电的 Electropositive基态 Ground state 氢键 Hydrogen bond 离子键 Ionic bond 同位素 Isotope金属键 Metallic bond 摩尔 Mole 分子 Molecule 泡利不相容原理 Pauli exclusion principle 元素周期表 Periodic table 原子 atom 分子 molecule 分子量 molecule weight极性分子 Polar molecule 量子数 quantum number 价电子 valence electron范德华键 van der waals bond 电子轨道 electron orbitals 点群 point group对称要素 symmetry elements 各向异性 anisotropy 原子堆积因数 atomic packing factorAPF 体心立方结构 body-centered cubic BCC 面心立方结构 face-centered cubic FCC布拉格定律bragg’s law 配位数 coordination number 晶体结构 crystal structure晶系 crystal system 晶体的 crystalline 衍射 diffraction 中子衍射 neutron diffraction电子衍射 electron diffraction 晶界 grain boundary 六方密堆积 hexagonal close-packed HCP 鲍林规则 Paulin g’s rules NaCl型结构 NaCl-type structureCsCl型结构Caesium Chloride structure 闪锌矿型结构 Blende-type structure纤锌矿型结构 Wurtzite structure 金红石型结构 Rutile structure萤石型结构 Fluorite structure 钙钛矿型结构 Perovskite-type structure尖晶石型结构 Spinel-type structure 硅酸盐结构 Structure of silicates岛状结构 Island structure 链状结构 Chain structure 层状结构 Layer structure架状结构 Framework structure 滑石 talc 叶蜡石 pyrophyllite 高岭石 kaolinite石英 quartz 长石 feldspar 美橄榄石 forsterite 各向同性的 isotropic各向异性的 anisotropy 晶格 lattice 晶格参数 lattice parameters 密勒指数 miller indices 非结晶的 noncrystalline多晶的 polycrystalline 多晶形 polymorphism 单晶single crystal 晶胞 unit cell电位 electron states化合价 valence 电子 electrons 共价键 covalent bonding金属键 metallic bonding 离子键Ionic bonding 极性分子 polar molecules原子面密度 atomic planar density 衍射角 diffraction angle 合金 alloy粒度,晶粒大小 grain size 显微结构 microstructure 显微照相 photomicrograph扫描电子显微镜 scanning electron microscope SEM透射电子显微镜 transmission electron microscope TEM 重量百分数 weight percent四方的 tetragonal 单斜的monoclinic 配位数 coordination number材料科学基础专业词汇:第二章晶体结构缺陷缺陷 defect, imperfection 点缺陷 point defect 线缺陷 line defect, dislocation面缺陷 interface defect 体缺陷 volume defect 位错排列 dislocation arrangement位错线 dislocation line 刃位错 edge dislocation 螺位错 screw dislocation混合位错 mixed dislocation 晶界 grain boundaries 大角度晶界 high-angle grain boundaries 小角度晶界 tilt boundary, 孪晶界 twin boundaries 位错阵列 dislocation array位错气团 dislocation atmosphere 位错轴dislocation axis 位错胞 dislocation cell位错爬移 dislocation climb 位错聚结 dislocation coalescence 位错滑移 dislocation slip位错核心能量 dislocation core energy 位错裂纹 dislocation crack位错阻尼 dislocation damping 位错密度 dislocation density原子错位 substitution of a wrong atom 间隙原子 interstitial atom晶格空位 vacant lattice sites 间隙位置 interstitial sites 杂质 impurities弗伦克尔缺陷 Frenkel disorder 肖脱基缺陷 Schottky disorder 主晶相 the host lattice错位原子 misplaced atoms 缔合中心 Associated Centers. 自由电子 Free Electrons电子空穴Electron Holes 伯格斯矢量 Burgers 克罗各-明克符号 Kroger Vink notation中性原子 neutral atom材料科学基础专业词汇:第二章晶体结构缺陷-固溶体固溶体 solid solution 固溶度 solid solubility 化合物 compound间隙固溶体 interstitial solid solution 置换固溶体 substitutional solid solution金属间化合物 intermetallics 不混溶固溶体 immiscible solid solution转熔型固溶体 peritectic solid solution 有序固溶体 ordered solid solution无序固溶体 disordered solid solution 固溶强化 solid solution strengthening取代型固溶体 Substitutional solid solutions 过饱和固溶体 supersaturated solid solution非化学计量化合物 Nonstoichiometric compound材料科学基础专业词汇:第三章熔体结构熔体结构 structure of melt过冷液体 supercooling melt 玻璃态 vitreous state软化温度 softening temperature 粘度 viscosity 表面张力 Surface tension介稳态过渡相 metastable phase 组织 constitution 淬火 quenching退火的 softened 玻璃分相 phase separation in glasses 体积收缩 volume shrinkage材料科学基础专业词汇:第四章固体的表面与界面表面 surface 界面 interface 同相界面 homophase boundary异相界面 heterophase boundary 晶界 grain boundary 表面能 surface energy小角度晶界 low angle grain boundary 大角度晶界 high angle grain boundary共格孪晶界 coherent twin boundary 晶界迁移 grain boundary migration错配度 mismatch 驰豫 relaxation 重构 reconstuction 表面吸附 surface adsorption表面能 surface energy 倾转晶界 titlt grain boundary 扭转晶界 twist grain boundary倒易密度 reciprocal density 共格界面 coherent boundary 半共格界面 semi-coherent boundary 非共格界面 noncoherent boundary 界面能 interfacial free energy应变能 strain energy 晶体学取向关系 crystallographic orientation惯习面habit plane材料科学基础专业词汇:第五章相图相图 phase diagrams 相 phase 组分 component 组元 compoonent相律 Phase rule 投影图 Projection drawing 浓度三角形 Concentration triangle冷却曲线 Cooling curve 成分 composition 自由度 freedom相平衡 phase equilibrium 化学势 chemical potential 热力学 thermodynamics相律 phase rule 吉布斯相律 Gibbs phase rule 自由能 free energy吉布斯自由能 Gibbs free energy 吉布斯混合能 Gibbs energy of mixing吉布斯熵 Gibbs entropy 吉布斯函数 Gibbs function 热力学函数 thermodynamics function 热分析 thermal analysis 过冷 supercooling 过冷度 degree of supercooling杠杆定律 lever rule 相界 phase boundary 相界线 phase boundary line相界交联 phase boundary crosslinking 共轭线 conjugate lines相界有限交联 phase boundary crosslinking 相界反应 phase boundary reaction相变 phase change 相组成 phase composition 共格相 phase-coherent金相相组织 phase constentuent 相衬 phase contrast 相衬显微镜 phase contrast microscope 相衬显微术 phase contrast microscopy 相分布 phase distribution相平衡常数 phase equilibrium constant 相平衡图 phase equilibrium diagram相变滞后 phase transition lag 相分离 phase segregation 相序 phase order相稳定性 phase stability 相态 phase state 相稳定区 phase stabile range相变温度 phase transition temperature 相变压力 phase transition pressure同质多晶转变 polymorphic transformation 同素异晶转变 allotropic transformation相平衡条件 phase equilibrium conditions 显微结构 microstructures 低共熔体 eutectoid不混溶性 immiscibility材料科学基础专业词汇:第六章扩散活化能 activation energy 扩散通量 diffusion flux 浓度梯度 concentration gradient菲克第一定律Fick’s first law 菲克第二定律Fick’s second law 相关因子 correlation factor 稳态扩散 steady state diffusion 非稳态扩散 nonsteady-state diffusion扩散系数 diffusion coefficient 跳动几率 jump frequency填隙机制 interstitalcy mechanism 晶界扩散 grain boundary diffusion短路扩散 short-circuit diffusion 上坡扩散 uphill diffusion 下坡扩散 Downhill diffusion互扩散系数 Mutual diffusion 渗碳剂 carburizing 浓度梯度 concentration gradient浓度分布曲线 concentration profile 扩散流量 diffusion flux 驱动力 driving force间隙扩散 interstitial diffusion 自扩散 self-diffusion 表面扩散 surface diffusion空位扩散 vacancy diffusion 扩散偶 diffusion couple 扩散方程 diffusion equation扩散机理 diffusion mechanism 扩散特性 diffusion property 无规行走 Random walk达肯方程 Dark equation 柯肯达尔效应 Kirkendall equation本征热缺陷 Intrinsic thermal defect 本征扩散系数 Intrinsic diffusion coefficient离子电导率 Ion-conductivity 空位机制 Vacancy concentration材料科学基础专业词汇:第七章相变过冷 supercooling 过冷度 degree of supercooling 晶核 nucleus 形核 nucleation形核功 nucleation energy 晶体长大 crystal growth 均匀形核 homogeneous nucleation非均匀形核 heterogeneous nucleation 形核率 nucleation rate 长大速率 growth rate热力学函数 thermodynamics function 临界晶核 critical nucleus临界晶核半径 critical nucleus radius 枝晶偏析 dendritic segregation局部平衡 localized equilibrium 平衡分配系数 equilibrium distributioncoefficient有效分配系数 effective distribution coefficient 成分过冷 constitutional supercooling引领领先相 leading phase 共晶组织 eutectic structure 层状共晶体 lamellar eutectic伪共晶 pseudoeutectic 离异共晶 divorsed eutectic 表面等轴晶区 chill zone柱状晶区 columnar zone 中心等轴晶区 equiaxed crystal zone定向凝固 unidirectional solidification 急冷技术 splatcooling 区域提纯 zone refining单晶提拉法 Czochralski method 晶界形核 boundary nucleation位错形核 dislocation nucleation 晶核长大 nuclei growth斯宾那多分解 spinodal decomposition 有序无序转变 disordered-order transition马氏体相变 martensite phase transformation 马氏体 martensite材料科学基础专业词汇:第八、九章固相反应和烧结固相反应 solid state reaction 烧结 sintering 烧成 fire 合金 alloy 再结晶 Recrystallization 二次再结晶 Secondary recrystallization 成核 nucleation 结晶 crystallization子晶,雏晶 matted crystal 耔晶取向 seed orientation 异质核化 heterogeneous nucleation均匀化热处理 homogenization heat treatment 铁碳合金 iron-carbon alloy渗碳体 cementite 铁素体 ferrite 奥氏体austenite 共晶反应 eutectic reaction 固溶处理 solution heat treatment。
这是同济大学结构研一硕士上的《高等混凝土结构理论》期末考试的复习要点，希望对考博选考《混凝土结构基本理论》这门课的同学有所帮助。
1．Stress-strain curves of concrete under monotonic, repeated and cyclic uniaxial loadings. 单轴受力时混凝土在单调、重复、反复加载时的应力应变曲线。
2．Creep of concrete (linear and nonlinear) 混凝土的徐变（线性、非线性徐变）3．Components of deformation of concrete 混凝土变形的多元组成4．Process of failure of concrete under uniaxial compression 混凝土在单向受压时破坏的过程。
5．Strength indices of concrete and the relations among them 混凝土的强度指标及其之间关系6．Features of stress-strain envelope curve of concrete under repeated compressive loading. 混凝土单向受压重复加载时的应力应变关系的包络线的特征。
7．The crack contact effect of concrete and its representation in stress-strain diagram. 混凝土的裂面效应及其在应力应变关系图上的表示。
8．The multi-level two-phase system of concrete. 混凝土的多层次二相体系。
9．The rheological model of concrete. 混凝土的流变学模型。
10．Influence of stress gradient on strength of concrete. 应力梯度对混凝土强度的影响。
2011年技术物理学院08级（激光方向）专业英语翻译重点！！！作者：邵晨宇Electromagnetic电磁的principle原则principal主要的macroscopic宏观的microscopic微观的differential微分vector矢量scalar标量permittivity介电常数photons光子oscillation振动density of states态密度dimensionality维数transverse wave横波dipole moment偶极矩diode 二极管mono-chromatic单色temporal时间的spatial空间的velocity速度wave packet波包be perpendicular to线垂直be nomal to线面垂直isotropic各向同性的anistropic各向异性的vacuum真空assumption假设semiconductor半导体nonmagnetic非磁性的considerable大量的ultraviolet紫外的diamagnetic抗磁的paramagnetic顺磁的antiparamagnetic反铁磁的ferro-magnetic铁磁的negligible可忽略的conductivity电导率intrinsic本征的inequality不等式infrared红外的weakly doped弱掺杂heavily doped重掺杂a second derivative in time对时间二阶导数vanish消失tensor张量refractive index折射率crucial主要的quantum mechanics 量子力学transition probability跃迁几率delve研究infinite无限的relevant相关的thermodynamic equilibrium热力学平衡（动态热平衡）fermions费米子bosons波色子potential barrier势垒standing wave驻波travelling wave行波degeneracy简并converge收敛diverge发散phonons声子singularity奇点（奇异值）vector potential向量式partical-wave dualism波粒二象性homogeneous均匀的elliptic椭圆的reasonable公平的合理的reflector反射器characteristic特性prerequisite必要条件quadratic二次的predominantly最重要的gaussian beams高斯光束azimuth方位角evolve推到spot size光斑尺寸radius of curvature曲率半径convention管理hyperbole双曲线hyperboloid双曲面radii半径asymptote渐近线apex顶点rigorous精确地manifestation体现表明wave diffraction波衍射aperture孔径complex beam radius复光束半径lenslike medium类透镜介质be adjacent to与之相邻confocal beam共焦光束a unity determinant单位行列式waveguide波导illustration说明induction归纳symmetric 对称的steady-state稳态be consistent with与之一致solid curves实线dashed curves虚线be identical to相同eigenvalue本征值noteworthy关注的counteract抵消reinforce加强the modal dispersion模式色散the group velocity dispersion群速度色散channel波段repetition rate重复率overlap重叠intuition直觉material dispersion材料色散information capacity信息量feed into 注入derive from由之产生semi-intuitive半直觉intermode mixing模式混合pulse duration脉宽mechanism原理dissipate损耗designate by命名为to a large extent在很大程度上etalon 标准具archetype圆形interferometer干涉计be attributed to归因于roundtrip一个往返infinite geometric progression无穷几何级数conservation of energy能量守恒free spectral range自由光谱区reflection coefficient（fraction of the intensity reflected）反射系数transmission coefficient（fraction of the intensity transmitted）透射系数optical resonator光学谐振腔unity 归一optical spectrum analyzer光谱分析grequency separations频率间隔scanning interferometer扫描干涉仪sweep移动replica复制品ambiguity不确定simultaneous同步的longitudinal laser mode纵模denominator分母finesse精细度the limiting resolution极限分辨率the width of a transmission bandpass透射带宽collimated beam线性光束noncollimated beam非线性光束transient condition瞬态情况spherical mirror 球面镜locus（loci）轨迹exponential factor指数因子radian弧度configuration不举intercept截断back and forth反复spatical mode空间模式algebra代数in practice在实际中symmetrical对称的a symmetrical conforal resonator对称共焦谐振腔criteria准则concentric同心的biperiodic lens sequence双周期透镜组序列stable solution稳态解equivalent lens等效透镜verge 边缘self-consistent自洽reference plane参考平面off-axis离轴shaded area阴影区clear area空白区perturbation扰动evolution渐变decay减弱unimodual matrix单位矩阵discrepancy相位差longitudinal mode index纵模指数resonance共振quantum electronics量子电子学phenomenon现象exploit利用spontaneous emission自发辐射initial初始的thermodynamic热力学inphase同相位的population inversion粒子数反转transparent透明的threshold阈值predominate over占主导地位的monochromaticity单色性spatical and temporal coherence时空相干性by virtue of利用directionality方向性superposition叠加pump rate泵浦速率shunt分流corona breakdown电晕击穿audacity畅通无阻versatile用途广泛的photoelectric effect光电效应quantum detector 量子探测器quantum efficiency量子效率vacuum photodiode真空光电二极管photoelectric work function光电功函数cathode阴极anode阳极formidable苛刻的恶光的irrespective无关的impinge撞击in turn依次capacitance电容photomultiplier光电信增管photoconductor光敏电阻junction photodiode结型光电二极管avalanche photodiode雪崩二极管shot noise 散粒噪声thermal noise热噪声1.In this chapter we consider Maxwell’s equations and what they reveal about the propagation of light in vacuum and in matter. We introduce the concept of photons and present their density of states.Since the density of states is a rather important property，not only for photons，we approach this quantity in a rather general way. We will use the density of states later also for other(quasi-) particles including systems of reduced dimensionality.In addition,we introduce the occupation probability of these states for various groups of particles.在本章中，我们讨论麦克斯韦方程和他们显示的有关光在真空中传播的问题。
a rXiv:h ep-th/9362v22J u n1993RU-93-20BOUNDARY S-MATRIX AND BOUNDARY STATE IN TWO-DIMENSIONAL INTEGRABLE QUANTUM FIELD THEORY Subir Ghoshal †and Alexander Zamolodchikov ‡⋆Department of Physics and Astronomy Rutgers University P.O.Box 849,Piscataway,NJ 08855-0849Abstract We study integrals of motion and factorizable S-matrices in two-dimensional integrable field theory with boundary.We propose the “boundary cross-unitarity equation”which is the boundary analog of the cross-symmetry condition of the “bulk”S-matrix.We derive the boundary S-matrices for the Ising field theory with boundary magnetic field and for the boundary sine-Gordon model.1.INTRODUCTION In this paper we study two-dimensional integrable field theory with boundary.Exact solution to such a field theory could provide better understanding of boundary-related phenomena in statistical systems near criticality[1].Quantum field theory with boundary can be applied to study quantum systems with dissipative forces[2].From a more general point of view,studying integrable models could throw some light on the structure of the“space of boundary interactions”,the object of primary significance in open string field theory[3].An integrable field theory posesses an infinite set of mutually commutative integrals of motion 1.In the “bulk theory”(i.e.without a boundary)these integrals of motion follow from the continuity equations for an infinite set of local currents.These currents can be shown to exist in many 2D quantum field theories,both the ones defined in terms of an action functional (like sine-Gordon or nonlinear sigma-models[4])and those whichare defined as“perturbed conformalfield theories”[5].In presence of a boundary,the existense of these currents is not sufficient to ensure integrability.Integrals of motion appear only if particular“integrable”boundary conditions are chosen.In general,the boundary condition can be specified either through the“boundary action functional”or as the“perturbed conformal boundary condition”2.In Sect.2we show how the integrals of motion of the“bulk”theory get modified(in most cases destroyed)in presence of the boundary and how one canfind“integrable”boundary conditions.An important characteristic of an integrablefield theory is its factorizable S-matrix. In the“bulk theory”,the factorizable S-matrix is completely determined in terms of the two-particle scattering amplitudes,the latter being required to satisfy the Yang-Baxter equation(also known as the“factorizability condition”),in addition to the standard equa-tions of unitarity and crossing symmetry[4].These equations have much restrictive power, determining the S-matrix up to the so-called“CDD ambiguity”.At present many examples of factorizable scattering theory are known(see e.g.[4,9-12]),most of which are obtained by explicitely solving the above equations(eliminating the“CDD ambiguity”usually requires a lot of guesswork).It is known since long[13]that the concept of factorizable scattering can be generalized in a rather straightforward way to the case where a reflecting boundary is present.The S-matrix is expressed in terms of the“bulk”two-particle S-matrix and specific“boundary reflection”amplitudes,the latter,again,being required to satisfy an appropriate gen-eralization of the Yang-Baxter equation(which we call here the“Boundary Yang-Baxter equation”)3.Generalization of the unitarity condition is also fairly straightforward.What was not known was the appropriate analog of the crossing-symmetry equation.In Section3 wefill this gap by deriving what we call the“boundary cross-unitarity equation”.Together with this,the above equations have exactly the same restrictive power as the corresponding “bulk”system,i.e.they allow one to pin down the factorizable boundary S-matrix up to the“CDD factors”.We also study integrable boundary conditions in two particular models(off-critical Isingfield theory and sine-Gordon theory)andfind the associated boundary S-matrices. This is done in Sections4and5.2.INTEGRALS OF MOTIONConsider a2D Euclideanfield theory,inflat space with coordinates(x1,x2)=(x,y). There are basically two ways to define a2Dfield theory.In the Lagrangian approach one specifies the actionA= ∞−∞dx ∞−∞dy a(ϕ,∂µϕ)(2.1)whereϕ(x,y)is some set of“fundamentalfields”and the action density a(ϕ,∂µϕ)is a local function of thesefields and derivatives∂µϕ=∂ϕ/∂xµwithµ=1,2.Another approach is to consider the“perturbed conformalfield theory”;in this case one writes the“symbolic action”A=A CF T+ ∞−∞Φ(x,y)dxdy(2.2)where A CF T is the“action of conformalfield theory(CFT)”andΦ(x,y)is a specific relevantfield of this CFT.In both approaches one can define a symmetric stress tensor Tµν=Tνµwhich satisfies the continuity equations∂¯z T=∂zΘ;∂z¯T=∂¯zΘ(2.3) Here we use complex coordinates z=x+iy,¯z=x−iy and denote the appropriate compo-nents T=T zz,¯T=T¯z¯z,Θ=T z¯z of the stress tensor.To achieve Hamiltonian formulation one chooses an arbitrary direction,say the y-direction,to be the“euclidean time”,and as-sociates a Hilbert space H with any“equal time section”y=const.,x∈(−∞,∞).States are vectors in H and their“time evolution”is described by the Hamiltonian operatorH= ∞−∞dxT yy= ∞−∞dx[T+¯T+2Θ](2.4)Let us assume that thefield theory(2.1)or(2.2)is integrable.In particular,the equations(2.3)appear to be thefirst representatives of an infinite sequence∂¯z T s+1=∂zΘs−1∂¯z¯T s+1=∂z¯Θs−1(2.5) where T s+1,Θs−1(¯T s+1,¯Θs−1)are localfields of spins s+1,s−1respectively and the integrals of motion(IM)P s= ∞−∞(T s+1+Θs−1)dx;¯P s= ∞−∞(¯T s+1+¯Θs−1)dx(2.6)constitute an infinite set of mutually commutative operators in H.The spin s of IM(2.6) takes integer values s1,s2,...in the infinite set{s}which is an important characteristic of an integrablefield theory[5].In any case,s1=1;for this value of s(2.5)coincides with (2.3)andH=(P1+¯P1)(2.7) Now,let us consider thisfield theory in the semi-infinite plane,x∈(−∞,0],y∈(−∞,∞),the y-axis being the boundary.Again,the boundary conditions are specified in different ways in the two approaches,(2.1)and(2.2).In the lagrangian approach one chooses the“boundary action density”b(ϕB(y),dA B= ∞−∞dy 0−∞dxa(ϕ,∂µϕ)+ ∞−∞dy b(ϕB,ddyθ(y)(2.10)whereθ(y)is some local boundaryfield.As the theory(2.8)or(2.9)is still symmetric with respect to translations along the y-axis,the equation(2.10)can be easily derived as a consequence of this symmetry.In the“perturbed CFT”approach,it is relatively easy to relate thefieldθ(y)to the“boundary perturbation”ΦB(y)(see below).The continuity equations(2.3)guarantee that the contour integralsP1(C)= C(T dz+Θd¯z);¯P1(C)= C(¯T d¯z+Θdz)(2.11)do not change under deformations of the integration contours C.Consider the contour C=C1+C12+C2shown in Fig.1.Obviously,P1(C)=¯P1(C)=0.If we take the combination0=P1(C)+¯P1(C)=P1(C1)+¯P1(C1)+P1(C2)+¯P1(C2)+P1(C12)+¯P1(C12)(2.12) the integration over C12part of this contour is easily done in view of(2.10)P1(C12)+¯P1(C12)=θ(y1)−θ(y2)(2.13) and hence the integralH B(y)= 0−∞(T+¯T+2Θ)dx+θ(y)(2.14)is in fact y-independent,i.e.it is an integral of motion in(2.8)or(2.9).Even if the bulk theory(2.1)or(2.2)is integrable,in general the boundary conditions in the semi-infinite system(2.8)or(2.9)will spoil integrability.Suppose,however,that we can choose particular boundary conditions,such that the equation[T s+1+¯Θs−1−¯T s+1−Θs−1]|x=0=dB 0|0 B(2.17)where|0 B∈H B is the ground state of H B,and O i(x i,y i)in the r.h.s.are understood as the corresponding Heisenburg operatorsO i(x,y)=e−yH B O i(x,0)e yH B(2.18)and T y means the“y-ordering”.In an integrable theory the operators H(s)B ;s∈{s}B,constitute a commutative set of IMs.Alternatively,one could take x to be the“euclidean time”.In this case the“equal time section”is the infinite line x=const,y∈(−∞,∞).Hence the associated space of states is the same H as in the bulk theory(2.1)((2.2)),and the Hamiltonian operator is given by the same eq.(2.4)(with x and y interchanged).The boundary at x=0appears as the“time boundary”,or“initial condition”at x=0which is described by the particular “boundary state”4|B ∈H.It is the state|B that concentrates all information about the boundary condition in this picture.The correlation functions(2.17)are expressed as O1(x1,y1)...O N(x N,y N) = 0|T x(O1(x1,y1)...O N(x N,y N)|B4The notion of boundary state is discussed in the context of CFT in[7].where now|0 ∈H is the ground state of H and O i(x,y)in the r.h.s.are the Heisenberg field operatorsO i(x,y)=e−xH O i(0,y)e xH(2.20) corresponding to this picture;T x means“x-ordering”.In an integrable theory,the same equations(2.6)(again with x and y interchanged)define an infinite set of mutually com-mutative operators P s,¯P s;s∈{s},acting in H.As a direct consequence of(2.15)onefinds that the boundary state|B satisfies the equations(P s−¯P s)|B =0;s∈{s}B.(2.21) To expose an example of an integrable boundaryfield theory,let us cosider the per-turbed CFT(2.9)with A CF T taken to be any c<1minimal model,and the degenerate spinlessfieldΦ(1,3)taken as the bulk perturbation,Φ(x,y)=λΦ(1,3)(x,y).(2.22) whereλis a constant of dimension[length]2∆−2;∆=∆(1,3).The local integrals of motion in the corresponding bulk theory(2.2)are discussed in[5].Thefields T s+1are composite fields built up from T=T zz and its derivatives,for examplec+2T2=T;T4=:T2:;T6=:T3:−+regular terms(2.24)(z−w)kare particular conformal descendants ofΦ(1,3))with(Ψ(k)s−k+1Ψ(1)s(w,¯w)=∂w Q s−1(w,¯w),(2.25) where Q s−1are localfields.The“null-vector”equation(L−3−2(∆+1)(∆+2)L3−1)Φ(1,3),(2.26)satisfied by the degeneratefieldΦ(1,3),is used to prove(2.25).The equation(2.25)is sufficient to show that thefield T s+1satisfy(2.5)in the perturbed theory,up tofirst order inλ.Then dimensional analysis shows that(2.5)is exact.There are infinitely manyfields T s+1satisfying(2.24)-(2.25),one for each odd s.So,the set{s}in this theory contains all positive odd integers.Conformal boundary conditions in c=1−6a particular cell (r,s )of the Kac table.In each case possible boundary operators are degenerate primary boundary fields ψ(n,m )(plus their Virasoro descendants),such that the fusion rule coefficients N (r,s )(n,m )(r,s )are non-zero (the other degenerate boundary fields,with N (r,s )(n,m )(r,s )=0correspond to “juxtapositions”of different conformal boundaries,see[7]for details).The field Φ(1,3)satisfies this condition for any (r,s ).We take this field to be the boundary perturbation in (2.9),i.e.ΦB (y )=λB ψ(1,3)(y ).(2.27)Its conformal dimension ∆=∆(1,3)<1,so that it is a relevant perturbation.We want to show that under this choice the theory (2.9)is integrable.To warm up,let us consider the components T and ¯Tof the stress tensor itself.In the conformal limit λ=0,λB =0,these components satisfy the boundary condition[T (y +ix )−¯T (y −ix )]|x =0=0,(2.28)i.e.the field ¯T (¯z )is just the analytic continuation of T (z )to the lower half-plane Imz <0.Let us “turn on”the boundary perturbation,λB =0,still keeping λ=0,and consider the correlation function[T (y +ix )−¯T (y −ix )]X λB =Z −1λB [T (y +ix )−¯T (y −ix )]Xe −λB ∞−∞ψ(1,3)(y ′)dy ′ CF T ,(2.29)where X is any product of fields located away from the boundary x =0and Z −1λB = e −λB ∞−∞ψ(1,3)(y )dy CF T .In the limit x →0the contribution to (2.29)is controlled by OPE(T (y +ix )−T (y −ix ))λB ψ(1,3)(y ′)=λB {∆(1,3)(y −y ′−ix )+1∂y ′−1∂y ′}ψ(1,3)(y ′)→→λB {∆(1,3)δ′(y −y ′)+δ(y −y ′)∂T s+1(z)ψ(1,3)(y)=sk=11dyq s(y),(2.32)similar to(2.24),(2.25)where q s,χare descendants ofψ(1,3).Whencelimit x→0(T s+1(y+ix)−T s+1(y−ix))λBψ(1,3)(y′)=λB(δ(y−y′)d(k−1)!d k−1dyθs(2.34)withθs(y)=λB[q s(y)+sk=21dy k−1χk s+1−k(y)].(2.35)The higher powers ofλB can contribute to(2.34)through the“resonance terms”similar to those discussed in[5].It is plausible,however,that these do not spoil the general form of(2.34)but simply modify(2.35)by higher order terms inλB.It is also plausible(and can be supported to some extent by dimensional analysis of[5])that“turning on”the bulk perturbation,λ=0,converts(2.34)to(2.15).We realise that the above arguments do not constitute a rigorous proof.First,we did not solve the problem of the“resonance terms”(this problem remains open in the bulk theory,too).More importantly,we did not analyse possible effects of mixing between the boundary and the bulk perturbations.We have shown,however,that thefield theory (2.9)with(2.22)and(2.26)satisfies some very non-trivial necessary(but not sufficient) conditions of integrability and we conjecture that this boundaryfield theory with boundary is integrable.3.BOUNDARY S-MATRIXIf thefield theory(2.1)((2.2))is massive the space H is the Fock space of multiparticle states.After rotation y=it to1+1Minkowski space-time these states are interpreted as the asymptotic(“in-”or“out-”)scattering states.For an integrablefield theory the scattering is purely elastic and the corresponding S-matrix is factorizable.The factorizable scattering theory in infinite space is discussed in many papers and reviews(see e.g.[4,5,9-12]);below,we describe just some basics of the theory.The boundary theory(2.8)((2.9)) in Minkowski space is also interpreted as a scattering theory.For the integrable boundaryfield theory this scattering theory is again purely elastic and the corresponding S-matrix is the“factorizable boundary S-matrix”.The“factorizable boundary scattering theory”is developed in close parallel with the“bulk”theory(see[13]).However there are still some gaps in this parallel(most importantly,the boundary analog of crossing-symmetry condition of the“bulk”S-matrix is not absolutely straightforward).It is the aim of this Section tofill these gaps.We start with a brief description of the basics of the factorizable scattering theory in the infinite space(“bulk theory”).Assume that the theory contains n sorts of particles A a;a=1,2,...,n with the masses m a.As usual,we describe the kinematic states of the particles in terms of their rapiditiesθ,p0+p1=meθ;p0−p1=me−θ,(3.1) where pµare the components of the two-momentum and m is the particle mass.The asymtotic particle states are generated by the“particle creation operators”A a(θ) |A a1(θ1)A a2(θ2)...A a N(θN) =A a1(θ1)A a2(θ2)...A a N(θN)|0 .(3.2) The state(3.2)is interpreted as an“in-state”if the rapiditiesθi are ordered asθ1>θ2> ...>θN;if insteadθ1<θ2<...<θN(3.2)is understood as an“out-state”of scattering. The“creation operators”A a(θ)satisfy the commutation relationsA a1(θ1)A a2(θ2)=S b1b2a1a2(θ1−θ2)A b2(θ2)A b1(θ1)(3.3)which are used to relate the“in-”and the“out-”bases and hence completely describe theS-matrix.The coefficient functions S b1b2a1a2(θ)are interpreted as the two-particle scatteringamplitudes describing the processes A a1A a2→A b1A b2(see Fig.2).The asymptotic states(3.2)diagonalise the local IM(2.6),the eigenvalues being determined by the relations[P s,A a(θ)]=γ(s)a e sθA a(θ);[¯P s,A a(θ)]=γ(s)a e−sθA a(θ);(3.4)P s|0 =0;¯P s|0 =0,(3.5) whereγ(s)a are constants(γ(1)a=m a).These IM must commute with the S-matrix;itfollows in particular that the amplitude S b1b2a1a2(θ)is zero unless m a1=m b1and m a2=m b2(other concequences are discussed in[5]).Charge conjugation C acts as an involution of the set of particles{A a},i.e.C:A a↔A¯a,where A¯a∈{A a},so that each particle in{A a}is either neutral C A a=A a or belongs to the particle-antiparticle pair(A a,A¯a).In this Section we assume for simplicity that the theory under consideration respects C,P and T symmetries,i.e.S b1b2 a1a2(θ)=S¯b1¯b2¯a1¯a2(θ)=S b2b1a2a1(θ)=S¯a2¯a1¯b2¯b1(θ).(3.6)The two-particle S-matrix S b1b2a1a2(θ)is the basic object of the theory.It must satisfyseveral general requirements1.Yang-Baxter(or“factorization”)equationS c1c2 a1a2(θ)S b1c3c1a3(θ+θ′)S b2b3c2c3(θ′)=S c2c3a2a3(θ′)S c1b3a1c3(θ+θ′)S b1b2c1c2(θ);(3.7)here and below summation over repeated indices is assumed.This equation is illustrated in Fig.3.Formally,this equation appears as the associativity condition for the algebra (3.3).2.Unitarity conditionS c1c2 a1a2(θ)S b1b2c1c2(−θ)=δb1a1δb2a2.(3.8)Graphic representation of this equation is shown in Fig.4.It can also be obtained as the consistancy condition for the algebra(3.3)(one applies(3.3)twice).3.Analyticity and Crossing symmetry.The amplitudes S b1b2a1a2(θ)are meromorphicfunctions ofθ,real at Reθ=0.The domain0<Imθ<πis called the“physical strip”.The physical scattering amplitudes of the“direct channel”A a1A a2→A b1A b2are given bythe values of the functions S b1b2a1a2(θ)at Imθ=0,Reθ>0.The values of these functions atImθ=0,Reθ<0describe the amplitudes of the“cross-channel”A a2A¯b1→A b2A¯a1.Thefunctions S b1b2a1a2(θ)satisfy the crossing symmetry relationS b1b2 a1a2(θ)=S b2¯a1a2¯b1(iπ−θ)(3.9)(see Fig.5).Combining this and the Eq.(3.8)one can derive the“cross-unitarity equation”S c1b2 a1c2(iπ−θ)S c2b1a2c1(iπ+θ)=δb1a1δb2a2.(3.10)4.Bootstrap condition.The only singularities of S b1b2a1a2(θ)admitted in the physicalstrip are poles located at Reθ=0.The simple poles are interpreted as bound states, either of the direct or of the cross channel.As the bound states are stable particles they must be in the set{A a}.Let iu c a1a2be the position of the pole of S b1b2a1a2(θ)associated with the“bound state”A c of the direct channel.Then u c a1a2must satisfy the relationm2a1+m2a2−m2c=−2m a1m a2cos u c a1a2,(3.11)i.e.the quantity¯u c a1a2=π−u c a1a2can be interpreted as the internal angle of the euclideantriangle with the sides m a1,m a2,m c.The pole termS b1b2a1a2(θ)≃if c a1a2f b1b2cdirectly,by solving the Eq.(3.7-3.9)above.Following this approach one can pin down the S-matrix S b1b2a1a2(θ)up to the so-called“CDD ambiguity”S b1b2 a1a2(θ)→S b1b2a1a2(θ)Φ(θ),(3.14)where the“CDD factor”Φ(θ)is an arbitrary function satisfying the equationsΦ(θ)=Φ(iπ−θ);Φ(θ)Φ(−θ)=1.(3.15) The bootstrap equation may impose further restrictions on this function.Let us turn now to the semi-infinite system(2.8)((2.9)).Here again the states in H B can be classified as asymptotic scattering states.The scattering occurs in the semi-infinite 1+1Minkowski space-time(x,t),t=−iy,x<0.The initial state|A a1(θ1)A a2(θ2)...A a N(θN) B,in(3.16) (the subscript B indicates that|... B∈H B)of the scattering consists of some number (N)of“incoming”particles moving towards the boundary at x=0,i.e.all the rapidities θ1,θ2,...,θN are positive.In the infinite future,t→∞,this state becomes a superposition of the“out-states”|A b1(θ1′)A b2(θ2′)...A b M(θM′) B,out(3.17) each containing some number of“outgoing”particles moving away from the boundary with negative rapiditiesθ1′,θ2′,...,θM′.In integrable boundaryfield theory this process is constrained by the IM(2.16).Like in the“bulk”theory the operators H s are diagonal in the basis of asymptotic states andH s|A a1(θ1)A a2(θ2)...A a N(θN) B,in(out)=(Ni=12γ(s)aicosh(sθi)+h(s))|A a1(θ1)A a2(θ2)...A aN(θN) B,in(out)(3.18)where h(s)are some constants.The constraintsNi=1γ(s)aicosh(sθi)=Mj=1γ(s)b jcosh(sθj′)(3.19)which follow from(3.18)show that M=N and the set of rapidities{θ1′,θ2′,...,θN′}can differ only by permutation from{−θ1,−θ2,...,−θN},i.e.the boundary scattering theory is purely elastic.It is possible to argue that the S-matrix in this case has a factorizable structure.The factorizable boundary scattering theory can be described in complete analogy with the“bulk”scattering theory.Again,the asymptotic states(3.16),(3.17)are generated by the“creation operators”A a(θ)satisfying the same commutation relations(3.3).If θ1>θ2>...>θN>0the in-state(3.16)can be written asA a1(θ1)A a2(θ2)...A aN(θN)|0 B,(3.20)where|0 B is the ground state of H B.One can think of the boundary as an infinitely heavy impenetrable particle B sitting at x=0and formally write the state|0 B as|0 B=B|0 (3.21) in terms of“operator”B which we call the“boundary creating operator”(formally B: H→H B;it is an interesting question wether(3.21)makes any more than just formal sense).The“operator”B satisfies the relationsA a(θ)B=R b a(θ)A b(−θ)B,(3.22) the coefficient functions R b a(θ)being interpreted as the amplitudes of one-particle reflection offthe boundary,as shown in Fig.9.The Eq.(3.18)is reproduced if we assume(3.4),(3.5) and[H s,B]=h(s)B.It follows from(3.17)that R b a(θ)vanishes if m a=m b.By purely algebraic manipulations,with the use of relations(3.3)and(3.22),one can expand any in-state(3.20)in terms of the out-statesA b1(−θ1)A b2(−θ2)...A bN(−θN)|0 B,(3.23)(θ1>θ2>...>θN>0)thus expressing the N-particle S-matrixA a1(θ1)A a2(θ2)...A aN(θN)|0 B=R b1b2...b N a1a2...a N (θ1,θ2,...,θN)A b1(−θ1)A b2(−θ2)...A bN(−θN)|0 B(3.24)in terms of the“fundamental amplitudes”S b1b2a1a2(θ)and R b a(θ)which are basic objects ofthe factorizable boundary scattering theory.The amplitudes R b a(θ)have to satisfy several general requirements analogous to the requirements1−4of the“bulk”theory above.1′.Boundary Yang-Baxter equationR c2a2(θ2)S c1d2a1c2(θ1+θ2)R d1c1(θ1)S b2b1d2d1(θ1−θ2)=S c1c2a1a2(θ1−θ2)R d1c1(θ1)S d2b1c2d1(θ1+θ2)R b2d2(θ2)(3.25) (represented graphically in Fig.10)can be obtained as the associativiy condition of the alge-bra(3.22),(3.3).These equations have been introducedfirst in[13]and studied in relation with the quantum inverse scattering method for the integrable systems with boundary in many subsequent papers(see e.g.[15-17]).Note that(3.25)is the direct analog of(3.7).2′.Boundary Unitarity conditionR c a(θ)R b c(−θ)=δb a(3.26) is also an absolutely straightforward generalization of(3.8)(see Fig.11).One obtains(3.26) applying(3.22)twice.3′.The boundary analog of crossing-symmetry condition(3.9)is far less straightfor-ward.Note that without any additional conditions the equations(3.25)and(3.26)are not restrictive enough.The ambiguity in the solution isR b a(θ)→R b a(θ)ΦB(θ)(3.27) with arbitrary functionΦB which satisfy the equationΦB(θ)ΦB(−θ)=1(3.28) only,which does not even imply(contrary to(3.15))thatΦB is analytic(meromorphic) in the full complex plane ofθ.To reveal the analog of the“cross channel”of the scattering process(3.24)one has to use the alternative Hamiltonian picture for the boundaryfield theory mentioned in Sect.2. Consider1+1Minkowski space-time(τ,y)withτ=ix(τ>0)interpreted as time.The “equal time”section now is the infinite line−∞<y<∞and the space of states H is the same as in the“bulk”theory.The boundary condition at x=0appears in this picture as the initial condition atτ=0;it is described by the“boundary state”|B as explained in Sect.2.As|B ∈H this state is a superposition of the asymptotic states(3.2)of the “bulk”theory.In integrable theory with“integrable boundary”the states(3.2)admitted to contribute to|B are restricted to satisfy(2.21).Noting that the eigenvalue of the operator P s−¯P s on(3.2)isNi=12γ(s)aisinh(sθ)(3.29)we conclude that the particles A a can enter the state|B only in pairs A a(θ)A b(−θ)of equal-mass particles of the opposite rapidities5.Thus we can write|B =N ∞N=00<θ1<θ2<...<θNdθ1dθ2...dθN K a N a N−1...a1,b1b2...b N2N(θ1,θ2,...,θN)A aN (−θN)A aN−1(−θN−1)...A a1(−θ1)A b1(θ1)A b2(θ2)...A bN(θN)|0 (3.30)where we have chosen the expansion in terms of the out-states;K2N are certain coefficient functions which can be related to the amplitudes R in(3.24).The overall factor N in(3.30) is chosen in such a way that K0=1;it is possible to argue that in the massive theory with a non-degenerate ground state|0 B,which we consider here,one can choose N=1 by adding an appropriate constant term to the boundary action density b in(2.8)(or to the“perturbingfield”φB in(2.9));in what follows we assume this choice.By applying the standard reduction technique to the equality(2.17),(2.19)one can show that(under appropriate normalization of the operators A a(θ))the following equations holdK a N a N−1...a1,b1b2...b N(θ1,θ2,...,θN)=R b1b2...b N¯a1¯a2...¯a N (iπ2−θ2,...,iπ2−θ)(3.33) i.e.the“elementary reflection amplitude”R b¯a(θ)can be obtained by analytic continuation of the amplitude K ab(θ)to the domain Imθ=iπ2 ∞−∞K ab(θ)A a(−θ)A b(θ)+...)|0 (3.37)Using the Equation(3.31)one can express all the amplitudes K2N in(3.30)in terms of the two-particle boundary amplitudes K ab(θ)and the elements of the two-particle S-matrix S cd ab(θ).The result is in exact agreement with the following simple expression|B =Ψ[K(θ)]|0 =exp( ∞−∞dθK(θ))|0 (3.38)whereK (θ)=12−u c ab 2due to the diagram in Fig.14.The corresponding residue can bewritten as K ab (θ)≃iθ−iu c ab ,(3.41)where the “three-particle couplings”f are the same as in (3.12)but g c are new constants describing the “couplings”of the particles A c to the boundary (Fig.14).More precisely,the nonzero value of g c indicates that the boundary state |B contains a separate contribution of the zero-momentum particle A c ,i.e.|B =N (1+g c A c (0)+1[g c A c(0),K(θ)]=0,(3.43) where K(θ)is the bilinear operator(3.39),and so in the general case one can look for the“boundary wave function”in the formΨ[g c A c(0),K(θ)].The commutativity(3.43) follows from the relationg c′K a′b(θ)S ca c′a′(θ)=g c′K ab′(θ)S cb b′c′(θ)(3.44)(Fig.15)which is easily obtained if one considers the limitθ→iπ2u c a1b1in(3.25)andtakes into account(3.13).It is also possible to show that if g c1,g c2=0the amplitude K c1c2(θ)has the pole atθ=0with the residueK c1c2(θ)≃−iθ;(3.45)This pole term is illustrated by the diagram in Fig.16.Let us illustrate these bootstrap conditions with an example of the so-called“Lie-Yangfield theory”.Thisfield theory describes the scaling limit of Ising Model with purely imaginary externalfield,the critical point being the“Lie-Yang edge singularity”(see[18]). It is a massivefield theory which can be obtained by perturbing the c=−225.Mussardo and Cardy[11]have shown that the“bulk”theory is integrable;theyhave also found the corresponding factorizable“bulk”S-matrix.The theory contains only one species of particles,A,and the“bulk”S-matrix is described byA(θ1)A(θ2)=S(θ1−θ2)A(θ2)A(θ1)(3.46) withS(θ)=sinhθ+i sin2πsinhθ−i sin2π3(which has negative residue thus making non-unitarity of thistheory manifest)corresponds to the same particle A appearing as the“AA bound state”. Here we do not attempt to analyze the possible integrable boundary conditions in this theory;we just assume that such ones do exist.Then the factorizable boundary S-matrix is described by(3.46)andA(θ)B=R(θ)A(−θ)B,(3.48) where the boundary scattering amplitude R has to satisfy(3.26)and(3.35)(the boundary Yang-Baxter equation(3.25)is satisfied identically),i.e.R(θ)R(−θ)=1;K(θ)=S(2θ)K(−θ);K(θ)=R(iπ。
